Fluid Mechanics and Fluid Power, Volume 4: Select Proceedings of FMFP 2022 (Lecture Notes in Mechanical Engineering) [2024 ed.] 9819971764, 9789819971763
This book comprises select peer-reviewed proceedings of the 9th International and 49th National Conference on Fluid Mech
124 10 37MB
English Pages 973 [931] Year 2024
Contents
About the Editors
Combustion
Numerical Analysis on the Effect of Aspect Ratio in a Diesel Injector Using Diesel and Diesel–Ethanol Blend
1 Introduction
2 Literature Review and Objective
3 Physical Model and Domain
3.1 Governing Equations
3.2 Geometry Details
3.3 Grid Independence Study
3.4 Mesh Generation
3.5 Boundary Condition
4 Results and Discussion
4.1 Effect on Sauter Mean Diameter (SMD)
4.2 Effect on Spray Angle
4.3 Effect of Cavitation
5 Conclusions
References
Numerical Simulation of Gasification and Plasma Pyrolysis Process for Lignite Coal: A Comparative Study
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 CFD Modeling
3.2 Problem Setup
3.3 Boundary Conditions
4 Results and Discussion
4.1 Syngas Composition
4.2 Plasma Pyrolysis
4.3 Plasma Pyrolysis Versus Gasification
5 Conclusions
References
Availability Analysis of Diesel-Powered CI Engines with Single and Multiple Injection Strategies
1 Introduction
2 Material and Method
2.1 Experimental Setup
2.2 Methodology
3 Results and Discussion
3.1 Combustion Analysis
3.2 Performance Analysis
3.3 Emission Analysis
3.4 Availability Analysis
4 Conclusions
References
Change in Vortex Breakdown Mode and It’s Influence on Flame Shape of a Co/counter Concentric Swirling Streams
1 Introduction
2 Literature Review and Objective
3 Experimental Details
3.1 Experimental Setup
3.2 Operating Conditions
3.3 Particle Image Velocimetry (PIV)
4 Results and Discussion
4.1 Bubble and Conical Breakdown Mode for Non-Reacting Cases
4.2 Recirculation Ratio (R) for Non-Reacting Cases
4.3 BVB and CVB Modes for Reacting Cases
5 Conclusions
References
Entrained Dust Combustion in Pre-Heated Air
1 Introduction
2 Literature Review and Objective
3 Numerical Methodology
3.1 Numerical Methods and Sub-Models
3.2 Computational Domain
3.3 Boundary Conditions
3.4 Grid Independent Study
3.5 Solution Procedure
4 Results and Discussion
4.1 Effect of Equivalence Ratio
4.2 Effect of Particle Size
5 Conclusions
References
An Experimental Investigation into the GDI Spray Characteristics of Ethanol and Lemon Peel Oil
1 Introduction
2 Experimental Set-Up and Conditions for Constant Volume Study
3 Results and Discussion
3.1 Effect of Ambient Pressure and Temperature on Spray Morphology
3.2 Effect of Real-Time Engine States on Spray Morphology
4 Conclusions
References
Numerical and Experimental Performance Comparison of a Typical Swirl Co-Axial Injector for a Cryogenic Combustor
1 Introduction
2 Physical Problem and Methodology
2.1 Processes and Methods
2.2 Numerical Simulation Method
3 Results and Discussion
3.1 Experimental Results
3.2 Simulation Results
4 Conclusions
References
Analytical Modelling of Effect of Steam Dilution on Hydrogen Combustion and Application to a Typical Nuclear Reactor Containment
1 Introduction
2 Analytical Modelling
3 Validation with Binary H2–Air Mixture
4 Effect of Steam Dilution
5 Application to Containment Safety Analysis
6 Conclusions
References
Thermal Performance of a Single-Layer Porous Radiant Burner with Biogas as Fuel: A Numerical Study
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Governing Equations
3.2 Grid Independence Test
4 Results and Discussion
4.1 Axial Temperature Analysis
4.2 Radiation Efficiency
5 Conclusions
References
Numerical Validation and Benchmarking of Hydrogen Flame Propagation in a Vertical Acceleration Tube Experimental Facility
1 Introduction
2 Combustion Modelling
3 Description of Test Facility
4 Numerical Set-Up
5 Numerical Results
5.1 Validation Results
5.2 Benchmarking Results
6 Conclusions
References
Detailed Chemical Kinetics Mechanism for Condensed Phase Decomposition of Ammonium Perchlorate
1 Introduction
2 Literature Review and Objective
3 Computational Methods
4 Mathematical Model
5 Results and Discussion
6 Conclusions
References
Onset of Thermoacoustic Oscillations in an Annular Combustor with Flames Stabilized by Circular Discs
1 Introduction
2 Experimental Set-Up, Data Acquisition and Processing Methods
3 Results and Discussion
4 Conclusions
References
Development of Advanced Fuel Injector Concepts for Compact Lean-Burn Gas-Turbine Combustors
1 Introduction
2 Design of the Injector
3 Experimental Arrangements
4 Laser Dianostic Arrangements
5 Results andDiscussion
5.1 Flow Field at the Exit of Pilot Mixer
5.2 Flow Field at the Exit of Main Mixer
5.3 Flow Field at the Exit of Injector
5.4 Spray Characteristics of the Injector
6 Conclusions
References
Experimental Study on GDI In-Cylinder Combustion Quality of Ethanol and Lemon Peel Oil
1 Introduction
2 Methodology
3 Results and Discussion
3.1 Effect of Injection Timing on In-Cylinder Combustion Characteristics
3.2 In-Cylinder Combustion Characteristics of Gasoline–Ethanol–LPO Blends
4 Conclusions
References
Numerical Study on Soot Formation of Methyl Methacrylate Pool Flames with Coflow Air
1 Introduction
2 Literature Review and Objective
3 Numerical Methodology
4 Results and Discussion
4.1 Comparison of Temperature and Major Species Profiles of Coflow Cases with the Reference Case
4.2 Effect of Coflow Air on Soot Ratio and Mass Loss Rate
5 Conclusions
References
Impact of Computational Domain and Cell Type on Large Eddy Simulations in OpenFOAM for a Turbulent Partially Premixed Flame
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
4 Results and Discussion
4.1 LES of the Sandia Flame D in Two-Dimensional Axisymmetric Domain
4.2 Three-Dimensional Domain with Unstructured Mesh for LES
4.3 Two-Dimensional Planar Domain with Structured Mesh for LES
4.4 Three-Dimensional Domain with Structured Mesh for LES
5 Conclusions
References
Exergy Analysis of Deflagration Wave Propagating in Autoignitive H2 Mixture for Constant Pressure Boundary Conditions
1 Introduction
2 Literature Review and Objective
3 Numerical Methods
4 Results and Discussion
4.1 0D Simulations
4.2 1D Simulation
5 Conclusions
References
Numerical Investigation of Combustion Dynamics in a Multi-element Combustor Using Flamelet Approach
1 Introduction
2 Literature Review and Objective
3 Formulation and Methodology
3.1 LES Framework
3.2 Turbulent Combustion—Flamelet Approach
4 Multi-element Computational Model
5 Results and Discussion
5.1 Instantaneous Flow Field
5.2 Spectral Analysis and Mechanism
6 Conclusion
References
Experimental Investigations on Emissions and Performance of Spark Ignition Engine Fuelled with Butanol–Pentane–Gasoline Blends
1 Introduction
2 Literature Review and Objective
3 Methodology
4 Results and Discussion
4.1 Brake Thermal Efficiency (BTE)
4.2 Brake-Specific Energy Consumption (BSEC)
4.3 Carbon Monoxide Emissions
4.4 Oxides of Nitrogen Emissions
4.5 Unburnt Hydrocarbon Emissions
5 Conclusions
References
CFD Analysis of Afterburner with Convergent–Divergent Nozzle for Various Air–Fuel Ratios
1 Introduction
2 Literature Review and Objective
3 Description of Afterburner
4 Governing Equations
5 Methodology
6 CFD Modeling
7 Results and Discussion
8 Conclusions
References
Computational Analysis of the Thermo Hydrodynamic Characteristics in a Can-Type Gas Turbine Combustor
1 Introduction
2 Literature Review
3 Modeling, Meshing, and Boundary Conditions
3.1 Governing Equations
3.2 Transport Equations for Standard k- ε Model
3.3 Species Equation
3.4 Boundary Conditions
4 Results and Discussion
4.1 Contours of Outlet Temperature
4.2 Contours of Velocity Magnitude
4.3 Contours of Species
5 Conclusions
References
Experimental Study of Acoustic Phenomenon in a Closed Combustion Chamber
1 Introduction
2 Experimental Methodology
3 Results and Discussion
4 Conclusions
References
The Effect of Lean Premixed Combustion on Thermoacoustic Instability in a Swirl Combustor
1 Introduction
2 Modelling of Swril Combuster
3 Governing Equations
4 Numerical Details and Boundary Conditions
4.1 Validation of Numerical Model
5 Results and Discussions
5.1 Steady State Results
5.2 Unsteady Results
6 Conclusions
References
Computational Modelling of MMH/NTO Combustion in a Multi-element Triplet Injector Combustor
1 Introduction
2 Literature Review and Objective
3 Formulation and Methodology
3.1 Droplet Model
3.2 Combustion Model
4 Multi-element Computational Model
5 Results and Discussion
6 Conclusion
References
Microfluidics
Novel Tree Branching Microchannel Heat Sink Under Variable and Constant Fluid Volume Approaches
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Problem Description
3.2 Governing Equations
3.3 Boundary Conditions and Calculation Methods
3.4 Grid Independent Test
3.5 Numerical Model Validation
4 Results and Discussion
5 Conclusions
References
Two-Dimensional, Magnetic Actuation of Ferrofluid Droplet on an Open-Surface Microfluidic Platform
1 Introduction
2 Theoretical Formulation
2.1 Force Due to Magnetic Field of Current Loops of Regular Polygons
2.2 Spherical Cap Ferrofluid Droplet Motion on a Solid Hydrophobic Surface
3 Validation of the Droplet Transport Model
4 Results and Discussion
5 Conclusion
References
Numerical Analysis of Heat Transfer and Fluid Flow in Microchannel Heat Sinks Designed for Uniform Cooling
1 Introduction
2 Geometry and Methodology
3 Validation
4 Results and Discussion
4.1 Turbulent Flow
4.2 Laminar Flow
5 Conclusions
6 Future Scope
References
Numerical Investigation on Hydrodynamics of Lubricant-Infused Hydrophobic Microchannel with Transversely Oriented Cavities
1 Introduction
2 Methodology
2.1 Governing Equation and Multi-phase Modelling
2.2 Validation
3 Results and Discussion
3.1 Effect of Geometry
3.2 Effect of Viscosity
4 Conclusion
References
Effect of Microstructures in the Flow Passage on the Flow Dynamics of Microchannel
1 Introduction
2 Methodology
2.1 Grid Independence Study
2.2 Validation
3 Results and Discussion
4 Conclusions
References
Combined Effect of Heterogeneous Zeta Potential on Microchannel Wall and Conductive Link in Induced Charge Electrokinetic Micromixing
1 Introduction
2 Problem Description
3 Mathematical Model
3.1 Electric Field
3.2 Flow Field
3.3 Concentration of Mixing Species
3.4 Micromixing Efficiency
3.5 Boundaries Conditions
4 Numerical Simulation
4.1 Grid Independence Study
4.2 Validation
5 Results and Discussion
5.1 ICEK Micromixer Configuration
5.2 Effect of Conductive Link Orientation (Angle)
5.3 Effect of Applied Electric Field and Zeta Potential on Heterogeneous Patch
6 Conclusions
References
Analysis of Sperm Cell Kinetics in Newtonian and Non-Newtonian Fluid Medium Within a Microfluidic Channel
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Design and Fabrication
3.2 Semen Collection and Sample Preparation
3.3 Experimental Setup and Procedure
3.4 Image Analysis
3.5 Simulation Procedure and Setup
4 Results and Discussion
4.1 Effect of Viscosity on Motility Parameters of Sperm
4.2 Variation in Drag Force in Newtonian and Non-Newtonian Fluid
4.3 Power Output of the Sperm
4.4 Power Required to Pull the Sperm
4.5 Efficiency of the Sperm Motion
5 Conclusion
References
Conjugate Heat Transfer Analysis of U-Bend/Turn Microchannel: A Computational Approach
1 Introduction
2 Literature Review and Objective
3 Computational Method
3.1 Problem Description
3.2 Governing Equations
3.3 Boundary Conditions
3.4 Method of Calculation
3.5 Data Acquisition
3.6 Grid Independence Study
3.7 Model Validation
4 Results and Discussion
4.1 Effect of Radius of Curvature
4.2 Overall Performance Analysis
5 Conclusions
References
Experimental Investigation of Fluid Flow Behaviour in Parallel Microchannel Using Micro-PIV
1 Introduction
2 Literature Review and Objective
3 Experimental Procedure
4 Results and Discussion
4.1 Effect of Seeding Particle
4.2 Effect of Flow Rates
4.3 Comparison of Experimental and Analytical Velocity Profiles
5 Conclusions
References
Study of Path Selection of a Droplet in a Symmetric Y-Microchannel Using a Uniform Electric Field
1 Introduction
2 Literature Review and Objective
3 Problem Description
4 Mathematical Modelling and Simulation Details
4.1 Grid Independence Study
4.2 Model Validation
5 Results and Discussion
6 Conclusions
References
Microfluidic Solute Transport by Interference of Oscillatory Thermal Marangoni Effect and Patterned Wall Slip
1 Introduction
2 Analytical Analysis
2.1 Problem Description
2.2 Hydrodynamics Characteristics
2.3 Solute Transport
3 Results and Discussion
3.1 Flow Characteristics
3.2 Solute Transport
4 Conclusions
References
Analysis of Micro-nozzle Flow Using Navier–Stokes and DSMC Method and Locating the Separation Plane Based on Modified Knudsen Number
1 Introduction
2 Literature Review and Objective
3 Methodology
3.1 Continuum/Navier–Stokes Approach
4 Results and Discussion
5 Conclusions
References
Parametric Study on the Primitive Lattice Using the Pore-Scale Simulation to Characterize the Flow and Heat Transfer Performance
1 Introduction
2 Literature Review and Objective
3 Computational Methodology
3.1 TPMS Lattice as the Porous Structure and Different Treatments of Two Subdomains
3.2 Governing Equations and Boundary Conditions
3.3 Calculations of Hydrodynamic and Thermal Performance Parameters
4 Results and Discussion
4.1 Grid Dependency Test
4.2 Validation of the Computational Model
4.3 Hydrodynamic Results
4.4 Heat Transfer Results
4.5 Thermo-mechanical Performance
5 Conclusions
References
Experimental and Numerical Studies on Liquid Bridge Stretching in Uni-port Lifted Hele-Shaw Cell for Spontaneous Fabrication of Well-Like Structures
1 Introduction
2 Problem Specification
3 Numerical Methodology
4 Results and Discussion
4.1 Simulation Results
4.2 Experimental Results
5 Conclusions
References
Numerical Investigation on Inertial Migration of Spherical Rigid Particle in the Entrance Region of a Microchannel
1 Introduction
2 Methodology
2.1 Validation
3 Results and Discussion
4 Conclusions
References
Dynamics of Electrically Actuated Carreau Fluid Flow in a Surface-Modulated Microchannel
1 Introduction
2 Problem Formulation
3 Governing Equations
4 Numerical Methods
5 Model Validation
6 Result Discussion
7 Conclusion
References
Heat Transfer Analysis of Peltier-Based Thermocycler for a Microfluidic-PCR Chip
1 Introduction
2 Literature Review
3 Methodology
3.1 Determination of Peltier Heat Flux
3.2 Numerical Model for Thermal Cycling
3.3 Grid Independence Study
3.4 Validation
4 Results and Discuss
4.1 Temperature–Time Response of PMMA Microchip in One Complete Cycle
4.2 Effect of Materials on Temperature Time Response in One Complete Cycle
4.3 Heat Transfer in Microchannel
5 Conclusions
References
Effect of Viscosity on the Margination of White Blood Cells in an Inertial Flow Microfluidic Channel
1 Introduction
2 Materials and Methods
2.1 Microfluidic Device Design and Fabrication
2.2 Experimental Set-up
2.3 Sample Preparation
3 Results and Discussion
3.1 Margination Using Whole Blood and Diluted Blood
3.2 Margination Using Different PVP Concentrations
3.3 Margination Using Different Dilution Factors
3.4 Discussion
4 Conclusions
References
Experimental Investigation of Two-Phase Immiscible Liquid Flow Through a Microchannel
1 Introduction
2 Literature Review and Objective
3 Experimental Setup
3.1 Experiment with Aq-Glycerol and Si-Oil (50 cSt)
3.2 Experiment with De-ionized Water and Si-Oil (20 cSt)
4 Summary
References
Elastohydrodynamics of Electromagnetically Actuated Deformable Microfluidic Systems
1 Introduction
2 Problem Formulation
2.1 Electrostatics
2.2 Hydrodynamics (Fluid Domain)
2.3 Deformation (Solid Domain)
3 Results and Discussion
3.1 Selection of Physical Parameters
3.2 Constant Flow Rate
3.3 Constant Pressure Gradient
4 Conclusions
References
Experimental and Numerical Analysis of Ferrofluid in Partially Heated Closed Rectangular Microchannel Tube Under Non-uniform Magnetic Field
1 Introduction
2 Theoretical Formulation
3 Magnetostatic Equation
3.1 Boundary Condition
3.2 Grid Independence and Validation
4 Simulation Methodology
5 Experimental Setup
6 Results and Discussion
7 Conclusions
References
Numerical Investigation on the Effect of Reynolds Number on the Droplet Bypass Through T-Junction Using Lattice Boltzmann Method
1 Introduction
2 Methodology
3 Results and Discussion
4 Conclusions
References
Bio-fluid Mechanics
Blood Flow Modeling in Stenosed Arteries Using CFD Solver
1 Introduction
2 Literature Review and Objective
3 Navier Stokes and Artificial Compressibility Method
4 Results and Discussion
4.1 Study of Velocity Profile Along the Cardiac Cycle for 50% Stenosis
4.2 Wall Shear Stress Calculation for 50% Stenosis
4.3 Study of Velocity Profile Along the Cardiac Cycle for 80% Stenosis
4.4 Wall Shear Stress Calculation for 80% Stenosis
5 Conclusions
References
Highlighting the Importance of Nasal Air Conditioning in Septoplasty Using Virtual Correction Tools: A Numerical Study
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 CT Scan Acquisition and 3D Reconstruction of the Nasal Cavity
3.2 Grid Generation
3.3 Mathematical Model
3.4 Numerical Methodology
4 Results and Discussion
4.1 Effect of Deviation and Virtual Correction on Nasal Airflow
4.2 Effect of Deviation and Virtual Correction on Nasal Humidification
5 Conclusions
References
Thrombosis Modelling in a Stenosed Artery
1 Introduction
2 Mathematical Modelling
2.1 Geometry and Boundary Conditions
2.2 Clotting Model
2.3 Governing Equations
2.4 Implementing the Clotting Model in Ansys Fluent
3 Results and Discussion
4 Conclusions and Future Work
References
Gold Nanoparticle-Antibody Bio-Probe Analysis: Synthesis, Conjugation, Characterization and Dot Blot Assay on Paper
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Materials and Reagents
3.2 Apparatus
3.3 Preparation of AuNPs and AuNP-Ab Bioprobes
4 Results and Discussion
4.1 Characterization of AuNPs
4.2 Characterization of AuNP-Ab Bioprobes
4.3 Immunoblotting Analysis
5 Conclusions
References
A Computational Analysis of the Impact of Blood’s Viscoelastic Properties on the Hemodynamics of a Stenosed Artery
1 Introduction
2 Literature Review and Objective
3 Methodology
3.1 Model Parameters for Blood
3.2 Dimensionless Numbers and Boundary Conditions
3.3 Numerical Details
4 Results and Discussion
4.1 Flow Pattern Prediction
4.2 Haemodynamic Descriptors Based on Wall Shear Stress
4.3 Wall Pressure Distribution
4.4 Steady Versus Pulsatile Flows
5 Conclusions
References
Effect of Induced Helicity on the Hemodynamics of Carotid Artery Passage
1 Introduction
1.1 Scope of Present Work
2 Literature Review and Objectives
2.1 Role of Hemodynamic Factors
2.2 Effect of Helical Flow
2.3 Objectives
3 Materials and Methods
3.1 Computational Model Generation
3.2 Flow Physics
3.3 Grid Independence Study
3.4 Validation
3.5 Novel Stent Design
4 Results and Discussion
4.1 Helicity Indicators
4.2 Flow Field Analysis
4.3 Relative Residence Time (RRT) Analysis
4.4 Pressure Drop
5 Conclusions
References
Numerical Simulation of Flow in an Idealized Intracranial Aneurysm Model to Study the Effect of Non-newtonian Blood Flow Rheology
1 Introduction
2 Literature Review and Objective
3 Model Geometry and Computational Methodology
3.1 Computational Domain and Governing Equations
3.2 Computational Mesh and Details of Grid Convergence Study
3.3 Numerical Modelling
4 Results and Discussion
4.1 Numerical Validation
4.2 Flow Features
4.3 Newtonian vs Non-Newtonian Blood Rheology
5 Conclusions
References
On the Replication of Human Skin Texture and Hydration on a PDMS-Based Artificial Human Skin Model
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Fabrication Method
3.2 Characterization
4 Results and Discussion
4.1 Surface Characterization
4.2 Hydration Characterization
5 Conclusions
References
Simulation of Lateral Migration of Red Blood Cell in Poiseuille Flow Using Smoothed Particle Hydrodynamics
1 Introduction
2 Literature Review and Objective
3 Methodology
3.1 Discretization Using SPH
3.2 Spring Network Model of RBC Membrane
3.3 GPU Acceleration Using NVIDIA FORTRAN
3.4 Numerical Simulation
4 Results and Discussion
4.1 Time Step Independency Study
4.2 Particle Independency Study
4.3 Validation of the Numerical Model
4.4 Effect of Initial Position on Shape of RBC
4.5 Effect of Initial Position of Migration Time
5 Conclusions
References
Effect of Stenosis Severity on the Hemodynamics of an Idealized Straight Arterial Tube
1 Introduction
2 Literature Review and Objective
3 Methodology
3.1 Geometry and Grid Configuration
3.2 Governing Equations
3.3 Boundary Conditions
3.4 Numerical Scheme
4 Results and Discussion
4.1 Validation
4.2 Flow Patterns
4.3 Pressure Drop Across the Stenosis
4.4 Time-Averaged Wall Shear Stress (TAWSS) and Oscillatory Shear Index (OSI)
5 Conclusions
References
Microdevice for Plasma Separation and in Vitro Quantification of Plasma Proteins
1 Introduction
2 Materials and Methodes
2.1 Design of Blood Plasma Separation Microdevice
2.2 Fabrication of Microchannels
2.3 Experimental Setup
3 Results and Discussion
3.1 Detection of Albumin
3.2 Methodology for Detection of Albumin
4 Conclusions
References
White Blood Cell Separation and Blood Typing Using a Spiral Microdevice
1 Introduction
2 Objective
3 Methods and Materials
3.1 Microchannel Design and Fabrication
3.2 Working of the Microdevice
3.3 Experimental Setup and Procedure
4 Results and Discussion
4.1 WBC Separation
4.2 Blood Group Test
4.3 Cell Counting and Quantification
5 Conclusions
References
Effect of Arterial Flow on Heat Transfer During Magnetic Hyperthermia Application
1 Introduction
2 Literature Review and Objective
3 Methodology
3.1 Physical Modeling Methods
3.2 Mathematical Modeling
3.3 Boundary Conditions
4 Results and Discussion
4.1 Temperature–Time Graph for Different Artery Positions
4.2 Temperature Contour for Different Artery Positions
4.3 Maximum Temperature Graph for Different Artery Positions
5 Conclusions
References
Flow Separation and Pressure Drop Analysis for Blood Flow in Symmetric Stenosed Arteries of Various Shapes
1 Introduction
2 Literature Review and Objective
3 Problem Description and Numerical Strategy
4 Results and Discussion
4.1 Local Velocity and Pressure Profiles
4.2 Separated Zones
4.3 Pressure Drop Analysis
5 Conclusions
References
Comparative Study of Uniform and Pulsatile Blood Flow Through Single Stenosed Carotid Artery
1 Introduction
2 Literature Review and Objective
3 Methodology
3.1 Governing Equations
3.2 Geometry and Boundary Conditions
3.3 Numerical Technique and Grid Independence Study
3.4 Validation
4 Results and Discussion
4.1 Effect of Severity on the Steady and Unsteady Flow
4.2 Effect of Pulsatile Inlet Velocity
4.3 Oscillatory Shear Index
5 Conclusions
References
Image-Based Retinal Haemodynamics Simulation of Healthy and Pathological Retinal Vasculature
1 Introduction
2 Literature Review and Objective
3 Methods
3.1 Fundus Image Acquisition and Processing
3.2 CAD Geometry Construction
3.3 Mathematical Modelling
3.4 Numerical Methodology
4 Results and discussion
4.1 The Geometry of the Retinal Arterioles
4.2 Retinal Haemodynamics
4.3 Wall Shear Stress (WSS)
5 Conclusions
References
Numerical Study on the Effect of Exercise on Various Configurations of Stenosis in Coronary Artery
1 Introduction
2 Materials and Methods
2.1 Geometry
2.2 Mathematical Model and Boundary Conditions
2.3 Validation and Grid Independence Study
3 Results and Discussion
3.1 Variation in Velocity Profiles
3.2 Variation in Vortex Structures at Peak Velocity
3.3 Tawss
3.4 OSI
3.5 Pressure Drop
4 Conclusions
References
Effect of Aging on Passive Drug Diffusion Through Human Skin
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Grid and Time Independence Test
3.2 Code Validation
4 Results and Discussion
4.1 Temporal and Spatial Variations of Trans-cinnamic Acid
4.2 Temporal and Spatial Variations of Caffeine
5 Conclusions
References
Computational Investigation on the Empirical Relation of Murray’s Law
1 Introduction
2 Literature Review and Objective
3 Computational Fluid Dynamics (CFD) Methodology
3.1 Computational Domain
3.2 Governing Equations and Boundary Conditions
3.3 Mesh Independence Study
4 Results and Discussion
5 Conclusions
References
Investigation of Impulse Jet Dispersion Mechanism of Needle-Free Drug Delivery Device
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Jet Injector Device
3.2 Polyacrylamide Gel
3.3 Numerical Simulation for Microjet
4 Results and Discussion
4.1 Impact Jet Pressure Measurement
4.2 Numerical Simulation
4.3 Jet Dispersion Study in Polyacrylamide Gel
5 Conclusions
References
Analysis of 2D Human Airway in Laminar and Turbulent Flow Model
1 Introduction
2 Geometry of the Respiratory Tract
3 Grid Generation
4 Grid Independency Test
5 Governing Equations
6 Computational Validation
7 Results and Discussion
8 Conclusion
References
Effects of Stenosis Profile on Hemodynamic and Mass Transport in Axisymmetric Geometries: A Numerical Study
1 Introduction
2 Methodology
2.1 Validation
3 Results and Discussion
4 Conclusions
References
Experimental and Numerical Study of Flow Through Ventilator Splitter
1 Introduction
2 Literature Review and Objective
2.1 Previous Study on Ventilator Splitter
2.2 Previous Study on General Flow Splitter
2.3 Medical Data for Mechanical Ventilation
3 Materials and Methods
3.1 Ventilator Splitter CAD Models Design
4 Experimental Setup
4.1 Measurement Technique
5 Computational Model
5.1 Boundary Conditions
5.2 Mesh Independence Study and Sampling Studies
6 Results and Discussion
6.1 Comparison of Simulated Results with Experimental Data
6.2 Flow Distribution Ratio of Y-Splitter for Various Split Angles
6.3 Pressure and Velocity Contours for Flow Through Y-Splitter
7 Summary
References
Bioconvective MHD Flow of Micropolar Nanofluid Over a Stretching Sheet Due to Gyrotactic Microorganisms with Internal Heat Generation/Absorption and Chemical Reaction
1 Introduction
2 Mathematical Formulation
3 Solution Methodology
4 Results and Discussion
5 Conclusion
References
Machine Learning in Fluid Mechanics
Application of Machine Learning for Forced Plume in Linearly Stratified Medium
1 Introduction
2 Methodology
2.1 Governing Equations
2.2 Machine Learning Methodology
2.3 Computational Methodology
3 Results and Discussions
3.1 Validation of the Model
4 Conclusion
References
Comparative Study of Future State Predictions of Unsteady Multiphase Flows Using DMD and Deep Learning
1 Introduction
2 Methods
2.1 PRS Simulations
2.2 Dynamic Mode Decomposition
2.3 CAE-LSTM
3 Results and Discussion
3.1 Flow Field Characterization
3.2 Prediction Accuracy
3.3 Future State Predictions for Case A
3.4 Future State Predictions for Case B
4 Conclusions
References
Deep Learning Approach to Predict Remaining Useful Life of Axial Piston Pump
1 Introduction
2 Dataset
3 NARX NN
4 Training Algorithm
4.1 Levenberg–Marquardt (LM)
4.2 Scaled Conjugate Gradient (SCG)
4.3 Bayesian Regularization
5 Performance Parameter
6 Results
7 Conclusion
References
Machine Learning-Assisted Modeling of Pressure Hessian Tensor
1 Introduction
2 Governing Equations
3 Machine Learning-Based Modeling
4 Evaluation of the Trained Models
5 Conclusions
References
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Lecture Notes in Mechanical Engineering
Krishna Mohan Singh Sushanta Dutta Sudhakar Subudhi Nikhil Kumar Singh Editors
Fluid Mechanics and Fluid Power, Volume 4 Select Proceedings of FMFP 2022
Lecture Notes in Mechanical Engineering Series Editors Fakher Chaari, National School of Engineers, University of Sfax, Sfax, Tunisia Francesco Gherardini , Dipartimento di Ingegneria “Enzo Ferrari”, Università di Modena e Reggio Emilia, Modena, Italy Vitalii Ivanov, Department of Manufacturing Engineering, Machines and Tools, Sumy State University, Sumy, Ukraine Mohamed Haddar, National School of Engineers of Sfax (ENIS), Sfax, Tunisia Editorial Board Francisco Cavas-Martínez , Departamento de Estructuras, Construcción y Expresión Gráfica Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain Francesca di Mare, Institute of Energy Technology, Ruhr-Universität Bochum, Bochum, Nordrhein-Westfalen, Germany Young W. Kwon, Department of Manufacturing Engineering and Aerospace Engineering, Graduate School of Engineering and Applied Science, Monterey, CA, USA Justyna Trojanowska, Poznan University of Technology, Poznan, Poland Jinyang Xu, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China
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Krishna Mohan Singh · Sushanta Dutta · Sudhakar Subudhi · Nikhil Kumar Singh Editors
Fluid Mechanics and Fluid Power, Volume 4 Select Proceedings of FMFP 2022
Editors Krishna Mohan Singh Department of Mechanical and Industrial Engineering Indian Institute of Technology Roorkee Roorkee, Uttarakhand, India
Sushanta Dutta Department of Mechanical and Industrial Engineering Indian Institute of Technology Roorkee Roorkee, Uttarakhand, India
Sudhakar Subudhi Department of Mechanical and Industrial Engineering Indian Institute of Technology Roorkee Roorkee, Uttarakhand, India
Nikhil Kumar Singh Department of Mechanical and Industrial Engineering Indian Institute of Technology Roorkee Roorkee, Uttarakhand, India
ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISBN 978-981-99-7176-3 ISBN 978-981-99-7177-0 (eBook) https://doi.org/10.1007/978-981-99-7177-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.
Contents
Combustion Numerical Analysis on the Effect of Aspect Ratio in a Diesel Injector Using Diesel and Diesel–Ethanol Blend . . . . . . . . . . . . . . . . . . . . . . Aiswarya A. Satheesan, Nikhil Prasad, Nevin Nelson, S. Niranjan, and Anjan R. Nair Numerical Simulation of Gasification and Plasma Pyrolysis Process for Lignite Coal: A Comparative Study . . . . . . . . . . . . . . . . . . . . . . Sidhartha Sondh, Darshit S. Upadhyay, Sanjay Patel, and Rajesh N. Patel Availability Analysis of Diesel-Powered CI Engines with Single and Multiple Injection Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ketan V. Warghat, Aditya Tiwari, B. Yogesh, G. M. Nayak, B. Saravanan, and Pankaj S. Kolhe Change in Vortex Breakdown Mode and It’s Influence on Flame Shape of a Co/counter Concentric Swirling Streams . . . . . . . . . . . . . . . . . . Atanu Dolai, Prasad Boggavarapu, and R. V. Ravikrishna Entrained Dust Combustion in Pre-Heated Air . . . . . . . . . . . . . . . . . . . . . . . Mohd. Tousif, A. Harish, and V. Raghavan An Experimental Investigation into the GDI Spray Characteristics of Ethanol and Lemon Peel Oil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. M. Nayak, B. Abinash, B. Yogesh, V. W. Ketan, P. S. Kolhe, and B. Saravanan Numerical and Experimental Performance Comparison of a Typical Swirl Co-Axial Injector for a Cryogenic Combustor . . . . . . . R. Sujithkumar, K. Chenthil Kumar, K. R. Anil Kumar, T. Jayachandran, and Kowsik Bodi
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Analytical Modelling of Effect of Steam Dilution on Hydrogen Combustion and Application to a Typical Nuclear Reactor Containment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aditya Karanam, Vishnu Verma, and J. Chattopadhyay
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Thermal Performance of a Single-Layer Porous Radiant Burner with Biogas as Fuel: A Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Ayush Painuly and Niraj K. Mishra Numerical Validation and Benchmarking of Hydrogen Flame Propagation in a Vertical Acceleration Tube Experimental Facility . . . . . 119 Aditya Karanam, Vishnu Verma, and J. Chattopadhyay Detailed Chemical Kinetics Mechanism for Condensed Phase Decomposition of Ammonium Perchlorate . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Jay Patel, Prathamesh Phadke, Rohit Sehrawat, Arvind Kumar, Arindrajit Chowdhury, and Neeraj Kumbhakarna Onset of Thermoacoustic Oscillations in an Annular Combustor with Flames Stabilized by Circular Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Balasundaram Mohan and Sathesh Mariappan Development of Advanced Fuel Injector Concepts for Compact Lean-Burn Gas-Turbine Combustors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Ayush Divyansh, Preetam Jamod, and K. P. Shanmugadas Experimental Study on GDI In-Cylinder Combustion Quality of Ethanol and Lemon Peel Oil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 B. Abinash, B. Yogesh, G. M. Nayak, V. W. Ketan, P. S. Kolhe, and B. Saravanan Numerical Study on Soot Formation of Methyl Methacrylate Pool Flames with Coflow Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Argha Bose, D. Shanmugasundaram, and V. Raghavan Impact of Computational Domain and Cell Type on Large Eddy Simulations in OpenFOAM for a Turbulent Partially Premixed Flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Sandeep Lamba and Krishna Kant Agrawal Exergy Analysis of Deflagration Wave Propagating in Autoignitive H2 Mixture for Constant Pressure Boundary Conditions . . . . . . . . . . . . . . 213 Rahul Patil and Sheshadri Sreedhara Numerical Investigation of Combustion Dynamics in a Multi-element Combustor Using Flamelet Approach . . . . . . . . . . . . . . . . . . 225 Abhishek Sharma, Ashoke De, Varghese M. Thannickal, T. John Tharakan, and S. Sunil Kumar
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Experimental Investigations on Emissions and Performance of Spark Ignition Engine Fuelled with Butanol–Pentane–Gasoline Blends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Parag P. Mangave, Vishal V. Patil, Nilesh D. Pawar, and Ranjit S. Patil CFD Analysis of Afterburner with Convergent–Divergent Nozzle for Various Air–Fuel Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Gurrala Srinivasa Rao Computational Analysis of the Thermo Hydrodynamic Characteristics in a Can-Type Gas Turbine Combustor . . . . . . . . . . . . . . . 269 Mohit Bansal, Satyam Dewivedi, and Abdur Rahim Experimental Study of Acoustic Phenomenon in a Closed Combustion Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 A. Ananthakrishnan, Siba Prasad Choudhury, S. Syam, and Ratan Joarder The Effect of Lean Premixed Combustion on Thermoacoustic Instability in a Swirl Combustor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Subhash Kumar, Sanjeev Kumar, and Sheshadri Sreedhara Computational Modelling of MMH/NTO Combustion in a Multi-element Triplet Injector Combustor . . . . . . . . . . . . . . . . . . . . . . . 301 Abhishek Sharma, Varghese M. Thannickal, T. John Tharakan, and S. Sunil Kumar Microfluidics Novel Tree Branching Microchannel Heat Sink Under Variable and Constant Fluid Volume Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Sangram Kumar Samal and Sandip Kumar Saha Two-Dimensional, Magnetic Actuation of Ferrofluid Droplet on an Open-Surface Microfluidic Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Debiprasad Chakrabarty, Niladri Chakraborty, and Ranjan Ganguly Numerical Analysis of Heat Transfer and Fluid Flow in Microchannel Heat Sinks Designed for Uniform Cooling . . . . . . . . . . . . 345 Shivayya C. Hiremath, Rohit Kumar, Arman Mohaddin Nadaf, and Manmohan Pandey Numerical Investigation on Hydrodynamics of Lubricant-Infused Hydrophobic Microchannel with Transversely Oriented Cavities . . . . . . . 357 Adarsh R. Nair, K. Nandakumar Chandran, and S. Kumar Ranjith Effect of Microstructures in the Flow Passage on the Flow Dynamics of Microchannel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 A. Rajalingam and Shubhankar Chakraborty
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Combined Effect of Heterogeneous Zeta Potential on Microchannel Wall and Conductive Link in Induced Charge Electrokinetic Micromixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Anshul Kumar Bansal, Ram Dayal, and Manish Kumar Analysis of Sperm Cell Kinetics in Newtonian and Non-Newtonian Fluid Medium Within a Microfluidic Channel . . . . . . . . . . . . . . . . . . . . . . . . 395 Dhiraj B. Puri, Vadiraj Hemadri, Arnab Banerjee, and Siddhartha Tripathi Conjugate Heat Transfer Analysis of U-Bend/Turn Microchannel: A Computational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Jyoti Ranjan Mohapatra and Manoj Kumar Moharana Experimental Investigation of Fluid Flow Behaviour in Parallel Microchannel Using Micro-PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Rohit Kumar, Chandan Nashine, Arman Mohaddin Nadaf, Mohd Sakib Hussain, and Manmohan Pandey Study of Path Selection of a Droplet in a Symmetric Y-Microchannel Using a Uniform Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Satya P. Pandey, Sandip Sarkar, and Debashis Pal Microfluidic Solute Transport by Interference of Oscillatory Thermal Marangoni Effect and Patterned Wall Slip . . . . . . . . . . . . . . . . . . 449 Shubham Agrawal, Prasanta K. Das, and Purbarun Dhar Analysis of Micro-nozzle Flow Using Navier–Stokes and DSMC Method and Locating the Separation Plane Based on Modified Knudsen Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 Ashok Kumar, Manu K. Sukesan, and Shine S. R. Parametric Study on the Primitive Lattice Using the Pore-Scale Simulation to Characterize the Flow and Heat Transfer Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 Surendra Singh Rathore, Balkrishna Mehta, Pradeep Kumar, and Mohammad Asfer Experimental and Numerical Studies on Liquid Bridge Stretching in Uni-port Lifted Hele-Shaw Cell for Spontaneous Fabrication of Well-Like Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 Makrand Rakshe, Sachin Kanhurkar, Amitabh Bhattacharya, and Prasanna Gandhi Numerical Investigation on Inertial Migration of Spherical Rigid Particle in the Entrance Region of a Microchannel . . . . . . . . . . . . . . . . . . . . 501 K. K. Krishnaram and S. Kumar Ranjith
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Dynamics of Electrically Actuated Carreau Fluid Flow in a Surface-Modulated Microchannel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 Subhajyoti Sahoo and Ameeya Kumar Nayak Heat Transfer Analysis of Peltier-Based Thermocycler for a Microfluidic-PCR Chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 Nikhil Prasad, B. Indulakshmi, R. Rahul, and Ranjith S. Kumar Effect of Viscosity on the Margination of White Blood Cells in an Inertial Flow Microfluidic Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Dhiren Mohapatra, Rahul Purwar, and Amit Agrawal Experimental Investigation of Two-Phase Immiscible Liquid Flow Through a Microchannel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 Rohit Kumar, Chandan Nashine, Arman Mohaddin Nadaf, Harish Kumar Tomar, and Manmohan Pandey Elastohydrodynamics of Electromagnetically Actuated Deformable Microfluidic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Apurba Roy and Purbarun Dhar Experimental and Numerical Analysis of Ferrofluid in Partially Heated Closed Rectangular Microchannel Tube Under Non-uniform Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 Ramesh Kumar, Shivam Raj, and S. K. Dhiman Numerical Investigation on the Effect of Reynolds Number on the Droplet Bypass Through T-Junction Using Lattice Boltzmann Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 T. Sudhakar, Arup K. Das, and Deepak Kumar Bio-fluid Mechanics Blood Flow Modeling in Stenosed Arteries Using CFD Solver . . . . . . . . . . 605 Priyambada Praharaj, Chandrakant Sonawane, and Vikas Kumar Highlighting the Importance of Nasal Air Conditioning in Septoplasty Using Virtual Correction Tools: A Numerical Study . . . . . 619 Kartika Chandra Tripathy and Ajay Bhandari Thrombosis Modelling in a Stenosed Artery . . . . . . . . . . . . . . . . . . . . . . . . . . 633 Prateek Gupta, Rakesh Kumar, Sibasish Panda, and Mohammad Riyan Gold Nanoparticle-Antibody Bio-Probe Analysis: Synthesis, Conjugation, Characterization and Dot Blot Assay on Paper . . . . . . . . . . 643 Prateechee Padma Behera, Shubham Kumar, Monika Kumari, Pranab Kumar Mondal, and Ravi Kumar Arun
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A Computational Analysis of the Impact of Blood’s Viscoelastic Properties on the Hemodynamics of a Stenosed Artery . . . . . . . . . . . . . . . . 655 Sourabh Dhawan, Pawan Kumar Pandey, Malay Kumar Das, and Pradipta Kumar Panigrahi Effect of Induced Helicity on the Hemodynamics of Carotid Artery Passage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 L. Rakesh, Arun Kadali, K. Prakashini, and S. Anish Numerical Simulation of Flow in an Idealized Intracranial Aneurysm Model to Study the Effect of Non-newtonian Blood Flow Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 Suraj Raj, S. Anil Lal, and Anjan R. Nair On the Replication of Human Skin Texture and Hydration on a PDMS-Based Artificial Human Skin Model . . . . . . . . . . . . . . . . . . . . . . 699 Aditya Ranjan, Vijay S. Duryodhan, and Nagesh D. Patil Simulation of Lateral Migration of Red Blood Cell in Poiseuille Flow Using Smoothed Particle Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . 709 Justin Antony and Ranjith Maniyeri Effect of Stenosis Severity on the Hemodynamics of an Idealized Straight Arterial Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 Pawan Kumar, Somnath Roy, and Prasanta Kumar Das Microdevice for Plasma Separation and in Vitro Quantification of Plasma Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 Tony Thomas, Neha Mishra, and Amit Agrawal White Blood Cell Separation and Blood Typing Using a Spiral Microdevice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Sanjay Mane, Vadiraj Hemadri, Sunil Bhand, and Siddhartha Tripathi Effect of Arterial Flow on Heat Transfer During Magnetic Hyperthermia Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 Subeg Singh and Neeraj Kumar Flow Separation and Pressure Drop Analysis for Blood Flow in Symmetric Stenosed Arteries of Various Shapes . . . . . . . . . . . . . . . . . . . . 767 Anamika Maurya, Janani Srree Murallidharan, and Atul Sharma Comparative Study of Uniform and Pulsatile Blood Flow Through Single Stenosed Carotid Artery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781 Swapnil Rajmane and Shaligram Tiwari Image-Based Retinal Haemodynamics Simulation of Healthy and Pathological Retinal Vasculature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797 Shivam Gupta and Ajay Bhandari
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Numerical Study on the Effect of Exercise on Various Configurations of Stenosis in Coronary Artery . . . . . . . . . . . . . . . . . . . . . . . 809 Siddharth D. Sharma, Piru Mohan Khan, Suman Chakraborty, and Somnath Roy Effect of Aging on Passive Drug Diffusion Through Human Skin . . . . . . . 823 Aditya Ranjan, Vijay S. Duryodhan, and Nagesh D. Patil Computational Investigation on the Empirical Relation of Murray’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 Mudrika Singhal and Raghvendra Gupta Investigation of Impulse Jet Dispersion Mechanism of Needle-Free Drug Delivery Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 Priyanka Hankare, Sanjeev Manjhi, and Viren Menezes Analysis of 2D Human Airway in Laminar and Turbulent Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 Vivek Kumar Srivastava and Aman Raj Anand Effects of Stenosis Profile on Hemodynamic and Mass Transport in Axisymmetric Geometries: A Numerical Study . . . . . . . . . . . . . . . . . . . . 865 Ankani Sunil Varma and K. Arul Prakash Experimental and Numerical Study of Flow Through Ventilator Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 Aniruddh Mukunth, Raj Shree Rajagopalan, and Naren Rajan Parlikkad Bioconvective MHD Flow of Micropolar Nanofluid Over a Stretching Sheet Due to Gyrotactic Microorganisms with Internal Heat Generation/Absorption and Chemical Reaction . . . . . 891 P. Vimala and R. Dhivyalakshmi Machine Learning in Fluid Mechanics Application of Machine Learning for Forced Plume in Linearly Stratified Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909 Manthan Mahajan, Nitin Kumar, Deep Shikha, Vamsi K. Chalamalla, and Sawan S. Sinha Comparative Study of Future State Predictions of Unsteady Multiphase Flows Using DMD and Deep Learning . . . . . . . . . . . . . . . . . . . . 923 Neil Ashwin Raj, Danesh Tafti, Nikhil Muralidhar, and Anuj Karpatne Deep Learning Approach to Predict Remaining Useful Life of Axial Piston Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937 Md Adil and Pratik Punj Machine Learning-Assisted Modeling of Pressure Hessian Tensor . . . . . . 949 Deep Shikha and Sawan S. Sinha
About the Editors
Prof. Krishna Mohan Singh is Professor in the Department of Mechanical and Industrial Engineering at Indian Institute of Technology (IIT) Roorkee. His research interests include the areas of computational mechanics, development of novel parallel algorithms, meshfree methods, shape and topology optimization, fluid dynamics, DNS/LES of turbulent flows, CAE, computer-aided analysis and design of thermo-fluid and multi-physics systems, computational fluid dynamics, modeling and simulation of flow and heat transfer in turbomachines, transport and energy systems. Prof. Sushanta Dutta is Professor in the Department of Mechanical and Industrial Engineering at Indian Institute of Technology (IIT) Roorkee. His research interests are in the areas of experimental fluid mechanics, experimental heat transfer, optical measurement techniques, active and passive control of flow field, wake dynamics, turbulence study, Schlieren, HWA, PIV, LCT, PSP, microfluidics and heat transfer augmentation using phase change material. Prof. Sudhakar Subudhi is Professor in the Department of Mechanical and Industrial Engineering at Indian Institute of Technology (IIT) Roorkee. His research interests are in the area of experimental heat transfer and fluid mechanics, heat transfer enhancement of natural and forced convection in water/nanofluids, natural ventilation and unconventional energy systems. Dr. Nikhil Kumar Singh is Assistant Professor in the Department of Mechanical and Industrial Engineering at Indian Institute of Technology (IIT) Roorkee. His broad research interests include direct numerical simulations of two-phase flows and phase change, computational fluid dynamics and heat transfer, numerical methods and turbulent flows.
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Combustion
Numerical Analysis on the Effect of Aspect Ratio in a Diesel Injector Using Diesel and Diesel–Ethanol Blend Aiswarya A. Satheesan, Nikhil Prasad, Nevin Nelson, S. Niranjan, and Anjan R. Nair
Abstract In direct injection diesel engines, spray optimization greatly enhances efficiency and low emissions combustion. The flow inside an injector impacts the process of spray, combustion, and exhaust. The nozzle shape and spray determine the atomization and the outlet engine emissions. The results were obtained for spray characteristics of diesel and ethanol–diesel blend in a nozzle injector with aspect ratios varying from 1, 1.2, 1.4, and 1.6. Parameters, such as spray penetration length, spray angle, and spray characteristics including the Sauter mean diameter (SMD), the De Brouckere diameter, the mean diameter and volume, and particle velocity, were investigated and revealed a strong dependence on modifications in the aspect ratio of the nozzle orifice. Simulation of atomization model was carried out and compared using discrete phase model (DPM) using computational fluid dynamics (CFD) modeling. Additionally, validation from the experiment finding results is also provided. Elliptical C was observed to have a minimum SMD up to 28.04% and a minimum De Brouckere diameter up to 28.63%. Ethanol–diesel blend showed best spray parameters when considering the macroscopic spray properties and the drop size distribution. Moreover, under non-evaporative conditions, the tested fuel ethanol–diesel Blend exhibited better spray characteristics and better cavitation phenomenon of 12.13% at higher aspect ratios than at lower ones. In addition, elliptical nozzle spray had a higher spray cone angle than circular nozzle spray. Keywords Aspect ratio · Spray simulation · Elliptical nozzle
A. A. Satheesan · N. Prasad · N. Nelson · S. Niranjan · A. R. Nair (B) Department of Mechanical Engineering, College of Engineering, Trivandrum 695016, India e-mail: [email protected] N. Nelson Department of Mechanical Engineering, Bishop Jerome Institute (Affiliated to A P J Abdul Kalam Technological University), Kollam, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_2
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1 Introduction Diesel engines are frequently utilized as the primary power source for the road transportation sector. Because of their outstanding thermal efficiency, operational dependability, and durability, the greater understanding of effective fuel use and automotive pollution reduction, which led to enhanced modern direct injection engines like strengthening the spray breakup and generating smaller droplets, has greatly assisted research on the fluid behavior of fuel injection nozzles [1]. Atomization and fuel spray properties in direct injection engines are critical, particularly for gas emissions and combustion efficiency; these factors significantly impact the spray’s shape, atomization quality, engine performance, and emission characteristics. So, the jet breakup inside the chamber also influences the subsequent processes of ignition, combustion, and pollutant generation. Therefore, it’s crucial to consider the fuel injector nozzle effect and the features of the spraying technique with different fuel types. The injector nozzle is a crucial component in a diesel engine. The elliptical orifice diesel nozzle has the potential to improve spray quality and air– fuel mixing [2]. Liquid sprays have been the subject of extensive research due to their actual relevance and the challenges in predicting their behavior from basic principle. While some sprays are composed of several short pulses and may never reach a steady state, others are continuous and stable, at least after a brief start-up transient. Alcohols, like other oxygenated fuels, enhance complete combustion and reduce particulate matter (PM), carbon monoxides (CO), and unburned hydrocarbon emissions (HC) [3]. Reduced SMD and larger spray angle was achieved by implementing elliptical-shaped sprays. Further study can be done on the impact of alternative fuels on the spray, performance, and regarding diesel engines’ emission characteristics, which affect engines parameters performance and emissions
2 Literature Review and Objective Many researchers and pioneers worldwide have investigated diesel fuel injectors and their influence. The discrete phase model (DPM) was developed to investigate the cavitation process in fuel injectors and the macro spray characteristics of three different types of nozzle spray shapes using diesel and hybrid biofuel blends at various injection pressures and backpressures. The findings of the nozzle simulation study showed that the nozzle spray morphology had a greater influence on the cavitation area than the fuel type [4]. A numerical analysis on the fuel spray behavior and fluctuation of spray characteristics in internal combustion engines were investigated, and it was observed that the fuel spray is impacted by the cavitation phenomena in diesel engines. More bubbles are generated when cavitation is severe [5].
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An experimental study on the biodiesel spray liquid-phase behaviors of elliptical and circular nozzles revealed that under steady-state conditions, the elliptical nozzle spray liquid-phase penetration is smaller than the circular one [6]. The elliptic orifice diesel nozzle can improve spray and air–fuel mixing quality, significantly impacting diesel engine combustion and emissions. In all view planes, the elliptical spray had a wider spread of particles than the circular spray, and the circular orifice’s spray cone angle was consistently smaller than that of the elliptical orifice [7]. The spray liquid breakdown behavior of a diesel nozzle with non-circular crosssectional geometries was investigated experimentally under evaporative conditions, and the impact of varied injection pressures and bulk temperatures. In both geometric cases, the study demonstrated that injection pressure has less impact on the penetration of liquid spray. Increasing the ambient temperature, on the other hand, can reduce spray- liquid penetration [8]. Since ethanol is an oxidized fuel, the oxygen level of the mix fuel rises, increasing the thermal efficiency of the engine’s brakes. The thermal efficiency increased by 3.63% while the cylinder pressure increased by 0.46%, when the ethanol content reached 20% at full load [9]. An efficient approach for determining the true extent of vapor zones and turbulence intensity was devised using a comprehensive model for cavitating flow in conjunction with the CFD-ACE+ code was introduced. Cavitation flow involves phase transition. And was shown to be sensitive to the development and motion of vapor bubbles, turbulent oscillations in pressure, velocity, and the quantity of non- condensable gases dissolved or consumed in the operating liquid [10]. Numerical simulation of spray was modeled to study the effect of cavitation on the quality and characteristics of spray, such as penetration length and Sauter mean diameter of the nozzle’s specific geometry. Smaller droplets produced by this spray will improve and help accelerate combustion, enhance power and torque, and reduce outlet emissions [11]. The CFD-programmed software CONVERGE incorporates a recently developed primary breakdown model (KH-ACT) for detailed engine simulations. KH-ACT takes into account the effects of the turbulence and cavitation created inside the injector nozzle. The conical and hydroground nozzle inner nozzle flow impacts of orifice geometry were analyzed. The analysis indicated that the reduced vaporization rate and air–fuel mixing could cause an earlier ignition of the nozzle downstream [12]. The aspect ratio of the elliptical nozzle improved the aerodynamic and penetration characteristics differently, but the optimum/maximum allowable aspect ratio for better aerodynamic characteristics was not reported. Only two types of fuel (diesel/ biofuel) were used to characterize the fuel injector nozzle effect. The mechanism of the liquid fuel breaking up, atomization, and size of the droplet is unclear near the nozzle’s exit. The objective of the study is to investigate the effect of fuel spray characteristics and variation for two types of fuels: Diesel and the combination of diesel and ethyl alcohol (ethanol), using numerical simulation approaches and to numerically evaluate the relationship between the Sauter mean diameter (SMD), De Brouckere diameter
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D [3, 4], mean diameter, and volume spray parameters relation to the aspect ratio of the nozzle and the cavitation phenomenon.
3 Physical Model and Domain The project aims to understand the spray characteristics inside a diesel injector nozzle with preliminary assumptions of unsteady 3D incompressible turbulent nozzle flow and obeying no-slip conditions (fluid velocity at the walls equals the wall velocity) were run with a commercial fluid dynamic code. The discrete phase model (DPM) was introduced to study the fuel injector process and the macro spray characteristic of the injector. The Ansys Design modeler does the 3D model of the elliptical diesel injector. The commercial CFD software Ansys Fluent 2020 R1 performs the numerical simulation. The Standard k − ε is chosen as the viscous model.
3.1 Governing Equations The problem considered is the spray simulation of a diesel injector by varying the aspect ratios of the orifice and also different fuels are used. The analysis is going to be carried out on an incompressible fluid with unsteady-state condition. The governing equations for the 3D continuous flow of the fuel in the injector consist of the continuity, momentum, and energy equation that solved the Navier–Stokes equations. The equations are listed as follows: Continuity equation Dρ + ρ∇ · υ = 0 Dt
(1)
D υ = −∇ p + μ∇ 2 υ + ρg Dt
(2)
Conservation of momentum ρ
3.2 Geometry Details The injector is coupled with an injection chamber (exit diameter = 5.1925 mm) with a nozzle hole to length diameter ratio of 0.280. The 3D design was drawn using ANSYS Workbench 20.0 using design modeler.
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Diesel fluid with a density of 730 kg/m3 and a viscosity of 0.0024 kg/ms is chosen as the fuel. For diesel–ethanol blend, the viscosity is 0.0018 kg/ms, and the density is 807 kg/m3 . The droplet surface tension is 0.026 and 0.0306 N/m for diesel and diesel–ethanol blends, respectively. A singular spray jet is modeled, and the injection takes from the center of the inlet.
3.3 Grid Independence Study The optimum number of grids must be specified in order to execute additional research and calculations. The calculated results ought to be grid-independent and never fluctuate as the number of cells changes (Table 1). For four distinct body sizes, grid independence research was conducted. From 0.2 and 0.02 body sizing onwards, the penetration length is steady. In the case of SMD, there was no significant modification when the number of nodes and elements were increased beyond 336,176 and 323,752, respectively. As a result, body sizes of 0.1 and 0.01 were found to be appropriate (Figs. 1 and 2). Table 1 Variation of penetration length and SMD with number of cells Body sizing → 1 (mm)
Body sizing → 2 (mm)
No. of nodes
No. of elements
Penetration length (m)
Overall SMD (m)
0.4
0.025
11,506
10,200
0.008035
2.253e − 7
0.2
0.02
61,321
57,780
0.00814
2.7743e − 7
0.1
0.01
336,176
323,752
0.00814
2.971e − 7
0.05
0.005
2,065,186
2,021,865
0.00814
3.00e − 7
Fig. 1 Variation of SMD with no. of elements
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Fig. 2 Mesh generation
From the above figures, it is clear that SMD does not vary when number of elements is increased from 323,752. Therefore, further calculations and analysis, body sizing of 0.1 and 0.01 is taken for the geometry.
3.4 Mesh Generation Mesh is generated using inbuilt meshing program inside ANSYS 20.0 in three dimensions. Cells are used to create a structured mesh that becomes finer as it moves from the cylinder’s edge to its core. The mesh quality was found to be 0.95 which implies the model is having a good mesh quality. The number of nodes and elements in the geometry after meshing are 336,176 and 323,752, respectively, chosen after obtaining results from the grid independence study plotted for penetration length versus the number of elements.
3.5 Boundary Condition In the present geometry, the left side is defined as the inlet and the right side is defined as the outlet. The remaining surface is defined as the wall (Tables 2 and 3). Table 2 Boundary conditions
Inlet pressure
100 MPa
Outlet pressure
1 MPa
Wall
No slip condition
Working fluid
i Diesel ii Combination of diesel and ethanol
Numerical Analysis on the Effect of Aspect Ratio in a Diesel Injector … Table 3 Settings for spray simulation
Parameter
Quality
Injection pressure
100 MPa
Outlet pressure
1 MPa
Mass flow rate
3e − 6 kg/s
Injection duration
1s
Injection type
Surface
4 Results and Discussion See Graphs 1, 2, 3 and 4. Graph 1 Comparison of SMD for aspect ratio 1
Graph 2 Comparison of SMD for aspect ratio 1.2
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Graph 3 Comparison of SMD for aspect ratio 1.4
Graph 4 Comparison of SMD for aspect ratio 1.6
4.1 Effect on Sauter Mean Diameter (SMD) The smaller the SMD, the evaporation and atomization process accelerates also it resulting in uniform size distribution and increased number of droplets. Therefore, it is of benefit to mixture formation. Due to diesel’s higher density, stronger intermolecular forces produce poor atomization. The difference in fuel viscosity and density is mostly responsible for the SMD variations between the fuels. Diesel exhibits larger droplet sizes than ethanol–diesel mixtures. Ethanol–diesel blends always have lower SMD and De Brouckere values than pure diesel. They get smaller as the quantity of diesel increases, while it randomly varies for variation in aspect ratio (Table 4).
Numerical Analysis on the Effect of Aspect Ratio in a Diesel Injector … Table 4 Spray angle obtained for various aspect ratios
Aspect ratio
Spray angle
1
12.32°
1.2
14.05°
1.4
15.33°
1.6
16.30°
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Fig. 3 Spray angle for aspect ratio 1
Fig. 4 Spray angle for aspect ratio 1.2
4.2 Effect on Spray Angle An important parameter of fuel sprays is the angle of the spray’s edge as it leaves the injector hole. For single sprays, the two lines tangent to the spray’s margins, extending from the injection point, constitute the spray angle. Lower aspect ratios result in smaller spray angles, while higher aspect ratios, in comparison, result in wider spray angles. The particle residence time is tracked to determine the spray angle for the cases of a Circle, Elliptical A, B, And C, respectively is shown in Figs. 2, 3, 4 and 5. The circle’s spray angle was found to be 12.32°, whereas the maximum spray angle was found to be 16.30° for Elliptical C.
4.3 Effect of Cavitation Figures 6, 7, 8 and 9 show the variation of pressure contour for circular injector nozzle and Elliptical A, B, and C cases, respectively. The pressure contour shows
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Fig. 5 Spray angle for aspect ratio 1.4
that in all cases of aspect ratio, cavitation bubbles first have been generated, close to the nozzle inlet’s sharp corners. Then, the flow of spray transfers these bubbles downward in both an axial and radial direction. The main cause of this phenomena is the development of low-pressure zones. Because of the abrupt change in flow direction near sharp corners, even negative values were detected (Fig. 10). Fig. 6 Spray angle for aspect ratio 1.6
Fig. 7 Pressure contour for aspect ratio 1
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Fig. 8 Pressure contour for aspect ratio 1.2
Fig. 9 Pressure contour for aspect ratio 1.4
The formation of cavitation inside the nozzle can be enhanced by an increase in aspect ratio. The cavitation intensity was more intensive for Elliptical B and C as compared to other nozzle shapes for the same injection time
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Fig. 10 Pressure contour for aspect ratio 1.6
5 Conclusions The present study aims to investigate the spray characteristics and fuel droplet atomization performance of the test fuels—diesel and biodiesel, by varying the aspect ratios. The spray characteristics of diesel and ethanol–diesel blend were determined numerically. The investigation led to the following conclusions: i. The variation of aspect ratio in diesel injector is recognized to play an important role in spray characteristics and formation. ii. Increasing the aspect ratio enhances turbulence, which causes cavitation in the chamber, hence, increasing the spray angle. iii. Due to lower viscosity and density, a lower SMD reduction of up to 28.04% for the ethanol–diesel blend is observed. De Brouckere Diameter also showed a similar trend, declining by 28.63%. iv. The spray cone angle was observed to be influenced by the aspect ratio of the elliptical nozzle shape with minimum spray angle in circle being 12.32° and maximum spray angle of 16.30° in case of Elliptical C. v. Fuel with higher viscosity, i.e., diesel, does not easily breakup in to smaller droplets. The smaller size of the droplet can improve spray atomization and air–fuel mixing, which is possible in the case of ethanol–diesel blend.
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References 1. Som S, Longman DE (2012) Influence of nozzle orifice geometry and fuel properties on flow and cavitation characteristics of a diesel injector. Fuel Inject Autom Eng 14:112–126 2. Chen PC, Wang WC (2013) Spray and atomization of diesel fuel and its alternatives from a single-hole injector using a common rail fuel injection system. Fuel 103:850–861 3. Iliev S (2018) Comparison of ethanol and methanol blending with gasoline using engine simulation. Biofuels Challenges Opport 4. Bannikov M (2015) Effect of alcohol additives on diesel engine performance and emissions. Mater Methods Technol 9:8–19 5. Shervani-Tabar MT, Sheykhvazayefi M, Ghorbani M (2013) Numerical study on the effect of the injection pressure on spray penetration length. Appl Math Model 37:7778–7788 6. Yu S, Yin B, Deng W, Jia H, Ye Z, Xu B, Xu H (2018) Experimental study on the spray characteristics discharging from elliptical diesel nozzle at typical diesel engine conditions. Fuel 221:28–34 7. Yin B, Xu B, Jia H, Yu S (2020) The effect of elliptical diesel nozzles on spray liquid-phase penetration under evaporative conditions. Energies 13:2234 8. Wang Z, Li L (2020) Effects of different ethanol/diesel blending ratios on combustion and emission characteristics of a medium-speed diesel engine. Processes 9. Singhal AK (2002) Mathematical basis and validation of the full cavitation model. J Fluids Eng CFD Res Corp J Fluids Eng 124:617–624 10. Shervani-Tabar MT et al (2012) Numerical study on the effect of the cavitation phenomenon on the characteristics of fuel spray. Math Comput Modell 56:105–117 11. Som S, Ramirez AI et al (2010) Effect of nozzle orifice geometry on spray, combustion, and emission characteristics under diesel engine conditions. Fuel 90:1267–1276 12. Sun Y, Hooman ZGK (2019) Cavitation in diesel fuel injector nozzles and its influence on atomization and spray. Chem Eng Technol 42:6–29
Numerical Simulation of Gasification and Plasma Pyrolysis Process for Lignite Coal: A Comparative Study Sidhartha Sondh, Darshit S. Upadhyay, Sanjay Patel, and Rajesh N. Patel
Abstract Computational fluid dynamics is a special tool for modeling thermochemical processes for process parameter optimization. The present study is a comparative study of the gasification and plasma pyrolysis process of lignite coal. Three temperatures (1023, 1123, 1223 K) are selected for the gasification process and a similar is done for the plasma pyrolysis (1223, 1323, 1423 K). The obtained results are compared with the experiment literature available. The RMSE approach was used for checking the accuracy of the model. The accuracy was observed to be appreciable. The composition of the syngas is compared for all the cases. It was observed that the concentration of hydrogen and carbon monoxide is found to be rich in plasma pyrolysis with an average of 43.4% as compared to 13.5% for gasification. The plasma pyrolysis process offered better results compared to the gasification process as it offered a higher H2 /CO ratio and (H2 + CO) factor. The CO/CO2 ratio also increased for the plasma pyrolysis process with an increase in temperature. Keywords Computational fluid dynamics · Gasification · Pyrolysis · Plasma pyrolysis · Thermochemical process
1 Introduction Thermochemical processes such as gasification and pyrolysis are commonly known for energy generation and waste treatment. Due to the huge initial investment and complex process, it is not feasible to carry out experimental research on all the thermochemical processes together. In such an instance, computational fluid dynamics (CFD) emerges as a potential tool for researchers [1]. It also helps in optimizing S. Sondh · D. S. Upadhyay (B) · R. N. Patel Department of Mechanical Engineering, Institute of Technology, Nirma University, Ahmedabad, Gujarat, India e-mail: [email protected] S. Patel Department of Chemical Engineering, Institute of Technology, Nirma University, Ahmedabad, Gujarat, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_3
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the designs and other parameters for such processes without involving any major investments [2]. The CFD can also be a useful tool in making the thermochemical processes environment-friendly. Different cases can be simulated to find an effective method for limiting pollutant emissions and improving the overall health of the environment. Gasification is a widely used thermochemical process for energy production using biomass, coal, municipal solid waste (MSW), etc. [3]. The pyrolysis process also offers the option of energy generation from the above-mentioned feedstocks [4]. The absence of oxygen in the pyrolysis process makes it a more suitable option due to the limited formation of harmful products such as carbon dioxide (CO2 ), SOx , NOx , PAHs. ANSYS Fluent V17.0 software is used to carry out the simulations of the lignite coal gasification and pyrolysis at different temperatures. The experimental results of lignite gasifications are compared with the CFD simulation results for both processes. The syngas or producer gas obtained from these thermochemical processes is a mixed gas comprising carbon monoxide (CO), hydrogen (H2 ), CO2 , methane (CH4 ), etc. This mixed gas is very valuable and can be used as fuel for cooking and energy generation in the form of electricity and heat [5].
2 Literature Review and Objective Thermochemical processes are the new way of handling wastes and obtaining useful products. The processes are effective options for meeting the energy demand of the country. Gasification is a globally used technology for generating energy from coal [6]. In the gasification process, the coal is partially oxidized due to the controlled presence of air, oxygen, steam, and CO2 . Since the presence of oxygen is limited, the process is always under the control and can be solved for different equivalent ratios [7]. The other process considered in this research is pyrolysis. The process of pyrolysis is a new technology that is used for purposes such as waste treatment, energy generation, and oil generation. Pyrolysis is majorly subdivided into three major categories: slow pyrolysis, fast pyrolysis, and flash pyrolysis [8]. However, another category of thermal plasma pyrolysis is also practiced in the industry [9]. The type of feedstock and reactor also influences the thermochemical process. The feedstock can be any waste, biomass, coal, plastics, etc. There are many types of reactors which include downdraft, updraft, fluidized bed, etc. [10]. In this research, fixed bed downdraft reactor is chosen for the analysis. CFD is an effective tool that is widely used to predict the results of thermochemical processes. Much research focusing on thermochemical processes has been effectively modeled using the CFD tools for optimizing various process parameters. The present research is focused on modeling the two thermochemical processes—gasification and plasma pyrolysis of lignite coal. The processes are modeled for three temperatures 1023, 1123, and 1223 K for gasification whereas that of plasma pyrolysis is 1223, 1323, and 1423 K. The operating temperature range of plasma pyrolysis is higher than
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the gasification due to the high working temperature. The mixed gas obtained from both processes is analyzed and compared with the experimental data available. The study highlights the importance of CFD in the optimization of process parameters.
3 Materials and Methods The fuel for the gasification process was chosen to be lignite coal. The ultimate and proximate analysis for the coal was also conducted and it is mentioned in Table 1. The experiments on lignite coal gasification were carried out at three different temperatures 1023, 1123, and 1223 K. The composition of the syngas was analyzed using the gas chromatography facility for the syngas sample for each temperature run. These sample data are used to compare and verify the simulation results obtained from the ANSYS Fluent software.
3.1 CFD Modeling The geometry of the reactor was modeled using the Parametric CREO 3.0 software. The next step in the simulation process is to create the mesh in the reactors. The meshing is done on the model to make a problem more approachable and convenient using the finite element techniques. It breaks the whole domain into small elements and solves the problem at each node. The meshing of the reactor is done in ANSYS ICEM software. For the surface mesh, all triangular elements are used (23,256 elements) Fig. 1, whereas, for the volume generation, hexahedral elements are used (179,821 elements) Fig. 2. The orthogonal quality of all the elements was duly found to be acceptable (> 0.3). Table 1 Lignite coal: ultimate and proximate analysis data
Ultimate analysisa
Proximate analysisb
Carbon
Volatile matter
42.07
Hydrogen
37.80 4.93
Ash
15.11
Nitrogen
1.625
Moisture
Sulphur
0.141
Oxygen
40.394
a b c
Fixed
Test method IS 1350 (Part II)-1970 Test method IS 1350 (Part I)-1984 By difference
carbonc
11.79 31.03
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Fig. 1 Surface mesh
Fig. 2 Cut-section of volume mesh
3.2 Problem Setup The ANSYS Fluent Package was used to model and set up the problem. The process of gasification is complex involving thermochemistry input. For defining a problem in Fluent, suitable boundary conditions and operating conditions are to be identified. The process temperature, turbulence model, species model, and reactions involved are a few of the parameters that need to be properly defined for obtaining real-life cases.
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Table 2 Operating parameters for thermochemical processes Parameters
Operating condition Gasification
Plasma pyrolysis
Temperature (T ) K
1023, 1123, 1223
1223, 1323, 1423
Pressure (P) Pa
101,325
101,325
Gravity (g) m/s2
9.81
9.81
Turbulence model
k − E Turbulence model (realizable)
k − E Turbulence model (realizable)
Species model
Species transport (chemkin mechanism import)
Species transport (chemkin mechanism import)
Reaction type
Volumetric reactions/particle reactions
Volumetric reactions/particle reactions
Power input (kW)
–
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There were three runs each carried out for both the thermo-chemical processes process. The operating parameters for the processes are shown in Table 2. The species transport model (STM) is used for defining the chemistry of the thermochemical processes. A chemkin mechanism is defined as consisting of 9 species and 5 elements for the gasification process as shown in Fig. 3. All the standard gasification reactions are used, and the activation energy is provided. The reaction rate is taken as default due to insufficient data.
3.3 Boundary Conditions An important step in the modeling process is defining the boundaries of the domain. Also, the input parameter at that boundary is defined for obtaining a real-life problem environment. Table 3 shows the boundary conditions added for the gasification case, whereas Table 4 represents the boundary condition for the plasma pyrolysis runs.
4 Results and Discussion 4.1 Syngas Composition The CFD simulation results were compared with the experimental results obtained from the literature [11]. The major parameter for the validation was the syngas composition as obtained at different temperatures. The gas composition as obtained from CFD simulations is CO2 , CO, H2 , CH4 , and N2 in the case of gasification, whereas CO2 , CO, H2 , and CH4 were obtained in the plasma pyrolysis process.
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Fig. 3 Chemkin mechanism for lignite coal gasification
Table 3 Boundary conditions for gasification
Surface
Boundary condition Input parameter
Fuel inlet Mass flow inlet
M = 10 kg/h/0.00277778 kg/s
Outlet
Pressure outlet
Pgauge = 0 Pa
Walls
Stationary wall
No slip boundary condition
Air inlet
Mass flow inlet
M = 17 kg/h/0.00472222 kg/s
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Table 4 Boundary condition for plasma pyrolysis Surface
Boundary condition
Input parameter
Fuel inlet
Mass flow inlet
M = 10 kg/h/0.00277778 kg/s
Outlet
Pressure outlet
Pgauge = 0 Pa
Walls
Stationary wall
No slip boundary condition
Electrodes
Wall
Electric potential (ON) V = 60 V R = 0.2 Ω/m2
Table 5 RMSE for gasification simulation results at different temperatures
Syngas
1023 K
1123 K
1223 K
CO2
0.043
0.002
0.020
CO
0.081
0.052
0.055
H2
0.040
0.007
0.014
CH4
0.011
0.002
0.005
N2
0.084
0.023
0.008
The accuracy of the results was calculated by the root mean square error (RMSE) approach with reference to the experimental literature available. The results were acceptable and are shown in Table 5. Figure 4 shows the comparison of the results obtained from the CFD simulation of lignite coal gasification with the experimental literature. The values obtained closely match the experimental literature available. The H2 and CO concentration is observed to be increasing with the increase in temperature, whereas the concentration of CO2 is observed to be decreasing as the temperature increase. The (H2 + CO) parameter determines the flammability of the syngas and it also increases with the increase in temperature. Fig. 4 Lignite coal gasification simulation versus experimental literature results
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4.2 Plasma Pyrolysis The plasma pyrolysis process was simulated for three temperatures and it was observed that the quality of syngas increases with the increase in temperature as shown in Fig. 5. At a higher temperature, the (H2 + CO) factor increases. Apart from the CO2 , CO, H2 , and CH4 , there is a small percentage of a group of higher order hydrocarbons such as C2 H2 , C4 H4 , etc., are found. The pattern of some of the important ratios such as CO/CO2 and H2 /CO is also observed for both the gasification and plasma pyrolysis process. It is found that both these parameters increase with the increase in temperature. The (H2 + CO) parameter also increases with the increase in temperature. From Table 6, it is quite evident that the quality of syngas from the plasma pyrolysis process is much better than that of the gasification. The value of all three parameters is much higher than that of the gasification process.
Fig. 5 Plasma pyrolysis simulation results
Table 6 Parameters obtained from CFD simulation
H2 /CO
(H2 + CO)
CO/CO2
Gasification simulation 1023 K
0.48
0.28
0.76
1123 K
1.09
0.26
0.69
1223 K
0.96
0.27
0.76
Plasma pyrolysis simulation 1223 K
8.07
0.88
1.43
1323 K
7.09
0.89
3.06
1423 K
4.76
0.84
2.23
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Fig. 6 Syngas composition at a temperature of 1223 K
4.3 Plasma Pyrolysis Versus Gasification The results obtained from the simulation of the plasma pyrolysis process and gasification at the same temperature of 1223 K are shown in Fig. 6. From the figure, it is visible that there is a major variation in the H2 and CO concentration for the two processes. Due to the absence of air and oxygen in the pyrolysis process, the N2 content is observed to be zero in the result. The concentration of the CO2 is also less in the plasma pyrolysis process as compared to the gasification process which makes it comparatively more environment-friendly. Since plasma pyrolysis majorly occurs at a higher temperature, the concentration of CO2 will be further limited.
5 Conclusions The thermochemical processes can be used effectively for syngas generation which is an alternative fuel. The CFD simulation offered comparable results with the experimental literature which validates the modeling approach used for simulation. The concentration of H2 is found to be more than 60% in the plasma pyrolysis simulations. The (H2 + CO) parameter increased with an increase in temperature, also the CO/CO2 ratio increased with an increase in temperature. The syngas performance parameter H2 /CO was observed to be 6.64 for the plasma pyrolysis process and 0.844 for the gasification process. These parameters define the quality of syngas and it was noted to be better for the plasma pyrolysis process. Acknowledgements The authors will like to thank the Gujarat Council of Science and Technology (GUJCOST), Department of Science and Technology, Gujarat, India, for funding the project (GUJCOST/2020-688 21/880).
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References 1. Lu L et al (2022) Multiscale CFD simulation of biomass fast pyrolysis with a machine learning derived intra-particle model and detailed pyrolysis kinetics. Chem Eng J 431:133853. https:// doi.org/10.1016/j.cej.2021.133853 2. Sharma D et al (2020) Thermal performance analysis and experimental validation of primary chamber of plasma pyrolysis system during preheating stage using CFD analysis in ANSYS CFX. Therm Sci Eng Prog 18:100525. https://doi.org/10.1016/j.tsep.2020.100525 3. Prakash PV (2016) Modelling of coal devolatilization. Indian Institute of Technology, Hyedrabad 4. Chhabra V, Bhattacharya S, Shastri Y (2019) Pyrolysis of mixed municipal solid waste: Characterisation, interaction effect and kinetic modelling using the thermogravimetric approach. Waste Manag 90:152–167. https://doi.org/10.1016/j.wasman.2019.03.048 5. He M et al (2010) Syngas production from pyrolysis of municipal solid waste (MSW) with dolomite as downstream catalysts. J Anal Appl Pyrolysis 87(2):181–187. https://doi.org/10. 1016/j.jaap.2009.11.005 6. Isabel Suarez-Ruiz JCC (ed) (2008) Chapter 5: coal gasification. In: Applied coal petrology: the role of petrology in coal utiliztion. Elsevier Science, Amsterdam, pp 119–144 7. Upadhyay DS, Panchal KR, Sakhiya AKV, Patel RN (2020) Air-steam gasification of lignite in a fixed bed gasifier: influence of steam to lignite ratio on performance of downdraft gasifier. Energy 211:8187. https://doi.org/10.1016/j.energy.2020.118187 8. Chen D, Yin L, Wang H, He P (2014) Pyrolysis technologies for municipal solid waste: a review. Waste Manag 34(12):2466–2486. https://doi.org/10.1016/j.wasman.2014.08.004 9. Vyas DS, Dave UB, Parekh HB (2011) Plasma pyrolysis : an innovative treatment to solid waste of plastic material. Natl Conf Recent Trends Eng Techonol 5:574 10. Upadhyay DS, Khosla A, Chaudhary A, Patel RN (2019) Effect of catalyst to lignite ratio on the performance of a pilot scale fixed bed gasifier. Energy 189:116229. https://doi.org/10.1016/ j.energy.2019.116229 11. Upadhyay DS (2019) Investigations on influence of steam injection and catalyst on producer gas quality in a fixed bed gasifier with lignite as feedstock. Nirma University, Ahmedabad
Availability Analysis of Diesel-Powered CI Engines with Single and Multiple Injection Strategies Ketan V. Warghat, Aditya Tiwari, B. Yogesh, G. M. Nayak, B. Saravanan, and Pankaj S. Kolhe
Abstract Injection timing heavily influences the diesel engine performance and emissions. The present study utilizes a various injection strategies such as single injection, 30° BTDC and 50° pilot injection, paired pilot injection, and split injection on the performance and emissions. There are two conditions for a single pilot injection: the first is a 20% pilot injection at 30° BTDC, and the second is a 20% pilot injection at 50° BTDC. A twin injection approach uses a pilot of 5% at 50° BTDC and another 15% at 30° BTDC. Performance metrics like BTE, BSFC, and IMEP are determined at a compression ratio of 18:1 for 1000 RPM. The split injection condition produces a lower NOx , CO, and UHC emission. The pilot operation at 50° produces more CO and NOx emissions. Applying the second law of thermodynamics analysis to the CI engine, exergetic efficiency is assessed for various injection strategies, with split injection exhibiting the most optimal engine performance along with controlled emissions. Keywords IC engine · Injection timing · Performance analysis · Emission and availability
Abbreviation Ain Acw Aefficiency Aexhaust BSFC BTDC BTE CI
Input availability Availability at cold water Second law efficiency Availability at exhaust Brake-specific fuel consumption Before top dead centre Brake thermal efficiency Compression ignition
K. V. Warghat · A. Tiwari · B. Yogesh · G. M. Nayak · B. Saravanan · P. S. Kolhe (B) Department of Mechanical and Aerospace Engineering, IIT Hyderabad, Telengana 502284, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_4
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CO CRDI ECU HRR LHV NOx RPM SOI TDC TO UHC
K. V. Warghat et al.
Carbon monoxide Common rail direct Electronic control unit Heat release rate Low heating value Nitrogen oxide Revolution per minute Start of injection Top dead centre Throttle opening Unburned hydrocarbon
Nomenclature Cpex cpw mex mf mw Po Pexo To T exo T wi T wo
Specific heat of exhaust [J/Kg K] Specific heat of water [J/Kg K] Mass flow rate of exhaust [Kg/s] Mass flow rate of fuel [Kg/s] Mass flow rate of water [Kg/s] Ambient pressure [bar] Exhaust pressure [bar] Ambient temperature [K] Exhaust temperature [K] Inlet water temperature [K] Outlet water temperature [K
1 Introduction CI engines are essential to society’s needs, be it public transport, goods vehicle, or power generator for power backup. However, as the population increases, the necessity for automobiles rises, resulting in rising pollution, which needs to be controlled. Performance and emissions are affected by various factors; injection timing and strategy are one of them could be optimized. In general, diesel engines operate primarily in lean conditions, resulting in increased thermal efficiency and higher exhaust pollutants, such as smoke and particulate matter. The lean burning condition gives higher unburned hydrocarbons in diesel engines. Higher combustion temperature leads to the breakage of nitrogen bonds to monoatomic, resulting in more NOx . In multiple injection techniques, a small amount of fuel is injected as one or two pilot injections during the compression stroke prior to the main injection. This results in
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a substantially better fuel–air mixture than the conventional single main injection strategy. Several studies investigated the effect of injection timing on the combustion process. MacMillan et al. [1] experimentally investigated the effect of pilot injection timing and fuel quantity in a single cylinder of a multi-cylinder engine at cold idle conditions. It was observed that increasing the number of pilot injections results in proper stability and ruggedness at low-temperature conditions, with the highest stability at the triple pilot condition at different speeds and injection timing. Single pilot and twin pilot injection conditions show almost similar heat release rates. Suh [2] reported the effect of multiple injection strategies on low CR engines using different emissions and performance parameters in a single-cylinder CI engine. It was revealed that the two pilot injections give higher pressure data with a maximum heat release rate reduction. Multiple injections improved combustion efficiency with lower UHC and a slight increase in CO emissions. In a heavy duty 6-cylinder water cooled engine, Yuo et al. [3] conducted experiments on some injection techniques, including pilot and post injection with a blend of n-butanol. They concluded that with a blend condition of 10%, both single and multiple injections give similar performance result, whereas pilot injection reduces soot with an increase in CO emission. Post injection also reduces soot, but the main injection and pressure must be adjusted carefully. Liu et al. [4] experimentally studied the effects of injection timing and quantity in a six-cylinder engine using diesel/CNG. CO emissions are higher than single diesel combustion, and UHC and soot particles reduce significantly as the diesel quantity increases. The maximum useful work that could be extracted by the interaction of a system with its surrounding considered as a reversible process to achieve thermal, mechanical, and chemical equilibrium is defined as the system’s availability in a given state [5]. Sahoo et al. [6] performed the availability study on a four-cylinder diesel engine to calculate the ideal throttle opening (TO) at various load and RPM conditions. They concluded that the ideal engine operating conditions for 70, 80, and 90% engine loads are 2000 rpm at 50% TO, 2300 rpm at 75% TO, and 3250 rpm at 100% TO respectively. Ismail and Mehta [7] studied the availability of various fuels with their chemical composition and found that availability destruction decreases with an increase in oxygen content in the fuel. The preheating of fuel helps in reduction of availability destruction. Therefore, the qualitative information of a system could be utilized to comprehend the engine performance and emission in detail. From the literature, it can be inferred that the timing, quantity, and number of the pilot injections all play a significant role in the combustion process such as performance, emissions, and power. The effectiveness of any process can be evaluated by its availability which gives maximum energy that can be extracted. This study investigates different injection strategies on engine performance and emission. All the strategies show a slight difference on performance, with a significant impact in emission parameters.
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Fig. 1 Experimental setup schematic
2 Material and Method 2.1 Experimental Setup The present study utilizes a twin-cylinder optical research engine with an operational range of 400–1300 RPM. One of the twin-cylinder is a thermodynamic cylinder, whereas the other is optical access to study the inside combustion. In this experiment, only a thermodynamic cylinder with a toroidal bowl piston top is used, which helps in compact and faster burning. The engine has a common rail direct injection system with a CRDI driver module and CRDI kit, which controls injection pressure, timing, and duration. The compression ratio ranges from 6.7 to 18. The schematic diagram of experimental study is shown in Fig. 1. Fuel injection pressure ranges from 200 to 1000 bar. CRDI module is operated by an open ECU system provided by legions brothers, which helps with the injection timing and pressure variation. Data acquisition software shows all the output parameters, such as air–fuel ratio, in-cylinder pressure, exhaust gas temperature, and fuel consumption. A Kistler made piezoelectric pressure transducer monitors in-cylinder pressure connected at the cylinder head. The detailed engine specification is provided in Table 1. For the performance analysis, different loading conditions employed on engine with hydrodynamic dynamometer. Engine exhaust is connected to an AVL gas analyzer to read the exhaust emissions like NOx , CO, and unburned hydrocarbon (UHC).
2.2 Methodology The experiments are carried out at 50 and 80% of maximum load conditions of constant 1050 RPM. Table 2 represents the different injection techniques at different
Availability Analysis of Diesel-Powered CI Engines with Single … Table 1 Engine specification
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Parameters
Values
No. of cylinders
1 of 2
Stroke (mm)
100
Bore (mm)
94
Compression ratio
18:1
Speed range (RPM)
1050
Injection pressure (bar)
500
Table 2 Injection timings Injection type
Injection timing
1
Single main injection
100%@9° BTDC
2
One pilot 30
20%@30° BTDC
3
One pilot 50
20%@50° BTDC
4
Twin pilot
5%@50° BTDC and 15%@30° BTDC
5
Split injection
50%@5° BTDC and 50%@5° ATDC
injection angles investigated in this study. An injection timing of 9° BTDC is considered optimal among test cases run at various injection timings. The performance data such as BTE, BSFC, and IMEP are collected at a steady engine condition for several cycles to determine the performance. A piezoelectric pressure sensor is used to acquire in-cylinder pressure. Inside combustion pressure is recorded for 100 consecutive cycles to average on each test point. Heat release rate (HRR) and pressure rise rate are computed using the pressure data. The engine’s exergetic efficiency at various injection strategies is evaluated by availability analysis. The exhaust emission data are collected from the gas analyzer (AVL DIGS 444N).
3 Results and Discussion 3.1 Combustion Analysis The combustion performance of an engine can be evaluated based on pressure and HRR of the run test conditions. Figures 2 and 3 depict the combustion pressure and HRR under different injection strategies at different loading conditions. The split injection exhibits the lowest pressure curve with two peaks at 50% load condition in Fig. 2 because of late injection with a lesser amount of fuel injected, which results in a delay in the combustion process resulting in lesser pressure. It should be noted that the similar peak pressure is observed in the pilot conditions. The burning of pilot fuel raises the temperature and pressure inside the combustion
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Fig. 2 Pressure and HRR versus Crankangle at 50% load condition
chamber before the main injection, which reduces the ignition delay. The maximum HRR is observed during early pilot injection over single injection. The reason is that the accumulation of the pilot fuel and combined burning with the main injection results into a rapid combustion phase. However, a lower HRR is observed in 30° BTDC and twin pilot compared to single injection and single pilot at 50° BTDC due to increased pressure and temperature prior to the main injection. Furthermore, the combustion pressure and the HRR are lowest with the split injection technique because of the retardation in the SOI timing and the discontinuous combustion. The SOI timing in the split injection technique is retarded to limit the combustion noise from the engine.
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Fig. 3 Pressure and HRR versus Crankangle at 80% load condition
The pressure curve and HRR for the 80% load condition are shown in Fig. 3. The twin pilot injection shows a higher maximum pressure for high load conditions compared to the single pilot and single injection. The pressure is expected to be higher at high load due to the higher temperature, which results in a lower ignition delay. Single main injection results in higher HRR than other strategies as it performs diffusion mode combustion, in contrast to other strategies which integrate premixed and diffusion combustion processes. Two HRR peaks are shown in the split injection system, where one peak shows a rapid combustion phase, and the other shows a mixing controlled combustion phase, which happens due to the second injection after TDC.
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3.2 Performance Analysis It is well known that the Brake thermal efficiency (BTE) indicates the conversion of chemical energy into work. Figure 4 shows BTE at different injection strategies for two load conditions. It should be noted that the improved BTE can be seen at high loads, as it generates more heat during combustion. For high load conditions, a single main injection gives maximum BTE, and single pilot at 30° BTDC shows a minimum; BTE decreases as the pilot move toward TDC, where early pilot injection provides the proper mixing and combustion. A slight variation in BTE is observed for all injection strategies for medium load conditions. Similarly, BSFC represents the amount of fuel is utilized to produce per KW of brake power. Figure 5 shows the BSFC at different loading and injection conditions. It should be noted that the BSFC corroborates with the BTE in Fig. 4. Single main injection gives better BSFC at medium load compared to other condition. A reduced BSFC is observed in advanced pilot injection for all the load conditions. The indicated mean effective pressure can be referenced to the pressure acting on the piston during its stroke to produce the same amount of work. At higher IMEP, a better performance could be expected. Figure 6 shows the IMEP at different injection and load conditions; it shows that the twin pilot condition gives maximum mean adequate pressure compared to other conditions as suffice fuel mixture is expected, resulting in a higher mean pressure. In general, IMEP decreases as the injection timing advances toward TDC, where main heat release is generated during the compression stroke. Note that the split injection condition gives the least effective pressure.
Fig. 4 Brake thermal efficiency at different injection and load condition
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Fig. 5 Brake-specific fuel consumption efficiency at different injection and load condition Fig. 6 IMEP at different injection strategy and load condition
3.3 Emission Analysis Emissions are calculated in terms of NOx , CO, and HC. Figure 7 shows the variation of CO emission at different injection and loading conditions; a single injection gives lower CO emission over multiple injections because of the multi-stage combustion events. Thus, a lower combustion temperature causes a reduced CO oxidation rate. Early single pilot injection shows the highest CO emissions at higher load conditions. However, at medium load, a minimal variation in emissions is observed for all injection strategies. HC emissions are primarily due to unburned fuel escaping after combustion due to wall quenching or lower in-cylinder temperatures. Early pilot injection at 50° BTDC
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Fig. 7 Variation of CO emission at different injection strategy and load condition
gives higher HC emissions over other conditions, as shown in Fig. 8. Higher HC at early pilot injection is due to wall quenching, which remained unburned during the combustion process. The twin pilot injection gives lower HC than the single pilot due to the amount of fuel injected at two stages, which helps in the combustion process. It is known that the NOx emission significantly depends on the combustion temperature. The higher the temperature, the higher the NOx production could be expected. Fig. 9 shows NOx emissions for different injection and loading conditions. A single main injection gives maximum NOx emissions compared to other conditions due to the higher in-cylinder temperature. For injection at single pilot injections, early pilot injection gives slightly higher NOx compared to late pilot injection. Twin pilot injection gives better NOx compared to single pilot as dividing fuel injection into two parts promotes homogeneity of charge with lower combustion temperature. Furthermore, spilt injection gives minimum NOx due to late first injection resulting in lower in-cylinder pressure and temperature. Fig. 8 Variation of HC emission at different injection strategy and load condition
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Fig. 9 Variation of NOx emission at different injection strategy and load condition
3.4 Availability Analysis The performance analysis carried out in this study is based on 1st law of thermodynamics. The injection strategy of single pilot and split injection gives the promising results in terms of performance and emission. Therefore, the qualitative information of both the strategies needs to be investigated. The 2nd law of thermodynamic determines the exergetic efficiency of the system, where optimum injection strategy could be evaluated. The availability of in-cylinder is known to rise in terms of chemical exergy during the injection period. However, the exergetic losses such as exhaust heat and engine cylinder convective losses, and exergetic destruction brought on by chemical reaction cause the total in-cylinder availability to drop. Figures 10 and 11 represent the total in-cylinder availability for 50 and 80% load condition at different injection strategies, respectively. Figure 12 represents the second law efficiency at different injection strategies, 80% load condition gives significantly higher exergetic efficiency than the 50% load condition. In addition, BSFC in Fig. 5 emphasizes the exergetic efficiency for higher load. The exergetic efficiency in split injection is higher compared to the single pilot injection, where chemical exergy destruction is expected to be lower due to low emission. Ain = 1.033 ∗ m f ∗ LHV /3600 Acw = (m w /3600) ∗
cpw ∗ (Two − Twi ) + T0 ∗ cpw ∗ ln(Twi /Two )
(1) (2)
Aexhaust = Q ex + [(m ex /3600) ∗ T0 ∗ {(ceex ∗ ln(To /Texo )) − (Rex ∗ ln(P0 /Pexo )}]
(3)
Adestroyed = Ain − (Ashaft + Acw + Aexhaust )
(4)
Aefficiency = 1 − Adestroyed /Ain ∗ 100
(5)
38 Fig. 10 Availability at different load condition 50% load condition
Fig. 11 Availability at different load condition at 80% load condition
Fig. 12 Exergy efficiency at different injection strategy and load condition
K. V. Warghat et al.
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4 Conclusions This paper investigated the different injection strategies in a CRDI combustion engine. The performance, emission, and availability analysis discussed in detail. The following observation is withdrawn from this study. • The CRDI engine test is performed at five different injection strategies with different injection timing for two load conditions. • Single pilot at 30° BTDC shows higher pressure at medium load conditions, whereas twin pilot injection gives higher in-cylinder pressure at high load. HRR is highest at single and single pilot injection at 50° BTDC for both load conditions. • BTE is lowest at single pilot at 30° BTDC. As the pilot injection angle gets closer to TDC, BTE lowers. In tested injection strategies, a single injection delivers a lower BSFC with improved BTE. However, an early single pilot injection results in a greater IMEP. • Single pilot injection at 50° BTDC shows higher CO and UHC for all loads. Besides, a split injection produces lower emissions compared to other strategies. • Thermodynamic second law efficiency at different injection strategies are studied. Though single pilot injection shows better BTE and lower BSFC but due to emission losses, split condition gives better exergetic efficiency compared to other strategies. Acknowledgements Authors would like to thank Indian Institute of Technology, Hyderabad and Ministry of Education, India, for their constant support and financial assistance. We also thank Jagadish for his assistance in the IC Engine Laboratory.
References 1. MacMillan D, LaRocca A, Shayler PJ, Morris T, Murphy M, Pegg I (2020) Investigating the effects of multiple pilot injections on stability at cold idle for a di diesel engine. SAE Int J Engines 2:14 2. Suh HK (2011) Investigations of multiple injection strategies for the improvement of combustion and exhaust emissions characteristics in a low compression ratio (cr) engine. Appl Energy 88(12):5013–5019 3. Yao M, Wang H, Zheng Z, Yue Y (2010) Experimental study of n-butanol additive and multiinjection on HD diesel engine performance and emissions. Fuel 89:2191–2201 4. Liu J, Yang F, Wang H, Ouyang M, Hao S (2013) Effects of pilot fuel quantity on the emissions characteristics of a CNG/diesel dual fuel engine with optimized pilot injection timing. Appl Energy 110:201–206 5. Rakopoulos CD, Giakoumis EG (2006) Second-law analyses applied to internal combustion engines operation. Prog Energy Combust Sci 32:2–47
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6. Sahoo BB, Saha UK, Sahoo N, Prusty P (2009) Analysis of throttle opening variation impact on a diesel engine performance using second law of thermodynamics. In: Internal combustion engine division spring technical conference, vol. ASME 2009 internal combustion engine division spring technical conference 7. Ismail S, Mehta PS (2011) Evaluation of the effects of fuel and combustion-related processes on exergetic efficiency. Fuel 90(5):1818–1825
Change in Vortex Breakdown Mode and It’s Influence on Flame Shape of a Co/counter Concentric Swirling Streams Atanu Dolai, Prasad Boggavarapu, and R. V. Ravikrishna
Abstract The present study experimentally investigates the flow field of co/ counter-swirling stream under non-reacting and reacting conditions by varying the momentum ratio (M ratio ). We utilized two-dimensional particle image velocimetry (2D-PIV) to examine the flow field. For non-reacting cases, co-swirl shows slender recirculation zone (RZ) at low M ratio . The volume of RZ increases slowly till M ratio = 1.1. However, a drastic change in flow field is observed after M ratio = 1.1. In comparison, the counter-swirl configuration has a small ellipsoidal RZ at low M ratio . Similar to co-swirl, a drastic change in flow topology occurs after M ratio = 1.7. This change in flame topology indicates the change in breakdown mode. PIV images indicate two types of breakdowns: bubble (BVB) and conical (CVB) vortex breakdown. This breakdown modes can be observed in reacting cases also. It is observed that the flame shape strongly depends on breakdown mode. The present study unravels few interesting features of the flow field of co/counter-swirling streams. Keywords Co/counter-swirl · Vortex breakdown · Bubble breakdown · Conical breakdown · Recirculation ratio · PIV
Abbreviation M ratio R
Momentum ratio Recirculation ratio
A. Dolai (B) · P. Boggavarapu · R. V. Ravikrishna Department of Mechanical Engineering, IISc Bangalore, Karnataka 560012, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_5
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1 Introduction Swirling flows can be observed in the natural flows, such as typhoons, and tornados. In combustion systems, such as gas turbines, combustors, boilers, and industrial burners, swirling flows are utilized to improve the mixing between fuel and air. Moreover, the swirling flows provide robust flame stabilization mechanism, which increases the operating range. Thus, swirl flows are investigated for several years. Swirl flows are often categorized by a non-dimension number which is the swirl number. It is defined by the axial flux of angular momentum divided by axial flux of linear momentum multiplied by characteristic radius [1]. If the swirl number is high, the axial pressure gradient created by swirling motion can cause flow reversal. This formation of recirculation zone is a form of vortex breakdown. Several parameters affect the vortex breakdown, such as swirl number, and configuration. Compared to single swirl configuration, the twin-swirl configuration exhibits improved mixing. In the present study, the flow fields of twin-swirl or co/counter-swirl flows are investigated.
2 Literature Review and Objective Lucca-Negro et al. [2] provides an extensive review on different types of breakdown mode. Bubble and spiral breakdown are often found in the swirling flows. Bubble vortex breakdown (BVB) is characterized by a small ellipsoidal recirculation zone [3]. Although spiral breakdown (SB) is considered as a separate bread down mode, BVB and SB both share several features qualitatively. The conical breakdown is first identified by Billant et al. [4]. It is characterized by a flow field with large opening angle. Oberleithner et al. [5] studied the breakdown mode with turbulent swirling jets. Pradeep et al. [3] numerically studied the change in breakdown mode in laminar flows. By reviewing the literature, it is found that breakdown mode and its transition is discussed in the context of single swirl configuration. To the best of the authors’ knowledge, there is no prior study discussing the effect of momentum ratio on the breakdown modes in a twin-swirl configuration. The present study tries to fill this gap in the literature. The paper is organized in the following way. First, a detailed description of experimental setup, flowlines, operating conditions are provided. Then, a few key results are discussed. Finally, key findings are summarized in the conclusion.
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3 Experimental Details 3.1 Experimental Setup Experiments were carried out in a square chamber with complete optical access. The illustration and schematic of the chamber with necessary dimensions are shown in Fig. 1a, b. The cross-section of the chamber is 96 mm × 96 mm. The walls are made of quartz (thickness = 5 mm) which are supported by four stainless steel columns. A conical section and an exhaust tube are placed on the square chamber. The exhaust tube has a smaller cross-sectional area (60 mm × 60 mm). The purpose of these two parts is to prevent any entrainment from the environment. The injector configuration is shown in detail in Fig. 2a, b. Figure 2a shows the inner and outer tubes and two axial swirlers are placed inside the tube to impart swirling motion. For the outer tube, an axial swirler with 12 straight vanes and a vane angle (θ out ) of 45° is used. The vane angle is kept the same for all operating conditions. The axial swirler can be interchanged for the inner tube depending on the configuration. For the co-swirl configuration, the vane angle for the inner swirler (θ in )
Fig. 1 a Illustration of the square chamber with conical and exhaust section, b schematic of the chamber where Di = 28 mm, Do = 48 mm, S chamber = 96 mm, S exit = 60 mm, L1 = 200 mm, L2 = 50 mm, and L3 = 50 mm (acronyms: I → inner stream, O → outer stream)
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is 45°, whereas the angle is changed to − 45° for the counter-swirl configuration. Here, the negative vane angle implies that the inner and outer streams are swirled in the opposite direction. The geometric swirl number for inner and outer tubes are calculated using Eq. 1 [6], 3 D1 1− D 2 2 SN = tanθ 2 3 D1 1− D 2
(1)
where D1 and D2 are the inner and outer diameters of the tubes, and θ is the vane angle. The geometric swirl number of the outer stream (SNout ) is 0.83. For the inner stream, the swirl number (SNin ) is 0.67 and − 0.67 for the co and counter-swirl configuration, respectively. Figure 3 shows the schematic of the flowlines. For non-reacting cases, air is supplied through the inner and outer tubes. The flow rates of air passing through each tube are precisely controlled using mass flow controllers (MFC). All MFCs (Make: ALICAT) range from 0 to 800 SLPM and have an accuracy of ± 0.8%. In the present study, only the air flowing through the outer tube is seeded with particles. The flowlines are slightly modified for reacting cases. A preheated fuel is passed through the inner tube, and oxidizer is provided through the outer tube. Similar to the non-reacting case, the outer stream is seeded with particles.
Fig. 2 a Illustration of the injector configuration showing the axial swirlers placed inside inner and outer tube, b schematic of the injector showing inner and outer tube, where L4 = 50 mm and L5 = 20 mm
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Fig. 3 Schematic of the flowlines
3.2 Operating Conditions The experiments are divided into two parts: non-reacting and reacting. The effect of outer to inner momentum ratio (M ratio ) on co/counter-swirl flow field is investigated under both reacting and non-reacting conditions. To change M ratio , the flowrates of air passing through inner and outer tube are changed for non-reacting cases. For reacting cases, the flowrate of outer stream is changed to achieve desired momentum ratio. Table 1 lists the flow rates of inner and outer streams at different momentum ratios (M ratio ) for non-reacting cases. For reacting cases, a fuel with a temperature of 860 K is supplied through the inner tube. The fuel comprises CO (13.745%), CO2 (12.595%), CH4 (0.98%), H2 (11.82%), H2 O (5.485%), O2 (0.29%), and N2 (55.085%). All percentages represent the mole fraction of individual species. This is the typical gas composition at the catalytic stage exit operating at fuel-rich conditions [7]. Table 2 lists the operating conditions for reacting cases. Table 1 Operating conditions for studying the effect of momentum ratio (M ratio ) for non-reacting cases Configuration
Inner stream flowrate (slpm)
Outer stream flowrate (slpm)
Momentum ratio (M ratio )
Co/counter-swirl
300
250–600
0.43–2.45
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Table 2 Operating conditions for studying the effect of momentum ratio (M ratio ) for reacting cases Configuration
Inner stream mass flowrate (g/s)
Outer stream mass flowrate (g/s)
Momentum ratio (M ratio )
Co/counter-swirl
5.172
7.81–11.72
0.4–0.9
3.3 Particle Image Velocimetry (PIV) Two-dimensional particle image velocimetry (2D-PIV) is utilized to capture several features of flow fields for various conditions. The schematic of the PIV experimental setup is shown in Fig. 4. The PIV experimental setup consists of a duel-pulsed Nd:YAG laser, sheet optics, and a CCD camera equipped with a Nikkor lens (focal length = 105 mm, f 5.6) and a bandpass filter of 532 nm (FWHM = 10 nm). The seeding particles are illuminated using a laser with 532 nm wavelength, which is the second harmonic of the Nd:YAG laser. The output of the laser is expended into a sheet using sheet optics, and the interval between two pulses is kept at 10 µs. The interrogation area is 86 mm × 95 mm. Alumina with a mean diameter of 1 µm is selected as seeding particles. The relaxation time and Stokes’ number corresponding to the particles are ~ 1.2 × 10−5 s and ~ 3 × 10–4 , which implies that the particles are suitable for the present study. The resolution of the CCD is 2048 pixels × 2048 pixels, and the velocity vectors are calculated with a window size of 32 pixels × 32 pixels (overlap = 75%) using Davis software. For each condition, 200 images are captured to evaluate the mean, standard deviation, and other statistics of the flow field.
Fig. 4 Schematic of PIV setup
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4 Results and Discussion 4.1 Bubble and Conical Breakdown Mode for Non-Reacting Cases Figure 5 shows the streamlines at different momentum ratios (M ratio ), where the dashed line indicates zero axial velocity. For brevity, three momentum ratios are shown for co/counter-swirl configuration. Streamlines are colored by the magnitude of the axial velocity (U x ) multiplied by its sign. For co-swirl configuration, the flow consists of a Y-shaped recirculation zone (RZ) at M ratio = 0.43. The strength of the recirculation is relatively higher near the dump plane. The volume of the recirculation zone progressively increases till M ratio = 1.1, maintaining a similar flow topology. However, at M ratio = 1.7, a drastic change in the flow field is observed. Downstream of the dump plane, the incoming jet expands radially till it encounters the chamber wall. The flow field comprises a large recirculation zone (RZ) which spans more than 100 mm from the dump plane. The volume of RZ increases, and it moves toward the dump plane when M ratio is increased further to 2.45. For counter-swirl configuration, the flow field shows some interesting features. For M ratio ≤ 1.7, flow consists of a small ellipsoid-shaped recirculation zone (RZ) near the dump plane. This RZ is generated possibly because of the vigorous mixing
Fig. 5 Streamlines for co/counter-swirl flows at different momentum ratios (M ratio ) where the dashed line indicates zero axial velocity (streamlines are colored by the axial velocity)
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between the streams as the swirl directions are opposite to each other. The tangential momentum becomes negligible after mixing, and no recirculation zone is present beyond x ~ 30 mm. The corresponding streamlines also indicate the columnar nature of the flow. In contrast to the co-swirl configuration, the shape and the volume of RZ do not change with M ratio . A transition of flow topology is observed across M ratio = 1.7. The axial velocity of the inflow swirling jets decelerates, and the flow expands radially with a large opening angle (~ 60°). The volume of the RZ increases significantly, and the flow field closely resembles the co-swirl configuration at that same condition (see Fig. 5). Thus, Fig. 5 shows several interesting flow topologies and transitions for co and counter-swirl configurations. We can observe two prominent modes of vortex breakdown (VB) modes: bubble (BVB) and conical (CVB) mode. The presence of a small ellipsoidal recirculation zone (RZ) is a distinct feature of bubble (BVB) breakdown mode [3], whereas, in conical (CVB) breakdown, the inflow jet expands radially after the stagnation point [3, 4]. For co-swirl, it is inconclusive whether the breakdown mode is a bubble or conical when M ratio ≤ 1.1. However, a conical breakdown mode is evident for M ratio > 1.1. For counter-swirl, a bubble breakdown is observed when M ratio ≤ 1.7. The bubble consists of a stagnation point followed by a pair of counter-rotating recirculation zone. Figure 5 suggests a change in breakdown mode from BVB to CVB across M ratio = 1.7.
4.2 Recirculation Ratio (R) for Non-Reacting Cases To understand the change in swirl strength, we estimated the recirculation ratio (R) for co/counter-swirl configurations at different momentum ratios (M ratio ). The recirculation ratio (R) represents the recirculated volume flow rate (Qre ) divided by the inlet flow rate (Qin ). The calculation of Qre is not straightforward from twodimensional (2-D) PIV images. However, if we assume that the mean axial velocity profile (Ux ) is axisymmetric, then Qre can be estimated using the following relation [8], +r Q re =
−π × r × Ux dr
(2)
−r
where the mean axial velocity (U x ) is negative within a radius of r from the center. The volume of recirculated gas varies with axial location. It is possible to calculate maximum recirculated volume at various M ratio . Then, we can construct a bifurcation diagram shown in Fig. 6, where the input parameter in momentum ratio (M ratio ) and the output parameter is the maximum recirculated flow rate (Qre_max ).
Change in Vortex Breakdown Mode and It’s Influence on Flame Shape …
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Fig. 6 The variation of maximum recirculated volume (Qre_max ) with momentum ratio (M ratio )
Figure 6 shows the variation of Qre_max with M ratio for the co/counter-swirl configuration. For co-swirl, Qre_max increases steadily till M ratio = 1.1. The recirculated flow rate increases by twofolds when M ratio changes to 1.7. This sudden increment marks the change in vortex breakdown mode. For M ratio ≥ 1.7, the flow field shows the characteristics of conical breakdown mode (CVB). An interesting trend is observed for the counter-swirl configuration (see Fig. 6). The recirculated volume stays nearly constant when M ratio ≤ 1.7. As discussed before, the flow field consists of a spindleshaped small recirculation zone which indicates a bubble form of breakdown (BVB). When the M ratio is increased slightly from 1.7, the flow field abruptly changes, and the recirculated flow rate increases by nearly three times. The recirculated flow rate is even higher than that observed in the co-swirl configuration. For M ratio > 1.7, Qre_max increases slightly. Thus, it can be concluded that the flow exhibits different vortex breakdown modes, and Qre_max can capture the transition from one mode to another.
4.3 BVB and CVB Modes for Reacting Cases Figure 7 shows the effect of momentum ratio (M ratio ) on co/counter-swirl reacting flows. It should be noted that the equivalence ratio changes with M ratio . The velocity fields are more symmetric compared to non-reacting cases. At M ratio = 0.4, for co-swirl, the velocity field resembles the non-reacting case at a similar momentum ratio. A slender recirculation zone is present near the dump plane. A ‘M’-shaped flame is observed in this condition. For counter-swirl, similar to non-reacting cases, a small recirculation bubble is found near the dump plane. At M ratio = 0.62, the flame shape suddenly changes to a ‘V ’-shaped flame, and the corresponding flow consists of a large RZ. The shape of RZ indicates that the vortex breakdown mode
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Fig. 7 Streamlines combined with flame luminosity at different M ratio for co/counter-swirl flames where arrow denotes the flow direction
is CVB. However, no change is observed for counter-swirl configuration. At M ratio = 0.9, both configuration show the characteristics of conical breakdown mode, and the corresponding flame shape is ‘V ’-shaped. Thus, a transition in breakdown mode is also observed for reacting cases. However, the transition occurs at a much lower momentum ratio as compared to the non-reacting cases. Moreover, the flame shape depends on the breakdown mode.
5 Conclusions The present study investigated the flow field of concentric co/counter-swirling streams by varying the momentum ratio (M ratio ) under non-reacting and reacting conditions using particle image velocimetry (PIV). The flow field shows two types of breakdown modes: bubble (BVB) and conical (CVB) breakdown modes. A transition of breakdown mode is observed for both configurations. Particularly, the counterswirl configuration exhibits a transition from BVB to CVB at M ratio beyond 1.7. This transition occurs at different momentum ratios for co and counter-swirl configuration. The recirculation ratio (r) is calculated from the velocity data. It is found that a sharp change in recirculated volume flowrate is suitable for capturing the transition.
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A change in breakdown mode is also observed for reacting cases. The flame shape strongly depends on the breakdown mode. A ‘M’-type flame is observed for BVB, whereas CVB creates a ‘V ’-shaped flame. Thus, the present study unravels few interesting features of the flow field of co/counter-swirl and encourages to investigate such flows by varying parameters such as density ratio, heat release rate. Acknowledgements The authors would like to thank Dr. Pabitra Badhuk and Dr. Pradeep Moise for several useful discussions.
References 1. Syred N, Beer JM (2017) Combustion in Swirling flows: a review. J Phys Conf Ser 891:12237. https://doi.org/10.1088/1742-6596/891/1/012237 2. Lucca-Negro O, O’Doherty T (2001) Vortex breakdown: a review. Prog Energy Combust Sci 27:431–481. https://doi.org/10.1016/S0360-1285(00)00022-8 3. Moise P, Mathew J (2019) Bubble and conical forms of vortex breakdown in swirling jets. J Fluid Mech 19:322–357. https://doi.org/10.1017/jfm.2019.401 4. Billant P, Chomaz JM, Huerre P (1998) Experimental study of vortex breakdown in swirling jets. J Fluid Mech 376:183–219. https://doi.org/10.1017/S0022112098002870 5. Oberleithner K, Sieber M, Nayeri CN, Paschereit CO, Petz C, Hege HC, Noack BR, Wygnanski I (2011) Three-dimensional coherent structures in a swirling jet undergoing vortex breakdown: stability analysis and empirical mode construction. J Fluid Mech 679:383–414. https://doi.org/ 10.1017/jfm.2011.141 6. Huang Y, Yang V (2009) Dynamics and stability of lean-premixed swirl-stabilized combustion. Prog Energy Combust Sci 35:293–364. https://doi.org/10.1016/j.pecs.2009.01.002 7. Dolai A, Ravikrishna RV (2021) An experimental investigation of syngas combustion using a 26 kW two-stage combustor. In: Experimental setup, p 4 8. Degeneve A, Vicquelin R, Mirat C, Caudal J, Schuller T (2021) Impact of co- and counterswirl on flow recirculation and liftoff of non-premixed oxy-flames above coaxial injectors. Proc Combust Inst 38:5501–5508. https://doi.org/10.1016/j.proci.2020.06.279
Entrained Dust Combustion in Pre-Heated Air Mohd. Tousif, A. Harish, and V. Raghavan
Abstract In many practical applications, such as in blast furnaces and boilers, pulverized coal is used as a reducing agent and energy source. This chapter presents the predicted characteristics of naturally entrained coal dust flames in a sudden expansion region. Further, it also investigates the effect of particle size and equivalence ratio on the flame dynamics. Commercially available ANSYS Fluent has been used for the numerical simulations. Discrete phase modelling is used for sub-bituminous coal particles tracking. Variable thermo-physical properties are employed. A skeletal chemical kinetic mechanism for methane consisting of 25 species and 121 reactions are used. A User Defined Function (UDF) is used to include the radiation due to species and soot particles. For a given coal flow rate, mass flow rate of air is varied to maintain the required global equivalence ratio at a constant temperature of 1273 K. Particle size distribution (1–25 and 53–63 μm) and equivalence ratio (0.65–0.95) are varied. Steady-state governing equations are solved till convergence. The flame dynamics is investigated through the contours of temperature, devolatilization zones, velocity vectors, mass fraction distribution of important species soot formation and reduction. Keywords Pulverized coal · Discrete phase modelling · User defined function · Equivalence ratio · Flame dynamics
Abbreviation r z Tg Tp φ
Radial distance [m] Axial distance [m] Maximum gas temperature [K] Particle temperature [K] Equivalence ratio
Mohd. Tousif · A. Harish · V. Raghavan (B) Department of Mechanical Engineering, IIT Madras, Chennai 600036, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_6
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1 Introduction Direct pulverized coal combustion has applications in boilers, steel plants, furnaces and fire hazards in coal mines [1]. The combustion of coal is also associated with safety and health hazards. During mining, methane gas is released into the mine and micron-sized coal particles also exist, as reported by Luo et al. [2] (around 40 g coal dust per ton of coal mined). These coal particles, called coal-dust, remains suspended in air. This suspended coal can create explosions in the presence of high-temperature air. Further, in a blast furnace, pulverized coal particles are injected into a hot air blast of around 1473 K. Therefore, there is a need to understand the combustion of coal dust in high-temperature air and this forms the motivation of the present work.
2 Literature Review and Objective The combustion of dust particles in air has been a topic of flammability research over the years. These studies were conducted to determine the fundamental characteristics of a wide range of pulverized coal dust particles. A review paper by Smooth and Horton [3] and one by Krazinski et al. [4] have reported several studies on this topic. Essenhigh and Csaba [5] developed a one-dimension flame front propagation model in a premixed dust cloud formed by finely ground coal particle suspended in air. This model was extended by Bhaduri and Bandyopadhyay [6] to include heat generation due to the chemical reactions and the data on burning velocity, gas concentration and flame length predictions were compared with experimental data from literature. Graves and Wendt [7] reported their experimental investigation of laminar opposed jet diffusion flame formed by pulverized coal. Wang et al. [8] used ANSYS Fluent to numerically investigate the coal dust explosions in a spherical explosion vessel and reported that dust particles of 1 micron size were uniformly distributed throughout the vessel. Tan et al. [9] compared the explosion characteristics of micron and nano sized coal particles. Their results showed that the characteristics of explosion pressure were strongly related to the concentration and size of the particle. Coal dust or pulverized coal particle combustion is encountered in many practical applications. But there is not much literature available on the naturally entrained coal combustion dynamics in the presence of a hot air blast. Therefore, the main objective of the present work is to investigate the flame characteristics of sub-bituminous pulverized coal or coal dust in a high temperature air blast of 1273 K. The effect of equivalence ratio and the particle size distribution on the flame structure is also studied.
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3 Numerical Methodology 3.1 Numerical Methods and Sub-Models Details of the numerical model is available in Tousif et al. [10]. ANSYS Fluent 16.1 is used for the numerical simulations of particle-laden reactive flow, where governing equations are solved in a segregated manner with a double precision solver. Volumetric particle loading is < 10%, therefore, Discrete Phase Modeling (DPM) is employed for predicting the particle transport and its interaction with the continuous phase. For convection terms, an upwind scheme of second-order and Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm for pressure velocity coupling are used. Further, gradient based least square cell method is used for the spatial discretization of pressure. Arrhenius form based finite rate chemistry is used to calculate the reaction rates of individual species. Moreover, species transport including the stiff chemistry solver and volumetric reactions are employed. Species diffusion due to concentration gradient and temperature gradient is also included. Reduced skeletal kinetic mechanism consisting of 25 species and 121 emementary reactions, has been employed. All thermo-physical properties have been claculated based on temperature and species concentration using appropriate polynomial fits and kinetic theory-based approach. These are achieved by the inputting three CHEMKIN files chemical kinetic mechanics, thermodynamic data and transport data files. Molecular weight, standard state enthalpy, entropy and Lennard–Jones parameters of all the species have been taken from thermal and transport files in CHEMKIN format [11]. Pulverized coal particles are tracked using transient DPM, which is employed for the particle phase prediction. However, particle–particle interactions are neglected. Appropriate submodels for the particles (inert heating, moisture removal, char combustion) are employed. All the governing equations to handle the mass transfer, momentum transfer and energy transfer such as particle heating and heat release due to char combustion are also included. Coal particle undergoes various process during combustion. Initially, the particle undergoes sensible heating (inert heating) till a temperature of 373 K. At this temperature, moisture is removed. Further, particle undergoes inert heating till it reaches its pyrolysis temperature (600 K), where volatiles are released at a specified rate. In this present study, the devolatilization rate is calculated using Arrhenius type singlestep equation [12]. These have been modelled systematically. Heat transfer due to convection and radiation are included for the particle energy balance (ANSYS Fluent Theory Guide [13]). Volatiles consist of a mixture hydrocarbon species over a wide range of molecular weights. Here, it is assumed that volatiles are composed of CO, N2 , H2 , C2 H4 , CH4 and tar [14], and the concentration of each species is calculated using the method prescribed by Peterson and Werther [14]. The fractions of CO, N2 , H2 , C2 H4 , CH4 , and C2 H2 are 35.1%, 8.1%, 6.8%, 33.9%, 4.8% and 11.3%, respectively. DPM submodel in Fluent can release only one devolatilizing species from the pulverized coal. Therefore, to include multi-species devolatilization, a User Defined
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Function (UDF) is employed. Complete devolatilization is immediately followed by char combustion which is modelled by kinetic/diffusion limited rate [15].
3.2 Computational Domain The computational domain employed in this study is similar to that used by Lee et al. [16] and it consists of an axisymmetric model of a typical Bunsen burner. Geometry creation and meshing are carried out using ANSYS Workbench. Computational domain has a burner made up of mild steel of 10 mm diameter and 251 mm in height as shown in Fig. 1. The burner wall has a thickness of 1.5 mm. There is an orifice of 1 mm diameter and thickness at the upstream of the burner. The computational domain is extended by 30 and 250 mm in the radial and axial direction, respectively from the burner rim. Multi-block structured grid is used with spacing which is nonuniform in both axial as well as in the radial directions. Near the burner exit, fine cells are used to capture the steep gradients. Fig. 1 Computational domain
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3.3 Boundary Conditions Imposed boundary conditions are presented below. Axis: Here, the first gradient of velocity, species mass fraction and temperature in the radial direction are set to zero. Radial component of velocity is zero at this boundary. Velocity inlet: A uniform velocity of air (normal to the boundary) calculated based on the equivalence ratio for a given coal flowrate, at a temperature of 1273 K, with oxygen and nitrogen mole fractions are 0.21 and 0.79, respectively. Pressure inlet: Atmospheric pressure is specified at the far boundary location. If the gas mixture leaves the domain depending upon the pressure gradient, all the flow parameters at this boundary are extrapolated from the interior. When the flow enters the domain, mole fraction of oxygen and temperature are specified as 0.21 and 300 K, respectively. Pressure outlet: Through this boundary, bulk of the products and particles leave the computational domain. This is also a pressure specified boundary. During adverse pressure gradient, air enters the domain at a temperature of 300 K. Coal dust injection: Pulverized coal particles of required size distribution with a mass flow rate of 1E−06 kg/s are injected at zero velocity in the immediate downstream of the orifice plate as indicated Fig. 1. DPM particles escape through fluid boundaries and reflect at walls. Coal dusts of sizes 1–25 μm and 53–63 μm are injected into the domain at ambient temperature (300 K). Rosin–Rammler distribution is used for particle size distribution. Burner wall: Coupled condition is imposed at the solid–fluid interface, where heat transfer between solid and fluid considered. A temperature of 300 K is imposed at the outer walls.
3.4 Grid Independent Study Grid independent study is carried out for a hybrid flame, where coal particles are naturally entrained into the premixed methane-air flame. Grid independence study is carried out with non-uniform grids having minimum cell sizes of 0.25 mm × 0.25 mm (35,176 cells), 0.1 mm × 0.1 mm (73,260 cells) and 0.05 mm × 0.05 mm (123,020 cells). Profiles of average temperature, species mass fraction and velocity magnitude are plotted along axial and radial directions. It is observed that the location of the peak values of average temperature and mean CO2 mass fraction does not change notably on changing the total number of grids from 73,260 to 123,020. Therefore, a grid having 73,260 cells with the smallest grid size of 0.1 mm has been used for further study.
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3.5 Solution Procedure The computational domain is first initialized using the default option available in ANSYS Fluent after specifying required boundary conditions. First, non-reactive steady state simulations without particle phase are carried out. After convergence, coal particles are injected at a required rate and tracked using a time step of 1 × 10–4 s. Simultaneously, steady-state homogenous gas-phase volumetric reactions are solved using stiff chemistry solver. The solution convergence criteria is set as the normalized residual values being < 10–3 for continuity, momentum, turbulence and species equations, whereas 10–6 is set for energy convergence. Required number of iterations is executed accordingly. All the variables are averaged for at least one flow time and they are presented in this chapter.
4 Results and Discussion This section presents the effect of the particle size distribution (1–25 and 53–63 μm) and equivalence ratio (0.65–0.95) on the flame dynamics. Coal dust and air are injected at the same time and the simulations are carried out for around one flow time, since the focus is only on volatile combustion and to understand the location of the start of char combustion.
4.1 Effect of Equivalence Ratio Figure 2 depicts the contours of temperature and mass fraction of oxygen at three equivalence ratios of 0.65, 0.8 and 0.95. The reaction zones are constituted by combustion of volatiles and subsequently combustion of char (fixed carbon). Based on the equivalence ratio, the locations of formation of these zones differ. Due to high temperature of air, the devolatilization is expected to occur almost instantly, close to the orifice exit from which air is injected. As the coal particles are injected as a coflow to the air jet coming out of the orifice, air is able to entrain the coal particles into it. Therefore, volatiles and char can burn at different zones based on the entrainment characteristics of coal dust into hot air stream. It is clear from the isotherms in Fig. 2 that the extent of the reaction zone closer and parallel to the axis reduces, by following the isotherms of 1800, 1600 and 1400 K. These isotherms are closer to the axis. On the other hand, higher temperature zone, 2000 K, is present in the zone away from the axis and its extent clearly depends upon the equivalence ratio. It is clear that an intermediate value of equivalence ratio results in distinct outer reaction zone embedded by the 2000 K isotherm at the particular time instant. The maximum gas temperatures (T g ) for φ = 0.65, 0.80 and 0.95 are 2123 K, 2143 K and 2099 K, respectively and the corresponding maximum particle temperatures (T p )
z (m)
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Fig. 2 Contours of mass fraction of oxygen (flooded) along with temperature isotherms for 53–63 μm particle size for (left) φ = 0.65, (centre) φ = 0.80 and (right) φ = 0.95
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are 2064 K, 2020 K and 1946 K, respectively (not shown in Fig. 2). It is observed that for the particle size range in this case, T g is always higher than T p . This is because of the lag in the particle heat up process. With an increase in the equivalence ratio, the mass fraction of oxygen close to the orifice exit decreases as observed from Fig. 2 (flooded region), indicating its consumption. A relatively higher mass fraction of oxygen is present around the orifice exit in the recirculation zone for φ = 0.65. These distributions are due to the air velocity values as well as the strength of recirculation zones formed around the orifice exit. Figure 3 presents the contours of devolatilization (lines) along with the contours of heat release rate (flooded). As the coal particles entrain into the hot air, volatiles are almost instantaneously released, for all lean equivalence ratio values considered in this study. Air stream at 1273 K is able to entrain coal dust particles and heat them to devolatilization temperature much near to the orifice exit. Based on the availability of air (equivalence ratio) and strength of recirculation zone, mixing of volatiles and air takes place and heat is released upon combustion. It is clear that around the region of volatile release, major heat release also occurs. It is apparent that the case of φ = 0.8 displays distinct features around one flow time of particle injection, as shown by some regions of higher heat release rate occurring much away from the orifice exit but closer to the axis. It should also be noted that the maximum values of heat of reaction for φ = 0.65, 0.80 and 0.95, are 0.044 W, 0.068 W and 0.080 W, respectively, and the volatile release slightly increases (varying in the range of 1.27E−09 to 1.29E−09 kg/ s). As the mixture moves from lean towards the stoichiometric value, the heat of reaction increases. However, its distribution depends upon the air velocity and the strength of recirculation zone formed besides the orifice.
60 0.07 0.05 0.03 0.01
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Fig. 3 Contours of devolatilization rate (lines) along with heat of reaction (flooded) for 53–63 μm particle size for (left) φ = 0.65, (centre) φ = 0.80 and (right) φ = 0.95
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Figure 4 presents the contours of soot volume fraction along with mass faction of soot precursor (C2 H2 ) at three equivalence ratios of 0.65, 0.8 and 0.95. The air inlet velocities corresponding to the three equivalence ratios are 88.8 m/s, 74.9 m/ s, and 64.8 m/s, respectively. There is a rise in the maximum mass fraction of C2 H2 with an increase in the equivalence ratio as observed from Fig. 4. With an increase in equivalence ratio, the velocity in the coal injection region decreases. Therefore, for the same devolatilization rate, the concentration of soot precursor is higher. The mass fraction of acetylene for case φ = 0.95 is around 15 times higher than that at φ = 0.65 case. Eventually, there is similar increase in the soot production as observed in Fig. 4. Further, reduced oxygen at higher equivalence ratios, also assists the soot production apart from the abstraction of hydrogen from C2 H2 . Moreover, majority of the soot is produced in the recirculation zone close to the orifice exit, where oxygen penetration is limited (Fig. 2).
4.2 Effect of Particle Size Particle size also has an important impact on coal dust combustion due to varying drag force, heat capacity, heating rate and surface area. As the particle size reduces, the heating rate of particle increases, which will lead to an earlier release of volatiles, and a similar effect will be observed during char oxidation. Currently, the effect of particle size distribution on the flame characteristics has been studied using particles in the size range of 1–25 and 53–63 μm, and the results are presented systematically in this section. Here, the overall equivalence ratio is kept constant at 0.75. Figure 5 presents the contours of temperature and mass fraction of oxygen for particle size distribution of 1–25 μm (left) and 53–63 μm (right) at φ = 0.75.
Entrained Dust Combustion in Pre-Heated Air
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Fig. 4 Contours of mass fraction of C2 H2 (lines) along with soot volume fraction (flooded) for 53–63 μm particle size for (left) φ = 0.65, (centre) φ = 0.80 and (right) φ = 0.95
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Coal and air flow rates are maintained same for both cases. For the same coal dust injection rate, the surface to volume ratio of 1–25 μm particles is much higher than that of 53–63 μm particles. Case with particles of 1–25 μm has a maximum temperature of 2210 K and that with 53–63 μm has a lower maximum temperature of 2137 K. The difference between gas-phase temperature, T g , and particle temperature, T p , is higher for 53–63 μm particles (around 132 K) than that for 1–25 μm particles (about 20 K). This is mainly due to lower rate of heating of larger sized particles. Further, in the case with 1–25 μm particles, the 2000 K isotherm is present in the recirculation zone much closer to the orifice exit and then it extends in the axial direction, much closer to the axis. Here, the consumption of oxygen around the orifice exit region is much higher. On the other hand, for particles of size 53–63 μm, the 2000 K isotherm occupies a larger area in the recirculation zone, as well as along the axial direction as shown in Figs. 5, 6 and 7. The contours of mass fraction of oxygen indicate higher penetration of oxygen in to the recirculation zone, increasing the volatile combustion area for this case. Figure 6 presents line contours of devolatilization rate (kg/s) and flooded contours of heat of reaction (J/s) for cases with particle size in the range of 1–25 μm and 53– 63 μm at φ = 0.75. The flame characteristics depends on how fast volatiles are released and their gas phase combustion kinetics. The devolatilization zone for the case of 1–25 μm occurs over a broader area with larger magnitude, indicating the higher rate of volatile release, when compared to the case with 53–63 μm particles. This is due to enhanced heat and mass transfer process facilitated by larger surface to volume ratio. For the given overall equivalence ratio, it is clear that the heat release
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occurs over a smaller area with much higher magnitude for the particles in 1–25 μm range, when compared to that in 53–63 μm range, as observed in Fig. 6. Further, for the case with 53–63 μm particles, the heat release is much distributed, almost uniformly distributed in the recirculation zone. The char combustion starts much closer to the orifice exit (around 20 mm) for 1–25 μm particles and it starts at a farther distance (around 80 mm) for 53–63 μm particles. Figure 7 presents the flooded contours of soot volume fraction along with the contour lines of mass fraction of soot precursor, C2 H2 , for particle sizes of 1–25 and 53–63 μm at φ = 0.75. The mass fraction of C2 H2 varies between 0.01 and 0.03 for 1–25 μm case and it is much lower (0.0002–0.002) for 53–63 μm case. A higher quantity of soot precursor species is produced due to the pyrolysis of the volatiles in the fuel rich region close to the orifice for the case with smaller particles. This in turn leads to formation of a higher soot volume fraction (around 40 times higher) for 1–25 μm particles as compared to the case with larger particles. The axial extent to which soot is formed is slightly lower for the lower particle case and its radial extent is higher. Figure 8 presents the flooded contours of soot oxidation rate and line contours of mass fraction of OH for particle sizes of 1–25 and 53–63 μm at φ = 0.75. Maximum soot oxidation rate is around 100 times higher for the case with 1– 25 μm particles when compared to the case with 53–63 μm particles. Soot oxidation occurs over a much larger area for the smaller particle case, as the higher volume of soot produced is oxidised based on the availability of oxygen, temperature and residence time. Volatile combustion consumes the oxygen at a faster rate much closer to the orifice exit and the remaining oxygen and OH radical are able to oxidize the soot subsequently.
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Fig. 8 Contours of soot oxidation rate (flooded) along with mass fraction of OH (lines) for φ = 0.75 and particle size of 1–25 μm (left) 53–63 μm (right)
0.012
0.008
0.002 0.0014 0.0008 0.0002
1E-05 8E-06 4E-06 2E-06
0.0015
z (m)
0.0017 0.005 0.004
0.004
0 0.003
0.003 0 r (m)
0
5 Conclusions The effects of equivalence ratio and particle size distribution on coal dust combustion in hot air flames are presented. Hot air is supplied at a temperature of 1273 K. Coal particle size is varied in the range of 1–25 μm (left) and 53–63 μm (right). Coal is injected at a rate of 1E−6 kg/s. A scenario, where coal dust is injected along with hot air flow (1273 K), has been considered with a focus of understanding devolatilization, volatile combustion, start of char oxidation, soot formation and its oxidation. An increase in overall equivalence ratio leads to a reduction in reaction zone height around the orifice. For a given particle size distribution, with an increase in overall equivalence ratio, the devolatilization rate remains almost a constant, but the heat release rate increases due to reduced air inlet velocity. The difference between the particle and gas temperatures is observed to be higher for bigger particles as compared to that of smaller particles. In the case of smaller particles, the maximum temperature, devolatilization rate and heat of reaction are quite higher. For smaller particles, soot formation and its oxidation are much higher compared to those in larger sized particles due to the production of higher of volatile species that contain soot precursors. The OH in the domain is almost three times lower for smaller particles when compared to that of larger particles, indicating its consumption for that case.
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Acknowledgements The authors thank the P. G. Senapathy centre for computing resources, IIT Madras for providing the computational facility.
References 1. Suda T, Takafuji M, Hirata T, Yoshino M, Sato J (2002) A study of combustion behavior of pulverized coal in high-temperature air. Proceed Combust Instit 29(1):503–509 2. Luo Y, Wang D, Cheng J (2017) Effects of rock dusting in preventing and reducing intensity of coal mineexplosions. Int J Coal Sci Technol 4:102–109 3. Smooth LD, Horton MD (1977) Propagation of laminar pulverized coal-air flames. Prog Energy Combust Sci 3(4):235–258 4. Krazinski JL, Buckius RO, Krier H (1979) Coal dust flames: a review and development of a model for flame propagation. Prog Energy Combust Sci 5(1):31–71 5. Essenhigh RH, Csaba J (1963) The thermal radiation theory for plane flame propagation in coal dust clouds. Symp Int Combust 9(1):111–125 6. Bhaduri D, Bandyopadhyay S (1971) Combustion in coal dust flames. Combust Flame 17(1):15–24 7. Graves DB, Wendt JOL (1982) Flammability characteristics and structure of a pulverized coal. Laminar Opp Jet Diff Flame Symp Int Combust 19(1):1189–1196 8. Wang ZH, Weng WB, He Y, Li ZS, Cen KF (2015) Effect of H2 /CO ratio and N2 /CO2 dilution rate on laminar burning velocity of syngas investigated by direct measurement and simulation. Fuel 141:285–292 9. Tan B, Shao Z, Xu B, Wei H, Wang T (2020) Analysis of explosion pressure and residual gas characteristics of micro-nano coal dust in confined space. J Loss Prevent Process Ind 64:104056 10. Tousif M, Harish A, Kumaran SM, Raghavan V (2020) Numerical study of interaction of coal dust with premixed fuel-lean methane-air flames. Adv Powder Technol 31:3833–3844 11. Kee RJ, Rupley FM, Miller JA (1989) Chemkin-II: a Fortran chemical kinetics package for the analysis of gas-phase chemical kinetics (No. SAND-89-8009). Sandia National Lab.(SNL-CA), Livermore, CA 12. Silaen A, Wang T (2010) Effect of turbulence and devolatilization models on coal gasification simulation in an entrained-flow gasifier. Int J Heat Mass Transf 53(9–10):2074–2091 13. ANSYS Inc (2016) ANSYS fluent theory guide 16.0 14. Petersen I, Werther J (2005) Experimental investigation and modeling of gasification of sewage sludge in the circulating fluidized bed. Chem Eng Process Process Intensif 44(7):717–736 15. Saxena SC (1990) Devolatilization and combustion characteristics of coal particles. Prog Energy Combust Sci 16(1):55–94 16. Lee M, Ranganathan S, Rangwala AS (2015) Influence of the reactant temperature on particle entrained laminar methane–air premixed flames. Proceed Combust Instit 35(1):729–736
An Experimental Investigation into the GDI Spray Characteristics of Ethanol and Lemon Peel Oil G. M. Nayak, B. Abinash, B. Yogesh, V. W. Ketan, P. S. Kolhe, and B. Saravanan
Abstract The fuel injection phenomenon significantly influences engine performance and emissions in a Gasoline Direct Injection (GDI) engine. Before using biofuel in a GDI engine, it is critical to understand its spray structure and combustion quality. The current study seeks to deepen the understanding of the spray behaviour of Lemon Peel Oil (LPO) and ethanol in a controlled environment under different pressure, temperature, and engine-like conditions. The thermophysical properties of the fuels help to comprehend the spray characteristics. The liquid spray morphology of fuels is captured using a standard Mie-scattering technique, and the spray penetration length is compared to the baseline fuel isooctane. The constant volume spray study revealed that the Isooctane has the shortest penetration. The higher boiling point of LPO allows for a longer spray tip penetration. Besides, ethanol has a longer penetration than Isooctane despite its low boiling point. Spray collapsing is observed at 1 bar chamber pressure and 453 K temperature due to flash boiling, which reduced the overall spray cone angle, resulting in greater LPO and ethanol penetration, where Isooctane behaves differently in this chamber condition. Keywords Lemon peel oil · Ethanol · Spray characterization · GDI optical engine · Mie scatter imaging · Flame luminosity
G. M. Nayak · B. Yogesh · V. W. Ketan · P. S. Kolhe · B. Saravanan (B) Department of Mechanical and Aerospace Engineering, IIT Hyderabad, Hyderabad 502285, India e-mail: [email protected] B. Abinash College of Engineering, Design and Physical Sciences, Brunel University, London, UK © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_7
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1 Introduction In recent years, there has been an increase in interest in renewable biofuels due to depleting fossil fuel resources, rising costs, and the need for energy security. As a result, numerous renewable and biodegradable alternative fuels have been considered energy sources worldwide [1–3]. Because of their superior evaporation characteristics, alcohol-based biofuels are becoming increasingly popular. Research on alternative fuels must now focus on proving the value of these fuels as alternatives to conventional fuel in order to encourage the mass production and, consequently, their affordability. Therefore, adapting the bio diesel and biofuels like methanol, ethanol etc. are suitable to options [4–9]. Because transportation is such an essential part of modern life, the automotive industry has grown to be one of the largest consumers of petroleum-based fuels. As a result, there is a dual problem of rapid fossil fuel depletion and environmental degradation. Given these concerns, the conditions on the spray structure of these biofuels is also progress of advanced engine technology is critical. Developing a gasoline direct injection (GDI) engine is required because it provides greater thermal efficiency and power output than traditional engines [10]. Furthermore, because stratification enables lean combustion, a GDI engine can improve transient response, more precise air–fuel ratio control, improved fuel economy, and lower tailpipe emissions [11]. Besides, an increase in the compression ratio leads to increased output efficiency due to a lower risk of knocking [12]. Understanding the spray, the atomization properties of various fuels, and the complex and varied physical chemistry phenomena in the GDI engine is essential in improving the GDI combustion system [13]. As a result, numerous researchers have studied the GDI spray behaviour of different fuels under a variety of environmental circumstances. Spray characterization of ethanol-gasoline blended fuel was carried out by Gao et al. [14]. The spray tip penetration decreases as the ethanol fraction increases, and the cone angle increases at low ambient pressure. Nonetheless, an insignificant difference in spray tip penetration was reported under high chamber pressure conditions, and the cone angle of all the blended fuel was found to be nearly the same for fully developed spray. However, at the start of the injection time, gasoline exhibits a larger spray cone angle. Arora et al. [15] studied the effect of different ambient pressures and fuel temperatures on spray tip penetration using ethanol, n- heptane, and ethanol and isooctane blending. Results show that the spray tip penetration significantly decreased with increased ambient pressure for each fuel. According to their findings, each fuel’s spray tip penetration dramatically decreased when ambient pressure rose. Nava Igual et al. [16] tested a multi-hole injector on a DISI engine where gasoline and n-butanol are injected at different bars in a constant volume spray chamber at standard operating conditions. They reported that the gasoline has a smaller droplet size than n-butanol due to its low viscosity and higher volatility. Similar findings were reported in the previous study [17]. Knorsch et al. [18] investigated how fuels
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thermophysical characteristics influence atomization and evaporation for different fuels and their blends at different ambient conditions. They concluded that fuel with a higher latent heat of vaporization and boiling point tends to have a more significant drop size. The present study investigates the spray behaviour of LPO and ethanol with baseline fuel isooctane in a constant volume chamber. The influence of thermophysical properties on fuel atomization is studied. The effect of individual pressure, temperature and various engine-like conditions on the spray structure of these biofuels is also thoroughly investigated.
2 Experimental Set-Up and Conditions for Constant Volume Study The primary goal of the constant volume spray characterization section is to comprehend the characteristics of LPO and ethanol at various injection timings and compare it to isooctane (assumed as a single component surrogate of gasoline). Pressure and temperature influence on spray plume structure is carried out in this study. In addition, the crank resolved motoring curve is used to determine the pressure and temperature at three different SOI timing. Table 2 shows the chamber pressure and temperature for the selected injection timing. A six-hole GDI injector with a constant injection pressure of 90 bar is used in the present study; more details reported in our previous study [19]. A naturally aspirated GDI engine’s in-cylinder engine conditions are simulated using a constant volume chamber in Fig. 1. This chamber has four optical windows made up of sapphire for visualization purposes having a thickness of 25 and 100 mm of diameter. The multi-hole injector, a thermocouple (to monitor chamber temperature), and two sensitive pressure gauges (to monitor chamber pressure) are mounted on the chamber’s top. The chamber is pressurized to the required value with nitrogen gas bottles, and a coil-type heater is used to achieve the desired temperature inside the spray chamber. When the required chamber conditions (P ± 0.1 bar and T ± 2 K) are met, an electronic injection con trol unit is used to start the injection process of the various biofuels. Table 1 shows the thermo-physical properties of all the fuels tested in this study. The injected spray is acquired using a high-speed camera (Phantom VEO 710) operating at 10,000 frames per second and with an exposure rate of 99 µs. The acquired images are post-processed using in-house MATLAB code to determine liquid penetration length. The Mie scattering and image processing techniques are discussed in our previous study [19].
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Fig. 1 Snapshot of constant volume spray chamber
Table 1 Various thermophysical properties of gasoline, ethanol and LPO [19, 20] Properties
Gasoline
Isooctane
Ethanol
LPO
Molecular formula
C5 –C12
C8 H18
C2 H5 OH
C10 H16 O0.045
Octane No
87
100
109
80
LHV (MJ/kg)
44
44.3
27
45
Kinematic viscosity
(mm2 /s)@313
0.6
0.6
1.08
1.06
Surface tension (mN/m)@293 K
K
–
14.7
22.4
22.1
Density (kg/m3 )@298 K
750
695.8
790
843
Latent heat of vaporization (kJ/kg)
380–400
305.4
938
290
Final boiling point (K)
498
372
351
449
Stoichiometry ratio (A/F)
14.7
12.5
9
14.1
3 Results and Discussion Spray characterization of GDI injectors is essential since it influences engine performance and emissions. This section looks into the effect of ambient conditions and the thermo-physical properties of various fuels on spray morphology. The fuels used in this study are isooctane, LPO, and ethanol. The spray morphology and liquid tip penetration length of these fuels are studied inside a constant volume chamber un- der varying pressure and temperature conditions. In addition, an attempt is made to simulate engine-like conditions inside the chamber to comprehend the spray behaviour further.
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3.1 Effect of Ambient Pressure and Temperature on Spray Morphology The atomization behaviour of various fuels is examined individually under various pressure and temperature conditions. The spray morphology of isooctane at a constant temperature (298 K) and variable pressure (1, 1.5, 2.5, and 6 bar) is shown in Fig. 2. It depicts the temporal evolution of spray inside the chamber after injection throughout 0.8 ms with a constant time interval of 0.2 ms. The length of the liquid spray decreases as the chamber pressure increases. The drag on the liquid droplets increases with ambient density, which reduces liquid penetration as chamber pressure increases (Table 2). All three tested fuels had a similar impact as pressure increased. Figure 3 depicts the penetration of the liquid spray tip following the commencement of injection for all three fuels. The penetration length increases monotonically overtime before becoming constant. It should be noted that the effect is limited between 1 and 2 bar at 298 K due to the small density change. However, the penetration length is significantly reduced when the pressure is increased to 6 bar. Due to the reduced axial flow of the jet caused by the larger drag force in this chamber state, there is more fuel dispersion in the radial direction. The inside temperature of the constant volume spray chamber considerably influences liquid penetration more than pressure. Figure 4 represents the spray structure of isooctane at different temperatures (i.e., 298, 329, 371, and 453 K) at a constant pressure of 1 bar. It can be noted from the spray images that the change in chamber temperature influences the development of the spray. As the temperature rises, the Fig. 2 Spray morphology of isooctane for different pressure at a constant temperature of 298 K. Columns show the results for different chamber pressures, and the rows are showing for different timings after the start of injection
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Table 2 Experimental conditions Type of injector
Bosch 6 hole solenoid GDI injector
Duration of injection
Four milliseconds
Injected pressure
90 bar
2*Chamber operating conditions:
Injection timing (bTDC)
Chamber pressure (bar)
Chamber temperature (K)
Chamber density (kg/m3 )
Ambient
–
1.0
298
1.13
Engine like condition
90 CAD
1.5
329
1.53
60 CAD
2.5
371
2.27
30 CAD
6.0
453
4.46
spray width and length decrease significantly. Higher temperatures promote droplet evaporation, resulting in shorter liquid penetration due to reduced momentum flux. A close examination reveals that plume merging occurs at the highest operating temperature condition in Fig. 4. 1 bar at 453 K, it is evident that at 0.2 ms, two plumes emerge from the injector and subsequently merge, leaving just a single plume visible. It should be noted that the chamber condition is above the boiling point of isooctane. Furthermore, the chamber pressure is below its saturation pressure, which is expected that the liquid droplet will begin to boil after injection, culminating in catastrophic liquid droplet disintegration. This can be referred to as flash boiling of liquid spray. The catastrophic breakdown of spray plumes causes all of the momentum to be lost, resulting in many tiny droplets that evaporate quickly. All three fuels behave differently in this chamber condition due to flash boiling. Further discussion, Fig. 5 presents the liquid spray morphology of isooctane, LPO, and ethanol at 1 bar and 453 K up to 3 ms after injection start. Please note that when the fuel is under flash boiling conditions, the spray structure changes significantly. Although all fuels exhibit spray collapse in this study, the spray cone angle and penetration length vary due to different thermophysical characteristics. Isooctane is more vulnerable to flash boiling due to its lower boiling point. Because of the catastrophic break-up, the Mie-scattering. Images of spray temporal evolution for isooctane are hardly visible. On the other hand, LPO and ethanol behave differently under these conditions. Although the spray cross-sectional area and cone angle increased for flashing spray in single-hole injectors [21, 22], this was not the case for multi-hole injectors, particularly for LPO and ethanol. Jet-to-jet interaction is believed to occur in multi-hole GDI injectors due to greater evaporation and spray plume expansion. As a result, a low-pressure area forms around the spray plumes, causing the spray structure to collapse [23]. The total spray cone angle is reduced due to spray collapse caused by flash boiling, affecting air entrainment and fuel–air mixing. In addition, LPO and ethanol jet penetrated longer due to this merge of multiple sprays. Despite
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Fig. 3 Effect of chamber pressure on liquid spray tip penetration for isooctane, LPO, and ethanol at 298 K constant chamber temperature
having a lower boiling point than isooctane, ethanol has a higher evaporation enthalpy, which causes droplets to cool more quickly and reduces evaporation [18]. A small portion of a low boiling point component in the fuel could be the catalyst for spray collapse even while most of the fuel elements are not in the flash boiling region in terms of temperature. A similar observation is reported by Aleiferis et al. [24]. The penetration length is depicted in Fig. 6 under constant pressure and varied temperature conditions. As the temperature rises, the penetration length of all tested fuels decreases significantly. For LPO, penetration length is negligible at 298 and 329 K under the constant pressure of 1 bar due to the higher boiling point. On the other hand, Isooctane and ethanol are more sensitive to small temperature gradients due to their high evaporative nature. Therefore, the appreciable reduction in penetration length is observed with a further rise in temperature. The merging of multiple sprays
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Fig. 4 Spray morphology of isooctane for the different temperatures at a constant pressure of 1 bar. Columns show the results for different chamber temperatures, and the rows are showing for different timings after the start of the injection
Fig. 5 Spray morphology of spray collapsing conditions for the chamber pressure of 1 bar and temperature of 453 K. Columns are showing the results for different fuels, and the rows are showing for different timings after the start of the injection
in the multi-hole GDI injector results in extended spray tip penetration for LPO and ethanol at 1 bar, 453 K.
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Fig. 6 Effect of temperature on liquid spray tip penetration for isooctane, LPO and ethanol. The chamber pressure is kept constant at 1 bar
3.2 Effect of Real-Time Engine States on Spray Morphology The previous section reports spray behaviour under constant pressure or temperature conditions. To further comprehend the spray atomization, the effect of real-time engine states on the isooctane spray plume is shown in Fig. 7 with relation to injection time, chamber pressure, and temperature conditions. It demonstrates that injection time influences spray progression due to chamber conditions. As the injection timing is delayed, the pressure and temperature rise, and the plume become shorter due to increased evaporation and drag, which reduces the overall spray cone angle. The liquid’s jet outer edge is populated with comparatively smaller droplets and a relative large velocity gradient between the ambient atmosphere and the fuel droplets, which describes a better understanding of the spray behaviour. Hence, a faster evaporation
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Fig. 7 Effect of real-time engine states on spray morphology of isooctane. Columns are showing the results for different chamber pressure and temperature, and the rows are showing for different timings after the start of injection
rate with more mass transfer can be expected. In addition, the liquid breakup is significantly increased on the outer periphery due to the development of shear forces between the liquid outer surface and the orifice wall. For quantitative analysis, the liquid spray tip penetration length of isooctane, LPO, and ethanol are shown in Fig. 8. There is a minimal variation in penetration length between the three fuels under nominal surrounding conditions. However, as chamber pressure and temperature increased the difference in thermophysical properties became significant. Due to lower viscosity, surface tension, and density, isooctane has the shortest spray tip penetration in each chamber condition. LPO has the most extended penetration length due to momentum flux, while boiling point prevents it from evaporating. Despite having a lower boiling point than isooctane, ethanol has a more extended penetration than isooctane due to the charge cooling effect.
4 Conclusions • The atomization and structure of liquid spray plumes are greatly influenced by chamber pressure, temperature, and the thermophysical characteristics of the fuel. • According to the experiment, the temperature significantly impacts spray morphology more than chamber pressure.
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Fig. 8 Effect of real-time engine states on spray penetration for isooctane, LPO and ethanol
• At a chamber pressure of 1 bar and a temperature of 453 K, spray collapsing was observed due to flash boiling. Spray collapse caused the overall spray cone angle to decrease, resulting in more extended penetration for LPO and ethanol. Isooctane, on the other hand, behaves differently in this chamber condition. • In all of the conditions tested, isooctane has the lowest spray tip penetration compared to LPO and ethanol. LPO had a longer penetration length because of its higher boiling point, viscosity, and surface tension. Although ethanol has a lower boiling point than isooctane, it evaporates slower due to its higher latent heat of vaporization. Acknowledgements This research is supported by the Science and Engineering Research Board (SERB) of India through grant No. CRG/2019/003325. The authors thank Mr. Pillai Madhu Shankar for his assistance in conducting experiments and the central workshop staff at IIT Hyderabad for their assistance in fabrication work.
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References 1. Ashok A, Gugulothu SK, Reddy RV, Burra B (2022) Influence of 1-pentanol as the renewable fuel blended with jatropha oil on the reactivity controlled compression ignition engine characteristics and trade-off study with variable fuel injection pressure. Sustain Energy Technol Assess 52:102215 2. Yildiz I, Caliskan H, Mori K (2022) Assessment of biofuels from waste cooking oils for diesel engines in terms of waste-to-energy perspectives. Sustain Energy Technol Assess 50:101839 3. Zare A, Bodisco TA, Verma P, Jafari M, Babaie M, Yang L, Rahman MM, Banks APW, Ristovski ZD, Brown RJ, Stevanovic S (2022) Particulate number emissions during cold-start with diesel and biofuels: a special focus on particle size distribution. Sustain Energy Technol Assess 51:101953 4. Chaudhari VD, Kulkarni A, Deshmukh D (2021) Spray characteristics of biofuels for advance combustion engines. Clean Eng Technol 5:100265 5. Dabi M, Saha UK (2022) Performance, combustion and emissions analyses of a single-cylinder stationary compression ignition engine powered by mesua ferrea linn oil–ethanol–diesel blend. Clean Eng Technol 8:100458 6. Guan W, Wang X, Zhao H, Liu H (2021) Exploring the high load potential of diesel–methanol dual-fuel operation with miller cycle, exhaust gas recirculation, and intake air cooling on a heavy-duty diesel engine. Int J Engine Res 22(7):2318–2336 7. Kumar N, Singh K (2022) Study of combustion, performance and emissions characteristics of oxygenated constituents and methanol fumigation in the inlet manifold of a diesel engine. Sustain Energy Technol Assess 49:101748 8. Verma TN, Nashine P, Chaurasiya PK, Rajak U, Afzal A, Kumar S, Singh DV, Azad AK (2020) The effect of ethanol-methanol-diesel- microalgae blends on performance, combustion and emissions of a direct injection diesel engine. Sustain Energy Technol Assess 42:100851 9. Yusuf AA, Inambao FL, Farooq AA (2020) Impact of n-butanol-gasoline-hydrogen blends on combustion reactivity, performance and tailpipe emissions using tgdi engine parameters variation. Sustain Energy Technol Assess 40:100773 10. Park SH, Kim HJ, Suh HK, Lee CS (2009) Atomization and spray characteristics of bioethanol and bioethanol blended gasoline fuel injected through a direct injection gasoline injector. Int J Heat Fluid Flow 30:1183–1192 11. Shen K, Xu Z, Chen H, Du J (2021) Combined effects of high energy ignition and tumble enhancement on performance of lean combustion for gdi engine. Exper Therm Fluid Sci 129:110464 12. Zhao F, Lai MC, Harrington DL (1999) Automotive spark-ignited direct-injection gasoline engines. Progr Energy Combust Sci 25:437–562 13. Attar MA, Herfatmanesh MR, Zhao H, Cairns A (2014) Experimental investigation of direct injection charge cooling in optical gdi engine using tracer-based plif technique. Exp Thermal Fluid Sci 59:96–108 14. Gao J, Jiang D, Huang Z (2007) Spray properties of alternative fuels: a comparative analysis of ethanol–gasoline blends and gasoline. Fuel 86:1645–1650 15. Arora R, Morgan CJ, Naber JD, Lee SY (2011) Flash boiling spray characterization of a gasoline multi-hole injector in a heated pressure vessel. In: Proceedings of the 2011 ILASS Americas 23rd annual conference on liquid atomization and spray systems paper, no. 162 16. Nava Igual S, Marchitto L, Merola SS, Tornatore C, Valentino G (2015) Characterization of n-butanol and gasoline spray from a multihole injector using phase Doppler anemometry. Atom Sprays 25:1047–1062 17. Li Y, Guo H, Wang JX, Xu H (2014) The comparative study of gasoline and n-butanol on spray characteristics. In: SAE technical paper, SAE International 18. Knorsch T, Heldmann M, Zigan L, Wensing M, Leipertz A (2013) On the role of physiochemical properties on evaporation behavior of disi biofuel sprays. Exper Fluids 54:1522 19. Biswal A, Kale R, Balusamy S, Banerjee R, Kolhe P (2019) Lemon peel oil as an alternative fuel for gdi engines: a spray characterization perspective. Renew Energy 142:249–263
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20. Ashok B, Thundil Karuppa Raj R, Nanthagopal K, Krishnan R, Subbarao R (2017) Lemon peel oil: a novel renewable alternative energy source for diesel engine. Energy Conver Manag 139:110–121 21. Allocca L, Montanaro A, Di Gioia R, Bonandrini G (2014) Spray characterization of a singlehole gasoline injector under flash boiling conditions. In: Technical report, SAE Technical Paper 22. Liu Y, Pei Y, Peng Z, Qin J, Zhang Y, Ren Y, Zhang M (2017) Spray development and droplet characteristics of high temperature single-hole gasoline spray. Fuel 191:97–105 23. Zhang G, Hung DLS, Xu M (2014) Experimental study of flash boiling spray vaporization through quantitative vapor concentration and liquid temperature measurements. Exper Fluids 55:1–12 24. Aleiferis PG, Malcolm JS, Todd AR, Cairns A, Hoffmann H (2008) An optical study of spray development and combustion of ethanol, iso-octane and gasoline blends in a disi engine. In: Technical report, SAE Technical Paper
Numerical and Experimental Performance Comparison of a Typical Swirl Co-Axial Injector for a Cryogenic Combustor R. Sujithkumar, K. Chenthil Kumar, K. R. Anil Kumar, T. Jayachandran, and Kowsik Bodi
Abstract A swirl co-axial injector is designed, fabricated and tested for a cryogenic combustor using liquid oxygen and gaseous hydrogen. Performance of this injector arrived at through experimental and numerical studies are compared in terms of chamber pressure. The numerical model developed for this injector and combustor is based on a Eulerian–Eulerian framework. It considers the two-phase flow of liquid oxygen and gaseous hydrogen with the assumption of local mechanical and thermal equilibrium, but with a local finite evaporation rate. Chemical reaction of gaseous oxygen and hydrogen is modelled with a modified eddy dissipation concept combustion model. It is found that the chamber pressures calculated by this model matches reasonably well with the test results. The steady-state chamber pressures obtained are 38 bar and 40 bar for experimental and numerical studies, respectively. Further, the numerical results are analysed to understand the flow field characteristics, such as evaporation and mixing, in the tested injector. It is found that the evaporation of the liquid oxygen is complete well within the first quarter and the combustion is almost complete in the three quarters, respectively, of the injector length. Keywords Cryogenic combustor · Swirl co-axial injector · Two-phase flow · Combustion · CFD
R. Sujithkumar (B) Vikram Sarabhai Space Centre, Thiruvananthapuram, Kerala 695022, India e-mail: [email protected] K. C. Kumar · K. R. A. Kumar Fluidyn Consultancy Private Limited, Banglore 560078, India T. Jayachandran Indian Institute of Technology Madras, Chennai 600036, India K. Bodi Indian Institute of Technology Bombay, Mumbai 400076, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_8
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Abbreviations − → J p − → q S i, g Sl ST t Tm um Y α β Φ ρ − → τ→
Species diffusion flux, kg/(m2 s) Pressure, Pa Conduction heat flux, W/m2 Species source term due to chemical reaction, kg/(m3 s) Mass source term due to evaporation, kg/(m3 s) Energy source term due to reaction, W/m3 Time, s Mixture temperature, K Average mixture velocity, [m/s] Species mass fraction Volume fraction of liquid Coefficient of thermal expansion, K− 1 . Energy source due to viscous dissipation, W/m3 Density of fluid, [kg/m3 ] Shear stress tensor
Subscripts g i l m
Gas Species Liquid Mixture
1 Introduction Injectors are the key components controlling liquid propellant atomization, mixing and combustion within a rocket engine and play a vital role in stability and also in the efficiency in converting the chemical energy into the thermal energy. Two major types of injectors used in cryogenic engines are shear co-axial and swirl co-axial injectors. In both these injectors, energy is transferred to the liquid inside the injector through pressure difference by passing it through orifices. In case of a swirl injector, a swirling chamber is responsible for imparting the swirl mostly by admitting the liquid through tangential holes/helical vanes. This results in a tangential velocity component to the flow, forming a hollow and thin liquid cone out of the orifice of the injector. Shear co-axial injectors generates a solid cone with narrow spray angle going into droplets by the shear\momentum exchange of the central liquid jet with co-flowing annular gas flowing at much higher velocities than
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the liquid jet. The details of design of these types of injectors are not covered in this paper and the same can be obtained from design literature such as that by Mazheri et al. [1]. For LO2 -GH2 combustors, injector element developed initially is the shear coaxial injector element. In these injectors, liquid oxygen is injected through the central tube and the gaseous fuel through the annular slit. Turbulence in the liquid jet as well as the adaptation of its velocity profile to the conditions when the liquid jet leaves the central tube to the wider space of the injector may result in distortions of the liquid jet surface. These surface distortions are amplified by the aerodynamic forces imposed by the high-speed annular gas flow and along with liquid jet destabilization this will result in final jet disintegration. The complexities associated with disintegration and the basic mechanism of jet break-up and atomization are not well modelled. Hence, most of the injector design is based on empirical correlations that use nondimensional parameters such as Weber and Reynolds numbers and the momentum flux ratio. On contrast, a swirl co-axial injector makes use of the Gas-to-Liquid Ratio (GLR) in addition to the non-dimensional numbers in deciding the droplet size distribution after the atomization. Figure 1 shows the schematic of shear and swirl co-axial injectors. Figure 2 and 3 and Table 1 show the dimensional details and important parameters such as Weber Number, Reynolds number and momentum flux ratio and GLR. Even though all the droplet estimation studies are carried out using simulant fluids such as water/GN2 /GHe or LN2 /GHe, these cold flow results cannot be transferred to the hot fire conditions. In addition, the sub-critical and super critical behaviour of the cryogenic fluid (LO2 ) makes the cold flow and hot test conditions drastically different where the direct change of liquid oxygen from liquid to vapour owing to lower latent heat mainly makes the mixing characteristic of the injector play a major role than the atomization and evaporation. At sub-critical pressure, mixing is dominated by atomization and evaporation of droplets whereas at supercritical pressure, mixing is mainly governed by turbulent mixing and diffusion. Hence, the development cycle of an injector for a high-pressure liquid rocket engine involves large number of iterations with different configurations and operating conditions. It is beneficial to have a set of comparison data for similar cases and a validated simulation tool for use in such future development activities. Even though efforts
Fig. 1 Schematic of the shear and swirl co-axial injectors (2)
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Fig. 2 Modular combustor and swirl co-axial injector Fig. 3 Injector head insert details for swirl injector
Table 1 Flow parameters
Parameters
Swirl co-axial
Momentum ratio GH2 /LO2
66.4491
Reynolds number for LO2
1.4431 × 106
Rel. Weber number
3.3112 × 109
Ohnesorge no. for LO2
0.0024
LO2 swirl number
1.4
GH2 swirl number
3.7
Type of swirl
Tangential holes
were taken to model the various phenomena taking place in a liquid rocket engine such as atomisation, mixing, combustion etc. way back in 1976 by Vingert et al. [2], because of the extreme complexity of physical phenomena at high pressure, only limited data are available for the validation of the initial designs and the simulation methods at the device relevant conditions. One of the objectives of the present work is to generate test data for high pressure rocket engine combustion with LO2 being
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injected at subcritical condition for various designs and operating conditions. The second objective is to develop and use a CFD based numerical method for this system.
2 Physical Problem and Methodology 2.1 Processes and Methods 2.1.1
Experimental Set-Up
Figure 2 shows the schematic of the combustor that consists of a copper sink type combustion chamber with ablative nozzle that can be assembled with either the shear co-axial or the swirl co-axial injector element. Figure 3 shows the inlet geometry of the swirl injector used to impart swirl velocity to the oxidizer and fuel flow. The liquid oxygen flow rate is 225 g/s and the gaseous hydrogen flow rate is 37.5 g/ s. LO2 side swirl is provided by four numbers of tangential holes, while gaseous hydrogen side swirl is provided by helical vanes. The swirl numbers for LO2 and gaseous hydrogen (GH2 ) are 1.4 and 3.7, respectively. This is a liquid-centred injector with low-temperature gaseous hydrogen flowing through the annular space. The entry conditions of the propellants are 52.0 bar injection pressure for gaseous hydrogen at 140 K temperature and liquid oxygen at 48 bar pressure at 90 K. The nozzle geometry is shown in Fig. 2 with semi-divergence angle of 15° and contour geometry for convergent section is arrived at through contour optimisation for smooth entry with nozzle throat diameter of 13.8 mm. The sequence of operation is as follows: 1. At T + 0s, that is, at 0 s injection of gaseous hydrogen starts. The temperature of hydrogen is 140 K and the flow rate is about 37.5 g/s. The pressure is about 55 bar. In actual test, the flow rate is not constant; it increases from 0 to 37.5 g/s in 10 s. 2. At T + 8s, that is, 2s before ignition, injection of liquid oxygen starts. Its temperature is 100 K and the maximum flow rate is about 225 g/s. 3. At T + 10 s the system is ignited. 4. At T + 16 s the oxygen flow is stopped. 5. Hydrogen injection continues for another three seconds. The objective of the experiment is to generate there acting phase performances of swirl co-axial injector using a copper sink–type combustion chamber. The parameters measured are the chamber pressure build-up during combustion phase along with mass flow rate of propellants. Figure 4 shows the photo of the experimental set-up during one of the tests. The experimental set-up is a pressure-fed facility that can admit a metered quantity of LO2 by forcing LO2 stored in polyurethane foam insulated storage tank using high pure gaseous nitrogen. Hydrogen flow is facilitated by thermally conditioning the gaseous hydrogen stored in standard 50L water capacity
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Fig. 4 Experimental set-up
Department of Transportation (DOT) cylinders. LN2 is used as the shell side fluid and the pressure control is achieved using spring-loaded pressure regulators for LO2 tank pressurisation as well as for gaseous hydrogen. All parametric measurements such as injection pressure and temperature of both the propellants and the mass flow rates along with chamber pressure are measured using strain gauge type pressure sensors, E-type/C-type thermo-couples, and Coriolis-type mass flow meters, respectively. The chamber pressure is measured at two locations, 30 and 150 mm away from the injector, using strain gauge type pressure sensors (Fig. 5).
2.2 Numerical Simulation Method The flow field inside the combustor consists of a mixture of two phases: (a) the liquid phase, which is made of a single component, in the present case, liquid oxygen (LO2 ) and (b) the gas phase, which is a mixture of oxygen (GO2 ), hydrogen (GH2 ), nitrogen (which is initially present) and H2 O. Generally, two approaches, one Eulerian–Lagrangian and the other Eulerian–Eulerian, are used to describe the two-phase flow. Present work uses the latter approach, where both the phases are modelled as continua and they are assumed to be in mechanical and thermal equilibrium locally. This model is similar to the reduced model (six equation model) described by Saurel et al. [3] with the additional assumption of thermal equilibrium between the phases. The basic conservation equations solved are: (1) mixture mass conservation, (2) mixture momentum equations in three directions, (3) mixture energy equation and (4) the governing equation of the LO2 volume fraction. In addition to these, the species conservation equations for GO2 , GH2 and H2 O are solved in the gas phase. The mass conservation equation for the liquid phase is
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Fig. 5 Chamber pressure measured during experiment
∂(αρl ) + ∇ · (αρl u→m ) = Sl ∂t
(1)
A similar equation can be written for the gas phase, with a source term for phase change that is equal and opposite the corresponding source term for the liquid phase. ) ( ( ) ∂ (1 − α)ρg + ∇ · (1 − α)ρg u→m = −Sl ∂t
(2)
The individual mass conservation equations are combined and recast as pressure correction equation using the formulation given by Darwish et al. [4]. ) ( −1 −1 {1 − α}ρg ∂ ρl,ref αρl + ρg,ref ∂t
[( ) ] −1 −1 {1 − α}ρg u→m + ∇ · ρl,ref αρl + ρg,ref ( ) −1 −1 = ρl,ref Sl − ρg,ref
(3)
Conservation of momentum and energy are given by
[ cpm
∂(ρm u→m ) + ∇ · (ρm u→m u→m ) = ∇ · τ→→m − ∇ p + ρm g→ ∂t ]
∂(ρm Tm ) + ∇ · (ρm u→m Tm ) = −∇ · q→m ∂t
(4)
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Table 2 LO2 properties
Molecular weight, kg/kmol
31.999
Boiling point @ 1atm, °C
− 183.0
Critical temperature, °C
− 118.4
Critical pressure, bar
50.5
Density of liquid @ BP, 1 atm, kg/m3
1143
Specific gravity, gas @ 20 °C, 1 atm
1.11
Latent heat of vapourisation, kJ/kg
214
] [ ] [ ∂p + ∇ · ( p u→m ) − p(∇ · u→m ) + αβl + (1 − α)βg Tm ∂t + Φ + ST
(5)
The gas phase has multiple species (oxygen and hydrogen), and hence the following species conservation equations are used to obtain the gaseous species mass fractions in the gas phase: ) ( ( ) ∂ {1 − α}ρg Yi,g + ∇ · {1 − α}ρg u→m Yi,g = −∇ · J→i,g + Si,g ∂t
(6)
In addition to the above conservation equations, the governing equations for the turbulent kinetic energy and its dissipation rate, in the framework of the standard k − ε formulation for the isotropic Reynolds averaged Navier–Stokes (RANS) turbulence model, are solved to compute the turbulent viscosity. All the components of the gas phase are assumed to be ideal gases. LO2 properties used are obtained from NIST database [5] and Streng [6] and Table 2 lists some of them. The governing equations are solved using a finite volume CFD solver, FluidynMP. This is a pressure-based solver, which can model unsteady and steady flows in both the incompressible and compressible regimes along with phase change and chemical reactions [7, 8]. All the simulations in this study are done using the unsteady compressible version of the solver. The convection terms are discretized using a hybrid scheme, which is a combination of the first order upwind differencing scheme and higher (second and third) order flux-limiter schemes. The diffusion and viscous terms are discretized using a second order central differencing scheme. The base flow solver already had Homogeneous Equilibrium Model built-in for trans-critical two-phase flow simulation and had been used for flows with large pressure gradients [9, 10]. It also had been used previously to study flows involving mixing of hydrogen and air in highly compressible, chemically reacting flow fields [11]. The additional models required for the two-phase flow, property variations and phase change with respect to the current simulations are incorporated through user-defined functions. Combustion of gaseous hydrogen and oxygen in the gas phase is modelled using a modified eddy dissipation concept (EDC) approach. This modification considers local pre-mixing and the reciprocal of the turbulent time scale ε/k is combined with
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Table 3 Performance parameters Parameters
Units
Swirl co-axial
O2 flow rate
g/s
215.3630
H2 flow rate
g/s
38.35568
Face plate GH2 flow rate
g/s
3.18770
Total mass flow rate of fuel
g/s
41.54338
Total flow rate
g/s
256.9070
Inject. temp. of LO2 , Ti, o
K
116.3752
Inject. pressure of LO2 , Pi,O
Bar
47.01124
Inject. temp. of GH2 , Ti, f
K
99.47717
Inject. pressure of GH2 , Pi, f
Bar
51.59991
O/F
5.1840
Cstar_theoretical
m/s
2387.8
Pc_measured
Bar
37.8716
Pc_numerical
Bar
40
Cstar_actual
m/s
2276.773
Cstar_efficiency
m/s
95.35
Cstar numerical
%
2328.7995
Cstareff.num
%
97.52
Diff
m/s
59.0005
% Difference
%
2.5335
the product of the local flame surface area and the turbulent burning velocity of the reacting mixture [12–14].
3 Results and Discussion 3.1 Experimental Results Table 3 lists the performance data for swirl co-axial injector. We find that the experimentally measured characteristic velocity (c*) of the engine is close to the theoretical prediction, indicating good performance.
3.2 Simulation Results Flow, phase change and combustion for a swirl injector is simulated. Table 4 lists the conditions used in the simulations. Figure 6 shows the computed chamber pressure.
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The time averaged chamber pressure is reaching a steady-state value of 36.9 bar, which is nearly within 2% of the measured value. Figures 7, 8, 9, 10 and 11 show the spatial distributions of temperature, LO2 volume fraction, GH2 mass fraction in the gas phase and H2 O mass fraction in the gas phase for swirl co-axial injector. It shows that the LO2 evaporates completely well within the first quarter of the injector. However, the mixing of the gaseous species and subsequent combustion continues throughout the injector. Most of the combustion is completed in at three quarters of the injector. Table 4 Propellant injection conditions in numerical simulations Parameters
GH2 inlet
LO2 inlet
Mass flow rate of LO2, g/s
0
225
Mass flow rate of hydrogen, g/s
37.5 g/s
0
Injection Temperature, K
140
90
Density, kg/m3
9.45
1150
Velocity, m/s
648.43
10.78
Turbulence intensity
2%
2%
Turbulent length scale
0.09 × (D0 − D1 )
0.09 × D2
Fig. 6 Simulated chamber pressure variation Fig. 7 Temperature variation within chamber
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Fig. 8 LO2 volume fraction within chamber
Fig. 9 GO2 volume fraction within chamber
Fig. 10 H2 mass fraction within chamber
Fig. 11 H2 O mass fraction within chamber
4 Conclusions A swirl co-axial injector for cryogenic LO2 -GH2 combustor is designed, fabricated and tested. Measurements of chamber pressure as well as numerical prediction of chamber pressures were made. This 3D, finite volume method (FVM) based numerical model developed gave reasonably accurate results in predicting average chamber pressure for numerical and experimental data. It showed that the evaporation takes place within one fourth of the injector length and the performance is limited by mixing and gas phase combustion. Acknowledgements The authors wish to thank LPSC and VSSC engineers for the support given for fabrication and testing.
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References 1. Mazaheri K, Morad MR, Shakeri AR (2012) A parametric study using two design methodologies for pressure jet and swirl injectors. In: IEEE aerospace conference, big sky, Montana, USA, 3–10 March, 2012 2. Vingert L, Gicquel P, Ledoux M, Care I, Mcci M, Glogowski M (2004) Atomisation in coaxial-jet injectors. [book auth.] Yang V, Habiballah M, Liquid rocket thrust chambers: aspects of modeling. Anal Des AIAA 3. Saurel R, Petitpas F, Abgrall R (2008) Modelling phase transition in metastable liquids: application to cavitating and flashing flows. J Fluid Mech 607:313–350 4. Darwish M, Moukalled F, Sekar B (2001) A unified formulation of the segregated class of algorithms for multifluid flow at all speeds. Numer Heat Transf B 40:99–137 5. NIST Chemistry WebBook (2021) SRD 69. NIST chemistry webbook, SRD 69. NIST 6. Streng AG (1971) Miscibility and compatibility of some liquid and solidified gases at low temperature. J Chem Eng Data 16:357–359 7. Shukla V, Tyagi D, Gera B, Varma S, Ganju S, Maheshwari NK (2021) Development and validation of CFD model for catalytic recombiner against experimental results. Chem Eng J 407:127216 8. Gera B, Ganju S, Chattophadhyay J (2021) Validation of CFD code FLUIDYN-MP for steam condensation at walls in presence of non-condensible gases. Ann Nucl Energy 152:107992 9. Kumar A, Levet M, Souprayen C, Kumar A, Worth L, Pearce R (2011) CFD analysis of twophase flow through micro-channel using Homogeneous Equilibrium Model (HEM). Fukuoka, Japan. In: Proceedings of the 4th international conference on heat transfer and fluid flow in microscale HTFFM-IV 10. Lankadasu A, Tripathi A, Saysset S, Yackow A, Roussiere BDL (2014) Numerical modeling of supercritical CO2 leaks and its subsequent dispersion in the ambient air. In: Proceedings of IUTAM symposium on multiphase flows with phase change: challenges and opportunities 11. Vyazmina E, Jallais S, Krumenacker L, Tripathi A, Mahon A, Commanay J, Kudriakov S, Studer E, Vuillez T, Rosset F (2019) Vented explosion of hydrogen/air mixture: an intercomparison benchmark exercise. Int J Hydr Energy 44:8914–8926 12. Chenthil Kumar K, Anil Kumar KR, Tripathi A (2015) A unified 3D CFD model for jet and pool fires. In: Symposium series No. 160, HAZARDS 25, Institution of Chemical Engineers
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13. Kuang Y, He B, Wang C, Tong W, He D (2021) Numerical analyses of MILD and conventional combustions with the Eddy Dissipation Concept (EDC). Energy 237:121622 14. Toosi AN, Farokhi M, Mashadi B (2015) Application of modified eddy dissipation concept with large eddy simulation for numerical investigation of internal combustion engines. Comput Fluids 109:85–99
Analytical Modelling of Effect of Steam Dilution on Hydrogen Combustion and Application to a Typical Nuclear Reactor Containment Aditya Karanam, Vishnu Verma, and J. Chattopadhyay
Abstract During postulated accident sequences in water-cooled nuclear reactors, steam and hydrogen may be released from the core and form a flammable mixture in the surrounding containment structure. Combustion of such mixtures and the subsequent pressure rise are an imminent threat for reactor containment integrity. Methods for evaluating combustion pressure rise are important for determining the design safety margins in such scenarios. Typically, combustion calculations are based on NS equations and CFD modelling, which are complex and time-consuming. A simpler and much faster approach is to use thermodynamic analysis to compute the final state after combustion for a given initial state. In the present work, thermodynamic modelling based on free energy minimization is presented. Predictions from the thermodynamic model have first been validated with published experimental data for binary hydrogen–air mixture. Then, parametric studies have been carried to compute combustion pressure rise in ternary mixture of hydrogen–steam–air as it represents more realistic mixture during accident. Finally, the model has been applied to a typical nuclear reactor containment to determine design safety margin as well as margin with respect to functional and structural failure of the containment. Keywords Hydrogen · Containment safety · Steam dilution · Element potential method · Pressure magnification
1 Introduction Containment is an ultimate barrier, which is designed to enclose the whole reactor core and other systems and to prevent the spread of active air-borne fission products during normal and accident conditions. During severe accident conditions like Loss of Coolant Accident (LOCA), large quantity of steam is released from the coolant circuit of water-cooled reactor into the containment. Initially, due to release of steam, A. Karanam (B) · V. Verma · J. Chattopadhyay Reactor Safety Division, Bhabha Atomic Research Centre, Mumbai 400085, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_9
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the containment pressure and temperature rises. Steam volume fraction may be as high as 50–80%. Steam condensation starts on cooler surfaces as well as in the bulk if suitable conditions are present. Typical containments are designed to withstand LOCA peak pressure. If the accident sequence gets more severe due to Loss of Emergency Core Cooling System (LOECCS), the core may get heated up leading to metal-water reaction and hydrogen production. The generated hydrogen is released into the containment consisting of air and steam (i.e. steam diluted containment atmosphere). Being a lighter gas, hydrogen has a potential of forming buoyancy induced vertically stratified distribution [1]. Moreover, hydrogen is a highly flammable gas with very low ignition energy (~ 0.02 mJ) and a wide flammability range in air: 4% (v/v) on the lower side to 75% (v/v) on the higher side. Hydrogen combustion can lead to large pressure loads on the containment walls and may threaten containment integrity. In case of such an eventuality, ground level release will be much higher than the prescribed regulatory limits. As containment is the last barrier of safety between the highly radioactive gases inside and the surrounding atmosphere, maintaining the integrity of the reactor containment is of critical importance during accident scenarios. Combustion of binary hydrogen air mixtures (absence of steam) have been studied in detail in the past [2]. However, in a real accident scenario, steam is present and leads to a ternary mixture with hydrogen and air. Just after release, steam condensation starts on the cooler surface as well as in the bulk if suitable conditions are present. As hydrogen is a Non-Condensable-Gas (NCG), it reduces local steam condensation rate. This may lead to prolonged steam dilution and helps in keeping the hydrogen concentration at lower level. A detailed review on the effect of non-condensable like hydrogen and air on steam dilution has been presented in [3]. Although steam dilution and hydrogen distribution are closely interlinked, issues related to steam dilution and hydrogen combustion risk have been separately studied in the past. Due to complex interaction of steam and hydrogen mixture, different concentrations may be expected in different regions in the containment. In some conditions, the local steam–air–hydrogen mixture may reach the flammability/detonability limit in the containment as depicted in the Shapiro-Moffette diagram [4]. The objective of the present work is to study the effect of steam dilution (i.e., steam addition) on hydrogen combustion. Analytical (thermodynamic) modelling based on free-energy minimization using element potential formulation has been used. Compared to a CFD simulation based on detailed NS equations which can take several days to weeks to give a detailed flow and thermal field, an analytical model provides the advantage of yielding solution almost instantaneously. This can be applied to quickly screen out results from various accident conditions. Important parameter of interest is the pressure magnification factor defined as the ratio of final pressure after combustion to initial pressure in the domain. It is important to mention that models presented and validated in this work are simplified models and do not take into account several factors like flammability, preferential flame propagation due to buoyancy, turbulence, 3D effects, spatial gradients, combustion time scale, containment pressurisation time scale and flame transition.
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However, they can be employed to get a quick and conservative estimate on pressures that can be expected from a hydrogen combustion event in the containment.
2 Analytical Modelling Combustion of hydrogen takes place through a network of elementary reactions which are essentially bimolecular collisions between reactive species. The kinetic models utilize a detailed reaction mechanism involving 19 elementary reactions and 8 reactive species [5]. The final state of the system after combustion can be determined using the principal of minimization of free energy. The free energy of relevance is the Helmholtz free energy which is applicable to a constant volume system. This method allows for dissociation effects and provides for a simultaneous solution of the composition and thermodynamic variables. Following the method of element potentials [6], the minimization is carried out by constructing a Lagrangian of the Helmholtz free energy. For a system comprising of s number of species, the Helmholtz free energy, A, is defined in Eq. 1 in terms of the chemical potentials μ j . The chemical potential can be physically interpreted as the molar Helmholtz free energy at constant temperature, volume and composition of species. P and V are the state variables pressure and volume of the system, respectively. A=
s
μ j n j − PV
(1)
j=1
The distinguishing feature of the element potential method is to use the mass balance of atoms instead of the species. Since atomic changes are not involved, the number of atoms of a particular element remains constant. Using this condition, the element potentials of the distinct elements constituting the reactive species are obtained and are then used to construct the species chemical potentials. Let ai j be i in species j. The number of atoms of element the number of atoms of element i can be calculated using bi = sj=1 ai j n j ; (i = 1 . . . l), where for each of the l distinct elements, summation over s species containing that element is taken. If a certain species does not contain an element, then ai j = 0 for that species–element combination. For each element, if the initial number of atoms is bio , then at equilibrium, atom balance constraint requires that bi − bio = 0; (i = 1 . . . l). Following the Lagrangian method for minimization, the Lagrangian of A, that is., L(A), subjected to atom balance constraint can be constructed as shown in Eq. 2, where λi are the Lagrange multipliers. L( A) = A +
l i=1
λi bi − bio
(2)
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At equilibrium, the condition to be satisfied is given by δ(L(A)) = 0. Substitution of Eq. 2 in this condition leads to the system of non-linear equations in index j given by Eq. 3. This equation relates the chemical potential μ j of each of the jth species to the atomic makeup of its molecule ai j and to a set of undetermined multipliers λi . The multiplier λi is nothing but the element potential of the ith element. μj +
l
λi ai j = 0
(3)
i=1
For solution purpose, a Taylor series expansion followed by Newton–Raphson linearization is sought; to convert Eq. 2 to a system of linear equations. After substantial and non-trivial algebraic manipulations, the final equations describing the equilibrium state are elaborated in Eqs. 4 and 5. These are a system of linear equations which have to be solved to obtain the final state (species concentrations and temperλi . Equation 4 is having the free ature) of the system. In both equations, πi = − RT index k = 1 . . . l for each of the l elements. l s
ak j ai j n j πi +
i=1 j=1
s ak j n j u oj j=1
RT
(lnT ) = bko − bk +
j=1
⎛ s l i=1 j=1
ai j n j u oj RT
⎜ πi + ⎝
s j=1
o n j cv, j
R
+
s ak j n j μ j
s j=1
RT
(4)
2 ⎞ n j u oj Uo − U ⎟ ⎠(lnT ) = 2 2 R T RT +
s n j u oj μ j j=1
R2 T 2
(5)
A flow diagram of solution procedure is illustrated in Fig. 1. The problem was solved in an iterative manner. The numerical library available in the open-source platform Cantera [7] has been used for solving the resulting matrix equations. The thermodynamic variables like specific heat were implemented for multi-component mixtures and with temperature dependent formulations available in the JANAF [8] database. The converged solution corresponds to a relative tolerance within a value of 10–8 or when the number of iterations exceeds 103 . Each run comprises giving the initial conditions (temperature, pressure and species mole fraction). The output is the equilibrium condition at which free energy of the system is minimized. Since energy is released in a constant volume system, the temperature will increase; and correspondingly, the pressure too will in accordance with the perfect gas EoS. It may be noted that thermodynamic analysis does not solve detailed flow equations. Thus, it does not give the transient pressure response; as it is concerned with computing only the final state for a given initial state. Also, unlike detailed NS analysis, thermodynamic analysis does not require transport properties like viscosity, thermal conductivity and mass diffusivity.
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Fig. 1 Flow diagram of analytical model based on element potential equations
3 Validation with Binary H2 –Air Mixture For validation purpose, a binary mixture of hydrogen and air has been taken due to availability of experimental data in the open literature and validation carried out by the authors [2]. Figure 2 presents a comparison of the pressure magnification predicted by the model along with the experimental data taken from open literature. The experimental set-up is a spherical chamber with internal volume of 120 L [9]. Initial mixture is at 25 °C and 1 atm and ignited with spark mounted at the centre of the vessel. It can be observed that the pressure magnification predicted by the model increases as the hydrogen mole fraction increases and peaks at nearly the stoichiometric composition of 29.5%. For higher mole fraction, the mixture becomes lean (i.e. air is in excess) and pressure magnification starts decreasing. This trend is corroborated by the experimental data as well. The decrease for lean mixture may be attributed to excess air acting as a heat sink. For hydrogen mole fractions higher than 10%, where flame propagation is rapid and isotropic, there is very good agreement between the measured pressure magnifications and those predicted by the analytical model, thus validating the model. Also, the peak pressure magnification of ~ 8.1 at stoichiometric condition, predicted by the model is found to be in good agreement with the data. However, a significant deviation can be observed for hydrogen mole fractions lower than 10%. This could be due to preferential flame propagation of hydrogen in which lower parts of the domain do not participate in combustion leading to lower energy release and hence lower pressure magnification. As the effect of selective flame propagation cannot be considered in thermodynamic modelling, the peak pressures predicted are higher than the actual values. To accommodate the effect of varying initial conditions that may be present in the containment, a parametric study covering a wide range of initial pressure and temperature has been carried out. The experimental data of Shroeder and Holtappels [10] has been used for validation. These experiments are from a spherical vessel with internal volume of 14 L and spark ignition at centre. Figure 3 shows the pressure magnification at an initial temperature of 20 °C and in the pressure range of 1–30 bar. The continuous lines represent the predictions from the present analytical model whereas the discrete symbols represent the corresponding experimental data points.
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Fig. 2 Validation at ambient conditions [2]
The model prediction shows no impact of initial pressure on the pressure magnification, except a small increase with initial pressure, in a narrow range around the stoichiometric composition. In the vicinity of the Lower Flammability Limit (LFL) of ~ 5%, a cluster of data points can be found with unit pressure magnification. This corresponds to no burning at all and corroborates the reasoning suggested in Fig. 2. At initial pressures up to 10 bar, the analytical prediction and data points (square, round and straight triangles) agree over a wide range of composition (10–60%) thus validating the model up-to 10 bar pressure. However, for initial pressures of 30 bar (inverted triangles), the predicted pressure magnification deviates significantly from experimental values throughout the spectrum of compositions considered. During accident scenario in containment, hydrogen concentrations higher than 60% and initial pressure higher than 10 bar may be seldom encountered. Therefore, the range of conditions for which model has been validated is applicable for containment safety applications. The effect of initial temperature on the pressure magnifications at an initial pressure of 10 bar and in an initial temperature range of 20–250 °C is depicted in Fig. 4. The corresponding experimental results are from Shroeder and Holtappels [10]. Unlike the initial pressure, the initial temperature has a marked effect on the pressure magnification. A higher initial temperature leads to a lower pressure magnification across the full spectrum of compositions from lean to rich mixtures. This may be attributed to a nearly constant adiabatic flame temperatures for hydrogen air flames irrespective of initial conditions [2]. The corresponding data also follow the same trend and also agree quite well with the model predictions, thus validating the model for higher initial temperature application.
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Fig. 3 Validation at higher initial pressure [2]
Fig. 4 Validation at higher initial temperature [2]
4 Effect of Steam Dilution As mentioned in the introduction, during accident scenario, a ternary mixture of hydrogen–steam–air needs is present in the containment. Thus, steam addition (referred to as steam dilution in this paper) to a binary hydrogen–air mixture needs to be considered. An important effect of steam dilution is that it changes the stoichiometry of a binary mixture. The stoichiometric composition of binary hydrogen–air
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mixture corresponds to 29.6% of hydrogen. When the mixture is diluted with 40% steam, the composition of the resulting mixture changes and is shown in Table 1. Due to steam dilution, the stoichiometric composition has changed to 17.8% of hydrogen. Thus, it is possible that for a given hydrogen mole fraction, there is lesser or nonavailability of air. For example, in the binary hydrogen–air mixture, 15% of hydrogen has access to 17.9% of oxygen available for burning, whereas when the mixture is diluted with steam upto 40% (Table 1), the oxygen availability reduces to 9.5% for the same hydrogen mole fraction. This can contribute to different energy release with and without steam dilution. Another factor is that steam has high specific heat capacity compared to air, leading to lower temperature rise for given heat released during reaction. Keeping the other parameters same, as steam fraction is increased, more of the energy released from combustion will be absorbed by steam due to higher specific heat and lead to lower rise in sensible internal energy and hence lower final temperature. Thereby final pressure will also be lower. A parametric study is conducted to study the combined effect of both factors described above. The initial pressure of the domain is kept fixed at 1 atm and initial temperature is 180 °C. This is a hypothetical scenario, since higher steam addition will also lead to higher initial temperature. However, to study the effect of single parameter of steam dilution, the temperature in all cases considered is kept fixed at 180 °C; and the resulting pressure magnification with steam dilution range from 0 to 40% is shown in Fig. 5. The 0% steam case (binary mixture) is similar to the result depicted in Fig. 4. As the steam mole percent is increased the stoichiometric composition of H2 shifts towards the leaner side which can be seen as graph peak shifts towards left. At low hydrogen fraction there is excess of available oxygen, even at higher steam mole fraction. Steam which acts as neutral component like N2 has very small Table 1 Mixture composition diluted with 40% steam X_H2 (%)
X_H2 O (%)
X_O2 (%)
Phi
4.0
40.0
11.8
0.17
10.0
40.0
10.5
0.48
Mixture type Lean
15.0
40.0
9.5
0.79
17.8
40.0
8.9
1.00
Stoic
20.0
40.0
8.4
1.19
Rich
25.0
40.0
7.4
1.70
29.6
40.0
6.4
2.32
30.0
40.0
6.3
2.38
35.0
40.0
5.3
3.33
40.0
40.0
4.2
4.76
45.0
40.0
3.2
7.14
50.0
40.0
2.1
11.90
55.0
40.0
1.1
26.19
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Fig. 5 Isolated effect of steam dilution on pressure magnification
effect on the pressure magnification. Due to higher specific heat, the temperature is lower and hence the final pressure magnification is also slightly lower. However, this effect is not very significant so the pressure magnification line can be seen coinciding at lower hydrogen mole fraction. But at higher hydrogen concentration, significant difference can be observed. In this region for given hydrogen mole percent there is oxygen deficit due to steam dilution. In this region, as the steam mole percent is increased, the value of pressure magnification decreases significantly.
5 Application to Containment Safety Analysis In Fig. 5, the mole fraction of different species has been varied keeping fixed initial pressure and temperature. However, during accident scenario, the net mass and energy of the species injected in the containment has to be taken into account. During accident scenarios like LOCA + LOECCS, large amount of steam is first injected in the initial 100 s as the coolant is discharged into the containment from break location. Then, hydrogen is slowly added over a period of ~ 10 h in the steam diluted containment atmosphere. Both steam and hydrogen addition will change the composition, pressure and temperature of the containment due to additional moles and enthalpy that are injected in the containment. To model this scenario, initially, only steam is added and pressure and temperature increase due to steam addition is calculated. Then, hydrogen is injected in the steam diluted mixture and the incremental pressure and temperature increase due to hydrogen injection is computed. Pressure and temperature increase due to injection only can be calculated using ideal gas equation of state and Daltons law of mixing. Also, due to hydrogen injection, mixture changes from binary to ternary; the resulting mole fraction of both steam and hydrogen are
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altered. Finally, in the resulting mixture at elevated pressure and temperature and ternary composition, the combustion state is calculated using the analytical model. Another factor to be taken into account is the incompleteness of hydrogen combustion for hydrogen concentration lower than 10%. From Fig. 2, it can be observed that for hydrogen mole fraction higher than 10%, the analytically predicted combustion pressure matches with the experimental value due to isotropic flame propagation in all directions. For lower than 10%, the analytical model estimates higher combustion pressure than the experimental value. To account for this, a factor called the Extent of Combustion (EoC) defined as the ratio of actual combustion pressure to analytical combustion pressure is defined. For concentration below 4%, the EoC is zero as the mixture is not in the flammable range. From Fig. 2, for concentration higher than 10% the analytical model predicts exact value as that of experimental data and hence EoC can be taken as 1. Using linear interpolation between two limits the value of EoC is calculated for hydrogen fractions between 4 and 10%. The combustion pressure obtained from the analytical model is multiplied with EoC to obtain the final pressure. In order to apply the analytical model to the reactor containment safety, all the parameters like LOCA pressure, design pressure, pressure corresponding to the functional failure and pressure corresponding to structural failure for containment have been adopted from [11] and reproduced in Table 2 for a typical concrete containment. Functional failure is considered at the formation of through and through crack width of minimum 0.2 mm and structural failure is considered at excessive cracking and spreading of rebar yielding zone [11]. The different pressures can be considered in terms of Load Factor which is defined as the ratio of internal gauge pressure inside the containment to the design gauge pressure. The load factor calculation at different conditions in the containment is shown in Table 2. It can be seen that normal operating condition corresponds to load factor of 0 and at design pressure, load factor is 1. At LOCA peak pressure, the load factor is 0.61. LOCA is a Design Basis Accident (DBA) and hence the containment structure is designed to withstand LOCA peak loads. For load factor higher than 1, ground level release may be higher than permissible limits. At much higher load factor, functional failure will occur at load factor of 1.85 and structural failure will occur at load factor higher than 1.97. Table 2 Load carrying capacity of containment structure of typical concrete containment Condition
Gauge pressure (kg/cm2 )
Load factor
Normal operating condition
0
0.00
LOCA peak pressure
1.06
0.61
Design pressure
1.73
1.00
Functional failure
3.20
1.85
Structural failure
3.41
1.97
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Fig. 6 Load factor calculation for a typical concrete containment
Initially, the calculation has been performed with steam mole fraction of 40% and LOCA initial conditions corresponding to pressure of 1.06 kg/cm2 (g) and temperature of 120 °C. The hydrogen mole fraction has been varied from 4 to 15%. The EoC varies linearly from 0 to 1 from 4 to 10% of hydrogen concentration and remains at 1 for higher than 10%. Figure 6 shows the result of the calculations done to get load factor. The figure is divided into three colour zones. The green zone represents conditions which is within the design margin (i.e., Load Factor < 1). The yellow zone is higher than the design margin but within the functional failure limit (1 < Load Factor < 1.85). The red zone is higher than the functional failure limit (Load Factor > 1.85). For these conditions, significant ground level release from the containment may be expected. The results calculated above shows that at a fixed steam concentration of 40%, at H2 mole fraction of 7.7%, the design limit is reached. At around 9.1% mole fraction of H2 , functional failure is predicted. When the steam in the containment condenses it reduces the containment initial pressure but increases the effective hydrogen mole fraction. To study this effect, a parametric study for steam concentration has been carried out for 30, 20 and 10%. It can be observed that as the steam fraction is decreased the corresponding hydrogen fraction required to cross design limit and functional failure limit is increasing. This is mainly because the reduction in initial pressure due to condensation has more effect compared to increase in pressure due to increase in effective mole fraction of hydrogen. It is important to mention that calculations presented here are hypothetical and mainly to determine the trend in Load Factor and impact of various parameters. In water cooled reactors, the amount of hydrogen released in accident conditions depends upon the type of reactor, severity of accident and containment geometry. In the Indian context, for Pressurized Heavy Water Reactor (PHWR), around 130 tonnes of steam and 40 kg of hydrogen will be discharged during DBA in a containment volume of 40,000 m3 . This results in steam concentration of 84.4% and
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hydrogen concentration of only 0.19%. Within half an hour of the initiation of the accident, steam condenses and reduces to 20% and effectively, hydrogen concentration changes to 0.97%. It can be seen that these are well within the design conditions of the containment. Such concentration levels are easily managed by hydrogen mitigation systems such as recombiners which are working in the containment. Higher quantity of hydrogen will be released only in severe accident such as core melt with vessel failure [resulting from Molten Corium Concrete Interaction (MCCI)]; which is not possible in PHWR due to large quantity of water and sufficient time to initiate SAMG action. The underlying assumption in this analysis is that the whole domain is uniformly mixed at this composition, pressure and temperature. But in the actual containment, there may spatial gradients. Also, for different containment volumes, the time scale of pressure and temperature increase will be different. Higher the containment volume, higher will be the time required for pressurization. These effects are not considered in the present analytical model and hence the model results are highly conservative.
6 Conclusions In this work, post combustion state and the effect of steam dilution on hydrogen combustion has been studied using analytical modelling. Modelling is based on free energy minimisation principle implemented using the method of element potentials. The following conclusions can be drawn from this study: i. For binary hydrogen–air mixture, a good agreement has been observed with experimental data for hydrogen concentration higher than 10% for both ambient and elevated initial conditions. ii. Below 10% of hydrogen concentration, the model over-predicts compared to experimental data and this may be attributed to preferential flame propagation. iii. With steam dilution, the final result will be combined effect of higher heat capacity of steam and non-availability of oxygen due to steam dilution. At low mole faction of steam, effect on pressure magnification is not very significant. But at higher steam fraction the pressure magnification decreases significantly because of non-availability of oxygen due to dilution effect. iv. For a typical concrete containment, the critical hydrogen concentration for design failure and functional failure at a given steam fraction have been derived. It has been observed that as the steam fraction is decreased the corresponding hydrogen fraction required to cross design limit and functional failure limit has increased.
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Nomenclature A P V U, u T R o cv, j nj ai j bi bio λi μj
Helmholtz free energy (J) Pressure (bar) Volume (m3 ) Internal energy, specific internal energy (J), (J/kg) Temperature (K) Universal gas constant (J/mol/K) Specific heat at constant volume of species j (J/kg/K) Number of moles of species j (mol) The number of atoms of element i in species j (–) The number of atoms of element i (–) Initial number of atoms of element i (–) Element potential of element i (J/mol) Chemical potential of species j (J/mol)
References 1. Karanam A, Gera B, Verma V, Chattopadhyay J (2021) Numerical studies on hydrogen distribution patterns in nuclear reactor containment. In: Paper #385, international heat and mass transfer conference, IIT Madras 2. Karanam A, Sharma PK, Ganju S, Singh RK (2016) Equilibrium based analytical model for estimation of pressure magnification during deflagration of hydrogen air mixtures. Kerntechnik 81:655–661 3. Yadav MK, Khandekar S, Sharma PK (2016) An integrated approach to steam condensation studies inside reactor containments: a review. Nucl Eng Des 300:181–209 4. Valdepenas M, Jimenez JM, Martin-Fuertes MA, Vernandez F (2007) Improvements in a CFD code for analysis of hydrogen behaviour with containments. Nucl Eng Des 237:627–647 5. ÓConaire M, Curran HJ, Simmie JM, Pitz WJ, Westbrook CK (2004) A comprehensive modeling study of hydrogen oxidation. Int J Chem Kinet 36:603–622 6. Gordon S, McBride BJ (1994) Computer program for calculations of complex chemical equilibrium compositions, rocket performances, incident and reflected shocks, and Chapman-Jouguet detonations. In: Technical report, NASA Reference Publication 1311 7. Goodwin DG (2001) Cantera user’s guide. California Institute of Technology, Pasadena 8. Chase MW (1998) NIST-JANAF thermochemical tables, 4th Edition. Thermochemical Tables 2 Volume-Set (Journal of Physical and Chemical Reference Data Monographs) 9. Cashdollar KL, Zlochower IA, Green GM, Thomas RA (2000) Hertzberg, M: Flammability of methane, propane, hydrogen gases. J Loss Prevent Process Ind 13(3):327 10. Shroeder V, Holtappels K (2011) Explosion characteristics of hydrogen-air and hydrogenoxygen mixtures at elevated pressures. Proc Int Conf Int Conf Hydr Saf 11. Roy R (2018) Behavior of containment structures of Indian pressurized heavy water reactor under severe accidents including validation. IAEA-GCNEP RTW
Thermal Performance of a Single-Layer Porous Radiant Burner with Biogas as Fuel: A Numerical Study Ayush Painuly and Niraj K. Mishra
Abstract Raw biogas has the ability to minimise waste and greenhouse gas emissions while also helping to meet the world’s insatiable energy needs. In this chapter, a circular Porous Radiant Burner (PRB) consisting of silicon carbide (SiC) foam is examined for its ability to substitute raw biogas. Raw biogas mixtures were examined on PRB using ANSYS 19 at firing rates ranging from 1 to 2 kW, and stable combustion was noted over an equivalence ratio range of 0.7–0.95. The increase in fire rate and equivalence ratio increased surface temperature, whereas radiation efficiency increased primarily as an outcome of the increase in equivalence ratio. Within the range of fire rates of 1–2 kW and equivalency ratio of 0.7–0.95, the PRB functions steadily with radiation efficiency of 12.4–37.5%. Furthermore, the newly developed PRB delivers more radiation efficiency for raw biogas than prior systems used in submerged mode. Keywords Raw biogas · Porous radiant burner · Radiation efficiency · Combustion · Numerical analysis
1 Introduction The efficiency of fossil fuel combustion is always a big challenge, despite the fact that it is currently the fuel of choice for power generation and many other uses. Since traditional combustion has relatively poor efficiency, the porous radiant burner may overcome this issue. Because of porous material has a larger emissive power from the gas, flames passing through it transmit heat differently than when they are open. Although biomass has a significant potential for energy, improper burning results in waste. The next stage at which biomass energy can be used more effectively is gasification. Numerous scholars worked on the biomass gasification and
A. Painuly · N. K. Mishra (B) Department of Mechanical Engineering, NIT Uttarakhand, Srinagar, Uttarakhand 246174, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_10
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radiant porous burner separately. Khanna [1] performed an experiment on doublelayer porous burner with different porosity with NOx and CO emission. A twodimensional porous burner with the pseudo homogeneity analysed by Brenner et al. [2]. They do not consider any radiation model in their work so that the radiative part has been affected by the experimental value of the effective thermal conductivity.
2 Literature Review and Objective Echigo et al. [3] investigated the combustion within the porous media with theoretically as well as experimentally. They predicted the solid temperature profiles by the analytical method and have good fit with the experimental data. Tong et al. [4] analysed the porous media burner having the radiative heat transfer and they concluded that the porous layer must be optically thick for the maximization of radiant heat transfer. Barra et al. [5] performed numerical simulation on the Khanna’s experimental set-up. Researchers recreated a one-dimensional investigation on the stability of the flame in a two-section porous burner. They concluded from their analysis of the simulation’s results that the porous matrix qualities have a substantial impact on the stable operating range. Liu et al. [6] did a numerical simulation on premixed methane air of two section porous media burner. They used single methane equation for simulation and showed that at the interface of porous burner temperature and velocity significantly changes. A numerical simulation on premixed combustion using the coal mine methane with the addition of water vapour had done and showed that temperature decrease at downstream section gradually during the water vapour addition [7]. Three-dimensional porous burner simulated numerically in Hoda’s work. The numerical simulation contained a five-step methane air reaction mechanism, and it was analysed to show that as the air ratio increased, the porous burner’s downstream temperature and pollution emission decreased. The combustion efficiency has major concern and a high efficiency is achieved by Dutta et al. [8] up to 82% at 6.8 AFR. A two-dimensional rectangular porous radiant burner analysed by Misha et al. [9]. The separate energy equations used for the solid phase and gas phase. The temperature profiles and concentration profiles were analysed by changing some parameters. A two-layer burner with an alumina bed at the upstream part has been the subject of an experimental inquiry. Analysis was done on the temperature profile of the flame, pollutant emission, and flame stability. The parameters are the porous material’s porosity and pore density. They demonstrated how the flame stability limitations grew by lowering pore density and raising material conductivity. The CO formation is majorly affected by temperature of flame [10]. Mishra et al. [11] performed an experiment in two-layer porous radiant burner having the LPG fuel. Many dependent parameters such as temperature, pollutant emission and thermal efficiency were reported. Their result showed that thermal efficiency is high and pollutant emission is less compared to the conventional LPG gas stove. Hashemi and Hashemi [12] investigated numerically two-layer porous burner. The simulation has been done on two-dimensional porous radiant burner. The result showed
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that the inlet fuel’s equivalence ratio primarily affects the both flame stability and temperature and can control by controlling the equivalence ratio of fuel. Sahraoui and Kaviany [13] performed a comparison of their two-dimensional model and one 1-D volume averaged techniques. They concluded that two-dimensional model can predict better local temperature distribution and species concentration on a pore scale but overall flame thickness, speed and solid and gas temperature are better determined by the one-dimensional approach. Mathis and Ellzey [14] studied the two section porous media burner having the different porosity and different material. The porosity are 10 and 60 PPI and materials are PSZ (Partially Stabilized Zirconia) and ZTM (Zirconia toughened mullite). According to their result they concluded that CO and NOx emission get lower than 15 and 10 ppm respectively. It is evident from the aforementioned literature that there hasn’t been much numerical research on the performance of raw biogas with a porous radiant burner. The burning of raw biogas in the porous radiant burner is the subject of the current investigation. It was already said, operating parameters also affect PRB performance. In order to examine the effects of fire rate and on temperature distribution and radiation efficiency, the combustion of raw biogas is carried out in the constructed PRB utilising ASNSYS FLUENT 19 for the firing rate of 1–2 kW.
3 Materials and Methods A schematic diagram of PRB used in the current study is shown in Fig. 1a. ANSYS FLUENT 19.0 is used to examine the performance of raw biogas with a porous radiant burner. A 3D model of porous burner has been considered for the numerical simulation. The geometry is axially symmetric so half section of porous burner was considered for the computational domain. The thickness of computational geometry is 20 mm and diameter are 90 mm. It is a single porous burner model and have a constant porosity of 90%. The burner material was also assumed to be a uniform, grey medium. The wall is radiative grey and adiabatic and the no slip condition has been assumed there. Gas-related radiation is disregarded. It is assumed that the chemical reactants and products are incompressible ideal gases. The impacts of possible catalysts for the solid are not taken into account at very high temperatures. The gas’s body forces were disregarded. ANSYS FLUENT 19 is used to perform each simulation. The equations in this algorithm, including the energy equation, are each solved separately and sequentially using a point implicit (Gauss–Seidel) linear equation solver. The Green-Gauss cellbased method is used to compute the gradient of a given variable, which is then utilized to discretize the convection and diffusion terms. The equations for turbulent kinetic energy and specific dissipation rate are discretized using the first-order upwind approach. However, the spatial discretization of the other equations employs the second-order upwind approach. It is anticipated that the solution will converge when the absolute sum of residuals is less than 10–6 . Additionally, the convergence of the
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Fig. 1 a Isometric view of PRB for axial temperature measurement and b mesh diagram of PRB
(a)
(b)
mean temperature at the exit of the solution domain is taken into account as another convergence criterion.
3.1 Governing Equations The model includes species diffusion, solid radiation, and heat transfer between solid and gas. The following equation, which defines all of the models mentioned above, can written as: Continuity Equation: ∂ ∈ ρg + ∇. ∈ ρg = 0 ∂t
(1)
∂ ερg uu + ∇. ερg uu = −ε∇ p + ∇.(ετ ) + si ∂t
(2)
Momentum equation:
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where si represent the pressure loss due to viscosity resistance and can be expressed ρg μ u | u in which k 1 is penetrability coefficient and k 2 is ineras si = − k1 + k2 | tial coefficient, p denotes the pressure of mixture of gases, τ defines the kinematic viscosity. Gas-phase equation: ∂ ερg cg Tg + ∇. εu ρg cg Tg = h v Ts − Tg + ∇. ε(K g + ρgi cgi DT i )∇Tg ∂t i
− ε h i ρYi (u i − u) + ε(τ u) + ε Q˙ (3) i
where p denotes the pressure, Q˙ denotes the heat transfer rate of the chemical reac tions and can be expressed as Q = i· ωi h i Wi , Cg represent the gas mixture specific heat, K g denotes the thermal conductivity of gaseous mixture, DT i denotes the thermal diffusion coefficient. h v represent the volumetric heat transfer coefficient between the solid and the gas. Tg is the gas temperature. Solid-phase energy equation: ∂[(1− ∈)ρs Cs Ts ] = ∇. K es ∇Ts + h v Tg − Ts ∂t
(4)
where K es denotes the heat transfer coefficients of porous media, Cs denoted the specific heat capacity of porous media and Ts represent the temperature of the solid. Species transport equation: ∂[ερYi ] + ∇. ερg uYi = −∇. ερg Yi Vi + εωWi ∂t
(5)
where Yi represents the i th species fraction of mass, Vi is the diffuse velocity of i th species and expressed as Vi = u i − u. Wi is the molecular mass of corresponding species and ω is the rate of reaction. State equation: ρg =
Wp RTg
(6)
where W is molecular mass, R is universal gas constant and Tg is the temperature of gas mixture. For the chemical reaction, a one-step reaction is used for the study. It is single step combustion reaction of methane air mixture. The pre-exponential factor for the reaction is 2.119e +11, the activation energy of the reaction is 2.027e + 0.8 J/ kmol and exponent of temperature considered to be zero.
114 Table 1 Grid independence test at the top surface of the PRB
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Case
1
2
3
No. of elements
11,302
23,605
46,667
Temp. (K)
818.3
804.5
800.8
3.2 Grid Independence Test Using a structural mesh, the discretization of the solution domain is performed. Each zone is divided into a number of blocks, each of which resembles a bent, mesh-like rectangle. In Fig. 1b, the mesh structure is displayed. To demonstrate the grid’s independence, the mesh elements has been doubled and quadrupled. There are 11,302, 23,605, and 46,667 mesh elements in cases 1, 2, and 3, respectively. In cases 2 and 3, the temperature at the middle top surface of the PRBis, respectively, 804.5 and 800.8 K, displaying relative errors of 0.0168 and 0.0214%. It was decided to employ case 1 for the remaining simulations due to the modest relative changes between cases 1 and 2 and case 1’s relatively low computational expenses. Table 1 represents the grid independence test for the temperature at the top surface of the PRB.
4 Results and Discussion For the investigated raw biogas mixes, the designed PRB operates steady combustion at the of 0.7–0.95 for the firing rate of 1–2 kW. The performance of PRB has been assessed for the firing rate of 1–2 kW within this stable range of 0.7–0.95. In the PRB, temperatures at various axial points were measured at a 30 °C ambient temperature. This section examines the PRB’s axial temperature and radiation efficiency. Figure 2 shows the comparison of present numerical study with the experimental data of Panigrahy et al. [15].
4.1 Axial Temperature Analysis To understand about the combustion characteristics and stability of the burner, temperatures inside the burner were measured at various axial points. Figure 3 displays the temperatures in the axial direction of the designed PRB at firing rates of 1–2 kW and ϕ of 0.7–0.95. For firing rates of 1–2 kW, the temperature at the base of the PZ (position 1 in Fig. 1a) of the PRB ranged from 980 to 1028 °C and 1041 to 1088 °C at ϕ of 0.7 and 0.95, respectively. For all cases of firing rate and, the continuous maximum temperature appears at position 1 and then gradually decreases toward the downstream region
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Fig. 2 Comparison of the present work with experimental data in the literature at ϕ = 0.7
of the PRB (position 3). This indicates that the biogas-air combustions at PZ have stabilised. Submerged combustion is the term given to this type of combustion. Since a higher firing rate during combustion generates more heat flux than a lower firing rate would, the pattern of axial temperature values at all places of the PRB is rising as a result. Additionally, it is discovered that the increase in at the specific firing rate causes the temperatures at different axial points to rise. It is thus so the airflow rate is higher at lower pressures, requiring more energy to heat the air than at higher pressures.
4.2 Radiation Efficiency Following equation is used to compute the PRB’s radiation efficiency. ηrad =
εσ Ts4 − T04 Ab ·
(7)
m f LC V SiC causes a very high emissive coefficient of 0.9, which has been taken into account for the PRB’s calculation of radiation efficiency, is well known. The ambient temperature during the simulation was taken to be 30 °C. The effect of surrounding temperature on the change of radiation efficiency is lower since it is way lower than the surface temperature of the PRB. According to Eq. 7, the radiation efficiency has an exponential relationship with surface temperature and a linear relationship
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(a)
(b)
Fig. 3 Axial temperature variation in PRB at a ϕ = 0.7 and b ϕ = 0.95
with fuel mass flowrate. As a result, surface temperature is an important factor in influencing the PRB’s radiation efficiency. Figure 4 depicts the effects of firing rate and ϕ on radiation efficiency. It shows how radiation efficiency drops as ϕ reduces from 0.95 to 0.7 at a set firing rate. It happens since the airflow rate is higher at lower ϕ, resulting in lower PRB surface temperatures. Additionally, it is observed that as the firing rate rises at a fixed ϕ, the radiation efficiency falls. Any porous media combustors have the fundamental property that radiation efficiency and firing rate are inversely proportional. An increase in firing rate causes an increase in the temperature of porous medium and the power of radiated flux (according to Stefan–Boltzmann law).
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Fig. 4 Radiation efficiency of PRB at different equivalence ratio
5 Conclusions This analysis utilized ANSYS FLUENT to provide a numerical analysis of the combustion of raw biogas in a PRB. 1. The stable combustions of the investigated raw biogas combinations took place at firing rates between 1 and 2 kW. 2. The temperature profile in the axial direction indicates that the modelled PRB can burn raw biogas across a wide working range. 3. Only the rise in equivalency ratio may be related to the increase in radiation efficiency. 4. With a radiation efficiency range of 12.4–37.5%, the PRB functions steadily. Acknowledgements The authors would like to gratefully acknowledge the science and engineering research board, DST Government of India. This work was supported by core research grant (CRG/ 2021/007295) funded by SERB, DST Government of India.
Nomenclature PRB m˙ Ab Ts
Porous radiant burner Mass flowrate (kg/s) Burner surface area (m2 ) PRB surface temperature (K)
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T0 ε ηrad σ ϕ PPI
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Ambient temperature [K] Emissivity [–] Radiation efficiency (–) Stefan-Boltzmann constant (W/m2 K4 ) Equivalence ratio (–) Pores per inch (–)
References 1. Khanna V, Goel R, Ellzey JL (1994) Measurements of emissions and radiation for methane combustion within a porous medium burner. Combust Sci Technol 99(1–3):133–142 2. Brenner G, Pickenäcker K, Pickenäcker O, Trimis D, Wawrzinek K, Weber T (2000) Numerical and experimental investigation of matrix-stabilized methane/air combustion in porous inert media. Combust Flame 123(1–2):201–213 3. Echigo R, Yoshizawa Y, Hanamura K, Tomimura T (1986) Analytical and experimental studies on radiative propagation in porous media with internal heat generation. In: International heat transfer conference digital library. Begel House Inc. 4. Sathe SB, Peck RE, Tong TW (1990) Flame stabilization and multimode heat transfer in inert porous media: a numerical study. Combust Sci Technol 70(4-6):93–109 5. Barra AJ, Diepvens G, Ellzey JL, Henneke MR (2003) Numerical study of the effects of material properties on flame stabilization in a porous burner. Combust Flame 134(4):369–379 6. Liu H, Dong S, Li BW, Chen HG (2010) Parametric investigations of premixed methane–air combustion in two-section porous media by numerical simulation. Fuel 89(7):1736–1742 7. Hoda SN, Nassab SAG, Ebrahim JJ (2019) Three dimensional numerical simulation of combustion and heat transfer in porous radiant burners. Int J Therm Sci 145:106024 8. Dutta PP, BorpatraGohain R, Dutta PP (2022) Performance assessment of an efficient biomass fired cook stove as a standby unit for community cooking. Biomass Conver Biorefinery 1–12 9. Mishra SC, Steven M, Nemoda S, Talukdar P, Trimis D, Durst F (2006) Heat transfer analysis of a two-dimensional rectangular porous radiant burner. Int Commun Heat Mass Transfer 33(4):467–474 10. Huang R, Cheng L, Qiu K, Zheng C, Luo Z (2016) Low-calorific gas combustion in a two-layer porous burner. Energy Fuels 30(2):1364–1374 11. Mishra NK, Mishra SC, Muthukumar P (2015) Performance characterization of a medium-scale liquefied petroleum gas cooking stove with a two-layer porous radiant burner. Appl Therm Eng 89:44–50 12. Hashemi SM, Hashemi SA (2017) Flame stability analysis of the premixed methane-air combustion in a two-layer porous media burner by numerical simulation. Fuel 202:56–65 13. Sahraoui M, Kaviany M (1994) Direct simulation versus volume-averaged treatment of adiabatic, premixed flame in a porous medium. Int J Heat Mass Transf 37(18):2817–2834 14. Mathis WM, Ellzey JL (2003) Flame stabilization, operating range, and emissions for a methane/air porous burner. Combust Sci Technol 175(5):825–839 15. Pantangi VK, Mishra SC, Muthukumar P, Reddy R (2011) Studies on porous radiant burners for LPG (liquefied petroleum gas) cooking applications. Energy 36(10):6074–6080
Numerical Validation and Benchmarking of Hydrogen Flame Propagation in a Vertical Acceleration Tube Experimental Facility Aditya Karanam, Vishnu Verma, and J. Chattopadhyay
Abstract During severe accident in a water-cooled nuclear power plant, hydrogen can be generated, leading to risks of possible hydrogen combustion and may threaten containment integrity. In this context, it is important to know the flame speed in a combustible mixture containing hydrogen. The ENACCEF test setup is a vertical acceleration tube with periodic obstacles and a dome of large volume at the top. It is a uniquely designed setup to study flame propagation under severe accident conditions in containment atmosphere. In this work, numerical validation has been carried out for one of the tests conducted at the ENACCEF facility using experimental data available in the open literature. A mechanistic combustion modelling framework especially suitable to fast flame propagation has been evolved and implemented using OpenFOAM. It has been observed that the numerical simulations are able to accurately capture a gamut of observed combustion phenomena like slow deflagrations, flame–turbulence interaction and fast flame acceleration. Moreover, benchmarking with numerical results obtained from other research groups suggests that the present results are more in line with the experimental data. The combined validation and benchmarking studies thus affirm the high fidelity of the adopted modelling approach and its numerical implementation with OpenFOAM. Keywords Hydrogen · Containment safety · Flame acceleration · CFD · OpenFOAM
A. Karanam (B) · V. Verma · J. Chattopadhyay Reactor Safety Division, Bhabha Atomic Research Centre, Mumbai 400085, India e-mail: [email protected] A. Karanam Homi Bhabha National Institute, Mumbai 400094, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_11
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1 Introduction During severe accident in water-cooled nuclear reactors, the reaction between steam and zirconium present in the fuel rods at high temperature can generate large quantity of hydrogen and released into the containment. Hydrogen is a highly flammable gas with very low ignition energy (~ 0.02 mJ) and a wide flammability range in air: 4% (v/v) to 75% (v/v). As a result, any small electrostatic discharge, heated surface, mechanical friction or minute increase in local temperature may lead to ignition. The resulting formation and propagation of combustion wave will lead to pressurization of the containment. This may damage the safety equipment inside the containment and jeopardize the structural integrity of the containment. The nuclear accident at Fukushima [1] is a pertinent example of the risk associated with hydrogen combustion. Therefore, to ensure safe operation of nuclear reactors, it is necessary to develop methods and tools that can be used to quantify the risks involved in hydrogen combustion during accidents. Under the international collaborative framework of the SARNET network (Severe Accident Research NETwork) [2], several hydrogen combustion experiments were performed at the ENACCEF test facility [2]; to obtain experimental data for the purpose of validation of numerical codes and inter-code benchmarking. ENACCEF operated by CNRS (French National Centre for Scientific Research) is a vertical acceleration tube with periodic obstacles and a dome of large volume at the top. These features are important due to their similarity to actual containment of a watercooled nuclear reactor. Considering these factors, one of the tests conducted at the ENACCEF facility available in open literature has been considered in this study for validation and benchmarking. Combustion can be modelled using fundamental reactive flow governing equations or through mechanistic correlations based on phenomenology. In the context of nuclear safety, considering large containment volume and long accident transients, practical cognizance mandates affordable simulations in a reasonable time-frame. Thus, phenomenology-based modelling such as Flame Surface Density (FSD), Probability Density Functions (PDF), G-equation, Renormalization Group Theory (RNG) and Turbulent Flame Closure (TFC) are generally considered [3]. Such models are integrated into special purpose CFD codes such as GASFLOW-MPI [4] or FLACS [5], which are widely used for explosion simulations; and also, in general purpose CFD codes like ANSYS Fluent and CFX. An international benchmarking report [6] gives in a tabulated form, the details of the numerical studies carried out by different research groups using these codes. The results demonstrate that the quality of prediction has reached a moderately high level of accuracy. However, such proprietary codes are closed source and impede the wider development and validation of practically relevant simulation approaches. More recently, the open-source alternatives have been adopted for combustion modelling in the ENACCEF facility [7, 8] and encouraging results were obtained. In continuity with this trend, the present work aims to leverage the OpenFOAM [9] platform; in an effort to implement and adequately validate an equivalent in-house
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solution. The objectives of the present work are: (i) evolve combustion model that is applicable especially for fast flame acceleration, (ii) its numerical implementation using OpenFOAM, (iii) associated validation with different turbulence closure models and (iv) benchmarking with results obtained by other researchers.
2 Combustion Modelling The paradigm for combustion modelling is decided based on the predominant combustion regime and assessed using the Damkohler number (Da). At high values Da ≫ 1, chemical reactions occurring faster than turbulent mixing. This implies that species are getting consumed by reactions faster than they are being made available by turbulent mixing. Thus, internal structure of the flame is intact; and the effect of turbulence is limited to stretching or wrinkling of the flame front depending on the local turbulent properties (intensity and direction). On the other hand, for Da « 1, the turbulent eddies can penetrate the flame zone and modify its internal structure such that the flame thickness may get extended and become comparable to the length scale of the domain. For hydrogen combustion in an acceleration tube type set-up, the authors have determined variation of the Damkohler number during flame propagation. This entails in situ computation of turbulent time scale and various chemical time scales and mapping of variation of the Damkohler number on the Borghi diagram. It has been determined that the Damkohler number varies in the range of 102 –104 , thus confirming that Da ≫ 1 and the combustion does indeed take place in the flamelet regime [10]. In flamelet combustion, the geometric approach [3] is widely used modelling strategy. It is primarily based on the premise that turbulence increases surface area of the flame by wrinkling. An increased flame surface has higher propensity to entrain and engulf the unburned gases, thereby increasing the rate of energy deposited in the unburned mixture. A higher rate of energy release can cause stronger mean flow gradients that promotes turbulence. This can in-turn lead to a further growth in flame surface area and the associated burning rate. Thus, a positive feedback loop between turbulence, flame surface area and burning rate is established. The net effect is that the effective flame speed keeps increasing, leading to Flame Acceleration (FA). Considering these effects, the flame front is identified to be a surface convected and wrinkled by the turbulent flow field. The burning rate (energy release rate) is then modelled in terms of turbulent flame speed (ST ) and the available flame surface area. To quantify this effect, the flame-wrinkling factor χ = ST /SL is defined, where ST and SL are the turbulent and laminar flame speeds respectively. For computing of χ , the algebraic closure model available in OpenFOAM, shown in Eq. 1 may be used. Here k is the turbulent kinetic energy and Reη is the Reynolds number based on the Kolmogorov scales. These parameters are computed using an appropriate turbulence model. SL is computed using polynomial correlation developed by Konnov [11] and adjusted for temperature and pressure variation.
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/ 1 ( ) k2 χ = 1 + 2(⟨c⟩ − 0.5) χeq − 1 ; χeq = 1 + 0.62 Reη SL
(1)
The non-dimensional parameter ⟨c⟩ i.e., progress variable (Favre-averaged), can be interpreted as the density weighted probability of finding burned gas at a particular instant and spatial location. The associated transport equation is given in Eq. 2. The source term (ω˙ c ) represents the heat release due to combustion. The turbulent Schmidt number Sc T = 1.0 has been used. The turbulent viscosity μT is computed using an appropriate turbulence model. ) ∂ ∂ ( ∂ ρ⟨u j ⟩⟨c⟩ = (ρ⟨c⟩) + ∂t ∂x j ∂x j
(
μT ∂⟨c⟩ Sc T ∂ x j
) + ω˙ c
(2)
For source term closure, Zimont [12] identified that the mean turbulent flame surface has a proclivity to propagate in the local normal direction. Moreover, the rate of propagation should be proportional to the turbulent flame speed (ST ) and the availability of unburned mixture (ρ u ). Considering these, Zimont proposed the closure relation shown in Eq. 3. The model recognizes the relation between flame wrinkling factor and the heat release rate. Higher the wrinkling of the flame, its surface area increases and promotes further heat release. The wrinkling factor is in turn related to the flow turbulence through its dependence on the turbulent flame speed (by definition). In this way, the Zimont model very concisely accounts for different physical mechanisms. ω˙ c = ρ u S T |∇⟨c⟩|
(3)
It is important to mention that the model assumes turbulent flame propagation throughout, due to its linear dependence on ST . Hence, the model is strictly applicable when the flow is fully turbulent. The laminar and quasi-laminar regimes that are expected in the initial part of flame propagation (when flame or geometry induced turbulence is low) will also be treated as fully turbulent flames. This shortcoming has to be kept in mind when analysing the numerical results.
3 Description of Test Facility The ENACCEF is a vertical and completely enclosed facility to study flame propagation under accident specific conditions. The facility consists of two sections: an acceleration tube at the bottom and a dome at the top. A schematic of the facility is depicted in Fig. 1. The acceleration tube is 3.2 m long and has an internal diameter of 0.154 m. It consists of nine annular baffles to create obstacles in the flow and increase turbulence. The first baffle is installed at 0.776 m from the bottom and subsequent baffles are located at a distance of 0.154 m. The opening diameter of the baffle is 0.093 m and
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Fig. 1 Schematic of the experimental facility
this leads to a blockage ratio of 63% (based on percentage of flow area blocked). The thickness of the baffle is 2 mm. The dome is located above the acceleration tube. The dome is 1.7 m long and has an internal diameter of 0.738 m. The internal volume of the acceleration tube is 62.1 L and that of the dome is 658 L. The mixture is ignited using two thin tungsten electrodes which are linked to a high-voltage source. The spark energy delivered is estimated to be around 20 mJ. The ignition point is located 0.138 m from the bottom of the facility. To measure flame position, 16 UV-sensitive photomultiplier tubes were used. The presence of the flame was detected based on the total emission of OH radical because of its high concentration in the flame front. Several hydrogen-air flame propagation experiments were performed of which test-153 [13] has been considered in the present study. This corresponds to a uniform hydrogen concentration of 13% by volume with a blockage ratio of 0.63. The initial pressure and temperature used in the experiment are 1 bar and 296 K respectively. The experimental data on measured flame speed versus axial distance for test-153 has been obtained from published literature available in the open domain [13]. Measured flame speed is based on linear interpolation between successive PMT locations. The test was conducted five times to check for repeatability and the average values have been digitized for validation with numerical results.
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4 Numerical Set-Up For the present problem, no formation of shock waves was reported from the experimental work. However, density changes are expected to be significant considering flame speed of the order of 500 m/s were observed. Thus, a Pressure-based Solver (PISO) formulated for a weakly compressible flow is most applicable. The pressure correction equation is solved in conjunction with equation of state for density computation in an inner or corrector loop. The conservation equations are solved in a segregated manner in an outer or predictor loop. The inner and outer loops are embedded in a separate time loop for transient computations. Implicit time stepping based on maximum Courant number of 0.2 is adopted for robust and efficient simulation framework. All convective terms have been discretized with a second order TVD upwind scheme and solved with a tolerance of 10–6 . The Preconditioned Bi-Conjugate Gradient (PBiCG) solver with convergence residual of 10–10 has been used for matrix inversion. The initial conditions were set according to the experiment (test-153) i.e., pressure of 105 Pa, temperature of 296.15 K and hydrogen mole fraction of 0.13. The mixture is assumed to be uniformly mixed. Since the initial turbulence levels were not measured, it is assumed to be low values for a quiescent mixture. The facility is not vented during the experiment and therefore modelled as a closed system. An adiabatic boundary condition at walls for temperature and no-slip boundary condition for velocity field have been applied. Neumann zero-gradient boundary conditions have been used for pressure and progress variable. For turbulence, standard wall functions have been used. Turbulence modelling is carried out based on the RANS standard k − ε model [14] and SST k − ω model proposed by Menter [15]. The SST model combines advantages in both the free stream and near wall regions. Moreover, the eddy viscosity formulation accounts to a large extent the transport of anisotropic principal turbulent shear stress, without deviating from the first-order closure models. The spatial discretization features perfectly orthogonal 3D hexahedral cells with zero skewness and unit aspect ratio. Grid size used is 7 mm and results in 2,370,160 cells. It may be noted that this is relatively coarse grids as compared to requirements for flame and shock resolving models. Since the main objective is to get the overall flame propagation, the present grid selection is suitable. Ignition is modelled by setting the progress variable to one in a region of diameter 40 mm at the ignition location and zero elsewhere. Initially, the flame propagates in the laminar and quasi-laminar regimes, which are not accurately modelled by TFC models. However, this discrepancy is expected to be limited to the initial part of the acceleration tube (before first obstacle), until the onset of the turbulent flame propagation regime in the large majority of the setup. Adiabatic flame temperature of 1616.03 K corresponding to 13% hydrogen has been patched in the ignition region assuming complete combustion. The cross-section of grid along with the ignition region is depicted in Fig. 2.
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Fig. 2 Spatial discretization and ignition region
5 Numerical Results 5.1 Validation Results Before comparing the numerical results, some trends from the experimental data are explained. Experimental result is shown in Fig. 3. Region A corresponds to flame propagation before the first obstacle and can be categorized as slow deflagration. Flame speed is of the order of ~ 10 m/s. Region B corresponds to fast flame acceleration, primarily owing to turbulence generation on interaction with the obstacles. Flame speed increases to ~ 500 m/s. After the final obstacle (region C), flame speed decreases due to reduced turbulence from the absence of obstacles. However, the flame speed is still higher than 300 m/s. Both regions B and C can be categorized as fast deflagration. Towards the end of the acceleration tube and the initial part of dome (region D), the flame speed again increases due to jet effect associated with the flame transitioning from the narrow acceleration tube to the much wider dome part. This corresponds to jet flame propagation. After the jet effect subsides in the aft end of the dome, flame propagation continues with a much-reduced flame speed. Validation of numerical results obtained with standard k − ε model is shown in Fig. 3. Combustion modelling is based on closure relations described in Sect. 2. It can be observed that, qualitatively, the numerical trends of flame acceleration in sections A and B, followed by flame deceleration in section C are in accordance with the measured values. Since initial turbulence levels were not measured in experiment, it is required to vary turbulence parameters in the numerical simulation to determine the optimum value. In this study, turbulence dissipation rate is varied and result is shown in Fig. 3.
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Fig. 3 Numerical validation with standard k − ε model
While the initial turbulent kinetic energy can also be varied, since the mixture is initially in a quiescent state, a constant value of k = 0.1 is assumed in all simulations. Moreover, the mean flow evolution depends on the ratio of turbulence dissipation rate to kinetic energy; and hence, varying one of the parameters is sufficient. Physically, dissipation occurs as a sink term in the turbulent kinetic energy equation and a higher initial dissipation should lead to lower TKE and lower flame acceleration. A higher dissipation also has the property of enhanced mixing at the lower scale thereby increasing the effect flame speed. It can be observed that as the dissipation rate is increased, flame acceleration in the sections A and B decreases, as expected. In the slow deflagration part (section A), ε = 25 or higher dissipation, seems to be a better match with experimental data. But, in the obstacle part (section B), ε = 5 seems to be optimum value. A lower value significantly increases flame acceleration and vice-versa. Since the combustion model is mainly applicable for fully turbulent flames, which are expected in the obstacle section, it is prudent to tune it in only its domain of applicability, that is, section B. The laminar and quasi-laminar regimes that are expected in section A are not entirely accounted for. Thus, optimum value of ε = 5 is established. This finding in conjunction with the adopted combustion model leads to an average absolute error of as low as 6% with respect to experimental data in section B. The flame deceleration predicted in the initial part of Section C is also satisfactory. However, in the aft end of the tube, the model predicts anomalous flame
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acceleration (highlighted in red). In the dome region, jet flame propagation in the trailing part is seen to be inline with the experimental data. The anomalous flame acceleration seems to be a short coming of the turbulence model. To improve prediction, the numerical simulation is repeated with the SST k − ω model and result is shown in Fig. 4. Combustion modelling strategy is exactly the same as used earlier. The numerical settings, spatial discretization and other numerical parameters are the same for both cases to get a back-to-back comparison. Both the models were initialized with the same turbulent kinetic energy. The initial specific dissipation rate required for the SST model is calculated based on turbulence dissipation rate ε = 5, optimized earlier. The quality of prediction from the SST k − ω is immediately evident. Performance of both turbulence models is comparable in the slow deflagration (section A) and flame acceleration (section B) regimes. The important difference is observed in section C. While the standard k − ε model predicts the flame deceleration better than the SST k − ω model, the former leads to anomalous FA at the aft end of section C (as noted earlier also). The SST k − ω model predict lower flame deceleration but does not lead to any spurious flame acceleration. In fact, the FA predicted by the SST k − ω is very comparable to the experimental data towards the end of section C. In
Fig. 4 Comparison between standard k−ε (Numerical #4) and SST k−ω (Numerical #5) turbulence models
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the dome region (section D), both turbulence models are comparable. Most importantly, flame acceleration in the obstacle section (B) are quantitatively well predicted in accordance with the experimental data by both turbulence models. From these observations, it may be deducted that the implemented combustion model is able to capture turbulence-flame interaction mechanism, that is mainly responsible for fast flame acceleration. Considering the complex propagation mechanism, the results are very encouraging. Between the turbulence closure models, while there is no clear winner, the SST k − ω model is a better candidate for further tuning, qualification and study.
5.2 Benchmarking Results Benchmarking studies are required to gauge present capabilities with independent simulations carried out by other research groups. Two such studies have been identified for benchmarking: i. In the first study carried out by Takasuo and Huhtanen [16], the commercial code Fluent has been used. Combustion is modelled with Eddy Break-up model. The mixture is assumed to be premixed and turbulence is considered to wrinkle the flame front and mix the active radicals to the unburned combustion mixture. The EBU approach does not model laminar combustion phase in detail. Turbulence is calculated using the two equation k − ε model with standard wall functions. Ignition is described by introducing an initial turbulence to the ignition location and a higher temperature. The simulations are highly sensitive to initial values of turbulence used at the ignition location. Calculations were done using full 3D grid with 195,312 cells. No-slip and adiabatic boundary conditions have been assumed. Calculation was run in transient manner with second order space and time discretization. ii. FLACS simulation by Bleyer et al. [13] uses compressible RANS formulation and k − ε model for turbulence. The numerical model uses second order schemes for flux computations. Time stepping is done using first order backward Euler scheme. The SIMPLE pressure correction algorithm is applied and extended to handle compressible flows with additional source terms for the compression work in the enthalpy equation. FLACS uses distributed porosity concept which enables detailed representation of complex geometries using a Cartesian grid. Combustion model assumes that the flame can be regarded as a collection of flamelets. One step reaction kinetics is assumed, with the laminar burning velocity being a measure of the reactivity of a given mixture. Burning velocity model consists of laminar, quasi-laminar and turbulent sub-models. Mesh is generated with a size of 10 mm and comprises 1,022,625 cells. The benchmarking result is shown in Fig. 5. The best result from present numerical simulation, that is, using the Zimont combustion model with SST k − ω turbulence closure on a 7 mm grid using OpenFOAM.
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Fig. 5 Benchmarking results
Comparison indicates that FLACS simulation [13] over predicts flame speed in the slow deflagration and initial part of the obstacle section. The peak velocity at the end of obstacle section is under predicted. Surprisingly, the trend in the smooth section is opposite to measured values. While the measured flame speed decreases, FLACS predicts increasing trend. Fluent calculation [16] significantly over predicts peak flame speed and shows spurious flame acceleration in the jet flame region. In the dome section, all codes seem to have comparable behaviour. Overall, the present simulation with OpenFOAM fares better, unambiguously, compared to other codes both qualitatively and quantitatively.
6 Conclusions In this study, a combustion model based on the geometric approach with Zimonts turbulent flame closure was presented. Numerical simulation was carried out using OpenFOAM for one of the tests conducted at the ENACCEF experimental facility. The numerical prediction has been validated with experimental data. Benchmarking studies were carried out to compare the present numerical capability with other research groups. Following conclusions can be drawn:
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i. The implemented combustion model leads to good qualitative prediction with respect to experimental data for both slow and fast deflagration. ii. In the fully turbulent region for which the model is applicable, the average absolute error is 6%. Hence the model is validated. iii. The initial turbulence dissipation rate (ε) was found to have a strong effect on flame propagation. Consistent with physical reasoning, a higher initial dissipation led to lower flame acceleration. Optimum value of ε has been established. iv. The standard k − ε model predicted spurious flame acceleration at the aft end of the acceleration tube, while the SST k − ω model does not suffer from this drawback. v. Benchmarking results indicate that the present simulation fares better than special purpose and commercial codes, both qualitatively and quantitatively. The validation and benchmarking study thus affirm the high fidelity of the adopted modelling approach and its numerical implementation with OpenFOAM.
Nomenclature c k Re SL ST uj ε ω˙ c μT Sc T χ ρ ⟨a⟩ a
Progress variable (–) Specific turbulent kinetic energy (m2 /s2 ) Reynolds number (–) Laminar flame speed (m/s) Turbulent flame speed (m/s) Velocity vector (m/s) Specific Turbulent dissipation rate (m2 /s3 ) Source term of progress variable transport equation (represents average heat release rate) (kg/m3 /s) Eddy viscosity (kg/m/s) Turbulent Schmidt number (–) Flame wrinkling factor (–) Density (kg/m3 ) Favre averaged value of a Time averaged value of a
References 1. Holt M, Campbell RJ, Nikitin MB (2012) Fukushima nuclear disaster. In: CRS Report, R41694 2. Bentaib A, Bleyer A, Meynet N, Chaumeix N, Schramm B, Hohne M, Kostka P, Movahed M et al (2014) SARNET hydrogen deflagration benchmarks: main outcomes and conclusions. Ann Nucl Energy 74:143–152
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3. Veynante D, Vervisch L (2002) Turbulent combustion modeling. Prog Energy Combust Sci 28:193–266 4. Li Y, Xiao J, Zhang H, Breitung W, Travis J, Kuznetsov M, Jordan T (2021) Numerical analysis of hydrogen release, dispersion and combustion in a tunnel with fuel cell vehicles using allspeed CFD code GASFLOW-MPI. Int J Hydrog Energy 23:12474–12486 5. Lucas M, Skjold T, Hisken H (2020) Computational fluid dynamics simulations of hydrogen releases and vented deflagrations in large enclosures. J Loss Prev Process Ind 63:103999 6. NEA (2011) ISP-49 on hydrogen combustion. In: Nuclear Safety Report NEA/CSNI/R 7. Kim GH, Woo M (2011) Simulation of hydrogen flame acceleration by using XiFoam. Sixth OpenFOAM workshop, PennState University 8. Povilaitis M, Jaseliunaite J (2021) Simulation of hydrogen-air-dilutents mixture combustion in an acceleration tube with FlameFoam solver. Energies 14:5504 9. OpenFOAM (Developed and distributed by CFD Direct). http://openfoam.org 10. Karanam A, Ganju S, Chattopadhyay J (2020) Time scale analysis, numerical simulation and validation of flame acceleration and DDT in hydrogen-air mixtures. Combust Sci Technol. https://doi.org/10.1080/00102202.2020.1732363 11. Konnov AA (2008) Remaining uncertainties in the kinetic mechanism of hydrogen combustion. Combust Flame 152:507–528 12. Zimont VL (2000) Gas premixed combustion at high turbulence: turbulent flame closure combustion model. Exp Thermal Fluid Sci 21:179–186 13. Bleyer A, Taveau J, Chaumeix N, Pallard CE, Bentaib A (2012) Comparison between FLACS explosion simulation and experiments conducted in a PWR steam generator casemate scale down with hydrogen gradients. Nucl Eng Des 245:189–196 14. Launder BE, Sharma BI (1974) Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc. Lett Heat Mass Transf 1(2):131–138 15. Menter FR (1994) Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J 32(8):1598–1605 16. Takasuo S, Huhtanen M (2007) Application of TONUS V2006 and FLUENT 6.2.16 CFD codes to ENACCEF hydrogen combustion tests
Detailed Chemical Kinetics Mechanism for Condensed Phase Decomposition of Ammonium Perchlorate Jay Patel, Prathamesh Phadke, Rohit Sehrawat, Arvind Kumar, Arindrajit Chowdhury, and Neeraj Kumbhakarna
Abstract The current study aims to improve the understanding of the complex thermal decomposition mechanism of Ammonium Perchlorate (AP-NH4 ClO4 ). The absence of detailed reaction mechanisms for AP with accurate rate kinetics data in the condensed phase is the main challenge of thermal decomposition and combustion modelling. In the present study, AP-condensed phase elementary reactions were investigated using Ab-initio molecular modelling, in which the polarizable continuum model using integral equation formalism variant (IEFPCM) was used to account for the condensed phase and transition state theory calculations are used for calculating kinetic parameters of the reactions. Gas-phase species evolution profiles were obtained from fast pyrolysis experiments conducted isothermally at three temperatures (420, 440, and 460 °C) at 10 bar gauge pressure. The same are used to validate the reaction mechanism. To simulate the fast pyrolysis experiments, computational model based on mass and species conservation in the condensed phase and gas phase control volumes was developed. The predicted gas phase mole fraction profiles of the decomposition products agree well with the experimental results, indicating that the proposed reaction mechanism captures the liquid phase decomposition of AP. Keywords Propulsion · Ammonium perchlorate (NH4 ClO4 ) · Thermal decomposition · Condensed phase
J. Patel · P. Phadke · A. Chowdhury · N. Kumbhakarna (B) Department of Mechanical Engineering, IIT Bombay, Mumbai 400076, India e-mail: [email protected] R. Sehrawat · A. Kumar High Energy Material Research Laboratory (HEMRL), Defence Research and Development Organisation, Pune 411021, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_12
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1 Introduction Solid propellants are one of the most widely used class of propellants for propulsion systems. Compared to liquid and cryogenic propulsion systems, solid propulsion systems do not require complicated engines. Currently, solid rocket motors are consistently used in launching space vehicles as well as strategic and tactical missiles. Compounds such as, Ammonium Perchlorate (AP) have been used in solid propellant compositions since 1960s. Numerous studies have been dedicated to studying the decomposition mechanism and structural as well as thermodynamic properties of AP and the products of its decomposition. Decomposition of AP is a complex process, as it has four different elements N, H, Cl and O. The propellant community is interested in the thermal decomposition of AP and the associated chemical kinetic parameters for two main reasons: first, the ability to simulate the one-dimensional combustion of such strands in order to predict the performance of composite propellants that contain potential fuels and burning rate enhancers, and second, the stability of various propellant mixtures that contain AP as an ingredient under thermal stress. As a result, modelling fast combustion at heating rates of approximately 106 K·s−1 and accelerated ageing at heating rates of approximately 5 K·s−1 , respectively, requires knowledge of AP’s decomposition behaviour. It was observed that decomposition of AP occurred simultaneously with sublimation under atmospheric conditions in a heterogeneous medium, resulting in a porous structure [1]. It has been reported that AP particles burn individually as monopropellants in composite and composite modified double base solid propellant formulations.
2 Literature Review and Objective AP has been consistently used in the solid propellant formulations because of its superior oxidizing characteristics, suitability with other solid propellant components, and ease of availability. Several studies have been conducted on the thermal decomposition mechanism of AP using experimental techniques such as TGA-FTIR spectroscopy [2]. However, a detailed chemical kinetics mechanism has not been determined till date owing to the inherent complexities of the reactions in condensed phase possibly involving hundreds of reactions. As compared to other oxidizers, AP forms chlorine. Whereas, other energetic materials consist of only carbon, hydrogen, nitrogen and oxygen. Beckstead et al. [3] developed the classical Beckstead-DerrPrice (BDP) model to simulate the combustion of composite solid propellants. The BDP model does not focus on the condensed phase chemistry, but assume a zeroth order surface pyrolysis law with a pre-exponential factor of 3 × 106 kg m−2 s−1 and an activation energy of 92 kJ·mol−1 . BDP model was expanded by the Jeppson et al. [4] for AP by including a first order semi global reaction mechanism. In order to get the kinetics of AP decomposition in condensed phase Mallick et al. [5] used TGAFTIR-MS set-up, in which AP sample is heated at different slow heating rate (5, 10
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and 15 K min−1 ). In that study it was concluded that AP decomposition takes place in three different stages: low temperature decomposition (260–330 °C), intermediate temperature decomposition (330–370 °C) and high temperature decomposition (370–420 °C). The species detected from the FTIR and the mass spectra were O2 , Cl2 , N2 O, HNO3 , H2 O, HCl, ClO and HOCl. In addition to the experimental and theoretical studies, computational approach based on quantum mechanics was used by researches for the development of reaction mechanism for AP as experiments alone are not adequate for study of decomposition. Zhu and Lin investigated the decomposition of AP computationally using CCSD(T) and B3LYP methods along with the 6–311 + G(3df,2p) basis set and compared gas phase and condensed phase results [6]. Later the kinetics for sublimation/decomposition of AP were studied by firstprinciple calculations [7] and the activation energy for sublimation process desorbing H3 N···HOClO3 as a pair was reported to be 28.8 kcal/mol. Recently, Prathamesh et al. [8] used the rapid electrical pyrolyzer to study the condensed phase reactions and to find kinetics parameters of AP decomposition. In their experiments heating rate was found to be much higher than the maximum heating rate of a commercial TGA system, which is similar to those observed during actual combustion in solid motors (~ 105 K·s−1 ). Even though there has been considerable research on AP decomposition leading to the formation of a detailed gas phase mechanism for combustion of AP at various pressures [9], decomposition of AP in the condensed phase is relatively unexplored, with a detailed chemical kinetic mechanism not available till date. Previous studies suggested that in an AP based composite solid propellant, the burning rate modifiers are active mainly in the condensed phase. Therefore, having a reliable condensed phase reaction mechanism is important for the modeling of combustion phenomena, study of burn rate characteristics and for the selection of appropriate additives. Hence the aim of this work is to developed the detailed chemical reaction mechanism of AP in the condensed phase.
3 Computational Methods Quantum mechanics calculations offer a way to corroborate already-measured data from experiments and to supply data that might otherwise be unavailable through experiments. Hence, quantum mechanics-based Density Functional Theory (DFT) calculations were carried out using the Gaussian 16 package and the Becke threeparameter hybrid density function approximation using B3LYP technique [10] to study AP decomposition. The structures of reactants, products and transition states are optimized using 6-31G(d,p)/6-31G++(d,p) basic set. An already optimized structures were used as a starting point for higher-order techniques like the CBS-QB3 compound method. Because it offers a superb balance of accuracy and computational effort, this approach was chosen. Various input geometries were used for ground state optimization and frequency computations, and the molecular structure with the lowest energy and no imaginary frequencies was selected as the stable configuration for the
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compound of interest. The relaxed potential energy scan uses the same methodology and offers a rough estimate of the pathway between reactant and product. A relaxed scan allows the geometry to optimize independently of the coordinates being scanned at each stage. The initial guess geometry for transition state optimization was chosen to be the highest point on the scan calculation curve. The imaginary frequencies are used to identify stationary and transition states. The number of imaginary frequencies is zero for the stationary state and one for the transition state. To confirm that the identified transition state connects the reactants and expected products, Intrinsic Reaction Coordinate (IRC) calculations were performed. An IRC calculation begins at the maximum energy point on the potential energy surface corresponding to the TS and proceeds to optimize the geometry at each point by following the minimum energy path in both directions. The reaction rate constants were determined using data from the potential energy surface obtained with Gaussian 16 and conventional Transition State Theory (TST) [11], This allows an unique type of equilibrium between the reactants and activated complex. TST was used to calculate the forward and backward rate constants for reactions: kB T k(T ) = k e hP
(
ΔS ‡ Ru
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e
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where ΔS ‡ and ΔH ‡ are, respectively, the activation entropy and activation enthalpy. The units of Î were appropriately determined for each reaction by multiplying with a suitable conversion factor depending on the type of reaction for which it was calculated. Aside from that, k B stands for Boltzmann’s constant, hP stands for Planck’s constant, and T stands for temperature in Kelvin. The Polarizable Continuum Model (PCM) with the Integral Equation Formalism Variant (IEFPCM) was used for all quantum chemical calculations in Gaussian 16 to reflect the assumption that condensed phase reactions can be treated as occurring in a solution phase. This model takes into account the effects of continuum solvation. The cavity in PCM was built using the UFF (universal force field) radius, which is the default option in Gaussian 16. Ammonium Perchlorate (NH4 ClO4 ) was specified as the solvent in all the optimization, frequency and IRC calculations to represent the solution-phase medium. All the calculations are performed at the default value of temperature in Gaussian 16 which is 298.15 K.
4 Mathematical Model The developed liquid-phase reaction mechanism is validated using rapid pyrolysis of AP experiment results. Experimental set-up and methodology are in our previous work [8]. In order to simulate such a system, an appropriate mathematical model is required and it was formulated on the same lines as that of Patidar et al. [12]. Because of chemical production and consumption, as well as vaporization, the mass
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of various species in the decomposing liquid phase AP changes. As a result, the mass conservation of liquid phase species takes the form, Nspec ∑ dYl.i = ω˙ l,i − Yl,i kevap,i + Yl,i Yl, j kevap, j dt j=1
(2)
Here Yl,i is the mass fraction, ω˙ l,i is the net production rate of species i in the liquid phase. kevap is the rate constant of vaporization, which is modelled using the Arrhenius equation. The stable species formed during the decomposition of AP such as NH3 , H2 O, O2 , HCl, NO2 , N2 O, Cl2 , ClO2 , HNO3 and HClO4 desorb from the liquid phase and evolve into the gas phase. These decomposition products quickly mix with the purge gas N2 and evolve into a cooler region. As a result, it is assumed that they will not undergo further reactions in the gas phase. As compare to the mass of purge gas in the control volume (Brill cell), the total mass evolving in the gas phase is very small so the pressure and the temperature in the gas phase remain constant. In the experimental setup there is no continuous flow of purge rate, but the cell is filled with it. The conservation of mass of species i in the gas phase control volume is takes the form, ⎡ ⎤ Nspec ∑ m l kevap Yl, j dX g,i Ru Tg ⎣ m l kevap,i Yl,i ⎦ − Xp (3) = dt Pg Vg M Wi M W j j=1 The mole fraction of species i in the gas phase is represented by X i,g . The gas phase region’s temperature, pressure, and volume, T g , Pg , and V g , are related to the Brill cell’s temperature, pressure, and volume, respectively, in the current experiments. Ru is the universal gas constant. During the solving gas phase equation, purge gas mole fraction is initialled with value one, while others with zero. The overall mass loss in the liquid phase take place due to the evaporation of stable species. So, the equation for the total liquid mass takes the form, ∑ dm l = −m l Yl,i K evap,i dt i=1 Nspec
(4)
Since it was known how temperatures varied experimentally, the energy equation could not be solved. By assuming uniform temperature and a very small amount of AP sample during the experiment, the model was made simpler. Governing equations given above were solved with the DVODE solver developed by Brown et al. [13] which employs Gear’s method to integrate stiff ordinary differential equations. At t = 0, 200 µg of AP are present at the start of the simulation. which is the starting mass used in the experiment, and the temperature is 298 K.
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5 Results and Discussion The detailed chemical kinetics reaction mechanism for condensed phase decomposition of AP derived through Ab initio calculations as described in the computational method consists of 160 elementary reactions of 59 species. The liquid phase decomposition of AP begins with unimolecular decomposition of ClO4 − into ClO2 − and O2 after that proton transfer between NH4 + and ClO2 − take place. Validation of the reaction mechanism is an important part of the mechanism’s development. In general, mass loss data and species evolution profiles obtained from various experiments such as TGA and fast pyrolysis are used for validation. The current mechanism is validated using a fast pyrolysis experiment performed at 10 bar gauge pressure at three different isothermal temperatures: 460, 440 and 420 °C. Experimental methodology is mentioned in detail in Prathamesh et al. [8] work. Currently, only species profiles are available for the validation of mechanism. As it has been well established that AP decomposition is a heterogenous process during which some part of AP directly sublimates to gas phase while the remaining part liquifies and undergoes reactions in the liquid phase resulting in the formation of stable species that evolve into the gas phase. However, it is not known accurately what fraction of AP sublimates, and it is assumed that these proportions change with physical conditions. Four steps semi global mechanism developed by Jeppson et al. [4] resulting in far more NH3 and HClO4 , Whereas Guirao and Williams [14] assumed 30% sublimation and 70% condensed phase surface reaction which gives better prediction to the burn rates between 20 to 100 atm pressure. In the current model 30% of AP is considered to be sublimating and remaining 70% to be reacting in liquid phase. Zhu and Lin [7] used a generalized gradient approximation with plane-wave density functional theory to calculate the kinetic parameters for the sublimation of AP using first-principles calculations that is k sub = 6.53 × 1012 exp (− 28.8 kcal/ mol/RT) s−1 . In the model, sublimation is simulated through the Arrhenius equation with E a = 28.8 kcal/mol, which is the same as calculated by Zhu and Lin and A = 1 × 108 . Arrhenius equations are also used to model the evaporation of species entering the gas phase. Because the activation energy and pre-exponential factor for evaporation for the majority of the species are unknown, those parameters are chosen based on the heat of formation, hydrogen bonding, and molecular weight. Figure 1 represents the mass loss profile obtained by the current model at three temperatures that were the same as those used in pyrolysis experiments. At 460 °C, we have almost total mass loss, whereas at 440 and 420 °C, we have around 90% and 86% mass loss, respectively. Figures 2, 3 and 4 depict species evolution profiles predicted by the model for various isothermal temperatures. At all three temperatures, H2 O is the major product and O2 comes first as compared to other species as it is the characteristic of AP being used as an oxidizer. However, the O2 mole fraction increases as temperature decreases, possibly because O2 consumption reactions are more active at higher temperatures at 440 and 460 °C. The mole fractions of NH3 and HClO4 are the same in this case, and both species enter the gas phase due to AP sublimation. Cl-containing
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Fig. 3 Species mole fraction profiles in the gas phase at 440 °C
Fig. 4 Species mole fraction profiles in the gas phase at 420 °C
Form the fast pyrolysis experiments we have species profiles of H2 O, HCl, NO2 and N2 O which are measured through a Fourier Transform Infrared (FTIR) spectrometer used for the validation of the current reaction mechanism. To compare these profiles, all of the species mole fractions are normalized using the mole fraction value of H2 O at 10 s for the temperatures studied. Normalization is done to achieve the same scale for comparison; it has no effect on the rate at which these species evolve
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in the gas phase. Figure 5 compares the evolution of species normalized mole fractions in the gas phase at isothermal 460 °C. Figure 5a shows that the H2 O evolution rate agrees well with experimental results, whereas the HCl rate is underestimated by the current mechanism maybe because the current mechanism lacks some important reactions that contain HCl specie and are active at higher temperatures Fig. 5b compares the normalized mole fraction evolution of NO2 and N2 O, demonstrating that both species are in good agreement with experimental results within the error bars. Figure 6 presents a comparison of the normalized mole fraction evolution profiles at 420 °C which also match well with the experimental results. The error bars in the figure indicate the errors related to the experiment, such as repeatability, capability of the FTIR set-up, etc. The comparison of simulated and experimental results at 440 °C is not shown here due to page limit constraints, but results at this temperature are also in good agreement. To gain insight into the chemical processes occurring during AP decomposition, it is important to identify the critical pathways for major species entering the gas phase. for that sensitivity analysis to be carried out at various points in time during the thermal decomposition of AP. Kumbhakarna and Thynell [15] go into great detail about the formulation for sensitivity analysis, which is used here. In the proposed mechanism, H2 O and N2 O mass fraction are found to be most sensitive to the pathways that proceeds through the intermediate NHHNO2 , Whereas the majority of O2 is formed in the first step to participate in the mechanism, only a small portion of which enters the gas phase. The mass fractions of HCl and NO2 are found to be sensitive to the HOCl-mediated pathway.
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6 Conclusions Using quantum mechanics-based molecular modelling calculations, a detailed chemical kinetics mechanism consisting of 59 species and 160 elementary reactions was developed for the condensed phase decomposition of AP. Based on these calculations, major reaction pathways were identified. The AP decomposition process was numerically simulated by solving a system of ordinary differential equations that represented mass loss, reversible reactions, and stable species evaporation. The simulated results were found to match the experimental data obtained by Fast pyrolysis satisfactorily. Currently, the mechanism has been validated at a single pressure; in the future, it will be validated across a wide range of pressures and temperatures. The validated liquid phase mechanism will be a valuable for use in combustion models for predicting the performance of AP-based composite propellants.
References 1. Beckstead MW, Puduppakkam K, Thakre P, Yang V (2007) Modeling of combustion and ignition of solid-propellant ingredients. Prog Energy Combust Sci 33(6):497–551 2. Brill TB, Brush PJ, Patil DG (1993) “Thermal decomposition of energetic materials 60: major reaction stages of a simulated burning surface of NH4 ClO4 . Combust Flame 94(1–2):70–76 3. Beckstead MW, Derr RL, Price CF (1970) A model of composite solid-propellant combustion based on multiple flames. AIAA J 8(12):2200–2207 4. Jeppson MB, Beckstead MW, Jing Q (1998) A kinetic model for the premtxed combustion of a fine AP/HTPB composite propellant. In: Proceedings of the 36th AIAA aerospace science and meeting exhibition 5. Mallick L, Kumar S, Chowdhury A (2015) Thermal decomposition of ammonium perchlorate: a TGA-FTIR-MS study: part I. Thermochim Acta 610:57–68
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6. Zhu RS, Lin MC (2006) A computational study on the decomposition of NH4 ClO4 : comparison of the gas-phase and condensed-phase results. Chem Phys Lett 431(4–6):272–277 7. Zhu R, Lin MC (2008) Mechanism and kinetics for ammonium perchlorate sublimation: a first-principles study. J Phys Chem C 112(37):14481–14485 8. Prathamesh P, Sehrawat R, Kumar A, Kumar S, Kumbhakarna AC (2022) Kinetic parameters govering the rapid pyrolysis of ammonium perchlorate. J Aerosp Sci Technol 74:10–17 9. Ermolin NE (1993) Kinetic parameters of overall gas-phase reactions for propellants based on ammonium perchlorate and polybutadiene binder. Combust Explos Shock Waves 29(4):508– 515 10. Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Petersson GA, Nakatsuji H, Li X, Caricato M, Marenich AV, Bloino J, Janesko BG, Gomperts R, Mennucci B, Hratchian HP, Ortiz JV, Izmaylov AF, Sonnenberg JL, Williams DF, Lipparini F, Egidi F, Goings J, Peng B, Petrone A, Henderson T, Ranasinghe D, Zakrzewski VG, Gao J, Rega N, Zheng G, Liang W, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Vreven T, Throssell K, Montgomery JA, Peralta JE, Ogliaro F, Bearpark MJ, Heyd JJ, Brothers EN, Kudin KN, Staroverov VN, Keith TA, Kobayashi R, Normand J, Raghavachari K, Rendell AP, Burant JC, Iyengar SS, Tomasi J, Cossi M, Millam JM, Klene M, Adamo C, Cammi R, Ochterski JW, Martin RL, Morokuma K, Farkas O, Foresman JB, Fox DJ (2016) G16_C01. p. Gaussian 16, Revision C.01, Gaussian, Inc., Wallin 11. Laidler KJ (1983) The development of transition-state theory. J Phys Chem A 87(15):2657– 2664 12. Patidar L, Khichar M, Thynell ST (2020) “A comprehensive mechanism for liquid-phase decomposition of 1,3,5,7-Tetranitro-1,3,5,7-Tetrazoctane (HMX): thermolysis experiments and detailed kinetic modeling. Combust Flame 212:67–78 13. Brown PN, Byrne GD, Hindmarsh AC (1989) VODE, a variable-coefficient ode solver prepared for submission to the SIAM journal on scientific and statistical computing. SIAM J Sci Stat Comput 31:40–91 14. Guirao C, Williams FA (1971) A model of ammonium perchlorate deflagration between 20 and 100 Atm. AIAA J 9(7):1345–1356 15. Kumbhakarna N, Thynell ST (2014) Thermochimica acta development of a reaction mechanism for liquid-phase decomposition of guanidinium 5-Amino tetrazolate. Thermochim Acta 582:25–34
Onset of Thermoacoustic Oscillations in an Annular Combustor with Flames Stabilized by Circular Discs Balasundaram Mohan and Sathesh Mariappan
Abstract This chapter examines the onset of thermoacoustic oscillations in an annular combustor. The combustor consists of 12 burners with flames stabilized in the flow path using circular discs. The premixed fuel and air are used in the present investigation to scrutinize the observed thermoacoustic oscillations. The acoustic pressure fluctuations acquired in this combustor exhibit two dominant peaks. These peaks occur at 664 and 331 Hz, associated with the first azimuthal-first longitudinal (1A-1L) and first longitudinal acoustic modes, respectively. The combustor dynamics are examined by keeping the fuel flow rate constant with varying the air flow rate. The combustor exhibits non-monotonic variation in acoustic pressure amplitude, with intermediate air flow rates showing relatively large amplitude acoustic pressure fluctuations. Further, we observed a difference in amplitude envelope between the increasing and decreasing airflow rates, illustrating the occurrence of subcritical bifurcation. In addition, the phase space, first return map and recurrence plots are used to characterize the transition from relatively low amplitude to large amplitude acoustic pressure fluctuations. Keywords Combustion instability · Annular combustor · V-gutters · Nonlinear time series analysis
1 Introduction Understanding thermoacoustic instability is of crucial importance in industrial and aero-derivative gas turbine engines operating with lean premixed, hydrogen blended hydrocarbon fuelled and pure hydrogen combustion systems [1]. This helps in achieving clean energy by minimizing pollutant emissions with revised combustor design or retrofitting the existing combustors. The presence of combustion instability leads to component level and whole system failure [1]. This instability exhibits B. Mohan (B) · S. Mariappan Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh 208016, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_13
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fluctuations in flow properties such as pressure, velocity and heat release rate inside the combustor. When the acoustic pressure fluctuations at the flame is in- phase with heat release rate fluctuations, acoustic energy is added to the thermoacoustic system, which promotes self- excited thermoacoustic oscillations [1]. Out-of-phase fluctuations lead to a stable operation of the combustor. Primarily frequency of this self-excited oscillation is determined by the characteristic length of the combustor with a speed of sound in the case of duct acoustic modes and convective time delay for intrinsic thermoacoustic modes. For instance, combustor geometries with a characteristic length of longitudinal and transversal nature typically exhibit dominant instability frequency corresponding to that geometry and other modes are cut-off [2]. The annular nature of combustion liners in modern gas turbine engines leads to the manifestation of azimuthal combustion instability [1]. Hence, numerous investigations were carried out to understand, predict and model the azimuthal combustion instability [3–5]. For instance, Karchmer [6] performed an experimental investigation of a fullscale YF-102 turbofan engine com- bustor in a test rig and found unstable circumferential and radial modes. Further, the author observed a similarity in the pressure field between the test rig and engine combustors. Worth and Dawson [7] experimentally investigated the modal dynamics in a model annular combustor using un- steady pressure and phase averaged OH∗ chemiluminescence measurements. Continuous transitions between standing and clockwise and counterclockwise spinning modes occur with the same resonant frequency. In another experimental study [8], the authors showed that peak (axisymmetric flame fluctuations) and negligible (anti-symmetric transverse flame fluctuations) heat release rate fluctuations occur at burners encountering pressure node and anti-node, respectively, in a standing wave. Self-excited azimuthal combustion instability of liquid n-heptane spray flames was studied experimentally by Prieur et al. [9]. The instability exhibits bursts in pressure and heat release rate fluctuations with predominantly standing mode. In addition, they observed at pressure anti-node strong longitudinal oscillation in flame and at pressure node large transversal velocity led to transversal flame oscillation which sometimes leads to partial blow-off. This partial blow-off of flames in the same combustor with different swirler configurations was characterized by Vignat et al. [10] using transversal acoustic velocity and Dynamic Mode Decomposition (DMD) of high-speed images obtained without any optical filters. They have shown that the blow-off (at pressure nodes in a standing wave) was caused by large amplitude pressure oscillations inside the combustor. Further, this leads to non-uniformity in the temperature field, causing deformity in acoustic mode pressure distribution. The experimental investigations of azimuthal combustion instability reported in the past mainly focuses on swirl burners due to their potential for industrial applications. However, few investigations involve flames stabilized either by matrix or bluff-body burners. In order to improve the understanding of self-excited thermoacoustic oscillations in annular combustors, we have developed a laboratory scale annular combustor with 12 burners. The flame stabilization is achieved using circular discs, similar to the longitudinal combustor facility available in our laboratory [11].
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This article examines the nonlinear features of self-excited thermoacoustic oscillations observed for circular disc bluff bodies using tools from nonlinear time series analysis. This article is arranged as follows. Section 2 discusses annular combustor, data acquisition processes and tools from nonlinear time series analysis. Results are discussed in Sect. 3. In Sect. 4, we sum up the key observations made in the present experimental investigation.
2 Experimental Set-Up, Data Acquisition and Processing Methods In our combustor, the flames are stabilized by circular discs and operated with a fully premixed fuel and air mixture. Liquefied Petroleum Gas (LPG) is used as fuel comprising 60% butane and 40% propane by volume. The experimental setup consists of three major parts, namely a plenum, injection tube and combustion chamber, as depicted in Fig. 1a. Burner cut section is shown in Fig. 1b. The injection tube and flame holder specifications are adapted from the single burner Rijke tube experimental facility available in our laboratory [11]. Twelve such burner configurations are azimuthally arranged (axis symmetrically with 30° apart) to mimic the annular combustor. Isometric view of the experimental set-up is shown in Fig. 1a. The combustor components, combustion chamber radii, injection tube length and microphone locations are marked in Fig. 1a. The premixed fuel and air are supplied at the bottom of the plenum. The plenum has provisions for acoustic forcing. Silencers (Festo 2130 U-1/2) are placed in the diverging portion to reduce upstream flow noise. A honeycomb is placed inside the plenum downstream of silencers which acts as a flow straightener. This premixed fuel-air mixture is fed through injection tubes into the combustion chamber. The 12 injection tubes with length (linj ) and inner diameter (d inj ) of 200 mm and 14 mm, respectively, connect the plenum and combustion chamber. Figure 1b shows a cut
Fig. 1 Schematic of a the annular chamber, b cut section of a burner, and c top view of the combustion chamber. a–c Illustrate various parts of the test rig with all the dimensions are in mm. The circular discs flame holders and pressure (MC) sensors are shown in (c), numbered counter clockwise from the horizontal axis
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section of an injection tube with components and dimensions marked. Constrictor is mounted over each injection tube as shown in Figure 1b with an inner diameter of 8 mm. Constrictor increases the lateral extent of recirculation zone, local flow acceleration and strong vortex shedding [11]. The circular discs are placed in injection tubes protruding into the combustion chamber. The length of protrusion (hbj = 20 mm) is measured from combustion chamber base as depicted in Figure 1b. For more details about this experimental setup, the reader is referred to [12, 13] Unsteady pressure data is acquired using PCB ICP condenser microphones and pressure transducer of model numbers 130A24 and 103B02, respectively. The nominal sensitivities of microphones and pressure transducers are 10 mV/Pa and 218 mV/kPa, respectively. The voltage signals are conditioned by PCB 482C15 and connected to a National Instruments data acquisition (NI-DAQ) unit to acquire data. Analog input module NI 9215 with cDAQ 9178 are used for the data acquisition. The data are sampled at 215 samples/s and stored for 15 s. The microphones are mounted in combustion chambers at four azimuthal locations 90° apart from each other and 20 mm from the combustor base, as depicted in panels (a,c). The burners and microphones are numbered from microphone location one (MC1) counterclockwise (CCW) direction from downstream (refer to top view schematic shown in panel c). The burner numbers 1, 4, 7 and 10 correspond to azimuthal locations (θ ) of 0°, 90°, 180° and 270°, respectively. The pressure transducer placed in burner 4 close to the plenum is used for further data processing. Fuel and air volume flow rates are controlled using Eureka rotameters, model SSRS-MGS-13 and CIVF-PG-16(M), respectively. The fixed fuel flow rate 16 slpm is used in the present study leading to a power output of 27 kW. The air volume flow rate (Qa ) is varied from 300 to 650 slpm with an increment of 25 slpm. It is relatively easier to alter Qa by keeping the fuel flow rate (or thermal power) constant to understand the combustor dynamics. Therefore, in the present study, Qa is used for illustration instead of equivalence ratio in the following section. However, it is straight forward to estimate the equivalence ratio from Qa as suggested by the reviewer.
3 Results and Discussion Figure 2a shows acoustic pressure time series obtained during self-excited thermoacoustic oscillation. This operating condition corresponds to Qa of 500 slpm, yielding a maximum root mean square (RMS) value of acoustic pressure fluctuations associated with the increasing Qa . It is seen from the figure that pressure fluctuations exhibit instability bursts, featuring low amplitude aperiodic fluctuations amidst relatively large amplitude periodic oscillations. This feature is typically known as intermittency in nonlinear dynamical systems theory and is also observed experimentally in turbulence and longitudinal combustors [14]. This feature is further examined in this article. The PSD corresponding to the pressure time series is shown in Fig. 2b.
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Fig. 2 For Qa = 500 slpm, a acquired pressure time series and associated b power spectral density (PSD) respectively. The first and second dominant peak in frequency occurs at 664 and 331 Hz respectively. The frequencies correspond to 1A-1L and 1L acoustic modes respectively (also refer [13]) (Colour online)
Two dominant peaks in sound pressure level are observed at 664 and 331 Hz, respectively, associated with a frequency of first azimuthal-first longitudinal ( f 1A1L ) and first quarter-wave ( f 1L ) acoustic modes. For further details on flame dynamics and instability frequencies, the reader is referred to [12, 13], respectively. As mentioned in the previous section, the Qa is used as a bifurcation parameter. ' Root Mean Square (RMS) of pressure amplitude (p rms ) normalized with maximum ' RMS value (max(p rms )) over Qa is shown in Fig. 3a. Filled and open triangles correspond to the paths, where Qa is increased and decreased, respectively. In both ' the paths, p rms achieves a peak value for an intermediate Qa . This trend is attributed to the following. For low values of Qa (say 300 slpm, φ = 1.15), fuel–air mixture is rich. As a consequence, flame possesses a significant wake (flame images not shown here). Low CH∗ intensity values concentrated around the high CH∗ intensity values denoting the presence of wake. The wake extends up to the combustion chamber length in the axial direction. This distributed heat release rate leads to weak acoustic feedback. As a consequence, no ' rigorous flame oscillation is observed (flame images not shown) and hence the p rms is relatively low (panel a). On increasing Qa further, the flame wake becomes short, leading to a more concentrated heat release rate and strong acoustic feedback. A reduction in spatial extent of the wake is attributed to the decreasing equivalence ratio associated with the increasing air flow rate. Further increment in Qa leads to a leaner fuel- air mixture, causing flame blow-off in some burners. On the whole, this reduces the acoustic feedback and therefore causes low amplitude fluctuations
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Fig. 3 a Normalized RMS of acoustic pressure amplitude and b normalized occurrence time period (d ) associated with dominant 1A-1L mode with threshold set at 15% [refer Eq. (1) or [13] for mathematical expression of d ]. Filled and open triangles correspond to the increasing and decreasing Qa (Colour online)
seen in panel (a). Though the non-monotonic trend is observed in panel (a) for both ' increasing and decreasing Qa , the maximum in normalized p rms occurs for different Qa . Hence, there is a hysteresis in the system with the parameter Qa (refer to the red dashed arrows in panel a). This hysteresis shows presence of subcritical bifurcation. Further, the drastic change in amplitude closer to bifurcation is attributed to the onset of bursts in intermittent oscillation. For clarity refer time series shown in Fig. 5 first row. In addition, the non-zero values in panel (a) points out the influence of turbulence induced noise effects on pressure time series. Furthermore, the reduction in amplitude after Qa = 500 slpm shows another bifurcation point, which is not detailed in this article. The d variation is shown in panel (b). The d also exhibits non-monotonic trend with Qa and follows closely with prms .
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Low values of d indicate more contributions from aperiodic fluctuations, 1L mode and other higher harmonics. A similar trend is observed between increasing and decreasing Qa for T d in panel (b). The T d is defined as follows [13]. Td =
N p −1 1 ∑
ttot
f j−1
(1)
j=1
where f j is the frequency falling within a certain percentage of dominant mode instability frequency. The f j is estimated from the time period between successive peak-to-peak amplitude of the acquired pressure time series. The t tot and N p denotes the total time and total number of local peaks in the pressure time series respectively. Figure 4a, b shows contours of PDF of acoustic pressure fluctuations normalized by its corresponding maxi-mum value and PSD, respectively. In panel (a), the PDFs for all Qa exhibit a maximum probability of occurrence close to 0 Pa. These PDFs are similar to the normal distribution. For intermediate values of airflow rates (Qa = 500 525 slpm), there is an increase in the variance due to the frequent occurrence of large amplitude acoustic fluctuations. The ‘normal distribution’ like PDFs observed in this chapter indicates a linearly or marginally stable stochastic thermoacoustic system. This behaviour is displayed in the broadband spectrum of PSD, shown in panel (b). For low air flow rates (say Qa = 300 slpm), no significant peak is observed, while two dominant peaks at about 660 and 330 Hz, corresponding to 1A-1L and 1L modes, respectively, emerge for large air flow rates (Qa > 325 slpm). In the intermediate flow rates (Qa = 500 525 slpm) d > 75% (refer panel b in Fig. 3), indicating dominant coherent oscillations. These flow rates, therefore, correspond to marginally stable systems. The other flow rates are associated with linearly stable systems. In the above paragraphs, we have deduced the stochastic nature of the observed acoustic pressure fluctuations with its stability envelope and dominant instability frequency. In the ensuing discussion, we characterize intermittent oscillation using tools from nonlinear time series analysis. Figure 5 shows the pressure time series (first row), phase space (second row), first return map (third row) and recurrent plot (fourth row) for three different air flow rates. The left, middle and right columns of panels in Fig. 5 correspond to Qa of 300, 425 and 500 slpm, respectively. Note that the phase space and recurrence plots are constructed as discussed in Sect. 2. The acoustic pressure fluctuations in the first row of panels (a1, b1, c1) show a zoomed-in portion of a time series acquired in the experiments. For instance, the pressure fluctuations shown in panel (c1) is a zoomed-in portion in the vicinity of a burst marked by a red dashed line in Fig. 2a. These time series are used for the nonlinear characterization of the respective flow rates to understand the onset of thermoacoustic oscillations. The first row of panels shows the transition from relatively low amplitude to large amplitude acoustic pressure fluctuations as Qa increased. Panel (a1) corresponds to a stable operation, having a relatively less number of 1A1L instability cycles compared to the other operating Qa considered in the present study (refer Fig. 3b). In addition, panel (a1) shows a noisy signal without any instability bursts. Note in the present study that the instability bursts are characterized
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Fig. 4 a Probability density function (PDF) and b power spectral density (PSD) associated with increasing Qa . In a, the legend shows a normalized PDF of individual maximum and in b sound pressure level in dB (Colour online)
by relatively large amplitude acoustic oscillations comprising of 1A-1L mode instability cycles. On the other hand, panels (b1,c1) exhibit intermittent oscillations. This signal comprises low amplitude aperiodic fluctuations amidst relatively large amplitude periodic oscillations. Again, this is clear from the Td indicator shown in Fig. 3b. Further, panel (b1) manifests a relatively less number of instability bursts compared to panel (c1).
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Fig. 5 In the panels, (1) pressure time series (first row), (2) phase space diagram (second row), (3) first return map (third row) and (4) recurrence plot (fourth row) are drawn to characterize the dynamic features. The panels a, b and c corresponds to Qa of 300 slpm (left column), 425 slpm (middle column) and 500 slpm (right column), respectively. The time series, phase space diagram, first return map and recurrence plot shows the dynamic transition from fixed stable point at 0 Pa with background noise to the intermittent oscillations. (Colour online)
The phase space diagrams are shown in panels (a2, b2) and (c2). The panel (a2) shows densely populated trajectories concentrated around the origin (0 Pa), representing a stable fixed point with background noise. Panel (b2) shows again densely populated trajectories concentrated around the origin with scarcely aligned trajectories for the large amplitude. These scarcely aligned trajectories are attributed to the instability bursts. Similarly, panel (c2) shows densely populated trajectories corresponding to low amplitude aperiodic fluctuations and scarcely aligned outer trajectories associated with relatively large amplitude 1A-1L oscillations. In this case, the two attractors are relatively distinguishable compared to panel (b2).
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The first return maps are reconstructed from the local maxima of the pressure fluctuations [15, 16]. The horizontal and vertical axes correspond to the consecutive maxima in a time series. The first return maps having a single point on, a circular loop around and scattered points along the main diagonal denotes a limit cycle, quasi-periodic and intermittent oscillations, respectively. Panel (a3) shows points distributed around the main diagonal illustrating background noise. In panel (b3), the points slightly drift away along the diagonal line (marked by the solid red line) compared to panel (a3). This trend, points drifting along the main diagonal, is clearly visible in panel (c3). This behaviour is a characteristic feature of intermittent oscillations. Note, panel (b3) also features intermittent oscillation. However, the influence of background noise is high and hence not clearly discernible. Further, the recurrence plots are shown in panels (a4, b4, c4). Panel (a4) shows the small-scale square and rectangular structures. As the air flow rate increases, these small-scale square and rectangular grow in size. The observed square and rectangular structures in the recurrence plot illustrate the presence of intermittent oscillation. Thus, the phase space, first return map and recurrence plot show that as the air flow is increased, the combustor dynamics transition from a stable fixed point with background noise to intermittent oscillations. This shows the onset of thermoacoustic oscillations observed in the present experimental investigation.
4 Conclusions In this article, we examined the onset of thermoacoustic oscillations in an annular combustor. The flame is stabilized by 12 circular discs. The air flow rate is varied by keeping the fuel flow rate as constant. For a given air flow rate, the combustor exhibits a first dominant peak corresponding to 1A-1L acoustic mode at about 664 Hz. A second dominant peak is also observed at about 331 Hz corresponding to a first quarter wave longitudinal mode. A non-monotonic variation of acoustic pressure amplitude is observed for increasing air flow rates with maximum amplitude occurring for intermediate air flow rate. Further, an increasing and decreasing air flow rates lead to a different amplitude envelopes with maximum amplitude occurring at different air flow rates. A subcritical bifurcation is observed in this burner configuration. In addition, the acquired acoustic pressure fluctuations exhibits intermittent oscillation comprising of aperiodic fluctuations amidst bursts of periodic oscillations. The phase portraits, first return map and recurrence plots points out the occurrence of transition from aperiodic fluctuations to intermittent oscillation. This transition is associated with the low to intermediate air flow rates. The burner configuration investigated mimics the afterburner and ramjet burner configuration. Hence, the dynamics reported here is of practical relevance. Acknowledgements Financial support: Science and Engineering Research Board (SERB) India with CRG/2019/000500 and YSS/2015/000351. The authors greatly acknowledge the assistance
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provided by Manmoman vishwakarma (IIT Kanpur) and Khurshid Ahmad (Eaton) while performing the experiments.
References 1. Lieuwen T, Yang V (2005) Combustion instabilities in gas turbine engines: operational experience, fundamental mechanisms, and modelling. Prog Astron Aeron 210:547 2. Lieuwen TC (2012) Unsteady combustor physics. Cambridge University Press, Cambridge 3. Bauerheim M, Nicoud F, Poinsot T (2016) Progress in analytical methods to predict and control azimuthal combustion instability modes in annular chambers. Phys Fluids 28(2):315–386 4. O’Connor J, Acharya V (2015) Lieuwen T (2015) Transverse combustion instabilities: acoustic, fluid mechanic, and flame processes. Prog Energy Combust Sci 49:1–39 5. Vignat G, Durox D, Schuller T, Candel S (2020) Combustion dynamics of annular systems. Combust Sci Technol 192(7):1358–1388 6. Karchmer AM (1983) Acoustic modal analysis of a full-scale annular combustor. In: Proceedings of the 8th aeroacoustics conference, p 760 7. Worth NA, Dawson JR (2013) Modal dynamics of self- excited azimuthal instabilities in an annular combustion chamber. Combust Flame 160(11):2476–2489 8. Dawson JR, Worth NA (2014) Flame dynamics and unsteady heat release rate of self-excited azimuthal modes in an annular combustor. Combust Flame 161:2565–2578 9. Prieur K, Durox D, Schuller T, Candel S (2017) Strong azimuthal combustion instabilities in a spray annular chamber with intermittent partial blow off. J Eng Gas Turb Power 128:1–13 10. Vignat G, Durox D, Renaud A, Candel S (2019) High amplitude combustion instabilities in an annular combustor inducing pressure field deformation and flame blow-off. Proc ASME Turbo Expo 148:GT2019-90738 11. Singh G, Mariappan S (2019) Experimental investigation on the route to vortex-acoustic lock-in phenomenon in bluff body stabilized combustors. Combust Sci Technol 128:1–29 12. Mohan B, Mariappan S (2018) Azimuthal combustion instability: characterization of laboratory scale annular test rig. Proc Asian Cong Gas Turb TS64:1–8 13. Mohan B, Mariappan S (2022) Nonlinear characterization of azimuthal combustion instability exhibitng flame transient phenomena. In: Proceedings of the 51st international congress and exposition on noise control engineering 14. Nair V, Thampi G, Sujith RI (2014) Intermittency route to thermoacoustic instability in turbulent combustors. J Fluid Mech 756:470–487 15. Kabiraj L, Sujith RI (2012) Nonlinear self-excited thermoacoustic oscillations: intermittency and flame blowout. J Fluid Mech 713:376–397 16. Kabiraj L (2012) Intermittency and route to chaos in thermoacoustic oscillations. PhD thesis, IIT Madras
Development of Advanced Fuel Injector Concepts for Compact Lean-Burn Gas-Turbine Combustors Ayush Divyansh, Preetam Jamod, and K. P. Shanmugadas
Abstract The present work investigates the aerodynamics and atomization characteristics of a lean burn fuel–air mixer assembly using laser flow diagnostic techniques. Two experimental configurations are designed with different pilot fuel injection mechanisms and different swirl configurations. The injector assembly is realized using additive manufacturing methods. Direct laser sheet imaging and particle image velocimetry experiments are conducted to capture the air velocity field and atomization characteristics. The efficacy of the coaxial annular fuel injection mechanism on the atomization characteristics and high swirl number configurations on the external aerodynamics are presented qualitatively. Keywords Lean burn fuel injector · Particle image velocimetry · Central toroidal recirculation zone · Atomization · Spray shear layer
1 Introduction One of the current major requirements in the gas turbine industry is the development of compact low-power micro gas turbine engines for drones and localized power production [1–3]. This requires advanced low-emission combustor concepts which are compact, durable, and fuel-flexible. Aero-engine combustor designs are primarily based on either rich burn (RQL SAC, DAC) [4–6, 12] or lean burn (TAPS, LPP, LDI) [7–11] combustion technologies. Out of the existing systems, lean burn systems are of primary interest because of their low emission level at a wide range of operating conditions and low fuel burn-rate levels. In a typical lean burn combustion system, 70–80% of the combustor air is admitted through the fuel–air mixer assembly, and the remaining air is used for dilution and liner cooling. The fuel–air mixer assembly A. Divyansh Department of Materials Engineering, IIT Jammu, Jammu 181121, India P. Jamod · K. P. Shanmugadas (B) Department of Mechanical Engineering, IIT Jammu, Jammu 181121, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_14
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involves a complicated swirler assembly that distributes the incoming air to a set of coaxial annular swirlers to obtain the desired primary zone aerodynamics and flame stabilization characteristics. The fuel will be admitted using dual-orifice fuel nozzles or through multiple annular passages around the swirlers. Typical geometries include pilot and main mixer assemblies which are coaxially arranged with appropriate fuel injection passages [17]. At IIT Jammu, an initiative is launched to develop compact combustor concepts for a 1 KN drone engine. Development of the fuel injector is taken up as the first step, adopting a lean-burn fuel injector technology similar to the existing aero engine injector concepts. The injector is realized using an additive manufacturing technique considering the geometrical complexities and required accuracy. The design of the injector focuses on two important aspects: (a) design of the swirler assembly to achieve the desired injector aerodynamics that will help for proper flame stabilization and atomization. (b) design of the fuel injection passages that provide proper atomization and mixing at all power levels with appropriate fuel staging. The present work focuses on the development of a compact swirl injector assembly for a micro gas turbine combustor. Different versions of the injector hardware are developed using the existing injector design guidelines, and investigations are conducted to capture the injector’s external aerodynamics and flow characteristics using laser flow diagnostic techniques.
2 Design of the Injector The injector design methodology focuses on lean burn injector concepts with the pilot and main combustor zones [13, 14]. Two injector configurations are designed as shown in Fig. 1. They are named G1 and G2. The G1 injector pilot assembly consists of one co and counter-rotating axial swirlers separated by a curved prefilming passage. A simplex nozzle of flow number 1.1 is used as the pilot injector to generate a 90 ◦ hollow cone spray which supplies the initial droplets to the pilot mixer. These droplets get atomized by the primary air and part of it undergoes wall filming on the prefilmer. A counter-rotating shear layer forms the prefilmer tip which atomizes the accumulated liquid rim and provides a uniform spray at the exit. The main mixer consists of an axial and radial swirler assembly. A liquid jet ensues from the inner wall of the radial swirler, and it mixes with the axial and radial airflow. This forms a jet in swirl cross-flow atomization mechanisms inside the main swirler. Major design parameters are listed below (Table 1). The geometrical swirl number is calculated using the approach given in [15, 16]. Axial swirl number Sna is estimated as 2 × (1 −
3
) 2 × tanθ 3 × (1 − DDht )
Dh Dt
(1)
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1- Primary Axial swirler, 2-Secondary Axial swirler ,3Tertiary Axial swirler, 4-Primary Radial swirler, 5-Primary venturi, 6-Secondary venturi, 7- Exit Flare, 8- Nozzle, 9- Fuel outlet hole for main, 10- Radial fuel inlet for main, (2,6,5,1,8) Pilot, (10,4,7,3,6)- Main. Fig. 1 Cut section view of injector G1
Table 1 Flow rates and design parameters of G1 and G2
G1
G2
m˙ tot
0.1258 kg/s
0.1167 kg/s
m˙ m
0.0854 kg/s
0.08 kg/s
m˙ p
0.0404 kg/s
0.0367 kg/s
m˙ f p
0.001 kg/s
0.001 kg/s
m˙ f m
0.01 kg/s
0.01 kg/s
Flow split (main: pilot)
60:40
70:30
Snp primary
0.85
0.98
Snp secondary
0.95
0.47
Snp tertiary
–
1.01
Snm primary
1.05
0.484
Snm secondary
1.026
1.01
Radial swirl number Snr is estimated as 1 R × tanαm × × [1 − 1−ϕ 2×L
Rp R
2 ]
(2)
In the G2 injector, the pilot nozzle is replaced with an annular fuel passage and axial swirler assembly. Liquid fuel ensuing from the annular passage is atomized by the co-flowing counter-rotating air swirl flows. The pilot is a fuel-rich region where
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1-Primary Axial swirler, 2-Secondary Axial swirler ,3Primary radial swirler, 4-Tertiary Axial swirler,5-secondary radial swirler 6-Primary venturi, 7-Secondary venturi, 8- Exit Flare, 9-axial pilot fuel inlet, 10- Radial Fuel inlet for main, 11exit fuel pilot, 12- exit holes outlet main, (9,1,11,2,6,3,7)- Pilot, (8,5,4,7,10)- Main. Fig. 2 Cut section view of injector G2
the air is squeezed by shear action such that pressure drop happens and the normal air at exit flows towards it, forming a recirculation zone which main also does but the excess of air mixes through main, causing the air and fuel mixture dilution and thus burns in a lean way. The third swirler is made radial for the pilot region in G2 compared to the G1 pilot where swirlers are limited to two. The pressure drop across the pilot and main swirlers is estimated as 2–4% (Fig. 2). The injector is realized using additive manufacturing methods. Different 3D printing techniques are implemented and MJP is identified as the better option to print the geometry. An M2R-WT material is used to print the final hardware, and geometrical inspection is done to verify the passage and swirler dimension accuracies.
3 Experimental Arrangements Experiments are conducted to capture the gas phase velocity field and spray structures. An atmospheric spray test facility is developed to conduct the experiments. A schematic of the spray test rig is shown in Fig. 3. The test rig contains an air supply line and a liquid supply line. A 10-bar compressed air facility delivers the air at a flow rate of 0.5 kg/s, and the flow rate is controlled using gate valves and a pressure regulator. The flow rate is measured using a thermal mass flow meter and the line pressure and injector air pressure are measured using dedicated strain-gauge-type pressure sensors. The liquid line is fed by a pressure vessel which is pressurized
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Fig. 3 Schematic of the experimental set-up
using nitrogen cylinders. The fuel passages and simplex nozzle are flow-checked and the injection pressure is measured using strain gauge-type pressure sensors. A nozzle holder is developed to fix the injector assembly with inlet connections for fuel and airlines. For PIV experiments, an oil seeder is used as a bypass line to seed the airflow. A collection tank is kept downstream of the injector assembly to collect the spray. The tank contains honeycomb and wire mesh arrangements to prevent the spray splashback. The experimental conditions are listed in Table 2. The airflow rate is varied from 0.0980 to 0.1019 kg/s with 0.1248 kg/s flow rate is selected as the baseline. Experiments are conducted with configurations namely pilot, main and pilot + main. The flow split through each of the configurations is selected based on the area variation, and individual sections are blocked as per the experimental configurations for each experiment.
4 Laser Dianostic Arrangements The air velocity field is captured using low-speed PIV measurements. An Nd:YAG laser, along with a sheet optics set is used to generate a 1 mm thick light sheet covering a measurement region of 100 mm × 100 mm. A CCD camera (PCOPixelfly make) along with a 100 mm Nikkon lens is used to capture the images. The imaging resolution is 1392 × 1040 pixel (Fig. 4).
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Table 2 Varying test flow rates and Reynolds number for G1 and G2
0.098 0.111 0.125 0.140
0.066 0.076 0.085 0.095
0.031 0.036 0.040 0.045
106111 121210 136249 151588
72025 82273 92481 102893
111124 126936 142684 158748
0.153
0.104
0.049
166627
113100
174497
0.101
0.069
0.032
132868
91066
86818
0.116
0.080
0.036
152056
104218
99356
0.131
0.090
0.041
171244
117369
111894
0.145
0.100
0.045
190070
130272
124195
0.160 0.110 0.050 G1 G2
209258
143423
136733
Fig. 4 Experimental arrangements
The laser and camera are synchronized using a sequencer unit and double images are captured at a frequency of 10 Hz. The delta T between the images varied from 0.02 ms to 0.04 ms based on the test conditions. A schematic of the PIV experimental arrangements is shown in Fig. 5. The spray structure is visualized using the direct laser sheet imaging techniques using the same equipment arrangements mentioned above. Images are processed using an open source PIV post-processing software PIV from Matlab. The error in PIV measurements is around ~ 5% and the cross correlation coefficient is in the range of 0.4–0.6.
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Fig. 5 Optical arrangements for PIV experiments
5 Results andDiscussion 5.1 Flow Field at the Exit of Pilot Mixer Experiments are conducted to capture the airflow characteristics of the pilot mixer configuration alone by blocking the main mixer assembly. The fuel passages are also blocked to prevent any airflow back to the tube passages. The air velocity field at the exit of the pilot mixer configurations for mixer G1 and G2 are investigated by PIV experiments. A sample time-averaged velocity field for G1 configuration is shown in Fig. 6. The primary air flow undergoes vortex breakdown forming recirculating flow pattern at the exit. Maximum velocity is observed in the shear layer region and it varies from 17 to 41 m/s as the flow rate is increased from 0.0137 to 0.0387 kg/s. However, strong central toroidal recirculating patterns (CTRZ) is not observed in both G1 and G2 injector cases. This could be due to the low swirl number of the axial-radial swirler assembly of the pilot mixer. The pilot fuel nozzle (simplex nozzle) occupied the upper portion of the pilot assembly and the obstructions to the incoming flow due to the tubing arrangements resulted in an asymmetry of the downstream swirl flow field.
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Fig. 6 Average velocity field at the exit of the pilot mixer of G1 injector (m˙ p = 0.0387 kg/s)
5.2 Flow Field at the Exit of Main Mixer The air velocity field at the exit of the main mixer assembly is captured using similar experimental arrangements as in the previous case. The pilot mixer assembly passages are blocked in this case. Figure 7 shows the time-averaged velocity field at the mixer exit. In the case of G1 injector, the formation of a strong recirculation zone is observed. As the flow rate increases, the CTRZ becomes predominant and expands radially. Whereas, in the case of G2, strong recirculation patterns are not formed as seen in Fig. 7a, b, respectively. The low swirl number of the secondary swirler of the main mixer assembly in the case of G2 resulted in weaker swirl strength. The length of the recirculation zone from the exit plane is measured as 29 mm for G1. G2 injector formed an open recirculating zone without a downstream stagnation point. As the airflow rate increases, the size of the recirculation zone is also increased.
5.3 Flow Field at the Exit of Injector Finally, the flow field at the injector exit with combined configuration (main + pilot) is captured. The time-averaged velocity field is shown in Fig. 8. The G1 injector forms a compact central recirculation zone at the exit which is originated from the pilot injector. The recirculation zone is slightly elongated due to the shearing action of the main mixer flow. An annular recirculating pattern is formed by the vortex breakdown of the main mixer and it can be seen in Fig. 8a. A thick shear layer is formed at the exit which will help for the atomization of the liquid fuel droplets (Fig. 9).
Development of Advanced Fuel Injector Concepts for Compact … Fig. 7 a Time averaged velocity field at the exit of main mixer of a G1 injector (m˙ m = 0.1189 kg/s), b G2 injector (m˙ m = 0.1197 kg/s)
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(a)
Recirculating pattern
(b)
In G2 injectors case, the pilot and main recirculation zones are not strong as compared in the G1. Smaller recirculation zones are formed in these cases as marked in Fig. 8b.
5.4 Spray Characteristics of the Injector The spray structure at the exit of G1 and G2 injectors is characterized using direct laser sheet imaging. In the case of pilot mixer, both G1 and G2 provided similar
166 Fig. 8 a Time-averaged velocity field at the exit of main mixer of a G1 injector (m˙ t = 0.1189 kg/s), b G2 injector (m˙ t = 0.1189 kg/s)
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(a)
Recirculating pattern
(b)
Weak recirculating pattern
spray structures. A sample image showing the pilot mixer spray structure is shown in Fig. 10. Droplets are finely atomized and ligaments are shedding from the venturi tip of the injector. It is observed that the coaxial fuel passage assembly (Fig. 2) in the case of G2 and pilot nozzle (simplex nozzle) in the case of G1 injector are performing similarly. This shows the efficacy of the prefilming atomization at the venturi since most of the droplets from the pilot nozzle and coaxial fuel passage impinge on the venturi and undergo successive atomization.
Development of Advanced Fuel Injector Concepts for Compact … Fig. 9 For m˙ t = 0.1189 kg/s plot of a V component, b vorticity for G1 along the radial axis
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(a)
(b)
The droplet formation inside the main mixer assembly is by the jet in swirl crossflow atomization. The liquid jets ensuing from the 1 mm passage holes interact with the swirl streams formed by the annular radial swirler and axial swirler. Droplets get atomized and ensue from the annular passages as shown in Fig. 11a. Finer droplets are formed in the shear layer region, and bigger droplets are ejected radially. Most of the droplets align with the outer shear layer region (as seen in Fig. 11b), with smaller droplets recirculating to the centre region.
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(a)
(b)
Spray shear layer
Fig. 10 Spray characteristics of the pilot mixer of G1 injector a Instantaneous image, b timeaveraged image
(a)
(b)
Spray shear layer
Fig. 11 Spray characteristics of the main mixer of G2 injector a Instantaneous image, b timeaveraged image
6 Conclusions Two novel fuel injection concepts are developed for compact lean burn gas turbine engines, following the existing lean injector design guidelines. The air velocity field and atomization characteristics of both injectors are investigated using PIV and direct laser sheet imaging techniques. The G1 injector features a pilot pre-filming concept using a simplex nozzle and the pilot fuel injection is done in G2 injector using coaxial annular fuel passages. G1 injectors feature radial swirler assemblies with high swirl numbers, whereas the G2 features weak swirl numbers with an elongated recirculation zone. Velocity field data shows that G1 injector provides good recirculation zone characteristics as compared to the G2 injector. Further, coaxial passage fuel injection is found to be effective since the atomization was similar in both pilot mixtures. So, it is decided to conduct further investigations with a modified G1 injection in which the pilot nozzle will be replaced by an annular fuel injection.
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Acknowledgements The experimental work is conducted at the National Centre for Combustion Research and Development (NCCRD), IIT Madras. The technical support and facility provided by NCCRD is greatly acknowledged. We also thank Mr. Saket Singh and Mr. Rachit Bundiwal for their support in experiments and injector design activities.
Nomenclature Dh Dt L m˙ fp m˙ fm m˙ m m˙ p m˙ tot R Rem Rep Ret Rp Sna Snm Snp Snr θ ϕ αm
Hub diameter (m) Outer diameter (m) Blade length (m) Fuel mass flow rate through pilot (kg/s) Fuel mass flow rate through main (kg/s) Air mass flow rate through main (kg/s) Air mass flow rate through pilot (kg/s) Air mass flow rate through main and pilot (kg/s) Radius of injection tube (m) Reynolds number main Reynolds number pilot Reynolds number main + pilot Radius of central rod or injector Axial swirl number Main swirl number Pilot swirl number Radial swirl number Vane angle (°) Blockage factor Average angle of flow at blade exit (°)
References 1. Nascimento MAR, Rodrigues LO, Santos EC, Gomes EEB, Dias FLG, Velásques EIG, Carrillo RAM (2013) Micro gas turbine engine: a review. Prog Gas Turb Perform 14:107–141 2. Pilavachi PA (2002) Mini- and micro-gas turbines for combined heat and power. Appl Thermal Eng 22(18):2003–2014 3. Sychenkov VA, Limanskii AS, Yousef WM (2019) Micro gas turbine engine for unmanned aerial vehicles. Russ Aeronaut 62:651–660 4. Mongia HC (2011) Engineering aspects of complex gas turbine combustion mixers part II: HighT3. In: AIAA Aerospace Science Meeting, p 107 5. P&W and GE, Critical propulsion components, NASA/CR—2005-213584 6. McKinney R, Cheung A, Sowa W, Sepulveda D (2007) The Pratt and Whitney TALON X low emissions combustor: revolutionary results with evolutionary technology. In: Proceedings of the 45th AIAA aerospace sciences meeting and exhibit, p 386
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7. Foust MJ, Thomsen D, Stickles R, Cooper C, Dodds W (2012) Development of the GE aviation low emissions TAPS combustor for next generation aircraft engines. In: Proceedings of the 50th AIAA aerospace science meeting, Nashville 8. Mongia HC (2003) TAPS: a fourth generation propulsion combustor technology for low emissions. In: AIAA international air and space symposium and exposition: the next 100 years, p 2657 9. Eggels R (2018) Gas turbine combustion modelling: lean combustion and fuels, Rolls Royce, combustion meeting presentation 10. Leonard G, Stegmaier J (1994) Development of an aeroderivative gas turbine dry low emissions combustion system. J Eng Gas Turbine Power 116(3):542 11. Liu Z, Sun X, Sethi S, Li N, Yi-Guang D, Wang L (2017) Review of modern low emissions combustion technologies for aero gas turbine engines. Prog Aerospace Sci 2017:12–45 12. Shanmugadas KP, Chakravarthy SR, Chiranthan RN, Jayanth S, Sundar K (2018) Characterization of wall filming and atomization inside a gas turbine fuel injector. Exp Fluids 59:151 13. Timothy AF, Steven W, Ulrich W, Jaap VK (2022) Multi-swirler mixers with plasma generating nozzle US patent 6455660 14. Crocker DS, Nickolaus DA, Smith CE (2001) Piloted airblast lean direct fuel injector: U.S. Patent 6272840 15. Bhupendra K, Dong L, Vishal S (2014) Design and study on performance of axial swirler for annular combustor by changing different design parameters. J Energy Instit 2014:4–6 16. Sebastien C, Daniel D, Thierry S, Jean-Franc OB, Jonas PM (2014) Dynamics of Swirling flames, pp 7–9 17. Alfred AM, Mongia HC (2012) Mixture assembly for gas turbine engine combustor, Patent No: US 8171735 B2, p 3
Experimental Study on GDI In-Cylinder Combustion Quality of Ethanol and Lemon Peel Oil B. Abinash, B. Yogesh, G. M. Nayak, V. W. Ketan, P. S. Kolhe, and B. Saravanan
Abstract A Gasoline Direct Injection (GDI) engine’s performance and emissions are greatly impacted by the fuel injection phenomena. It is critical to understand biofuel’s spray structure and combustion quality before using it in a GDI engine. This study investigates the combustion quality at different Start of Injection (SOI) timings using an optical GDI engine. Lemon Peel Oil (LPO) and ethanol are prepared on a volume basis, and three different SOI timing are selected to study the spray effect on combustion. Two binary blends and one ternary blend are prepared in different volume bases for this purpose: G60L40 (60% gasoline and 40% LPO), G80E20 (80% gasoline and 20% ethanol), and G40E20L40 (40% gasoline, 20% ethanol, and 40% LPO), and the effect of the thermophysical properties of the fuel blends on combustion characteristics are evaluated. In-cylinder combustion imaging revealed that injection timing significantly impacts combustion. The addition of ethanol improves the combustion, but LPO blends showed higher residence of diffusion burning due to poor evaporation characteristics. Keywords Lemon peel oil · Ethanol · GDI optical engine · Flame luminosity
1 Introduction In recent years, there has been an increase in interest in renewable biofuels due to depleting fossil fuel resources, rising costs, and the need for energy security. As a result, numerous renewable and biodegradable alternative fuels have been considered energy sources worldwide [1, 19, 21]. Because of their superior evaporation characteristics, alcohol-based biofuels are becoming increasingly popular. Demonstrating
B. Abinash College of Engineering, Design and Physical Sciences, Brunel University, London, UK B. Yogesh · G. M. Nayak · V. W. Ketan · P. S. Kolhe · B. Saravanan (B) Department of Mechanical and Aerospace Engineering, IIT Hyderabad, Kandi, Sangareddy, Telangana 502284, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_15
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the utility of renewable resource-based alternative fuels as a replacement for conventional fuel has become an essential part of alternative fuel research to promote mass production and, thus, cost-effective alternative fuel. In this context, liquid biofuels such as methanol, ethanol, and biodiesel are the most valuable and versatile options [5, 6, 9, 14, 18, 20]. Kim et al. [12] investigated spray and combustion visualization of gasoline and diesel at high pressure and temperature. They simulated the three combustion regimes for CI engines: homogeneous charge compression ignition, premixed charge compression ignition, and standard combustion using different gas densities, gas temperature, and oxygen concentration. Under all experimental conditions, they observed that gasoline spray had a shorter liquid penetration length than diesel fuel. Ignition delay and lift-off length were found to be higher for gasoline. Their optical engine studies revealed that gasoline spray had significantly shorter liquid penetration than diesel, owing to gasoline’s superior evaporation characteristics [13]. Because of its higher evaporation properties, gasoline avoided wall wetting and fuel impingement on the piston bowl. Kale and Banerjee [10] investigated the effect of engine-like conditions on alcohol spray behavior in a constant volume spray chamber using a six-hole GDI injector. Their findings revealed that thermo-physical properties such as latent heat of vaporization, viscosity, surface tension, and density significantly impact spray behavior in engine-relevant chamber conditions [15]. The literature review of alternative fuels for SI engines focuses primarily on alcohol-based fuels, which suffer from lower calorific value and phase separation issues. Few alternative fuels for gasoline engines have been researched to date, and lemon peel oil is a promising fuel since its properties, such as LHV and A/ F ratio, are similar to baseline gasoline, as opposed to some other alternatives, such as ethanol and methanol. The molecular structure of LPO constituents is dominated by cyclo-paraffins, which have an excellent anti-knock tendency. LPO is derived from waste lemon rinds, and India is the world’s largest producer of lemon and lime, producing over 2600 thousand tons in 2016. These considerations point to LPO being a promising alternative fuel for gasoline engines yet to be thoroughly studied. Velavan et al. [17] investigated the effect of combustion, performance, and emission characteristics of LPO blend in a twin-cylinder PFI engine using flame visualization photography and combustion diagnostic tools. Compared to gasoline, they found that a 10% LPO blend produced a low diffusion flame and prompted more premixed combustion. Biswal et al. [3, 4] ran two separate experiments in a single-cylinder PFI engine with binary and ternary blends of gasoline, ethanol, and LPO. They reported that up to 40% LPO could be blended with gasoline without any hardware modifications to the PFI engine. A theoretical chemical kinetic analysis of one-dimensional laminal flames reveals that LPO surrogates have a higher flame speed than gasoline surrogates due to the higher reactivity of cyclic hydrocarbons in LPO versus branched alkanes in gasoline. Adding ethanol in the gasoline-LPO blend increases the fuel-bound oxygen, improving the combustion chemistry, leading to higher combustion pressure and heat release rate than dual blends and baseline gasoline. In addition, for ternary blends, an EGR of 10% is recommended.
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The spray phenomenon has a significant impact on the performance and emissions of GDI engines. Previous studies demonstrated how fuel thermo-physical properties and chamber conditions significantly impact the atomization process. The combustion quality of spray atomization must be addressed to understand engine performance and emissions fully. This study analyzes the flame luminosity of combustion reactions for various gasoline, LPO, and ethanol blends to better understand the combustion process in an optical GDI engine with the same injector and injection pressure. Soot natural flame luminosity is analyzed using high-speed imaging inside an optically accessible GDI engine under various operating conditions.
2 Methodology The optical engine study is carried out to comprehend the combustion quality of the proposed oxygenated ternary blends and baseline fuel gasoline at various injection timings. Two binary blends and one ternary blend are prepared in different volume bases for this purpose: G60L40 (60% gasoline and 40% LPO), G80E20 (80% gasoline and 20% ethanol), and G40E20L40 (40% gasoline, 20% ethanol, and 40% LPO), and the effect of the thermophysical properties of the fuel blends on combustion characteristics are evaluated. There was no evidence of phase separation in the samples. All blends are thoroughly stirred before the experiments to maintain fuel homogeneity. Table 1 shows the thermo-physical properties of all the fuels tested in this study. An optical GDI engine with twin cylinders is used for in-cylinder combustion visualization. Each cylinder has a displacement capacity of nearly 694 cm3 . Both cylinders have a bore and stroke length of 94 mm and 100 mm, respectively. The optical access cylinder is one of the two, while the other is a regular thermodynamic cylinder with a metallic piston, which is commonly used for gross performance and emission tests. The cylinder head has two overhead camshafts, four valves per Table 1 Various thermophysical properties of gasoline, ethanol and LPO [2, 3] Property
Gasoline
Isooctane
Ethanol
LPO
Molecular formula
C5 –C12
C8 H18
C2 H5 OH
C10 H16 O0.045
Research octane number
87
100
109
80
Lower calorific value (MJ/kg)
44
44.3
27
45
Kinematic viscosity (mm2 /s) @ 313 K
0.6
0.6
1.08
1.06
Surface tension (mN/m) @ 293 K
–
14.7
22.4
22.1
Density
(kg/m3 )
750
695.8
790
843
Latent heat of vaporization (kJ/kg)
@ 298 K
380–400
305.4
938
290
Final boiling point (K)
498
372
351
449
Stoichiometric air/fuel ratio
14.7
12.5
9
14.1
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Fig. 1 Photograph of the GDI optical engine is shown on the left, where the rectangular dotted marks show the optical access. Middle: schematic diagram of the experimental setup. Right: cylinder head showing the inlet and exhaust valves
cylinder, a fuel injector, and a spark plug. The injector is vertically situated in the cylinder’s center, while the spark plug is off-center and close to the cylinder wall. The typical optical GDI engine used for the study is depicted in Fig. 1. A 1° resolution crank angle encoder is used to get the information about the crank position. The optical engine is driven at a constant speed of 1000 rpm. The compression ratio is set to 11. A fixed medium load throttle position with a stoichiometric air-fuel ratio is maintained for all test conditions. A Bosch six-hole GDI injector with a 90 bar injection pressure is used for all blended fuels. The equivalence ratio is tracked using a Bosch Lambda sensor. Three injection timings are chosen: 30°, 60°, and 90° crank angle after suction top dead center (ASTDC), and the spark timing is kept constant at 30° before top dead center. Different injection durations are used for different blends to maintain the equivalence ratio. Axial and orthogonal optical access is provided via an expanded hollow piston window and a cylinder liner window. For in-cylinder combustion visualization, a sapphire window with a diameter of 40 mm is installed in the hollow piston. A reflective mirror is mounted at 45° to the plane of the piston surface to visualize the in-cylinder process. A colored high-speed CMOS camera (Photron SA2.1) is employed during the combustion process to capture the natural flame luminance. The high-speed camera is equipped with a Nikon 50 mm lens. The camera is set to collect two images at each crank angle at a frame rate of 12,000 frames per second (0.5° CA resolution). The experimental arrangement is depicted schematically in Fig. 1. Due to intense combustion light, longer exposure times and larger camera lens apertures can result in CMOS camera sensor saturation. For low intensity, a larger lens aperture is required. Hence, a camera setting was determined that is ideal for all of the test conditions. A fixed camera aperture of F-stop 8 and an exposure time of 0.16 µs are used with a spatial resolution of 0.17 mm/pixel. The average spatially integrated natural flame luminosity (SINL) of each tested fuel is plotted and compared to baseline gasoline at injection time of 30°, 60°, and 90° ASTDC.
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The images of the combustion event acquired by the color CMOS camera are post-processed using Matlab code created in-house. The RGB image is converted to grayscale to calculate the intensity of the combustion light from each pixel location. After converting each image to grayscale, the spatially integrated natural flame luminosity (SINL) is calculated using the following equation. SINL =
Ix,y
(1)
where, x and y are the pixel coordinates in the grayscale image, and I represent the intensity of that pixel. In addition, the luminosity of the flame is measured based on a chemiluminescence signal using a photomultiplier tube (PMT, Thorlabs, PMM01) with a bandpass filter. In the present study, the CH* signal is acquired through an optical filter at wavelengths of 430 ± 2 nm. The CH* signal is acquired at a similar sampling rate of a highspeed camera for 1 s, converted to a voltage signal using the LabVIEW 12 virtual instrument. Matlab is used to compare the amplitude of the real-time signal under different conditions about the crank angle.
3 Results and Discussion This study chose various binary and ternary blends of gasoline, ethanol and LPO for combustion analysis. It should be noted that due to engine limitations, using 100% biofuel for the combustion study inside the GDI engine is challenging. In our previous study [3], we investigated different binary and ternary blends of gasoline, ethanol, and LPO in a PFI engine, and the results were promising. In this study, a similar blends are prepared for in-cylinder combustion analysis of the GDI engine. Fuel impingement is well known to cause wall wetting and is a substantial source of soot and unburned hydrocarbon emissions. The impact of potential wall impingement conditions must be thoroughly investigated. This study experiments with an optical GDI engine to visualize in cylinder combustion at various injection timings for binary and ternary blends of gasoline–ethanol–LPO. This section analyzes soot natural flame luminosity using high-speed imaging with optical access to the combustion chamber. Estimating the actual distribution of soot within the cylinder is a difficult task. Hence, the current study focuses on interpreting direct images of soot luminosity by analyzing how combustion from fuel impingement is predicted to vary as it passes through the combustion chamber. In addition, a real-time CH* chemiluminescence signal is acquired using a PMT at various injection timing. Chemiluminescence is the emission of photons by electronically stimulated molecules when they return to a lower energy state, and it can be used to assess the quality of combustion [8]. It is an essential parameter in the flame study, occurring in UV, VIS, and IR regions. The spectrum and magnitude of these regions depend on the flame’s characteristics, such as flame temperature and thermal radiation of the species. An excess of fuel
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or a shortage of combustion air induces a luminous/yellow flame due to black body soot radiation, which occurs in the visible spectrum, whereas blue color is emitted by excited radicals limited in the visible spectrum. For cameras’ spectral response curves, a direct correlation between visible combustion luminance and internal airfuel mixing can determine in-cylinder combustion quality.
3.1 Effect of Injection Timing on In-Cylinder Combustion Characteristics The spray characterization described in the previous section provides insight into the spray atomization of isooctane and LPO under various chamber conditions. However, turbulence caused by piston movement and flow past intake and exhaust valves during suction and compression strokes cause in-cylinder conditions to differ significantly from constant volume chambers. Therefore, optically accessible in-cylinder combustion is captured as a function of crank angle. Figure 2 depicts the crank resolved natural flame luminosity of gasoline at three different starts of injection timings: 30°, 60°, and 90° ASTDC. The instantaneous direct images are shown at crank angles of 20°, 40°, 60° and 80° following compression TDC. It clearly illustrates the impact of injection time on combustion mode. The flame luminosity is higher at 30° ASTDC and lowers with delayed injection time.
Fig. 2 Crank-resolved natural flame luminosity of gasoline at three different start of injection timings (CA is after compression TDC)
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It should be noted that gasoline impingement on the piston head is more frequent with early injection, and a thicker fuel film builds on the piston top [7, 16]. At 30° ASTDC injection, the distance between the injector tip and piston head is approximately 9 mm, which is expected to be small compared to the fuel penetration length and increases the possibility of fuel impingement. However, the distance between the fuel injector tip and the piston head is expected to increase with delayed injection timing to minimize the wetness of the piston head. The impinged fuel layer burns in a diffusion mode, resulting in higher hydrocarbon and soot emissions [11]. Note that the premixed combustion mode is feasible in PFI due to the sufficient fuel-air mixture. However, this notion is more difficult to pursue in the case of GDI engines. There is less residence time for fuel droplets to evaporate and mix with air, whereas fuel injection occurs instantly inside the cylinder. Therefore, a higher soot emission can be expected from GDI engine combustion. The spatially integrated natural flame luminosity (SINL) is examined at various crank angles to determine the wall wetting impact in sooty pool combustion. Figure 3 displays the average SINL of five successive combustion cycles at three different fuel injection timings. The SINL peak is evident at the injection of 30° ASTDC due to the enhanced sooty combustion. A significant drop in peak SINL value for gasoline combustion is seen with delayed injection time. At the start of injection 90° ASTDC, the gap between the injector tip and the piston top is approximately 56 mm, reducing wall impingement and enhancing air-fuel mixing, lowering the diffusion mode of burning. The peak SINL value for gasoline combustion is reduced by 91% when the injection is delayed from 30° to 90° ASTDC. Furthermore, a real-time CH* chemiluminescence signal is plotted (Fig. 4) to a crank angle using a photo-multiplier tube to support the in-cylinder imaging result. The amplitude of the CH* signal is highest at the start of injection 30° ASTDC, and it falls significantly with delayed injection start. It should be noted that the luminance inside the combustion chamber and the amplitude of the chemiluminescence signal Fig. 3 Spatially integrated natural flame luminosity (SINL) of gasoline at different start of injection timings
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Fig. 4 Real time CH* chemiluminescence signal for gasoline at different start of injection timing
follow a similar trend with late injection. In addition, increasing the injection time from 30° to 90° ASTDC reduces the amplitude of CH* for gasoline combustion by 92%, similar to SINL analysis. Again, all blended fuels behave similarly when the injection time is delayed.
3.2 In-Cylinder Combustion Characteristics of Gasoline–Ethanol–LPO Blends Figures 5, 6 and 7 show the crank resolved natural flame luminance of all the tested fuels at start of injection time of 30°, 60° and 90° ASTDC, respectively. As previously discussed, a shorter distance between the injector and the piston head causes the combustion to be dominated by the wall wetting effect for all fuel blends, increasing brightness in diffusion burning. Figure 5 shows that the burning of localized heterogeneous mixtures causes all tested fuels to emit more soot and illuminate more vividly. As the start of combustion is delayed, the yellow flame decreases significantly for all fuel blends. The crank resolved instantaneous image shows that the G80E20 blend has the lowest luminance of all tested fuels. The addition of LPO to gasoline enhances the propensity of wall wetting and results in diffusion burning. However, as compared to binary gasoline-LPO blends, the combustion quality of the ternary blend improves with less sooty combustion. At SOI 90° ASTDC in Fig. 7, the light intensity is hardly visible in gasoline and G80E20 blends, while diffusion burning remained noticeable for LPO blends owing to poor evaporation characteristics. Figure 8 shows the SINL for the binary and ternary blends at different SOI timing. A significant reduction in peak SINL value is observed with delayed injection start across all tested fuel blends. The thermophysical properties of the fuel are critical in addressing the SINL of blends at a specific injection start. The higher sooty
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Fig. 5 Crank-resolved natural flame luminosity of different fuel blends at start of injection 30° ASTDC (CA is after compression TDC)
combustion occurs at SOI 30° ASTDC; hence, all fuels have a higher SINL peak. G80E20, on the other hand, exhibits the lowest SINL peak in all three injection start timings. Because of its high vapor pressure, ethanol is very evaporative and improves air-fuel mixing. Furthermore, in ethanol, fuel-bound oxygen can form a localized premixed zone, reducing diffusion burning and, as a result, flame luminosity. In contrast, the addition of LPO to gasoline raises the peak in SINL for all tested conditions. The higher boiling point, lower vapor pressure, increased surface tension, and viscosity of LPO inhibit evaporation, resulting in enhanced diffusion burning. As a result, higher SINL and longer pool fire residence time are observed for LPO blends. However, adding ethanol to gasoline-LPO blends reduces the peak in SINL and remains more significant than gasoline. We hypothesize that preheating the present ternary blend with a late injection approach would enhance combustion quality due to faster evaporation of the impinged fuel film layer at high temperatures, requiring further investigation.
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Fig. 6 Crank-resolved natural flame luminosity of different fuel blends at start of injection 60° ASTDC (CA is after compression TDC)
4 Conclusions Fuel injection timing has a considerable impact on combustion quality because of the impingement of fuel in the piston head. For all of the studied fuels, an early start of injection resulted in noticeably higher flame luminance. By burning as a diffusion flame, the fuel film contributes to higher hydrocarbon and soot emissions. However, as the injection timing is delayed, the possibility of piston head wetting decreases due to the increased distance between the fuel injector and the piston head. Thermo-physical properties of the fuel are important in impinged film evaporation. LPO blends had a longer pool fire residence time due to the thicker fuel film on the piston head. Adding ethanol leads to a reduction in the SINL value. A late injection strategies with fuel preheating for the proposed ternary blends can improve the combustion quality.
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Fig. 7 Crank-resolved natural flame luminosity of different fuel blends at start of injection 90° ASTDC (CA is after compression TDC)
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Fig. 8 Spatially integrated natural flame luminosity of different fuel blends at different start of injection (SOI). a SOI-30° ASTDC b SOI-60° ASTDC, and c SOI-90° ASTDC
Acknowledgements This research is supported by the Science and Engineering Research Board (SERB) of India through grant No. CRG/2019/003325. The authors thank Mr. Pillai Madhu Shankar for his assistance in conducting experiments and the central workshop staff at IIT Hyderabad for their assistance in fabrication work.
References 1. Ashok A, Gugulothu SK, Reddy RV, Burra B (2022) Influence of 1-pentanol as the renewable fuel blended with jatropha oil on the reactivity controlled compression ignition engine characteristics and trade-off study with variable fuel injection pressure. Sustain Energy Technol Assess 52:102215 2. Ashok B, Thundil Karuppa Raj R, Nanthagopal K, Krishnan R, Subbarao R (2017) Lemon peel oil: a novel renewable alternative energy source for diesel engine. Energy Conver Manag 139:110–121 3. Biswal A, Gedam S, Balusamy S, Kolhe P (2020) Effects of using ternary gasoline-ethanol-lpo blend on pfi engine performance and emissions. Fuel 281:118664 4. Biswal A, Kale R, Teja GR, Banerjee S, Kolhe P, Balusamy S (2020) An experimental and kinetic modeling study of gasoline/lemon peel oil blends for pfi engine. Fuel 267:117189
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Numerical Study on Soot Formation of Methyl Methacrylate Pool Flames with Coflow Air Argha Bose, D. Shanmugasundaram, and V. Raghavan
Abstract Polymethyl methacrylate (PMMA) is a polymer, and it finds its application in construction, transportation and domestic sectors. However, it is flammable. The combustion and soot characteristics of this polymer can be better understood by studying its monomer methyl methacrylate (MMA) that forms as a dispersed layer at the PMMA surface when heated to 250–300 °C. MMA is prone to produce soot. The effect coflow air towards mitigation of soot is analysed numerically considering MMA pool combustion. A parametric study is carried out with 30 mm pool diameter and 3 mm ullage for different amounts of coflow air (50, 100 and 150% theoretical air). Mass loss rate is observed to be almost the same for all the cases. However, the soot ratio is more for the reference case without coflow air, and it decreases for 50 and 100% theoretical air cases. The soot ratio increases when the coflow air increases from 100 to 150% theoretical air. The reasons for these trends are explained with the help of temperature, soot volume fraction, soot precursor, soot oxidation rate and OH contours. Keywords MMA · Pool flames · Coflow flames · Soot formation · MMA kinetics
1 Introduction Polymers find its widespread applications in various sectors such as construction, transportation, domestic and hybrid propellants. One such polymer mostly used for the above applications is polymethyl methacrylate (PMMA) [1]. It is indispensable to understand its combustion and emission characteristics in view of fire safety aspect. When PMMA is heated to 250–300 °C, its monomer—methyl methacrylate (MMA)—forms as a dispersed layer at its PMMA surface and it vaporizes. MMA is the main fuel species that predominantly oxidizes in the gas phase [2]. Most importantly, this monomer is an unsaturated methyl ester and more prone to produce soot during the gas-phase reactions. Therefore, it is important to understand the A. Bose · D. Shanmugasundaram · V. Raghavan (B) Department of Mechanical Engineering, IIT Madras, Chennai 600036, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_16
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soot characteristics of MMA flames in order to mitigate its formation. Fuel dilution, oxygen-enriched combustion, coflow air and addition of water vapour are various methods used to control the soot formation and oxidation. In this work, a coflow air stream is fed around a pool of MMA of 30 mm diameter and an ullage of 3 mm. Its burning characteristics for various coflow rates of air have been studied using a comprehensive numerical model.
2 Literature Review and Objective Several studies related to PMMA and MMA combustion are available in the past literature. Modak and Croce [3] studied the combustion phenomena of horizontal, thick (51 mm) solid PMMA and found that the mass burning rate per unit surface area increases as the size of the pool is increased. Bard and Pagni [4] measured the soot volume fraction in PMMA flames and reported that the soot volume fraction increases as the PMMA surface area is increased. Radiation properties in PMMA flames were measured by Markstein [5], and a correlation was developed between emissivity and burning rates. Gore et al. [6] observed that the sooting tendency of MMA pool flames was independent of radiative heat loss fraction. Experimental work of Hamins et al. [7, 8] calculated the heat feedback from MMA pool fires and also reported the radiative heat loss fraction. Research works of Wang et al. [9] and Lin et al. [10] analysed the soot production in lean and rich MMA-air flames. Numerical study of Rakesh Ranga et al. [11] used global kinetic step to determine the burning rate of PMMA and MMA flames using Fire Dynamics Simulator (FDS). Researchers also developed well-validated detailed and reduced kinetic schemes for MMA. Recently, Shanmugasundaram et al. [12] studied the effect of ullage and burner diameter on combustion and soot characteristics of MMA pool flames. They used a comprehensive numerical model, which consisted of interface coupling conditions for steady burning of liquid pool, a short kinetic mechanism which predicts the gas-phase combustion and simple soot and radiation models. Though several experimental and numerical studies are reported in literature for understanding MMA combustion and soot characteristics, no studies are available in literature related to soot characteristics in MMA pool flames using coflow air. Therefore, in the present work, the effect of coflow air towards soot formation and oxidation in MMA pool flames is numerically studied in detail. Shanmugasundaram et al. [12] reported experimental data for a steady flame over an MMA pool of diameter 30 mm and an ullage of 3 mm, in terms of temperature and species profiles. Ullage is the distance between the pool surface and burner rim. Their comprehensive numerical model has also been validated against these results. This case, without coflow air, is used as the reference case in this work. The same validated numerical model has been used in this study. Further simulations have been carried out with coflow air having varying flow rates. The objectives of the present work are: (1) To carry out the parametric study for analysing the effects of coflow air (50, 100 and 150% of theoretical air) on flame and soot characteristics. (2) To gain
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more physical insights on soot characteristics among different cases with the help of temperature, fuel, soot precursors and soot volume fraction contours.
3 Numerical Methodology A heterogeneous laminar diffusion flame over an MMA pool flame of diameter 30 and 3 mm ullage has been simulated using a numerical model in Ansys Fluent 15.0 framework. The model is incorporated with well-validated multi-step mechanism consisting of 49 species and 376 reactions. Figure 1 shows the schematic of 2D axisymmetric computational domain. The burner wall (hatched region in Fig. 1) is a solid region that is meshed, and heat transfer inside it is solved. Inner wall surface in the computational domain is solved in a conjugate manner, allowing heat transfer from the flame to the wall. Wall surface next to the MMA pool is specified isothermal at 321 K. The temperature at the bottom surfaces is kept at 298 K for all cases. The outer wall is exposed to atmosphere, allowing the convective heat transfer to ambient air (h = 10 W/m2 K, T = 298 K) for coflow case. Velocity and the normal derivative of the species mass fractions are set to zero at wall surfaces. In coflow cases, the mass flow rate of air through the annulus is varied 50, 100 and 150% of theoretical air, which is calculated from the reference case [12]. The stoichiometric (theoretical) air required to burn steady mass loss rate of MMA (1.17 × 10–5 kg/s) for reference case is 9.641 × 10–5 kg/s. In reference to this case, the mass flow rates of coflow of air are calculated, and they are reported in Table 1. At the axis of symmetry, normal velocity and the gradients of all other variables are set to zero. The far field (inlet and outlet in Fig. 1) is considered as pressure specified boundary. Multi-block, non-uniform and structured grids are used to discretize the domain. A uniform cell size of 0.5 mm is used along radial and axial directions up to 25 and 125 mm, respectively. The total number of cells used for the remaining distances along the radial and axial directions is 35 and 75, respectively. For solid region, the uniform cell size of 0.5 mm is used. After a grid independence study, the total number of cells is fixed as 25,770. All the simulations are carried out by setting the normalized energy and continuity residuals as 10–6 and 10–3 , respectively. Mass imbalance [absolute (outgoing mass-incoming mass)/incoming mass] is kept below 1%. Moss–Brookes [13] model is used to model the source terms for formation and growth of soot particles. The soot precursors considered are C2 H2 and C2 H4 . Soot oxidation rate is calculated using Fennimore and Jones model [14] with OH radical as the oxidation parameter. Model constants used in the Ansys Fluent [15] are not altered, since a relative comparison between different cases is attempted. SIMPLE algorithm and second-order upwind schemes are implemented appropriately. To solve for the coupling conditions at the interface [12], customized userdefined function (UDF) is used to iteratively solve the Clausius–Clapeyron equation, Fick’s Law of Diffusion and energy balance at the interface with an objective to determine the MMA mass loss rate and mole fraction of MMA. The detailed information
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on numerical methodology followed in the present work can be found elsewhere [12].
4 Results and Discussion Numerical results of radial profiles of temperature and major species profiles at several axial locations for varying coflow cases are compared in Figs. 2 and 3. The mass loss rate and soot ratio (defined as mass of soot produced when unit mass of MMA is burnt) are compared in Fig. 4. Soot mass is calculated as the volume integral of soot volume fraction in the entire domain multiplied by the soot density. The effects of coflow air towards these parameters are discussed below.
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4.1 Comparison of Temperature and Major Species Profiles of Coflow Cases with the Reference Case Comparison of radial variation of temperature between the reference and the coflow cases at different axial locations are shown in Fig. 2. As evident from Fig. 2a, the radial variation of temperature of all the coflow cases follows the reference case at z = 15 mm. The location of peak temperatures for all the cases is nearly same at all axial locations, and it is seen in Fig. 2a–d. At the downstream, the temperatures of 50 and 100% theoretical air cases are significantly lower (with maximum temperature difference of ~ 200 K) than the reference case. The 150% theoretical air case, as shown in Fig. 2b–d, follows the reference case at all the axial locations except at z = 60 mm. The radial variation of major species profiles at several axial locations is plotted in Fig. 3. It is observed from Fig. 3 that the MMA mole fraction for the reference case is higher than the coflow cases. When compared to the reference case, the other species profiles of coflow cases do not show significant differences in their amounts at all axial locations as evident from Fig. 3c–h.
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4.2 Effect of Coflow Air on Soot Ratio and Mass Loss Rate The average mass loss rate of fuel and soot ratio for the reference case and coflow cases are shown in Fig. 4a, b. It is observed that mass loss rate for the reference case is maximum and it decreases slightly (around ~ 8.5%) for coflow cases. As the
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percentage theoretical air increases, the mass loss rate remains almost constant for all the coflow cases. The soot ratio is observed to be maximum for the reference case when compared with all the coflow cases. When the coflow air (50% theoretical air) is introduced, nearly 236% reduction in soot ratio is observed. Further increase of coflow air (100%) decreases the soot ratio to 238% when compared with the reference case. The soot ratio is observed to be minimum for this case (100% theoretical air). Thereafter, the soot ratio shown an increasing trend as the amount of coflow air increases from 100% theoretical air to 150% theoretical air. The reasons for these trends are explained with the aid of contours of temperature, soot precursor and soot parameters. Figure 5 depicts the contours of temperature and MMA species mass fraction for the reference and coflow cases. The extent and location at which the high-temperature zone is formed for all the cases are nearly the same (~ z = 0.08 m), and it is shown in Fig. 5. Thus, the mass loss rate shown in Fig. 4 is almost same for all the cases. The peak temperature is nearly same for the reference case and 150% theoretical air case, and it is slightly more (~ 50 K) than the other coflow cases. The increase in peak temperature for coflow case with 150% theoretical air and reference case is due to more oxygen entrainment into the flame zone when compared to other two coflow cases (50 and 100% theoretical air). Figure 6 shows the velocity vectors of air entrainment into the flame zone and oxygen contour lines for all the cases. It is observed from Fig. 6b–d, the amount of air penetrates into the flame zone is more for 150% theoretical air case when compared with 50 and 100% theoretical air cases. This fact is confirmed by the vector lengths of the 150% theoretical air (Fig. 6d) close to the burner rim are more compared to other two cases. It is also evident from Fig. 6a that the amount of air entrainment into the flame zone without coflow is more when compared with all the coflow cases. In summary, as the availability of oxygen in the flame zone is more for reference and 150% theoretical air cases compared to other two cases (50 and 100% theoretical air), the peak temperature is more for these cases.
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Fig. 5 Contours of temperature (lines) and MMA mass fraction (greyscale) for 30 mm pool diameter with 3 mm ullage cases of a reference case b 50% theoretical air c 100% theoretical air d 150% theoretical air. Peak temperature for each case is also indicated alongside
The trend shown in Fig. 4 for soot ratio is better explained with the contours of soot volume fraction, C2 H2 , soot oxidation rate and OH radical. Figure 6 shows the distributions of soot volume fraction (grey scale) and mass fraction of soot precursor (C2 H2 ) for all the cases. The contours of soot oxidation rate and mass fraction of OH radical are shown in Fig. 7. It is evident from Fig. 7a, d, the distribution of mass fraction of C2 H2 and maximum value of C2 H2 is more for reference case and 150% theoretical air cases, when compared with other two cases. From Fig. 8a, d, it is also observed that the extent of maximum OH mass fraction contour line is more for the reference and 150% theoretical air cases. The corresponding peak values are also more for the reference and 150% coflow cases when compared to other two cases. It is due to more air entrainment, which increases the availability of oxygen in the flame zone (shown in Fig. 6). Thus, the peak temperature is higher for these two cases (shown in Fig. 5) and produces more OH radical through reactions of H + O2 = > O + OH and H2 O + O = > OH + OH. It is also to be noted that these chain branching reactions are more active as the
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temperature increases. As the production of OH radicals is more in the flame zone for these two cases, it favours the important exothermic reaction of CO + OH = > CO2 + H. It also explains the reason for increase in flame temperature for reference and 150% theoretical air cases. In summary, the soot formation rate (such as nucleation rate and surface growth rate) and soot oxidation rate increase as the flame temperature increases. It is due to the fact the kinetics of these two processes are faster as the temperature increases, which promotes both formation and oxidation simultaneously. It is observed from the Fig. 7a, d that the distribution and peak value of soot volume fraction is more for the reference and 150% theoretical air cases. Thus, it indicates the net effect promotes soot formation as the flame temperature increases for these cases. This explains the reason for the soot ratio trend followed in the Fig. 4b.
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5 Conclusions A parametric study is carried out with 30 mm burner diameter and 3 mm ullage for different amounts of coflow air (50, 100 and 150% theoretical air), and comparison is made with the without coflow cases (reference case). The fuel mass loss rate remains almost constant. It is due to the extent and location of maximum temperature is nearly same for all the cases. The soot ratio is observed to be more for the reference case and 150% theoretical air cases when compared to other two cases. It is due to the fact that the peak temperatures are more for these two cases. The kinetics of soot production and oxidation rates are faster as the temperature increases which promotes both formation and oxidation simultaneously for these cases. Thus, net effect promotes soot production as the flame temperature increases.
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Fig. 8 Contours of OH mass fraction(lines) and rate of soot oxidation (greyscale) for 30 mm pool diameter and 3 mm ullage with a reference case b 50% theoretical air c 100% theoretical air d 150% theoretical air
Acknowledgements Authors acknowledge P.G. Senapathy centre for computing resources, IIT Madras, for providing the computational facility required for the present work.
References 1. Harper CA (2000) Modern plastics handbook. McGraw-Hill, Lutherville, p 64 2. Seshadri K, Williams FA (1978) Structure and extinction of counter flow diffusion flames above condensed fuels: comparison between poly (methyl methacrylate) and its liquid monomer, both burning in nitrogen–air mixtures. J Polym Sci Polym Chem 167:1755–1778 3. Modak AT, Croce PA (1977) Plastic pool fires. Combust Flame 30:251–265 4. Bard S, Pagni PJ (1986) Spatial variation of soot volume fractions. Fire Saf Sci 1:361 5. Markstein GH (1979) Radiative properties of plastics fires. Int Symp Combust 17(1):1053–1062 6. Gore J, Klassen M, Hamins A, Kashiwagi T (1991) Fuel property effects on burning rate and radiative transfer from liquid pool flames. Fire Saf Sci 3:395–404 7. Hamins A, Klassen M, Gore J, Kashiwagi T (1991) Estimate of flame radiance via a single location measurement in liquid pool fires. Combust Flame 86(3):223–228
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8. Hamins A, Fischer SJ, Kashiwagi T, Klassen ME, Gore JP (1994) Heat feedback to the fuel surface in pool fires. Combust Sci Technol 97(1–3):37–62 9. Wang T, Li S, Lin Z, Han D, Han X (2008) Experimental study of laminar lean premixed methyl methacrylate/oxygen/argon flame at low pressure. J Phys Chem 112(6):1219–1227 10. Lin Z, Wang T, Han D, Han X, Li S, Li Y, Tian Z (2009) Study of combustion intermediates in fuel-rich methyl methacrylate flame with tunable synchrotron vacuum ultraviolet photo ionization mass spectrometry. Rapid Commun Mass Spectr 23(1):85–92 11. Rakesh Ranga HR, Korobeinichev OP, Raghavan V, Tereshchenko AG, Trubachev SA, Shmakov AG (2019) A study of the effects of ullage during the burning of horizontal PMMA and MMA surfaces. Fire Mater 43(3):241–255 12. Shanmugasundaram D, Muthu Kumran S, Trubachev SA, Bespalova A, Korobeinichev OP, Stanislay AG, Raghavan V (2020) Burning characteristics and soot formation in laminar methyl methacrylate pool flames. Combust Theory Model 26(6):1153–1178 13. Brookes SJ, Moss JB (1999) Prediction of soot and thermal radiation in confined turbulent jet diffusion flames. Combust Flame 116:486–503 14. Fenimore CP, Jones GW (1967) Oxidation of soot by hydroxyl radicals. J Phys Chem 71:593– 597 15. ANSYS Fluent, Release 15.0, Help System, Theory Guide. ANSYS, Inc.
Impact of Computational Domain and Cell Type on Large Eddy Simulations in OpenFOAM for a Turbulent Partially Premixed Flame Sandeep Lamba and Krishna Kant Agrawal
Abstract Large eddy simulations (LES) are becoming state-of-art for industrial combustion applications due to its capability to better capture fluid mixing and flame front by resolving large-scale turbulent eddies. However, this increased sensitivity to the physics can become more demanding on the numerical and meshing methods and boundary condition requirements. This paper highlights some challenges in performing LES of a canonical turbulent partially premixed flame (Sandia flame D) in open-source CFD platform OpenFOAM. Both RANS and LES approaches were evaluated, and it was found that time-averaged flow and combustion properties were captured by RANS with reasonable ease and accuracy. However, LES was not able to capture the flame eddies and turbulent diffusion at the first attempt. LES results were found quite sensitive to the computational mesh and modelling methods. As the flow unsteadiness is captured in LES, the turbulence and acoustic boundary conditions also become important. Explorations on all these aspects are made, and finally, conclusion is made on the importance of using structured mesh over unstructured mesh and avoidance of two-dimensional axisymmetric domains in the OpenFOAM CFD solver, specifically in the LES context. Keywords LES · CFD · OpenFOAM · Turbulent combustion · Structured and un-structured meshing
1 Introduction Studies in industrial combustion devices such as gas turbines are important for controlling pollutants emissions such as NOx and carbon-monoxide (CO), understanding flame stability and increasing part-load flexibility[1, 2]. Flow and combustion in industrial applications is inherently turbulent due to large mass throughput, which requires understanding and modelling of turbulent combustion [3, 4]. This S. Lamba · K. K. Agrawal (B) Department of Mechanical Engineering, IIT Delhi, New Delhi 110016, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_17
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is because the local conditions for chemical reactions, which decide burn rates and pollutant formation routes, become highly transient and random in a turbulent flow. This requires a statistical description of the flow field where all flow variables such as velocity components, temperature, pressure and species concentration are resolved in a mean and a fluctuating component as shown in Fig. 1. In the traditional Reynolds average Navier Stokes (RANS) approach, the mean is taken over the entire time history, and hence, there is single description of the mean flow. All the fluctuations are described in statistical deviation of individual parameters such as turbulent kinetic energy for velocity and variances of temperature, mass fraction. In the LES approach, however, the time averaging is done over a small timewindow only, due to which the mean flow quantities are also continuously changing with time in a turbulent flow. A schematic of comparison of velocity field modelled with RANS and LES is shown in Fig. 1. Only the small-scale fluctuations now form the variances, which are much smaller than their RANS counterparts. This is expressed in terms of spatial filtering in LES where large-scale turbulent eddies or flow transientness is resolved (time varying mean component) and the small-scale fluctuations are modelled (sub-grid scale turbulence). Although the mean quantities in RANS approach provide sufficient description of global features such as the turbulent flame structure, stabilization and average temperatures, it is insufficient to describe transient flame behaviour such as combustion dynamics or physics needing accurate local interactions such as prediction of CO emissions. Even for the long duration time-averaged “mean” flow quantities, LES predictions are better representation of reality due to explicit resolution of major anisotropic large scale of turbulence. In this way LES modelling is more accurate than RANS (where all turbulent scales are modelled) and less expensive than DNS (where all the scales are resolved) [5]. With the advancement in computational resources, for turbulent closure the main contribution in literature is of LES approach over other main counterpart such as RANS and DNS. Among the various tools available for the numerical study, OpenFOAM is widely gaining attention among the academicians as well as in industrial applications. The main benefits with selecting OpenFOAM is the availability of source code and no licensing limitations. While the former allows inspection, modification and addition of models and numerical schemes, the latter solves limitations with mesh size, CPU
Fig. 1 Example of resolution extent of RANS and LES approach
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cores and other restrictions. Also, its source code libraries are well constructed for parallel computation on multiple threads which make it faster by utilizing all available threads in a computing device. Along with many advantages, OpenFOAM also has some downside especially for the beginners. It is not based on GUI, and user should be familiar with code structure (libraries organization) of OpenFOAM and also with basic of programming language (C++). There is insufficient documentation for setting up case and the mesh generation and postprocessing utility is also not advanced unlike its commercial counterparts and hence for these one has to rely on third-party software. Due to these, the learning curve in OpenFOAM is steep, and there is need of significant troubleshooting. In a challenging application such as LES of turbulent reacting flows, where finer resolution of physics is attempted, the numerical errors can easily amplify. The present paper describes some challenges faced in terms of geometry, mesh grid type, boundary conditions, and numerical schemes during setting case for canonical partially premixed Sandia piloted flame D [6] in OpenFOAM. It is hoped that this experience will help a new user in avoiding pitfalls in similar studies and enrich documentation of this open-source software.
2 Literature Review and Objective For the turbulent combustion modelling, various approaches such as laminar finite rate (LFR), eddy dissipation models/concept (EDM/EDC) [7] flamelet-generated manifolds (FGM) [8–10], transported probability density functions (PDF) [11] and conditional moment closure (CMC) [12, 13] are present in literature. Out of these the FGM approach has been found to capture important details of combustion which being versatile and computationally affordable, which is important for LES simulations [14, 15]. In this approach, the detailed chemical kinetics is solved and then parameterized in terms of few controlling variables such as mixture fraction and reaction progress variable. Then, the transport equations are solved only for mean and variances of these controlling variables, which result in significantly less computations in CFD due to lesser equations as compared to solving for all chemical species. Regarding LES of turbulent reacting flows in OpenFOAM, there is some literature available [16, 17]. In these studies, however, the challenges related to mesh, boundary conditions and numerical schemes in LES are not highlighted, and only focus is on discussion of combustion physics. The present work is exploring more on the potential issues and challenges, on which the literature is not systematic. This has been observed in discussions on various online CFD collaboration platforms and many personal communications where such issues are being extensively discussed [18, 19]. Based on this, it was felt that there is a dire need for such documentation for CFD community specifically dealing with an open-source tool.
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3 Materials and Methods The appropriate version of OpenFOAM solver for simulating turbulent flames is reactingFoam (available in standard version of OpenFOAM and FGMFoam [20]). The reactingFoam solver uses EDC combustion model [7] where finite rate chemistry is also solved during the simulation and FGMFoam uses the FGM tabulated chemistry-based approach [21], where flamelets solutions are parametrized in the form of few controlling variables. During simulation only solution of the controlling variables along with mass and momentum equations is performed. In this study, both the RANS and LES turbulence models are simulated for both the EDC and FGM combustion models using OpenFOAM-V7 version [22]. The burner geometry is taken from turbulent non-premixed flames (TNF) [23] workshop which is a database of detailed experimental data on turbulent flames. The burner is termed as Sandia piloted flame burner, the current flame configuration is termed as Sandia flame D, and its geometry is shown in Fig. 2. The central jet [diameter (d) of 7.2 mm] issues a mixture of CH4 and air with a volumetric ratio of 1:3. Surrounding the central jet is pilot jet which has burnt product of lean (Φ = 0.77) mixture of C2 H2 , H2 , air, CO2 , and N2 having nominally the same equilibrium composition and enthalpy as CH4/air. The burner and other experimental details are well documented in [6]. Results in terms of temperature, species mass fraction, velocity components, and turbulence levels are available for validation of present simulation results.
Fig. 2 Schematic of Sandia burner[24]
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4 Results and Discussion As a first step RANS simulation was done with a two-dimensional axisymmetric geometry with both FGMFoam and reactingFoam. Total of 46 thousand rectangular structured grid cells with single cell in third direction are used for a 20d × 70d domain. There are ten cells across the main fuel jet radius and 15 cells across pilot jet diameter. The cell size is refined in the critical mixing zone just downstream of main and pilot injection. The courant number is fixed to 0.9, and corresponding time step during the simulation remains in the order of 10−5 s. The standard k − ε turbulence model is used for both the reactingFoam and FGMFoam solver. The numerical scheme for time step derivative used is Euler which is a first-order accurate implicit scheme. For gradient terms cellLimited gauss linear scheme is used which is second-order accurate. For divergence and Laplacian term, upwind and gauss-limited-linear schemes are selected, respectively, which are first-order accurate. The interpolation scheme is linear. For pressure velocity coupling, PIMPLE algorithms is used. The velocity boundary condition at all inlets outlets and walls is used fixedValue pressureInletOutletVelocity and no-slip, respectively. For pressure the boundary conditions at inlets and wall zeroGradient and outlets totalPressure is used. For all other scalars such as progress variable and mixture fraction, the boundary condition at all inlets are fixedValue, and at outlets and wall zeroGradient is used. For FGM, temperature and other thermophysical properties such as viscosity, density and thermal viscosity are updated by looking up from tables corresponding to controlling variable solution. However, in EDC separate energy equation is solved to get temperature and other thermophysical properties updated. The turbulent Schimdt and Lewis numbers are taken as unity. Temperature contours for both EDC and FGM models in RANS are shown in Fig. 3. The comparison of simulation results at three different axial locations (marked in Fig. 3) and at centreline with the experimental data is shown in Fig. 11. It is observed that the temperature profiles are predicted with reasonable accuracy by RANS simulations against experimental data. There is still some scope for improvement at downstream locations, which will be explored with the help of LES in next section.
4.1 LES of the Sandia Flame D in Two-Dimensional Axisymmetric Domain The same mesh as used in RANS simulation is utilized for LES simulations. In this step only the turbulence model is changed from k − ε to LES with single k-Eqn for modelling sub-grid scale turbulence. Temperature contours with both EDC and FGM combustion models in LES are shown in Fig. 4a.
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Fig. 3 Temperature contours for RANS simulation with reactingFoam (EDC) and FGMFoam (FGM)
Fig. 4 Temperature contours for LES simulations with reactingFoam and FGMFoam with a first order, and b second-order numerical schemes (for colour bar see Fig. 3)
As can be noticed in case of EDC, there is no spread in the flame, and in FGMFoam, the flame is appearing totally extinguished. It was found in literature that LES is more demanding in terms of numerical schemes [8, 15, 24]. Hence, all numerical schemes were changed to second-order accurate. The results are shown in Fig. 4b. As can be seen there is a bit improvement in case of the EDC combustion model in terms of spread in flame, but no improvement in case of FGMFoam. The results remained same when same simulations are repeated with Smagorinsky LES subgrid model.
4.2 Three-Dimensional Domain with Unstructured Mesh for LES After trying higher-order schemes with the two-dimensional mesh, it was explored to create a three-dimensional domain and mesh for LES simulations. This is because
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turbulence is inherently a three-dimensional phenomenon, and development of turbulent eddies development requires all spatial degrees of freedom. In this step, a cut section (45° wedge shape) of entire geometry is taken, and unstructured mesh with tetrahedral elements with total of 6 lakh cells is created. The mesh element size is varied with more refinement in the mixing zone. The minimum cell size is of the order of 0.5 mm. Mesh clip is shown in Fig. 5a with zoomed in view in Fig. 5b. The numerical schemes remained second-order accurate. Only the FGMFoam solver is used in this step, as multi-step chemistry in EDC becomes computationally prohibitive. Both RANS and LES simulation are carried out, and the results in terms of temperature contours are shown in Fig. 5c. The RANS simulation is run only for small time where it shows good spread of flame in the span-wise direction, indicating results similar to the two-dimensional simulations which agree well with experiments. In case of LES, as can be observed in Fig. 5c, there is slight improvement as compared to the two-dimensional geometry for the span-wise spread. The flame is appearing bit more brushy and broad as compared to the earlier LES case. However, still the flame appears to be laminarized, which means the eddies are not visible as they should be in a LES simulation. To check the turbulence (fluctuations) four velocity monitor probes are created at inlet and outlet of fuel and pilot jets (shown in Fig. 5d). The run time fluctuations are plotted in Fig. 6. From the probe plots for velocity, it is clear that the turbulence is actually diminishing from the pipe’s inlet towards the outlet. However, the expectation would be development of turbulent eddies in LES and increase in velocity fluctuations. This diminishing turbulence is visible in the laminarization of flame in present LES cases. After seeing the diminishing turbulence some artificial turbulence with the help of white noise turbulent inlet boundary condition turbulentInlet [25] which uses
Fig. 5 a 3D unstructured mesh clip with 45° cut section along with the zoomed in view (b). Temperature contours for RANS and LES simulations with FGMFoam (c). Probe location at inlets and outlets of fuel and pilot tubes (d) (for colour bar see Fig. 3)
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Fig. 6 Run time velocity plots at four different probe location for unstructured 3D mesh
a set of pseudo-random numbers for creating field for any scalar and boundary condition turbulentDFSEM [26] which uses synthesized eddies to create divergencefree turbulence field are employed at the inlets. But still the turbulence was found to be diminishing downstream. Also based on advice from various CFD experts many other things are also tried at this stage such as: RANS initialization, varying number of correctors in PIMPLE loop, different courant number, finer mesh, different type of boundary conditions (symmetric, AMI, cyclicAMI) [27] for symmetric cut section. But none of these changes were able to result in large-scale turbulent eddies behaviour.
4.3 Two-Dimensional Planar Domain with Structured Mesh for LES After trying higher-order schemes and three-dimensional mesh, still development of flow eddies was not obtained. Based on online resources and available case studies of LES in OpenFOAM, a two-dimensional planar (and not axisymmetric as in case of Sandia flame D) with structured mesh is created.
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Fig. 7 a Structured 2D mesh for rectangular channel, b temperature contour of LES simulation (for colour bar see Fig. 3)
This geometry has planar rectangular channel with passage dimensions similar to the Sandia flame D (Fig. 7a). This was maintained to keep similar flow velocity and length scales while investigating only the impact of mesh and domain type. With simulations, it was found that large-scale turbulent eddies were indeed getting formed in two-dimensional planar LES simulations as shown in temperature contours (Fig. 7b). Also, the velocity monitor probes are created at four locations across the main and pilot tubes. From the velocity plots for probes locations in Fig. 8, it was found turbulence is amplifying while passing through the fuel and pilot passages. However, the structured two-dimensional mesh, which was already tried earlier for the Sandia flame, in axisymmetric domain did not result in turbulent eddies development. For three-dimensional wedge shape geometry with unstructured mesh, the eddies did not develop either. It is concluded that the axisymmetric domain in two-dimensions and unstructured mesh in three-dimensions is causing the issue. Both of these configurations create a corner region in the computational cell, which might be not compatible with the current level of implementation of LES in OpenFOAM. Based on this conclusion, a full 360° three-dimensional domain with structured mesh will be used for LES in the next section.
4.4 Three-Dimensional Domain with Structured Mesh for LES In this section, a three-dimensional structured mesh is created with a total of 17 lakh cell count. Again, more refinement is done at the central core to capture large gradients at periphery of the flame. To reduce the computational cost, the length of the domain is reduced to 55d. The mesh clip along with zoomed in view is shown in Fig. 9a and b, respectively.
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Fig. 8 Run time velocity plots at four different probe location for structured 2D mesh
The temperature contour shown in Fig. 9c shows flame is appearing and eddies are also visible hence suggesting that turbulence is no longer diminished. But pressure contours (Fig. 9d) suggest there are pressure wave present which can affect the flame and hence causing large-scale flame extinction (visible in Fig. 9c). The pressure wave seen was doubted as an attribute of acoustically reflecting boundary conditions for pressure and velocity. Hence, based on OpenFOAM manual, the velocity outlet boundary conditions are changed from inletOutlet to zeroGradient, and simulation is repeated. The resultant contours for temperature, pressure, time-averaged temperature and mixture fraction are shown in Fig. 10a, b, c and d, respectively. It is clear from the Fig. 10 that the pressure waves are no more present. The LES time-averaged temperature at different axial distances and along centreline against experimental data is plotted in Fig. 10 which shows a good agreement. Hence, it can be concluded that a three-dimensional domain with structured mesh and appropriate turbulence inlet and velocity boundary conditions are necessary to achieve satisfactory results from LES for turbulent flame.
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Fig. 9 a Structured 3D mesh for Sandia burner with zoomed in view (b), temperature (c) and pressure (d) contour of LES simulation
Fig. 10 Contours for instantaneous temperature (a), pressure (b), time-averaged temperature (c) and instantaneous mixture fraction
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Fig. 11 Comparison of RANS and LES simulation results against the experimental data at centreline and various axial locations
5 Conclusions A canonical piloted turbulent partially premixed flame (Sandia flame D) is simulated using both RANS and LES approaches in OpenFOAM CFD platform using EDC and FGM turbulent combustion models. While RANS provided reasonable accuracy with a structured mesh in two-dimensional axisymmetric domain, it was found that with this approach LES produced almost a laminarized flame with no turbulent eddies or diffusion. Many computational strategies such as switching to higher-order numerical schemes, RANS initialization, varying number of correctors in PIMPLE loop, varying courant number ceiling, finer mesh and different type of boundary conditions (symmetric, AMI and cyclicAMI) for symmetric section were tried and did not help. Since turbulence is inherently three-dimensional and highly anisotropic, a threedimensional computational domain with unstructured mesh (to have reasonable mesh count) was created and used for both RANS and LES simulations. While RANS is working fine as before, for LES there is only slight improvement in terms of spread of flame, but eddies are not forming as should be for a LES simulation. The velocity monitor probes plots suggest the turbulence is actually diminishing which should otherwise be amplified. A solution to this issue was attempted by providing artificial turbulence at velocity inlet but still the problem persisted. To troubleshoot the issue further, a simple two-dimensional planar geometry is created with dimension and flow parameters similar to Sandia flame D, and LES
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simulation is performed with same simulation parameters. This is to investigate effect of the nature of computational cells while having similar flow and length scales of the Sandia flame D. Results showed that “LES like” large eddy formation behaviour is indeed reflected in simulation. These results suggest a two-dimensional planar structured mesh without wedge shape geometry is working fine for LES. Hence, a two-dimensional axisymmetric domain or a unstructured mesh creates the issue of no turbulent eddies formation in LES. Both of these configurations create a corner region in a computational cell, which might be not compatible with the current level of implementation of LES in OpenFOAM and hence the issue. Based on this, a full three-dimensional geometry with all hexahedral structured mesh cells is tested for LES, and the outcome is satisfactory in terms of presence of turbulence eddies. However, some presence of large length scale pressure waves is also visible in the eddy formation behaviour, which is dependent on acoustics of the domain. To correct for the domain exit velocity boundary condition is changed from inletOutlet to zeroGradient, which helped in detuning of pressure oscillations in the domain. With these the LES cases were run for 3–4 flow through times in the domain, and then time averaging of results for further time duration is done to get mean flow statistics in LES. These are compared against the experimental data and found to be in satisfactory agreement. With these the effect of computational domain and mesh cell type on LES results are observed, and appropriate strategy for LES simulations in OpenFOAM (structured mesh in three-dimensions) is recommended. Acknowledgements We are thankful to Chalmers University of Technology [20], for the availability of FGMFoam solver on their Website. We are also thankful to various CFD experts from CFD Online [18, 19] and LinkedIn for their fruitful advice.
Nomenclature GUI LES RANS EDC FGM DFSEM AMI Z ε Φ PV d
Graphical user interface Large eddy simulation Reynold averaged Navier Stoke Eddy dissipation concept Flamelet-generated manifold Divergence-free synthetic eddy method Arbitrary mesh interface Mixture fraction Epsilon Equivalence ratio Progress variable Fuel jet diameter (mm)
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References 1. Gonzalez-Salazar MA, Kirsten T, Prchlik L (2018) Review of the operational flexibility and emissions of gas- and coal-fired power plants in a future with growing renewables. Renew Sustain Energy Rev 82:1497–1513. https://doi.org/10.1016/J.RSER.2017.05.278 2. Abdin IF, Zio E (2018) An integrated framework for operational flexibility assessment in multiperiod power system planning with renewable energy production. Appl Energy 222:898–914. https://doi.org/10.1016/J.APENERGY.2018.04.009 3. Peters N (2000) Turbulent combustion. Turbul Combust. https://doi.org/10.1017/CBO978051 1612701 4. Pope SB (2022) Turbulent flows. https://pope.mae.cornell.edu/TurbulentFlows.html. Accessed 06 Oct 2022 5. Pitsch H (2005) Large-eddy simulation of turbulent combustion. Annu Rev Fluid Mech 38:453– 482. https://doi.org/10.1146/annurev.fluid.38.050304.092133 6. Meier W, Barlow RS, Chen YL, Chen JY (2000) Raman/Rayleigh/LIF measurements in a turbulent CH4 /H2 /N2 jet diffusion flame: experimental techniques and turbulence-chemistry interaction. Combust Flame 123(3):326–343. https://doi.org/10.1016/S0010-2180(00)00171-1 7. Magnussen BF (2005) The eddy dissipation concept: a bridge between science and technology. In: ECCOMAS thematic conference on computational combustion, vol 21, pp 24. Libson, Portugal 8. Ma L (2016) Computational modeling of turbulent spray combustion. [Dissertation (TU Delft), Delft University of Technology]. https://doi.org/10.4233/uuid:c1c27066-a205-45f4-a7b4-e36 016bc313a 9. van Oijen JA, de Goey LPH (2007) Modelling of premixed laminar flames using flameletgenerated manifolds. Combust Sci Technol 161(1):113–137. https://doi.org/10.1080/001022 00008935814 10. Zhang Y, Wang H, Both A, Ma L, Yao M (2019). Effects of turbulence-chemistry interactions on auto-ignition and flame structure for n-dodecane spray combustion. Combust Theory Model 23(5):907–934. https://doi.org/10.1080/13647830.2019.1600722 11. Xu J, Pope SB (2000) PDF calculations of turbulent nonpremixed flames with local extinction. Combust Flame 123(3):281–307. https://doi.org/10.1016/S0010-2180(00)00155-3 12. Garmory A, Mastorakos E (2011) Capturing localised extinction in Sandia Flame F with LES– CMC. Proc Combust Inst 33(1):1673–1680. https://doi.org/10.1016/J.PROCI.2010.06.065 13. Roomina MR, Bilger RW (2001) Conditional moment closure (CMC) predictions of a turbulent methane-air jet flame. Combust Flame 125(3):1176–1195. https://doi.org/10.1016/S0010-218 0(01)00237-1 14. Ihme M, Pitsch H (2008) Prediction of extinction and reignition in nonpremixed turbulent flames using a flamelet/progress variable model: 2. Application in LES of Sandia flames D and E. Combust Flame 155(1–2):90–107. https://doi.org/10.1016/J.COMBUSTFLAME.2008. 04.015 15. Sheikhi MRH, Drozda TG, Givi P, Jaberi FA, Pope SB (2005) Large eddy simulation of a turbulent nonpremixed piloted methane jet flame (Sandia Flame D). Proc Combust Inst 30(1):549–556. https://doi.org/10.1016/J.PROCI.2004.08.028 16. Zhang H, Zhao M, Huang Z (2020) Large eddy simulation of turbulent supersonic hydrogen flames with OpenFOAM. Fuel 282:118812. https://doi.org/10.1016/J.FUEL.2020.118812 17. Ottino GM, Fancello A, Falcone M, Bastiaans RJM, de Goey LPH (2015) Combustion modeling including heat loss using flamelet generated manifolds: a validation study in OpenFOAM. Flow Turbul Combust 96(3):773–800. https://doi.org/10.1007/S10494-015-9666-5 18. Requesting help in LES set up for SandiaD flame—CFD online discussion forums. https:// www.cfd-online.com/Forums/openfoam-solving/202302-requesting-help-les-set-up-sandiadflame.html. Accessed 04 Oct 2022 19. Best-practices for LES—CFD online discussion forums. https://www.cfd-online.com/Forums/ openfoam-solving/196958-best-practices-les.html. Accessed 04 Oct 2022
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20. Ph.D. course in CFD with OpenSource software. http://www.tfd.chalmers.se/~hani/kurser/OS_ CFD/. Accessed 07 Oct 2022 21. van Oijen JA, Donini A, Bastiaans RJM, ten Thije Boonkkamp JHM, de Goey LPH (2016) State-of-the-art in premixed combustion modeling using flamelet generated manifolds. Prog Energy Combust Sci 57:30–74. https://doi.org/10.1016/J.PECS.2016.07.001 22. Download v7 | Ubuntu | OpenFOAM. https://openfoam.org/download/7-ubuntu/. Accessed 06 Oct 2022 23. Sandia/TUD piloted CH4 /air jet flames | TNF workshop. https://tnfworkshop.org/data-archives/ pilotedjet/ch4-air/. Accessed 06 Oct 2022 24. Mahmoud R, Jangi M, Fiorina B, Pfitzner M, Sadiki A (2018) Numerical investigation of an oxyfuel non-premixed combustion using a hybrid Eulerian stochastic field/flamelet progress variable approach: effects of H2 /CO2 enrichment and Reynolds number. Energies 11(11):3158. https://doi.org/10.3390/EN11113158 25. OpenFOAM: API guide: turbulentInletFvPatchField class template reference. https:// www.openfoam.com/documentation/guides/latest/api/classFoam_1_1turbulentInletFvPatch Field.html. Accessed 06 Oct 2022 26. OpenFOAM: user guide: turbulence DF-SEM. https://www.openfoam.com/documentation/gui des/latest/doc/guide-bcs-inlet-velocity-dfsem.html. Accessed 06 Oct 2022 27. OpenFOAM v8 user guide—5.2 boundaries. https://doc.cfd.direct/openfoam/user-guide-v8/ boundaries. Accessed 07 Oct 2022
Exergy Analysis of Deflagration Wave Propagating in Autoignitive H2 Mixture for Constant Pressure Boundary Conditions Rahul Patil and Sheshadri Sreedhara
Abstract Exergy analysis has been performed on propagation of deflagration wave in autoignitive H2 -air medium. Various one-dimensional (1D) direct numerical simulations (DNS) have been performed with constant pressure boundary conditions. The analysis is performed with multiple initial pressures and equivalence ratios. During autoignition in stratified mixture, flame like structure (deflagration wave) originates from thermal or compositional inhomogeneity inside the domain. In present study, the deflagration wave propagating from the hotspot is allowed to interact with the autoigniting mixture. Three major components are identified for the exergy loss from the system: heat conduction, mass diffusion, and chemical reactions. The results show that the 1D simulation shows larger ignition delays than the homogeneous simulations (0D). The irreversibilities associated with the reactions are found to be major contributors to entropy generated during combustion. During the interaction of deflagration wave and autoignition spot, the contribution of conduction and diffusion irreversibilities is observed to reduce. Whereas the contribution of reaction irreversibilities increases. For low-pressure combustion (10–20 bar), the ratio of exergy loss and heat release rate is observed to reduce during the interaction of deflagration wave and autoignition spot. Keywords 1D · DNS · Exergy analysis · Autoignition · Combustion · Deflagration waves
1 Introduction The diesel engines and gas turbine combustors operate with a constant pressure combustion cycle. The working principle involves injecting fuel into a hot oxidizer environment. The fuel injection into the oxidizer environment creates inhomogeneity and stratification inside the combustion chamber. In case of stratified combustion, R. Patil (B) · S. Sreedhara Department of Mechanical Engineering, IIT Bombay, Mumbai 400076, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_18
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the flame like structures are observed to originate from thermal or compositional inhomogeneity. In distributed stratified mixture, the flame like structure (deflagration wave) originates at multiple locations at once, and interaction between deflagration wave and autoignition site can be observed. The interaction between deflagration waves and autoignition sites is a complicated phenomenon to study. In the case of experimental studies, difficulties lie as the propagation of deflagration waves is observed only for the duration of ignition delay. In past decades, combustion research has been mainly focused on improving the efficiencies and performance of combustion devices. The first law of thermodynamics deals with the quantification of energy balance. In comparison, the second law analysis deals with the qualitative transfer of energy. The second law analysis provides the framework for measuring entropy production in process. The various irreversibilities associated with energy conversion reduce a system’s ability to extract maximum work (exergy). The usable work is lost in the form of entropy. In the case of combustion, four types of irreversibilities, namely heat conduction, mass diffusion, chemical reactions, and viscous dissipation, are identified. The irreversibilities during combustion process cause a reduction in potential and useful work extracted from combustion products. Understanding the local sources of irreversibilities is essential to minimize them. The effect of initial pressure and chemical composition on exergy loss rate during interaction of deflagration wave and autoignition spot is analyzed in this study. The contribution of viscous dissipation toward exergy is negligible compared to other components of entropy generation, so it is not included in the analysis. The analysis is performed with a 1D domain using direct numerical simulations (DNS). The fuel used for this study is hydrogen (H2 ), and constant pressure boundary conditions are used. A variety of initial pressures and equivalence ratios are used to perform analysis.
2 Literature Review and Objective The combustion devices convert the chemical energy of fuel into useful work. During this conversion, a large part of the fuel’s chemical potential gets converted into entropy or destroyed. Dunbar and Lior [1] performed an analysis on methane and H2 combustion. The results showed that about one-third of the total useful work was destroyed during this energy conversion. Som and Datta [2] provided the theoretical framework for quantifying combustion’s irreversibilities. Four components of irreversibly responsible for exergy loss were identified by Nishida et al. [3]. These components are heat conduction, mass diffusion, chemical reactions, and viscous dissipation. The contribution of these four components varies for various combustion applications. The contribution of viscous dissipation toward exergy loss is negligible compared to the other component, so the present analysis is only presented for heat conduction, mass diffusion, and chemical reactions. For autoignition combustion, irreversibilities associated with chemical reactions were major contributors to
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exergy loss [4]. In comparison, for propagating flame, the contribution of the heat conduction component was observed to comparable with reaction irreversibilities and tend to with the increase in pressure and lowering of initial temperature [5]. The rate of exergy loss can be calculated as sum of all individual rates generation of irreversibilities. sgen = sgen |conduction + sgen |diffusion + sgen |reaction ) ] ∂T 2 ∂x [ ∑ ρ Rg,i Di ∂ X i ∂Yi ] Sgen |diffusion = Xi ∂x ∂x i λ Sgen |conduction = 2 T
[(
(1)
(2)
where T is temperature, λ is thermal conductivity, ρ is density, X is a mole fraction, Y is a mass fraction, Rg,i is the specific gas constant for ith species, and Di is the binary diffusion coefficient of ith species in the mixture. The standard chemical exergy of a reaction can be defined as ψr0xn =
∑
0 υi ψc,i
(3)
where υ i is the stoichiometric coefficient, and ψ 0 C,i is the standard chemical exergy of ith species in a reaction. The standard chemical exergy can be defined in terms of chemical potential ) ( ψr0xn = μi0 − μˆ i,e
(4)
where μi0 is standard chemical potential, and μˆ i,e is the mixture chemical potential of species i in the reference environment. The chemical potential of ith species in jth reaction, μˆ i,e , can be defined as [6] ( μi, j = h i, j − T si, j + RT ln
xi P Patm
) (5)
where h i, j and si, j are enthalpy and entropy and are calculated from NASA polynomials. x i is the mole fraction of ith species, and P is pressure. Sgen |reaction =
∑ ωi μi i
T
where ωi is the production rate of ith species in reactions.
(6)
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The rate of destruction of exergy {exergy loss rate (ELR)} of inside the domain can be defined as ELR = T0 sgen
(7)
where subscript 0 corresponds to the dead state, for this particular case, 1070 K is used as the temperature of the dead state. In this study, the ELR corresponding to irreversibility is presented as ELR|reaction for reaction irreversibility, ELR|conduction for irreversibilities associated with thermal conduction, and ELR|diffusion for mass diffusivity-related irreversibilities. The present work’s objective is to analyze the effect of initial pressure and equivalence ratio on ELR during the propagation of a deflagration wave. The analysis includes the ELR during the interaction between deflagration wave and autoignition spot.
3 Numerical Methods Combustion phasing is an important mechanism for controlling the heat release rate of fuel inside the combustion chamber. By adding the source of ignition, uncontrolled combustion can be avoided. Here, combustion phasing was achieved with the introduction of thermal inhomogeneity. The thin wave-like structures originating from temperature inhomogeneity can be identified as deflagration waves [7]. In this study, the interactions between the deflagration wave initiating from the hot spot and the evolving autoignition site are analyzed with the help of numerical simulations. The initial pressure conditions inside the domain are selected such that it should be able to depict the behavior of autoignition for H2 in Zone-II and Zone-III. Since H2 autoignition is explosive in nature, lean equivalence ratios are chosen for the analysis to save computational cost. One-dimensional constant pressure simulations are performed using direct numerical simulations (DNS) with PENCIL code [8]. A 9 species and 21 reactions chemical mechanism is used to perform simulations [9]. The gas turbine and internal combustion engine-specific operating conditions, such as the initial temperature of 1070 K and 10, 20, and 40 bar pressure, are used to perform simulations. The Lean combustion conditions are simulated by equivalence ratios of 0.1, 0.2, 0.3, and 0.4 inside the domain. As the equivalence ratio increases, flame speed also increases. In order to see the complete interaction of deflagration wave with the ignition site, the domain lengths are increased accordingly. The length of the domain is chosen such that flame propagating with laminar flame speed should stay in the domain for twice delay period. The deflagration wave is resolved in at least 15 grid points for all cases. In order to save computational cost case symmetrical about the central axis was chosen with normal boundaries to the domain selected as periodic. The boundary
Exergy Analysis of Deflagration Wave Propagating in Autoignitive H2 … Table 1 Parameters of simulation
Pressure (bar)
Equivalence ratio
Length (cm)
10
0.1, 0.2, 0.3, 0.4
0.6, 1.25, 3.0, 4.0
217
20
0.3, 1.25, 1.55, 2.56
40
0.15, 0.625, 0.53, 1.4
opposite the central axis is selected as the constant pressure boundary condition. The constant pressure boundary has been subjected to Navier Stoke characteristics boundary condition [10]. The details of simulated cases are given in Table 1. The schematic diagram of the simulated domain can be seen in Fig. 1. The domain is initialized by a uniform premixed mixture and temperature across the domain. At x = 0, the domain is closed, and a wall boundary condition is imposed. The deflagration wave is initiated with a hot spot given by Eq. (8). Domain, as shown in Fig. 1, is simulated with the help of planar 1D simulations. [ ( )2 ] − 2n 2 x − L2 A A Tx = exp − π L2 2.5n
(8)
where A represents the amplitude of the hot spot, n width of the hot spot, L is the length of the domain, and x is the location. In this case, the temperature peak is imposed on the mean temperature, so the lowest temperature is 1070 K. The adiabatic flame temperature is selected for the maximum temperature for a particular chemical mixture. This temperature profile was adjusted to the flame profile of laminar burning of the H2 -air mixture, and the profile of completely combusted products is imposed near the symmetric plane to achieve stable deflagration wave. The combustion products are assumed to be H2 O and unburnt fuel and oxidizer. The mass fraction of burnt products is decided by a temperature-based progress variable. As the domain is subjected to high temperature, the part of the domain away from the deflagration wave propagation (region near L, in Fig. 1). The analysis of the interaction of propagating deflagration wave and autoignition site is performed at the interface, which is situated toward the right side of the domain.
4 Results and Discussion 4.1 0D Simulations A zero dimensional (0D) study is performed for the H2 -air autoigniton for various pressure and equivalence ratio conditions. The results obtained from the study are shown in Fig. 2. From this figure, three zones for autoignition of H2 can be observed. In Zone-I, ignition delay (t ig ) decreases with an increase in pressure, whereas for Zone-II, an increase in t ig is observed with an increase in pressure. And again, in
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AXISYMMETRIC BOUNDARY FLAME
CONSTANT PRESSURE BOUNDARY
BURNT GAS
T0 U0 = 0 m/s Yfuel
0
L
Fig. 1 Schematics diagram of initial conditions for simulation
Zone–III, t ig decreases with an increase in pressure. In case of reactivity of mixtures, the t ig shows monotonic behavior, i.e., lower t ig is observed for a highly reactive mixture. The non-monotonic nature of ignition delay with pressure is mainly due to the evolution of species during the delay period for various pressure conditions. In Fig. 2, the dotted black lines indicate the initial pressure conditions for 1D analysis. From this figure, it can be observed that, cases corresponding to initial pressure = 10 bar, cases fall in Zone-II where t ig increases with pressure, whereas the cases corresponding to initial pressure 20 and 40 bar are in zone-III. The species evolve during the delay period for a 0.1 equivalence ratio at various initial pressure cases are shown in Fig. 3. In this figure, the time is normalized by ignition delays of respective cases. From this figure, it can be observed that the formation and consumption of OH species are observed mainly at the time of autoignition when temperatures are higher in the domain. The higher OH formation
Equivalence Ratio 0.1
Ignition delay (ms)
3.20 1.60
0.2
0.80
0.3
0.40
0.4
0.20 0.10 0.05
1
2
4 8 16 Pressure (bar)
32
Fig. 2 Variation of ignition delay across pressure conditions and equivalence ratios. Dotted lines correspond to initial pressures (10, 20, and 40 bar) used for this analysis
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Fig. 3 Species evolve during the delay period of H2 -air autoignition a OH, b HO2 , and c H2 O2
is observed for lower pressure conditions than higher domain pressure. The species built up during autoignition was mainly observed for HO2 and H2 O2 species. Both species obtain higher peak values for 20 bar pressure conditions. And higher t ig also is observed for 20 bar cases. For initial pressure equal to 10 bar, more H2 O2 formation is observed compared to 40 bar case. Whereas for 40 bar cases, more evolution of HO2 is observed compared to 10 bar cases. Due to the difference in the formation pattern of species, the ignition delay shows non-monotonic behavior with an increase in pressure.
4.2 1D Simulation The results corresponding to the exergy analysis of H2 -air combustion are presented in this section. The t ig observed for the autoignition spot in a 1D domain is higher than the corresponding 0D case. In the present simulation, burn gases push the unburned mixture with deflagration wave’s speed. The gas velocities from propagating deflagration wave might be affecting the residence period of auto igniting mixture, resulting in delayed autoignition. The increase in t ig for 1D simulation is observed to decrease with an increase in the equivalence ratio. The evolution of the exergy loss rate (ELR) of propagating deflagration wave toward the autoignition site can be seen in Fig. 4. From this figure structure of ELR across the deflagration wave interface can be observed. ELR|reaction is observed to be a major contributor to total ELR. ELR|reaction is observed to be distributed all over the deflagration wave interface and peaked near the location of maximum reaction rate. ELR|conduction and ELR|diffusion peak is observed in preheat zone where higher temperature gradient and cracking fuel and oxidizer species are observed. Due to higher mass diffusivity of H2 broader profile for ELR|diffusion is observed compared to ELR|conduction. This figure shows a gradual increase in temperature of the autoignition site (right corner). As deflagration wave propagates toward the autoignition site, the temperature gradient is also observed to be reduced. The reduction in the gradient of
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Fig. 4 Evolution of ELR across deflagration wave with time, for t/t ig = 1.08 (dash dotted lines), t/t ig = 1.13 (dashed lines), and t/t ig = 1.15 (solid lines) for equivalence ratio = 0.2 and initial pressure = 20 bar
temperate results in reduction in ELR|conduction. For initial deflagration wave propagation, the peak of ELR|conduction is higher than ELR|diffusion. As deflagration wave propagates toward the autoigniton spot, the ELR|conduction peak reduces compared to ELR|diffusion. The ELR|reaction increases as deflagration wave propagates toward the autoignition front. The increase in ELR|reaction may be due to increased temperatures. As increase in temperature results in higher reaction rates. The temporal evolution of ELR across the domain with an increase in pressure for equivalence ratio = 0.1 can be seen in Fig. 5. The summation of all fractions of ELR is termed as net ELR. From this figure, the increase in HRR with an increase in pressure can be seen. The ELR shown in this figure is normalized by HRR. During propagation of deflagration wave, around 30% of total HRR is observed to be lost in the form of entropy. The ELR|reaction is observed to be the largest contributor to ELR. The contribution of ELR|diffusion and ELR|conduction is observed across all cases as deflagration wave propagates into autoignition spot. During the propagation of deflagration wave, with an increase in pressure, the ratio of ELR|reaction to HRR is observed to reduce. In Fig. 4, we have observed that the ELR|reaction increases as deflagration wave interacts with the autoignition spot, but the rate of increase in ELR is observed to be smaller than the rate of increase in HRR. The difference in rate of increase in ELR and HRR results in decreasing trends for ELR/HRR lines in this figure. With the increase in pressure, the rate of increase in ELR|reaction is observed to be comparable to the rate of HRR. The difference between ELR|conduction and ELR|diffusion is observed to reduce with increase in pressure. The temporal evolution of ELR across domains with an increase in equivalence ratio for initial pressure = 40 bar can be seen in Fig. 6. This figure shows the increase in HRR due to the increased reactivity of the chemical mixture. The difference between ELR|conduction and ELR|diffusion is also observed to reduce with an increase in equivalence ratio. The net ELR is also observed to increase with an increase in equivalence ratio. The slope of net ELR/HRR is also observed to become shallow with an increase in equivalence ratio. The rate of increase in ELR|reaction is nearly equal to the HRR.
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Fig. 5 Temporal evolution of total ELR across domain with time, for equivalence ratio = 0.1 and Initial Pressure a 10 bar, b 20 bar and 40 bar
Fig. 6 Temporal evolution of total ELR across domain with time, for initial pressure = 40 bar and equivalence ratio a 0.1, b 0.2, c 0.3, and d 0.4
From these results, it can be concluded that lower ELR is observed for lowpressure, low equivalence ratio cases. Indicate more exergy can be utilized in these operating points. This indicates that more useful work can be extracted from low pressure and low reactivity mixture.
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5 Conclusions A 1D DNS is performed propagating of deflagration wave in autoignitive H2 -air mixture configuration. The exergy analysis is performed on this DNS database. The effect of initial pressure and chemical composition is analyzed on the temporal evolution of exergy loss rate (ELR). The three major irreversibilities contributing to ELR are identified, namely thermal conduction (ELR|conduction), mass diffusion (ELR|diffusion), and chemical reactions (ELR|reaction). The results show that the ignition delay in the case of the 1D study is observed to be larger compared to homogeneous reactor (0D) solutions. The ELR|reaction is a major contributor to the ELR. During the interaction between propagating deflagration wave and the ELR|reaction is observed to increase, whereas ELR|conduction is observed to reduce. Temporal evolution of ELR shows a reduction in total ELR/HRR as deflagration wave interacts with autoignition spot. For lower equivalence ratio and pressure cases, the reduction in ELR|reaction/HRR is observed. With increase in equivalence ratio, ELR|reaction/ HRR remained constant.
Nomenclature DNS ELR HRR H2 t ig
Direct numerical simulations Exergy loss rate (GW/m3 ) Heat release rate (GW/m3 ) Hydrogen Ignition delay (s)
References 1. Dunbar WR, Lior N (1994) Sources of combustion irreversibility. Combust Sci Technol 103:41– 61 2. Datta A, Som SK (1999) Thermodynamic irreversibilities and second law analysis in a spray combustion process. Combust Sci Technol 142:29–54 3. Nishida K, Takagi T, Kinoshita S (2002) Analysis of entropy generation and exergy loss during combustion. Proc Combust Inst 29:869–874 4. Zhang J, Luong MB, Pérez FEH, Han D, Im HG, Huang Z (2021) Exergy loss characteristics of DME/air and ethanol/air mixtures with temperature and concentration fluctuations under HCCI/SCCI conditions: a DNS study. Combust Flame 226:334–346 5. Acampora L, Marra FS (2020) Second law thermodynamic analysis of syngas premixed flames. Int J Hydrogen Energy 45:12185–12202 6. Zhang J, Huang Z, Min K, Han D (2018) Dilution, thermal, and chemical effects of carbon dioxide on the exergy destruction in n-Heptane and Iso-octane autoignition processes: a numerical study. Energy Fuels 32:5559–5570
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7. Roy RN, Muto M, Kurose R (2017) Direct numerical simulation of ignition of syngas (H2 /CO) mixtures with temperature and composition stratifications relevant to HCCI conditions. Int J Hydrogen Energy 42:26152–26161 8. Babkovskaia N, Haugen NEL, Brandenburg A (2011) A high-order public domain code for direct numerical simulations of turbulent combustion. J Comput Phys 230:1–12 9. Li J, Zhao Z, Kazakov A, Dryer FL (2004) An updated comprehensive kinetic model of hydrogen combustion. Int J Chem Kinet 36:566–575 10. Poinsot TJ, Lelef SK (1992) Boundary conditions for direct simulations of compressible viscous flows. J Comput Phys 101:104–129
Numerical Investigation of Combustion Dynamics in a Multi-element Combustor Using Flamelet Approach Abhishek Sharma, Ashoke De, Varghese M. Thannickal, T. John Tharakan, and S. Sunil Kumar
Abstract This paper investigates combustion dynamics in a complex multi-injector element combustor using a flamelet approach in a large eddy simulation (LES) framework. The capability of computationally less expensive chemistry tabulation method to capture the interaction between unsteady heat release and acoustics is investigated. A non-adiabatic steady flamelet-based tabulated chemistry closure is invoked to simulate hydrogen–oxygen reactions in mixture fraction space. The model incorporates flow-induced non-equilibrium flame effects through scalar dissipation rate and the turbulence-chemistry interaction using a probability density function (PDF). A multi-element combustor dynamic study captures the first tangential mode close to 4000 Hz and corresponding high-frequency harmonics appropriately. Spectral analysis of the pressure variation displays similar frequency features in chamber and injector sections, suggesting the possibility of injector-chamber coupling. The coupling of the transverse pressure waves in the combustion chamber with the longitudinal pressure oscillations in the oxidizer post was probed as the reason for the pressure dynamics observed in the combustor. Keywords Combustion instability · Large eddy simulation · Steady flamelet · Multi-element · Injector-coupled
1 Introduction An understanding of combustion dynamics is essential for a liquid rocket engine program. The unsteady combustion dynamics in a rocket engine can damage hardware and hamper the design and realization cycle of the rocket engine. An important A. Sharma (B) · V. M. Thannickal · T. John Tharakan · S. Sunil Kumar Liquid Propulsion Systems Centre, ISRO, Thiruvananthapuram, Kerala 695547, India e-mail: [email protected] A. De Department of Aerospace Engineering, IIT Kanpur, Kanpur, Uttar Pradesh 208016, India Department of Sustainable Energy Engineering, IIT Kanpur, Kanpur, Uttar Pradesh 208016, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_19
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facet of combustion dynamics is the interaction of pressure oscillations with unsteady heat release in the chamber. Combustion instability originates from the coupling of unsteady heat release with acoustic eigenmodes of the combustion chamber and amplifies when heat release is in phase with the acoustic pressure, as first explained by Lord Rayleigh [1]. Studying combustion instability is imperative to refine the design process and safely operate a high-pressure rocket engine system. The combustion instability mechanism is more complex in a multi-element combustion chamber than in a single injector configuration [2–5]. The interaction among injectors and acoustic wave interaction with injectors, manifold, and feed systems is difficult to study through experiments [6]. High-fidelity reactive flow simulations can capture large-scale unsteady phenomena and provide deep insight into combustion instability mechanisms [7, 8]. The capability of LES to capture complex unsteady flows, heat release, and acoustic wave propagation makes it an ideal technique for combustion dynamic simulations. Generally, such high-fidelity simulations are performed with only a few step chemical mechanism in a finite rate chemistry framework. Recently, the flamelet [7–10]-based approach has been explored to simulate combustion dynamics. This approach incorporates detailed chemistry and the effect of turbulence-chemistry interaction. This approach is computationally less expensive and does not require the direct solution of multiple species transport equations. In the current work, a flamelet-based LES is performed on a multi-element combustor to capture combustion instability. Till now, no study is reported in the literature that highlights the combustion dynamics in actual rocket-scale combustor. This study focuses on capturing and understanding the mechanism of unsteady combustion dynamics at near ignition conditions of a typical rocket engine. This work briefly presents the computational framework and mechanism of combustion dynamics observed in the multi-element combustor. The future goal is to predict the combustion dynamic features at steady state operating conditions in similar multi-element rocket-scale combustors.
2 Literature Review and Objective Recent development in computational capabilities has led to extensive research on combustion dynamics based on high-fidelity numerical methods. Detailed numerical simulations are a robust approach to capturing and understanding the driving mechanisms of combustion instability. The self-excited unstable case of continuously variable resonant combustor (CVRC) [11] from Purdue University is widely studied using LES. Garby et al. [12] studied CVRC using AVBP LES code [2]. A dynamically thickened flame (DTF) model is used in this study with a two-step mechanism. In this approach, the flame is artificially thickened to be resolved on the LES mesh, and the sub-grid reaction rate is recovered through an efficiency function. Flame stabilization and evolution of triple-flame structure were identified as the source of self-excited instability. Harvazinski [13] explained combustion dynamics in CVRC
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using GEMS code with a hybrid detached eddy simulation (DES) approach. A singlestep methane-oxygen reaction was used to represent chemistry. Sardeshmukh et al. [14] elaborated in detail the role of chemistry in predicting combustion dynamics in CVRC. The case with detailed chemistry captured higher oscillation amplitude than a single-step reaction simulation. The recent work by Harvazinski et al. [10], using the flamelet model for CVRC, showed that both finite rate and flamelet computations provide similar results. Pant et al. [9] used flamelet methodology to examine the role of transient flame dynamics/ignition delay as a mechanism of self-excited instability in CVRC. A multi-element combustor, BKD [6], developed by the German Aerospace Center (DLR), was studied numerically by Urbano et al. [2]. Self-excited combustion instabilities occur due to frequency coupling/interaction between the injector’s chamber transverse and longitudinal modes. LES was employed with a four-species H2 –O2 reacting mechanism in the flamelet framework. Philo et al. [3] also investigated injector-coupled instabilities in an experimental multi-element rig at Purdue University. Zhan et al. [4] investigated the combustion instability in a 10injector combustor using a flamelet approach. Results highlighted higher amplitude prediction in the flamelet case compared to a single-step reaction. The resonance in injectors due to longitudinal mode oscillations in the chamber was presented briefly. Guo et al. [5] analysed a multi-element rectangular combustor with a hybrid RANS/ LES model using a flamelet approach. The self-sustained instability mechanism was the coupling of longitudinal mode oscillations in the GOX post with the transverse pressure oscillations in the combustion chamber. High-fidelity LES provides a robust platform to study complex dynamic features in a multi-element chamber, which is extremely difficult to study through experimentation. This work aims to study the complex process of combustion dynamics in a multi-element rocket engine at a specific operating condition. The complex process of multi-element flame interaction, flame-wall interaction, and interaction of chamber acoustics with injectors is studied. A three-dimensional LES is performed with a steady laminar flamelet combustion model. A suitable spatial and temporal resolution is used to capture the coupling of unsteady heat release and acoustic waves. The results are presented in peak-to-peak pressure amplitude and power spectrum density (PSD) variation at different locations over the combustor.
3 Formulation and Methodology This section presents the details of the numerical framework used in this study and the multi-element computational domain with input and boundary conditions used in LES.
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3.1 LES Framework The coupling of heat release and acoustic coupling in combustion dynamics hinges on the resolution of turbulent flow. An LES framework is employed in this study which can appropriately resolve turbulence and combustion scales. In this study, the best-known practices in resolving the LES turbulent and chemical scales are used as in study on the simulation of self-excited instability in CVRC [8]. Appropriate mesh resolution captures large-scale eddies, while sub-grid scales are modelled with the dynamic version of the Smagorinsky-Lilly model by Germano et al. [4]. Below is a brief overview of the LES model used in this study. Favre averaged continuity, momentum, energy, and other governing equations are given as: Filtered Continuity: ∂ ∂ρ + (ρ u˜ i ) = 0 ∂t ∂ xi
(1)
∂τi j ∂ ∂ ∂p ∂ σ~ ρ u˜ i u˜ j = − (ρ u˜ i ) + ij − ∂t ∂x j ∂x j ∂ xi ∂x j
(2)
Filtered Momentum:
where σi j is the stress tensor due to molecular viscosity, and τi j is sub-grid stress tensor. Energy/Enthalpy Equation: ∂ρu i h s ∂ρh s ∂p ∂p ∂ ∂T ∂ + − uj ρ ui hs − ui hs − − λ =− ∂t ∂ xi ∂t ∂ xi ∂ xi ∂ xi ∂x j
(3)
3.2 Turbulent Combustion—Flamelet Approach In this study, fuel and oxidizer enter the reaction zone in distinct streams, and a non-premixed combustion modelling formulation is employed. The model uses a conserved scalar approach, assuming unity Lewis number. Literature [7] highlights the utility of the flamelet approach when turbulent time and spatial scales are bigger than chemical scales. The flamelet [15] approach incorporates realistic chemical kinetic effects, including non-equilibrium due to aerodynamic straining of the flame by turbulence. The chemistry is preprocessed, solving the flamelet equations, and a look-up table is generated for mean quantities related to the conserved scalar mixture fraction. In this study of H2 –O2 reactions, a non-adiabatic version of the steady diffusion flamelet model is employed. Two transport equations for the mean mixture
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fraction and its variance are solved along with basic governing equations representing non-premixed combustion. The Favre-averaged transport equation for the mixture fraction is given by:
∂ μeff ρ f + ∇ . ρ→ν f = ∇ . ∇ f + Sm ∂t σt
(4)
where Sm is only due to the mass transfer into the gas phase from liquid droplets. μeff is the effective viscosity, composed of a laminar (μl ) and a turbulent contribution (μt ). In this framework, the transport equation for mixture fraction variance is not solved; instead, it is modelled by an algebraic equation given as: | |2 f '2 = Cvar L 2s |∇ f |
(5)
The constant Cvar is computed dynamically based on the procedure of Germano [16] for the dynamic SGL model. A validated H2 –O2 Chemkin mechanism was used to generate non-adiabatic flamelets. The steady flamelet model incorporates turbulence-chemistry interaction with a presumed PDF. The β-PDF function is used in this study, which is computed from the mean and variance of the mixture fraction at each point in the flow field. The mean scalars can be related to instantaneous scalars and expressed as: ¨ φi =
ϕi f, χ , H P( f )P(χ )d f dχ
(6)
where P( f ) and P(χ ) denote the probability density function decomposed from the joint PDF P( f, χst ). The flamelet profiles are convoluted with the assumed-shape PDF, and the mean values for species mass fraction and temperature are tabulated for look-up. In this study, the PDF table has an extra dimension of mean enthalpy H to consider the non-adiabatic steady diffusion flamelets. It is assumed that the heat loss or gain by the system only has a negligible effect on species mass fraction. The non adiabatic PDFtable generated in this study can be expressed as ϕ f , f '2 , H , χst .
4 Multi-element Computational Model This section details the computational domain and input conditions used for LES. A multi-injector element domain truncated at the throat location is utilized for this study. The computational domain represents a typical rocket engine operating on hydrogen–oxygen propellants. Multiple injectors are utilized in this configuration for appropriate mixing and higher combustion efficiency. The propellants come into the chamber as non-premixed separate streams. Propellants exhibit rapid turbulent mixing at the injector exit region. The injector elements are bi-directional swirl
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coaxial with oxygen stream at the centre and hydrogen flow at the periphery. The bidirectional swirl creates overlapping flow cones, which lead to rapid mixing along the axial direction. In this study, LES is performed at flow conditions close to the ignition condition of the engine. This specific condition is chosen to understand the combustion dynamics in the transient phase of engine operation. The ignition flow conditions reveal the thermodynamic condition of oxygen and hydrogen as an ideal gas. The identified condition allows analysing complex multi-element domains in reactive flow LES framework without invoking the stiff real gas thermodynamics. Figure 1 displays the computational domain of multiple swirl coaxial injector elements. A close-up view of injector element is displayed in the model, with colour marking of fuel and oxidizer entry. Hydrogen enters the injector element in the axial direction and passes through helical vanes to acquire swirl motion, whereas tangential entry is provided to oxygen. Mass flow inlet boundary condition is imposed at hydrogen and oxygen injector inlets. The mass flow rate corresponding to the ignition condition is used in this simulation. The overall oxidizer to fuel mixture ratio is maintained close to 2.6. Hydrogen purge through an igniter port is also considered in this study. Walls are treated as an adiabatic and no-slip condition imposed. The domain is truncated at the throat, automatically providing acoustically fully reflective boundary conditions due to the choked throat condition. The hydrogen injection condition is maintained close to 18 bar and 170 K, whereas oxygen injection parameters are 15 bar and 280 K. The corresponding injection density is 2.5 and 20 kg/m3 for hydrogen and oxygen, respectively. A significant difference in sound speed is noticed in hydrogen (987.25 m/s) and oxygen (323.95 m/s) streams, which can impact the interaction of chamber acoustic waves with upstream injectors. An appropriately refined mesh with a higher number of cells close to the injector outlet region is generated with an overall size of 19.3 million cells. The best quality mesh, which can resolve more than 80% of turbulent scales, is generated, which can be executed on available compute resources. The LES computational methodology developed in our earlier work [8] is adopted for this case. All governing equations are implicitly filtered by the finite-volume methodology of ANSYS Fluent [16]. Spatial discretization is carried out using second-order bounded schemes, while time integration is carried out with a bounded second-order implicit method. An appropriate Fig. 1 Computational domain-multi-element combustor
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temporal resolution was used in this study to retrieve unsteady combustion dynamics without significant numerical dissipation. LES was performed for sufficient acoustic cycles to ensure initial condition independent results and to collect sufficient samples of unsteady pressure statistics for spectral analysis. The perturbed RANS data is used as an initial turbulent condition for the LES. The adequacy of the mesh was checked by the variation of turbulent kinetic energy (TKE), which displayed a slope of − 5/3, indicating that the mesh to distinguish between the integral and Kolmogorov length scales. The grid resolution calculated by the ratio of resolved TKE to total TKE also displayed an overall value above 0.8, which can resolve more than 80% of turbulent scales. The acoustic CFL number is based on the flame region’s acoustic velocity and mesh size. It is necessary to keep a lower acoustic CFL (ACFL) value to capture pressure oscillations in LES and minimize wave dissipation. The earlier CVRC study [8] indicates that an ACFL value below 6 is reasonable to capture instability and should be maintained for this case also. In this study, an appropriate ACFL and hence time step size is deduced considering the overall computational time required to capture the unsteady dynamics originating in this compute intensive multi-element combustion chamber.
5 Results and Discussion 5.1 Instantaneous Flow Field LES is conducted for more than 44 acoustic cycles to capture unsteady physics in the chamber and the injector section. Instantaneous LES contours are discussed initially to find the flow and flame features in the combustor. This is followed by pressure fluctuations and evolution of dominant frequency modes in the combustor. Significant simulation features are highlighted by the instantaneous contour plots at a time of 0.04 s. Figure 2 displays the instantaneous flame temperature contour at the cut plane. It displays the high-temperature turbulent eddies breaking from the injector outlet, where the flame is anchored at the oxygen post location. The sound speed variation in the duct controls the propagation of pressure waves. Figure 3 displays the variation of sound speed from the injectors to the chamber region. A homogenous sound speed of 1800 m/s is visible in the flame region, whereas 300 m/s is noticed in the oxidizer path of injectors. Figure 4 displays the hydrogen mass fraction at the chamber axial plane. It displays a sudden reduction in mass fraction as hydrogen enters the chamber. The sudden reduction is attributed to an intense turbulent diffusion and reaction of hydrogen. Figure 5 displays the instantaneous O2 mass fraction at the cut plane with oxygen emanating from the centre tube. The concentrated oxygen stream is consumed farther from the faceplate with a potential core of 3 times oxygen post diameter. The oxygen eddies diffuse completely in the cylindrical section of the chamber, indicating complete combustion close to the faceplate.
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Fig. 2 Flame temperature, K
Fig. 3 Sound speed, m/s
The combustion features are presented as the instantaneous concentration of H2 O and OH at the cut plane. Figure 6 displays the H2 O mass fraction. It displays the characteristics of classical diffusion flame with higher product concentration in the stoichiometric shear layer location. The strong swirl component of both fuel and oxidizer leads to rapid mixing of the shear layer region. OH, concentration is regarded as the marker of heat release. Figure 7 shows the OH mass fraction at the axial plane, which displays a higher mass fraction in the shear layer and immediately downstream of injectors. The OH contour also shows that the flame anchors in the recess region of injectors. The injector recess is considered to aid in mixing and flame stability and is included in the current combustion dynamic analysis. The combustion dynamics is assessed through the evolution of fluctuating pressure waves in the combustor. The temporal evolution of pressure can reveal the onset of any
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Fig. 4 H2 mass fraction
Fig. 5 O2 mass fraction
dynamic activity in the combustor. The validation study [8] demonstrated the onset of longitudinal pressure fluctuations in a long combustor. Figure 8 displays pressure fluctuations in the multi-element combustor. The pressure fluctuation exhibits the transverse movement of a pressure wave with high pressure close to the left wall at one instant and at right wall at another time duration. The dominant features of this pressure wave are extracted through spectral analysis of pressure probe data and are presented in the next section.
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Fig. 6 H2 O mass fraction
Fig. 7 OH mass fraction
5.2 Spectral Analysis and Mechanism The evolution of pressure waves is analysed through the pressure time history in the combustor, aiming to capture dominant frequencies in LES. The time histories of pressure are collected at probes placed at different locations at chamber circumference and in the oxygen path of the injector. The pressure data in the injector region is collected to reveal the mechanism of injector associated combustion dynamics in the combustor. The total simulation time for LES is 12 ms, sufficient to capture 44 acoustic cycles of the first tangential frequency of the chamber. Absolute pressure is collected at a sampling frequency of 1 MHz. Figure 9 displays the pressure probe locations at the chamber wall and in the injector region of the oxygen post. Figure 10a, b shows raw absolute pressure time histories in the chamber.
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Fig. 8 Pressure wave evolution in chamber
Figure 10a displays the peak-to-peak pressure variation above the mean pressure of 8.2 bar at locations Ch1 and Ch2 and the centre point. It presents higher fluctuating pressure at circumferentially opposite near-wall points and lower amplitude at the centre. Locations Ch3 and Ch4 also display similar trends as of 1, 2, as shown in Fig. 10b. A closer view of combustor probe data reveals the acoustic wave dynamics through pressure time history at diametrically opposite points. Figure 11 displays the pressure variation for channel pair 1, 2 and channel pair 3, 4, respectively. There is no phase change observed in the pressure trace of axially located channels 1, 3 and 2, 4, whereas a 180° out-of-phase pressure variation is seen between channel 1 and 2 and same with channel 3 and 4, respectively. This indicates the evolution of tangential standing wave in the combustor, as visualized in Fig. 9. Figure 12 presents pressure fluctuations at three injector probes. It displays a peak-to-peak pressure fluctuation of 0.9 bar in the oxygen path of injectors.
Fig. 9 Pressure probe location
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Fig. 10 a, b Pressure variations at chamber probes Fig. 11 Phase variation
Before spectral analysis of the accumulated LES data, an analytical estimate on acoustic frequencies ( f ) is done with: / f m,n.q
c = 2π
2 βm,n q 2π 2 + Rc2 L 2c
(9)
where c denotes the speed of sound in the chamber, Rc and L c represent the radius and length of the combustor, βm,n are roots of the Bessel function, and m, n, and q are the mode numbers. The theoretical dominant frequencies of the combustor are calculated as the first tangential (1 T) frequency of 3865 Hz, first longitudinal mode of 3982 Hz, and first radial mode of 8044 Hz, respectively. Figure 13 displays the FFT of the LES data. A frequency-power spectral density (Pa2 /Hz) plot of near-wall
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Fig. 12 Injector probes
combustor probes is presented. Figure 13a displays 4 kHz as the first dominant mode, followed by 6, 8, 10 kHz as the frequency harmonics. The PSD plot of channels 3 and 4 also presents a similar trend, as shown in Fig. 13b. The out-of-phase pressure probe data and frequency captured in FFT of LES data indicate the dominant modes as 1Tand its harmonics. The LES mode values are higher than the analytical values, as the simulation is conducted at no heat loss condition. Such a difference in resonant frequency was also seen in the CVRC validation study [8] conducted earlier. The effect of combustor dynamics on injectors is probed briefly. Pressure trace analysis and FFT of probes placed in the injector section are performed to understand acoustic wave movement between the chamber and upstream injectors. Figure 14a displays pressure traces at the inlet and exit of the injector. It shows out-of-phase pressure variation at inlet and exit, indicating the presence of a longitudinal wave
Fig. 13 FFT-Chamber, a 1, 2 and b 3, 4
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in the oxygen path. The oxygen path of injectors will exhibit longitudinal modes and behave as a quarter-wavelength tube. They will also exhibit transverse modes, although transverse frequencies are invariably very high for small diameter oxygen tube. Figure 14b presents a PSD plot for injector points, which displays a discrete tone of 4 kHz and subsequent harmonics in the injector region. The injector entry can behave as a closed (choked) end and the injector exit as a partially open boundary. The theoretical first longitudinal mode (1 L) frequency at an oxygen sound speed of 323 m/s in open-close boundary condition is 2900 Hz. The realistic acoustic condition in chamber can be close to partially open-close condition which provides 1 L frequency of 3750 Hz. A close match can be observed between analytical and LES calculated frequency. The FFT trend indicates the same frequency features in the chamber and injector section, with a 4 kHz frequency trace visible in both injector (oxygen post) and chamber near-wall probes. The above analysis demonstrates the mutual influence of chamber and injector acoustics. The same frequency features in both combustor and injector region are observed because of frequency coupled system. It suggests the possibility of acoustic coupling between the combustor’s transverse mode and the oxygen post’s longitudinal mode as the frequencies of the modes are quite similar. The oxygen path acts as a source of fluctuations (resonator) and drives the instability as seen in the experimental BKD combustor [2, 6]. A similar coupling phenomenon is recently reported in Refs. [3–5]. The frequency coupling of longitudinal mode oscillations in the oxygen post with the transverse mode of the combustor can periodically impede the oxygen flow leading to mass flow rate oscillations, further leading to unsteady heat release and resultant pressure fluctuations in the chamber. The frequency match-up can eventually control the combustion dynamic process in the combustor.
Fig. 14 Pressure trace (a) and FFT injector (b)
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6 Conclusion A complex LES of a multi-element combustor was performed to capture combustion dynamic features near ignition conditions. The reactive flow LES captured instantaneous flow and flame features in a multi-element rocket-scale combustor accurately. A transverse pressure wave movement was captured appropriately indicating the complex interplay of turbulence, reactions, and acoustics. The spectral analysis showed the dynamics in the combustor and injector region. The LES reproduced dominant frequencies in chamber and injector region in close match with the analytical values. Pressure oscillations corresponding to the 1 T mode of chamber and 1 L mode of injectors are identified in the spectral analysis. The PSD spectrum displayed the same frequencies in the combustor and injector region indicating the possibility of combustion dynamics led by frequency coupling. Further analysis to reveal the impact of coupling between the 1 T mode of the combustor with the 1 L mode of the injector post, injector recess, injection temperature, and pressure will be the scope of future studies. The underlying factors impacting the frequency and amplitude prediction in this complex framework are being studied and will be further refined as understanding of this complex topic evolves. Acknowledgements The technical help provided by ANSYS, India team, to conduct this study is kindly acknowledged.
Nomenclature ∼ ρ u p t x T i, j, k
Favre average (–) Density (kg/m3 ) Velocity (m/s) Pressure (Pa) Time (s) Dimension (m) Temperature (K) Index (–)
References 1. Rayleigh J (1878) The explanation of certain acoustical phenomena. Nature 18:319–321 2. Urbano A, Selle L, Staffelbach G, Benedicte C, Schmitt T, Ducruix S, Candel S (2016) Exploration of combustion instability triggering using large eddy simulation of a multiple injector liquid rocket engine. Combust Flame 169:129 3. Philo JJ, Gejji RM, Slabaugh CD (2020) Injector-coupled transverse instabilities in a multielement premixed combustor. Int J Spray Combust 12:1756827720932832
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Experimental Investigations on Emissions and Performance of Spark Ignition Engine Fuelled with Butanol–Pentane–Gasoline Blends Parag P. Mangave, Vishal V. Patil, Nilesh D. Pawar, and Ranjit S. Patil
Abstract Current energy demand has led to an increase in fossil fuel consumption. Sources of fuels have been depleting over decades. Hence, there is a need to find alternatives to gasoline and diesel in the automotive industry. Alcohols are promising oxygenates and octane boosters for gasoline. The current investigation deals with blending butanol in 20% volume along with 10% pentane to gasoline in spark ignition engine in variation with spark timing from 15°, 18°, 21°, 24°, 27°, 30° bTDC. The performance and emission characteristics of fuel have been compared with base gasoline with 1800 rpm engine speed. Here, the results show a decrease in carbon monoxide emissions and a decrease in oxides of nitrogen emissions due to the addition of butanol, brake-specific fuel consumption increases for a butanol– pentane–gasoline blend. The optimum performance of the engine is at a spark timing of 24° bTDC. Keywords Gasoline · Butanol · Emissions · Brake thermal efficiency
1 Introduction Energy security and emissions are the major challenges the automobile industry encounters in the twenty-first century. The current demand for fossil fuels such as gasoline and diesel is continuously growing to fulfil the needs in the agriculture and automobile sector. As a result, the limited sources of fossil fuels have been depleting over the years [1]. Moreover, the combustion of fossil fuels in internal combustion engines produces undesirable emissions, including hydrocarbons (HC), carbon P. P. Mangave · N. D. Pawar (B) · R. S. Patil Department of Mechanical Engineering, Birla Institute of Technology and Science Pilani, K. K. Birla Goa Campus, Zuarinagar, Goa, India e-mail: [email protected] V. V. Patil Department of Mechanical Engineering, Sharad Institute of Technology College of Engineering, Yadrav, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_20
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monoxide (CO), oxides of nitrogen (NOx ), and particulate matter, which contribute to climate change and environmental pollution [2]. Thus, strict regulations concerning fuel efficiency and emissions, for instance, Bharat Stage VI, are imposed on the new vehicles. Therefore, exploring sustainable and environment-friendly alternate fuels is necessary to reduce dependence on fossil fuels and emissions. Furthermore, their emissions are considerably lower than fossil fuels, which makes them environmentfriendly. On the other hand, recently, electric vehicles have gained much attention as a promising solution. However, several shortcomings of EVs, such as electricity generation from coal, the limited raw material to manufacture batteries, recycling of waste batteries, battery size, and energy density, limit the usage of EVs. Hence, biofuels are a tempting alternative to tackle global challenges [1]. Therefore, EVs and vehicles powered by IC engines can be utilized collectively to meet the power requirements. Several blend stocks or groups have been previously used by different researchers in spark ignition engines. Alcohols, esters, alkanes, alkenes, furans, and ethers have been previously tested in spark ignition engines [3]. Current gasoline consists of various chemical groups in it. Aromatics, ethers, and alkanes are found in different percentages in it [4]. Screening of these fuel blend stocks is done on the basis of boiling point, oxygen content, research octane number, lower heating value, density, and viscosity [5]. So selection of the alternatives based on availability, the abundance of resources, and properties of fuels would be a major consideration for testing fuels.
2 Literature Review and Objective Alcohols are chemical groups with properties similar to that of gasoline. Methanol, ethanol, propanol, and butanol are better oxygenates in spark ignition engines than gasoline [6]. Ethanol is used in India and Brazil on a wider basis, and up to 20% V of blending is permitted by the Indian government to blend with gasoline. Butanol is used in a variety of industrial applications like solvents, fertilizers, cosmetics, plasticizers, and gasoline blending [7]. Low volatility and higher flash points make butanol as fuel more feasible than gasoline [7]. To reduce the dependency on fossil fuels, new methods or eco-friendly resources of the production of butanol are necessary. One efficient method of converting biomass to biofuels. Corn, starch, sorghum, bagasse, sugar cane, and wheat can be fermented using clostridium species. These microorganisms are spore-forming anaerobes. Acidogenic phase results in the formation of butanol with desired properties as fuels in Spark ignition engines [7]. Blending 15% volume butanol with gasoline decreases the efficiency of the engine as compared to gasoline [8]. Among all the isomers of butanol, only n-butanol and iso-butanol are qualified blend stocks for gasoline. Ter-butanol and sec-butanol have very low octane ratings and boiling points. Hence cannot be used for blending with gasoline [8]. As the carbon number increases, there is an increase in the lower heating value of the fuel. The lower heating value of n-butanol is greater than ethanol, and the higher stoichiometric ratio of butanol is compared to ethanol. Hence butanol can be
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blended at a higher percentage than ethanol [9]. Butanol shows a low tendency of separation if it comes in contact with water [10]. The lower heating value of butanol is 33.1 MJ/kg, which is greater than other alcohols and almost nearer to gasoline. At 70% wide open throttle, position carbon monoxide emissions are 0.4% which is lesser than base gasoline [11]. Blending butanol by 30% volume with gasoline brake-specific fuel consumption increased by 20% [12]. Butanol is also blended in gasoline because of the higher research octane number i.e. 96 [13]. Fuel consumption of 10% butanol–gasoline blend is 1.23 L, for neat gasoline is 1 L to produce the same power output [14]. The stoichiometric air-to-fuel ratio is 11.2, almost near to gasoline [15]. Model gasoline contains an alkanes group in its composition. Alkanes like pentane and hexane are present in the higher composition. As per this composition, alkanes are hydrocarbons with higher lower heating values and hence can be used for blending gasoline. Current gasoline composition also contains a 10% volume of higher alkane in it. Hence, to increase the energy content of base butanol–gasoline blends, alkanes can be better alternatives than aromatics [5]. Pentane is an alkane group with a boiling point (36 °C) greater than room temperature; hence, it exists in a liquid state [16]. Pentane composition in gasoline varies from 10 to 40% in gasoline worldwide [17]. The current objectives of the investigation aim at determining the performance and emission characteristics at various spark timing for a spark ignition engine at 1800 rpm. Here addition of pentane to gasoline by 10% volume and further addition of 20% volume of butanol to resemble the current gasoline octane rating [18].
3 Methodology Commercially available gasoline was used as a base fuel for all the experimental investigations. The selected test fuels used were purchased from a local commercially available chemical laboratory with 98% of purity. The blends were prepared on a volumetric basis in a measuring cylinder. These blends are stirred mechanically with the help of a glass stirrer rod. These blends are observed for 24 h in a transparent glass cylinder to ensure no separation. Butanol was added by 20% to gasoline to form a butanol–gasoline blend and was denoted by B20. Pentane was added by 10% volume to base gasoline, and this base fuel was then blended with 20% volume of butanol denoted by B20PG. The reason behind adding pentane group is its boiling point, calorific value, and current gasoline composition. As lower alkane groups are in gaseous phase, they cannot be used for blending in gasoline. Properties of fuel are identified as listed in Table 1. A schematic diagram of the experimental setup of the spark ignition engine is shown in Fig. 1. An open programmable electronic control unit is used to vary the different spark timings. Specification of the engine is listed in Table 2. An “Engine soft LV” software records the data for performance characteristics taken from each pressure and temperature sensor fitted at various locations to record various parameters. The engine was run at a speed of 1800 rpm with 14 kg full load condition.
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Table 1 Properties of fuels Gasoline
Pentane [16]
n-butanol
Chemical formula
–
C5 H10
C4 H10 O
Research octane number (RON)
91
61
96
Oxygen %
–
–
21.5
Latent heat of vaporization 25 °C (kJ/kg)
380–500 [15]
–
716 [15]
Boiling point
(o C)
225
36
117 [19]
Lower heating value (kJ/kg)
44,000
45,400
33,100 [19]
Density (g/cm3 )
0.746 [19]
0.630
0.810 [19]
For accuracy of observations, the data was not recorded till a steady state was not achieved. To ensure the repeatability of observation, each recorded data is repeated for five cycles. Average readings of these observations are taken into consideration for calculations. The accuracy of various parameters measured is listed in Table 3. An AVL 444 gas analyser is used to record exhaust emissions. Initially, the probe of the analyser is inserted in the tailpipe, and observations are not recorded till stable data is found.
Fig. 1 Schematic diagram of spark ignition engine
Experimental Investigations on Emissions and Performance of Spark … Table 2 Specifications of spark ignition engine
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Details
Specification
Type
Single cylinder, four-stroke, gasoline
Bore × stroke
87.5 mm × 110 mm
Capacity
661 cc
Rated power
4.5 kW@1500 rpm
Speed
1800 rpm
Compression ratio
10:1
Dynamometer
Eddy current, water-cooled
Table 3 Accuracy and uncertainty of various SI engine parameters Parameters
Uncertainties (%)
Accuracy
Load sensor
Load
± 1.39
± 0.1 kg
Speed measuring unit
Engine speed
± 0.06
± 1 rpm
Crank angle encoder
Crank angle
± 0.14
± 1°
Pressure transmitter
Cylinder pressure
± 0.95
± 0.25 bar
Flow transmitter
Fuel flow
± 1.01
± 0.01 kg/h
Air flow transmitter
Air flow
± 0.09
± 0.02 kg/h
Temperature transmitter
Cylinder gas temperature
± 0.29
±2K
4 Results and Discussion Performance characteristics such as brake thermal efficiency, brake-specific fuel consumption, and emission characteristics such as carbon monoxide emissions, nitrogen oxide emissions, and unburnt hydrocarbons were calculated. They are discussed in detail below.
4.1 Brake Thermal Efficiency (BTE) Variation of brake thermal efficiency against various spark timing is shown in Fig. 2. For regular gasoline 24° bTDC is optimum spark timing, where brake thermal efficiency is found to be 22.55%. As the spark timing is retarded a general decreasing trend is found for gasoline. A similar trend is found for the B20 butanol–gasoline blend, at 24° bTDC maximum brake thermal efficiency of 22.1%. A decrease in brake thermal efficiency of B20 butanol–gasoline blend is due to a decrease in the calorific value of butanol. For B20PG blends, brake thermal efficiency, i.e. 22.2% is more than B20 butanol–gasoline blends. Adding pentane helps increase the calorific value of the gasoline butanol blend. The brake thermal efficiency for gasoline is 21.44%, 22.05%, 21.95%, 20.76%, 19.8% at 15°, 18°, 21°, 27°, 30° bTDC, respectively. There
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Fig. 2 Variation of brake thermal efficiency with spark timing
is a decrease in brake thermal efficiency for various spark timing as the maximum brake power generated is less than other spark timings. This decrease in BTE is due to the time taken by the fuel to atomize or attain its auto-ignition temperature.
4.2 Brake-Specific Energy Consumption (BSEC) Variation of brake-specific energy consumption of various spark timings is shown in Fig. 3. The lower heating value of butanol plays an important role. The lowest brake-specific fuel consumption is for 24° bTDC for all blends. Here, the calorific value of the blend indicated that less fuel is required to generate the same brake power at existing engine conditions. Above various spark timing, fuel and oxidizer mix and react to produce the power output. But advancing the spark timing leaves most of the fuel particles unburnt and thereby consumes more fuel to generate brake power. For gasoline BSEC is 14.52, 14.63 MJ/kWh for B20 butanol–gasoline, and 14.59 MJ/kWh for B20PG blend.
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Fig. 3 Variation of brake-specific energy consumption with spark timing
4.3 Carbon Monoxide Emissions Variations of carbon monoxide emission versus spark timing from the exhaust are shown in Fig. 4. Here, carbon monoxide in the exhaust denotes incomplete combustion and deficit oxygen molecules available to react with fuel molecules. At existing spark timing of 24° bTDC gives minimum carbon monoxide emissions, i.e. 2.13 g/ kWh for gasoline, 2 g/kWh for B20 butanol–gasoline, and 2.05 g/kWh for B20PG blend. Adding pentane increases carbon monoxide emissions as only hydrocarbon molecules are added, and no excess particles of oxygen are available to react and convert the same into carbon dioxide molecules. As a further change in spark timing from 15° to 30° except 24° bTDC additional fuel particles remain in the combustion chamber and hence lead to an increase in carbon monoxide emissions.
4.4 Oxides of Nitrogen Emissions Variation in oxides of nitrogen emission from the exhaust is shown in Fig. 5. A decrease in nitrogen oxide emissions is obtained in all butanol blends. This is due to the charge cooling effect of the addition of butanol to gasoline. The heat of vaporization of butanol helps to decrease the in-cylinder temperature of butanol–gasoline blends. A similar trend is observed for the B20 pentane–gasoline blend. For spark timing at 24° bTDC lesser oxides of nitrogen are found as sufficient oxygen is available to react with fuel particles, decreasing the in-cylinder temperature. Variation in
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Fig. 4 Variation of carbon monoxide emission with spark timing
spark timing before and after 24° bTDC leads to more fuel particles, as seen in more fuel consumption in Fig. 3. Hence, an increase in the cylinder pressure causes an increase in nitrogen oxide emissions. An increase in emissions by 22.01%, 19.95%, and 21.42% is seen if spark timing increases from 24° to 27° bTDC for gasoline, B20, and B20PG blends, respectively.
4.5 Unburnt Hydrocarbon Emissions Unburnt hydrocarbons versus spark timing variations are shown in Fig. 6. Here, it is clear from the figure that unburnt hydrocarbon decreases on the addition of butanol in gasoline by 20% as oxygen is available to react with fuel particles, thereby decreasing its content in the exhaust emissions. Lesser fuel consumption and decreased pressure in the combustion chamber have led to a decrease in unburnt hydrocarbon emissions. But as the spark timing is increased beyond 24° bTDC, fuel particles do not combust thus and adequately are clearly reflected in the exhaust. Also, as pentane has carbon and hydrogen atoms, this also leads to an increase in unburnt hydrocarbon emissions in exhaust in the case of B20PG than the B20 blend. 0.27 g/kWh, 0.22 g/kWh, and 0.25 g/kWh are emissions at 24° bTDC for gasoline, B20, and B20PG blends, respectively.
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Fig. 5 Variation of oxides of nitrogen with spark timing
Fig. 6 Variation of unburnt hydrocarbons with spark timing
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5 Conclusions The conclusions from the experiments are as follows: • Optimum engine performance occurs at the existing spark timing of 24° bTDC for all blends. • Change in spark timing leads to lesser performance, and incomplete combustion of fuel has led to an increase in emissions. • The addition of 20% V butanol to gasoline decreases the brake thermal efficiency by 2.036, 1.67% for the B20PG blend than gasoline. • Brake-specific fuel consumption increases by 3.03% for B20 and by 1.51% for the B20PG blend. • Carbon monoxide emissions decrease by 6.5% for B20, and 3.90% for B20PG blend as compared to gasoline. • As compared to gasoline oxides of nitrogen emissions decrease by 3.89% for B20 blend, 1.67% for B20PG blend. • Unburnt hydrocarbon emissions decreased by 22.72% for B20, and the addition of pentane to butanol and gasoline has decreased emissions by 9.09% compared to base fuel. The addition of pentane to butanol–gasoline has led to an increase in brake thermal efficiency decrease in brake-specific fuel consumption as compared to butanol–gasoline blends. Also, decrease in emissions from the exhaust as compared with gasoline for all spark timings.
Nomenclature bTDC BTE BSEC CO NOx HC
Before top dead centre (–) Brake thermal efficiency (–) Brake-specific energy consumption (–) Carbon monoxide (–) Oxides of nitrogen (–) Unburnt hydrocarbons (–)
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3. Badia JH, Ramírez E, Bringué R, Cunill F, Delgado J (2021) New octane booster molecules for modern gasoline composition. Energy Fuels 35(14):10949–10997. https://doi.org/10.1021/ acs.energyfuels.1c00912 4. Patil V, Singh P, Sonage S, Kumbhakarna N, Kumar S (2022) Experimental investigation to assess the efficacy of gasoline surrogates with engine testing. Fuel 324(PA):124493. https:// doi.org/10.1016/j.fuel.2022.124493 5. McCormick RL et al (2017) Selection criteria and screening of potential biomass-derived streams as fuel blendstocks for advanced spark-ignition engines. SAE Int J Fuels Lubr 10(2):442–460. https://doi.org/10.4271/2017-01-0868 6. Gökta¸s M, Kemal Balki M, Sayin C, Canakci M (2020) An evaluation of the use of alcohol fuels in SI engines in terms of performance, emission and combustion characteristics: a review. Fuel 286. https://doi.org/10.1016/j.fuel.2020.119425 7. Jin C, Yao M, Liu H, Lee CFF, Ji J (2011) Progress in the production and application of nbutanol as a biofuel. Renew Sustain Energy Rev 15(8):4080–4106. https://doi.org/10.1016/j. rser.2011.06.001 8. Cooney C et al (2009) Effects of blending gasoline with ethanol and butanol. In: Proceedings of the Spring technical conference of the ASME internal combustion engine division, pp 1–9 9. Regalbuto C, Pennisi M, Wigg B, Kyritsis D (2012) Experimental investigation of butanol isomer combustion in spark ignition engines. SAE Technical Papers. https://doi.org/10.4271/ 2012-01-1271 10. Chen Z, Yang F, Xue S, Wu Z, Liu J (2015) Impact of higher n-butanol addition on combustion and performance of GDI engine in stoichiometric combustion. Energy Convers Manag 106:385–392. https://doi.org/10.1016/j.enconman.2015.09.051 11. Huynh TT, Le MD, Duong DN (2019) Effects of butanol–gasoline blends on SI engine performance, fuel consumption, and emission characteristics at partial engine speeds. Int J Energy Environ Eng 10(4):483–492. https://doi.org/10.1007/s40095-019-0309-9 12. Farkade HS, Pathre AP (2012) Experimental investigation of methanol, ethanol and butanol blends with gasoline on SI engine. Int J Technol Adv Eng 2(4):205–215 13. Energy Efficiency (2019) Top ten blendstocks for turbocharged gasoline engines 14. Varol Y, Öner C, Öztop HF, Altun S¸ (2014) Comparison of methanol, ethanol, or n -butanol blending with unleaded gasoline on exhaust emissions of an SI engine. Energy Sources Part A Recover Util Environ Eff 36(9):938–948. https://doi.org/10.1080/15567036.2011.572141 15. Wallner T, Miers SA, McConnell S (2009) A comparison of ethanol and butanol as oxygenates using a direct-injection, spark-ignition engine. J Eng Gas Turbines Power 131(3):1–9. https:// doi.org/10.1115/1.3043810 16. Cheng S, Yang Y, Brear MJ, Kang D, Bohac S, Boehman AL (2017) Autoignition of pentane isomers in a spark-ignition engine. Proc Combust Inst 36(3):3499–3506. https://doi.org/10. 1016/j.proci.2016.08.042 17. Burri J, Crockett R, Hany R, Rentsch D (2004) Gasoline composition determined by 1H NMR spectroscopy. Fuel 83(2):187–193. https://doi.org/10.1016/S0016-2361(03)00261-8 18. Singh E, Badra J, Mehl M, Sarathy SM (2017) Chemical kinetic insights into the octane number and octane sensitivity of gasoline surrogate mixtures. Energy Fuels 31(2):1945–1960. https:// doi.org/10.1021/acs.energyfuels.6b02659 19. Elfasakhany A (2014) Experimental study on emissions and performance of an internal combustion engine fueled with gasoline and gasoline/n-butanol blends. Energy Convers Manag 88:277–283. https://doi.org/10.1016/j.enconman.2014.08.031
CFD Analysis of Afterburner with Convergent–Divergent Nozzle for Various Air–Fuel Ratios Gurrala Srinivasa Rao
Abstract Military jet aircraft fitted with afterburner gets additional thrust in extraordinary circumstances like combat. The jet aircraft must work in wide-ranging temperatures with different air–fuel ratios under varying circumstances. However, many researchers observed that jet aircraft operating with the lean air–fuel configuration is associated with instabilities, which is not under the scope of this paper. In the present paper, the afterburner, consisting of a convergent–divergent nozzle and essential components, is simulated for varying air–fuel ratios from 16 to 45 to check the effect of the increase in the fuel supply. Accordingly, the afterburner is modeled with liner, diffuser, V-gutter, fuel manifolds, and casing with a convergent–divergent nozzle. Computational fluid dynamics analysis is carried out with the help of Ansys Fluent® using SIMPLE algorithm, realizable k − ε turbulence model, energy equation, species transport, and discrete phase with finite-rate/eddy-dissipation model for combustion. The simulations were carried out for various air–fuel ratios 16, 19, 23, 30, and 45. Out of these different afterburner models, the afterburner with the minimum air–fuel ratio of 16 is found to attain the maximum velocity and maximum thrust. These results also match the experimental results of Useller et al. (Influence of combustion chamber length on afterburner performance. Lewis Flight Propuslion Laboratory, Cleveland, 1954, [1]). Keywords Afterburner · Air fuel ratio · V-gutter · Fuel manifolds · Combustion
1 Introduction The provision of an afterburner is the best option for the gas turbine to supply additional thrust during take-off, combat, maneuvers, and emergencies. The other way of providing this extra thrust is by using a bigger engine, which is associated with an additional weight penalty and increased specific fuel consumption. The use of a G. Srinivasa Rao (B) Faculty of Mechanical Engineering, Indian Naval Academy, Ezhimala, Kannur, Kerala 670310, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_21
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larger engine will also affect its maneuverability. The amount of oxygen in the exhaust gases coming out of the turbine is sufficient to assist the burning of an additional amount of fuel supplied in the afterburner. The combustion of this extra fuel in the afterburner will help increase thrust augmentation. The subsequent passage of these combustion products through the convergent–divergent nozzle will help improve the velocity of exhaust gases coming out of the afterburner to the supersonic speeds of jet aircraft. Due to the complex and very expensive experimental setups involved in testing the jet aircraft, not everyone can establish such facilities. However, few papers are available on afterburners in which the experimental and computational results were published. Some of the papers are discussed below.
2 Literature Review and Objective Useller et al. [1] carried out experiments, with erstwhile NACA, on different afterburner models with a convergent nozzle with various combustion chamber lengths and different air–fuel ratios. It was found that the afterburner with an equivalence ratio of 1 and combustion chamber length of 5 ft gives the best performance. Kampa [2] conducted experiments on jet aircraft afterburners to mitigate screech instabilities consisting of liners with different porosities 1.5, 2.5, and 4. He found appreciable attenuation of these screech instabilities with the liner porosity factor of 2.5. Unaune and Ganesan [3] carried out CFD analysis of reacting flows in an aero-engine afterburner. The variations of pressure, velocity, O2 , and CO2 contours were presented. The afterburners were also simulated for different air–fuel ratios of 30 and 46. They observed better performance for the afterburner model with an air–fuel ratio of 30. Gurrala and Shaija [4] carried out computational analysis of reacting flows in the afterburner. SIMPLE algorithm was used for computations. k − ε model was used for turbulence calculations, and kerosene (C12 H23 ) is selected as fuel and virtual injectors for fuel injection with the help of Ansys Fluent® software. The results were compared with the earlier computational and experimental values. Dowling [5] investigated a simple geometry of thermoacoustic oscillations in which heat is supplied to an acoustic resonator systematically to determine the importance of various flow parameters on the frequency of the oscillations. The coupling between the heat input and the unsteady flow affected the oscillation frequency. Eldredge and Dowling [6] investigated the effectiveness of a cylindrical perforated liner. They found that the liner can absorb a significant fraction of incoming energy and even prevent all the energy produced by an upstream source in specific frequency ranges from reflecting. Hughes and Dowling [7] examined a liner’s effect on a cylinder’s resonances. The effectiveness of the well-designed liner in suppressing the resonances at all the frequencies was established. Stow and Dowling [8] described a linear model for thermoacoustic oscillations in LPP combustors. It was not only found that entropy and vorticity waves have a significant effect even when the outlet is open but also found to influence the flow at regions of the area change. The resonant modes obtained numerically were also compared with the experimental results.
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Moeck et al. [9] proposed a bilateral coupling, where the heat release region is described by a finite-difference zero Mach number solver, for the low Mach number, long wavelength case. He found that the coupling of heat release to higher mode frequencies, in the case of shorter tubes, is less efficient and the energy gain through the Rayleigh-integral does not exceed the loss at the boundaries. Durox et al. [10] examined the application of the Rayleigh criterion in radiant burners for domestic or industrial processes, and the results are found satisfactory. Cuquel et al. [11] experimentally determined the flame transfer function (FTF), to analyze acoustic induced combustion instabilities. Gregory et al. [12] invented a method to control the thermoacoustic instabilities in a combustor. Oliver [13] invented a method and device for reducing combustor screech relating to a flame holder that generates a radial phase variation. George [14] invented a combustion instability reduction device, wherein a screech liner formed with holes includes a swirl device. A system and method of reducing noise caused by jet engines, by using the afterburner to heat the exhaust gas flow while simultaneously reducing the power to core engine, to reduce the pressure of exhaust gas in the nozzle area while holding the exhaust gas velocity constant by maintaining engine thrust while decreasing engine noise, was invented by Alfred [15]. A thermal afterburning system and a method to operate such a system, in which method or system of burner gas is fed in a conventional manner to a burner heating the combustion space of a combustion chamber, and exhaust air with pollutant load is fed to the combustion space of a combustion chamber, is invented by Katefidis [16]. Instabilities are common in afterburners that too particularly as they operate with lean air–fuel ratios. Useller et al. [1] also experimentally verified the afterburner, with convergent nozzle, for different air–fuel ratios and found that the afterburner with minimum air–fuel ratio or maximum equivalence ratio is producing maximum thrust. However, in this paper the afterburner, with convergent–divergent nozzle angle and with combustion chamber length of 5 ft, is computationally verified for the five different air–fuel ratios of 16, 19, 23, 30, and 46 to find out the afterburner, which produces maximum velocity or thrust.
3 Description of Afterburner Afterburner is located between the turbine and exhaust nozzle. The main combustor is placed between the compressor and turbine, and the exhaust from the turbine will enter the afterburner. The exhaust gases from the turbine still contain some oxygen which can help burn the fuel supplied through the fuel manifold of the afterburner. These products of combustion will pass through the convergent–divergent nozzle and attains the supersonic speed. Figure 1a shows the schematic diagram of the afterburner consisting of diffuser with struts, fuel manifolds, V-gutter, anti-screech liner, and casing including the convergent–divergent nozzle as explained by Gurrala
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Fig. 1 Cut section model of the afterburner
and Shaija [4]. Figure 1 shows the cut section model of the afterburner. However, 60° model of the afterburner is used for the computational fluid flow analysis, due to symmetry.
4 Governing Equations The equations used for solving the computational domain as per the Ansys Fluent® Theory Guide [17] are given below. Mass Conservation Equation The mass conservation equation [17] is as given below: ∂ρ + ∇ . (ρ v→) = Sm ∂t
(1)
The above equation is valid for compressible and incompressible flows. The source Sm is the mass added to the continuous phase from the dispersed second phase, and any user-defined sources, v→, is the velocity vector, and t is the time. Momentum Conservation Equation The momentum conservation equation [17] is given below: ∂ → g + F→ (ρ v→) + ∇ . (ρ v→v→) = −∇ p + ∇ . τ + ρ − ∂t
(2)
→ where p is the static pressure, τ is the stress tensor, ρ − g and F→ are the gravitational force and the external body forces (forces that arise from interaction with dispersed
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phase), respectively. F→ also contains other model-dependent source terms such as porous media and user-defined sources. The stress tensor [17] is defined by 2 T τ = μ ∇ v→ + ∇ v→ − ∇ . v→ I 3
(3)
where μ is the molecular viscosity, I is the unit tensor, and the second term on the right side is the effect of volume dilation. Energy Conservation Equation The energy conservation equation [17] can be written as ⎞ ⎛ ∂ h j j j + τ eff . v→ ⎠ + Sh (4) v (ρ E + p)) = −∇ . ⎝keff ∇T − (ρ E) + ∇ . (→ ∂t j where keff is the effective conductivity (K + K t ), K is the thermal conductivity, K t is the turbulent thermal conductivity, and j j is the diffusion flux of species J . The first three terms on the right-hand side of the equation represent energy transfer due to conduction, species diffusion, and viscous dissipation respectively. Sh includes heat of chemical reaction and any other volumetric heat resources, and τ eff is the deviatoric stress tensor which is obtained after substituting the effective viscosity (μeff ) in the stress tensor Eq. (3). Realizable k − ε Model The realizable k − ε contains the alternative formulation for the turbulent viscosity, and a modified transport equation for the dissipation rate ε has been derived from an exact equation for the transport of mean-square velocity fluctuation. The term realizable means that the model satisfies certain mathematical constraints on the Reynolds stresses, consistent with the physics of turbulent flows. In this model, the Boussinesq relationship and the eddy viscosity definitions have been combined to obtain the following expression for the normal Reynolds stress [17]. u2 =
∂U 2 k − 2νt 3 ∂x
(5)
Using Eq. (5), for νt ≡ μρt , one obtains the result that the normal stress, u 2 , which by definition is a positive quantity, becomes negative, that is “non-realizable”, when the strain is large enough to satisfy 1 k ∂U > ≈ 3.7 ε ∂x 3Cμ
(6)
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The realizable k − ε model was intended to address the deficiencies of traditional k − ε models by adopting a new eddy viscosity formula involving a variable Cμ and the new model equation for dissipation (ε) based on the dynamic equation of the mean-square vorticity fluctuation. Transport Equations for Realizable k − ε Model The applicable equations [17] are as given below: μt ∂k μ+ + G k + G b − ρε − Y M + Sk (7) σk ∂ x j μt ∂ε ∂ ε2 ∂ ∂ μ+ + ρC1 Sε − ρC2 ρεu j = √ (ρε) + ∂t ∂ xi ∂x j σε ∂ x j K + νε
∂ ∂ ∂ ρku j = (ρk) + ∂t ∂ xi ∂x j
ε ε2 + C1ε (G k + C3ε G b ) − C2ε ρ + Sε (8) k k
√ n where C1 = max 0.43, n+5 , n = S kε , S = 2Si j Si j where G k is the generation of turbulence kinetic energy due to the mean velocity gradients, G b is the generation of turbulence kinetic energy due to buoyancy, Y M is the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, and C1ε , C2ε , and C3ε are constants. σk and σε are the turbulent Prandtl numbers for k and ε. Sk and Sε are user-defined source terms. Turbulent (or eddy) 2 viscosity is computed as μt = ρCμ kε where Cμ is a constant.
C1ε = 1.44, C2ε = 1.92, Cμ = 0.09, σk = 1.0, σε = 1.3 and C3ε = tan h uv , where v is the component of flow velocity parallel to the gravitational vector, and u is the component of flow velocity perpendicular to the gravitational vector. Hence, C3ε will become 1 for buoyant shear layers for which the main flow direction is aligned with the direction of gravity and become 0 for buoyant shear layers that are perpendicular to the gravitational vector. Species Transport Equations Ansys Fluent® predicts the local mass fraction of each species, Yi , through the solution of a convection–diffusion equation for the ith species. The conservation then can be written as [17] ∂ − → (ρYi ) + ∇ . (ρ v→Yi ) = −∇ . Ji + Ri + Si ∂t
(9)
where Ri is the net rate of production of species I by chemical reaction, and Si is the rate of creation by addition from the dispersed phase plus any user-defined sources. Discrete Phase Model Ansys Fluent® predicts the trajectory of a discrete phase particle (or droplet or bubble) by integrating the force balance on the particle, which is written in a Lagrangian reference frame. This force balance equates the particle inertia with the forces acting
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on the particle [17]: → − g ρp − ρ d→ up + F→ = FD u→ − u→ p + dt ρp
(10)
where F→ is an additional acceleration (force/unit particle mass) term, FD u→ − u→ p is D Re the drag force per unit particle mass and FD = 18μC , where u→ is the fluid phase ρ p d 2p 24 velocity, u→ p is the particle velocity, μ is the molecular viscosity of the fluid, ρ is the density, C D is the drag coefficient, ρ p is the density of the particle, d p is the particle ρd |u→ −→ u| diameter, and Re is the relative Reynolds number which is defined as Re ≡ p μp .
5 Methodology A 60° full-scaled model of afterburner, with the components as shown in Fig. 1a, is created in SolidWorks with the nozzle angle 6° and area ratio (ratio of aperture area to the cross-sectional area of the afterburner) of 5.2. The extended domain was also created for imposing the proper boundary conditions and to obtain better results. Computational fluid flow analysis of results using Ansys Fluent® . This model was then imported to Fluent for computational analysis of fluid flow, and the results for the five models with air–fuel ratios of 16, 19, 23, 30, and 45, by changing the air–fuel ratio with varying the fuel, were obtained.
6 CFD Modeling The afterburner model consists of diffuser, struts, fuel manifolds, V-gutter, and liner are created in SolidWorks. An extended domain with diameter twice the outlet diameter and length thrice the outlet diameter is created to analyze this complex flow in the afterburner. The extended domain is created to actually analyze the afterburner with perfect outlet boundary conditions. A 60° sector of the afterburner is taken for study which captures 03 radial Vgutters (2 outer + 1 inner) and 01 Strut. Unstructured grid with tetrahedral cells is used for discretizing the domain. A total of 7,135,549 tetrahedral elements with 1,410,033 nodes are considered for the computational analysis. Fine mesh is adopted near the flame stabilizer, where large gradients in flow can be expected and course mesh where not much variation in the flow is expected. The injection points are defined virtually. A total of 24 spray bars are considered each consisting of 9 injection points for the full model. Hence, for a 60° sector model, a total of 4 spray bars and a total 9 × 4 = 36 injection points are created. The spray bar was considered as line geometry with 9 equidistant points on each spray bar. Each injection point was chosen as a solid-cone injector with fuel as C12 H23 . The
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diameter of injector is taken as 1e−06 m, and mass flow rate was calculated and set as per the air–fuel ratio to be analyzed. The boundary conditions used are inlet, outlet, wall, and periodic as explained by Gurrala and Shaija [4]. Mass and pressure inlet and pressure outlet boundary conditions are chosen for inlet and outlet boundary conditions. Wall boundary conditions are chosen for Vgutter, fuel manifold, diffuser, inner wall, and top wall of bypass flow. Periodic/cyclic boundary conditions are assigned for the side faces. The boundary conditions for main inlet are specified with mass flow of 11.67 kg/s, pressure 353 kPa, temperature as 873 K, whereas for bypass inlet with mass flow rate of 2.33 kg/s, pressure 396 kPa and temperature as 373 K. The holes provided in the liner for anti-screech and cooling rings have been applied with interior boundary conditions. The numerical calculations are performed using SIMPLE algorithm, and k − ε model has been used for turbulence. Energy equation is used for temperature calculations. Species transport with eddy dissipation model is used to model multiple simultaneous chemical reactions. Discrete phase model is selected for the computations. Finite-rate/eddy-dissipation model is applied for computation of chemical reactions occurring due to combustion of fuel, as explained by Gurrala and Shaija [4]. Grid independence was also carried out by creating one afterburner model with 3,000,000 elements, 7,000,000 elements, and 12,000,000 elements for one air– fuel ratio. Finally, the model with 7,000,000 elements is selected for study of all these models with different air–fuel ratios as not much deviation in the results were observed between the model with 7,000,000 elements and 12,000,000 elements.
7 Results and Discussion Five afterburner models with different air–fuel ratios of 16, 19, 23, 30, and 45 were computationally analyzed for the attainment of maximum velocity and thrust. The results are presented below. The variation of velocity for the five models, along the axis (centerline) of afterburner, is shown in Fig. 2. Maximum velocity above 1400 m/s at the exit of the afterburner in the case of air–fuel ratio of 16 is observed. This indicates the maximum amount of thrust production due to the additional fuel supplied. The results also match with Fig. 4 of Useller et al. [1]. The variation of temperature along the axis of afterburner for different air–fuel ratios along the length of afterburner is shown in Fig. 3. Maximum temperature is observed in the case of afterburner with air–fuel ratio of 16. The temperature variation matches with Fig. 5 of Useller et al. [1] as increase in combustion temperature observed with increase in fuel injection or equivalence ratio. The temperature contour of afterburner with air–fuel ratio of 16 is shown in Fig. 4.
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Fig. 2 Variation of velocity along the axis
Fig. 3 Variation of temperature along the axis
The variation of pressure along the axis of afterburner for different air–fuel ratios along the length of afterburner is shown in Fig. 5. Minimum pressure at the exit of the afterburner is observed for the afterburner with air–fuel ratio of 16. The pressure contour for afterburner with air–fuel ratio of 16 is also shown in Fig. 6. The variation of CO2 mass fraction is shown in Fig. 7, for the different air–fuel ratios along the axis of afterburner. Figure 8 shows the contours of CO2 mass fraction for the afterburner with air–fuel ratio of 16.
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Fig. 4 Temperature contour of afterburner (AF 16)
Fig. 5 Variation of pressure along the axis
Figure 9 shows the variation of O2 mass fraction along the axis for different air– fuel ratios along the length of afterburner. Figure 10 shows the contour of O2 mass fraction for the area ratio of 16 for ease of appreciation. Maximum CO2 and minimum O2 at the exit is observed for the air–fuel ratio of 16, which indicates complete combustion, and production of maximum thrust.
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Fig. 6 Pressure contour of afterburner (AF 16)
Fig. 7 Variation of CO2 along the axis
8 Conclusions Five afterburners models are computationally analyzed for different air–fuel ratios of 16, 29, 23, 20, and 45 to compare the CFD results with the experimental results obtained by Useller et al. [1]. It is found that the afterburner with air–fuel ratio of 16 produces maximum thrust in accordance with results published Useller et al. [1].
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Fig. 8 Contour of CO2 mass fraction (AF 16)
Fig. 9 Variation of O2 along the axis
Hence, the present CFD analysis and assumed boundary conditions are correct. This computational analysis can be safely used to predict the suitability of further models for experimental verification. Once the simulated model is predicted to give better performance using CFD analysis, and then only further experimental verification can be recommended. This saves considerable cost and time in preparation of the models and their testing.
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Fig. 10 Contour of O2 mass fraction (AF 16)
Nomenclature cp C1ε C2ε C3ε dp f F→ Gb Gk h jj k keff Kt p q0 Re Ri Si Sm Sε Sk Sh
Specific heat at constant pressure (J/kg K) Constant (–) Constant (–) Constant (–) Particle diameter (m) Body force (N/kg) External body force (N) Generation of turbulence kinetic energy due to buoyancy (kg/m s3 ) Generation of turbulence kinetic energy due to the mean velocity gradients (kg/m s3 ) Sensible enthalpy (J/kg) Diffusion flux of species J (kg/m2 s) Turbulence kinetic energy (M2 s− 2 ) Effective conductivity (W/m K) Turbulent thermal conductivity (W/m K) Static Pressure (Pa) Time averaged flow value Relative Reynolds number (–) Net rate of production of species I by chemical reaction (kg/m3 s) Rate of creation by addition from the dispersed phase (kg/m3 s) Mass added to the continuous phase (kg/m3 s) User-defined term (kg/m s4 ) User-defined source term (kg/m s3 ) Heat of chemical reaction (kg/m s3 )
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T u v w Yj YM
Temperature (K) Fluid velocity in x direction (m/s) Fluid velocity in y direction (m/s) Fluid velocity in z direction (m/s) Mass fraction of species j (–) Contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate (kg/m s3 ) Fluid phase velocity (m/s) Particle velocity (m/s)
u→ u→ p
Greek Symbols ε λ μ ρ ρp ρT σε σk τ
Turbulence eddy dissipation (m2 /s3 ) Material conductivity (W/m K) Molecular viscosity (pa-s) Density (kg/m3 ) Density of the particle (kg/m3 ) Total density (kg/m3 ) Turbulent Prandtl numbers for ε (–) Turbulent Prandtl numbers for k (–) Stress tensor (Pa)
References 1. Useller JW, Braithwaite WM, Rudy CJ (1954) Influence of combustion chamber length on afterburner performance. Lewis Flight Propulsion Laboratory, Cleveland, Ohio, USA, Rep. NACA RM E54E06 2. Kampa A (2007) Combustion instability screech in gas turbine afterburners. Ph.D. thesis. Department of Aerospace Engineering, Indian Institute of Science, Bangalore, July 2007 3. Unaune SV, Ganesan V (2005) Analysis of reacting flows in an aero-engine afterburner using computational fluid dynamics. Indian J Eng Mater Sci 11:31–37 4. Gurrala SR, Shaija A (2020) Computational analysis of reacting flows in afterburner. Heat Transf Eng 41(6–7):576–594 5. Dowling AP (1995) The calculation of thermoacoustic oscillations. J Sound Vib 180(4):557– 581 6. Eldredge JD, Dowling AP (2003) The absorption of axial acoustic waves by a perforated liner with bias flow. J Fluid Mech 485:307–335 7. Hughes IJ, Dowling AP (1990) The absorption of sound by perforated linings. J Fluid Mech 218:299–335 8. Stow SR, Dowling AP (2001) Thermoacoustic oscillations in an annular combustor. In: Proceedings of ASME Turbo Expo 2001, New Orleans, Louisiana, USA 2001-T-0037, June 2001
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9. Moeck J, Schmidt H, Oevermann M, Paschereit C, Klein R (2007) An asymptotically motivated hyrodynamic-acoustic two-way coupling for modeling thermoacoustic instabilities in a Rijke tube. In: 14th international congress on sound and vibration 2007, ICSV 2007, vol 1, Cairns, Australia, pp 623–631 10. Durox D, Schuller T, Noiray N, Birbaud AL, Candel S (2009) Rayleigh criterion and acoustic energy balance in unconfined self-sustained oscillating flame. Combust Flame 156:106–119 11. Cuquel A, Durox D, Schuller T (2011) Experimental determination of flame transfer function using random velocity perturbations. In: Proceedings of ASME Turbo Expo 2011 GT2011, Vancouver, British Columbia, Canada, 6–10 June 2011 12. Gregory SH, Andrzej B, Prasanth GM, Jeffrey MC, William P (2011) Method for control of thermoacoustic instabilities in a combustor. US8037688B2 13. Oliver VA (2011) FlameHolder for minimizing combustor screech. US7954328B2 14. George DL (1976) Combustion instability reduction device having swirling flow. US3974647 15. Alfred MS (2015) Method of using an Afterburner to reduce high-velocity Jet Engine Noise, US 9,140,214 B2 16. Katefidis A (2016) Thermal Afterburning system, and method of operating such a system, US 9,523,500 B2 17. ANSYS Inc (2013) ANSYS fluent theory guide, Canonsburg, PA, USA, Release 15.0
Computational Analysis of the Thermo Hydrodynamic Characteristics in a Can-Type Gas Turbine Combustor Mohit Bansal, Satyam Dewivedi, and Abdur Rahim
Abstract The present work performs numerical simulations of a can-type gas turbine combustor. The fuel in the present study is taken as methane. The stoichiometric ratios for the correct amount of fuel–air mixture required in the combustor are calculated using a Python program. The number of fuel injector inlets is six, which are symmetric to the center of the combustor. The number of secondary air inlets is kept as four and symmetric to the center. The standard k − ε model is used because of the strong turbulence after the injector nozzle. This is also clearly observed from the findings that as we move along the length of the combustor toward the outlet, the profiles of velocity and static pressure and temperatures become symmetric and more uniform and steadier. Mass fractions of O2 and CH4 increase along the axis, and the overall pressure loss coefficient values reach nearly 5%. Keywords Total pressure loss coefficient · Mass fraction · Primary and secondary air
1 Introduction A combustor of a gas turbine engine constitutes different cans. These cans are generally cylindrical and are pretty long. These cans consist of an outer casing with a perforated combustion chamber liner. Along the engine axis, these can combustors are placed radially. The compressed gases flow through the cans, where the fuel is first sprayed by fuel injectors and then burned to add thermal energy. Primary air flows through the inner liner holes to keep an optimum temperature and avoid damage.
M. Bansal (B) · S. Dewivedi · A. Rahim Department of Mechanical Engineering, JMI, New Delhi 110025, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_22
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2 Literature Review Chaouki Ghenai [1] has analyzed the combustion of fuel mixture of syngas in gas turbine can combustor, focusing on the composition of fuel variation and its variation on emissions. McGuirk and Palma [2] discuss water flow inside a can-type combustor in their experiments. Their results showed a better understanding of the mathematical modeling conditions and the flow patterns during combustion. Bicen et al. [3] discussed the combustion characteristics of a model can-type combustor at the inlet temperatures of 315 and 523 K at atmospheric pressures. It mentioned the influence of inlet temperatures which led to flatter scalar profiles. Jaafar et al. [4] found that with a high swirl number of 2.29 for flat vane and 1.57 for curve vane, an apparent reversal mass flow rate occurs, and higher swirl strength lowers the corner recirculation zone size and improves the combustion process. The analysis found that a swirl angle of 50° is suitable for generating an appreciable recirculation zone with lesser pressure drop. Lefebvre [5, 6] mentions the ideal configuration for the combustor as it should reduce the total pressure loss. Can-type combustors are well suited to engines with centrifugal compressors where the flow is divided into separate streams in the diffuser. The significant advantage of a can or tubular combustor is that the development could be carried out on a single can using only a fraction of the overall airflow and fuel flow. Crocker et al. [7] emphasize the implicit description of flow splits and flow conditions for openings into the combustor liner and liner wall temperature. Mattingly [8], A well-designed combustor will produce complete combustion and minimize total pressure loss. Green and Whitelaw [9] stated the standard k − ε model gives better results than the other turbulence models.
3 Modeling, Meshing, and Boundary Conditions In this work, a can combustor of 0.7 m in diameter and 5 m in length is taken in which the primary air is coming from the lining, and the fuel is injected via the injectors, which are 6 in number. There are four holes in the geometry for the secondary air entry. The dimensions are kept large as the combustion simulations work perfectly (Fig. 1). The element size for the mesh is kept at 0.003, and the mesh quality chosen is fine with high smoothing. The mesh is generated with 368,006 elements and 72,428 nodes. The governing equations are solved in these subdomains (Fig. 2).
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Fig. 1 Geometry of can-type combustor
Fig. 2 Meshing diagram of the geometry
3.1 Governing Equations Turbulence in fluids must satisfy the laws of classical physics, viz. the law of mass conservation, energy conservation Newton’s second law of motion, and equation of state, etc., and these laws form the fundamental governing equations.
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The continuity Eq. (1), momentum Eq. (2), and energy Eq. (3) for a compressible fluid are shown as follows: δ[ρu] δρ + =0 δx δx
(1)
δ(ρu i ) δ[ρu i v j + pδi j − τ ji ] = 0, i = 1, 2, 3 + δt δx j
(2)
δ(ρe0 ) δ[ρu j e0 + u j p + q j − u i τi j ] + =0 δt δx j
(3)
In our present work, commercial code in FLUENT is used because it consists of abundance of modeling capabilities. The solution domain in FLUENT is divided into control volumes, and the above-mentioned Eqs. (1), (2) and (3) are applied to each control volume.
3.2 Transport Equations for Standard k − ε Model Due to the fluctuations in the velocity of fluids, the flow generated inside the domain is of turbulent nature. With the presence of these fluctuations, the momentum and energy also start fluctuating, and it becomes quite complicated to simulate the same. Therefore, the modified form of these equations is used to remove the small scales of variations. In the modified form, there comes terms like turbulent kinetic energy and turbulence dissipation rate. For the calculation of turbulent eddy viscosity, the following relation is used μt = ρCμ
k2 ε
(4)
For turbulent kinetic energy k, δ δ(ρk) δ(ρku i ) + = δt δxi
[( μ+
μt σk
δx j
)
δk δx j
] + Pk + Pb − ρε − Y M + Sk
(5)
For turbulence dissipation rate ε, δ(ρε) δ(ρεu i ) + = δt δxi
δ
[(
μ+
μt εk
δx j
)
δε δx j
] ε ε2 + C1ε (Pk + C3ε Pb ) − C2ερ + Sε (6) k ρ
where Pk is the generation of turbulent kinetic energy due to the mean velocity gradient and is calculated by the following relation
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(7)
and S is the mean rate of shear stress tensor given as S=
√
2Si j Si j
(8)
where ) ( ∂u j 1 ∂u i Si j = + 2 ∂ xi ∂x j the above-mentioned governing equations of mass, momentum, and other turbulent quantities are solved by the finite volume-based technique using the quadrilateral mesh.
3.3 Species Equation Along with the mass and momentum equations, the interphase species transport equations are also solved in the domain. The species equation is shown in (9) δ(ρYi ) − → → + ∇.(ρ − v Yi ) = −∇ . Ji + Ri + Si δt
(9)
→ where ρ is the density of the mixture, Y mass fraction of species i, ρ − v Yi convection − → term which depends on current velocity, ∇ . Ji is the diffusion term, and Ri and S i are reaction terms.
3.4 Boundary Conditions We have taken non premixed combustion with mass fractions of methane and air in our calculations. We have calculated the mass fractions with an equivalence ratio of 1; the percentage of methane is taken as 5.5%, and the rest is the air. The lower heating value of methane is taken as 50 MJ/kg, and the power desired is 35 kW. The boundary conditions for the air inlet are the mass flow of air is 0.0095 kg/s, the temperature of 300 K with a turbulent intensity, and a turbulent viscosity ratio of 10% each. The secondary air mass flow rate is 0.00079545 kg/s with an inlet temperature of 300 K through the four secondary air holes. The inlet conditions for fuel inlet-like mass flow rate for fuel in the individual injector are 0.000167 kg/s, and the inlet temperature is 300 K. The finite volume standard k − ε model is used with P1 radiation model with energy transfer.
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Fig. 3 Temperature across the geometry
4 Results and Discussion The current work results are presented in the form of the contours of static temperatures, and the contours of mass fractions are plotted in the Ansys Fluent post-processor. The semi-implicit method for pressure-linked equations (SIMPLE) algorithm is utilized in the domain to apply the conservation equations and to obtain the pressure fields.
4.1 Contours of Outlet Temperature The maximum temperature generated inside the can combustor with the combustion of air and methane is 1953 K which depicts the appropriate combustion of the reactants (Fig. 3). The total temperature at exit comes out to be 697.5 K through the simulations, whereas the temperature value from the empirical relations is 613 K.
4.2 Contours of Velocity Magnitude The velocity contours represent the variations along the geometry of the profile. The velocities at the inlet just after the nozzle are increasing. Along the lengths, the velocity’s magnitude is steadier, as depicted in post-processing results as the blue region (Fig. 4).
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Fig. 4 Velocity across the geometry
4.3 Contours of Species Due to high turbulence intensity, methane and air mixing in the can combustor are better. Figures 5, 6, and 7 indicate that the fuel gets mixed with the secondary air just after exiting from the injector nozzles, and the temperature inside starts to slow down with the presence of primary air. So, if there are very high temperatures along the length of the combustor, then the dilution holes can also be added to the geometry. The presence of a steady flame envelope can also be observed in Fig. 5. It is apparent that by variations in the geometry like the number of injector holes, number secondary air holes and by changing the mass fractions of the various gases we can easily regulate the air being provided in the primary part. It is visible from Fig. 5 that the concentrations of CH4 drastically decrease along the combustor axis toward the outlet. At the inlet, just after the nozzle, the value of methane is 0.99, and the value at the outlet is 0.06.
Fig. 5 Mass fraction of CH4
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Fig. 6 Mass fraction of O2
Fig. 7 Mass imbalance
The oxygen mass fraction is substantial at the inlet because the air is injected inside through the secondary air holes and from the primary perspective. The amount of direct air entering the liner is 75%, and the rest is entered in the combustor from the secondary holes. Figure 7 shows the nature of mass imbalance inside the combustor. The entry region is perfectly mixed. Due to turbulence, better combustion rates are achieved, and along the length after a few distance from the entry, there is a slight jump in the mass imbalance due to the momentum of the mixture caused by the combustion. The mass fraction of O–h from Fig. 8 represents the combustion reactants. Likewise, the various concentrations of other pollutants can also be plotted along the length of the combustor.
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Fig. 8 Mass fraction of O–h
5 Conclusions As per the findings mentioned by Lefebvre [5], combustor pressure loss parameters are vital in the design of combustor pressure. Out of these two, one is overall pressure loss and is represented in %. ΔPinlet−outlet ΔPinlet−outlet R = Pinlet qref 2
(
m 3 T30.5 Aref Pinlet
) (10)
The relation represents the loss in the pressure, and in our study, the value of overall pressure loss is 5%. The other one is the pressure loss factor which denotes the resistance to flow between the outlet of the compressor and the inlet of the turbine, and sometimes this loss is also known as the “drag coefficient”. ΔPinlet−outlet ΔPdiffuser ΔPL = + qref qref qref
(11)
The value of the pressure loss factor as found out using the above relation comes out to be 0.4. It depicts the efficiency in the combustion, but loss factors must still be minimized in designing the combustors, so for future calculations, further scope for improvement will be there. Also, the steady flame temperature produced indicates effective combustion modeling. For future work, varying fuel concentrations can be taken, and a complete analysis can also be performed for varied geometry and concentrations.
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Nomenclature ρ ω ΔPL ΔPdiffuser S ε k P ν μ R qref m3 T3 A C 1 ε , C 2 ε , σk , σε , C μ
Density of air (kg/m3 ) Rotor rotational speed (rad/s) Liner pressure loss (Pa) Diffuser pressure loss (Pa) Modulus of the mean rate-of-strain tensor (–) Turbulence dissipation rate (m2 /s3 ) Turbulent kinetic energy (m2 /s2 ) Static pressure (Pa) Kinematic viscosity (m2 /s) Dynamic viscosity (kg m− 1 s− 1 ) Gas constant (J/kg k) Dynamic pressure reference value (Pa) Air mass flow rate at the inlet (kg/s) The total temperature at the inlet (K) Total reference area (m2 ) Constants of turbulence model (–)
References 1. Ghenai C (2010) Combustion of syngas fuel in gas turbine can combustor. In: Advances in mechanical engineering. Hindawi Publishing Corporation, p 13 2. McGuirck JJ, Palms JMLM (1995) Experimental investigation of the flow inside a water model of a gas turbine combustor. Part 1—mean and turbulent flow field. J Fluids Eng 117:450–458 3. Bicen AF, Tse DGN, Whitelaw JH (1990) Combustion characteristics of a model can-type combustor. Combust Flame 80(2):111–125 4. Mohd Jaafar MN, Jusoff K, Osman MS, Ishak MSA (2011) Combustor aerodynamic using radial swirler. Int J Phys Sci 6(13):3091–3098 5. Lefebvre AH (1999) Gas turbine combustion chamber, 2nd edn. Taylor and Francis, USA 6. Lefebvre AH (1985) Fuel effects on gas turbine combustion—ignition, stability, and combustion efficiency 7. Crocker DS, Nickolaus D, Smith CE (2015) CFD modeling of a gas turbine combustor from compressor exit to turbine inlet. J Eng Gas Turbines Power 121(89):89–95 8. Mattingly JD (1996) Elements of gas turbine propulsion. McGraw-Hill Inc., USA; ANSYS Inc (2015) ANSYS fluent theory guide 12.0 9. Green AS, Whitelaw JH (1983) Isothermal models of gas turbine combustor. J Fluid Mech 126:399–412
Experimental Study of Acoustic Phenomenon in a Closed Combustion Chamber A. Ananthakrishnan, Siba Prasad Choudhury, S. Syam, and Ratan Joarder
Abstract This study is aimed at obtaining data from pressure waves generated by the flame front and studying their effects on the formation of tulip flame. A closed rectangular chamber made of acrylic sheets is used for the experiment with premixed propane-air mixture for the combustion process. Equivalence ratio of 0.9, 1.0, and 1.1 are tested for the combustion. Microphones are used to measure pressure signals. In order to see the variation of the pressure wave characteristics at different instances of the combustion process, the microphone is placed at different locations. Data acquisition from the microphone is done through an oscilloscope. The signals are then analyzed to see different aspects of the recorded pressure signals. Keywords Thermo-acoustics · Pressure wave · FFT · Premixed flame
1 Introduction The combustion of premixed fuel in a closed chamber shows some interesting features. Initially, the shape of the flame front is spherical, but as the flame expands further and touches the boundaries of the chamber, the shape of the flame front transitions into a capsule or a finger-shaped flame. In further propagation, this fingershaped flame front flattens and suddenly curves inward with a cusp at the center pointed toward burned gas. This process is called flame inversion. This particular flame front pattern is called “tulip” flame because it resembles the petals of a tulip flower. A series of flame propagation in the closed chamber with liquefied petroleum gas (LPG)–air mixture is shown in Fig. 1 (left). Tulip flame is a phenomenon that occurs during premixed combustion in a closed or semi-open chamber of aspect ratio greater than two [1]. Even though many theories have been proposed to explain the formation of the tulip flame, no concrete theories exist till date. Some theories suggest that the sudden decrease in the flame surface area is a prime cause and also theories like hydrodynamic instability are also used to justify the formation of A. Ananthakrishnan (B) · S. P. Choudhury · S. Syam · R. Joarder Department of Aerospace Engineering, IIT Kharagpur, Kharagpur, West Bengal 721302, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_23
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Fig. 1 Series of high-speed images of flame propagation in the closed chamber from the present experiment and the computational result from [3]
tulip flame [2]. But several numerical studies have indicated the presence of strong acoustic waves in the chamber right after ignition and its interaction with the flame front [3–5]. Figure 1 (right) shows one of the computational results from [3]. We have also observed the presence of acoustics, and its interaction with the flame front using optical techniques in a closed combustion chamber [6]. In an enclosed chamber, the pressure waves generated by the flame front can reflect from the chamber walls and interact with the flame front itself. This can result in flame instabilities which can be characterized as thermo-acoustic instability. If left unchecked these instabilities are potentially hazardous for the combustion chamber as the heat release and pressure wave oscillations can form a feedback loop and result in high amplitude pressure oscillations and uneven heat release. This paper aims to study the effects of the interaction of the pressure waves on the formation of tulip flame in premixed LPG–air combustion. Microphones were used for pressure wave detection as they have a dynamic range from 20 Hz to 20 kHz and hence can detect a wide range of pressure oscillation frequencies. These data are analyzed to see the frequency components, their magnitudes, and their evolution with time. This is particularly important for the detection of pressure wave interaction with the flame front.
2 Experimental Methodology The experimental setup consists of a combustion chamber of dimension 40 mm × 40 mm × 400 mm, made of acrylic sheets (8 and 15 mm thick), mass flow controllers, microphone (CZN-15E), oscilloscope, phantom v7.3 high-speed camera
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with 52 mm lens, digital CMOS camera storage tank for gaseous propane, air compressor, solenoid valves, ignition system, digital delay generator, and a computer to control all these apparatuses. All the experiments were conducted in a closed chamber. One side of the chamber is closed with an 8 mm acrylic sheet, and the other side of the chamber is sealed with a cellophane sheet. The sheet ruptures as the pressure in the chamber increases to some threshold pressure value. This methodology is sufficient to capture the evolution of the flame front from the spherical flame front to the tulip inversion. Details of the experimental setup can be found in [7]. The schematic diagram of the setup is shown in Fig. 2. Initial measurements were done with a microphone placed at 105 mm from the location of ignition, and data was taken for an equivalence ratio (ϕ) of 0.9, 1.0, 1.1. The experiments are repeated with the microphone placed at 155, 200, and 345 mm from the location of the ignition for the same equivalence ratio. The microphone is connected to the external circuit with thin 0.1 mm diameter coated copper wire, as this aid in preventing leakage in the setup. The wire is selected such that it will not disturb the flow dynamics or flame front propagation. An oscilloscope is used to acquire the signal from the microphone. The oscilloscope is triggered by the microphone as it detects the acoustics generated by the spark. The microphone was placed perpendicular to the axis of the chamber in both cases. Flame propagation in the combustion chamber is captured with the high-speed camera.
Fig. 2 Schematic diagram of the experimental setup
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3 Results and Discussion The microphone is placed at 105, 155, 200, and 345 mm from the spark location. Microphone which is placed at 105 mm from the ignition source faces the fingertipshaped flame front, while the microphone placed at other locations will be at a region close to the flame front flattening and inversion. The difference in the signal captured by the microphones at these locations can hint at the change in the behavior of the pressure wave fluctuations. Figure 3 shows the oscilloscope reading from the microphone for the combustion process done for ϕ = 1.0. It is a voltage versus time plot for the microphone placed at 155 mm from the spark location. The obtained signal can be categorized into three, the region of the acoustics of the spark, the region of pressure wave interaction and the flame passage which is represented in the dotted box, and the region of acoustics due to the rupture of the chamber. The data from the microphone captured certain interesting features. The first one is the difference in the spark signal (Fig. 4). The spark signal corresponds to different conditions (spark in quiescent air and spark in the fuel–air mixture) were compared, and it was observed that when the spark was triggered in quiescent air, the acoustics of the spark was observed but as the spark was triggered in the fuel–air mixture, another voltage pulse was observed along with the acoustic signal of the spark. From the computational studies, we have observed the existence of a blast wave or pressure pulse during the initial phase of flame evolution and this is what was detected as the voltage pulse. It was observed that as the ignition happened immediately after the spark, a voltage pulse was observed along with the acoustics of the spark, and when the initial phase of combustion occurred after some time lag the voltage pulse was detected at a later instance. Figure 5 shows the comparison of the time instances of the occurrence of the initial phase of the combustion process and the time of detection of the voltage pulse from the microphone data. The time of initiation of
Fig. 3 Voltage versus time plot for the flame propagation at ϕ = 1.0 with microphone placed at 155 mm from the spark location
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Fig. 4 Comparison of the microphone signal obtained from the spark triggered in quiescent air (red) and spark triggered in fuel–air mixture which resulted in combustion (black)
the first phase of combustion matches closely with the time of the detection of the voltage pulse, and from this observation, the presence of a blast wave in the initial phase of the combustion can be confirmed. From [3], the author has observed the evolution of expansion wave as the flame skirts (flame surface close to the chamber side walls) touch the side walls. This expansion wave are formed in the burned gas region and propagates to the unburned region crossing the flame front at some point. The author claims that this causes
Fig. 5 Comparison plot of microphone data and time instance of initiation of the first phase of combustion
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Fig. 6 Flame tip position versus time
a reduction in flame speed. From the experiments conducted, we found that the microphone detected a voltage dip soon after the flame skirt touched the side walls of the chamber. The flame tip position was calculated using MATLAB, and from the data obtained, we observed that the flame decelerated once during its propagation. The time of this deceleration can be obtained from the flame tip position versus time plot (Fig. 6). Correlating both the time observed from the flame tip position plot and the time of the occurrence of the voltage dip from the microphone data for multiple experiments (Fig. 7), we were able to conclude that the expansion wave had some effect on the deceleration of the flame front. This hypothesis is not completely proved as there exist multiple expansion waves during the combustion process but the flame deceleration occurs only once during the combustion process. Complex pressure wave interactions are present in the chamber which result in the presence of many frequency components in the chamber, and the FFT of the signal shows the same (Fig. 8). In order to analyze the major frequency component present in the chamber, FFT analysis was done, and also, spectrogram analysis was done to see at what time instance these frequency components exist (Fig. 9). From the analysis, we observed that high amplitude, low frequency pressure oscillations exist during the combustion process and these oscillations were prominent during the passage of pressure waves and expansion waves. Different experiments provided different frequency components with the same behavior, and from this analysis, we concluded that random noises are present during the combustion process but frequencies corresponding to the resonance and the harmonics of the chamber were not observed. The observed pressure fronts might induce flow in their wake which might affect the flow field in the combustion chamber and hence alter the flame dynamics.
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Fig. 7 Comparison plot for the time instance of detection of expansion wave and flame speed reduction
Fig. 8 FFT of combustion process
4 Conclusions Flame propagation for premixed LPG–air mixture in a closed combustion chamber was analyzed to understand the acoustic phenomenon during the combustion. Microphones were used at different locations to capture the acoustics. The data obtained from the microphone was analyzed using FFT and spectrogram. The microphones
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Fig. 9 Spectrogram analysis of combustion process
were able to capture the acoustics of the spark and also detect the pressure fronts generated during combustion. Processes like the generation of the blast wave and the presence of expansion waves were observed. It was also observed that the detection of expansion waves coincided with flame front flattening and tulip flame formation. FFT analysis showed that low frequency high amplitude oscillations exist during the combustion process and spectrogram analysis indicated that this frequency components are prominent during the passage of blast waves and expansion waves. This may be due to the flow generated by the wake of the pressure waves interacting with the microphone. In [3], acoustic waves generated by the flame front were observed in the unburned region during the initial phase of the combustion process but in the experiments conducted in the present paper, the microphone did not detect any acoustic signals.
References 1. Dunn-Rankin D, Sawyer RF (1998) Tulip flames: changes in shape of premixed flames propagating in closed tubes. Exp Fluids 24:130–140 2. Ponizy B, Claverie A, Veyssière B (2014) Tulip flame—the mechanism of flame front inversion. Combust Flame 161(12):3051–3062 3. Xiao H, Oran ES, Houim RW (2017) Effects of pressure waves on the stability of flames propagating in tubes. Proc Combust Inst 36(1):1577–1583 4. Joarder R, Choudhury SP, Syam S, Singh N, Biswas SK (2022) Thermo-acoustics and its detection in a premixed flame. arXiv preprint, arXiv:2208.02146
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5. Xiao H, Houim RW, Oran ES (2015) Formation and evolution of distorted tulip flames. Combust Flame 162(11):4084–4101 6. Choudhury SP, Syam S, Joarder R (2022) A study on capturing acoustic behaviour in confined tube combustion using optical technique. In: 16th international combustion symposium (INCOS). The Combustion Institute–Turkey Section, Aydin, Turkey 7. Choudhury SP, Joarder R (2022) High-speed photography and background oriented schlieren techniques for characterizing tulip flame. Combust Flame 245:112304
The Effect of Lean Premixed Combustion on Thermoacoustic Instability in a Swirl Combustor Subhash Kumar, Sanjeev Kumar, and Sheshadri Sreedhara
Abstract Combustion instability is a large amplitude oscillation of one or more natural acoustic modes of the combustion chamber. It is the outcome of advancement in the modern engine to reduce emission and fuel consumption by maintaining the lean equivalence ratio. It creates large amplitude pressure and velocity oscillation which results in thrust oscillation, severe vibrations, enhanced heat transfer, thermal stress, and failure of the system. In this work, a laboratory model swirl combustor with a lean mixture has been simulated to understand the mechanism of combustion instability. A 3D unsteady Reynolds-averaged Navier–Stoke numerical simulations are performed to understand the effect of lean premixed combustion on combustion instability. The results show that flame fluctuation due to lean premixed combustion leads to heat release oscillation. Heat release oscillation leads to pressure and velocity oscillation which results in combustion instabilities. The amplitude of pressure and velocity oscillation increases with time and finally become saturates. The numerical model is validated with experimental results. Keywords Swirler · Instability · Lean premixed · CFD
1 Introduction Instability is a very adverse condition for a combustion chamber, which causes noise and vibration and may reduce the performance and durability of the system. The basic reason for thermoacoustic instability is the coupling of two or more natural modes of acoustics in the combustion chamber. These coupling depends on the mixture inhomogeneity, upstream velocity fluctuation, fuel air equivalence ratio oscillation, vorticity wave, etc. It is also a challenging task to understand before operation of a system.
S. Kumar (B) · S. Kumar · S. Sreedhara Department of Mechanical Engineering, IIT Bombay, Mumbai 400076, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_24
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Cohen and Anderson [1] performed experimental work and observed the effect of the equivalence ratio (ϕ) modification on combustion. They found that the instabilities appeared as the equivalence ratio approached the lean mixture (< 0.75). They suggested that this behaviour was due to the interaction of flame with large-scale vortex. They also suggested that reason for an increase in the magnitude of combustion instability with decreasing ϕ is an increase in chemical time scale, and hence, blow off takes place. Michael et al. [2] have also investigated the lean premixed combustion system with respect to environmental conditions (different inlet conditions). They suggest that a change in inlet temperature changed the chemical reaction time scale and resulted in combustion instability. Lieuwen et al. [3] explained the experimental investigation of combustion instability for low NOx gas turbine. They studied the dependence of pressure amplitude upon combustor operating parameters like as mean pressure, mean velocity, and equivalence ratio. They suggested the combustion instability is the outcome of the interaction between pressure oscillations, heat release rate, and equivalence ratio fluctuations. They suggested instability can be avoided by the design a combustor which operating time should have out of instability time (Tconv,eff /T ). Broada et al. [4] performed an experimental investigation of the model gas turbine using radical chemiluminescence as an indicator of heat release. They suggested that the instability depends on the equivalence ratio and inlet temperature. They also developed a stability range for a gas turbine engine. Gejji et al. [5] performed an experimental investigation of a model gas turbine. They presented the effect of geometry, operating conditions with respect to the different modes of oscillation. They suggested that size change of inlet section has a strong effect on the interaction of acoustic mode to hydrodynamic mode and heat release rate mode and ultimately on pressure oscillation. They also suggested that the fundamental mode of pressure increases as the equivalence ratio decreases and also as inlet temperature decreases the instability increases. Huang et al. [6] have numerically investigated the combustion dynamics in a lean premixed swirl stabilized combustors using large eddy simulation (LES). They used level set flamelet library approach to model turbulent premixed combustion. They found that this model can predict premixed turbulent combustion with strong swirl motion with an acoustic. Literature review suggested that there is a need to understand the combustion instability and the parameters which are responsible for the combustion instability in swirl combustor. In the present work, unsteady Reynolds-averaged Navier– Stokes (URANS) have been performed to understand the generation and control of thermoacoustic acoustic instability in a model swirl combustor.
2 Modelling of Swril Combuster A laboratory model swirl combustor for premixed combustion has been investigated to generate and mitigate the mechanism of combustion instability. 3D unsteady Reynolds-Averaged Navier–Stokes numerical simulations are performed to understand the effect of lean premixed combustion on combustion instability. Further,
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Fig. 1 Schematic diagram of the set-up
Table 1 Dimensions of swirler Swirler
Di
Do
θ
No. of vanes
Swirl number
1
13
35
30
12
0.42
we have also validated our numerical model with experimental results. A numerical model of a swirl combustor with cylindrical shape with D = 70 mm and L = 200 mm is shown in Fig. 1. A swirler dimensions are given in Table 1.
3 Governing Equations The governing equations, turbulence models, combustion models, and acoustic models have been reported. The basic sets of balance equations used to model the Rijke tube and combustion modelling consist of the classical Navier–Stokes, species, and transport equations which are given as follows. Continuity Equation: ∂(ρu i ) ∂ρ + =0 ∂t ∂ xi
(1)
here, ρ is the mass density and u i is the velocity vector. Momentum Equation: ∂τi j ∂(ρu i ) ∂ ρu i u j ∂p + =− + + Fi ∂t ∂ xi ∂ xi ∂x j
(2)
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here, τi j means viscous stress tensor and F i is the body force. Viscous stress tensor for Newtonian fluid is defined as follows. ∂u j 2 ∂u k ∂u i −μ τi j = μ + δi j ∂ xi ∂ xi 3 ∂ xk
(3)
In below equation, viscous dissipation is neglected, where S rad is heat loss/gain due to radiation, h is mixture enthalpy per unit mass, hk is specific enthalpy of species ‘k’, σ h is the mixture Prandtl number, and S ck is Schmidt number for species ‘k’. Energy Equation: N ∂(ρh) ∂(ρu i h) ∂ μ ∂h 1 1 ∂Yk ∂p + + Srad = +μ − hk + ∂t ∂ xi ∂ x i σh ∂ x i Sck σh k=1 ∂ xi ∂t here, σ h =
μC p k
is the mixture Prandtl number and (Sck ) =
μ ρD
is the Schmidt number.
Ideal Gas Law: p = ρ
R T M
(4)
Species Transport Equation: The species transport equation for species ‘k’ is given as, ∂(ρYk ) ∂(ρu i Yk ) ∂Yk ∂ ρ Dk + ωk + = ∂t ∂ xi ∂ xi ∂ xx
(5)
here, Dk is the mass diffusion coefficient and Y k is the mass fraction. Equations (1)–(5) have been solved iteratively to simulate the 2D Rijke tube with a structured grid. In the above equations, viscous dissipation is neglected.
4 Numerical Details and Boundary Conditions The swirl combustor premixed combustion has been modelled using FLUENT 16.5 ANSYS that has been used for modelling geometry. The mesh has been developed ICEM CFD for 3D. The structured mesh has been developed for 3D geometry as shown in Fig. 2. To save computer time, we neglect the swirler vanes. Grid independence test is also performed to get better results. The swirl velocity and axial velocity components have been calculated with the help of velocity magnitude by swirl number relations presented in the below equations [6, 7]:
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Fig. 2 Structured 3D mesh for model combustor
3 2 1 − Di Do SN = tan θ 3 1 − Di Do 2 SN =
Gφ 2 vt = RG x 3 vx
(6)
(7)
R Gφ =
vt ρva 2 r dr 0
R Ga =
R va ρva 2 r dr +
p2 r dr
(8)
o
0
v 2 = vt2 + va2
(9)
here, SN is swirl number, Di and Do are swirler inner and outer diameter, respectively, Gϕ and Ga are the axial flux of tangential momentum, and axial flux of axial momentum respectively, and vt and va are tangential and axial velocity, respectively. RANS model has been used for swirling flow. Standard, RNG, and RKE k-epsilon model has been used for the turbulence model. It has been observed that the renormalization group (RNG) is best for this swirling flow. Eddy dissipation concept (EDC) model has been used for combustion model in the case involving a two-step global reaction mechanism. The combustion boundary conditions, flow boundary conditions, numerical setting, and schemes are summarized in Tables 2, 3, and 4, respectively. The species boundary (in mass %) has been calculated for equivalence ratio 0.6 as presented in Ref. [8].
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Table 2 Species boundary conditions for swirl combustor for equivalence ratio 0.6 Boundary
T (K)
CH4
O2
N2
CO2
H2 O
Inlet swirl
300
3.378
22.52
74.1
0
0
Outlet
600
0
9.008
74.10
9.29
7.601
Wall inlet
–
0
0
0
0
0
Outer wall
750
0
0
0
0
0
Table 3 Flow boundary conditions Boundary Inlet swirl
Velocity boundary conditions that are calculated from swirl number and inlet velocity magnitude
Outlet
Pressure outlet constant relative pressure, 0 Pa
Wall inlet
No-slip boundary condition and adiabatic wall condition
Outer wall
No-slip boundary and constant wall temperature 750 K
Table 4 Numerical setting and schemes Turbulence model
SST k − ω model
Combustion model
EDC model
Pressure–velocity coupling
COUPLE
Discretization scheme
Second-order implicit for temporal discretization Second-order upwind for turbulence dissipation Third-order MUSCL for others
Reaction mechanism
Two-step global mechanism
4.1 Validation of Numerical Model In this subsection, predictions of turbulent flow and mixing field and scalar quantities are discussed in detail. The axial and radial profiles of the mean quantities showed good agreement between predictions and experimental data. The axial and radial profiles of axial velocity and radial velocity are plotted in Fig. 3. The predicted results of axial and radial velocity are compared with the experimental data by Zhang et al. [8] at x = 30, 60, 90, 120, and 150 mm, where X indicates the location in the axial direction from the inlet. It may be seen that the predicted results are in excellent agreement with the experimental measurements.
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Fig. 3 Variation of C P with TSR of the rotors
5 Results and Discussions 5.1 Steady State Results Figure 4 shows the contour plots of axial velocity (m/s) distribution of flow along Z-plane by using the SST k − ω turbulence model. It shows one recirculation zones near the axis of the chamber and two external recirculation zone with symmetric along the axis. The contour clearly shows a good mixing of combusted products and burner streams. The temperature plot depicted in Fig. 5 may provide a visualization of qualitative flame at the axial Z-plane. The lean mixture has been taken for simulation; hence, the combustion takes place away from the outer wall. In the mixing region, combustion products with high temperatures are observed as expected.
Fig. 4 Axial velocity contour on Z-plane
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Fig. 5 Contour plot for the temperature on Z-plane
Fig. 6 Mass fraction of CH4 on Z-plane
Fig. 7 Mass fraction of CO2 on Z-plane
The species CH4 and CO2 concentration contour plots have been shown in Figs. 6 and 7, respectively. In the mixing region concentration of combustion products, CO2 is maximum, whereas CH4 is minimum as expected.
5.2 Unsteady Results We have also performed the unsteady simulation for lean combustion. Figure 7 shows the temperature contour with respect to time. It shows the flame oscillation in the combustion chamber due to lean premixed combustion. Further, this flame fluctuation or flame oscillation leads to heat release oscillation. Heat release oscillation leads to
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pressure and velocity oscillation which results in combustion instabilities (Fig. 8). The amplitude of pressure and velocity oscillation increases with time and finally become saturates. This flame oscillation fluctuation creates pressure oscillation as shown in Fig. 9. Velocity fluctuations is also noticed in swirl combustor due to this pressure oscillation as shown in Fig. 10.
Fig. 8 Temperature contour with respect to time for lean mixture
Fig. 9 Pressure oscillation at 0.02 mm from inlet
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Fig. 10 Velocity oscillation at 0.02 mm from inlet along axis
6 Conclusions In this work, swirl combustion flame at lean mixture has been simulated using FLUET to investigate the effect of lean premixed turbulent combustion and to further use it to predict combustion instability. The numerical implementation and solution procedure of the solver have been discussed in detail. A 3D steady CFD simulation has been carried out in conjunction with the RNG k − ε model for turbulence model and EDC model for combustion. FLUENT results were benchmarked by comparing these results with the experimental data [8] obtained by Barlow et al. [9]. Predicted results showed an excellent match with the experimental data for all scalars and vector fields. The unsteady results show the flame is unstable and is fluctuating; hence, the pressure is also varying with time which results in combustion instability in swirl combustor.
Nomenclature A AR CD α ρ ω
Frontal area of rotor (m2 ) Aspect ratio (−) Drag coefficient (−) Angle of attack (°) Density of air (kg/m3 ) Rotor rotational speed (rad/s)
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References 1. Cohen J, Anderson T (1996) Experimental investigation of near-blowout instabilities in a lean, premixed step combustor. In: 34th aerospace sciences meeting and exhibit 2. Janus MC, Richards GA, Yip MJ, Robey EH (1997) Effect of ambient conditions and fuel composition on combustion instability. In: ASME 1997 international gas turbine and aeroengine congress and exhibition, p 2 3. Torres H, Lieuwen T, Johnson C, Daniel B, Zinn B (1999) Experimental investigation of combustion instabilities in a gas turbine combustor simulator. In: 37th aerospace sciences meeting and exhibit 4. Broda JC, Seo S, Santoro RJ, Shirhattikar G, Yang V (1998) An experimental study of combustion dynamics of a premixed swirl injector. In: 27th symposium (international) on combustion, vol 27, pp 1849–1856 5. Gejji RM, Huang C, Fugger C, Yoon C, Anderson W (2018) Parametric investigation of combustion instabilities in a single-element lean direct injection combustor. Int J Spray Combust Dyn 1–16 6. Huang Y, Sung H-G, Hsieh S-Y, Yang V (2003) Large-eddy simulation of combustion dynamics of lean-premixed swirl-stabilized combustor. J Propuls Power 19:782–794 7. Boushaki T. Introductory chapter: swirling flows and flames. In: Swirling flows and flames, pp 1–7 8. Zhang M, Wei X, Wang J, Huang Z, Tan H (2021) The blow-off and transient characteristics of co-firing ammonia/methane fuels in a swirl combustor. Proc Combust Inst 38:5859–5868 9. Barlow RS, Frank JH, Karpetis AN, Chen JY (2005) Piloted methane/air jet flames: Transport effects and aspects of scalar structure. Combust Flame 143:433–449
Computational Modelling of MMH/NTO Combustion in a Multi-element Triplet Injector Combustor Abhishek Sharma, Varghese M. Thannickal, T. John Tharakan, and S. Sunil Kumar
Abstract This paper presents the computational methodology developed to simulate monomethyl hydrazine/nitrogen tetroxide (MMH/NTO) combustion. A threedimensional rocket scale combustor domain with multi-element triplet injectors is utilized to study hypergolic flow and flame features. A Eulerian–Lagrangian framework is invoked for continuous phase treatment of combustion gas and discrete phase treatment for both MMH and NTO droplets. A discrete particle-based method (DPM) with finite rate chemistry is employed to study droplet injection, evaporation, and combustion. A description of flow and flame characteristics in three-dimensional RANS framework is presented in this paper. The model captures impinging jets from multiple triplet injectors, and MMH film cooling injection appropriately. It presents physical trends on the core combustion process, as well as the global evolution of temperature, pressure, and droplet spray in the combustor. The focus of the study is to develop a hypergolic combustion model which can be used to predict combustion performance under off-nominal operating conditions. The aim is to extend the model to study the combustion instability aspects of MMH/NTO-based combustors. Keywords MMH–NTO · Hypergolic · Combustion · Mechanism · Triplet · Finite rate
1 Introduction There is an entrepreneurial boom in the world to develop launch vehicles to tap the growing market of lower earth observation satellites. Monomethylhydrazine (MMH) and nitrogen tetroxide (NTO) is still a widely used propellant combination for many launch vehicles. Its high energy content, excellent physical properties, and ignition capability even at low temperature and pressure conditions make it an ideal and irreplaceable candidate. Hypergolic thrusters are used in many propulsion modules, such as reaction control in launch vehicles, attitude control in satellites, and even A. Sharma (B) · V. M. Thannickal · T. John Tharakan · S. Sunil Kumar Liquid Propulsion Systems Centre, ISRO, Thiruvananthapuram, Kerala 695547, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_25
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in majority of interplanetary missions till date. There is an undiminishing need for hypergolic propulsion, which necessitates the requirement of physical understanding of the combustion processes occurring in such systems. There is a clear need for a computational model which can simulate hypergolic combustion and predict the performance under off-nominal operating conditions. The effect of physical and chemical processes on thruster performance is important for future thruster developments. A computational model to study combustion dynamics of MMH/NTO combustion is even more important in the scenario of off-nominal operation. Parallel experimental and computational investigation is essential for design optimization of hypergolic thrusters. Hypergolic combustion modelling poses challenges, as the numerical model should be capable of droplet modelling, droplet to gas/wall interaction, evaporation, and liquid–gas-phase reactions. The computational modelling has been impeded for a long time due to the lack of an MMH/NTO reaction mechanism. Due to the complex and hazardous operating conditions, only a few experimental studies are carried to extract chemical kinetic information on this hypergolic combination. Catoire et al. [1, 2] developed 403 equilibrium reactions and 82 species kinetic model for ignition of MMH/NTO gaseous combustion, but the associated kinetic and thermodynamic data are not available in public domain. However, it is always difficult to introduce such detailed mechanisms for CFD model development, as it entails enormous computational resources. Despite such difficulties, a hypergolic combustion model is imperative to study stability aspects of such combustors and requires a concerted effort for separate development of kinetic and CFD model.
2 Literature Review and Objective Experimental and numerical studies have been performed to understand the complex nature of MMH–NTO combustion. Numerical studies are performed with an aim to develop a CFD model for optimization of such hypergolic thrusters. Dedicated CFD codes [3] were written to simulate MMH–NTO combustion in rocket engines, such as PHEDRE [4], ROCKFLAM [5], and HPRECSA [6]. PHEDRE and HPRECSA employ one-step global mechanism, whereas ROCKFLAM utilizes a four-step global mechanism including decomposition of MMH and NTO. The unavailability of appropriate multi-step reaction has led to multiple CFD investigation [4–10] for MMH– NTO spray features, secondary droplet break-up, and even combustion dynamics [11] using single-step reaction model. Zhang et al. [12] developed combustion model for a 490N MMH–NTO apogee attitude and orbit control thruster. It uses a five-step reaction mechanism for CFD model development. A recent study by Hou et al. [8] highlights the development of a multi-step MMH–NTO mechanism consisting of 23 species and 20 reactions. A sensitivity analysis was performed to identify the important reactions in gaseous MMH/NTO combustion. The mechanism was valid under a wide range of temperature and pressure conditions. The multi-step mechanism was incorporated into CFD model of the MMH/NTO liquid rocket engine. The study highlights the difference between single-step and multi-step reaction model results.
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Ohminami et al. [13] developed combustion model for ISAS-500N with 61-step reduced mechanism in the EDC framework. The CFD model is used to investigate the effect of varying mixture ratio (oxidizer/fuel mass flow ratio), injection method, and droplet characteristics on the combustion process. Initial studies on combustion instabilities in a small MMH/NTO liquid rocket engine were carried out by Qin et al. [9]. A spray combustion model was developed to study the effect of varying droplet diameter on stability. The study captured self-excited combustion instability in the first tangential (1 T) and the first longitudinal (1 L) mode of the chamber. The aim of the current study is to develop a computational model which can simulate hypergolic combustion and is amenable to future modifications of multi-step kinetics, droplet diameter distribution, and secondary droplet break-up and combustion dynamic analysis with high-fidelity turbulence models. In the current study, a basic computational methodology is established on a multi-element rocket scale combustor. An attempt is made to simulate hypergolic combustion with a complex triplet injection system. The results are verified based on physical understanding of hypergolic combustion processes. A description of flow and flame characteristics in three-dimensional RANS framework is presented. The study has been carried as a precursor to complex combustion dynamic studies in the future.
3 Formulation and Methodology MMH–NTO combustors exhibit the processes of injection, primary and secondary atomization, vaporization, mixing, and reaction. It is quite complex to model these physical processes altogether in a single framework. It is further challenging to model the formation of liquid film and droplet interaction with the flow and with the chamber walls. The use of Eulerian–Lagrangian methodology to tackle this process along with gas-phase reaction of MMH and NTO is highlighted in the literature. This methodology has been satisfactorily employed on single and a few multi-element studies with doublet injectors. In these studies, also, similar methodologies have been adopted to solve two-phase reactive flow combustion. A three-dimensional RANS framework is employed for gas-phase equations and Lagrangian treatment of MMH–NTO liquid particles. The governing equations are represented in compact form, where ϕ, ⎡ϕ , and u j represent conservation variables, diffusion coefficient, and velocity in three dimensions, respectively. ( ) ( ) ∂(ρϕ) ∂ ρϕu j ∂(ϕ) ∂ ⎡ϕ + Sm + S M,i + Sh + SYk + SYk + = ∂t ∂x j ∂x j ∂x j
(1)
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The interaction between the two phases is expressed as source terms in the above governing equation. Sm represents the mass addition to continuous phase from the dispersed phase. S M,i is momentum loss/gain due to droplets, whereas, Sh is energy source term due to heat exchange with the dispersed phase. SYk accounts for generation or consumption of species due to MMH–NTO evaporation and combustion.
3.1 Droplet Model In this study, both MMH and NTO are injected in the form of liquid-phase droplets. A convection/diffusion-controlled model [14] is used in this study which considers droplet evaporation due to convective heating and concentration gradient. The complex primary break-up of propellants is not considered in this model. Discrete particle tracking from incipience of a droplet is performed by trajectory calculation via integration of force, mass transfer, and energy balance on MMH–NTO droplets. The force balance in the discrete particle model is solved as follows: ) ( ( ) g→ ρ p − ρ d→ up + F→ = FD u→ − u→ p + dt ρp
(2)
here, u→ is the fluid phase velocity, u→ p is the particle velocity, μ is the molecular viscosity of the fluid, and ρ p and dp are the density and diameter of the particle, respectively. FD is the drag coefficient and F→ is an additional acceleration term. A mathematical description of the DPM model is same as provided in our earlier work [15]. In this study, due to droplet entry into high-temperature ambient gas, the secondary break-up of MMH–NTO droplets due to aerodynamic drag is not considered. The change in droplet diameter occurs only due to phase change. The dispersed phase exchanges mass, momentum, and energy with the continuous phase, and the continuous flow field calculation is updated with volumetric sources.
3.2 Combustion Model Hypergolic combustion of MMH–NTO is known to be controlled by the finite rate of reactions [1, 2]. The finite rate physics dominate over turbulence effects. The finite rate model directly calculates the finite rate chemistry and is a recommended formulation even for turbulent flows with highly disparate chemistry timescales. In this study, the finite rate model of ANSYS Fluent [14] is used for combustion modelling. The most important requirement of finite rate mode is a kinetic mechanism along with thermodynamic and transport data. A large body of work was dedicated to develop a mechanism [1, 2, 8] which can represent the hypergolic reaction accurately.
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Most of the reported works are proprietary, and finer details of mechanism-associated thermodynamic data are not available in public domain. There is a need to introduce MMH–NTO liquid-phase reactions [16] in combustion models, but they are not incorporated due to the highly complex nature and unavailability of valid reaction mechanisms. Overall, a rigorous effort is required to develop such mechanisms and is not in the scope of this CFD-based study. In this work, a single-step gas-phase reaction [5] is used to describe MMH–NTO chemistry. The single-step reaction is considered appropriate for this model development, as it can eventually cut down on the computational time of large eddy simulations for stability assessment in the future. Harvazinski et al. [11] efficiently use a single-step reaction for combustion dynamic simulations in continuously variable resonance combustor. The single-step reaction used in this study is expressed as follows: k
4CH3 NHNH2 + 5N2 O4 → 4CO2 + 12H2 O + 9N2
(3)
The forward rate constant, K f,r of reaction is calculated using Arrhenius expression: K f,r = Ar T β exp(Er /RT)
(4)
here, Ar = 1.00e + 06 is pre-exponential factor, the temperature exponent, β = 0, Er = 0 is the activation energy, and an oxidizer rate constant of 0.825 is used. R is the gas constant. All thermodynamic conditions are considered at chamber operating condition of 8.3 bar.
4 Multi-element Computational Model A typical rocket engine consists of multiple injector elements, a combustion chamber, a converging–diverging part, and a throat. Multi-element triplet injectors are used to attain good mixing efficiency and atomization for sub-critical MMH–NTO combustion. The study is performed using multi-element triplet injectors, with a combustion chamber modelled till the throat section. Figure 1 displays computational domain of the multi-element combustor with multiple triplet injectors, faceplate, chamber walls, and throat. The triplet injectors are assisted by film coolant jets to maintain chamber wall temperature within thermal limiting conditions. The film cooling orifices are arranged in a circular pattern at the periphery of the triplet elements which cools the thrust chamber. 30% of the nominal fuel flow rate is used for film cooling of combustor. A full three-dimensional computational domain is utilized for the combustion simulations, which can be further used to evaluate inter-element mixing at off-nominal conditions. It shows the injector placement at the faceplate, with entry of MMH and NTO in a zoomed view. Figure 2a displays a single triplet
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injector configuration, which describes the entry and path of MMH and NTO. Fortyfive sets of such triplet injector and 36 numbers of film cooling orifices are geometrically modelled in this study. One injection element consists of two oxidizer and one fuel injection orifices, which are displayed in the zoomed view of injector faceplate as MMH and NTO inlets in Fig. 1. The MMH droplet stream is surrounded by two impinging NTO streams for atomization and reaction to take place efficiently. Injector angles are directly modelled in geometry. Figure 2b presents the meshed model of the combustor. It displays local refinement at the injector, film cooling inlets. A refined mesh is provided downstream of injector exit/combustion region, combustor walls, and throat to capture the high-gradient variations accurately.
Fig. 1 Computational domain of multi-element triplet combustor
NTO
MMH
NTO
(a) Fig. 2 Triplet injection (a) and meshed model (b)
(b)
Computational Modelling of MMH/NTO Combustion … Table 1 Boundary conditions and inputs
307
Boundary and input conditions Boundaries
Condition
Injector faceplate
Wall, no slip
Chamber walls
Wall, no slip
Throat
Pressure outlet
MMH injector inlet
Mass flow rate
Droplet injection
MMH film cooling inlet
Mass flow rate
Droplet injection
NTO injector inlet
Mass flow rate
Droplet injection
Table 1 displays the boundary and input conditions for simulation. DPM formulation is used for both MMH and NTO inlets to simulate injection of droplets into the combustion chamber a mass flow rate of 0.7147 kg/s for MMH, 1.429 kg/s for NTO, and 0.3053 kg/s for MMH film coolant. Simulation is conducted at an overall oxidizer to fuel mixture ratio of 1.4. Constant size spherical droplets are injected into chamber at 300 K. All walls in computational domain are considered as no slip. A finite volume method is employed to discretize continuous phase governing equations. A second-order upwind scheme is used for interpolation of convective fields. The two equation SST k − ω model is used for turbulence closure, which employs k − ω model near solid walls and uses a blending function to convert it to k − ε formulation for free stream flow. The turbulent eddies are modelled using turbulent kinetic energy and kinetic energy specific dissipation. The semi-implicit method for pressure-linked equations (SIMPLE) method is used for pressure–velocity coupling.
5 Results and Discussion The numerical framework is applied to carry out three-dimensional multi-phase combustion analyses. The combined methodology of droplet evaporation and gaseous phase combustion is applied to capture reactive flow features. The droplet flow features, MMH–NTO flow characteristics, and evolution of multi-element flames and combustion pattern are discussed here. Figure 3 displays the droplet pattern emanating from inlet faces of MMH and NTO. The droplets enter the combustion chamber at an angle and impinge on each other. The figure displays droplet temperature increasing from injection to the evaporation state. A sudden droplet evaporation above 350 K is noticed in the simulation. The droplet diameter reduces along the length of combustor as it undergoes evaporation in high-temperature ambient gases. The droplet lifespan corresponds to a distance of approximately 25 mm from the injection point. A physical nature of three jets impinging on each other has been captured in DPM. The droplet model invoked captures the droplet entry, movement, and evaporation of droplets in the chamber appropriately.
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Fig. 3 Droplet injection pattern
It is noticed that the use of single-step chemistry can accelerate the droplet consumption rate due to higher flame temperature. Figure 4 displays pressure evolution in the chamber using this model. The chamber pressure evolves to a value close to nominal operating pressure of 8.3 bar. The axial velocity trend is presented in Fig. 5, which shows higher velocity at a certain distance from the injector faceplate. The higher axial velocity map displays the effect of density change to combustion. A lower axial velocity region is seen close the high-density droplet injection location. The evolution of Mach number and speed of sound is displayed in Figs. 6 and 7, respectively. The throat is seen to be choked at the exit. Figure 7 displays a physical trend of sound speed, with higher intensity downstream of the injection point. The higher intensity is present at the higher temperature region of the chamber. The flow features of MMH and NTO are discussed before the hypergolic flame characteristics. Figure 8 shows MMH mass fraction at a cut plane along the axis of combustor. The MMH concentration is visible till the mid-section of the chamber and close to the wall. The MMH film coolant survives along the wall till throat providing an efficient cooling. The higher boiling point of MMH at chamber conditions makes it an ideal coolant, and this is displayed in the MMH concentration at different axial Fig. 4 Pressure contour
Computational Modelling of MMH/NTO Combustion … Fig. 5 Axial velocity
Fig. 6 Mach number
Fig. 7 Speed of sound, m/s
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locations. Figure 10 displays MMH mass fraction at radial planes from 6 to 76 mm downstream of injector faceplate. It shows that the MMH film is intact till 76 mm from injection location. Figures 9 and 11 show the variation of NTO mass fraction in the chamber. A higher NTO mass fraction is visible near the injector faceplate in the droplet evaporation region. The NTO mass fraction drastically reduces from an axial location of 16 mm indicating rapid evaporation and MMH–NTO gas-phase combustion. The lower boiling point and higher reactivity of NTO induce quick combustion on exposure to MMH. The kinetic rate of reaction and heat release are displayed in Figs. 12 and 13, respectively. It shows very high rate of reaction immediately downstream of injector faceplate at an axial location of 6 mm. A complex pattern of intense heat release is visible at the core region and less intense close to the walls. The impingement region created by outer impinging streams of NTO onto the central MMH stream can exhibit higher kinetic rate. A similar trend for heat of reaction is visible in Fig. 13. Fig. 8 MMH mass fraction
Fig. 9 NTO mass fraction
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Fig. 10 MMH mass fraction at axial locations
Fig. 11 NTO mass fraction at axial locations
The higher heat release is also concentrated in the core region of combustor. The rapid heat release seen in the simulation can be the manifestation of single-step reaction used in this study and can change with the use of a multi-step mechanism. The combustion model represents the physical nature of the flow characteristics in the combustor, leading to appropriate evolution of the multi-injector flame structure. Figure 14 displays the flame temperature at an axial slice of the combustor. The flame temperature contour is averaged over DPM iterations to avoid instantaneous
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Fig. 12 Reaction rate
Fig. 13 Heat of reaction
features in this steady state simulation. A physically plausible multi-element flame structure is observed, with a high-temperature combustion zone downstream of the MMH–NTO injection plane. A low-temperature un-reacted zone is visible close to the injector exit. It also shows a low-temperature zone close to the walls due to MMH film flow. The multi-flames merge 60 mm from the faceplate. Fig. 14 Temperature at axial slice of combustor
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Fig. 15 Temperature at radial planes along the axis
The radial plane distribution of temperature is shown in Fig. 15. It clearly presents the variation of temperature at different axial locations. A high-temperature zone is visible at the core region, whereas a lower temperature is visible at the periphery. The maximum temperature of 3800 K indicates that the one-step global mechanism leads to overprediction of temperature. The equilibrium calculation gives an adiabatic flame temperature of 3150 K at a core mixture ratio of 1.99. The complete conversion of gas-phase MMH and NTO to products like CO2 , N2 , and H2 O leads to higher heat release and overprediction of flame temperature. It is reported in the literature that the intermediate reactions control the heat release process in a multi-step kinetic mechanism, and consequently, the single-step mechanism simulation overpredicts the temperature. A variation of product mass fractions is shown in Figs. 16 and 17. The overall maximum mass fraction captured in the simulation is lower than the stoichiometric values, as it is operated in fuel-rich conditions. The product concentration of CO2 and H2 O at a cut plain along the axis of combustor has a physical trend. Higher product concentration is seen at higher temperature regions as seen by comparison with Fig. 14. The above results showcase the efficacy of the computational model developed in this study to capture complex combustion physics associated with hypergolic propellants. Flow and multiple flames originating from triplet injectors exhibited physical features. The prime focus was to develop a computational methodology which can easily be extended to incorporate multi-step kinetics and other effects in the future.
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Fig. 16 CO2 mass fraction
Fig. 17 H2 O mass fraction
6 Conclusion A RANS-based three-dimensional computational methodology is developed to simulate hypergolic combustion with a Lagrangian treatment of both MMH and NTO propellants. A single-step global mechanism is used in this study to represent gasphase combustion, with a future aim to induct multi-step detailed/reduced mechanism. It is necessary to develop a multi-step mechanism which can represent MMH– NTO combustion accurately and can be used in a CFD framework at lesser computational cost. The current model captured the basic combustion performance appropriately with an overprediction in flame temperature. The overestimation is attributed to the one-step global mechanism which neglects the decomposition of NTO and other potential intermediate species. This study fulfils the initial aim to set up a computational model for hypergolic combustion, which will be extended further for
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combustion dynamics studies using the large eddy simulation framework. The physical phenomena captured in this study reveal the importance of MMH inlet conditions on the combustion process. It is noticed that MMH evaporates slower than NTO, and evaporation rate of MMH can affect the ignition delay and stability of the combustor. A study to capture the effect of such factors on stability of this combustor is the scope for future work. Acknowledgements The technical help provided by ANSYS, India, team to conduct this study is kindly acknowledged.
Nomenclature ∼ ρ u p t x T i, j, k
Favre average (−) Density (kg/m3 ) Velocity (m/s) Pressure (Pa) Time (s) Dimension (m) Temperature (K) Index (–)
References 1. Catoire L, Chaumeix N, Paillard C (2004) Chemical Kinetic model for mono methyl hydrazine/ nitrogen tetroxide gas-phase combustion and hypergolic ignition. J Propul Power 20(1):87–92. https://doi.org/10.2514/1.9234 2. Catoire L, Chaumeix N, Pichon S, Paillard C (2006) Visualizations of gas-phase NTO/MMH reactivity. J Propul Power 22(1):120–126. https://doi.org/10.2514/1.10417 3. Jiang TL, Chiu H-H (1992) Bipropellant combustion in a liquid rocket combustion chamber. J Propul Power 8(5):995–1003 (1992) 4. Habiballah M, Lourme D, Pit F (1991) PHEDRE-numerical model for combustion stability studies applied to the Ariane viking engine. J Propul Power 7(3):322–329. https://doi.org/10. 2514/3.23330 5. Knab O, Preclik D, Estublier D (1998) Flow field prediction within liquid film cooled combustion chambers of storable bi-propellant rocket engines. In: 34th AIAA/ASME/SAE/ASEE joint propulsion conference and exhibit. AIAA-98-3370. Cleveland, OH 6. Zhuang FC, Nie WS, Zou Q (1999) Numerical simulation of MMH/NTO rocket engine combustion instability. In: 35th AIAA/ASME/SAE/ASEE joint propulsion conference and exhibit. AIAA-99-2779. Los Angeles, CA 7. Ishikawa Y, McQuaid MJ (2007) Reactions of NO2 with CH3 NHNH and CH3 NNH2 : a direct molecular dynamics study. J Mol Struct Theochem 818(1–3):119–124. https://doi.org/10.1016/ j.theochem.2007.05.014 8. Hou L, Fu P, Ba Y (2018) Chemical mechanism of MMH/NTO and simulation in a small liquid rocket engine. Combust Sci Technol. https://doi.org/10.1080/00102202.2018.1551214
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9. Qin J, Zhang H (2020) Numerical analysis of self-excited combustion instabilities in a small MMH/NTO liquid rocket engine. Hindawi Int J Aerosp Eng 2020:17. Article ID 3493214. https://doi.org/10.1155/2020/3493214 10. Xu KM, Cai GB, Zhou YT (2006) Flow field numerical study within thrust chamber of aerospace small liquid propellant rocket engine. In: AIAA 57th international astronautical congress, IAC, vol 10. Valencia, Spain, pp 6510–6519 11. Harvazinski ME, Huang C, Sankaran V, Feldman TW, Anderson WE, Merkle CL, Talley DG (2015) Coupling between hydrodynamics, acoustics, and heat release in a self-excited unstable combustor. Phys Fluids 27:045102 12. Zhang LB, Chu M, Xu X (2014) Performance prediction of apogee attitude and orbit control thruster for MMH/NTO hypergolic bipropellant. In: 50th AIAA/ASME/SAE/ASEE joint propulsion conference, Cleveland, OH, July 2014, p 3572 13. Ohminami K, Ogawa H, Uesugi KT (2009) Numerical bipropellant thruster simulation with hydrazine and NTO reduced kinetic reaction model. In: 47th AIAA aerospace sciences meeting including the 9ew horizons forum and aerospace exposition. AIAA-2009-452 14. Fluent ANSYS (2021) Ansys fluent 21 user’s guide. ANSYS, Inc., Canonsburg, PA 15. Sharma A, Tharakan JT, Kumar SS (2022) Analysis for design optimization of high thrust liquid engine hot test facility, Acta Astronaut 193:653–666. ISSN 0094-5765. https://doi.org/ 10.1016/j.actaastro.2021.07.047 16. Qiang W, Guozhu L (2019) Numerical simulation of ignition process for the monomethyl hydrazine–nitrogen tetroxide thrusters. J Propul Power 35(2):1–16. https://doi.org/10.2514/1. B37136
Microfluidics
Novel Tree Branching Microchannel Heat Sink Under Variable and Constant Fluid Volume Approaches Sangram Kumar Samal and Sandip Kumar Saha
Abstract In this work, two designs of radial tree branching microchannel heat sinks (TB-MCHSs) are proposed: (i) TB-MCHS with continuously increasing channel depth from the centre inlet to the radial outlet (TB-MCHS-DIV) and (ii) TBMCHS with continuously decreasing channel depth (TB-MCHS-CON). Thermohydrodynamic performance of TB-MCHS-DIV and TB-MCHS-CON is numerically evaluated and compared with TB-MCHS with constant channel depth (TB-MCHSST). The present work is carried out in two scenarios: In the first scenario, the fluid/ channel volumes of TB-MCHS-DIV and TB-MCHS-CON are significantly more than TB-MCHS-ST, whereas in the second scenario, the same is nearly equal to that of TB-MCHS-ST. The results reveal that in the second scenario compared to TBMCHS-ST, the proposed design shows slightly higher (~ 3%) average heat transfer coefficient, whereas it also shows higher (~ 7%) pressure drop. However, in the first scenario, the proposed design shows lower (~ 8%) average heat transfer coefficient than the TB-MCHS-ST, whereas it also shows lower (27%) pressure drop. The substantial reduction in the pressure drop dominated the reduction in heat transfer coefficient resulting in the performance evaluation criteria (PEC) value more than unity for the proposed design based on the variable fluid volume approach. Keywords Tree-like microchannel heat sink · Thermal management · Electronic chip cooling · Pumping power · Heat transfer coefficient
1 Introduction In the recent era of digitalisation, the demand for high-performance electronic devices has increased constantly. The continuous removal of high heat flux generated in these electronic devices is essential for maintaining the device temperature below its same operating limit; otherwise, this will lead to higher component failure and reduced longevity of these electronic devices. Due to the compact design, larger surface S. K. Samal (B) · S. K. Saha Department of Mechanical Engineering, IIT Bombay, Mumbai 400076, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_27
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area to volume ratio, and high heat flux dissipating capacity, the microchannel heat sink (MCHS) has become a promising solution for the thermal management of these high-performance electronic devices [1]. However, the conventional straight MCHSs are no more suitable for the thermal management of very high heat flux dissipating electronic devices. Therefore, the thermal performance of MCHSs is augmented by modifying the MCHS geometry, incorporating flow disruptions, using nanofluids as coolant, and modifying the MCHS surface [2]. The conventional straight MCHS with these techniques significantly augments its heat dissipation capacity; however, the pumping power requirement also increases. The natural branching networks such as leaf vein structure, plant root, and tree branching structure are believed to be efficient transport structures because, with the help of these branching systems, the plant spends less energy (i.e. less pumping power) during the transportation of water for survival [3]. Bejan and Errera [4] used the tree-like branching network as a heat sink for electronics cooling applications. They found that the tree-like branching network shows improved temperature distribution uniformity on the chip surface whilst reducing the pumping power requirement.
2 Literature Review and Objective After its introduction, the tree-like branching networks in the microchannel heat sink received immense recognition amongst researchers [5]. Pence [6] proposed a disc-shaped heat sink with fractal-like microchannel networks and found that the fractal-like MCHS shows a 60% lower pressure drop and reduced the wall temperature by 30 °C compared to the conventional straight MCHS. From the numerical investigations, Alharbi et al. [7] observed that the local pressure recovery at each branching level causes a lower total pressure drop in the fractal-like MCHS compared to the traditional straight MCHS. Wang et al. [8] numerically investigated the effect of bifurcation angle on the overall performance of tree-shaped MCHS. They found that temperature and pressure drop reduces at a lower bifurcation angle. Salimpour and Menbari [9] found that increasing the complexity (i.e. the number of branching levels) in a tree-shaped flow structure reduces thermal resistance and increases flow resistance. Huang et al. [10] proposed tree-shaped MCHS with ribs and cavities on sidewalls and compared their thermo-hydrodynamic performance with the traditional smooth tree-shaped MCHS. Their result demonstrated that continuous breaking and redeveloping of both the hydraulic and thermal boundary layers occur due to the presence of ribs and cavities in the tree-shaped MCHS, resulting in augmented thermal performance compared to the conventional smooth tree-shaped MCHS. Zhuang et al. [11] found that their proposed MCHS with a rhombus fractal-like structure augments the coefficient of performance by 7.9–68.7% compared to the traditional parallel MCHS under the same operating conditions. From the numerical investigation, Lu and Zhang [12] found that the fractal Y-shaped liquid cooling heat sink with three branching levels reduced the maximum and average surface temperature by 15.2 °C
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and 6.3 °C, respectively, and also reduced the pressure drop by 63% compared to the conventional S-shaped liquid cooling heat sink. Recently, Sakamatapan et al. [13] investigated the effect of the number of clusters (i.e. the number of fractal networks starting from the inlet plenum) on the thermo-hydrodynamic performance of a liquid-cooled heat sink based on fractal structures. Their result indicates that with increasing the number of clusters, the solid–fluid heat transfer area increases, resulting in enhanced Nusselt number, reduced surface temperature, and reduced pressure drop. The detailed literature review shows that the fractal-like tree-shaped MCHS is extensively studied due to its augmented heat transfer capability with lower pumping power requirement compared to the conventional parallel MCHS. However, all the studies on the fractal-like tree-shaped MCHS carried out so far considered constant channel depth throughout the fractal-like network. To the best of the authors’ knowledge, the effects of continuously varying channel depth on the overall performance of the fractal-like tree-shaped MCHS are seldom reported. Therefore, this work proposes a radial tree branching microchannel heat sink (TB-MCHS) with continuously varying channel depth from centre inlet to radial outlet that is generated using fixed length and width scaling ratios. The novelty of this study is to investigate the effect of continuously varying channel depth on the overall performance of TB-MCHS. For this purpose, two configurations of TB-MCHS (i) with increasing channel depth from inlet to outlet with a diverging angle (TB-MCHS-DIV) and (ii) with decreasing channel depth from inlet to outlet with a converging angle (TB-MCHS-CON) are developed.
3 Materials and Methods 3.1 Problem Description The TB-MCHS consists of a disc-type copper substrate with a diameter, d CS = 34 mm and a thickness, hCS = 1.5 mm on which six fractal-like microchannel networks are engraved, as shown in Fig. 1. Geometrical dimensions of the fractallike microchannel network are evaluated using the optimal width scaling ratio (β = wi+1 /wi = 2−1/2 ) and length scaling ratio (γ = L i+1 /L i = 2−1/2 ) [14], where i indicates lower-order branching level and i + 1 indicates higher-order branching level. The total flow length of a fractal-like microchannel network is 17.5 mm. The working fluid enters through the inlet at the centre of the disc, flows in the fractallike microchannel network, and exits through two outlets from the outlet manifold ring. All the geometrical dimensions are described in Table 1. To investigate the effects of geometrical configuration on the thermo-hydrodynamic performance of TB-MCHS, three different designs of TB-MCHS are considered: They are (i) straight (TB-MCHS-ST), (ii) diverging (TB-MCHS-DIV), and (iii) converging (TB-MCHSCON). All the dimensions of these three designs are the same except for the depth
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Fig. 1 Schematic diagram of TB-MCHS along with fluid domains of TB-MCHS-ST, TB-MCHSDIV, and TB-MCHS-CON
of the microchannel. In TB-MCHS-ST, the microchannel depth is kept constant (h = 0.7 mm) throughout the fractal network. However, in TB-MCHS-DIV and TBMCHS-CON, the microchannel depth varies along the flow length depending on the diverging and converging angle α = 1°. The present work is carried out in two scenarios: In the first scenario, the fluid/channel volumes of TB-MCHS-DIV and TBMCHS-CON are significantly more than TB-MCHS-ST, for which the channel depth is varied from 0.7 mm at the start of the tree-like network (i.e. h0,in ) and 0.97 mm at the end of the tree-like network (i.e. h0,out ) for TB-MCHS-DIV-1°, and vice versa for TB-MCHS-CON-1°. In the second scenario, the total fluid volume of TB-MCHSDIV-1° and TB-MCHS-CON-1° is within ± 3% of TB-MCHS-ST, for which the channel depth is varied from 0.57 mm at the start of the tree-like network (i.e. h0,in ) and 0.83 mm at the end of the tree-like network (i.e. h0,out ) for TB-MCHS-DIV-1°, and vice versa for TB-MCHS-CON-1°. It is to be noted that the x–y–z coordinate system is at the centre of the copper substrate’s top surface.
3.2 Governing Equations In this work, all the simulations are carried out using the commercially available finite volume method (FVM)-based computational fluid dynamics (CFD) software
Novel Tree Branching Microchannel Heat Sink Under Variable … Table 1 Geometrical dimensions of the TB-MCHS
323
Overall geometrical dimensions of TB-MCHS Diameter of copper substrate, d CS
34 mm
Total thickness of copper substrate, hCS
1.5 mm
Inlet diameter, d in
3.4 mm
Outlet diameter, d out
1.7 mm
Diameter of outlet ring, d OR
44 mm
Fractal-like microchannel network dimensions Level, i
Length, L (mm)
Width, w
Angle, θ (°)
0
7.93
1
60
1
5.61
0.7
60
2
3.96
0.5
45
ANSYS Fluent V2020, which is used to solve the governing differential equations like continuity, momentum, and energy equations. The fluid flow is assumed to be incompressible, steady, and laminar. It is also assumed that there is no internal heat generation in the substrate and heat loss from the MCHS through natural convection and radiation. The channel walls are considered to be smooth. Temperaturedependent thermophysical properties of water are considered to accurately analyse their effects on the fluid flow and heat transfer characteristics. However, the thermophysical properties of the copper substrate are considered to be temperature independent. Table 2 depicts the thermophysical properties of water and copper, where T is the fluid temperature in kelvin (K). Under the above assumptions, the continuity, momentum, and energy equations are as follows: ( − →) Continuity equation: ∇ · ρf V = 0
(1)
( ( − →− →) − →) Momentum equation: ∇ · ρf V V = − ∇ p + ∇ · μf ∇ V
(2)
Energy equations: ) ( − → Fluid domain: ∇ · ρf V Cp,f Tf = ∇ · (kf ∇Tf )
(3)
Solid domain: ks ∇ 2 Ts = 0
(4)
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Table 2 Thermophysical properties of water and copper Parameters
Water [15]
ρ (kg/m3 )
429.78566 + 4.99588T − 0.01349T 2
Copper [16]
+ 1.04448 × 10 k (W/mK)
−8
T
−5
T
−5
T
385
3
0.06111 − 4.96758 × 10−4 T + 1.36371 × 10−6 T 2 − 1.25788 × 10
401
3
6173.3346 − 15.78198T + 0.03966T 2 − 3.06353 × 10
μ (Pa s)
T
8933
3
− 1.50487 + 0.01422T − 2.99256 × 10−5 T 2 + 2.012 × 10
C p (J/kgK)
−5
–
3
3.3 Boundary Conditions and Calculation Methods Water at 300 K enters through the inlet at a uniform velocity v corresponding to the inlet volumetric flow rate range Q = 25–100 ml/min. Zero pressure outlet boundary condition is applied at the outlet. The substrate bottom surface is subjected to a constant heat flux of q'' = 10 W/cm2 . Both no-slip and conjugate heat transfer conditions are applied at the solid–fluid interface. As the TB-MCHS geometry is symmetry, half of it (i.e. about the y–z plane as indicated with the red dotted box in Fig. 1b) is considered as the computational domain by using symmetry boundary conditions. Other surfaces exposed to the atmosphere are considered to be insulated. The pressure equation is discretised using the second-order discretisation scheme in this work. The second-order upwind scheme is used to discretise both momentum and energy equations. The SIMPLE algorithm is used for the pressure–velocity coupling. The convergence criterion for continuity and momentum equations is set to 10−6 , whilst for the energy equation, it is set to 10−8 .
3.4 Grid Independent Test The grid independence test (GIT) is carried out by considering different numbers of unstructured polyhedral mesh elements to choose the optimum grid size for the reduced computational time. For the brevity of this paper, the GIT result of TBMCHS-ST is described in this section; however, the GIT is also carried out for TB-MCHS-DIV and TB-MCHS-CON. For the GIT, the inlet volumetric flow rate of Q = 25 ml/min and applied heat flux of q'' = 10 W/cm2 are considered. The pressure drop (Δp) is evaluated for all the mesh element sizes considered for the GIT and is depicted in Table 3. From Table 3, it can be seen that the deviation in Δp is 0.97% between Mesh-4 (1,831,549) and Mesh-5 (3,382,203). Therefore, the Mesh-4 is considered for the simulation in this work.
Novel Tree Branching Microchannel Heat Sink Under Variable … Table 3 Grid independent test
325 Δp (Pa)
%age diff.
1
557,847
46.295
5.46
2
750,015
45.700
4.11
3
1,124,312
44.973
2.45
4
1,831,549
44.322
0.97
5
3,382,203
43.898
–
Mesh
No. of elements
3.5 Numerical Model Validation The validity of the present numerical model is ascertained by comparing pressure drop and heat transfer in the silicon fractal-like microchannel heat sink for varying Reynolds numbers with the experimental work of Zhang et al. [17]. The pressure drop in the fractal-like microchannel heat sink is evaluated from the present numerical results and compared with the experimental results of Zhang et al. [17] and is depicted in Fig. 2. It is evident that the present numerical model agrees well with the experimental results of Zhang et al. [17]. The maximum and average deviations for pressure drop are 12.64% and 4.95%, respectively. This deviation in pressure drop may be attributed to inconsideration of surface roughness as well as the heat loss from the heat sink. Under the assumptions mentioned earlier, the present numerical model is reasonably validated with the experimental work [17]. Fig. 2 Comparison of pressure drop between the present model and the experimental results of Zhang et al. [17]
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4 Results and Discussion In this work, a three-dimensional numerical study is carried out to investigate the thermo-hydrodynamic performance of TB-MCHS with continuously varying channel depth along the flow length in terms of diverging (TB-MCHS-DIV) and converging (TB-MCHS-CON). The results for all the designs (i.e. TB-MCHS-ST, TB-MCHS-DIV-1°, and TB-MCHS-CON-1°) in both variable fluid volume and constant fluid volume scenarios are reported and discussed. It is to be noted that TB-MCHS-DIV-1° indicates diverging TB-MCHS with angle, α = 1°; similarly, TB-MCHS-CON-1° indicates converging TB-MCHS with angle, α = 1°. The thermo-hydrodynamic analysis is carried out by evaluating the average heat transfer coefficient (havg ), and pressure drop (Δp). The average convective heat transfer coefficient is defined as follows: h avg =
q '' × Ab Aw [Tw − Tf ]
(5)
Here, q'' is the applied heat flux at the substrate bottom surface, Ab is the substrate bottom surface area, Aw is the wetted surface area (i.e. solid–fluid interface area), T w is the area-weighted average solid–fluid interface wall temperature, and T f is the volume-averaged fluid temperature. The pressure drop is defined as follows: Δp = pin − pout
(6)
The variation of average heat transfer coefficient (havg ) with inlet volumetric flow rate (Q) for all the designs of TB-MCHS is depicted in Fig. 3. It can be seen from Fig. 3 that with increase in volumetric flow rate, the havg increases. This is because, at a higher volumetric flow rate, the fluid flows with a higher velocity which enhances the flow mixing and reduces the thermal boundary layer thickness, resulting in the enhanced convective heat transport by the fluid. In the variable fluid volume scenario, both the TB-MCHS-DIV-1° and TB-MCHS-CON-1° show lower havg compared to TB-MCHS-ST. As mentioned in Sect. 3.1, for TB-MCHS-DIV1°, the channel depth at the inlet of the tree-like channel network is constant (i.e. 0.7 mm) as of TB-MCHS-ST, and the channel depth at the outlet of the network is 0.97 mm corresponding to the diverging angle α = 1°, and vice versa for TBMCHS-CON-1°. Due to this constraint, the fluid volumes of TB-MCHS-DIV-1° and TB-MCHS-CON-1° are significantly different from that of TB-MCHS-ST. The total fluid volume in TB-MCHS-DIV-1° and TB-MCHS-CON-1° is increased by 22% and 16%, respectively, compared to TB-MCHS-ST. Similarly, the wetted surface area in TB-MCHS-DIV-1° and TB-MCHS-CON-1° is also increased by 17% and 9%, respectively, compared to TB-MCHS-ST. However, the wetted surface area to volume ratio in TB-MCHS-DIV-1° and TB-MCHS-CON-1° is lowered by 4.6% and 6.2%, respectively, compared to TB-MCHS-ST. Due to the lower wetted surface area to volume ratio, both the TB-MCHS-DIV-1° and TB-MCHS-CON-1° show reduced
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Fig. 3 Variation of average heat transfer coefficient (havg ) with inlet volumetric flow rate (Q) in TB-MCHS-ST, TB-MCHS-DIV-1°, and TB-MCHS-CON-1° for variable fluid volume and constant fluid volume scenarios
average heat transfer coefficient (i.e. maximum of 8.4% and 5.1%, respectively) compared to TB-MCHS-ST. Further, in the constant fluid volume scenario, the TB-MCHS-CON-1° shows higher havg and the TB-MCHS-DIV-1° shows lower havg compared to TB-MCHS-ST. To maintain the equivalent fluid volume as that of TB-MCHS-ST and the diverging and converging angle, α = 1°, the channel depth is varied from 0.57 mm at the start of the tree-like network and 0.83 mm at the end of the tree-like network for TB-MCHS-DIV-1°, and vice versa for TB-MCHS-CON-1°. Compared to the TBMCHS-ST, the total fluid volume of TB-MCHS-DIV-1° and TB-MCHS-CON-1° is within ± 3%, whereas the wetted surface area is within ± 3.7%. As the channel depth continuously varies along the flow length from the inlet to the outlet of a treelike network, the cross-sectional area and the hydraulic diameter vary in the TBMCHS-DIV-1° and TB-MCHS-CON-1°. Due to the smaller channel depth, the local flow cross-sectional area along the terminal channel is less in TB-MCHS-CON-1° compared to the TB-MCHS-ST and the TB-MCHS-DIV-1°, resulting in higher fluid velocity for a given inlet volume flow rate. The increase in the fluid velocity leads to thinning of the boundary layer, thereby enhancing the local heat transfer coefficient by a maximum of 3% and 6%, respectively, compared to the TB-MCHS-ST and the TB-MCHS-DIV-1°. The variation of the pressure drop (Δp) with the inlet volumetric flow rate (Q) for TB-MCHS-ST, TB-MCHS-DIV-1°, and TB-MCHS-CON-1° is illustrated in Fig. 4. It can be observed from Fig. 4 that the pressure drop increases with the increasing volume flow rate. This is attributed to the increased fluid velocity with volume flow rate. Figure 4 shows that compared to TB-MCHS-ST, both the TB-MCHS-DIV-1° and TB-MCHS-CON-1° show lower Δp in variable fluid volume scenario, whereas they show higher Δp in constant fluid volume scenario. In the variable fluid volume scenario, due to the larger fluid volume, the TB-MCHS-DIV-1° and TB-MCHSCON-1° have a larger local flow cross-sectional area than TB-MCHS-ST, due to
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Fig. 4 Variation of pressure drop (Δp) with inlet volumetric flow rate (Q) in TB-MCHS-ST, TB-MCHS-DIV-1°, and TB-MCHS-CON-1° for variable fluid volume and constant fluid volume scenarios
which the flow resistance is less than TB-MCHS-ST. Thus, both the TB-MCHSDIV-1° and TB-MCHS-CON-1° show a significant reduction in the pressure drop (i.e. maximum of 26.85% and 27.49%, respectively) than TB-MCHS-ST. Further, in the constant fluid volume scenario, the inlet flow cross-sectional area is less in the TB-MCHS-DIV-1° than in the TB-MCHS-ST, resulting in a higher fluid velocity at the inlet of the tree-like network for a given inlet volume flow rate. Due to the higher flow resistance in the level-0 channel, the pressure drop is higher by a maximum of 7.2% in TB-MCHS-DIV-1° than in the TB-MCHS-ST. Similarly, in TB-MCHS-CON-1°, the cross-sectional area at the exit of the terminal channel is less by 18.57% than the TB-MCHS-ST, resulting in a higher fluid velocity in the terminal channels for a given inlet volume flow rate. Due to the higher flow resistance in the terminal channels (i.e. level-2 channel), the pressure drop is higher by a maximum of 6.5% in the TB-MCHS-CON-1° than in the TB-MCHS-ST. From the variation of havg and Δp, it is observed that the in the variable fluid volume scenario, both the TB-MCHS-DIV-1° and TB-MCHS-CON-1° show lower havg (negative effect) and lower Δp (positive effect) compared to TB-MCHS-ST. Similarly, in the constant fluid volume scenario, the TB-MCHS-CON-1° shows higher havg (positive effect) and a higher Δp (negative effect) compared to the TBMCHS-ST. Therefore, to evaluate the overall thermo-hydrodynamic performance of the modified designs (here these are TB-MCHS-DIV-1° and TB-MCHS-CON-1°) based on the referenced design (here it is TB-MCHS-ST), the performance evaluation criteria (PEC) are considered and are defined as follows: ( PEC =
) ( )1 h avg Δp 3 / h avg,0 Δp0
(7)
Here, havg,0 and Δp0 are the average heat transfer coefficient and pressure drop of the referenced design (here it is TB-MCHS-ST), respectively. It is noteworthy
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Fig. 5 Variation of performance evaluation criteria (PEC) with inlet volumetric flow rate (Q) in, TB-MCHS-DIV-1°, and TB-MCHS-CON-1° for variable fluid volume and constant fluid volume scenarios
that the PEC value greater than unity indicates better overall thermo-hydrodynamic performance. The performance evaluation criteria (PEC) are evaluated for both the TB-MCHSDIV-1° and TB-MCHS-CON-1° with respect to the TB-MCHS-ST, and its variation with inlet volumetric flow rate is depicted in Fig. 5. From Fig. 5, it can be observed that for variable fluid volume scenario, both the TB-MCHS-DIV-1° and TB-MCHSCON-1° show PEC value more than unity. This is due to the substantial reduction Δp dominated the reduction in havg resulting in the PEC value more than unity for TBMCHS-DIV-1° and TB-MCHS-CON-1°. The TB-MCHS-DIV-1° shows a maximum PEC value of 1.09, whereas the TB-MCHS-CON-1° shows a maximum PEC value of 1.06. Further, for the constant fluid volume scenario, the TB-MCHS-DIV-1° shows a PEC value below unity. This is because the TB-MCHS-DIV-1° shows lower havg and higher Δp than the TB-MCHS-ST. However, the TB-MCHS-CON-1° shows the PEC value below unity for Q = 25 and 50 ml/min, and it shows slightly more than unity PEC values for Q = 75 and 100 ml/min. This is because in TB-MCHS-CON-1°, at a higher flow rate, the increase in heat transfer coefficient dominates the increase in pressure drop, which gives a slightly higher PEC value than unity at a higher volume flow rate.
5 Conclusions In this work, two novel configurations of radial tree branching microchannel heat sinks (TB-MCHSs) with continuously varying channel depths are proposed. One configuration has channel depth continuously increasing from the inlet to the outlet with a diverging angle (TB-MCHS-DIV), and the other configuration has channel
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depth continuously decreasing from inlet to outlet with a converging angle (TBMCHS-CON). The thermo-hydrodynamic analysis is carried out numerically for the TB-MCHS-DIV-1° and TB-MCHS-CON-1° and compared with the traditional TB-MCHS with constant channel depth (TB-MCHS-ST) for variable and constant fluid volume scenarios. The result reveals that for the variable fluid volume scenario, both the TB-MCHS-DIV-1° and TB-MCHS-CON-1° show lower havg compared to TB-MCHS-ST; however, the significant reduction in Δp (i.e. maximum of 26.85% and 27.49%, respectively) dominated the reduction in havg resulting in the PEC value more than unity. Similarly, for the constant fluid volume scenario, the TB-MCHSDIV-1° shows lower havg and higher Δp than TB-MCHS-ST, whereas TB-MCHSCON-1° shows higher havg and higher Δp, due to the dominance of the increase in havg over the increase in Δp resulting in the PEC value slightly more than unity for TB-MCHS-CON-1°. Overall, it can be concluded that the variable fluid volume TB-MCHS shows a significant reduction in pressure drop compared to the constant fluid volume TB-MCHS, thus resulting in a PEC value more than unity. Acknowledgements No specific grant is received for the research work presented herein.
Nomenclature Ab Aw Cp d CS d in d OR d out h havg hcs k L p Δp PEC q'' Q T Tf Tw v − → V w
Area of the substrate bottom surface (mm2 ) Wetted surface area (mm2 ) Specific heat (J/kgK) Diameter of copper substrate (mm) Inlet diameter (mm) Diameter of outlet ring (mm) Outlet diameter (mm) Microchannel depth (mm) Average heat transfer coefficient (W/m2 K) Total thickness of copper substrate (mm) Thermal conductivity (W/mK) Microchannel flow length (mm) Pressure (Pa) Pressure drop (Pa) Performance evaluation criteria (–) Applied heat flux (W/m2 ) Volumetric flow rate (ml/min) Temperature (K) Volume-averaged fluid temperature (K) Area-weighted average wall temperature (K) Fluid velocity at inlet (m/s) Velocity vector (m/s) Channel width (mm)
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α β γ μ ρ θ
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Diverging and converging angle (°) Width ratio (–) Length ratio (–) Dynamic viscosity (kg/ms) Density (kg/m3 ) Branching angle (°)
References 1. Kandlikar S, Garimella S, Li D, Colin S, King MR (2005) Heat transfer and fluid flow in minichannels and microchannels. Elsevier, Waltham 2. Alihosseini Y, Targhi MZ, Heyhat MM, Ghorbani N (2020) Effect of a micro heat sink geometric design on thermo-hydraulic performance: a review. Appl Therm Eng 170:114974-1–25 3. Bejan A (2000) Shape and structure, from engineering to nature. Cambridge University Press, Cambridge 4. Bejan A, Errera MR (2000) Convective trees of fluid channels for volumetric cooling. Int J Heat Mass Transf 43(17):3105–3118 5. Xu P, Sasmito AP, Yu B, Mujumdar AS (2016) Transport phenomena and properties in tree-like networks. Appl Mech Rev 68(4):040802-1–17 6. Pence D (2003) Reduced pumping power and wall temperature in microchannel heat sinks with fractal-like branching channel networks. Microscale Thermophys Eng 6(4):319–330 7. Alharbi AY, Pence DV, Cullion RN (2003) Fluid flow through microscale fractal-like branching channel networks. ASME J Fluids Eng 125(6):1051–1057 8. Wang XQ, Mujumdar AS, Yap C (2007) Effect of bifurcation angle in tree-shaped microchannel networks. J Appl Phys 102(7):073530-1–8 9. Salimpour MR, Menbari A (2014) Constructal design of cooling channels embedded in a ring-shaped heat-generating body. Energy 73:302–310 10. Huang P, Dong G, Zhong X, Pan M (2020) Numerical investigation of the fluid flow and heat transfer characteristics of tree-shaped microchannel heat sink with variable cross-section. Chem Eng Process Process Intensif 147:107769-1–10 11. Zhuang D, Yang Y, Ding G, Du X, Hu Z (2020) Optimisation of microchannel heat sink with rhombus fractal-like units for electronic chip cooling. Int J Refrig 116:108–118 12. Lu Z, Zhang K (2021) Study on the performance of a Y-shaped liquid cooling heat sink based on constructal law for electronic chip cooling. ASME J Therm Sci Eng Appl 13(3):034501-1–7 13. Sakamatapan K, Mesgarpour M, Kaew-On J, Dalkılıç AS, Ahn HS, Mahian O, Wongwises S (2022) Novel design of a liquid-cooled heat sink for a high-performance processor based on constructal theory: a numerical and experimental approach. Alex Eng J 61(12):10341–10358 14. Pence D, Enfield K (2004) Inherent benefits in microscale fractal-like devices for enhanced transport phenomena. In: Collins M, Brebbia CA (eds) Design and nature. WIT Press, Rhodes, GR, pp 317–328 15. Lemmon EW, Huber ML, McLinden MO (2010) NIST standard reference database 23, reference fluid thermodynamic and transport properties (REFPROP), version 9.0. National Institute of Standards and Technology. R1234yf. Fld file dated 22 Dec 2010 16. Bergman TL, Lavine AS, Incropera FP, Dewitt DP (2011) Fundamentals of heat and mass transfer. Wiley, New York 17. Zhang CP, Lian YF, Yu XF, Liu W, Teng JT, Xu TT, Hsu CH, Chang YJ, Greif R (2013) Numerical and experimental studies on laminar hydrodynamic and thermal characteristics in fractal-like microchannel networks. Part B: comparisons of two numerical analysis methods on friction factor and Nusselt number. Int J Heat Mass Transf 66:939–947
Two-Dimensional, Magnetic Actuation of Ferrofluid Droplet on an Open-Surface Microfluidic Platform Debiprasad Chakrabarty, Niladri Chakraborty, and Ranjan Ganguly
Abstract The role of non-contact manipulation of discrete droplets on surface microfluidic platforms has wide applications in development of low-cost biosensors and biomedical diagnostic systems. Magnetic digital microfluidic platforms utilize magnetic force to actuate ferrofluid droplets on open hydrophobic surface and offer distinctive benefits compared to other digital microfluidic actuation schemes. This allows droplets—containing different type of sample and reagents—to be actuated and controlled independently; this can be leveraged to achieve different bioanalytical protocols for point-of-care diagnosis and other different micro-total analytical systems. Here, investigate, through a physically realistic model, magnetic fieldactuated transport of a spherical cap ferrofluid droplet on a surface microfluidic platform. Manipulation of a microliter volume droplet in a sequence of rectilinear paths, leading to a guided transport, is achieved through an array of double-layer, planar, square-shaped electromagnetic micro-coils embedded in the substrate. Appropriate sequence of coil energization for attaining the desired trajectory of the droplet is described. The study paves the foundation of developing more complex digital microfluidic devices for different biomicrofluidic applications. Keywords Magnetic fluid · Microfluidics · Droplets · Surface tension · Numerical model
D. Chakrabarty (B) Department of Electrical Engineering, College of Engineering and Management, Kolaghat, West Bengal 721171, India e-mail: [email protected] N. Chakraborty · R. Ganguly Department of Power Engineering, Jadavpur University, LB 8, Sector 3, Salt Lake, Kolkata 700106, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_28
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1 Introduction Digital microfluidics (DMF) is an emerging fluid handling technology, which has a wide range of significant applications in the areas of automation, miniaturization and biochemical applications. In surface-type DMF, liquid is manipulated in the form of discrete droplets containing samples and reagents on an open, integrated microfluidic platform [1–4] instead of an enclosed micro-channel. Droplet-based surface-type DMF (SDMF) has many outstanding features such as portability, less sample consumption, shorter chemical reaction time and flexibility compared to the conventional flow-through microfluidics. Here, each droplet can be controlled independently without complex networks of channels, pumps, valves or mechanical mixers. Thus, numerous reaction processes can be achieved at the same time with a simple and compact design. Magnetic forced-based SDMF offers distinctive advantages where droplet filled with magnetic particles actuates under influence of magnetic force. Magnetic SDMF is easy to implement in terms of fabrication and integration for point-of-care diagnostics with flexible controllability and high accuracy. Here, flat surfaces of silica, PMMA or PDMA, coated with a low surface energy materials such as Teflon AF (amorphous fluoropolymer), may be used to support droplets with limited spreading, and transport those using localized magnetic field gradients using miniaturized permanent magnet [5] or electromagnet coils [6] or by combination of both permanent magnet and electromagnet. Prior works on flowthrough magnetic DMF [7, 8] have shown the feasibility of droplet actuation using permanent magnet or electromagnetic rail-based approaches, but they are limited in terms of flexibility, and sans the controllability, for example, the capability of droplet operation along predefined paths. To attain two-dimensional flexible and controllable operation of ferrofluid droplets on a hydrophobic surface, planar microcoil array [9, 10] or combination of planar coils coupled with bias field of permanent magnets [9] can be a more suitable substitute. Nguyen et al. [11] proposed a simple magnetic digital microfluidic device to drive ferrofluid droplet by four planar coils in combination with a pair of permanent magnets and a soft magnetic steel sheet. Manipulation of large amounts of individual droplets using a deformable magnetic membrane was achieved by Chen et al. [12]. In our previous study, we demonstrated a complex, two-dimensional manipulation of liquid droplets over a thin aqueous film on an open surface using a periodically switched array of electromagnetic micro-coils [13]. However, compatibility of the film fluid with the droplet liquid and the stability of the film pose constraints. So to achieve more precise, flexible and wide range two-dimensional droplet manipulation, solid surface with adequate repellence to the droplet fluid is preferred. Repellence of the surface to the droplet fluid (rendering a high sessile droplet contact angle) is a necessary condition to attain a distinct, transportable droplet as opposed to a liquid that wets, spreads and sticks to the surface. Here, we present a numerical model to analyse controlled manipulation of spherical
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cap ferrofluid droplets on a solid substrate. Periodically switched array of electromagnetic micro-coils is used to generate a spatio-temporal distribution of magnetic field that steers the ferrofluid droplet on the surface. We use square coil with different orientation and dimension to get better packing density (minimizing the dead space between the neighbouring coils).
2 Theoretical Formulation Figure 1a shows the arrangement of double-layer planar square coils of two different sizes (six 20 mm2 square coil in 1st layer and six 10 mm2 square coil in 2nd layer) in a two-stranded array of 12 coils (typical dimension of a stranded array is 40 mm × 60 mm) fabricated on a flat PDMS substrate of 50 mm × 70 mm. The smaller coils are positioned in the upper tier accurately in between two larger coils (20 mm2 ) which act as a booster field to accelerate the droplets as they pass over one coil to the other. The two layers of coils are separated by an insulating layer of 350 µm thickness. A thin (2000 µm) coating of hydrophobic material such as Teflon AF placed above the stranded array of micro-coil-fabricated PDMS substrate. Figure 1b shows a single square coil with detail specification. Each square spiral consists of 12 turn of conductors at 500 µm pitch. The conductors have 500 µm × 500 µm square cross section. Figure 1c shows the magnetic field (Tesla) computed on the hydrophobic surface (i.e. at z = 2 mm) for a coil carrying current 5 A. The magnetic field is computed using Biot–Savart law [14] following the approach of Santra et al. [15].
Fig. 1 Coil configuration: a arrangement of a 12 planar square coils of 20 mm × 20 mm (6 coils of 12 turn each) and 10 mm × 10 mm (6 coils of 6 turn each) size and in a two-stranded array. b Magnetic field (Tesla) computed on the hydrophobic surface (i.e. at z = 2 mm) for a coil (20 mm × 20 mm) carrying current 5 A
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2.1 Force Due to Magnetic Field of Current Loops of Regular Polygons Magnetic manipulation of the microliter volume spherical cap ferrofluid droplet driven by magnetic force field is described schematically in Fig. 2. The magnetic force F mag on such a droplet (considered homogeneous magnetization M induced due to an imposed field H) within the magnetic field is calculated from the general expression [15]. (| |) Fmag = 1/2μ0 Vsd χm ∇ | H 2 |
(1)
Here, H is the magnetic field at the centre of circular footprint of spherical cap droplet in consideration of the demagnetization effect caused by the magnetic material due to magnetic polarization. χ m the measured susceptibility of the droplet including demagnetization effects. It is assumed that the integrand is constant over the volume V sd of ferrofluid droplet with its radius RFF (valid for a very small drop size, such that the local field is almost constant over the entire volume of the droplet), and that H is curl free (since there are no free currents outside the copper coils). The volume of spherical cap ferrofluid droplet at rest or equilibrium condition is [16]. ( { )} Vsd = π D 3 2 − 3 cos θ + cos3 θ /24 sin3 θ
(2)
Here, D is the diameter of circular footprint of the spherical cap droplet and h is the height of droplet. Droplet equilibrium contact angle is θ. The volume of asymmetrical spherical cap ferrofluid droplet sliding on a solid substrate due to an external force (with effect of contact angle hysteresis) will vary as droplet will not retain in equilibrium shape due to contact angle hysteresis. The volume of asymmetrical spherical cap droplet is further calculated following the approach of Elsherbini et al. [17].
Fig. 2 Schematic of a spherical cap ferrofluid droplets transport model on a solid hydrophobic surface over planar electromagnet coil. F mag , F n and F hyst denote magnetic force, resistive surface frictional drag and contact angle hysteresis force, respectively
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2.2 Spherical Cap Ferrofluid Droplet Motion on a Solid Hydrophobic Surface Spherical cap ferrofluid droplet is assumed to move on the solid hydrophobic surface under actuation of magnetic force generated by planar electromagnet coil (shown in set of Fig. 2) where a resistive surface frictional drag and contact angle hysteresis force act against the sliding motion. The equation of motion for the spherical cap ferrofluid droplet of mass mFF in the x–y plane on a solid surface is obtained by balancing the net unbalanced forces due to frictional drag and contact angle hysteresis [18], i.e. m FF d2 S/dt 2 = Fmag (x, y) − Fhyst (x, y) − Fn (x, y)
(3)
Here, mFF and S denote the mass and position vector of spherical cap ferrofluid droplet on the x–y plane, respectively. Droplet mass m FF = Vsd ρ, where ρ is the density droplet. The resistive frictional drag is calculated as [19] follows: ∫xmax Fn = 3ηFF πrU xmin
dx ξ(x)
(4)
Here, ηFF and r are the viscosity and footprint radius of the cap volume droplet, respectively, U = dS/dt represents velocity of the droplet in x–y plane, x min = 10−9 mm, x max = D − 10−9 (D = 2r is footprint diameter of cap volume droplet) and ( ) √ ( ) ξ(x) = 0.5 − 2R cos θ + (2R cos θ )2 − 4 x 2 − 2x sin θ (5) Here, R is the radius of extended sphere profile of cap volume droplet at equilibrium position, and θ is the equilibrium contact angle. The contact angles formed by advancing and receding liquid fronts are denoted by θ A and θ R , respectively. Contact angle hysteresis is defined as difference between the advancing and the receding contact angles, although the term is also used to describe the expression (cos θ R − cos θ A ) [20]. The in-plane resistive force due to contact angle hysteresis experienced by the droplet against any motion is given by Fhyst = γlvr k(cos θR − cos θA )
(6)
Here, k is a constant, γ lv is the liquid–vapour surface tension, as recommended in prior work of Elsherbini [21, 22]. Initially, when U = 0 and no force applies on the droplet, both the advancing and receding contact angles are equal to the static contact angle (θ A = θ R = θ m ). When a small force (the magnetic force in this case) is
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imposed on the droplet, the droplet bulk tends to move, but its footprint stays pinned on the substrate; the droplet deforms asymmetrically, leading to an increase in θ A and a decrease in θ R . The droplet does not move until the net in-plane component of the driving force overcomes the static contact angle hysteresis force; droplets start moving and the difference (θ A − θ R ) increases with increasing the capillary number (Ca = ηU/γ ), which denotes the ratio of viscous to surface tension force [23]. This hysteresis is called dynamic contact angle hysteresis. A model [24] derived from a combination of hydrodynamic [25, 26] and molecular kinetic [27, 28] models is used to calculate the dynamic advancing contact angle θ A and the receding contact angle θ R as a function of the droplet capillary number. [ ( ) ( ( ))]3 L U 2kB T −1 −1 = cos sinh + 9Ca ln cos θm − 2 0 γlv λ 2k λ Ls ( ( ))]3 [ ( ) U L 2kB T θR3 = cos−1 cos θm + sinh−1 − 9Ca ln 2 0 γlv λ 2k λ Ls
θA3
(7)
(8)
Here, k B is the Boltzmann constant, L is the characteristic capillary length, L S is the slip length and T is the absolute temperature. Here, k 0 and λ are the fitting parameters. Rearranging the transport equation (Eq. 3), equations of motion of the spherical cap ferrofluid droplets take following forms (| |) γlv Rk d2 S 1 μ0 χm ∇ | H 2 | − = (cos θR − cos θA ) 2 dt 2 ρ Vsd ρ ∫xmax 3ηFF πr dx − U Vsd ρ ξ(x)
(9)
xmin
Equation (9) is a second-order ordinary differential equation for droplet position vector S, which is solved (using Euler explicit scheme of integration [29]) on a MATLAB platform with initial conditions of droplet released velocity at the point of their release, i.e. at t = 0, U = U 0 , S = S 0 and z = z0 . The time step of integration dt is adaptively chosen so that dt ≤ 0.01D/|U|.
3 Validation of the Droplet Transport Model The numerical model used to manipulate spherical cap volume droplets on solid substrate using periodically switching array of electromagnetic micro-coil was adopted from our earlier work [13], which was also validated experimentally. For further validation, the magnetic force computed by the present model was also
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Fig. 3 Validation of the droplet transport model: a 10 turn square coil configuration as considered as Fulcrand et al. [30]. b Variation of magnetic force on the magnetic particles versus height from the coil plane validation between Fulcrand’s experiment and our simulation. c Micro-bead velocity as a function of its distance from the coil centre obtained from the both experiment and simulation
compared with the magnetic field model by Fulcrand et al. [30], who used multiple electromagnetic micro-coils akin to the present work to manipulate magnetic particles suspended in laminar flow inside a microfluidic channel with flow rate not more than 1 µm/min. Magnetic force on magnetic microbeads—having diameter of 2.8 µm, particle density of 1.6 g cm−3 and an effective magnetic susceptibility of 0.17—was calculated using Biot–Savart law for different z-distance of from the coils. Figure 3a shows the square coil configurations (10 turn) considered from the work of Fulcrand et al. [30] adopted for validating the transport model. Figure 3b shows the comparison of magnetic force on the magnetic particles calculated as a function of height from the coil plane. Figure 3c shows the micro-bead velocity as a function of its distance from the coil centre obtained from the both experiment and our simulation. Both results indicate a good match between simulation and experiment.
4 Results and Discussion Having validated the droplet transport model, trajectory of a spherical cap ferrofluid droplet on the solid hydrophobic surface, under the actuation of magnetic force generated by the two-tier array of planar electromagnet coils implanted underneath the solid hydrophobic surface (shown in set of Fig. 2) is estimated. Figure 4a shows the droplet trajectory where a 3 mm [31] base diameter spherical cap volume ferrofluid droplet (viscosity of 0.005 Pa s [32]) is released on the solid surface at the edge of coil number 1 (see Fig. 1a). Initially, droplet starts from rest with very low acceleration as the net force acting on the droplet is very small due to its large distance from the coil centre ahead of it. As it slowly moves towards centre of the coil, the force and hence the acceleration increase and so does the droplet velocity. Here, coil 1 is switched on for 0.77 s when droplet is transported from the edge of this coil to the coil centre. As soon as droplet reach the centre of the first coil (coil 1), it is switched off with simultaneous switching on of the next coil (the coil 2). Right after this coil switching,
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the droplet velocity falls sharply—magnetic force on the droplet reduces suddenly because its large initial distance from coil 2 immediately after the coil switching event—but it again picks up as the droplet continues its journey towards the centre of coil 2. This sequential operation of coil 1 and coil 2 allows the droplet to move ahead in a straight line with a nearly saw-tooth type velocity profile. Figure 4a shows the droplet actuation starting from edge of coil 1 up to centre of the coil 5. Figure 4b–f shows the magnetic field (Tesla) plot computed on the hydrophobic surface (i.e. at z = 2 mm) for coil 1 to coil 5 whilst droplet actuates by sequentially switching through the mentioned electromagnetic micro-coils. Figure 4g shows variation of driving force (magnetic force, F mag ) due to electromagnetic micro-coil with frictional drag (F drag ) and contact angle hysteresis force (F hyst ) acting against the droplet motion. In the same way, to achieve two-dimensional transport, the droplet is further pulled by the next coils in sequence until the droplet finally reached coil 12. The complete droplet trajectory and detailed switching pulses of different coils used in this operation are shown in Fig. 5. There are three right-angle turns on the way of droplet transport (indicated by A, B and C in Fig. 5a) at the end of which, the droplet speed reduced to zero; this was achieved by switching off the particular coil (underneath the droplet) at an instant little sooner than when the droplet reached the coil centre (this is unlike the other coils which remain on until the droplet reaches its centre), and a time delay of one second is applied prior to switching on next coil (pointed by A, B and C in Fig. 5b). In this way, the droplet can be turned at right angle smoothly without over travelling in previous path. Figure 5b shows the coil pulses (coil 1 to coil 12) used to actuate droplet by sequentially switching through the mentioned electromagnetic micro-coils for the entire droplet trajectory. The time delay in between the consecutive coil switching events during every right turn is pointed by A, B and C. It has been observed that the pulse width for the coils after each right turn (coil 6, coil 8 and coil 12, as shown in figure) is higher compared to other coil pulses. Here, droplet takes longer time to reach next coil (smaller coil at upper tier) primarily due to its higher distance from the predecessor coil (larger coil at lower tier), and moreover, droplet starts from rest.
5 Conclusion Here, we proposed a transport model of a spherical cap ferrofluid droplet manipulation technique on a solid hydrophobic surface. The droplets were actuated by periodically switched array of a double-layer electromagnetic micro-coils. Magnetic force produced by an individual micro-coil implemented from our earlier work was validated experimentally. For the present model, the magnetic force by the micro-coil is further validated with the magnetic field model used in multiple electromagnetic micro-coils similar to the present work to manipulate magnetic particles suspended in
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Fig. 4 a Droplet actuation starting from edge of coil 1 up to centre of the coil 5. b–f Magnetic field (Tesla) plot computed on the hydrophobic surface (i.e. at z = 2 mm) for coil 1 to coil 5 during sequential coil switching to actuate droplet. g Variation of driving force (magnetic force) due to electromagnetic micro-coil (F mag ), frictional drag (F drag ), contact angle hysteresis force (F hyst ), net force (F net ) and droplet speed
Fig. 5 a Complete droplet trajectory starting from edge of coil 1 to its final destination, centre of the coil 12. b Sequential coil switching pulse corresponding to droplet actuation starting from coil 1 to coil 12
laminar flow inside a microfluidic channel. Velocity profile of the ferrofluid droplets as a function of its distance from the coil centre is also validated with experimental results. The two-dimensional manipulation of a microliter volume droplet on a solid substrate in a sequence of rectilinear paths indicates the prospect of multifaceted onchip handling of numerous reaction processes that will enable simple and compact designs of surface microfluidic manipulator.
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References 1. Jebrail MJ, Wheeler AR (2010) Let’s get digital: digitizing chemical biology with microfluidics. Curr Opin Chem Biol 14(5):574 2. Pollack MJ, Pamula VK, Srinivasan V, Eckhardt AE (2011) Applications of electrowettingbased digital microfluidics in clinical diagnostics. Expert Rev Mol Diagn 11(4):407 3. Cho SK, Moon H (2009) Electrowetting on dielectric (EWOD): new tool for bio/micro fluids handling. Bio Chip J 2(2):79 4. Malic L, Brassard D, Veres T, Tabrizian M (2010) Integration and detection of biochemical assays in digital microfluidic LOC devices. Lab Chip 10(4):418 5. Zhang K, Liang Q, Ma S, Mu X, Hu P, Wang Y, Luo G (2009) On-chip manipulation of continuous pico-liter-volume superparamagnetic droplets using a magnetic force. Lab Chip 20(9):2992 6. Surenjav E, Priest C, Hemminghaus S, Seemann R (2009) Manipulation of gel emulsions by variable microchannel geometry. Lab Chip 9(2):325 7. Teste B, Jamond N, Ferraro D, Viovy JL, Malaquin L (2015) Selective handling of droplets in a microfluidic device using magnetic rails. Microfluid Nanofluid 19:141 8. Long Z, Shetty AM, Solomon MJ, Larson RG (2009) Fundamentals of magnet-actuated droplet manipulation on an open hydrophobic surface. Lab Chip 9:1567 9. Assadsangabi B, Ali MSM, Takahata K (2012) Bidirectional actuation of ferrofluid using micro patterned planar coils assisted by bias magnetic fields. Sens Actuators A 173:219 10. Beyzavi A, Nguyen NT (2010) Programmable two-dimensional actuation of ferrofluid droplet using planar microcoils. J Micromech Microeng 20:015018 11. Nguyen NT, Beyzavi A, Ng KM, Huang X (2007) Kinematics and deformation of ferrofluid droplets under magnetic actuation. Microfluid Nanofluid 3:571 12. Chen G, Gao Y, Li M, Ji B, Tong R, Law MK, Wen W, Zhou B (2018) Rapid and flexible actuation of droplets via a low adhesive and deformable magnetically functionalized membrane. J Mater Sci 53:13253 13. Chakrabarty D, Dutta S, Chakraborty N, Ganguly R (2016) Magnetically actuated transport of ferrofluid droplets over micro coil array on a digital microfluidic platform. Sens Actuators B Chem 236:367 14. Griffiths DJ (2004) Introduction to electrodynamics, 4th edn. Prentice Hall India, New Delhi, p 217. 15. Santra A, Chakraborty N, Ganguly R (2009) Analytical evaluation of magnetic field by planar micro-electromagnet spirals for MEMS application. J Micromech Microeng 19:1–10 16. Elsherbini AI, Jacobi AM (2006) Retention forces and contact angles for critical liquid drops on non-horizontal surfaces. J Colloid Interface Sci 299:841 17. Elsherbini AI, Jacobi AM (2004) Liquid drops on vertical and inclined surfaces; II. A method for approximating drop shapes. J Colloid Interface Sci 273(2):566 18. Dupont JB, Legendre D (2010) Numerical simulation of static and sliding drop with contact angle hysteresis. J Comput Phys 229:2453 19. Daniel S, Chaudhury MK (2002) Rectified motion of liquid drops on gradient surfaces induced by vibration. Langmuir 189:3404 20. Elsherbini AI, Jacobi AM (2004) Liquid drops on vertical and inclined surfaces; I. An experimental study of drop geometry. J Colloid Interface Sci 273:556 21. Elsherbini AI (2003) Modelling condensate drops retained on the air-side of heat exchangers. Ph.D. thesis, University of Illinois at Urbana–Champaign, Urbana 22. Elsherbini AI, Jacobi AM (2006) Retention forces and contact angles for critical liquid drops on non-horizontal surfaces. J Colloid Interface Sci 299:841 23. Marmur A, Bittoun E (2009) When Wenzel and Cassie are right: reconciling local and global considerations. Langmuir 25(3):1277 24. Petrov PG, Petrov JG (1992) A combined molecular-hydrodynamic approach to wetting kinetics. Langmuir 8(7):1762
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25. Voinov OV (1979) Hydrodynamics of wetting. Fluid Dyn 11:714 26. Cox RG (1986) The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J Fluid Mech 168:169 27. Yarnold G, Mason B (1949) A theory of the angle of contact. Proc Phys Soc Lond B 62(2):121 28. Blake TD, Haynes JM (1969) Contact angle hysteresis. J Colloid Interface Sci 30:421 29. Sinha A, Ganguly R, De AK, Puri IK (2007) Single magnetic particle dynamics in a micro channel. Phys Fluids 19:117102 30. Fulcrand R, Bancauda A, Escribaa C, Hea Q, Charlota S, Boukabachea A, Guea AM (2011) On chip magnetic actuator for batch-mode dynamic manipulation of magnetic particles in compact lab-on-chip. Sens Actuators B 160:1520 31. Paulssen D, Hardt S, Levkin PA (2019) Droplet sorting and manipulation on patterned twophase slippery lubricant-infused surface. ACS Appl Mater Interfaces 11:16130 32. Rosensweig RE, Elborai S, Lee SH, Zahn M (2005) Ferrofluid meniscus in a horizontal or vertical magnetic field. J Magn Magn Mater 289:192
Numerical Analysis of Heat Transfer and Fluid Flow in Microchannel Heat Sinks Designed for Uniform Cooling Shivayya C. Hiremath, Rohit Kumar, Arman Mohaddin Nadaf, and Manmohan Pandey
Abstract Liquid convective heat transfer for miniature heat sinks has proved to be efficient in heat dissipation from electronic components of shrunk scales. The current numerical study presents a novel fin topography for microchannel heat sink which has been more efficient than conventional straight channels in uniform cooling. Unlike inlet plenums, the design includes flow to the heat sink (lower plate) entirely through jets with an optimized pattern, which has been designed to account for uniform cooling. The temperature distribution over the heating surface (the surface to be cooled) of the novel design has been compared to that of a straight microchannel with rectangular fins of 0.8 mm height with 1 mm wall height. Comparison between novel design and straight channels has been done using standard deviation for the same heating surface area of 645.16 mm2 . A constant heat flux of 50 × 10−2 W/mm2 has been used throughout the numerical analysis. Enhanced uniformity of cooling comes at the expense of increased pressure loss. Steady flow in both laminar and turbulent regimes was analyzed with appropriate models. The results suggested that the novel design would be much more suitable for turbulent flows of liquid-cooled systems than laminar. Keywords Microchannels heat sink · Uniform cooling · Numerical analysis · Fluid flow
1 Introduction The issue of effective heat management is brought on by the rising demand for accelerated computational speed and device compactness. Agwu Nnanna et al. [1] mentioned that the failure rate of electronic components increases by a factor of S. C. Hiremath Department of Aerospace Engineering, R.V. College of Engineering, Bengaluru 560059, India R. Kumar · A. M. Nadaf · M. Pandey (B) Department of Mechanical Engineering, IIT Guwahati, Guwahati 781039, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_29
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two for every 10 K rise in temperature, which implies the need for effective cooling. Alihosseini et al. [2] proposed a state-of-the-art liquid cooling design to accelerate the PCR procedure, emphasizing the cooling requirement in medical applications. Any cooling system used should be able to cool at the same rate and be as compact as the technological equipment it is cooling. Microchannel heat sinks (MCHSs) are the most efficient option for cooling electronic devices installed with high-power integrated circuit packages (microchips), proton exchange membrane fuel cells, laser diode arrays, combustors, evaporator/condenser in the miniature refrigerator, LASERs, solar energy utilization, etc. Conventional microchannel heat sinks have straight channels with geometric modifications of fins to enhance heat transfer; however, there always exists a temperature gradient within the flow path that is not desired in some applications. The novel design aims at minimizing the gradient of the contour, i.e., cooling the heating surface much more uniformly than straight microchannels. Prajapati [3] studied the effect of varying fin height on heat transfer in a straight microchannel. The numerical study has been done with varying Reynolds numbers from 100 to 400 at a constant heat flux of 50 × 10−2 and 100 × 10−2 W/mm2 . Seven different fin heights have been analyzed where 0.8 mm height of the fin yielded the most efficient results of average Nusselt number.
2 Geometry and Methodology The proposed idea was designed using SOLIDWORKS (2020 edition) software and was simulated on ANSYS FLUENT 2021 R2. The novel topography of the channel and fin arrangement was designed for a single-phase flow heat sink. The proposed heat sink configuration has been devised in two plates, placed one above another. Varying fin lengths aided by cylindrical jets were used. The coolant (water) flow in the proposed design happens through 18 inlets to the upper plate and then to the lower plate (heat sink) entirely through jets. The flow merges at the diagonal mainstream of the lower plate. The fin arrangement has been designed to achieve enhanced heat transfer and uniform cooling at constant heat flux conditions. The proposed geometry consists of eight fins laid on the heat sink in one triangular half of the square. The heat exchanger is a square of side 25.4 mm, with fin width and height being 1.5 mm and 1 mm, respectively. Every slot has a similar jet pattern impinging but with different jet diameters. The slots in the lower plate open to diagonal mainstream through which fluid flows to the outlet. The diagonal mainstream would be cooled effectively by successive addition of flow from each slot to its way to the outlet. Coolant is supplied to the upper plate through rectangular inlets and then to the lower plate by cylindrical jets whose diameters range from 1.4 to 0.7 mm. The best-cooling effect producing geometry yet has been presented in Figs. 1, 2, and 3. The jets have been projected to the slots from the upper plate placed upon the heat sink. There are 18 inlets, 9 each on the perpendicular side of the upper plate. A set of 9 inlets can be referred to as
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inlets normal to x and inlet normal to y. Figure 4 explains the inlet configuration. Eight out of 9 inlets in a direction (normal to x or y) are face inlets with slot cross section 1.5 mm in width and 0.5 mm in height, and the inlet at the corner is 1 mm in width and 0.5 mm in height (refer to Fig. 4). The width of the two corner inlets has been designed differently from the other 16 inlets to ensure the diagonal flow in the lower plate by providing sufficient down flow to the heat sink’s (lower plate’s) farthest corner from the outlet. The inlet velocities for 8 × 2 face inlets (Dh = 0.75 mm) and a corner inlet × 2 (Dh = 0.66 mm) have been set to 0.5353 and 0.6023 m/s to achieve Reynolds number of 400 (Laminar). The inlet pattern and cross section are similar in inlets normal to x and inlets normal to y; hence, 2 × number of inlets has been done. Fig. 1 Fluid domain of novel geometry
Fig. 2 Jet pattern on the fluid domain of novel geometry
Fig. 3 Fluid and solid domain of proposed design
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Fig. 4 a Inlets and fluid domain of the upper plate, b flow path of the coolant
In the case of turbulent flow, mass flow rate of 0.003838 kg/s through each of 8 × 2 face inlets and 0.002558 kg/s through each of 1 × 2 corner inlets have been used to achieve a 4 Lpm net flow rate. The Reynolds number for turbulent flow at the inlets hence has been 3829.403 (at each face inlet) and 3368.307 (at each corner inlet). It is to be noted that the Reynolds numbers have been calculated at the inlets (using the hydraulic diameters of rectangular inlets) in every case and not at any interior section of the heat sink. Element size has been iterated to reduce the residuals. Triangular mesh with an element size of 0.5 mm was employed with inflation up to 5 layers with a growth rate of 1.2. The total number of elements in the most optimum result providing mesh was 7,453,883 with 1,936,247 nodes. The fluid body has been sized with an element size of 0.09 mm. Laminar and k-omega SST models with SIMPLE solver have been used for the study. The temperature uniformity has been compared with that of a novel design in both laminar and turbulent conditions for a constant heat flux of 50 × 10−2 W/mm2 . For the comparison, a straight microchannel from the literature Prajapati [3] has been numerically analyzed for the heating surface temperature at the same boundary conditions. The heating surface area of 15 mm × 3.7 mm from literature has been increased to 25.4 mm × 25.4 mm (since the area of heating surface of novel design is 25.4 mm×25.4 mm). Water has been used as a coolant in all the simulated cases. The density and viscosity are defined as polynomial (quadratic) functions of temperature (from Prajapati [3]). ρ = 758.1214 + 1.8781T − 3.604 × 10−3 T 2
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μ = 0.02143 − 1.195 × 10−4 T + 1.699 × 10−9 T 2 The specific heat (C p ) and thermal conductivity have been set to 4182 J/(kg K) and 0.6 W/(m K), respectively.
3 Validation To validate the models used for laminar and turbulent flows, experimental studies of Qu and Mudawar [4], and Dalkılıç et al. [5] were reproduced using ANSYS FLUENT with the laminar and turbulent models similar to those used for novel fin geometry. The validation for laminar flow has been done by using the same boundary conditions for the unit cell of the microchannel heat sink. The pressure drop between the inlet and outlet for a constant heat flux of 100 × 10−2 W/mm2 was validated. Figure 5 represents the comparison between variation (pressure difference between inlet and outlet vs. Reynolds number) in the experimental and numerical study of Qu and Mudawar [4]. It can be inferred from graphs that the variation comes to close agreement with numerical and experimental values of the literature at lower Reynolds numbers but has been deviating from experimental values for higher Reynolds numbers. The model however is useful since the current study employs the Reynolds number of 400 for laminar flow regime study. The difference in experimental and numerical values can be accounted for the pressure loss due to plenums, sudden contraction, sudden expansion, and errors in the experimental procedure. The Nusselt number variation with the Reynolds number in the reproduced study came to close agreement with experimental data. Figure 6 depicts the variation of the Nusselt number with the Reynolds number in experimental and numerical analysis. The Reynolds number was varied from 3500 to 6000 with steps of 500 for an inlet temperature of 300 K. The models though deviate from experimental data at higher values of the Reynolds number are suitable for the current study as no huge deviation is found at the Reynolds number around 3000. The k-epsilon model was eliminated Fig. 5 Comparison of numerical results with experimental data [4] for laminar regime
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Fig. 6 Comparison of numerical results with experimental data [5] for turbulent regime
as there is extensive flow separation in the flow path. The Nusselt number variation with increasing velocity was almost linear.
4 Results and Discussion 4.1 Turbulent Flow The average temperature of the heating surface has been recorded to be 301.147 K with a standard deviation of 0.7335 K. The maximum temperature at the heating surface is 302.376 K, which is lower when compared to the geometry derived from the literature. Consequently, the pressure drop is higher, 5.0765 MPa. Uniform temperature contours are presented in Fig. 7. Note that the local temperature contour of the heating surface varies for only about 5 K. Figure 8 contains the velocity contour of fluid at the plane 0.1 mm below the jet openings to the lower plate.
4.2 Laminar Flow The average temperature of the heating surface was 324.345 K with a standard deviation of 2.9594 K. The maximum temperature at the heating surface is 328.476 K. The corresponding pressure drop has been very high, 0.6365 MPa. The contours of the temperature of the heating surface and velocity are presented in Figs. 9 and 10. Table 1 gives a comparison of the uniformity of cooling in both cases. Note that the heating surface is a square of side 25.4 mm and position. The graphs represent the temperature variation across the inlet and outlet for straight channels with rectangular fins (the heating surface area of the heat sink from Prajapati increased to square of
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Fig. 7 Global and local temperature contours of heating surface in turbulent flow
Fig. 8 Velocity contour at the lower plate
side 25.4 mm) and triangular fins (rows of equilateral triangular fins of 500 µm and 0.8 mm height, each row placed 0.5 mm apart). The temperature data in the plot for straight slots are the temperature readings along the y-coordinate of the line from the midpoint of the inlet to the midpoint of the outlet (top view of the square where the bottom and top sides of the square are inlet and outlet, respectively), and position 0 is the center of the square in all the cases. The curve data of the novel design correspond to the temperature along the y-coordinate of the line connecting the bottom-left corner to the top-right corner in the turbulent approach (refer to Fig. 7 to
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Fig. 9 Global and local temperature contours of heating surface in laminar flow
Fig. 10 Velocity contour at the lower plate
observe the temperature extremes) and from the bottom-right to the top-left corner in the laminar approach (refer to Fig. 9). The diagonal was considered to include extremes of the temperature hot spots and still prove that the variation including extremes of temperature is less than that of the straight channel approach.
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Table 1 Comparison Nature of flow at the inlets
Design
Average temperature of heated surface (in K)
Standard deviation (in K)
Pressure drop
Laminar (at Re = 400)
Rectangular fins of 0.8 mm ht
338.40
5.77
0.3877 kPa
Diagonally merging novel geometry
324.35
2.96
0.6365 MPa
Rectangular fins of 0.8 mm ht
303.98
1.38
0.0189 MPa
Diagonally merging novel geometry
301.15
0.73
5.0765 MPa
Turbulent (Re = 3829.403 at face inlets and 3368.307 at corner inlets)
5 Conclusions The topography was found to be appropriate for cooling high heat flux-producing surfaces that need to be maintained at a uniform temperature. It has been efficient in uniform cooling in comparison with straighter channels. Uniformity can be quantitatively measured by the standard deviation of the temperature data on the heating surface. The standard deviation from the mean temperature in the turbulent inlet flow is 0.73 K, which is comparatively cooler and less deviated than the straight channels’ 1.38 K. The data suggest a 47% lesser deviation. The laminar flow however does not promise much enhancement as the standard deviation of novel geometry and that of the straighter channels’ has been recorded to be 2.96 and 5.77 K. The cooling enhancement is not efficient in laminar flow, though a 48.7% lesser deviation of temperature is found. It is because of the very huge difference in pressure drop which demands greater pumping power which might render the novel approach uneconomical if practically realized. The pressure difference of the novel design increases by a factor of 1000 in comparison with the pressure drop at the same flow conditions in straighter channels. In turbulent flow, the factor is about 100. Additionally, lower inner wall temperature is assured in the proposed design. The continuity of the fluid in novel geometry has been hard to track because of the complex flow path (Figs. 11 and 12).
6 Future Scope The design is adaptable to geometric changes, while the basic fin layout is maintained similarly. In the proposed design, the inlet to the higher plate is through two inlet plenums. However, a single nozzle directed into the upper plate without any fins can be substituted for plenums, which can make the heat sink much more compact. Fin
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Fig. 11 Temperature variation on the heating surface for Re = 400 (laminar) at every inlet
Fig. 12 Temperature variation on the heating surface for Re > 3000 (turbulent) at every inlet
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geometry and channel size can be optimized for enhanced heat transfer. Fin width can be reduced for a significantly lower maximum base heating surface temperature. Inlet cylindrical jets can be modeled as nozzles which increases the impact velocity. Jet impingement angle can be changed to experiment with its effect. According to Pandey et al. [6], the effect of jet impingement angles has been analyzed by varying the angles between 30 and 90°, reported the optimum performance at 60°, i.e., the pressure drop for the same cooling effect of the proposed design can be reduced by analyzing different inlet angles and shapes. Acknowledgements This work was done during the internship of the first author at IIT Guwahati, which was supported by the Indian Academy of Sciences (IAS) under the Summer Research Fellowship Program (SRFP).
References 1. Agwu Nnanna AG, Rutherford W, Elomar W, Sankowski B (2007) Assessment of thermoelectric module with nanofluid heat exchanger. In: ASME international mechanical engineering congress and exposition, Jan 2007, vol 43025, pp 663–672 2. Alihosseini Y, Azaddel MR, Moslemi S, Mohammadi M, Pormohammad A, Targhi MZ, Heyhat MM (2021) Effect of liquid cooling on PCR performance with the parametric study of crosssection shapes of microchannels. Sci Rep 11(1):1–12 3. Prajapati YK (2019) Influence of fin height on heat transfer and fluid flow characteristics of rectangular microchannel heat sink. Int J Heat Mass Transf 137:1041–1052 4. Qu W, Mudawar I (2002) Experimental and numerical study of pressure drop and heat transfer in a single-phase micro-channel heat sink. Int J Heat Mass Transf 45(12):2549–2565 5. Dalkılıç AS, Mahian O, Yılmaz S, Sakamatapan K, Wongwises S (2017) Experimental investigation of single-phase turbulent flow of R-134a in a multiport microchannel heat sink. Int Commun Heat Mass Transf 89:47–56 6. Pandey J, Ansari M, Husain A (2022) Effect of nozzle inclination angle on the performance of hybrid jet impingement microchannel heat sink. In: Recent advances in manufacturing, automation, design and energy technologies. Springer, Singapore, pp 887–896
Numerical Investigation on Hydrodynamics of Lubricant-Infused Hydrophobic Microchannel with Transversely Oriented Cavities Adarsh R. Nair, K. Nandakumar Chandran, and S. Kumar Ranjith
Abstract Higher values of pressure drop due to the resistance offered in the microchannel can be dealt with the introduction of fluid within the microgroove surfaces of a microchannel. The non-uniformity in the groove shapes observed calls for an extensive study to determine which geometry contributes to higher drag reduction because of an increase in the volume of entrapped fluid. It is also to be noted that the inherent viscosity of the infused fluid can also govern the effective slip length value. Hence, the present study contributes to the enhancement of drag reduction of a lubricant-infused hydrophobic microchannel for isothermal conditions. In Stokes’s regime, the effect of varying the geometrical parameter of a lubricant-infused cavity showed a significant reduction in the flow resistance inside the microchannel. Present study recorded that the increase in cavity area and high aspect ratio reduces hydrodynamic resistance. Further, low viscosity ratio ribbed channels are suitable for internal fluid applications. Keywords Hydrophobic microchannel · Ribs and cavity · Transverse flow · Multi-phase flow · Volume of fluid · SLIPS · Stokes regime
1 Introduction In several microfluidic applications such as microreactors and electronic microcontrollers, micro-heat exchangers face challenges of having a large pressure drop due to the hydrodynamic resistance offered by the microchannel [15]. Hence, there is a major impetus to minimize the necessary pumping power in microfluidic applications, and as such, the study on dynamics of flow through microchannels has a vital role. Recent studies on the wetting behaviour of liquid droplets on superhydrophobic surfaces show a significant impact on the pressure drop [9] and add to the A. R. Nair (B) · K. Nandakumar Chandran · S. Kumar Ranjith Micro/Nanofluidics Research Laboratory, Department of Mechanical Engineering, College of Engineering Trivandrum (Government of Kerala), Thiruvananthapuram, Kerala 695016, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_30
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relevance of drag reduction studies in developing such microchannels. One amongst the various approaches proposed for reducing the frictional resistance to fluid flow is orienting transverse microribs and cavities on microchannel walls. As fluid is prevented from direct contact with the solid surface, the liquid interface is generated and it subsequently reduces the frictional resistance of the flow [18]. In the cavity region, a gas–liquid contact ensures a Cassie state instead of a liquid–solid contact which ensures a reduction in frictional resistance [7]. The hydrodynamics of flow through a hydrophobic microchannel with transverse ribs and cavities is studied in detail using experimental and numerical methods [14, 20]. Furthermore, numerous theoretical and experimental investigations indicated that the frictional resistance is largely influenced by the interface shape [6, 12], demonstrating predominant dependence on the apparent contact angle of the fluid interface and the initial volume of the cavity [10]. The lack of stability over a longer period is one of the major concerns with using gas–liquid interfaces. Recent studies provide better stability by replacing gas with denser fluids inside the cavity [13, 19]. The total reduction in frictional resistance with a lubricant-infused microchannel is not high as that of superhydrophobic surfaces, but compared to a normal microchannel, the effects are still significant [21]. The interface formed between two fluids is assumed slippery along the flow direction and non-deformable in the vertical direction, which is comparable to the ideal case of infinite surface tension [1]. It has been demonstrated that infusing these slippery interfaces reduces skin friction drag in microchannel flows to a large extend [8]. The fluid inside the cavity is restricted by the walls, and the resulting flow depends on the geometrical parameters of the cavity. Any factor that varies the fluid movement inside the cavity has an impact on frictional resistance [17]. The role of the fluid interface is significant as its effect on the effective slip is predominant when comparing with other geometrical parameters [4] but is limited by stability concerns. The present work deals with the effects of different geometrical parameters affecting the flow resistance and the effect of viscosity. The phenomenon is numerically studied, and effects of the parametric studies are discussed in detail. The article is structured as follows, with a discussion of the numerical methodology used in Sect 2. Section 3 provides the detailed explanations of results obtained on the effects of variation in viscosity and groove geometry on effective slip length values before the conclusion in Sect 4.
2 Methodology In this section, the details of numerical modelling of the multi-phase problem for the study of lubricant-infused microgrooved channels are covered. For defining the problem, the authors have modelled a two-dimensional domain of microchannel with non-uniform structured mesh, using finer grids near the wall surface and fluid interface region. The schematics of domain is illustrated with the help of Fig. 1, which explains the dimensional and physical variables associated with the study.
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Fig. 1 Schematic of computational domain a single-phase domain with combination of no-slip and shear-free surface, b multi-phase domain with ribs and cavities of microgrooves
The computational domain for validation purpose is modelled as the combination of a no-slip surface and a shear-free surface arranged alternatively with a periodicity L = lr + lg , and the flow domain is symmetric with respect to the half height h as depicted in Fig. 1a. Subsequently to model the lubricant-infused multi-groove condition, a multi-phase domain is modelled with ribs and cavities as depicted in Fig. 1b, that are arranged transverse to the flow direction. The aspect ratio of the cavity is represented using z c , where z c is the ratio of lg and height of cavity, b. The length of the shear-free surface can be represented using Fs , where Fs = lg /L. For simplification of the problem, the flow physics associated is assumed to be two-dimensional, incompressible, unsteady, Newtonian, laminar, isothermal and two phase. The nature of flow is generally related to Reynolds number and is given by the relation, Re = u ∞ Dh /ν. The hydrodynamic resistance of hydrophobic pipe wall is modelled using Navier slip model and is given by the relation, λ = u wall /(δu/δy)wall . For further explanations, concerned to numerical methodology employed, the governing equation for field variables and multi-components are explained along with the details of validation [2, 12].
2.1 Governing Equation and Multi-phase Modelling The finite volume method (FVM) and the VOF model [11] are used to solve this twophase problem, and to model the hydrodynamics of microchannel flow, a non-uniform structured mesh is used to discretize the computational domain. The continuum-based mass and momentum balance equations provide the governing equations that explain
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such physical phenomena, which are given by: ∇ · v→ = 0
(1)
1 1 1 ∂ v→ + ∇ · v→v→ = − ∇ P + ∇ · τ→ + g→ + F→ ∂t ρ ρ ρ
(2)
The continuous surface model (CSF), which could be established if the value of → surface tension remains same, has been used to include the surface tension term ( F) [3]. The Laplacian pressure jump between the fluids is depicted as surface tension acting along the interface, and also surface force is converted to volume force with divergence theorem applied as follows: F→ = σ κ∇α = σ (− ∇ · n)∇α
(3)
Here, α is the indicator function used to specify which portion of the cell is occupied by the corresponding fluid and is obtained from, {
αl dv ( ) v→∂ x 3 αl + αg dv ) ∂α ( → + V ·∇ α =0 ∂t
α={
v→x 3
(4) (5)
Note that, α = 1 corresponds to phase 1, α = 0 represents phase 2 and 0 < α < 1 for fluid interface. For an incompressible fluid, the conservation of mass is equivalent to the conservation of volume and there by conservation of the function α. The local values of fluid properties (ρ and μ) are weighted average of the fluid properties of both fluids and are given by ρ = ρ1 α + ρ2 (1 − α) and μ = μ1 α + μ2 (1 − α). In order to account for interface compression, a further artificial compression term is added to the phase fraction function. For additional details on VOF method implementation with regard to the contact line interface employed in the present work, refer to previous work [2, 12]. On details of both initial and boundary conditions, initially, the indicator function is assigned, α = 0 for phase 2 and α = 1 for phase 1 for the computational domain given in Fig. 1b. The walls of both ribs and cavities of the microcavities are given no-slip condition and top wall to be having symmetric condition. Also, at inlet and outlet boundary conditions, a velocity boundary condition of 0.025 m/s and pressure outlet condition Pgauge = 0 are defined. For the present numerical simulation, interFoam solver, available with OpenFOAM, a CFD toolbox based on finite volumes is employed. The pressure–velocity coupling in the solver is addressed by the pressure implicit with splitting of operators (PISO) technique. Using the Crank–Nicholson approach, transient terms included in the governing equations are discretized. To achieve better precision, a maximum Courant number of 0.4 is employed for the time step control. To discretize gradient terms, a second-order Gaussian integration with linear interpolation is used. In order
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to enhance the accuracy of the solution, the Gauss-linear corrected scheme is used to discretize the Laplacian terms, and the Gauss limited linear scheme is employed to discretize the convection term.
2.2 Validation To validate the numerical procedure employed to study the hydrodynamics of flow characteristics through microchannel, the simulation is compared against previously established analytical results [16]. The relation between effective slip length λ and length of shear-free region to the total length, Fs , is given as follows: ) ( 1 1 λ = ln L π cos(Fs π/2)
(6)
For validation purpose, both single-phase domain Fig. 1a and two-phase domain Fig. 1b were considered, respectively, as mentioned in Sect. 2. The fluid properties of phase 1 and phase 2 chosen for the present numerical study are as follows: density, ρ1 = 998 kg/m3 , ρ2 = 970 kg/m3 , viscosity, μ1 = 1 × 10−3 Pa s, μ2 = 1.45 × 10−3 Pa s and surface tension, σ = 0.07 N/m at a standard temperature of 27 °C. The value of slip length (λ) is determined in the numerical simulations using the given co-relation [2]. Numerical model is simulated under flow condition of Re = 1, L/Dh = 1 and z c = 0.5 for different values of Fs , where hydraulic diameter, Dh = 4h, where h is half height. h λ= 3
[(
ΔPno-slip ΔPslip
[
) −1
(7)
Q
Corresponding values of slip length for single-phase model and two-phase lubricant-infused model are compared with result from analytical model of Lauga and Stone [16] and is illustrated in Fig. 2. It is to be noted that the present study also shows the same nature as theoretical study, i.e. the flow rate will increase as the slip length increases for a specific pressure difference. The FVM approach combined with the VOF and CSF modelling used in this work is capable of predicting the hydrodynamics of the lubricant-infused microchannel, according to these quantitative comparisons carried out. In addition, the grid independent analysis is performed by monitoring the slip length to periodicity (λ/L) and noticed that the chosen grid size of having 48,000 cells is sufficient to produce accurate results. Also, the time step selected is Δt = 10−8 s, and the relative tolerance of the solver for each iteration is less than 10−6 .
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Fig. 2 Variation of slip length with respect to shear-free fraction. Analytical solution is plotted using Eq. (6) [16]
3 Results and Discussion In this section, detailed discussions on results obtained from the numerical study are illustrated, and the physics involved in the hydrodynamics is explained for the parametric studies performed. The study focuses on the effect of varying shapes and aspect ratio (3.1) of a chosen geometry for same fluid properties and flow conditions in a lubricant-infused microchannel aiming for drag reduction. Following, it studies for the effect of varying viscosity of phase 2 (3.2) under described geometric and flow conditions to understand the flow characteristics generated by the presence of microgrooves.
3.1 Effect of Geometry Hydrophobic nature of lubricant-infused surface very much depends on volume of fluid occupied inside the cavity leading to an effect on the frictional resistance offered [5]. As the volume of fluid inside the microgroove is a quantity governed by the geometrical parameters of the rib and the cavity, in general, we could say as more amount of fluid present inside the cavity, the less will be the frictional resistance offered. But effect of shapes does contribute for the hydrodynamics inside the cavity and from previous researches it is observed, as width of the cavity or the fluid interface increases, there is decrease in frictional resistance which predominates. So as to optimize the configuration of the cavity geometry, in this work, the aspect ratio and the shape are further studied considering a low viscosity lubricant-infused cavity. Looking into the simulation results for varying height of the cavity at Re = 1 flow condition, the volume fraction and stream lines provide insights into the flow characteristics within the two phases (refer Fig. 3a). It is observed that the increase in height of the cavity (aspect ratio) affects the reverse movement of phase 2 fluid in the cavity due to circulation. As a result of this, the effective slip velocity at the fluid interface increases as shown in Fig. 4a. From Fig. 4b, it can be deduced that for
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low aspect ratio say z c = 0.25, magnitude of reverse flow circulation is more and it reduces as aspect ratio increases. As a result, the velocity of fluid circulation inside the cavity increases which aids the flow of phase 1 resulting in the easiness of flow. As the aspect ratio increases, there is an increase in the effective slip length, and as the value of the aspect ratio is greater than z c = 0.75, the effects get diminished. As stated above, the effect of aspect ratio of the cavity by changing the height of cavity is limited for a microchannel, so as to further increase the volume of fluid inside the cavity is to increase the cavity area without varying height or the length of fluid interface. For understanding the effect of varying cross-sectional area, in this work, the geometry of the cavity is varied by incrementing the length of the bottom wall of the cavity. The different geometries are represented as CS-1: triangle, CS-2: inverted trapezium, CS-3: rectangle and CS-4: trapezium as in the order of increasing length of cavity bottom wall. Referring to Fig. 3b, it can be understood from the volume fraction and stream lines that as the area of the cavity increases, the fluid circulation shows further improvement. Changing the cavity shape has same significance as that of changing the aspect ratio, since there is considerable effect on frictional resistance. The volume of fluid circulation inside the cavity CS-1 is restricted by the reduced area, and from Fig. 5a, it can be observed that the flow circulation is not evident nearer to bottom of cavity wall. The fluid circulation inside the cavity CS-2, CS-3 and CS-4 is increased, Fig. 3 Qualitative analysis of hydrodynamics showing a volume fraction and flow streamlines for varying z c values; b volume fraction and flow stream lines for varying cavity shapes (CS)
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Fig. 4 Effect of varying z c values on a non-dimensional x-velocity profile along y-direction of microgroove; b non-dimensional x-velocity profile along fluid interface
which further increases the slip velocity at the fluid interface. Figure 5b illustrates how the slip velocity varies along the fluid interface as the area of the cavity increases. For a fixed fluid interface and height, an increase in cavity area can affect the slip velocity along the fluid interface which decreases the frictional resistance. The results incline to show that it is the level-off of the above discussed three parameters lead to an optimized geometry for minimum frictional resistance in case of liquid-infused microchannels.
3.2 Effect of Viscosity In the context of flow through a microchannel with lubricant-infused microcavity, the flow properties are directly related to the viscosity of the fluid inside the cavity as phase 1 slides over phase 2 maintaining a straight interface. As to study the effects of change in viscosity, for a constant value of surface tensions of both phases, viscosity ratio N , defined as the ratio of viscosity of fluid inside the cavity (phase 2) to fluid flowing inside the microchannel (phase 1), is varied. As the viscosity ratio increases, the effect of geometrical parameters on frictional resistance is lowered. The effect on fluid interface length, height of the cavity and the volume of fluid inside the cavity is studied for different viscosity ratio. From Fig. 6a,
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Fig. 5 Effect of varying cavity shapes on a non-dimensional x-velocity profile along y-direction of microgroove; b non-dimensional x-velocity profile along fluid interface
it is evident that for low values of Fs , increasing viscosity ratio has less effect on the slip length. As for higher values of viscosity ratio, the flow circulation is restricted and the increase in slip length due to Fs is low as compared to low viscosity ratio. For higher values of viscosity ratio, the effects on frictional resistance are insignificant which limits to increase the fluid interface region. This behaviour could be justified by the understanding that higher viscous fluid as phase 2 is imparting the resistance to free flow of lower viscous fluid as phase 1, which in turn reduces the effective slip length and vice versa. From Fig. 6b, increasing the aspect ratio, there is an initial increase in the value of slip length, and further, the variation in slip length is not evident. For higher viscosity ratio, the effect of increasing aspect ratio is reduced. So, for high viscosity ratio, using low aspect ratio for cavity is preferred. Effect of increase in the volume of fluid inside the chosen cavity shapes for different viscosity ratios is illustrated in Fig. 6c. As we increase the area of the cavity, there is a slight increase in the value of slip length, and the changes are more noticeable for low viscous fluids. For high viscosity ratio, effect of fluid circulation inside the cavity is restricted, so dominance of geometrical parameter in enhancing slip inside microchannel is less significant for cavity filled with high viscosity fluid.
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Fig. 6 Slip length variations for varying viscosity ratio a with respect to varying Fs values, b with respect to varying z c values, c with respect to varying cavity shapes (CS)
4 Conclusion The present study provides quantitative and qualitative remarks on the hydrodynamic analysis of flow through a lubricant-infused microchannel confined to flow condition of Re = 1 with varying cavity geometrical parameters and for different viscosity ratios. It is to be concluded from the studies that considerable variations in the frictional resistance prevail for varying the volume of aforementioned phase 2 inside the cavity in case of a microchannel flow. The rib and cavity representing the microscaled roughness over a microchannel surface with lubricant-infused conditions can act as
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a lubricating medium for the flow reducing pumping power consumption. Furthermore, the results illustrated that with a fixed fluid interface, the aspect ratio of the cavity has a significant effect; an increase in aspect ratio with respect to changing the height of the cavity can reduce the frictional resistance. Similarly, an increase in the cavity area with respect to an increase in the length of the bottom cavity wall affects the fluid circulation to increase the slip velocity. The concluding statement for the present study can be stated as a microchannel with high viscous fluid infused inside the cavity can opt for a low aspect ratio for a fixed fluid interface owing to minimal frictional resistance.
Nomenclature Re υ μ Dh u∞ λ L lg lr b h zc N v→ t P ρ τ→ g F→ σ κ α
Reynolds number (–) Kinematic viscosity (m2 /s) Dynamic viscosity (Ns/m2 ) Hydraulic diameter (m) Average flow velocity (m/s) Slip length (m) Periodicity (m) Width of cavity (m) Width of rib (m) Height of cavity (m) Half height of channel (m) Aspect ratio (–) Viscosity ratio (–) Velocity vector (m/s) Time (s) Pressure (N/m2 ) Density (kg/m3 ) Viscosity stress tensor (N/m2 ) Acceleration due to gravity (m/s2 ) Surface tension force (N) Surface tension (N/m) Curvature (–) Volume fraction (–)
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References 1. Arenas I, García E, Fu MK, Orlandi P, Hultmark M, Leonardi S (2019) Comparison between super-hydrophobic, liquid infused and rough surfaces: a direct numerical simulation study. J Fluid Mech 869:500–525 2. Arun MG, Dilip D, Kumar Ranjith S (2021) Effect of interface curvature on isothermal heat transfer in a hydrophobic microchannel with transverse ribs and cavities. Int J Therm Sci 167:107014 3. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. J Comput Phys 100(2):335–354 4. Davies J, Maynes D, Webb BW, Woolford B (2006) Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs. Phys Fluids 18(8):087110 5. Dilip D, Jha NK, Govardhan RN, Bobji MS (2014) Controlling air solubility to maintain “Cassie” state for sustained drag reduction. Colloids Surf A Physicochem Eng Asp 459:217– 224 6. Dilip D, Vijay Kumar S, Bobji MS, Govardhan RN (2018) Sustained drag reduction and thermohydraulic performance enhancement in textured hydrophobic microchannels. Int J Heat Mass Transf 119:551–563 7. Dong Z, Schumann MF, Hokkanen MJ, Chang B, Welle A, Zhou Q, Ras RHA, Xu Z, Wegener M, Levkin PA (2018) Superoleophobic slippery lubricant-infused surfaces 8. Fu MK, Arenas I, Leonardi S, Hultmark M (2017) Liquid-infused surfaces as a passive method of turbulent drag reduction. J Fluid Mech 824:688–700 9. Gogte S, Vorobieff P, Truesdell R, Mammoli A, van Swol F, Shah P, Brinker CJ (2005) Effective slip on textured superhydrophobic surfaces. Phys Fluids 17(5):051701 10. Hemeda AA, Tafreshi HV (2015) Instantaneous slip length in superhydrophobic microchannels having grooves with curved or dissimilar walls. Phys Fluids 27(10):102101 11. Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39(1):201–225 12. Joseph MP, Mathew G, Krishnaraj GG, Dilip D, Ranjith SK (2020) Numerical simulation of liquid–gas interface formation in long superhydrophobic microchannels with transverse ribs and grooves. Exp Comput Multiph Flow 2:162–173 13. Kim SJ, Kim HN, Lee SJ, Sung HJ (2020) A lubricant-infused slip surface for drag reduction. Phys Fluids 32(9):091901 14. Kim TJ, Hidrovo C (2012) Pressure and partial wetting effects on superhydrophobic friction reduction in microchannel flow. Phys Fluids 24(11):112003 15. Lauga E, Brenner MP, Stone HA (2005) Microfluidics: the no-slip boundary condition. arXiv preprint cond-mat/0501557 16. Lauga E, Stone HA (2003) Effective slip in pressure-driven Stokes flow. J Fluid Mech 489:55– 77 17. Maynes D, Jeffs K, Woolford B, Webb BW (2007) Laminar flow in a microchannel with hydrophobic surface patterned microribs oriented parallel to the flow direction. Phys Fluids 19(9):093603 18. Ou J, Perot B, Rothstein JP (2004) Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys Fluids 16(12):4635–4643 19. Ren L, Hu H, Bao L, Priezjev NV, Wen J, Xie L (2022) Two local slip modes at the liquid–liquid interface over liquid-infused surfaces. Phys Fluids 34(8):082017 20. Rothstein JP (2010) Slip on superhydrophobic surfaces. Annu Rev Fluid Mech 42:89–109 21. Schönecker C, Baier T, Hardt S (2014) Influence of the enclosed fluid on the flow over a microstructured surface in the Cassie state. J Fluid Mech 740:168–195
Effect of Microstructures in the Flow Passage on the Flow Dynamics of Microchannel A. Rajalingam and Shubhankar Chakraborty
Abstract The microchannels serve as a heat dissipation device in the miniaturized— high performance electronic components. The incorporation of microstructures in the flow passage of microchannel helps to increase the rate of heat dissipation due to the flow interruption, acceleration, increase of heat transfer surface area, boundary layer deformation, etc. However, the flow dynamics is the integrated part of the heat transfer. This manuscript deals with the effect of various arrangements and cross-sections of microstructure in the flow field. The pressure drop increases as the microstructure splits and the space between the split microstructure increases. The cross-section and velocity of fluid flow influence the skin friction coefficient on the surface of the microstructure. The trend of variation of pressure drop and drag coefficient is almost the same. Keywords Microchannel · Microstructure · Flow dynamics
1 Introduction The conceptualization of microchannel heat sink (MCHS) originated by Tuckerman and Pease [12]. It received more attention as a heat dissipation device in the miniaturized electronic chips and components due to its unique benefits such as ability to dissipate high heat flux, low weight, compactness, etc. It can be directly attached with the electronic chip or components to remove the heat. The non-uniform dissipation of heat and larger pressure drop is the significant concerns of the MCHS. Therefore, the researchers accelerated their research to sort out this concern and improve the thermohydraulic performance for competing with the present and future heat dissipating demand. Some mechanisms help to improve the thermal performance of MCHS such as generating the acceleration of fluid flow near the channel surface, deformation of boundary layers and interruption to the fluid flow [11]. All these mechanisms are A. Rajalingam (B) · S. Chakraborty Department of Mechanical Engineering, Indian Institute of Information Technology Design and Manufacturing Kancheepuram, Chennai 600127, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_31
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attainable by the incorporation of microstructure in the flow passage of the channel and it shows that the flow dynamics is an unavoidable parameter in heat transfer phenomenon. The formation of vortices influences the heat dissipation and pressure loss of the microchannel. The better enhancement in the heat dissipation is possible by the longitudinal vortices than the transverse vortices for the same pressure drop [4]. The longitudinal vortex generators enhance the hydrothermal performance of microchannel. The size of vortices enlarges as the Reynolds number (Re) and inclination angle of the vortex generator increase. The second pair of vortex generators (when two pairs of vortex generators are located in-line along the flow direction) deforms the vortices and mixes the flow which helps to improve the thermohydraulic performances [3]. The microstructure surface influences the flow field of the microchannel. The vortices can be formed by the grooved structure of the microchannel [8]. Both the cavities and fins help to enhance the hydrothermal behavior of heat sink. The cavities disturb the fluid flow and create the chaotic advection [7]. The finned microchannel ensures shorter distances to the development of boundary layer, steep velocity gradient near the heat transfer surface, flow mixing, etc. [5]. The rib significantly creates an impact on the mainstream fluid flow when it is placed in the transverse microchannel by flow separation, boundary layer deformation, recirculation of the vortices, etc. [2]. The dimensions (height, radius, and location) of pin fin have a role on the performance of the microchannel. Both fluid velocity and pressure loss in the channel increase as the channel height and diameter increase due to the decrease in the area of flow passage normal to the direction of the flow [6]. The slots in the oval pin fin (located within the trapezoidal cavity) can disturb the flow dynamics and improve the thermohydraulic performance. The fluid is also accelerated through the slots which disturbs the fluid in the wake region [1]. The cross-sectional shape of micro-pillars is influenceable on the thermohydraulic performance of the heat sink. From the existing study, it is noticed that the presence of microstructure inside the channel increases the pressure drop. However, the gradual cross-sectional variation of upstream half helps to decrease the pressure drop [10]. The incorporation of microstructure inside the microchannel helps to enhance the rate of heat dissipation significantly with the pressure loss penalty. Though there are several analyses investigating the flow dynamics and heat transfer mechanisms, there is a possibility to achieve the optimized hydrothermal performance (high heat transfer enhancement for low pressure loss). However, the heat transfer phenomenon is coupled with the flow dynamics. The existing studies evidence it. The optimization of the thermohydraulic performance is not conceivable without a detailed investigation of the flow dynamics. The present study is dedicated to reveal the flow dynamics as the flow passes across the configurations of types of microstructures.
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2 Methodology Figure 1 shows the actual three-dimensional microchannel. However, a 2D plane is selected to study the flow dynamics of the microchannel as depicted in the figure. The length and width of the channel are 3 cm and 1 mm respectively. Figure 2 describes the various microchannels with microstructures (MCMS) which are considered for the current computational analysis. The microstructure is incorporated at the center of the channel. In MCMS 1, the rectangular microstructure is placed. The study also attempts to reveal the variation in the flow dynamics due to the sudden and gradual variation in the cross-section of microstructure. Therefore, the semi-circular portion (width of the rectangular microstructure is equal to the diameter of the semi-circular portion) is incorporated in both upstream and downstream ends of the rectangular microstructure (converted as oval cross-section) in MCMS 2. In MCMS 3 and 4, the oval cross-sectional microstructure is split into two and located at the same axis (along the channel length) with distance. In MCMS 5, the two portions of microstructure (as in MCMS 3) are located with an offset distance along the channel width. For MCMS 6, the second microstructure is rotated 180° of MCMS 5. The oval microstructure is titled for 15° from the centerline of the channel in MCMS 7. In MCMS 8, the oval microstructure is split into two portions with an offset distance along the channel length and width and located with 15° deviation angle from the channel center line along the fluid flow. A further attempt needs to explore the influence of various tilting angles of microstructure on the flow dynamics. The water is considered as flowing through the channel with constant physical properties. The solutions for the governing Eqs. (1) and (2) are attained with the help of Ansys fluent software package.
Fig. 1 Schematic of the microchannel
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Fig. 2 Description of the considered microchannel configurations
Continuity equation, ∇·V =0
(1)
Momentum equation, ρf
DV = − ∇ P + μf ∇ 2 V Dt
(2)
The following are the boundary conditions which are incorporated in the present study as depicted in Fig. 3. • The velocity is taken as uniform at the inlet of the microchannel. • There is no penetration and no slip at the interface (outer surface of fluid domain) of solid and fluid. • The pressure outlet condition is considered at the outlet of the channel. Fig. 3 Computational domain and boundary conditions
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Fig. 4 Selection of grid
2.1 Grid Independence Study The selection of a suitable grid optimizes the computational time without any sacrifice in physics. For the present investigation, MCMS 1 is considered for the grid independence study with two parameters such as pressure drop (ΔP) and average skin friction coefficient (C f ) for inlet Re 1000. Figure 4 depicts the variation in ΔP and C f with the increase in the number of the grid. From the figure, it is observed that there is no significant effect of the number of elements in both the parameters after the number of elements 86,558. Therefore, such a grid has been used for all considered microchannel configurations and inlet Re.
2.2 Validation The considered schemes and selected grid are validated with an experimental result [9]. The validation study is performed to ascertain the authenticity of the numerical study. The considered governing Eqs. (1) and (2) are solved with their dimension using the schemes and grid which is used for the present analysis. Figure 5 displays the variation in the ΔP of present analysis to the experimental analysis [9]. Though both numerical and experimental analysis show the same trend, there is a variation at lower Re. However, the variation is marginal at the intermediate Re conditions.
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Fig. 5 Comparison of the present numerical schemes with existing experimental study (plain channel) [9]
3 Results and Discussion The objective of the study is to investigate the flow dynamics of microchannels with microstructure. There are nine various microchannel configurations considered including plain channel. The following parameters are taken for the detailed investigation of the present study—variation of velocity and pressure across the microstructure, formation of vortices in the wake region, pressure drop, average skin friction coefficient on the microstructure surface, and drag coefficient, etc. Figures 6 and 7 display the variation of velocity, pressure, and the formation of vortices due to the microstructure for lower and higher Re respectively. The figure interprets that the flow velocity increases (it causes the enhancement of rate of heat transfer) by the microstructure due to the decrease in the microchannel flow area (cross-section) and the formation of vortices is also observed at the wake regime (where the velocity of the working liquid is very less and it causes a lower heat transfer rate) of all the considered microstructure for both inlet Re conditions. In MCMS 1, the rectangular microstructure is placed at the channel. It creates a sudden variation in the cross-section of flow path. The flow separation is noticed at the corner of microstructure and fluid gets stuck in between the accelerated fluid stream and the microstructure surface at higher Re. The stuck fluid recirculates by the drag of the accelerated fluid stream. However, such a phenomenon is not observed in the lower Re condition of MCMS 1 and MCMS 2 due to the lower velocity and gradual change in the flow area (cross-sectional) of the channel respectively. However, in MCMS 2, it also causes smaller vortices than MCMS 1. In MCMS 2 and 3, the oval cross-sectional microstructure is split into two portions. Therefore, there is a pair of vortices which form in between them. In both cases, the dimensions of vortices are comparable with the space between two portions of microstructures due to the absence of the secondary flow in the space. However, such a secondary flow is observed in MCMS 4 and 5 due to the offset arrangement of two portions. There
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is a formation of vortices in the wake regime of both split portions. The second portion creates larger vortices than the first portion. The secondary flow disturbs the vortices (at the wake region). It is also influenceable on the heat transfer performance. The microstructure is located with a certain angle in MCMS 7 and 8 which causes more reduction in the cross-sectional area than all others. Therefore, it creates larger vortices than all other MCMS. Figures 6 and 7 also depict the variation of pressure in the microchannel due to the location of the microstructure. From the figure, it is important to note that there is a reduction in pressure in the flow field after passing the microstructure. The higher
Fig. 6 Variation of velocity, formation of vortices and pressure for Re 200
Fig. 7 Variation of velocity, formation of vortices and pressure for Re 1200
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pressure is observed near the stagnation point of the microstructure. In MCMS 7 and 8, the microstructure is located with an offset angle (to the axis of the channel along the flow direction) where the flow experiences gradual convergence and divergence at the same cross-section (normal to the direction of fluid flow). The pressure is higher and lower in the convergence and divergence portions respectively. The pressure drop (ΔP) is provided in Fig. 8. The ΔP of all the microstructured microchannels are more than the plain channel. It exposes that the incorporation of microstructures increases the ΔP of the channel. The ΔP of MCMS 8 and MCMS 7 is higher than all others due to the smaller area of cross-section when the flow passes the microstructure. However, the MCMS 8 attains a higher pressure drop than the MCMS 7. Similarly, the pressure drop of MCMS 4 is higher than MCMS 3 and both are higher than MCMS 2. These results show that the split of microstructure contributes to an increase in the ΔP and the ΔP also increases as the distance between both portions of microstructures increases. However, the offset arrangement of split microstructure portions also induces pressure loss due to flow disturbance. Therefore, the ΔP of MCMS 5 and 6 is higher than the MCMS from 1 to 4. The pressure drop of MCMS 2 and MCMS 6 is lower than MCMS 1 and MCMS 3. It shows that the gradual variation in the area of the cross-section of the microstructure helps to reduce the ΔP. Figures 9 and 10 provide the average skin friction coefficient (C f ) of the surface of the microstructure and the local skin friction coefficient on the surface of the microstructures respectively. From the figures, it was noted that both the cross-section of the microstructure and the velocity of the fluid flow influence the skin friction coefficient. In MCMS 1, the higher skin friction coefficient is observed at the corners of the rectangular microstructures where the flow velocity is maximum. However, there is no contact of the accelerated flow stream with the other surfaces of the rectangular microstructures. Therefore, the average skin friction (C f ) of MCMS 1 is the lowest. In the oval cross-sectional microstructures, there are no such corners and Fig. 8 Variation of pressure drop with Re of all considered MCMSs
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Fig. 9 Variation of average skin friction coefficient with Re on the microstructure surface
the C f of MCMS 2 is higher than MCMS 1. The same phenomenon is also observed in MCMS 5. The C f is higher than MCMS 8 for all Re due to its higher flow velocity. The MCMS 7 is next to MCMS 8. The C f of MCMS 3 and 4 are almost the same and lower than all others except MCMS 1. The variation of the average drag coefficient (C D ) of all the considered MCMSs is shown in Fig. 11. It almost reflects the variation of ΔP. The higher and lower C D is observed from MCMS 8 and MCMS 2 respectively. MCMS 7 and 5 are in the next consecutive higher places after MCMS 8. However, there is no significant variation among the C D of MCMS 2, 3, and 4 and these MCMSs are lower than MCMS 1 and 6 in C D .
4 Conclusions The effect of incorporation of various microstructures in the microchannel on the flow dynamics has been investigated in detail. The microstructure helps to accelerate the flow and form the vortices in the wake regimes. From the study, it is found that the cross-section of microstructure influences the pressure drop of the channel, skin friction coefficient, and drag coefficient. The gradual variation shows a lower pressure drop penalty than the sudden variation in the cross-sectional area of the microstructure. The split of the microstructure contributes to increase the pressure drop and the pressure drop also increases as the space between the two split portions of microstructure increases. The higher pressure drop, skin friction coefficient, and drag coefficient are observed when the microstructure is tilted with an angle reference to the axis of fluid flow. Further analysis is warranted on the heat transfer phenomena of microchannel with these microstructures to identify the effect of these flow dynamics on the thermal performance of microchannel.
378 Fig. 10 Variation of the local skin friction coefficient on the surface of the microstructure
Fig. 11 Variation of average drag coefficient with Re on the microstructure surface
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Nomenclature MCMS ΔP Re V ρ μ Cf CD
Microchannel with microstructure (–) Pressure drop (kPa) Reynolds number (–) Velocity vector (–) Density (kg/m3 ) Dynamic viscosity (Pa s) Skin friction coefficient (–) Drag coefficient (–)
References 1. Alfellag MA, Ahmed HE, Fadhil OT, Kherbeet AS (2019) Optimal hydrothermal design of microchannel heat sink using trapezoidal cavities and solid/slotted oval pins. Appl Therm Eng 158:113765 2. Chai L, Xia G, Zhou M, Li J, Qi J (2013) Optimum thermal design of interrupted microchannel heat sink with rectangular ribs in the transverse microchambers. Appl Therm Eng 51(1–2):880– 889 3. Datta A, Sanyal D, Das AK (2016) Numerical investigation of heat transfer in microchannel using inclined longitudinal vortex generator. Appl Therm Eng 108:1008–1019 4. Fiebig M (1995) Embedded vortices in internal flow: heat transfer and pressure loss enhancement. Int J Heat Fluid Flow 16(5):376–388 5. Foong AJL, Ramesh N, Chandratilleke TT (2009) Laminar convective heat transfer in a microchannel with internal longitudinal fins. Int J Therm Sci 48(10):1908–1913 6. Jia Y, Xia G, Li Y, Ma D, Cai B (2018) Heat transfer and fluid flow characteristics of combined microchannel with cone-shaped micro pin fins. Int Commun Heat Mass Transf 92:78–89 7. Li Y, Xia G, Jia Y, Ma D, Cai B, Wang J (2017) Effect of geometric configuration on the laminar flow and heat transfer in microchannel heat sinks with cavities and fins. Numer Heat Transf Part A Appl 71(5):528–546 8. Liu Y, Cui J, Jiang YX, Li WZ (2011) A numerical study on heat transfer performance of microchannels with different surface microstructures. Appl Therm Eng 31(5):921–931 9. Qu W, Mudawar I (2002) Experimental and numerical study of pressure drop and heat transfer in a single-phase micro-channel heat sink. Int J Heat Mass Transf 45(12):2549–2565 10. Rajalingam A, Chakraborty S (2021) Effect of shape and arrangement of micro-structures in a microchannel heat sink on the thermo-hydraulic performance. Appl Therm Eng 190:116755 11. Tao WQ, He YL, Wang QW, Qu ZG, Song FQ (2002) A unified analysis on enhancing single phase convective heat transfer with field synergy principle. Int J Heat Mass Transf 45(24):4871– 4879 12. Tuckerman DB, Pease RFW (1981) High-performance heat sinking for VLSI. IEEE Electron Device Lett 2(5):126–129
Combined Effect of Heterogeneous Zeta Potential on Microchannel Wall and Conductive Link in Induced Charge Electrokinetic Micromixing Anshul Kumar Bansal, Ram Dayal, and Manish Kumar
Abstract In this paper, a variation of induced charge electrokinetic (ICEK) micromixing is investigated numerically. The channel walls are subjected to heterogeneous zeta potentials. Further, conductive links are also employed inside the channel to enhanced mixing. The primary aim is to understand the influence of control parameters (i.e. zeta potential, number of conductive links, their orientation and applied electrical potential) on mixing efficiency. Heterogeneous zeta potential creates recirculation zone in the flow and increases the interferential contact between the fluids layer, hence improving the mixing. Further, the conductive links induce the micro vortices which enhance the mixing. The induced charge on the conductive link is calculated using correction model. Results show that the mixing efficiency increased by two times when conductive link inside the channel and heterogeneous zeta potential patch on wall is mounted. It is also observed that the mixing efficiency increased as the number of links and patches increased. Keywords Heterogeneous zeta potential · ICEK · Micromixer · Mixing efficiency · Conductive link
1 Introduction In recent years, microfluidics has been widely used in biological and chemical industries. Microfluidics systems are used in many applications such as DNA hybridization [1], drug discovery and delivery [2, 3], biological screening [4], Lab on a chip device [5], chemical synthesis [6], etc. In microfluidics devices, rapid and homogeneous mixture is required in a short time, therefore, micromixer play an important role in microfluidics system. Due to low Reynold number flow in microchannel, flow is laminar hence mixing is limited and required a long channel and more time. There exist various [7] active and passive techniques to enhance mixing inside the A. K. Bansal (B) · Ram Dayal · M. Kumar Department of Mechanical Engineering, MNIT Jaipur, Jaipur 302017, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_32
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microchannel. In passive micromixer, no external power source is required to perturb the flow and only geometrical modification creates chaotic advection to enhance the mixing. Aoki and Mae [8] experimentally investigated the geometrical parameter effect on the mixing performance and they concluded that the reducing size of microchannel increases the mixing efficiency. On the other hand, active micromixers require some external power source such as magnetic [9], electrokinetic [9, 10], acoustic [11] and thermal [12] to perturb the flow and enhance the mixing. Acoustic and thermal methods increase the sample temperature during mixing process, therefore unsuitable for biological processes which are sensitive to temperature variation. In this context, electrokinetics have major advantages such as easy to integrate with the microfluidics system, bidirectional flow, easy to control and low dispersive flow of sample. Chen and Cho [13] investigated the electrokinetically mixing in a grooved shaped channel. They observed that the mixing is increased as the wave amplitude of the grooved increases in the microchannel. Many researchers in the field of electrokinetic micromixer used heterogeneous zeta potential on the microchannel wall to enhance the mixing process. The flow recirculation near the heterogeneous zeta potential enhanced the mixing. Biddiss et al. [14] experimentally investigated the mixing enhancement using surface charge heterogeneity on microchannel wall and reported mixing efficiency 68% approximately. Jain and Nandakumar [15] optimised the mixing performance for micromixer with heterogeneous pattern of zeta potential on microchannel wall. They were able to increase mixing performance about 3 times compared to homogeneous zeta potential microchannel. Qaderi et al. [16] numerically simulated the mixing performance for a triangle shape hurdle with surface charge heterogeneity in a microchannel using electrokinetic flow and showed 48% increase in mixing as the zeta potential heterogeneity is applied on the hurdles. Induced charge electrokinetic (ICEK) flow phenomenon induces the micro vortices near the conductive object inside the micromixer. These micro vortices increase the chaotic advection resulting in higher mixing efficiency. Wu and Li [17] experimentally investigated the mixing performance in a ICEK micromixer with conductive triangular hurdle. Mixing efficiency of about 92% was reported with 2 cm long microchannel. Nazari et al. [18] numerically investigated the mixing performance for microchannel with conductive microchamber by applying DC electric voltage. Circular, rectangular, triangular and rhombus shapes mixing chambers are used to analyse the mixing performance. They reported better mixing performance for triangular and rhombus shaped mixer chamber inside the microchannel. Azimi et al. [19] investigated the mixing performance for a micromixer using deformable flexible conductive link inside the microchannel. They observe that mixing enhances due to the micro vortices generated near the conductive link and movement of the link inside the microchannel. Mixing efficiency of about 90% was achieved in this micromixer with a length of 5 cm. Najjaran et al. [20] numerically investigated the mixing performance in a ICEK micromixer using corrugated microchannel wall. They investigated the effect of conductive plate length and its position in the channel. They observe that mixing efficiency of 97% is reported when the conductive plate is installed near the inlet of micromixer.
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Fig. 1 Geometry of the micromixer
It is evident from the literature so far that the significant work has been undertaken to understand the effect of surface charge heterogeneity on microchannel wall and ICEK phenomenon separately. While studies considering the combined effect of these two phenomena are still rare, the objective of this work is to investigate the combined effect of surface charge heterogeneity and ICEK on the mixing for design an efficient micromixer. In this study, a 2-D microchannel with two inlets and one outlet section using conductive link and surface charge heterogeneity on wall is used for micromixing. DC electric field is applied to generate the electrokinetic flow. Effect of number of conductive links, angle (orientation), heterogeneous zeta potential and electrical potential is studied for mixing efficiency.
2 Problem Description Figure 1 shows the geometry of the micromixer used in this study. Microchannel contains two inlets for entering two different types of fluids for mixing and one outlet for outing the mix flow. Microchannel length (L) of 2 cm and width (W ) of 250 µm were used to mix the flow. A conductive link length (l) and width (w) are used inside the microchannel. Conductive link is placed to generate the micro vortices when external applied electric field is applied between inlet and outlet section. A heterogeneous patch (length L P ) of positive zeta potential is used to generate the recirculated flow. Micromixers with different numbers of conductive links and heterogeneous patches are used to analyse the mixing performance.
3 Mathematical Model A 2-D incompressible, steady state laminar flow model is numerically simulated. The governing equations for the electrical field, flow field and concentration field, which are used to evaluate the mixing performance, are described below.
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3.1 Electric Field The Poisson equation was used to evaluate the induced electrical field inside the microchannel due to electrical double layer. The equation is given by [21]: ∇2ψ = −
ρe ε0 ε
(1)
Laplace equation described the electrical potential distribution due to applied electrical voltage between the inlet and outlet and it is given by [21]: ∇ 2 φe = 0
(2)
3.2 Flow Field Navier–Stokes equation and continuity equation are used to describe the flow field in the microchannel and these equations are given by: ∇ ·V=0
(3)
ρ[(V · ∇)V] = − ∇ p + μ∇ 2 V + ρe E
(4)
In Eq. (4), the last term denotes the body force due to electrical field. For a thin layer of electrical double layer, this term dropped off by using Helmholtz–Smoluchowski slip velocity (uslip ) near the wall region [21]. u slip = −
ε0 εζ E μ
(5)
Due to induced charge electrokinetic effect, zeta potential (ζi ) is induced near the conductive plate which is given by [21]: ζi = − φ e + φ c
(6)
where φc is constant and given by ∫ φc =
s
φe dA A
(7)
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slip velocity at boundary of conductive plate is given by: u link = −
ε0 εζi E μ
(8)
3.3 Concentration of Mixing Species In the steady state condition, concentration of mixing species is governed by the convection–diffusion equation which is given by: (V · ∇)C = D∇ 2 C
(9)
3.4 Micromixing Efficiency Micromixing efficiency (σ ) is given by: ( σ = 1−
∫W
|C
|dy
− C∞ 0 ∫W |C0 − C∞ |dy 0
) × 100
(10)
3.5 Boundaries Conditions Boundary conditions for all the governing flow equations for this micromixer are given below: At inlet: φe = φin ; p = 0; ∇V · n→ = 0; ∇ψ · n→ = 0; C1 = C0 and C2 = 0. At outlet: φe = φout ; p = 0; ∇V · n→ = 0; ∇ψ · n→ = 0; ∇ · C · n→ = 0. At microchannel wall: ∇φ · n→ = 0; ∇ p · n→ = 0; V = 0; ψ = ζ or ζi ; ∇ · C · n→ = 0.
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Fig. 2 Effect of grid element size for concentration profile at channel outlet section
4 Numerical Simulation 4.1 Grid Independence Study All governing equations are solved numerically in the computational domain using COMSOL Multiphysics software. A grid independence test is performed achieving grid-independent results. Concentration profile at outlet section of the microchannel was drawn with 12,600, 15,980 and 24,170 grid number elements. Results (Fig. 2) show that the concentration profiles for 15,980 and 24,170 have negligible differences. Therefore, 15,980 grid number elements were used for this study to get grid-independent results.
4.2 Validation To validate the present study, numerical simulated results are validated with the experimental work done by Wu and Li [17]. Figure 3 shows that the numerical results for concentration profile are in good agreement with the experimental results and thus, validate the present numerical model.
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Fig. 3 Validation of present study with experimental study by Wu and Li [17]
5 Results and Discussion The input parameters and constants used for numerical simulation of ICEK micromixer in the present study are given in Table 1. Effect of different configuration of link and heterogeneous patches of zeta potential, applied electric field and conductive link orientation on mixing performance is discussed in the following section.
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Descriptions
Values
Relative permittivity, ε
80.2
Vacuum permittivity, ε0
8.85 × 10−12 C/Vm
Diffusion constant, D
4.157 × 10−11 m2 /s
Viscosity, μ
0.001 kg/m/s
Density, ρ
1000 kg/m3
Absolute temperature, T
300 K
Microchannel length, L
2000 µm
Microchannel width, W
250 µm
Inlet molar concentration, C0
10 mol/m3
Micro-link length, l
250 µm
Micro-link width, w
25 µm
Zeta potential of channel walls, ζ (mV)
50 mV
Applied electric potential φ
1500 V/m
5.1 ICEK Micromixer Configuration Figure 4 shows the streamline and concentration contour for a micromixer with one conductive link and one heterogeneous zeta potential patch on microchannel wall. It clearly shows that the mixing is increased due to generated micro vortices and recirculation of flow inside the micromixer. The use of conductive link in micromixer generates the micro vortices due to the induced charge electrokinetic effect whereas heterogenous zeta potential patch on the microchannel wall creates the recirculation zone near it because of reversed electroosmosis velocity near the patch. Therefore, the mixing is enhanced due to increase in chaotic advection and increases the interferential contact between the fluid layers. Mixing efficiency increases to 46% using conductive plate and heterogeneous patch. It is to be mentioned that mixing efficiency is 36% in a simple straight channel without any link and patch. Figure 5 shows the different geometrical configuration of microchannel with different numbers of conductive links and heterogeneous zeta potential patches. Table 2 summarises the mixing efficiency for these different micromixer configurations. It is clearly seen that the mixing efficiency increases as the number of conductive link and patches are increased. This happens due to increases in the number of recirculation zones and the micro vortices which facilitate more chaotic advection generated and mixing increases. Vertically aligned conductive link in micromixer increases the mixing efficiency more in comparison to horizontally aligned links. This is due to more micro vortices being generated at the same section of the micromixer and this increases the interferential contact between the fluid layers. But vertically aligned conductive links obstruct the flow and moreover, the presence of micro vortices decreases the volume flow rate.
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Fig. 4 Streamline and concentration contour for micromixer with one conductive link and one patch
Fig. 5 Configuration of micromixers with different numbers and orientations of conductive links and heterogeneous zeta potential on the microchannel wall. Red and black walls indicate positive and negative zeta potential respectively
Table 2 Mixing efficiency for different configurations of micromixers Case No.
Heterogeneous patch pair
Conductive link
Link alignment
Mixing efficiency
1
1
1
–
47.32
2
2
2
H
61.73
3
2
2
V
60.85
4
3
2
H
74.70
5
3
2
V
79.69
6
2
3
H
63.29
7
2
3
V
73.18
8
3
3
H
82.92
9
3
3
V
97.55
H horizontal; V vertical
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Fig. 6 Variation of mixing efficiency with conductive link orientation
5.2 Effect of Conductive Link Orientation (Angle) Figure 6 shows the effect of conductive link angle with horizontal axis with different numbers of conductive links and heterogeneous zeta potential patches on mixing performance. Mixing efficiency increases as the conductive link orientation angle increases (positive or negative) from horizontal position but up to a certain angle. It is due to obstruction in flow increases as the link orientation angle increases. Increasing number of link and patches generated more vortices near the link and recirculated zones. Therefore, an increase in angle of link above 35° and 5° in the case of 2–2 and 3–3 links-patches combination in micromixer reduces the mixing efficiency. Overall, increasing in orientation angle of conducting link up to a certain value increases the mixing efficiency and after that, it decreases the mixing efficiency.
5.3 Effect of Applied Electric Field and Zeta Potential on Heterogeneous Patch Figure 7 shows the effect of external applied electric field between inlet and outlet of micromixer on mixing performance. It clearly shows that the decreasing applied electric field increases the mixing efficiency since electroosmosis velocity inside the micromixer is directly proportional to the applied electric field. Decreasing electric field increases the time for diffusion mixing inside the channel due to lower electroosmosis velocity. This increases the mixing performance but increases the overall mixing time. Figure 7 also shows that the mixing efficiency is less affected with applied electric field as the links-patches combination increases in numbers. Since induced slip velocity is proportional to the square of the applied electric field, the vortices’ strength and flow velocity increased as the applied electric field
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Fig. 7 Variation of mixing efficiency with external applied electric field
increased. As a result, increasing velocity and vortex strength within the micromixer both determine mixing performance at higher potential fields. Figure 8 shows the effect of zeta potential on the heterogeneous patch on the microchannel wall on mixing performance. It clearly shows that the increasing zeta potential on the patches increases the mixing efficiency. Since higher zeta potential increases electroosmosis velocity therefore recirculation zone increases. Hence more interferential contact increases between the fluid layer and increases the mixing efficiency. As the number of patches increases, the total recirculation zone area also increases therefore more uniform mixing is found at the outlet. It is also shown in the figure that compared to micromixer with one and two patches, the micromixer with three patches increases the mixing efficiency in a more effective manner with an increase in zeta potential on patches.
6 Conclusions Micromixing in an induced charged micromixer using conductive plate and heterogeneous zeta potential wall is numerically simulated. Heterogeneous zeta potential on microchannel wall causes the recirculation of flow which enhances the mixing. The mixing efficiency is found to increase when using more than one conductive plate, due to corresponding increase in number of vortices generated inside the microchannel. Up to 97% mixing efficiency is achieved using 3 vertically aligned conductive links with 3 pairs of heterogeneous zeta potential patches on microchannel wall. Link orientation also affects the mixing performance significantly. Up to a certain angle orientation (45° for 1-patch and 1-link, 35° for 2-patch and 2-link and 5° for 3-patch and 3-link combination) mixing is increased but after that, it is decreased due to more obstruction of flow on one side. Results show a significant influence of applied
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Fig. 8 Variation of mixing efficiency with zeta potential on the microchannel wall patch
electrical voltage and patch zeta potential on mixing performance. Low applied electric field increases the mixing but it delays the mixing time as well. Overall, present study shows that the use of combined heterogeneous zeta potential patch and conductive link in a micromixer enhanced the mixing significantly in a short-length microchannel. Acknowledgements The author thanks to Material Research Centre at MNIT Jaipur for providing computational resources for this work.
Nomenclature σ ζ ψ ρe φe V p E ζi C C∞ φin φout n→
Mixing efficiency (–) Zeta potential (V) Electrical potential in the EDL (V) Free charge density (kg/m3 ) Applied electric potential (V) Velocity (m/s) Pressure (Pa) Applied electric field (V/m) Induced zeta potential on the conductive link (V) Concentration (mol/m3 ) Ideal concentration (mol/m3 ) Inlet applied potential (V) Outlet applied potential (V) Normal vector (–)
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References 1. Kastania AS et al (2016) Plasma micro-nanotextured polymeric micromixer for DNA purification with high efficiency and dynamic range. Anal Chim Acta 942:58–67. https://doi.org/10. 1016/j.aca.2016.09.007 2. Dittrich PS, Manz A (2006) Lab-on-a-chip: microfluidics in drug discovery. Nat Rev Drug Discov 5(3):210–218. https://doi.org/10.1038/nrd1985 3. Razzacki SZ, Thwar PK, Yang M, Ugaz VM, Burns MA (2004) Integrated microsystems for controlled drug delivery. Adv Drug Deliv Rev 56(2):185–198. https://doi.org/10.1016/j.addr. 2003.08.012 4. Lee C-Y, Lee G-B, Lin J-L, Huang F-C, Liao C-S (2005) Integrated microfluidic systems for cell lysis, mixing/pumping and DNA amplification. J Micromech Microeng 15(6):1215 5. Bockelmann H, Heuveline V, Ehrhard P, Barz DPJ (2012) An electrokinetic micro mixer for labon-chip applications: modeling, validation, and optimization. In: ASME 2011 9th international conference on nanochannels, microchannels, and minichannels, ICNMM 2011, vol 2, May 2012, pp 473–480. https://doi.org/10.1115/ICNMM2011-58278 6. Kawai Y, Yamamoto T (2019) Synthesis of dimpled and submicron-sized polymer particles of different morphologies using free micromixer. Colloids Interface Sci Commun 32:100193. https://doi.org/10.1016/j.colcom.2019.100193 7. Rashidi S, Bafekr H, Valipour MS, Esfahani JA (2018) A review on the application, simulation, and experiment of the electrokinetic mixers. Chem Eng Process Process Intensif 126:108–122 8. Aoki N, Mae K (2006) Effects of channel geometry on mixing performance of micromixers using collision of fluid segments. Chem Eng J 118(3):189–197 9. Lu LH, Ryu KS, Liu C (2002) A magnetic microstirrer and array for microfluidic mixing. J Microelectromech Syst 11(5):462–469. https://doi.org/10.1109/JMEMS.2002.802899 10. Kazemi Z, Rashidi S, Esfahani JA (2017) Effect of flap installation on improving the homogeneity of the mixture in an induced-charge electrokinetic micro-mixer. Chem Eng Process 121:188–197 11. Gao Y, Tran P, Petkovic-Duran K, Swallow T, Zhu Y (2015) Acoustic micromixing increases antibody-antigen binding in immunoassays. Biomed Microdevice 17(4):1–5. https://doi.org/ 10.1007/s10544-015-9987-0 12. Nayak AK, Haque A, Weigand B (2018) Analysis of electroosmotic flow and Joule heating effect in a hydrophobic channel. Chem Eng Sci 176:165–179. https://doi.org/10.1016/j.ces. 2017.10.014 13. Chen CK, Cho CC (2007) Electrokinetically-driven flow mixing in microchannels with wavy surface. J Colloid Interface Sci 312(2):470–480. https://doi.org/10.1016/j.jcis.2007.03.033 14. Biddiss E, Erickson D, Li D (2004) Heterogeneous surface charge enhanced micromixing for electrokinetic flows. Anal Chem 76(11):3208–3213 15. Jain M, Nandakumar K (2013) Optimal patterning of heterogeneous surface charge for improved electrokinetic micromixing. Comput Chem Eng 49:18–24 16. Qaderi A, Jamaati J, Bahiraei M (2019) CFD simulation of combined electroosmotic-pressure driven micro-mixing in a microchannel equipped with triangular hurdle and zeta-potential heterogeneity. Chem Eng Sci 199:463–477. https://doi.org/10.1016/j.ces.2019.01.034 17. Wu Z, Li D (2008) Micromixing using induced-charge electrokinetic flow. Electrochim Acta 53(19):5827–5835 18. Nazari M, Rashidi S, Esfahani JA (2019) Mixing process and mass transfer in a novel design of induced-charge electrokinetic micromixer with a conductive mixing-chamber. Int Commun Heat Mass Transf 108:104293 19. Azimi S, Nazari M, Daghighi Y (2017) Developing a fast and tunable micro-mixer using induced vortices around a conductive flexible link. Phys Fluids 29(3):32004
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20. Najjaran S, Rashidi S, Valipour MS (2020) A new design of induced-charge electrokinetic micromixer with corrugated walls and conductive plate installation. Int Commun Heat Mass Transf 114:104564. https://doi.org/10.1016/j.icheatmasstransfer.2020.104564 21. Wu Z, Li D (2008) Mixing and flow regulating by induced-charge electrokinetic flow in a microchannel with a pair of conducting triangle hurdles. Microfluid Nanofluid 5(1):65–76
Analysis of Sperm Cell Kinetics in Newtonian and Non-Newtonian Fluid Medium Within a Microfluidic Channel Dhiraj B. Puri, Vadiraj Hemadri, Arnab Banerjee, and Siddhartha Tripathi
Abstract The success of the internal fertilization process results from the successful migration of sperm through the female reproductive tract. During the course of motion, sperm must overcome the obstacles and chemical changes that occur in their path towards the oocyte. Herein, we conducted an experimental and simulation study of sperm in different fluid medium, whose rheological properties mimic the actual environment of the female reproductive tract. In this work, two surrounding mediums of sperm are prepared by diluting Polyvinylpyrrolidone (PVP) and Methylcellulose (MC) in Phosphate-buffered saline (PBS). They exhibit Newtonian and NonNewtonian behaviour, respectively. Sperm motility and kinetic parameters such as velocity, beat frequency, forces, power and swimming efficiency are calculated. The results indicate that sperm possesses high straight-line velocity, curvilinear velocity and beat frequency in MC 2% Non-Newtonian fluid. The Drag force, power output, the power required for sperm motion and sperm swimming efficiency are high in Non-Newtonian fluid compared to the Newtonian fluid of the same viscosity range. Keywords Microchannel · Motility · Sperm cells · Velocity · Viscosity
1 Introduction Infertility has been a major global health issue with significant psychological, economic and social consequences, thus affecting more than 48 million couples and 186 million individuals worldwide [1]. It has also been estimated that over 20– 30% of infertility problems are due to male factors, 50% of infertility problems are due to female factors and the remaining 20–30% of factors are contributed by males D. B. Puri · V. Hemadri · S. Tripathi (B) Department of Mechanical Engineering, Birla Institute of Technology and Science-Pilani, K K Birla Goa Campus, Zuarinagar, Sancoale, Goa 403726, India e-mail: [email protected] A. Banerjee Department of Biological Sciences, Birla Institute of Technology and Science-Pilani, K K Birla Goa Campus, Zuarinagar, Sancoale, Goa 403726, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_33
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and females [2]. The majority of male infertility issues are associated with low count and quality of sperm, morphology and a low percentage of motile sperm [3]. These male infertility factors generally arise due to genetic and lifestyle related issues. The rising of these infertility problems requires immediate attention, and there is a need to study the causes and to make diagnostic of the infertility problems easily accessible and affordable. The causes of infertility are still less known, and hence to improve the understanding of the cause of infertility, it is important to study the dynamics of mammalian sperm especially human sperm in female reproductive tract. Hence, a study of the human sperm migration in a Newtonian and Non-Newtonian fluids within microchannel is conducted, and the rheological properties of these fluids are maintained identical to female cervical mucus.
2 Literature Review and Objective The migration of sperm plays a vital role in successful fertilization of female eggs. Mammalian sperm such as human and bovine follow the internal fertilization process and travel a distance of nearly 1000 times their own length in female reproductive tract to reach the egg [4]. During its passage through the female reproductive tract, sperm must travel through the cervix, uterus, uterine tubal junction and oviduct by overcoming the obstacles in the path to reach the egg present in the oviductal ampulla [5]. Several studies have been carried out to investigate the behaviour of the cervical and oviductal mucus and they found that these fluids are made up of macromolecules and gelatinous material with Non-Newtonian shear thinning behaviour. The viscosities of these fluids are varying between 100 and 1000 cP [6]. The ability of a sperm to migrate in a such highly viscous and complex fluid environment decides the chances of the sperm to reach the oocyte for successful fertilization. The pioneering research work in the field of sperm migration involves the measurement of sperm motility in viscoelastic fluid similar to mucus and standard laboratory fluid used in Assisted reproductive techniques (ARTs) [7, 8]. It is observed that even with an increase in viscosity, human sperm overcome the resistance and maintain their speed in Non-Newtonian fluid environment [9]. However, opposite behaviour was observed for externally fertilizing sperm where the sperm failed to maintain the progressive velocity in Non-Newtonian fluid [10]. Several qualitative studies have been done related to modulation of sperm wave form and effect of amplitude and frequency on progressive velocity in viscoelastic fluid [8, 11]. Ishijima et al. [12] studied the three-dimensional beating of sperm here they, found the relationship between the sperm head rotation and flagella beat frequency. Smith et al. [13] performed detailed quantitative research on flagella bending and its modulation at low and high viscoelastic fluid. Mane et al. [14] conducted a study on behaviour of sperm cells in different low viscosity Newtonian and Non-Newtonian environments. They found that sperm possesses high motility in Non-Newtonian fluid medium. Tung et al. [15] studied the motion of the sperm in viscoelastic fluids in which sperm forms a cluster of different sizes. Hyakutake et al. [16, 17] showed
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that motility of sperm changes with a change in characteristics of surrounding fluid. Stehnach et al. [17] showed that the viscosity gradient in cervical mucus causes the cells swimming speed to decrease and causes them to collect in the high viscosity areas. Nosrati et al. [18] reported that the swimming behaviour of sperm in female reproductive tract depends on the chemical, physiological and rheological properties of the fluid. Few studies involve the use of mathematical model to study the hydrodynamics of sperm migration. Gillies et al. [19] developed a model of sperm using regularized stokeslet. They reported that the reduction in head width results increase in straightline velocity of sperm. Dresdner and Katz [20] developed a mathematical model to describe the relationship between the morphology and flagella beat frequency with power output and swimming velocity. Razavi and Seyed Ahmadi [21] performed parametric study using finite element model and observed the variation in the sperm velocity with flagella amplitude, wavelength and different head geometries. Our literature survey shows that limited studies are available which provide the quantitative analysis of the motility and kinetic parameters of sperm cells in female reproductive tract. Recent development in microfluidics enables to mimic the actual design and condition of female reproductive tract in order to study the behaviour and diagnostics possibilities of infertility [5]. In the present study, we performed the experimental and computational analysis of the motion of sperm in highly viscous Newtonian and Non-Newtonian fluid using straight microchannel. Here, we analyzed the effect of rheological properties on the motion of the sperm in which we measured various motility and kinetic parameters such as curvilinear velocity, straight-line velocity, average path velocity, beat frequency, drag force and swimming efficiency involved in the migration of sperm with similar environmental condition of the female reproductive tract.
3 Materials and Methods 3.1 Design and Fabrication A design and photograph of PDMS based simple straight microchannel having a rectangular cross-section are shown in Fig. 1. The microchannel consists of single inlet–outlet reservoirs. The channel width and height were kept at 200 µm and 50 µm respectively. The fabrication of the channel was carried out using photolithography and soft lithography techniques. A negative photoresist (SU-8 2050) was dispensed on the silicon wafer and rotated at 3000 RPM to obtain a uniform thickness of 50 µm. Before placing it in the mask aligner photoresist was prebaked and then exposed to a UV ray. Upon UV exposure, the unexposed portion of the photoresist was washed out using the developer solution. Finally, the silicon SU-8 mould was post-baked. Further, employing soft lithography technique, a 10:1 (w/w) mixture of PDMS (Polydimethylsiloxane) and a curing agent
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Fig. 1 Design and photograph of PDMS based straight microfluidic channel
was prepared and poured on the silicon mould. The mould was then placed in a hot air oven at 65 °C for 45 min. After curing, the PDMS was removed off the mould and holes were drilled at the inlet and outlet reservoirs. The fabricated PDMS chip was then bonded to the glass slide using an adhesive formed by a mixture of PDMS and curing agent in the 6:1 (w/w) ratio. To increase the bonding strength, the bonded PDMS chip was further heated in a hot air oven at 80 °C for 1 h [22, 23].
3.2 Semen Collection and Sample Preparation Semen sample was collected from a healthy, well-informed person in a sterile container after the abstinence period of three days. The collected sample is then allowed to liquefy at 37 °C for 25–30 min as per the guidelines of WHO [1]. The liquified semen is then added into the high viscosity Newtonian and Non-Newtonian fluid in a ratio of 2:5 (v/v) media to change the viscosity of the surrounding medium of the sperm. MC was (Methyl cellulose, Loba Chemie Pvt. Ltd., India) used to prepare Non-Newtonian medium with 1.5 and 2% concentration whereas PVP (Polyvinyl propamidine, Sigma Aldrich, USA) was used to prepare Newtonian fluid media with 12 and 14% concentration in PBS (Phosphate buffer saline, Sigma Aldrich, USA). The viscosity of all fluids was measured using a Brookfield Viscometer at different shear rates as shown in Fig. 2. It is observed that the viscosity of the PBS and PVP based viscous environment (12 and 14%) were constant for different shear rates which shows that these two fluids are Newtonian. However, the viscosity of methyl cellulose fluid is decreased with the shear rate showing Non-Newtonian fluid behaviour (Table 1).
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Fig. 2 Variation of viscosity with the shear rate. MC 1.5 and 2% viscosity decreases with the shear rate showing shear thinning behaviour or Non-Newtonian behaviour
Table 1 Viscosity of the different fluid medium and their behaviour Fluids
Viscosity (cP)
Newtonian (N)/Non-Newtonian (NN)
PBS
0.744
N
PVP 12%
138
N
PVP 14%
260
N
MC 1.5%
145
NN
MC 2%
280
NN
3.3 Experimental Setup and Procedure The experimental setup consists of an inverted microscope (CKX53, Olympus) equipped with a digital camera (C11440-36U, ORCA-spark Hamamatsu) to observe and record the sperm behaviour. The prepared high viscosity semen sample is then injected into the microchannel shown in Fig. 1, using syringe to study the behaviour of the sperm.
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Fig. 3 Various kinematic parameters of sperm trajectory
3.4 Image Analysis The behaviour of the sperm in a straight microchannel was recorded and monitored using HCImage Live software. To analyze the motion of the sperm, the locations of the sperm head and its orientation were manually tracked frame by frame using a manual tracking plugin in ImageJ software. Different sperm velocities such as straight-line velocity or progressive velocity (VSL), curvilinear velocity (VCL) and average path velocity (VAP) were calculated from the trajectories of the sperm cell. VCL is the curvilinear velocity calculated by averaging the velocity at each point, VSL is the time-average velocity calculated along the straight line between the initial and final point of the sperm head, and VAP is the velocity along the average path taken by the sperm that represents the actual path but in a smooth manner shown in Fig. 3. Flagella beat frequency is calculated by measuring the number of waves initiated per second.
3.5 Simulation Procedure and Setup For simulation of sperm migration ANSYS Fluent software (2020R1) was used. Initially, the required trajectories of the sperm flagella were tracked using ImageJ software and using those points, the required sperm cell geometry was prepared in Solidworks design software. Different flagella shape based on its motion during one complete beat cycle was used for simulation. Figure 4 shows the discretization of the domain with inflation layers at near-wall elements for the accurate measurement of drag force. After mesh independence study, 759,279 elements are considered for study. The simulation was performed with the individual shape at a time, and the surrounding fluid medium was set as per the experiments. For the simulation of Newtonian fluid, a simple laminar flow model with constant fluid viscosity was used, since the motion of sperm comes under the low Reynolds number study, and for Non-Newtonian fluid, power law model was utilized. Inlet velocity opposite to the sperm and pressure outlet (atmospheric) was considered as boundary conditions. To simulate the various forces on the sperm bodies, the sperm was considered as stationary while the velocity was given to the fluid. This velocity is comparable to the propulsive or straight-line velocity of the sperm obtained during experimentation.
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Fig. 4 a Discretization of domain and b magnified view of near-wall element of sperm cell at a particular instant during its beating
4 Results and Discussion In this section, the effect of viscosity on the motility and kinetic parameters of sperm cells is analyzed by quantifying them in terms of VSL, VCL, VAP, beat frequency, power and swimming efficiency.
4.1 Effect of Viscosity on Motility Parameters of Sperm Several experiments were carried out within a microchannel to investigate the role of fluid viscosity and fluid behaviour on the motion characteristics of the sperm cells. Figure 5 shows the variation in VSL, VCL and VAP of sperm cells. It is observed that the motion of the sperm cells in PBS is quite random, which results in a large difference between the VCL and VSL compared to the sperm cells in other fluid environments. The progression of sperm in Newtonian fluid i.e. PVP decreases with an increase in concentration. For Non-Newtonian fluid, i.e. MC, the velocities increase with the increase in the concentration. It shows the opposite behaviour to Newtonian fluid where velocities decrease with an increase in percentage concentration. It is also observed that sperm trying to maintain its speed in Non-Newtonian fluid even at high resistance during its migration. Another motility parameter shown in Fig. 6 i.e. beat frequency was analyzed to understand the behaviour of sperm migration. It is observed that sperm beating frequency is highest in MC 2% Non-Newtonian fluid while lowest in PVP 14%. With an increase in the concentration, sperm beats with more frequency in Non-Newtonian fluid. However, for Newtonian fluid beat frequency of the sperm decreases with the viscosity.
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Fig. 5 Variation of velocities of sperm in low and high viscosity Newtonian and Non-Newtonian fluid
Fig. 6 Variation of beat frequency of sperm in low and high viscosity Newtonian and NonNewtonian fluid
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4.2 Variation in Drag Force in Newtonian and Non-Newtonian Fluid The drag force on sperm was calculated experimentally and using numerical simulation for Newtonian fluid, while for Non-Newtonian fluid, only simulation was used. The theoretical relationship for the calculation of drag force considering both sperm head and flagella was given by Phan-Thien et al. [24]. Fd = (6π μR + K T L)U KT =
2π μ log(2L/a)
(1) (2)
where, F d is drag force on sperm, R is the equivalent radius, for standard head size R = 1.39 µm, μ is fluid viscosity, U is progressive velocity or straight-line velocity of sperm, L is length of the sperm, a is radius of the flagella. This drag force is same as drag force acting on the ellipsoidal head and straight flagella. Figure 7 presents the variation of drag force on sperm in different fluid environments. It is observed that sperm undergoes the highest drag in MC 2%. The lowest drag was observed in PBS, whose viscosity is equivalent to water. The important thing to notice here is that though the sperm experiences more drag in MC 2%, its velocity is still higher than the other fluids except PBS. This means that sperm can penetrate easily into Non-Newtonian fluid, or sperm can easily break the microstructure of the Non-Newtonian fluid.
Fig. 7 Drag force on sperm body using experiments and simulation
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Fig. 8 a Power output of the sperm cells in different fluid mediums. b Power required to move the sperm cells in different fluid mediums
4.3 Power Output of the Sperm The resistance to the motion of the sperm by the surrounding fluid requires the expenditure of hydrodynamic energy. The rate at which this energy is spent is hydrodynamic power output or rate of work done by the swimmer on adjacent fluid layer. The power output relationship was given by Dresdner and Katz [20] shown in Eq. (3). It is observed that sperm output power depends on the frequency of beating and the surrounding fluid viscosity. Po = μf 2 L 3 pˆ
(3)
where, 0.4 < pˆ < 0.6 is a dimensionless parameter that depends on the morphology of the sperm. Po is power output, μ is viscosity of surrounding fluid, f is beat frequency, L is sperm body length. Figure 8 shows the variation of power output of the sperm in Newtonian and NonNewtonian medium. Power output is proportional to the frequency of the beat and from Fig. 6, it is observed that the sperm beat with maximum frequency in MC 2%; hence the power output of the sperms in MC 2% is maximum (0.743 pW). However, for low viscosity fluids like PBS, sperm output power is minimal i.e. 0.00097 pW as the frequency and viscosity of the surrounding fluid are low. In both Newtonian and Non-Newtonian fluid sperm power output increases with concentration.
4.4 Power Required to Pull the Sperm The expression shown in Eq. (4) was given by Higdon [25], which gives the approximate power required to pull the sperm through Newtonian fluid. This expression is valid only for Newtonian fluid hence the pulling power required by the sperm in
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Non-Newtonian fluid is calculated using simulation. 2 Pp = (6π μR + K T L) U
(4)
This expression combines the effect of drag on both sperm head and the flagella. Figure 8b shows the power required to pull the sperm in both Newtonian and NonNewtonian fluid environments. It is observed that the power required to pull the sperm is more in Non-Newtonian fluid as the drag force acting on the sperm is more compared to the other Non-Newtonian and Newtonian fluid. For Newtonian fluid, the pulling power was calculated experimentally and using simulation. The pulling power through simulation is little more compared to the experimental reading.
4.5 Efficiency of the Sperm Motion The swimming efficiency of the sperm is the ratio of power required to pull the sperm through the fluid to the power output of the sperm i.e. (Pp /Po ) [24, 25]. 2 (6π μR + K T L) U η= Po
(5)
Figure 9 gives an idea about how efficiently sperm are moving in the fluid medium. It is observed that sperm move efficiently in high viscosity Non-Newtonian fluid i.e. MC 1.5%. Here we can say from the previous analysis of power output and pulling power in MC 2% is more meaning sperm does not operate at its full capacity and its behaviour depends on the type of fluid surrounding it. The Non-Newtonian fluids provide favourable environment for the sperm motion compared to Newtonian fluid environment.
5 Conclusion In this paper, we performed the analysis of human sperm motion in Newtonian and Non-Newtonian fluid using microfluidic channel. Different motility and kinetic parameters related to sperm motion were calculated. Sperm possesses maximum velocity in MC 2%. In Non-Newtonian fluid, the velocity and beating frequency of sperm increased with viscosity while for Newtonian fluid opposite behaviour was observed. Due to high viscosity and VSL, sperm in Non-Newtonian fluid undergoes the highest drag compared to Newtonian fluid. Due to the increase in drag, the power required to pull the sperm also increases. Sperm requires the lowest pulling power in PBS and the highest in MC 2%. It is observed that sperm generate more power in Non-Newtonian environment as they are beating at high frequency. Finally, the
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Fig. 9 Swimming efficiency of sperm cell in different fluid mediums
swimming efficiency of sperm is more in high viscosity Non-Newtonian fluid. The results in the mentioned study concluded that the Non-Newtonian fluid mediums are more suitable for sperm migration as they mimic the actual conditions in female reproductive tract. Acknowledgements The authors acknowledge support from the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India (Start-up Research Grant SRG/2019/000285). Ethical Clearance All experiments were conducted following the rules and regulations approved by an ethical committee, Birla Institute of Technology and Science-Pilani, K K Birla Goa Campus.
Nomenclature μ L a R U Po Pp f η
Dynamic viscosity of fluid (Pa s) Sperm body length (m) Radius of flagella (m) Equivalent radius of sperm head (m) Progressive velocity (m/s) Power output (pW) Pulling power (pW) Beat frequency of flagella (Hz) Efficiency of swimming (%)
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References 1. World Health Organization (2010) WHO laboratory manual for the examination and processing of human semen, 5th edn 2. Agarwal A, Mulgund A, Hamada A, Chyatte MR (2015) A unique view on male infertility around the globe. Reprod Biol Endocrinol 13(1):37 3. Nosrati R, Graham P, Zhang B, Riordon J, Lagunov A, Escobedo C, Jarvi K, Sinton D, Lu A et al (2017) Microfluidics for sperm analysis and selection. Nat Rev Urol 14(12):707–730 4. Zhang Z, Liu J, Meriano J, Ru C, Xie S, Luo J, Sun Y (2016) Human sperm rheotaxis: a passive physical process. Sci Rep 6 5. Suarez SS, Wu M (2017) Microfluidic devices for the study of sperm migration. Mol Hum Reprod 23(4):227–234 6. Tian FB, Wang L (2021) Numerical modeling of sperm swimming. Fluids 6(2) 7. Kremer J, Jager S (1976) The sperm-cervical mucus contact test: a preliminary report. Fertil Steril 27(3):335–340 8. Katz DF, Mills RN, Pritchett TR (1978) The movement of human spermatozoa in cervical mucus. Reproduction 53:259–265 9. Kirkman-Brown JC, Smith DJ (2011) Sperm motility: is viscosity fundamental to progress? Mol Hum Reprod 17(8):539–544 10. Woolley DM (2003) Motility of spermatozoa at surfaces. Reproduction 126:259–270 11. Katz DF, Brofeldt BT, Overstreet JW, Hanson FW (1982) Alteration of cervical mucus by vanguard human spermatozoa. Reproduction 65:171–175 12. Ishijima S, Oshio S, Mohri H (1986) Flagellar movement of human spermatozoa. Gamete Res 13:185–187 13. Smith DJ, Gaffney EA, Gadelha H, Kapur N, Kirkman-Brown JC (2009) Bend propagation in the flagella of migrating human sperm, and its modulation by viscosity. Cell Motil Cytoskelet 66(4):220–236 14. Mane N, Mane S, Shah K, Banarjee A, Tripathi S (2021) Effect of Newtonian and shear thinning medium on human sperm motion within a microchannel. In: 48th national conference on fluid mechanics and fluid power. Paper No. FMFP2021-08-179, 27–29 Dec 2021. BITS Pilani, Rajasthan 15. Tung CK, Lin C, Harvey B, Fiore A, Ardon F, Wu M, Suarez S (2017) Fluid viscoelasticity promotes collective swimming of sperm. Sci Rep 7(1) 16. Hyakutake T, Suzuki H, Yamamoto S (2015) Effect of non-Newtonian fluid properties on bovine sperm motility. J Biomech 48(12):2941–2947 17. Hyakutake T, Suzuki H, Yamamoto S (2015) Effect of viscosity on motion characteristics of bovine sperm. J Aero Aqua Bio-mech 4(1):63–70 18. Nosrati R, Driouchi A, Yip CM, Sinton D (2015) Two-dimensional slither swimming of sperm within a micrometre of a surface. Nat Commun 6 19. Gillies EA, Cannon RM, Green RB, Pacey AA (2009) Hydrodynamic propulsion of human sperm. J Fluid Mech 625:445–474 20. Dresdner RD, Katz DF (1981) Relationships of mammalian sperm motility and morphology to hydrodynamic aspects of cell function. Biol Reprod 25:923–930 21. Razavi SE, Seyed Ahmadi A (2015) An ALE-based finite element model of flagellar motion driven by beating waves: a parametric study. Comput Biol Med 66:179–189 22. Tripathi S, Prabhakar A, Kumar N, Singh SG, Agrawal A (2013) Blood plasma separation in elevated dimension T-shaped microchannel. Biomed Microdevices 15(3):415–425 23. Mane NS, Puri DB, Mane S, Hemadri V, Banerjee A, Tripathi S (2022) Separation of motile human sperms in a T-shaped sealed microchannel. Biomed Eng Lett 12(3):331–342 24. Phan-Thien N, Tran-Cong T, Ramiat M (1987) A boundary-element analysis of flagellar propulsion. J Fluid Mech 184:533–549 25. Higdon JJL (1979) A hydrodynamic analysis of flagellar propulsion. J Fluid Mech 90(4):685– 711
Conjugate Heat Transfer Analysis of U-Bend/Turn Microchannel: A Computational Approach Jyoti Ranjan Mohapatra and Manoj Kumar Moharana
Abstract To examine the fluid flow and heat transfer properties of a U-shaped bend microchannel, a three-dimensional single-phase numerical study involving the conjugate effect of heat transfer is analyzed. The hydraulic diameter of the geometry was constant (0.4 mm) and the radius of curvature was varied from 2 to 4 mm. The constant heat flux boundary condition was applied at the bottom of the substrate and a comparative study with a straight conventional microchannel design is presented in terms of the performance evaluation factor. Different flow rates were considered to evaluate the effect of secondary flow in the form of Dean Vortices. Nondimensional heat transfer parameter such as the average Nusselt number and frictional losses in terms of pressure drop was evaluated and the contours of temperature and velocity were analyzed to elaborate the effect of heat transfer enhancement in the bend microchannel. Higher radii of curvature improved the overall performance of the bend channel. At a lower radius of curvature, the performance of the bend microchannel can be improved with increased pumping power. Due to unavoidable pressure loss in the case of microchannel, it is advisable to use U-bend microchannel at relatively low flow rates and higher radius of curvature. Keywords Microchannel · Average Nusselt number · Total pressure drop · Performance evaluation factor · Heat flux · Transport process
1 Introduction There is a constant demand for high heat flux removal from miniaturized electronic devices for better performance and life. Enhancement of the heat transfer process is directly dependent upon the surface area as it occurs across the channel walls. As the diameter of the channel decreases, the surface area to the volume J. R. Mohapatra · M. K. Moharana (B) Thermal Systems Laboratory, Department of Mechanical Engineering, National Institute of Technology Rourkela, Rourkela, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_34
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ratio increases leading to efficient transport processes and lower coolant requirements than conventional-sized channels. Therefore, there is a need for a miniaturized channel to intensify the rate of heat transfer [1]. Researchers have found different microchannel-based cooling methods that have their advantages and limitations. While designing the micro flow channels, there should be a balance between the rate of heat transfer and pressure drop. Researchers have presented different channel modifications through experimental and numerical works to maintain this optimum balance.
2 Literature Review and Objective The revolutionary work of Tuckerman and Pease [2] explored the thermohydrodynamics of forced convective heat transfer through a simple and straight microchannel. Many passive and active enhancement techniques have been developed further to cater to the high heat flux dissipation requirement. These include introducing roughness structures in channel walls, flow disruptions, channel curvature, fluid additives, electrostatic field, flow pulsation, etc. [3]. The design modifications improved the wall average Nusselt number Nu and friction factor f . As demonstrated by Sturgis and Mudawar [4] for a hydraulic diameter of 3.33 mm, heat transfer enhancement in a curved channel resulted in a 26% improvement compared to the straight channel. But this range was within the conventional size limit. There is an added advantage for the microscale range where the radius of curvature is most practically achievable compared to the conventional-sized channels. Pradhan et al. [5] studied the heat transfer characteristics of a conventional U-bend pipe of different cross sections. They found the highest heat transfer rate for the non-circular cross sections compared to the circular ones. Nivedita et al. [6] investigated the dean flow in microchannels experimentally and numerically and observed the presence of secondary Dean Vortices that create secondary flows in curvilinear geometry. Zhang et al. [7] performed multi-objective optimization considering three different parameters maximum temperature, thermal resistance, and pressure drop. For optimization purposes, both the thermal resistance and pressure drop should be taken into account. Al-Neama et al. [8] did a numerical and experimental investigation of a flow-through serpentine microchannel with single, double, and triple paths. The single path channel provided the best heat transfer performance followed by double and triple paths. This paper uses a computational method to analyze heat transfer through a 180° Ubend microchannel with a rectangular cross-section. Thermo-hydraulic performance is evaluated using the performance evaluation factor (PEF) [9]. Geometric parameters with a hydraulic diameter maintained constant at 0.4 mm and radius of bend section varying from 2 to 4 mm were considered. Reynolds number was varied from 50 to 200. The axial variation of local Nusselt number, temperature, velocity, and pressure distribution, along with their contours, is presented in this study.
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3 Computational Method 3.1 Problem Description Serpentine microchannel typically uses multiple U-bends/turns as shown in Fig. 1. Figure 1 represents a three-dimensional computational domain of a simple microchannel (SMC) and U-bend microchannel (UMC). The corresponding dimensions are listed in Table 1. For the analysis of heat transfer characteristics in the bend section, the SMC is divided into three parts, with two parts being 15 mm each and varying the middle section according to the radius of curvature. The planes are created at the interval from the centreline with its centre as ‘o’ as shown in Fig. 1.
3.2 Governing Equations The modelling simplifies by the following assumptions (i) steady state laminar flow, (ii) single phase and incompressible, (iii) thermo-physical properties are constant for both solid and liquid, (iv) no internal heat generation, (v) no viscous dissipation. The governing equation is as follows: Continuity equation: − → ∇· V =0
(1)
Momentum equation: ] [ − → ( ∂V → − →)− − → + ∇ · V V = − ∇ p + μ∇ 2 V ρ ∂t
(2)
Energy equation: [ ρf cf
] ∂T − → + V · ∇T = ∇ · kf ∇T ∂t
(3)
Energy equation for the solid, ks ∇ 2 TS = 0
(4)
Another important consideration is the effect of axial back conduction in the solid substrate, which is taken into account by considering δsf = 1 and moderately high thermal conductivity of solid (ks ) [10].
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Fig. 1 Schematic of a serpentine microchannel; b the computational domain i.e. U-bend microchannel (UMC); c straight microchannel; d U-bend microchannel (UMC); cross sectional view of e straight microchannel (SMC) and f U-bend microchannel (UMC) Table 1 Geometrical parameters Parameters in (mm)
L
Ws
Wf
δf
δs
R
SMC
36.2, 39.4, 42.5
0.8
0.4
0.4
0.4
–
UMC
–
0.8
0.4
0.4
0.4
2, 3, 4
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Table 2 Thermo-physical properties of the coolant and substrate material at 300 K Material
ρ (kg/m3 )
k (w/mK)
Aluminium
2719
237
871
Water
998.2
0.6
4182
C P (kJ/kgK)
μ (kg/ms) – 0.001003
3.3 Boundary Conditions Following assumptions are considered for simplified numerical simulation which represents realistic situations, 1. The inlet velocity of fluid is assumed to be uniform. 2. No slip boundary condition at fluid solid interface and conjugate heat transfer. 3. A pressure outlet boundary condition is considered at the exit, ( pout = 0) (gauge pressure). 4. The initial temperature for both solid and fluid was 300 K. Constant heat flux (q'' = 50,000 W/m2 ) is applied at the bottom face of the substrate. This value is decided based on literature input and requirement of singlephase fluid flow throughout the channel. q'' = 0 on the other walls including sidewalls. All other faces exposed to the ambient are considered to be adiabatic. Water is used as a coolant and aluminium as the substrate material. The thermo-physical properties of water and aluminium are listed in Table 2.
3.4 Method of Calculation Finite volume-based commercially available CFD solver ANSYS Fluent V20 is used for solving the governing differential equations. The SIMPLE algorithm is used for pressure–velocity coupling. Second order upwind is used to discretize the momentum and energy equations. The residual levels for convergence were taken as 10−6 , 10−6 , and 10−8 for continuity equation, momentum equations, and energy equation respectively.
3.5 Data Acquisition The following dimensionless parameters are used for the purpose of analysis, which helps in evaluating performance evaluation factor. Local heat transfer coefficient, hz =
qz'' (Tw − Tf )
(5)
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Local Nusselt number, Nuz =
h Dh kf
(6)
Average Nusselt Number,
Nuavg
1 = L
∫L Nuz dz
(7)
0
Pressure Drop, f Lu 2avg ρ
Δ p =
2Dh
(8)
Thermal Resistance, Rth =
Tmax − Tmin Q
(9)
Performance evaluation factor (PEF), PEF =
Nuavg /Nuo (Δ p/Δ po )1/3
(10)
The dimensionless parameters used in this study are non-dimensional axial distance z ∗ = z/L and Reynolds number (Re).
3.6 Grid Independence Study Structured hexahedral elements as shown in Fig. 2 are considered for the simulation purpose due to simplicity of the geometry and to have a trade-off between computational time and accuracy. Grid independence study of the rectangular microchannel with aspect ratio one and δ sf = 1 is done for five different mesh sizes of 20 × 20 × 100, 24 × 24 × 600, 32 × 32 × 800, 40 × 40 × 1000, 48 × 48 × 1200 at a Re of 100 (Fig. 3). The local Nusselt number and their relative errors were compared with the finest mesh size. The local Nusselt number varied by 5.906, 3.876, 1.202, 0.768% on moving towards the finest grid. Relative error was less than 1% for mesh no. 4. So, in order to have unanimity between the computational time and accuracy, the intermediate mesh size of (40 × 40 × 1000) is selected for the simulation.
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Fig. 2 Structured mesh with quad elements of the U-bend section
Fig. 3 Axial variation of local Nusselt number Nuz
3.7 Model Validation The present computational model is validated with available correlations to establish the correctness of the discretization scheme. The meshing is created by considering inflation layers/prism layers. An error of less than 2% in Nuavg is observed for the mesh without consideration of inflation layers considering that there is no phase change and turbulence involved in this study. Figure 4 shows the variation of local Nusselt number (Nuz ) with the existing correlation by Lee and Garimella [11] and Shah and London [12] and the friction factor results with the correlation of Steinke
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Fig. 4 a Axial variation of local Nusselt number Nuz . b Variation of friction factor with flow Re
and Kandlikar [13]. The variation of local Nusselt number (Nuz ) in the fully developed region is close to the analytical value of 3.556 with relative error less than 1.5% and for friction factor ( f ) it is less than 0.9%.
4 Results and Discussion 4.1 Effect of Radius of Curvature Three different radii of curvature for the U-bend section are considered and local variation of Nusselt number is analyzed with varying Reynolds number for each case. It was observed that with an increase in radius of curvature of the U-bend, the value of local Nusselt number increases sharply in the U-bend section as shown in Fig. 6, which is compared with the SMC where there is no such sharp increase in Nuz . The increment in local Nusselt number extends to a greater length in case of R = 4 mm because of which the maximum increase in Nuavg was observed at R = 4 mm. This is attributed to better mixing and flow recirculation at the U-bend section which results from the Dean instability [6]. The description of the secondary flow (as reported in Fig. 5) behaviour can be attributed to shifting of maximum velocity from the channel centre towards the outer wall (concave wall). The velocity profile in the laminar flow as shown in Fig. 7 is disturbed due to the external centrifugal force and there is a shift in maximum velocity towards the concave channel wall because of which there is an increase in pressure [6]. The pressure at a Re of 200 increases from 8.80% to 17% when R changes from 3 to 4 mm respectively. Due to the redevelopment of thermal boundary layer in the bent section, there is an increase in heat transfer coefficient. As we move along the bend section of UMC from at each 30◦ interval in Fig. 8, the temperature is lower in the concave wall and
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Fig. 5 Velocity contour for Y-vorticity at R = 4 mm, for Reynolds number 50–200
the mass weighted average temperature in planes of bend portion is lower than the temperature in SMC corresponding to that particular.
4.2 Overall Performance Analysis The performance of the U-bend microchannel (UMC) is analyzed by comparing the smooth and bend microchannel based on average Nusselt number (Nuavg ) and total pressure drop (Δ p). Figure 9 represents the variation of (Nuavg ) with Reynolds number for UMC with R = 2 mm, 3 mm, 4 mm and SMC with length corresponding to the radius of curvature i.e. 36.20 mm, 39.40 mm, and 42.50 mm respectively. It is observed that (Nuavg ) increases with Re and it is higher in the case of UMC compared to SMC with maximum at R = 4 mm. This can be attributed to an increase in concavity of the bend, which promotes more disturbance in velocity gradient that results in secondary flows. There is 3.56% and 5.96% increase in Nuavg when the radius changes to 3 mm and 4 mm respectively. For a constant value of heat input, the thermal resistance (Fig. 10) is calculated between the maximum temperature at the bottom of the solid and minimum i.e. the
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Fig. 6 Variation of local Nusselt number with z* for i R = 2 mm, ii R = 3 mm, iii R = 4 mm, iv L = 42.5 mm at different values of Re
temperature at inlet. The thermal resistance decreases with an increase in Re. This can be attributed to an improvement in heat transfer coefficient, which signifies a high rate of heat transfer. Figure 11 represents the total pressure drop variation with Reynolds number for three different radii of curvature compared with the corresponding length of the microchannel. It is evident that with the improvement in heat transfer coefficient, there is a penalty of pressure drop but it should not be at the cost of overall performance which is shown in Fig. 12. The pressure drop is minimum for UMC at R = 2 mm and L = 36.2 mm for SMC. This is due to linear variation of pressure drop with heating length of the channel. The thermo-hydrodynamic performance is improved for the U-Bend microchannel depicted by the Performance Evaluation Factor (PEF). PEF of UMC is higher for different radii of curvature. This signifies better thermal performance of UMC than SMC. PEF is maximum i.e. 1.33 at Re = 200 and R = 4 mm as there is a significant improvement in average Nusselt number compared to reduction in pressure drop. But the increase is not linear i.e. there may be scope for performance degradation due to dominance of pressure drop at higher flow rate.
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Fig. 7 Velocity profile of the plane at i z = 15 mm for SMC with L = 42.5 mm and Re = 200, ii 30°, iii 60°, and iv 90° from centre of the bend section for R = 4 mm and Re = 200
Fig. 8 Temperature profile of the plane at i z = 15 mm for SMC with L = 42.5 mm and Re = 100, ii 30°, iii 60°, and iv 90° from centre of the bend section for R = 4 mm and Re = 200
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Fig. 9 Variation of thermal resistance with Reynolds number for UMC, R in ‘mm’
Fig. 10 Variation of Nuavg with Reynolds number for UMC and SMC, both R and L are in ‘mm’
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Fig. 11 Variation of Δ p with Reynolds number for UMC and SMC, both R and L are in ‘mm’
Fig. 12 Variation of PEF for different Reynolds number for UMC and SMC, R is in ‘mm’
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5 Conclusions Heat transfer and fluid flow characteristics of a U-Bend microchannel (UMC) are investigated through a numerical model that is capable of evaluating the performance of the proposed model compared to the simple and straight microchannel (SMC). The study was carried out for different values of radius of U-bend section with the corresponding length of SMC. The effect of radius of the bend channel on the performance of UMC is evaluated in terms of performance evaluation factor (PEF). Due to the presence of bend, there is an effect of centrifugal force because of which there is a shift in location maximum velocity due to the change in velocity profile. This induces extra pressure drop but there is an improvement in the rate of heat transfer also due to better mixing of fluid. This is quantified by the PEF that is greater than one for each parametric variation and there is up to 33% improvement in the thermo-hydrodynamic performance for 4 mm radius of bend and flow rate corresponding to Re = 200. At low radius of curvature, the performance of bend microchannel improves at higher pumping power; since frictional losses cannot be controlled, it is wise to employ channels at an increasing radius of bend section and a relatively low flow rate. The effect of axial conduction needs investigation in order to predict the optimal value of Nusselt number. This will be addressed in future work.
Nomenclature A Cp Dh f hz kf ks L Nuz Nuavg p Δ p Q Re Rth T Tf Ts Tw z*
Area of cross-section (m2 ) Specific heat at constant pressure (J/kgK) Hydraulic diameter (m) Darcy friction factor Local heat transfer coefficient (W/m2 K) Thermal conductivity of fluid (W/mK) Thermal conductivity of solid (W/mK) Length of the substrate (m) Local Nusselt number Average Nusselt number Pressure (N/m2 ) Total pressure drop (N/m2 ) Heat transfer rate (W) Reynolds number Thermal resistance (K/W) Temperature (K) Fluid temperature (K) Solid temperature (K) Wall temperature (K) Non-dimensional axial distance
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Total width (m) Width of solid substrate (m) Width of the fluid channel (m) Dynamic viscosity (Ns/m2 ) Radius of curvature (m)
References 1. Kandlikar SG, Garimella S, Li D, Colin S, King MR (2005) Heat transfer and fluid flow in minichannels and microchannels. Elsevier, Oxford 2. Tuckerman DB, Pease RFW (1981) High-performance heat sinking for VLSI. IEEE Electron Device Lett 2(5):126–129 3. Steinke ME, Kandlikar SG (2004) Single-phase heat transfer enhancement techniques in microchannel and minichannel flows. In: ASME international conference on nanochannels, microchannels, and minichannels, Paper No. ICMM2004–2328, Rochester, NY, 17–19 June 2004 4. Sturgis JC, Mudawar I (1999) Single-phase heat transfer enhancement in a curved, rectangular channel subjected to concave heating. Int J Heat Mass Transf 42(7):1255–1272 5. Pradhan HK, Sahoo AK, Roul MK, Awad MM, Barik AK (2020) Heat transfer characteristics of an 180° bend pipe of different cross sections using nano-enhanced ionic liquids (NEILs). SN Appl Sci 2(6):1–13 6. Nivedita N, Ligrani P, Papautsky I (2017) Dean flow dynamics in low-aspect ratio spiral microchannels. Sci Rep 7(1):1–10 7. Zhang J, Lin PT, Jaluria Y (2011) Designs of multiple microchannel heat transfer systems. In: ASME international mechanical engineering congress and exposition, Paper No. IMECE2011– 62539, Denver, CO, 11–17 Nov 2011 8. Al-Neama AF, Kapur N, Summers J, Thompson HM (2017) An experimental and numerical investigation of the use of liquid flow in serpentine microchannels for microelectronics cooling. Appl Therm Eng 116:709–723 9. Samal SK, Moharana MK (2019) Thermo-hydraulic performance evaluation of a novel design recharging microchannel. Int J Therm Sci 135:459–470 10. Moharana MK, Singh PK, Khandekar S (2012) Optimum Nusselt number for simultaneously developing internal flow under conjugate conditions in a square microchannel. J Heat Transf 134(7) 11. Lee PS, Garimella SV (2006) Thermally developing flow and heat transfer in rectangular microchannels of different aspect ratios. Int J Heat Mass Transf 49(17–18):3060–3067 12. Shah RK, London AL (2014) Laminar flow forced convection in ducts: a source book for compact heat exchanger analytical data. Academic Press, New York 13. Steinke ME, Kandlikar SG (2006) Single-phase liquid friction factors in microchannels. Int J Therm Sci 45(11):1073–1083
Experimental Investigation of Fluid Flow Behaviour in Parallel Microchannel Using Micro-PIV Rohit Kumar, Chandan Nashine, Arman Mohaddin Nadaf, Mohd Sakib Hussain, and Manmohan Pandey
Abstract With the miniaturization of the system, the demand for high heat flux removal technologies increases. Innovative technology is the use of microchannel using water as a coolant which can handle the high rate of heat fluxes. For a given pumping power, the rate of flow through the microchannel is low, therefore, many microchannels operating in parallel are required. In this present work, an experimental study of flow through three rectangular parallel microchannels has been carried out. The experiments are performed using micro-PIV at various flow rates and different concentrations of the seeding particles. The post-processing of raw images captured during the micro-PIV experiment is done by PIV Labs software. The experimental analysis is done for various parameters like the concentration of seeding particles and the effect of flow rates. Experimental results are compared with analytical solutions for the parallel microchannel. Keywords Microchannel · Micro-PIV · Flow distribution · Seeding particles
R. Kumar (B) · C. Nashine · A. M. Nadaf · M. S. Hussain · M. Pandey Department of Mechanical Engineering, IIT Guwahati, Guwahati 781039, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_35
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1 Introduction The practical application of microchannels is in compact heat exchangers and the biomedical field. To optimize these devices, there is a need for a good understanding of flow behaviour in these passages [1]. Microchannel heat exchangers have applications in several important fields including bioengineering, automotive, aerospace, cooling of gas turbine blades, refrigeration, and air conditioning and microelectronics. The advantages of microchannel heat exchangers include high volumetric heat flux, compactness for space-critical applications, effective flow distribution, and very less pressure drops. Micron-scale measurement has become a very important aspect in many areas of science and engineering [2]. Such advancements in science and technology have forced researchers to study the flow dynamics of these downscaled devices. One such technique is µ-PIV which can be used for microfluidic devices for studying their flow dynamics [3, 4]. Particle image velocimetry is a flow visualization technique. It is used to measure instantaneous velocity and related properties of fluids. What we do in this technique is to put some neutrally buoyant particles in the fluid flow and laser light is focussed on it. The scattered light is then captured with the help of a CCD camera. These images are further post-processed to get velocity vectors, velocity profiles, and other fluid properties [5]. µ-PIV is an extension of PIV for the measurement of micron range devices. It uses fluorescent particles for seeding which is not the case with PIV. In µ-PIV volume illumination technique is used as generation of the light sheet in micron order ranges is a tough task. µ-PIV is used for flow visualization for many applications [6]. One of the basic applications of micro-PIV was the measurement of the velocity profiles in a microchannel. Velocity measurements near the wall or interfacial regions are of—ten needed in microfluidic applications, especially due to the high surface-to-volume ratio. These measurements can be very useful to characterize quantities related to the fluid–solid structure interaction. Electrokinetic phenomena which are widely used to operate microfluidic devices, µ-PIV is nowadays widely used as a diagnostic tool in electrokinetic flows, either for the observation and understanding of the underlying physical phenomena or for the characterization of the functioning and performances of devices [7]. Various examples can be already found in nature such as the micro-flow in blood capillaries or the biochemical reactions and transport phenomena in a single cell [8]. As it is non-tactile, µ-PIV is a good technique for velocity measurements in living organisms with a very small impact on normal physiological activities.
2 Literature Review and Objective µ-PIV is used for the analysis of fluid flow in microchannel by visualization methods. It is now widely used and explored in microfluidics applications [9]. The concentration of particles for the micro-PIV experiment should be selected carefully to get
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the required spatial resolution and signal-to-noise ratio in the field of particle image [10]. The size of particles also plays a key role in depth measuring. The diameter of the particle and intensity of the particle image are dependent on the distance from the object plane. The particles away from the object plane created background noise that makes it difficult to see near-focus particles [11]. The effect of Brownian motion is negligible in the case of PIV but in the case of µ-PIV, a partial effect is initiated which affects measuring the velocity. Because of Brownian motion, the peak of the correlation signal spreads widely. Depth of correlation and interrogation volume is also affected in µ-PIV due to Brownian motion [12]. Lindken et al. [13] performed experiments by using stereo-Micro Particle Image Velocimetry which measures all three independent components of the velocity field in the plane of measurement. Pan et al. [14] experimentally investigated the flow uniformity of a microchannel plate with a rectangular manifold. It was concluded that flow uniformity decreases with higher entrance velocity while at low entrance velocity, the velocity distribution of the microchannel plate is almost symmetric. The velocity distribution increases in a plate with a larger inlet manifold while it decreases with a large outlet manifold. They also concluded that to obtain uniform flow distribution the following conditions should be there longer microchannel, the perpendicular direction of the inlet and outlet to the microchannel plane, smaller microchannel width, symmetric manifold structure, and larger manifold area. For a compact parallel flow heat exchanger, the tube-to-header area ratio is an important global parameter for controlling maldistribution. As the area ratio increases, flow maldistribution becomes more pronounced and more sensitive to increased Reynolds number and decreased parallel pipe length [15]. In inline flow inlet configuration, there is more maldistribution than in vertical flow inlet configuration. In the case of header shapes, trapezoidal and triangular headers give less flow maldistribution at a low flow rate while rectangular headers give less flow maldistribution at higher flow rates. On increasing the flow rate, maldistribution decreases [16]. The geometrical structure of the microchannels is an important factor in flow distribution between microchannels, which also affects the heat and mass transfer efficiency. Better uniformity of velocity and temperature distributions was achieved in the I-shape flow arrangement while the inlet/outlet flow arrangements showed a significant effect on velocity and temperature distributions among microchannels. Due to the vortices that occurred at the manifold, the behaviour of the flow distributions inside channels was non-uniform. The inlet and outlet area of the manifold on velocity and fluid temperature distributions was insignificant [17]. The variable height microchannels heat sinks approach mitigates flow maldistribution rapidly in comparison to the variable width microchannels heat sinks approach, almost 50% computational time can be saved by using the variable height microchannels heat sinks approach [18]. Gupta [19] did the experimental investigation on PDMS rectangular microchannel by using micro-PIV for a wide range of volume concentration and flow rate. The study showed that the concentration of seeding particles plays a crucial role in the accuracy of results. It has been observed that optimal concentration providing an accurate axial velocity profile is different from the one for the best cross-sectional velocity profile. For the range of flow rates, the best results were obtained for a volume concentration of 0.00078 and 0.0013%.
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Gupta et al. [20] experimentally investigated the behaviour of fluid flow through a PDMS rectangular microchannel of size 400 × 520 µm by using micro-PIV. The optimal concentration for different flow rates was observed differently. They also concluded that the velocity profile is uncertain for the axial velocity profile obtained from the experiment than the cross-sectional profile observed numerically. It is preferred to perform numerical simulations before conducting experiments in the microchannel. Literature available reveals that a large number of studies have been done and developments made in a micro-PIV setup like camera, laser, and high-performance computer. Therefore, analysis of fluid flow through microchannel can be done experimentally with the help of micro-PIV very easily and compared with the numerical results. Flow distribution and its visualization inside the parallel microchannel by using µ-PIV and post-processing the images to get velocity vectors and profiles. A comparative analysis is needed for the analytical and experimental velocity profile. The objective of this work is also to check for uniform flow distribution in parallel channels.
3 Experimental Procedure Figure 1 shows the schematic of the experimental setup used for the experimentation. The microchannel is placed over the stand of the microscope and all the connecting tubes are attached to the inlet and outlet ports of the microchannel. The prepared solution of seeding particles and deionized water is made to flow through the microchannel using a syringe pump. After the flow is established, the focus of the microchannel is done. After focussing on the desired plane, the Nd-YAG laser is fired. The images are captured using a CCD camera for the set flow rate which is stored in the system. The raw images obtained are further post-processed using the PIV lab to get the desired results. The seeding particles used were Fluospheres Polystyrene. The size of the particles was 1 µm and the density was 1.05 g/cm3 . Spectral data are as follows: λex = 540 nm (maximum excitation wavelength) and λem = 560 nm (maximum emission wavelength). The laser employed for the µ-PIV setup has a wavelength of 532 nm. So, theoretically, there would be 98.54% absorption of the laser light. The emission wavelength should pass through the epifluorescent prism/filter cube for the emission case. In the present case, the emission wavelength passes through the filter cube. According to the manufacturer’s specification, 1 ml of the supplied polystyrene solution contains 1 × 1010 microparticles, which gives us a volume concentration (v/v) of 0.52%. So if 10 µl of these seeding particles are mixed in the 20 ml of deionized water, it gives the final v/v concentration of 0.00026%. Similarly, 30 µl, 50 µl, and 100 µl of the seeding particles in deionized (20 ml each) give the resultant concentration of C = 0.00078%, C = 0.0013%, and 0.0026% respectively.
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Fig. 1 Experimental setup of micro-particle image velocimetry
4 Results and Discussion The experiments were carried out for 3 different concentrations 0.00078% (v/v), 0.00104% (v/v) and 0.00130% (v/v) and different flow rates (ml/h): 0.02, 0.04, 0.08, 0.1, 0.12, 0.15, 0.20, 0.5, 0.8, 1, 2, 4, 6, 8, and 10. Here velocity of flow are very small so flow is purely laminar flow. Velocity vectors and velocity magnitude is obtained after post-processing the raw images. Filters are used to enhance the images to reduce the number of erroneous velocity estimates to ensure the highest measurement quality possible before the actual image correlation takes place. CLAHE (Contrast limited adaptive histogram equalization) improves the probability of detecting valid vectors in experimental images. Sometimes due to reflections and inhomogeneous seeding particles, distribution correlation signals are affected. A high-pass Kernel filter is applied to remove these low-frequency backgrounds and preserve the high-frequency information. In intensity capping, an upper limit of the intensity is selected and all pixels that exceed the threshold are replaced by this upper limit. Therefore, in intensity capping, only a small amount of the pixel intensity information is adjusted, unlike CLAHE. Combining the intensity capping filter with the CLAHE filter increases the probability of valid vector detection in the PIV lab. The pre-processed image after applying CLAHE and high-pass Kernel filter is shown in Fig. 2. All the postprocessing results are obtained after applying these 2 filters in combination.
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Fig. 2 Pre-processed image after applying CLAHE and high-pass Kernel filter
4.1 Effect of Seeding Particle Experiments were performed while varying the concentration ratio of seeding particles and keeping the flow rate constant. From Fig. 3a–c, the velocity contour is smooth with a maximum velocity at the centre of each channel shown by the yellow colour and very low velocities at the wall of the channel shown by blue colour. From Fig. 4a–c, it can be observed that the velocity vectors, contour, and magnitude diagram for the flow rate of 0.12 ml/h and C = 0.00130% are not the smooth and maximum velocity is not present at the centre of each channel but present in chunks throughout the channel as shown by yellow colour distribution. The above differences in the two concentrations can be explained when the seeding particle concentration is too low then the particles are not enough for uniform distribution throughout the channel therefore the number of particles also becomes less for each interrogation window. When seeding particle concentration is too high, there is a possibility of an increase in the particle–particle interaction due to which the particles may not follow the path of fluid faithfully. Therefore, it can be concluded that the concentration of seeding particles plays an important role in the smoothness of contours for a particular flow rate.
4.2 Effect of Flow Rates Experiments were conducted with an 8.1 mm diameter syringe in which the minimum flow rate obtained was 0.02 ml/h. Flow rates used were: 0.02, 0. 04, 0.08, 0.10, 0.12, and 0.15 ml/h. From Fig. 5, it can be observed that as the flow rate is increasing, there is some intermingling of particles or the movement of the particles is too fast to be captured in the same frame which causes the error in correlation and interpolation.
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Fig. 3 Velocity vectors and magnitude for Q = 0.12 ml/h at a first channel, b middle channel, c last channel for C = 0.00078%
4.3 Comparison of Experimental and Analytical Velocity Profiles It can be seen from Fig. 6 that there exists a closer match between the analytical and experimental profiles for Q = 0.02 ml/h. The maximum velocity for both the profiles matches but the velocity at the right side walls and bottom walls does not match. Ideally, the velocities at the wall should be zero due to the no-slip condition but in the experiments, after doing post-processing of raw images, it was observed that the velocities at the wall were non-zero. The variation observed between the two profiles may have various reasons. One of the causes can be the visibility of seeding particles in the flow field which can be increased by increasing the depth of the device or by reducing the concentration of seeding particles. If the concentration of seeding particles is reduced, it will reduce the number of particles in the interrogation window, therefore, there exists a need to find out some optimum concentration to conduct experiments to get good results.
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Fig. 4 Velocity vectors and magnitude for Q = 0.12 ml/h at a first channel, b middle channel, c last channel for C = 0.00130%
5 Conclusions The experiments were performed for various flow rates and different concentrations of the seeding particles on parallel rectangular microchannels with the help of microPIV. Post-processing of the raw images is done using the PIV Labs software. The velocity gets reduced by moving towards the boundary in the velocity profiles. For any location, the velocity is maximum at the centre and gradually reduces to zero near the wall. For flow visualization using µ-PIV Fluorescent polystyrene, tracer particles have been prepared keeping all the parameters of µ-PIV setup in consideration. The experiments were performed to check various things like the effect of the concentration of seeding particles on the post-processed results, the effect of flow rates, and the comparison of experimental and analytical velocity profiles. The results show that at a concentration of 0.00078% (v/v), the velocity vectors and velocity profiles are smoother compared to the concentration of 0.00130% (v/v). The velocity vectors were smooth in case of lower flow rates than at higher flow rates. In the comparison
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Fig. 5 Velocity vectors at low flow rates of Q = 0.02 ml/h (a, b, c), Q = 0.04 ml/h (d, e, f), Q = 0.10 ml/h (g) and Q = 0.15 ml/h (h)
of experimental and analytical velocity profiles, it can be observed that there is a good match between them except at boundaries. In the flow distribution study, all the velocity profiles in the parallel channels take approximately the same shape which shows that flow is uniformly distributed in all three channels.
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Fig. 6 Velocity profiles comparison for a flow rate of Q = 0.02 ml/h in a first channel, b middle channel, c last channel
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Acknowledgements This work was partially funded through Grant No. EMR/2016/003513 of SERB, DST, Govt. of India. The experiments are conducted in Central Instruments Facility, IIT Guwahati.
Nomenclature CCD µ-PIV PDMS PIV C λem λex
Charge Coupled-Device (–) Micro Particle Image Velocimetry (–) Polydimethylsiloxane (–) Particle Image Velocimetry (–) A volume concentration (in %) (–) Maximum emission wavelength (nm) Maximum excitation wavelength (nm)
References 1. Zhao CY, Lu TJ (2002) Analysis of microchannel heat sinks for electronics cooling. Int J Heat Mass Transf 45(24):4857–4869 2. Kandlikar SG, Grande WJ (2003) Evolution of microchannel flow passages—thermohydraulic performance and fabrication technology. Heat Transf Eng 24(1):3–17 3. Lindken R, Rossi M, Große S, Westerweel J (2009) Micro-particle image velocimetry (µPIV): recent developments, applications, and guidelines. Lab Chip 9(17):2551–2567 4. Gao Q, Wang HP, Shen GX (2013) Review on development of volumetric particle image velocimetry. Chin Sci Bull 58(36):4541–4556 5. Santiago JG, Wereley ST, Meinhart CD, Beebe DJ, Adrian RJ (1998) A particle image velocimetry system for microfluidics. Exp Fluids 25(4):316–319 6. Meinhart CD, Wereley ST, Santiago JG (1999) PIV measurements of a microchannel flow. Exp Fluids 27(5):414–419 7. Devasenathipathy S, Santiago JG, Takehara K (2002) Particle tracking techniques for electrokinetic microchannel flows. Anal Chem 74(15):3704–3713 8. Lima R, Wada S, Tanaka S, Takeda M, Ishikawa T, Tsubota K-I, Imai Y, Yamaguchi T (2008) In vitro blood flow in a rectangular PDMS microchannel: experimental observations using a confocal micro-PIV system. Biomed Microdevice 10(2):153–167 9. Wereley ST, Meinhart CD (2010) Recent advances in micro-particle image velocimetry. Annu Rev Fluid Mech 42:557–576 10. Meinhart CD, Wereley ST, Gray MHB (2000) Volume illumination for two-dimensional particle image velocimetry. Meas Sci Technol 11(6):809 11. Olsen MG, Adrian RJ (2000) Out-of-focus effects on particle image visibility and correlation in microscopic particle image velocimetry. Exp Fluids 29(1):S166–S174 12. Olsen MG, Adrian RJ (2000) Brownian motion and correlation in particle image velocimetry. Opt Laser Technol 32(7):621–627 13. Lindken R, Westerweel J, Wieneke B (2006) Stereoscopic micro particle image velocimetry. Exp Fluids 41(2):161–171 14. Pan M, Shao X, Liang L (2013) Analysis of velocity uniformity in a single microchannel plate with rectangular manifolds at different entrance velocities. Chem Eng Technol 36(6):1067– 1074
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15. Camilleri R, Howey DA, McCulloch MD (2015) Predicting the flow distribution in compact parallel flow heat exchangers. Appl Therm Eng 90:551–558 16. Anbumeenakshi C, Thansekhar MR (2016) Experimental investigation of header shape and inlet configuration on flow maldistribution in microchannel. Exp Therm Fluid Sci 75:156–161 17. Sahar AM, Shaiful AIM (2017) Numerical study of the effect of area of manifold and inlet/outlet flow arrangement on flow distribution in parallel rectangular microchannel cooling system. J Teknol (Sci Eng) 79:81–87 18. Kumar A (2018) Numerical simulation and flow visualization of liquid flow in arrays of miniature channels. Master’s thesis, Indian Institute of Technology Guwahati, India 19. Gupta R (2019) µ-PIV flow visualization and mathematical modelling of liquid flow in microchannels. Master’s thesis, Indian Institute of Technology Guwahati, India 20. Gupta R, Sarma SK, Iqbal A, Pandey M (2019) m-PIV experiments in microchannels. In: Heat and mass transfer conference (IHMTC-2019). Paper id. IHMTC2019-MNT-857
Study of Path Selection of a Droplet in a Symmetric Y-Microchannel Using a Uniform Electric Field Satya P. Pandey, Sandip Sarkar, and Debashis Pal
Abstract The droplet dynamics in a symmetric bifurcating Y-microchannel under the influence of a direct current (DC) electric field imposed across only one daughter channel are investigated. The interface of the droplet has been captured using Cahn– Hilliard equation, while the effect of electric force at the droplet interface has been incorporated by modelling it as a body force term in the momentum equation. Two different sorting behaviours of the droplet have been observed depending on the relative permittivity ratio (εr ) of the droplet and carrier fluid and the choice of daughter channel where electric field is imposed. The droplet chooses the path of the channel where the electric field is applied if its permittivity is higher than that of the carrier fluid. The reverse phenomenon is seen to take place when droplet electrical permittivity is lower than that of the carrier. Furthermore, it is also observed that by altering the intensity of the electric field, it is possible to accurately regulate the daughter droplet breakup ratio in a Y junction and reach droplet of any size. We have also identified a critical electric Capillary number (Cae ) at which the droplet completely transcends from breakup to no-breakup regime and gets sorted in any one branch channel depending on εr and the branch channel where the electric field is imposed. The increase in Cae beyond its critical value doesn’t affect the no-breakup regime but increases droplet velocity and facilitates a bit faster sorting than the previous droplet. Keywords Droplet breakup · Electro-hydrodynamics · Multi-phase flow · Microfluidics · Y-junction · Phase field method
S. P. Pandey · D. Pal (B) Department of Aerospace Engineering and Applied Mechanics, IIEST Shibpur, Howrah 711103, India e-mail: [email protected] S. Sarkar Department of Mechanical Engineering, Jadavpur University, Kolkata 700032, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_36
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1 Introduction Droplet propagation and bifurcation within another immiscible liquid are commonly encountered phenomena in many industrial processes such as food processing, inkjet printing and pharmaceuticals. Droplet microfluidics has grown in popularity and had a big impact on many applications, including chemical processes, drug delivery, single cell analysis and many more due to its precise control and manipulation of droplets [1–5]. This is because it can handle individual monodisperse droplets and has the advantage of offering high surface to volume ratios. As a result, it has been a topic of immense interest for researchers to study the functions of droplets like formation, propagation, splitting and mixing in these devices. For the creation and manipulation of these droplets, a number of methods have been explored in the literature using both passive and active methods [6–13]. The electric field actuated droplet microfluidics has gained more popularity in recent years due to its flexibility in providing desired droplet sizes without contaminating the chemicals used in the process.
2 Literature Review and Objective Numerous theoretical and numerical studies [1–4] have been performed concerning the deformation of droplet due to an externally applied electric field. Taylor [1] demonstrated that the flow caused by tangential electric tensions at the drop surface might cause the droplets to deform in an oblate manner, whereas only prolate drop deformations are expected for perfectly conducting or perfectly dielectric fluids. The behaviour of an isolated droplet in an emulsion is well explained by the leaky dielectric theory, which is discussed in detail by Melcher and Taylor [5]. The relationship between the electro-kinetic and leaky dielectric theories is demonstrated in Saville [4] thorough overview of electro-hydrodynamics. The manipulation of droplet in a microchannel junction can be achieved by means of active or passive techniques. The passive method of droplet splitting comes with the limitation of providing the desired droplet size due to the lack of flexibility as droplet manipulation can only be done by varying flow rate if the geometry is kept fixed. This limitation can be overcome by using an active technique to split the droplets into desired size. This technique is achieved by applying an additional energy field near the junction like temperature field [6], magnetic field [7], valves [8] and electric field [9–12]. When compared to other alternative active control techniques, electrical field induced droplet manipulation has gained tremendous popularity due to its wide range of control and faster time response. Link et al. [9] was one of the pioneers to experimentally investigate the effects of electric field on droplet dynamics in a symmetric T junction. According to their findings, splitting parent drops into several droplets using the electric field creates a powerful droplet-fission module. In a symmetric cross-junction (T) microchannel, droplet breakdown was
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examined numerically by Jafari and Fallah [10]. They demonstrated that a droplet splits more quickly due to the applied electric field than it would in the absence of one under identical circumstances. The influence of mother droplet length, Capillary number and electric field strength on droplet breakup in a symmetric T junction has been performed by Fallah and Fattahi [11]. Wehking et al. [12] proposed that it is possible to steer a droplet in either direction of a T-junction by adjusting the respective strengths of the inertial and electric forces, which causes droplet deceleration and pinning. The transport and splitting of a droplet in an open Y channel with the means of only wettability gradient using the method of energy conservation is done by Chowdhury et al. [13]. The literature survey reveals that there is still a lack of comprehensive knowledge of the droplet breakup dynamics in Y-junctions under the influence of an externally applied electric field. In the present study, we have varied the relative permittivity of both phases in such a way that the two contrasting phenomena of droplet splitting could be investigated. The novelty of our work lies in the proposed new technique for getting a desired droplet breakup size in a Y microchannel by varying the intensity of the electric field and applying it across any one of the branch channels. Thus, the goal of the present work is to enhance such a system’s performance and adaptability by using the electric field. The effects of pertinent dimensionless parameters such as electric Capillary number (Cae ) and permittivity ratio (εr ) on the droplet splitting process are thoroughly investigated.
3 Problem Description The deformation behaviour of an elliptical shaped viscous droplet moving in a symmetric Y junction microchannel by externally applying an asymmetric electric field at one of the branching channel is studied here. Figure 1 demonstrates the 2D schematic figure of a symmetric Y-junction microchannel taken for the present study. The microchannel consists of a horizontal inflow channel and two bifurcating channels. The mother horizontal channel has a width (w = 100 µm). The centre of the droplet is taken as the origin and the distance between the centre of the droplet and the bifurcation point is 250 µm. The bifurcating daughter channels are both having identical dimensions with width equal to 0.7w. The angle of bifurcation is equal to α which is 40° and the Y channel is symmetrically bifurcated about the X axis. As shown in Fig. 1, the dispersed phase (drop) has an elliptical shape at time, t = 0 s, with semi-minor and semi-major radii as 40 µm and 100 µm, respectively. The physical properties of the fluid considered for our study are provided in Table 1. The surface tension (γ ) between carrier fluid and bubble is taken as 0.0048 N/m. The two fluids are considered incompressible, immiscible and non-reacting. Since the densities of the two fluids are close to each other and the length scale of the problem is quite small, we have ignored the gravity effects on the problem. The Reynolds number (Re) and Capillary number (Ca) values are 1 and 0.002, respectively, and they are kept constant throughout the simulation. The present study is
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Fig. 1 Schematic diagram of the set-up
Table 1 Properties of the fluids taken for study Fluid
Density (ρ) (kg/m3 ) Viscosity (μ) (Pa s) Relative permittivity (ε)
Carrier fluid (D.I. water)
1000
0.001
78.5
Dispersed phase (silicone oil)
960
0.05
2.8
focussed on the effects of droplet breakup dynamics due to an external electric field. Hence, the concerned parameters are primarily electrical permittivity ratio and electric capillary number. The electrical capillary number represents the relative strength of electric force to surface tension force and is expressed as Cae = μcγUe , ε E2 R
where, Ue = c μ∝c is the velocity scale associated with the applied electric field [9]. Permittivity ratio, εr = εεdc describes the fluid response to the applied electric field. The subscripts d and c represent droplet phase and carrier fluid phase, respectively.
4 Mathematical Modelling and Simulation Details Assuming pressure driven, incompressible, transient and immiscible two phase flow (laminar) in a bifurcating microchannel, we have modelled our numerical setup. The continuity and momentum governing equations for the problem can be written as. ∇ ·u =0
(1)
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∂u + ρ(u · ∇)u = ∇ · τ + FST + Fe ∂t
(2)
ρ
where, ρ, u, τ , FST and Fe represent density, velocity vector, stress tensor, surface tension force and electrical force, respectively. The droplet interface is tracked using phase field transport equation which is as follows: 3χ γ κ ∂ϕ ∇ψ (3) + u · ∇ϕ = ∇ · √ ∂t 2 2 ψ = − ∇ · κ 2 ∇ϕ + ϕ 2 − 1 ϕ (4) Here, ϕ, χ , ψ and κ are phase field variable, mobility tuning parameter, phase field help variable and parameter controlling the interface thickness, respectively. The phase field method is a one fluid formulation where two phase system is treated as a mixture of two components and the properties values are interpolated as a function of ϕ. Each property is interpolated as: P = 0.5 ∗ Pd (1 + ϕ) + 0.5 ∗ Pc (1 − ϕ)
(5)
Here, P is any property like density, viscosity, relative permittivity and electrical conductivity. The surface tension force in Eq. (2) will be as FST = G∇ϕ
(6)
where, G is referred to as a measure of phase [14]:
ϕ ϕ2 − 1 G = λ −∇ ϕ + κ2 2
(7)
The charge conservation can be written as: Dqv + ∇ · (σe E) = 0 Dt
(8)
where qv represents volume density of local free charges and σe is the conductivity of the fluid. The charge accumulation at the interface happens much faster compared to the time scale of fluid motion [10]. Hence, the first term of Eq. (8) can be ignored and it can be simplified as ∇ · (σe E) = 0
(9)
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For coupling the two phase flow interface with electrostatics module in Comsol Multiphysics [15], we have specified the electric force, F e as a volume force in Eq. (2). The electric force is given by the divergence of the Maxwell stress tensor (M). Fe = ∇ · M
(10)
1 M = E D T − (E · D)I 2
(11)
where, E(= − ∇ · V ) is the electric field and D = ε0 εr E is the electric displacement field. Here, V is the voltage applied. 1 Fe = − E 2 ε0 ∇ε 2
(12)
A fully developed velocity profile is set at the inlet. The outlets at both the branching channels are assigned zero gauge pressure. A constant electric field is applied along the outer wall of lower daughter channel V 0 as shown in Fig. 1, and ground conditions are applied to the inner wall of the bifurcated channels. All other domain boundaries are assigned with zero-charge condition. The droplet domain is given an initial phase volume fraction (ϕ) of − 1 whereas the remaining domain is given as 1. In the present study, first order discretization is performed for velocity and pressure (P1 + P1), linear discretization scheme is employed for the phase field variable and quadratic discretization scheme for electric potential. The default time dependent solver (BDF) method with maximum order set to two is used. The relative and absolute tolerance of 0.001 and 0.000001 are specified. PARDISO has been used as a linear system solver.
4.1 Grid Independence Study The mesh independence study is performed by tracking the droplet interface for different mesh sizes at time (t = 0.02 s) when the droplet has almost reached the bifurcation point. Now, we took the volume fraction contours of each mesh sizes and shown in Fig. 2. There are four triangular elements that are simulated, and they contain 12,023, 23,259, 37,734 and 54,849 triangular elements, respectively. When a grid contains more than 37,734 elements, it is found that the drop interface barely changes. For the purpose of the current simulations and future simulations, the grid with 37,734 triangular elements is taken into consideration.
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Fig. 2 The droplet profile at time = 0.02 s near the bifurcation point for four different mesh size
4.2 Model Validation For validating our numerical model for the aforesaid study, we have used a two-step validation procedure. Firstly, we validated our interface tracking method (phase field approach) with the analytical solution provided by Bretherton [16], which proposed an analytical relation for the droplet velocity as: Ud = Uc
2 μc u c 3 1 + 1.29 γ
(13)
where, Ud and Uc are the droplet velocity and carrier velocity respectively. This relationship holds good for low Ca flow. Figure 3 suggests that our results match quite perfectly with Eq. (13). Fig. 3 Comparison of results obtained in our study with Bretherton [16] analytical solution
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Fig. 4 Comparison of results obtained in our study with previously studied literature for droplet deformation for a wide range of conductivity ratio. In the inset, the schematic figure used for the validation has been shown. All the simulations are performed at Cae = 0.18 and ρr = 10
We also validated our simulation model used in this study by comparing the deformation of a droplet (D) studied previously theoretically [2, 3]. The simulation parameter used in this study is similar to those of Das et al. [17]. Due to different electrical properties of the two fluids, they tend to deform either into a prolate or an oblate shape when acted upon by an external applied electric field. Here, D = (L − W )/(L + W ) is the measure of the deformation [2], with L being the length along the direction of the applied electric field and W being length perpendicular to applied electric field. The relation provided by Taylor [2] holds good in the limit of small deformation and it is also the function of fluid properties and electrical properties of the working fluid. We observe in Fig. 4 that for small drop deformation, there is good agreement between our simulated results and Taylor [2] theory and only at larger D is there are deviations from the theory essentially due to the breakdown of the assumption of linearity [17]. Comparison with Ajayi [3] second-order theory shows good agreement with the simulation results.
5 Results and Discussion The breakup dynamics of a droplet under the effect of the applied electric field are examined as it travels through a Y-junction microchannel. Permittivity ratio (of the droplet and carrier fluid) and electrical capillary number varied widely and different splitting regimes are observed. In the subsequent section, we explore how the presence of the electric field influences the droplet splitting process and causes transition from breakup to no breakup regime. We have considered two different cases with different εr = 28 and 0.036, which imply permittivity of droplet being higher and lower than that of the carrier fluid, respectively. The Cae is varied from 0 to 0.45 by increasing the electric field.
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Figure 5 shows the time evolution of droplet having εr = 28 for different applied electric field at time = 0.05 s. The Cae is increased from 0 to 0.3 in steps of 0.01, 0.05, 0.1, 0.2 and 0.3. At Cae = 0, the droplet splits symmetrically but with an increase in Cae , droplet splitting changes significantly due to the influence of electrical force on the droplet when it comes in close vicinity of the lower channel, where E is applied. It is evident that with the increase in electric field, the bigger volume fraction of the droplet enters the branch channel where electric field is applied. At Cae = 0.305, the entire droplet goes into the lower channel where electric field is imposed. Here, εd > εc which creates an electrical field force in the outward direction of the droplet interface [1, 12]. This phenomenon causes an electrical stress jump at that part of the droplet interface which is located inside the lower channel, thereby, creating an additional force to drive the droplet more into the lower channel. It is also evident from the increase in droplet volume in the lower channel with Cae . Here, Cae is increased by gradually increasing F e . The main advantage is that F e is proportional to E 2 which means with an increase in electric field, we increase the F e significantly. This effect can be utilised in sorting different size of droplet with the use of electric field in the microchannel. The opposite sorting of droplet, i.e. gradual increase of volume of droplet in upper channel can be achieved by applying potential at upper channel instead of lower channel and we have got sorting pattern being the same with the only difference being the droplet with increasing Cae gradually goes in upper channel (where the E is applied). Next, we investigate the droplet breakup dynamics under the influence of a similar electric field for εr < 1. Results are presented in Fig. 6, which demonstrates droplet path selection opposite to that in Fig. 5. The reason for such behaviour can be attributed to the direction of electric field force on the interface. Sherwood [1] showed Fig. 5 Volume fraction contours of droplet for εr = 28 with increasing Cae at time = 0.05 s
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numerically that the direction of the electric field force is inward on the interface if εd < εc . Here, we observe that the droplet gradually moves more into the upper channel where no electric potential is applied. The sorting pattern is similar to that in Fig. 5, the only difference being the choice of path of the droplet. In Fig. 7, we have shown the quantitative variation of volume of droplet with the applied E. We have defined volume split ratio (V SR ) as the volume of the droplet going into the upper channel to the total volume of the droplet before breakup and plotted it against different Cae for two different scenarios. If we look at εr = 28, due to small increase in Cae from 0 to 0.1 we can see there is a significant changes in the volume of the droplet being dragged into the lower channel due to sudden increase in F e which overpowers the viscous and surface tension forces. For Cae = 0, bifurcation behaviour is symmetric and V SR = 0.5. If we further increase the Cae , the rate of reduction of V SR (slope of the curve) decreases, and beyond Cae = 0.3, V SR becomes zero. Any further increase in Cae doesn’t have any impact on sorting as the entire droplet has entered the lower channel. With further increase of Cae , the droplet in the lower branch (stronger F e due to applied E) will experience a slightly faster movement in that channel. For εr = 0.036, we observe the opposite trend of droplet sorting (droplet moving into the upper channel) as discussed above. The main difference is that the entire droplet now enters the upper channel at a lower Cae = 0.2, unlike the εr = 28 case. This is attributed to the difference in viscosity between the droplet and the carrier fluid. For same viscosity, it is expected to yield a similar trend of bifurcation with respect to the symmetric bifurcation line, εr = 1 (as there is no difference in electrical permittivity, there will be no F e effect on the droplet interface according to Eq. 12). Fig. 6 Volume fraction contours of droplet for εr = 0.036 with increasing Cae at time = 0.05 s
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Fig. 7 Variation of V SR with Cae for different permittivity ratio (εr )
After a particular value of Cae , increasing electrical field intensity won’t affect the droplet splitting behaviour as it gets sorted in one of the branch channels only, depending on relative electrical permittivity and the position of the applied field. This is known as a critical electric Capillary number Cae,cr since it quantitatively indicates when the transition from symmetric-splitting to no breakup is finished and sorting of the droplets prevails. Further increase in Cae beyond Cae,cr will only lead to an increase of droplet velocity with which it gets sorted in the same channel. In Fig. 7, we see that the increasing Cae beyond its critical value yields the same V SR .
6 Conclusions The present method of active control of droplet splitting in a microchannel Yjunction using an electric field demonstrates some interesting findings which can be summarised as follows. 1. The gradual increase of external applied electric field shifts the phenomena from equal breakup of droplets to no-breakup regime and eventually sorts the droplet into only one channel if the value of the Cae reaches the Cae,cr . Further, increase of Cae beyond the critical value results in more rapid movement of droplet into the same channel without altering the no breakup regime. 2. The sorting of droplets can be achieved by simply manipulating the magnitude of E and also switching the channel where E is applied. It is evident that the droplet follows the channel where E is imposed, if the electrical permittivity of the droplet is higher than that of the carrier fluid.
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3. For a low permittivity droplet (εr < 1), we observe that the droplet follows the path of the channel opposite to the branch channel where E is applied. This could also be useful in interpreting the path selection of droplet according to the relative electrical permittivity of the two fluids.
References 1. Sherwood JD (1988) Breakup of fluid droplets in electric and magnetic fields. J Fluid Mech 188:133–146 2. Taylor G (1966) Studies in electrohydrodynamics. I: the circulation produced in a drop by an electric field. Proc R Soc Lond Ser A Math Phys Sci 291(1425):159–166 3. Ajayi OO (1978) A note on Taylor’s electrohydrodynamic theory. Proc R Soc A 364(1719):499– 507 4. Saville DA (1997) Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu Rev Fluid Mech 29(1962):27–64 5. Melcher JR, Taylor GI (1969) Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu Rev Fluid Mech 1(1):111–146 6. Ting TH, Yap YF, Nguyen NT, Wong TN, Chai JCK, Yobas L (2006) Thermally mediated breakup of drops in microchannels. Appl Phys Lett 89(23) 7. Bijarchi MA, Dizani M, Honarmand M, Shafii MB (2021) Splitting dynamics of ferrofluid droplets inside a microfluidic T-junction using a pulse-width modulated magnetic field in micro-magnetofluidics. Soft Matter 17(5):1317–1329 8. Agnihotri SN, Raveshi MR, Bhardwaj R, Neild A (2020) Microfluidic valves for selective on-chip droplet splitting at multiple sites. Langmuir 36(5):1138–1146 9. Link DR et al (2006) Electric control of droplets in microfluidic devices. Angew Chem Int Ed 45(16):2556–2560 10. Jafari I, Fallah K (2020) Drop breakup in a symmetric T-junction microchannel under electric field. Microfluid Nanofluid 24(12) 11. Fallah K, Fattahi E (2022) Splitting of droplet with different sizes inside a symmetric T-junction microchannel using an electric field. Sci Rep 12(1):1–12 12. Wehking JD, Chew L, Kumar R (2013) Droplet deformation and manipulation in an electrified microfluidic channel. Appl Phys Lett 103(5) 13. Chowdhury IU, Mahapatra PS, Sen AK, Pattamatta A, Tiwari MK (2021) Autonomous transport and splitting of a droplet on an open surface. Phys Rev Fluids 6(9):1–23 14. Feng JQ, Scott TC (1996) A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field. J Fluid Mech 311:289–326 15. COMSOL (2017) COMSOL multiphysics reference manual. COMSOL Multiphysics®v. 5.3 a, Stockholm 16. Bretherton FP (1961) The motion of long bubbles in tubes. J Fluid Mech 10(2):166–188 17. Das SK, Dalal A, Tomar G (2021) Electrohydrodynamic-induced interactions between droplets. J Fluid Mech 915:1–27
Microfluidic Solute Transport by Interference of Oscillatory Thermal Marangoni Effect and Patterned Wall Slip Shubham Agrawal, Prasanta K. Das, and Purbarun Dhar
Abstract For quick sample processing, the reagents’ mixing is crucial in microfluidic devices. However, it is difficult to obtain efficient mixing at such small scales due to low Reynolds number (Re) flow within the micro-conduits. We study fluidictransport and mixing phenomenon in a binary-liquid system by the thermocapillary effect, actuated by the periodic wall thermal stimuli. Our study also demonstrates the impact of the interference of thermo-capillarity and wall slip on the behavior of the phenomena mentioned above, by employing patterned wettability at the wall surfaces. We semi-analytically solve the Navier–Stokes and the continuity equations to obtain the hydrodynamic characteristics of the system. We also solve the species transport equation to obtain the solute distribution in the microchannel for given inlet concentration conditions. Our study explores different mechanisms through which the flow pattern can be morphed to enhance the solute particles’ mixing. The present work illustrates the direct relationship between mixing dynamics and wall slip through the qualitative study of the stream function and solute distributions within the microchannel. Keywords Microfluidics · Binary-fluid · Thermo-capillarity · Marangoni effect
1 Introduction The surface forces which are usually negligible relative to the body forces at large scales govern the flow dynamics in microchannels owing to small length scales. Surface forces can be morphed to obtain the desired flow configuration by using electric and magnetic fields [1], acoustic excitation [2], capillary action [3], etc. Capillary S. Agrawal (B) · P. Dhar Hydrodynamics and Thermal Multiphysics Lab (HTML), Department of Mechanical Engineering, IIT Kharagpur, Kharagpur, West Bengal 721302, India e-mail: [email protected] P. K. Das Department of Mechanical Engineering, IIT Kharagpur, Kharagpur, West Bengal 721302, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_37
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forces are modulated by varying the interfacial tension in a multi-component system. The presence of the temperature gradient along the fluid–fluid interface creates a disturbance in the interfacial tension, which leads to the flow generation within the system; the phenomenon is called the thermocapillary effect. Thermocapillary induced Marangoni convection is observed in weld pool [4], crystal growth [5], drop coalescence [6], bubble interaction [7], etc. Thermocapillarity is extensively employed to generate a flow of discrete fluid elements like droplets [8] and bubbles [9], which are submerged in another carrier fluid. Lu et al. [10] numerically studied the thermo-capillarity-driven migration of deformable bubbles in an obstructed microchannel using the finite volume method. Thermocapillarity is coupled with other forces to morph the flow patterns and attain desired configurations. Luo et al. [11] explored droplet transport under the coupled influence of the thermal and surfactant-induced Marangoni flow. Alloui et al. [12] analytically studied the flow behavior of a power-law obeying non-Newtonian fluid, which is actuated by the buoyant thermocapillary effect. The present study explores the thermo-capillarity-driven binary-liquid system in a microchannel with patterned wettability at the walls. Pendse and Esmaeeli [13] studied the thermocapillary transport of the binary-fluid system under no-slip boundary conditions at the walls. Ghosh and Chakraborty [14] demonstrated the influence of patterned wall slip on the electro-osmotic flow of a liquid film sandwiched between the two parallel plates of a microchannel. They observed that the wall wettability leads to enhanced mixing efficiency in the system. However, the interaction of the periodic wettability and the thermo-capillarity is not studied. Therefore, we semi-analytically obtain the stream function distributions and the interfacial velocity profiles at the fluid–fluid interface for a thermally actuated binary-liquid system under periodic wall slip conditions. Our study explores the effect of the variation in the relative film thickness of the two fluid layers on the mixing hydrodynamics. We also demonstrate the solute transport phenomenon within the system for a periodic inlet concentration profile.
2 Analytical Analysis 2.1 Problem Description The binary-liquid system is sandwiched between two parallel plates of a microchannel, which are exposed to periodic thermal stimuli, as shown in Fig. 1. Pattern wettability conditions are applied at the wall surfaces, such that the slip velocities are given as: u a | y=−a
| ( ( )) ∂u a || x = l s 1 + ζ cos βl and ∂ y | y=−a lc
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Fig. 1 Flow domain of the binary-fluid system with applied wall thermal stimuli. T l and T u are the mean temperatures, T 0 the perturbation amplitude, αl and αu the frequencies, and δ the phase difference between the wall thermal stimuli
| ( ( )) x ∂u b || u b | y=b = −l s 1 + ζ cos βu + γ ∂ y | y=b lc
(1)
where, u a | y=−a and u b | y=b are the slip velocities at the lower and the upper wall, respectively, ζ the slip perturbation amplitude, l s the slip length, l c the problem’s characteristic length, βl and βu the frequency of the two slip perturbation, and γ the phase difference between them. Here, we represent the dimensional terms by the usual symbols with an overhead bar over it. Later, the normalized parameters are represented by the corresponding symbols without any overhead bar.
2.2 Hydrodynamics Characteristics We solve the continuity and the Navier–Stokes equation to obtain the stream function distribution within the microchannel and the interfacial velocity profiles, as: ∇ · u→ i = 0, and − ∇ pi + μi ∇ 2 u→ i = 0; where, i = a, b By introducing the stream function, which is defined as u = normalizing the length scales, Eq. (2) reduces to:
∂ψ , ∂y
(2)
v = − ∂ψ and ∂x
∂ 4ψ i ∂ 4ψ i ∂ 4ψ i + 2 + = 0; where, i = a, b ∂x4 ∂ x 2∂ y2 ∂ y4
(3)
The corresponding boundary conditions are given as: (a) Slip velocity at the walls: | | ∂ψ a | (x, y)| | ∂y
y=−a
| | ∂ψ b | (x, y)| | ∂y
| | ∂ 2ψ a | = ls (1 + ζ cos(βl x)) y) (x, | | ∂ y2
, y=−a
| | ∂ ψb | = −ls (1 + ζ cos(βu x + γ )) (x, y)| 2 | ∂y 2
y=b
(4) y=b
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(b) No penetration across the walls: | | ∂ψ a | (x, y)| | ∂x
| | ∂ψ b | = (x, y)| | ∂x
y=−a
=0
(5)
y=b
(c) No slip condition at the fluid-fluid interface: | | ∂ψ a | (x, y)| | ∂y
y=0
| | ∂ψ b | = (x, y)| | ∂y
(6) y=0
(d) No penetration across the fluid-fluid interface: | | ∂ψ a | (x, y)| | ∂x
y=0
| | ∂ψ b | = (x, y)| | ∂x
=0
(7)
y=0
(e) Shear stress balance across the fluid-fluid interface: ( μa
∂ 2ψ a ∂ 2ψ a − ∂ y2 ∂x2
)| | | | |
( − μb y=0
∂ 2ψ b ∂ 2ψ b − ∂ y2 ∂x2
)| | | | |
y=0
| ∂ T || = σ Tl c | ∂x |
(8) y=0
where, the interfacial temperature gradient required to in Eq. (8) is adopted from our earlier work [15], in which we solved the decoupled energy equation to obtain the temperature distribution in the flow domain, which gives the form of interfacial capillary stress, as: | ∂ T || σ Tl c | ∂x |
= f 1 (y) cos(αl x) + f 2 (y) cos(αu x + δ)
(9)
y=0
where, f 1 (y) and f 2 (y) are functions of y. We choose an approximate form of stream function by observing the interfacial and wall boundary conditions, as: ψ a (x, y) = f a (y)(sin(αl x) + ηa (1 + ζ cos(βl x))), ψ b (x, y) = f b (y)(sin(αu x + δ) + ηb (1 + ζ cos(βu x + γ )))
(10)
where, f a (y) and f b (y) are some arbitrary functions of y. ηa and ηb represent the significance of wall slip velocities relative to the thermal Marangoni velocity. Hence, the velocity scales due to wall ( their values are)chosen to be the ratio of ( ) ls slip ∼ Vc l , or ∼ Vcls and Marangoni effect Vc , which gives: ηa ∼ ηb ∼ ls . c
Now, substituting the expressions of the stream functions (Eq. 10) in the biharmonic differential equation (Eq. 3), and dropping the terms of lower order of magnitude, we obtain:
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f a'''' (y) − 2αl2 f a'' (y) + αl4 f a (y) = 0, f b'''' (y) − 2αu2 f b'' (y) + αu4 f b (y) = 0
(11)
The above ODEs are solved by assuming f a (y) = e py and f b (y) = eqy , which gives: f a (y) = (C1a + C2a y) cosh(αl y) + (C3a + C4a y) sinh(αl y), f b (y) = (C1b + C2b y) cosh(αu y) + (C3b + C4b y) sinh(αu y)
(12)
Now, we obtain the final form of the stream function distribution by combining Eqs. (10) and (12), as: ψ a (x, y) = ((C1a + C2a y) cosh(αl y) + (C3a + C4a y) sinh(αl y)) · (sin(αl x) + ηa (1 + ζ cos(βl x))), ψ b (x, y) = ((C1b + C2b y) cosh(αu y) + (C3b + C4b y) sinh(αu y)) · (sin(αu x + δ) + ηb (1 + ζ cos(βu x + γ ))) (13) The boundary conditions mentioned in Eqs. (4)–(8) are employed to determine the constants of Eq. (13). The values of the constants are omitted in the paper since they are complex functions of the thermophysical properties. The influence of different parameters on the flow patterns is later discussed in Sect. 3.1.
2.3 Solute Transport We also solve the species transport equation to understand the steady state distribution of immiscible solute particles for a given set of entry and exit conditions in a finite length microchannel: ( ) i i i i ∂ 2C ∂C ∂C ∂ 2C + vi = D ci ui + ; where, i = a, b ∂x ∂y ∂x2 ∂ y2
(14)
i
Here, C and D ci are the species concentration and the diffusion coefficient, respectively of the solute particles in the solvent fluids. The normalized form of Eq. (14) is given by: (
i
∂C ∂C + vi Pe u i ∂x ∂y
i
) i i ∂ 2C ∂ 2C + ; = ∂x2 ∂ y2 ( ) i i i i ∂C ∂C ∂ 2C ∂ 2C + vi = Pe u i ; or, − ∂x2 ∂x ∂y ∂ y2
)
(
(15)
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where, i = a, b and Pe = VDc lc is the Peclet number. ci The species transport equation (Eq. 15) is constrained to the following boundary conditions: (a) Species concentration at the microchannel entry (x = 0): | | C (x, y)|
(
Nπy = 1 + cos i
i
x=0
) (16)
where, i = a, b and N is an integer. (b) Species concentration at the channel exit (x = 1): | i | C (x, y)|
x=1
= 1; where, i = a, b
(17)
(c) Species flux across the microchannel walls: | a ∂C (x, y) || | | ∂y
y=−a
| b ∂C (x, y) || = | | ∂y
=0
(18)
y=b
(d) Species flux across the fluidic interface: | i ∂C (x, y) || | | ∂y
= 0; where, i = a, b
(19)
y=0
Now, integrating twice Eq. (15) partially with respect to x under the limits of x = 0 to x, we obtain: | i | ∂C | C (x, y) − x (x, y)| | ∂x
| | − C (x, y)|
i
i
x=0
x=0
) ∫ x ∫ x ( i i ∂C ∂C + vi = Pe ui dxdx ∂x ∂y 0
0
∫ x ∫ x − 0
0
i
∂ 2C dxdx; ∂ y2
(20)
We assume a series solution of the species distribution to obtain the solution of Eq. (20) by Adomian decomposition method [16], as: i
C (x, y) =
∞ ∑
i
C r (x, y);
(21)
r =0
where,
i C 0 (x,
| i | ∂C | y) = x (x, y)| | ∂x
| i | + C (x, y)| x=0
x=0
,
(22a)
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i C r (x,
and
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) i i ∫ x ∫ x ( ∂C r −1 ∂C r −1 ui y) = Pe + vi dxdx ∂x ∂y 0
0
∫ x ∫ x − 0
0
i
∂ 2 C r −1 dxdx for r ≥ 1; ∂ y2
(22b)
Here, i = a, b. Now, by using Eqs. (16)–(22a, 22b), we evaluate a series solution for the solute concentration distribution up to second order terms, as: i
C (x, y) = 1 + cos
(
Nπy i
)
i
+ xh i (y) + C 1 (x, y)
(23)
i
where, i = a, b. The expressions of h i (y) and C 1 (x, y) are complex functions of the problem parameters and are not mentioned in the manuscript to maintain brevity of the paper. However, their influence on the solute transport phenomenon is demonstrated through the contour plots in the result section.
3 Results and Discussion Periodic fluctuation of the interface (fluid–fluid) temperature creates disturbances in the interfacial tension, which leads to the vortical (circular) motion of the liquid molecules in the two layers of the binary-fluid system. The temperature distribution within the microchannel is taken from our earlier work [15]. The interfacial temperature obtained in the study is presented in Fig. 2 to demonstrate the basis of the thermocapillary effect in the domain. The fluctuation of the interfacial temperature generates periodic flow patterns in the microchannel, which is presented in Sect. 3.1.
Fig. 2 Interfacial (fluid–fluid interface) temperature profile for different film thickness ratios [15]
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3.1 Flow Characteristics The stream function distribution and the interfacial velocity profiles originated from the thermocapillary effect and their dependence on the film thickness ratio are presented in (Figs. 3 and 4. The plots are prepared at a fixed combination of other parameters: T0 = 0.5, αl = π, αu = 2π, δ = π2 , βl = π, βu = 2π, γ = π2 , ls = ) 0.05, ζ = 0.5, λ = λλab = 0.5, and μ = μμab = 0.5 . Vortex enveloping phenomenon is observed in Fig. 3 due to the interaction of the periodic thermal stimuli and patterned wettability at the walls, which results in better mixing of the reagents present in each fluid layer. It is noted that the periodicity of the wall temperatures is replicated in the stream function distribution of the adjoining fluid layer. ( )The amplitude of the interfacial velocity increases with the film thickness ratio ab (refer to Fig. 4). However, the frequency of the interfacial velocity is observed to be independent of the relative film thickness of the two fluid layers. This results in more ( ) localized mixing for the system configuration with a higher film thickness ratio ab . The phenomenon is also reflected in the stream function distribution shown in Fig. 3 (compare the vortex patterns in Fig. 3a, c). Knowing that the temperature perturbation frequency of the top wall is twice that of the bottom wall in the cases shown, it can be concluded that the fluid–fluid interface must be closer to the wall with higher temperature perturbation frequency to obtain better local mixing.
( Fig. 3 Stream function distribution for different film thickness ratios ψ =
ψ−ψmin ψmax −ψmin
)
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Fig. 4 Interfacial velocity profiles (along fluid–fluid interface) and its dependence on the film thickness ratio
3.2 Solute Transport The distribution of solute particles in a finite length microchannel for fixed entry and exit concentration profiles is demonstrated in the current section. The solute particles are assumed to be immiscible in the carrier fluids. Figure 5 illustrates the dependence of the solute particles’ distribution on the wall slip length (αl = π, αl = 2π, T0 = 0.5, λ = 0.5, a = b = 0.25, N = 1). Close observation of the contour plots for the solute concentration reveals that post entry, the solute particles tend to diffuse across the microchannel axis. After traversing a certain distance, it again accumulates near the interface (fluid–fluid) due to the immiscibility of the particles in the carrier fluids. It is also observed that the extent of diffusion of the solute particles across the channel axis increases with the wall slip length, which indicates a direct relation between the mixing efficiency of the system and the wall slip.
4 Conclusions A semi-analytical solution of the Navier–Stokes and continuity equation is obtained for thermo-capillarity-driven fluid transport in a binary-fluid system. The stream function distribution within the microchannel is presented for different film thickness ratios, demonstrating the system’s mixing dynamics. The interfacial velocity profiles reveal that better mixing is obtained for the system configuration, where the film thickness is lesser for the fluid layer adjacent to the wall with higher temperature perturbation frequency. Solute distribution within the microchannel is also discussed for a fixed set of entry and exit species concentration profiles. We solved the species transport equation by the Adomian decomposition method to obtain a series solution of the solute concentration. The mixing efficiency of the solute particles in the carrier fluids is observed to be proportional to the wall slip length. The outcomes of the work can
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( Fig. 5 Solute transport and its dependence on the wall slip length C =
C−Cmin Cmax −Cmin
)
play a pivotal role in improving the mixing efficiency of the present microdevices, which can further reduce the reagents’ processing cost. Acknowledgements We acknowledge IIT Kharagpur for providing the requisite facilities to perform this work. We also acknowledge the scholarship by the Ministry of Education, Govt. of India.
Nomenclature u ls ζ β γ T α p μ lc ψ
Axial velocity [m/s] Slip length [m] Slip perturbation amplitude [–] Slip perturbation frequency [–] Slip phase difference [–] Temperature [K] Temperature perturbation frequency [–] Pressure [N/m2 ] Dynamic viscosity [Ns/m2 ] Characteristic length [m] Stream function [m2 /s]
Microfluidic Solute Transport by Interference of Oscillatory Thermal …
σT C Pe Dc
459
Temperature coefficient of surface tension [N/m K] Solute concentration [mol] Peclet number [–] Species diffusivity [m2 /s]
References 1. Chakraborty S, Paul D (2006) Microchannel flow control through a combined electromagnetohydrodynamic transport. J Phys D Appl Phys 39:5364–5371 2. Wang T, Ni Q, Crane NB, Guldiken R (2017) Surface acoustic wave based pumping in a microchannel. Microsyst Technol 23(5):1335–1342 3. Zhao JF, Li ZD, Li HX, Li J (2010) Thermocapillary migration of deformable bubbles at moderate to large Marangoni number in microgravity. Microgravity Sci Technol 22:295–303 4. Wei PS, Lin CL, Liu HJ, Ting CN (2012) Transient thermocapillary convection in a molten or weld pool. J Manuf Sci Eng 134(1):011001 5. Cuvelier C, Driessen JM (1986) Thermocapillary free boundaries in crystal growth. J Fluid Mech 169:1–26 6. Xie H, Zong Z, Zhang L, Yokota Y, Kawazoe Y, Yoshikawa A (2016) Simulation on thermocapillary-driven drop coalescence by hybrid Lattice Boltzmann method. Microgravity Sci Technol 28:67–77 7. Sun R, Hu WR (2002) The thermocapillary migrations of two bubbles in microgravity environment. J Colloid Interface Sci 255:375–381 8. Jiao Z, Huang X, Nguyen NT, Abgrall P (2008) Thermocapillary actuation of droplet in a planar microchannel. Microfluid Nanofluid 5:205–214 9. Alhendal Y, Turan A, Hollingsworth P (2013) Thermocapillary simulation of single bubble dynamics in zero gravity. Acta Astronaut 88:108–115 10. Lu M, Lu J, Zhang Y, Tryggvason G (2019) Numerical study of thermocapillary migration of a bubble in a channel with an obstruction. Phys Fluids 31:062101 11. Luo X, Luo ZY, Bai BF (2020) Effect of thermal convection on thermocapillary migration of a surfactant-laden droplet in a microchannel. Phys Fluids 32:092009 12. Alloui Z, Ouzani R, Vasseur P (2020) Thermocapillary-buoyancy convection of a power-law fluid layer heated from below. J Non-Newtonian Fluid Mech 282 13. Pendse B, Esmaeeli A (2010) An analytical solution for thermocapillary-driven convection of superimposed fluids at zero Reynolds and Marangoni numbers. Int J Therm Sci 49(7):1147– 1155 14. Ghosh U, Chakraborty S (2012) Patterned-wettability-induced alteration of electro-osmosis over charge-modulated surfaces in narrow confinements. Phys Rev E 85:046304 15. Agrawal S, Das PK, Dhar P (2022) Thermo-capillarity in microfluidic binary systems via phase modulated sinusoidal thermal stimuli. Phys Fluids 34(3):032012 16. Wazwaz AM (2009) Partial differential equations and solitary waves theory. Non-linear physical science
Analysis of Micro-nozzle Flow Using Navier–Stokes and DSMC Method and Locating the Separation Plane Based on Modified Knudsen Number Ashok Kumar, Manu K. Sukesan, and Shine S. R.
Abstract The flow through a micro-nozzle was studied using the particle-based DSMC approach and continuum approach using Navier–Stokes. Slip and no-slip boundary conditions are examined for the Navier–Stokes approach. The results are compared with the full DSMC simulation of a micro-nozzle with a 2 µm diameter. Predictions from all three methods are very close in the converging section. However, due to an increase in rarefaction effects, there are variations in the parameters observed at the diverging section of the nozzle. The extent of rarefaction is plotted using the Knudsen number isocurves inside the nozzle. To account for the local variation in characteristics length, Knudsen number is derived based on different properties like pressure, temperature, and density. The Knudsen number isocurves show that the continuum assumption breaks down at the throat, and the effects of rarefaction increase toward the nozzle’s exit. Keywords DSMC · Species separation · Rarefaction · Micro-nozzle · Mole fraction
1 Introduction Numerical simulation is used in the most recent and significant research across many scientific and technical domains. Different numerical simulation techniques like continuum method, DSMC methods are used in analyzing a flow through micronozzle, which has a wide variety of applications like micro-thrusters, separation of species in a flow, nozzles used in 3D printers and industrial printers, and other medical applications. In this analysis, the DSMC approach, the Navier–Stokes approach, A. Kumar · M. K. Sukesan (B) · S. S. R. Department of Aerospace Engineering, Indian Institute of Space Science and Technology, IIST, Thiruvanathapuram 695547, India e-mail: [email protected] S. S. R. e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_38
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and the Navier–Stokes approach with slip are used to numerically examine the performance of a micro-nozzle.
2 Literature Review and Objective Rafi et al. [1] studied the impact of varying wall temperature conditions on the evolution of the boundary layer in the micro-nozzle and observed the changes in the structure of the exit plume using a hybrid Navier–Stokes-DSMC technique. It is found that as the wall temperature increases, the subsonic layer inside the micronozzle increased, lowering the effective nozzle area ratio. The effect of changing cone angle of the nozzle on exit plume structure showed that the thrust of the nozzle increased for 2° rise in the nozzle divergence angle. Yuan et al. [2] obtained micro-nozzle flow properties using Navier–Stokes approach with different slip models (Maxwell, Langmuir, Langmuir-Maxwell) compared with DSMC results. The study demonstrates that the flow in a transition regime may be reliably simulated by the Langmuir model. White et al. [3] described the usage of new solver dsmcFoam+, which is a direct simulation Monte Carlo (DSMC) solver for rarefied gases, is developed within the OpenFOAM software framework and parallelized with Message Processing Interface (MPI). In a different work, Rafi et al. [4] used numerical and experimental approaches to find that the abrupt change from slip-flow to free-molecular regime in the subsonic boundary layer causes considerable back flow effects at the nozzle exit. It also states, when the flow is in a transition regime, the flow features predicted by Navier–Stokes and DSMC approaches do not perfectly match. Louisos and Hitt [5] used numerical simulation to observe the impact of heat transfer and viscosity in 2D and 3D supersonic micro-nozzle flows. Numerical simulations show that heat loss from the micro-nozzle flow increases Mach number and thrust output close to the expander walls because of Rayleigh flow acceleration, and that heat addition to the micronozzle flow decreases these quantities. Complete DSMC simulation of a micronozzle was compared to a hybrid NS-DSMC simulation by La Torre et al. [6], who found that the hybrid approach had an inaccuracy of less than 2% and could be run in between 5 and 25% of the CPU time of the full DSMC method. Liu et al. [7] used the continuum and DSMC approach to study the micro-nozzle. Both DSMC and continuum simulations have been shown to accord well with one another for flows with relatively low Knudsen numbers, with the exception of the nozzle exit lip area. As the Knudsen number rises over 0.045, there are considerable variances. Torre et al. [8] analyzed the performance of micro-nozzle using DSMC, Navier–Stokes and Coupled DSMC/Navier–Stokes approaches. The location of separation of the DSMC and Navier–Stokes approach has a significant impact on the prediction of nozzle performance. The separation plane showed that the majority of the diverging portion must be modeled using the DSMC approach. Modified Knudsen number was derived using gradient of properties to account for the local variation in characteristics length. Also, a better agreement between DSMC and CFD was observed
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when a slip wall boundary condition was implemented. Hao et al. [9] studied the size effect on gas flow in micro-nozzle experimentally and numerically using NS approach. In a micro-nozzle, the Mach number in the downstream of the throat is substantially lower and have a supersonic area instead of clearly visible shockwaves compared to a conventional large nozzle. As the nozzle size decreases, the sonic point moves farther away from the throat and toward the outlet. Low-density nozzle flow was examined by Chung et al. [10] using the Navier–Stokes method and the DSMC method. The DSMC method’s findings were in good accord with the experimental data but influenced by the simulation parameters, such as the gas/surface interaction model and the energy exchange model. It is observed from the literature review that majority of the research papers analyzed flow through micro-nozzle using Navier–Stokes and DSMC method. Also, a hybrid Navier–Stokes-DSMC method saves computation cost. But the location of the separation plane is very important in hybrid method. Therefore, the Knudsen number based on different properties like pressure, temperature, and density is studied for the micro-nozzle to find out the location where the continuum breaks. So that from that location we can apply the DSMC method, which will reduce the computation cost. In the current study, centerline properties of three methods, namely NS, NS with slip, and DSMC are also compared.
3 Methodology 3.1 Continuum/Navier–Stokes Approach The OpenFoam CFD computer program, which is free and open-source software distributed under the GNU General Public License, implements a finite volume spatial discretization on a structured two-dimensional grid to achieve the numerical solution of the Navier–Stokes equations for viscous fluid flow. By solving the continuity (Eq. 1), momentum (Eq. 2), and energy equation in compressible form, the OpenFoam software simulates the flow via the converging diverging nozzle. ∂(ρu) ∂(ρv) ∂(ρw) ∂ρ + + + =0 ∂t ∂x ∂y ∂z
(1)
D V→ = −Δ p + ρ g→ + μΔ2 V→ Dt
(2)
ρ
A transient, compressible and density-based solver Rho-CentralFoam is used for solving the above equation based on Finite Volume Method (FVM). Sutherland viscosity model which calculates μ as a function of T from a Sutherland coefficient As and Sutherland Temperature T s according to Eq. 3:
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μ=
3.1.1
√ As T 1+
Ts T
(3)
Navier–Stokes Approach with Maxwell Slip
The influence of rarefaction on micro-nozzle gas flow is a major element, particularly as thrust is reduced to the µN range [6]. When the throat diameter and gas flow boundary layer thickness are comparable to the mean free path of the molecules at these low thrusts and small nozzle dimensions, the flow enters a new domain. An important consideration in this regard is the Knudsen number, which is the ratio of the mean free path of the gas molecules to the characteristic length of the flow, L (given by Eq. 4) Kn =
λ L
(4)
When Kn is more than 10, the flow is in the free-molecular regime, where molecule-to-molecule interactions are less important than molecule-to-wall interactions. Models of molecular dynamics are capable of accurately simulating this domain. Intermolecular interactions take over the flow when Kn is less than 0.01, and continuum equations like the Navier–Stokes equations can describe behavior. Knudsen numbers between 0.01 and 0.1 are referred to as the slip-flow regime, which calls for modified boundary conditions to account for momentum and energy transfer at the walls, and between 0.10 and 10 are referred to as the transition regime, which calls for particle-based simulation techniques like the DSMC method. Maxwell proposed the theory of jump in velocity at the wall as a result of scattering of molecules at the wall. Ideally, only specular and diffuse reflections should theoretically occur in the gas molecules at the surface of the wall. The tangential velocity of the molecules does not change during specular reflection, but their normal velocity changes because their normal momentum is transferred to the wall. The average tangential velocity of molecules in diffuse reflection is zero. A tangential momentum accommodation coefficient, σ v , was first presented by Maxwell. When σ v = 1, all molecules reflect on the wall in a diffuse manner. Considering the reflection of gas molecules in wall, Maxwell first order velocity and temperature jump [2] can be expressed as Eqs. 5 and 6, respectively. Us = C 1 λ
∂u x + Uw ∂n
where C1 = α
2 − σv σv
(5)
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Like a jump in velocity at wall due to scattering of molecules, there is jump in temperature also. Smolokowski temperature jump condition at wall is given by Eq. 6: Ts =
2γ 2 − σT ∂T + Tw λ σT (γ + 1)Pr ∂n
(6)
where C 1 and T represent the slip coefficient and the static temperature respectively. Both α (the constant coefficient) and σ v are set to one for Maxwell first order slip model. The wall and the direction tangential to the wall are denoted by the subscripts w and x, respectively.
3.1.2
DSMC Method
The direct statistical modeling of the molecular flows based on the kinetic theory is used to resolve rarefied regimes in the micro-nozzle flow field. The Boltzmann equation for a simple dilute gas is given by Eq. 7: ∂(n f ) ∂(n f ) − → ∂(n f ) = +→ c. + F. ∂t ∂ r→ ∂ c→
∫∞ ∫4π
[ ] n 2 f ∗ f 1∗ − f f 1 cr σ dΩd→ c1
(7)
−∞ 0
→ → cr , σ, dΩ, − where n, f , c→, F, c1 represents the number density, the velocity distribution function, molecular velocity, external force acting on the molecule, relative velocity of the collision pair, collision cross section, infinitesimal velocity space solid angle, and the infinitesimal velocity of field molecules, respectively. The probabilistic particle simulation approach provides the foundation for the Direct Simulation Monte Carlo (DSMC) method, a numerical tool. In the current study, DSMC is employed to resolve the non-continuum regimes of the flow field. The DSMC technique stochastically models intermolecular interactions while algorithmically simulating molecular mobility and wall collisions for the solution of rarefied gas flows. The conventional Bird’s DSMC approach is used by the open-source framework dsmcFOAM+ to simulate the rarefied regimes of the flow field for micro-nozzles. The micro-nozzle flow field’s non-continuum regime is discretized in a way that keeps the control volume’s size well within the mean free path (typically less than λ/3 for practical case) and allows for the creation of enough DSMC particles in it. For the purpose of simulating binary collisions, the variable hard-sphere (VHS) intermolecular collision model is employed.
3.1.3
Validation of Numerical Approach
Three different numerical simulation approach, namely Navier–Stokes approach, Navier–Stokes with slip approach and DSMC approach are validated by comparing
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Fig. 1 Micro-nozzle geometry
the numerical results with experimental results of Hao’s nozzle [9]. Inlet pressure and temperature of the nozzle are fixed at 100 kPa and 300 K, respectively. Air is used as the working fluid and the wall is assumed adiabatic. The geometry of the nozzle is shown in Fig. 1, where ht of Hao’s nozzle is 20 µm. Domain independence study and the grid independence study are done for the selected nozzle and then the corresponding numerical simulation data are measured. Domain Independence Study: The simulation of Hao’s micro-nozzle is done with a square exit domain to prevent a flow reflected back from the outlet. The exit domain size is increased in size from 10 ht (10 times the height of throat) to the point, where the properties at the exit are not affected by the reflected flow from the outlet. The inlet pressure and the outlet pressure are fixed at 100 kPa and 65 kPa for doing this domain independence study and the pressure at the exit of the nozzle is observed. It is observed that when the size of the domain is 50 ht, the pressure at the exit of the nozzle is unaffected due to reflection from outlet. The variation of pressure at the exit with the iteration time step for different size of the exit domain is shown in Fig. 2. Grid Independence Study: Grid independence study of Hao’s nozzle [9] is conducted by increasing the number of cells inside the nozzle from 5000 to 30,000 and observing the exit mass flow rate. Inlet pressure is fixed at 100 kPa and the outlet pressure is fixed at 65 kPa. It is observed that as the number of cells increased from 20,000 to 30,000, the mass flow rate at the nozzle’s exit hardly changed by 0.1%. Therefore, 30,000 cells inside the Hao’s nozzle is chosen for the further study of the nozzle. The results of the grid independence study are given in Table 1. Exit mass flow rate measurement: The outlet pressure of the nozzle is varied between 90 and 10 kPa and the corresponding mass flow rate at the exit is measured numerically and compared with the experimental results mentioned in the paper [9]. When compared to the experimental measurement, the largest numerically estimated mass flow rate error, which occurs at ΔP of 40 kPa, is − 4.1% for Navier–Stokes approach,
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Fig. 2 Exit pressure versus time step of iteration
Table 1 Comparison of exit mass flow rate
Number of cells
Exit mass flow rate (10−7 kg/s)
5000
4.7657
10,000
4.8148
20,000
4.7082
30,000
4.7027
and for DSMC, it is 2.1%. ΔP, the difference between inlet and outlet pressure is taken as x-axis and plotted against the experimental and numerical mass flow rate as seen in Fig. 3. The mass flow rate at exit is measured using all three approaches, namely Navier–Stokes, Navier–Stokes with Maxwell slip and DSMC method. Since the error between numerical and the experimental values are less than 4.1%, we are using the three approaches for further study of micro-nozzle.
4 Results and Discussion Three methods—Navier–Stokes, Navier–Stokes with slip, and DSMC method—are used in the numerical simulation of scaled Hao’s nozzle (one-tenth of its original size, i.e., ht = 2 µm in Fig. 1), with an inlet pressure of 100 kPa and an exit pressure of 10 kPa. Air is used as the working fluid with wall considered as adiabatic. The mass flow rate at the throat and exit, and the centerline properties like pressure, temperature, density, and Mach number based on three methods are measured and compared. Mass flow rate per unit width for 2 µm nozzle measured by three different methods is given in Table 2.
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Fig. 3 Experimental versus numerical mass flow
Table 2 Mass flow rate comparison
Method
Throat (10−11 kg/s)
Exit (10−11 kg/s)
Navier–Stokes
1.12126
1.12817
NS with slip
1.35889
1.3583
DSMC
1.38915
1.40657
The mass flow rate recorded at the throat and the exit of the nozzle, as calculated by DSMC simulation and Navier–Stokes simulation with Maxwell first-order slip, is quite close. At the throat and exit, the mass flow rate predicted by the Navier–Stokes simulation with Maxwell first-order slip approach is 2.1% and 3.43% lower than the mass flow rate predicted by DSMC. In comparison to the DSMC approach, the variance in mass flow rate measured by Navier–Stokes with no-slip is substantially larger. At the throat and exit, the mass flow rate predicted by the Navier–Stokes simulation with no-slip approach is 19.2% and 19.8% lower than the mass flow rate predicted by DSMC. In Navier–Stokes with no-slip boundary condition, the subsonic zone near the wall is significantly greater than in Navier–Stokes with slip boundary condition. This results in increased boundary layer thickness and reduced effective flow area and thus low mass flow rate in no-slip boundary condition. This can be seen in Fig. 4. The subsonic viscous layer thickness at the nozzle exit is measured to be 64.9% of the nozzle’s exit length with no-slip boundary condition and 51.8% with slip boundary condition. With slip velocity, the maximum Mach number is 1.41, while with no-slip, it is 1.34. Along with comparing the mass flow rate data provided using the DSMC, Navier–Stokes, and Navier–Stokes with slip numerical techniques, the centerline parameters of the scaled Hao’s nozzle, including pressure, temperature,
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Fig. 4 Mach contour-slip versus no-slip boundary condition
density, and Mach number, are also measured numerically and compared. The results are given in the graph in Figs. 5, 6, 7, 8. The centerline pressure and the temperature values are non-dimensionalized using the inlet pressure and temperature values, respectively. In the converging section, values, viz. pressure, temperature, density, velocity, and Mach number are almost same for DSMC simulation and Navier–Stokes simulation with Maxwell slip boundary condition. In the diverging section, the P, ρ, T, Ma values are nearly equal for DSMC and Navier–Stokes with Maxwell slip. However, the values are starting to diverge after Nozzle axis length of 0.8, which shows that the rarefaction effects are higher near the end of the nozzle for the given back pressure of 10 kPa. This is seen in the Knudsen number contour of the flow through CD nozzle shown in Fig. 9. When local variations in the characteristic length scales of the flow are taken into account using the gradient length scale of another physical variable, Knudsen number can be calculated more precisely [8]. This is shown in Eq. 8. Kn Q =
λ ∗ |∇ Q| Q
where Q can be any property, namely P, ρ, or T.
(8)
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Fig. 5 DSMC versus NS versus NS-slip Mach number Fig. 6 DSMC versus NS versus NS-slip non-dimentionalized pressure
Figure 10 shows modified Knudsen numbers based on gradient length scales of (a) pressure, (b) density, and (c) temperature. When the maximum of the three estimated values of Kn exceeds a value of 0.01 for the first time, which is the location of the separation plane, the continuum CFD technique is deemed to be incorrect. A sizeable portion of the diverging section of the CD nozzle is clearly in the non-continuum domain, as can be seen in Fig. 10. Kn based on pressure and density indicates that the continuum breaks just before the throat, which can be the location of separation
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Fig. 7 DSMC versus NS versus NS-slip non-dimensionalized temperature
Fig. 8 DSMC versus NS versus NS-slip density
Fig. 9 Knudsen number plot
plane. Therefore, we can conduct DSMC simulation from the location of separation plane (i.e., the throat). The convergent section of the micro-nozzle is in continuum regime in all three Knudsen number plots (pressure, density, and temperature), and the degree of rarefaction increases rapidly as the flow moves toward the exit, while the nozzle lip region is in transitional regime, as can be seen from the Knudsen
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Fig. 10 Kn number isocurve
number isocurves (Fig. 9). According to the Torre et al. [8] paper, the computation domain can be split into two sections, one where the Navier–Stokes technique can be used for regions with Kn numbers less than 0.01 and the other where the DSMC simulation is used for the remainder of the region. Accordingly the split is decided at the throat in our study.
5 Conclusions Analysis of flow through 2 µm nozzle was done for a back pressure of 10 kPa using Navier–Stokes (with slip and no-slip) and the DSMC method. Based on the comparison of the centerline properties, there is no appreciable difference in the properties measured by the three methods in the convergent section. Compared to the DSMC method, there is a divergence in the centerline Mach number assessed by Navier–Stokes with slip and no-slip as the flow approaches the nozzle exit. This is due to the rapid increase in the rarefaction effects toward the end of the divergent section of the nozzle. Also, when the slip boundary condition is applied, the maximum Mach number in the nozzle has increased from 1.34 to 1.41, and the subsonic boundary layer at the exit has decreased from 64.9 to 51.8%. The Knudsen number plot based on pressure, density, and temperature illustrate this. The Knudsen number isocurves are widely apart at the beginning of the divergent section. They are much closer to the nozzle’s exit, indicating an increase in the effects of rarefaction. Compared
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to the conventional Knudsen number, which is the ratio of the mean free path to characteristic length, the Knudsen number can be defined more precisely by taking into account local differences in the characteristic length scales of the flow using the gradient length scale of another physical quantity. This modified Knudsen number based on density and pressure reveals that the throat is where the separation plane is located, and the nozzle lip is in the transition region. On comparing centerline non-dimensionalized properties, it is observed that the differences between the three approaches are negligible in the convergent section. As a result, either NS with slip or no-slip can be employed in the convergent section, which is computationally less expensive compared to DSMC. Due to higher rarefaction effects and larger deviation in properties measured using NS with slip and no-slip, DSMC is preferred in the divergent section.
Nomenclature Kn λ ht Ld
Knudsen number (–) Mean free path (µm) Throat height (µm) Diverging length (µm)
References 1. Rafi KM, Deepu M, Rajesh G (2019) Effect of heat transfer and geometry on micro-thruster performance. Int J Therm Sci 146:106063 2. Li X, Yuan J, Ren X, Cai G (2022) Simulation applicability verification of various slip models in micro-nozzle. Acta Astronaut 192:68–76 3. White C, Borg MK, Scanlon TJ, Longshaw SM, John B, Emerson DR, Reese JM (2018) Dsmcfoam+: an openfoam based direct simulation Monte Carlo solver. Comput Phys Commun 224:22–43 4. Rafi K, Fahd B, Deepu M, Rajesh G (2017) Experimental and numerical studies on the plume structure of micro-nozzles operating at high-vacuum conditions. In: International symposium on shock waves. Springer, Berlin, pp 927–936 5. Louisos W, Hitt D (2007) Heat transfer & viscous effects in 2d & 3d supersonic micro-nozzle flows. In: 37th AIAA fluid dynamics conference and exhibit, p 3987 6. La Torre F, Kenjereš S, Moerel J-L, Kleijn C (2011) Hybrid simulations of rarefied supersonic gas flows in micro-nozzles. Comput Fluids 49(1):312–322 7. Liu M, Zhang X, Zhang G, Chen Y (2006) Study on micronozzle flow and propulsion performance using dsmc and continuum methods. Acta Mech Sin 22(5):409–416 8. Torre FL, Kenjeres S, Kleijn CR, Moerel J-L (2009) Evaluation of micronozzle performance through dsmc, Navier-stokes and coupled dsmc/Navier-stokes approaches. In: International conference on computational science. Springer, Berlin, pp 675–684
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9. Hao P-F, Ding Y-T, Yao Z-H, He F, Zhu K-Q (2005) Size effect on gas flow in micro nozzles. J Micromech Microeng 15(11):2069 10. Chung C-H, Kim SC, Stubbs RM, De Witt KJ (1995) Low-density nozzle flow by the direct simulation Monte Carlo and continuum methods. J Propul Power 11(1):64–70
Parametric Study on the Primitive Lattice Using the Pore-Scale Simulation to Characterize the Flow and Heat Transfer Performance Surendra Singh Rathore, Balkrishna Mehta, Pradeep Kumar, and Mohammad Asfer
Abstract In this study, a Primitive lattice based on Triply Periodic Minimal Surfaces (TPMS) is used to build a porous structure for performing a combination of pore-scale numerical simulation along with the porous media flow simulation. On the threedimensional lattice, numerical analysis is performed for single-phase fluid subjected to uniform heating at the walls. The void subdomain of the lattice is designated as the fluid zone, to perform the pore-scale numerical simulations; whereas, the solid subdomain of the lattice is designated as the microporous zone to perform porous transport simulations. The parametric studies for overall pressure drop and heat transfer coefficient are performed for a range of permeability of microporous zone. It is shown that when the micro-permeability is increased in the range 10−10 < Daμ < 10−5 , there is no significant change in pressure drop as well as the heat transfer coefficients. On the other hand, increase of micro-permeability in the range of 10−5 < Daμ < 10−1 causes a sharp drop in the pressure drop and a marginal drop in the effective heat transfer coefficient. Therefore, replacing the solid zone with porous zone for the solid subdomain of the lattice provides an improved thermo-mechanical performance for the mini-channel even in the low flow rate regime (Re = 10). Keywords TPMS Primitive lattice · Pore-scale simulation · Periodic boundary condition · Local thermal equilibrium
S. S. Rathore · B. Mehta (B) Department of Mechanical Engineering, Indian Institute of Technology Bhilai, Bhilai, Chhattisgarh 492015, India e-mail: [email protected] P. Kumar Numerical Experiment Laboratory (Radiation and Fluid Flow Physics), Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175075, India M. Asfer Department of Mechanical Engineering, College of Engineering, Shaqra University, Dawadmi Shaqra 11911, Saudi Arabia © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_39
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1 Introduction As fluid flows through a network of pores of varying size, shape, and orientation, it confronts varying levels of stagnation, detachment, recirculation, and attachment well within the pore-scale, which leads to increased heat mixing in porous media. Furthermore, a solid–fluid interface with an extraordinarily large specific surface area also improves thermal energy transfer between the solid matrix and saturated fluid. These two properties make porous medium an appealing option for small heat management systems, albeit at the expense of a significant pressure loss. Because of the large pressure drop, researchers are finding it challenging to implement alternate procedures that would provide increased heat transfer with reduced pressure drop. Due to the inherent contradiction between heat transfer improvement and pressure drop penalty, effective and innovative development of microchannel heat sinks is still necessary to fulfill the ever-increasing cooling requirements of microelectronic devices.
2 Literature Review and Objective A parametric study [1] performed on the square micro-channels inserted with different layouts of porous plugs in the laminar regime found heat transfer coefficients increased by 2–12 times. In one of the reviews, the benefits of using metal foam as porous medium in heat pipe applications are explained in details [2]. In the numerical work [3], the thermo-mechanical performance of a circular mini-channel with a metal foam annular porous plug with helical grooves resulted in TPF to be 1.21. The experimental and numerical studies [4] on rib configurations with localized heating and emphasized on using the dummy ribs for heat transfer enhancement. The experimental and analytical comparison on circular channels filled with two grain sizes, it was found that smaller grain packing increases pressure drop more than heat transfer [5]. Another numerical work [6] developed a wavy microchannel heat sink with porous walls and ascribed pressure loss to slip and permeation effects. A parametric study [7] performed on pin fins reveals the anisotropic nature of both permeability and thermal conductivity greatly affects the heat transfer rate. Another numerical work [8], the influence of porous plug configurations on the Nusselt number and observed enhancements based on flow, heating conditions, and porous media features. In most recent works [9, 10], using the ordered structures (such as TPMS lattices) in place of random porous structures are found to be resulting in enhanced thermosmechanical performances. While there are numerous studies that use different techniques to reduce pressure drop in porous media with improved heat transfer attributes, the use of Triply Periodic Minimal Surface (TPMS) lattice as the porous structure on the hydrodynamic and heat transfer performances is still in its early stages and requires further investigation.
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Fig. 1 Arrangement of Primitive lattice in the channel showing the flow and heating conditions
The objective of the present study is to numerically investigate the thermomechanical performance of a mini-channel, in which TPMS-based Primitive lattice is used as the porous insert. The schematic figure of the present case, showing the arrangement of Primitive lattice inside the square channel, boundary conditions, and treatment of two subdomains are shown in Fig. 1. The solid subdomain of this lattice is numerically assigned as a microporous zone, while the void subdomain as the fluid zone. The walls of square channel of size 4 mm are heated uniformly from all four sides with heat flux q '' = 10 KW/m2 , , while the inlet and outlet boundaries are provided with the translational periodic condition for a fixed value of mass flow rate (m˙ = 4 × 10−5 kg/s). The micro-scale porosity is kept constant at εμ = 0.395, and micro-scale permeability is varied in the range 10−16 to 10−7 m2 .
3 Computational Methodology 3.1 TPMS Lattice as the Porous Structure and Different Treatments of Two Subdomains Numerous attempts have been undertaken to construct porous structures, inspired by porous natural architectures. Due to its smooth surfaces, highly linked porous architectures, and mathematically predictable geometry, TPMS approach have emerged in recent years as an excellent method for creating porous structures. Therefore, in the present work, ordered porous structures are constructed by using one of the popular TPMS lattice, the Primitive lattice as shown in Fig. 1.
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The TPMS-based Primitive lattice originally created with 32% void fraction has two subdomains: solid and void. The void subdomain consists macro-scale porosity and is assigned as the fluid zone for pore-scale simulations. The solid subdomain, on the other hand, is numerically assigned as the microporous zone, and DarcyForchheimer porous transport simulations are implemented to it. Therefore, a combination of the pore-scale numerical simulation and the porous transport simulation is employed to handle the macro and micro-scale porous structures, respectively.
3.2 Governing Equations and Boundary Conditions The liquid water with constant material properties ( ) kg kg kJ W ρ f = 998.2 m3 ; μ = 0.001 m s ; c p f = 4.186 kg K ; k f = 0.6 m K is used in the present study as the working fluid. Moreover, for the conjugate heat transfer, steady state energy equation is solved for both subdomains, using constant thermal properties of copper material used for solid phase ( ) kg kJ W ( ρs = 8940 m3 ; c ps = 3.89 kg K ; ks = 387.6 m K ). Pore-scale simulations are performed on the void subdomain of the lattice. To achieve this, steady, incompressible, and three-dimensional Navier–Stokes equations are solved to calculate the momentum transport by the fluid in the void subdomain. Also, steady state energy equation is solved to calculate the heat transfer as given below. • Continuity equation ∇ · v→ = 0
(1)
)] [ ( ρ(→ v · ∇)→ v = −∇ p + ∇ · μ ∇ v→ + (∇ v→)T
(2)
( ) ρc p f (→ v · ∇)T f = −k f ∇ 2 T f
(2)
• Momentum equation
• Energy equation
Porous transport model simulations are performed on the solid subdomain, which is assigned as the microporous zone. Darcy-Forchheimer equations are solved to calculate the momentum transport and pressure drop; whereas, local thermal equilibrium (LTE) based energy equation is solved for the heat transfer calculations, as shown below.
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Table 1 Details of the boundary conditions Name of the boundary
Momentum equation
Inlet–outlet pair
u(x) = u(x + L); p(x) − p(x + L) =δ L
Walls (top, bottom, front and back)
u = 0; ∂p =0 ∂n
Energy equation q '' A ; mc ˙ pL T (x) − T (x + L) =γ L
γ =
Fluid zone walls −
∂T q '' = ∂n kf
Micro -porous zone walls −
∂T q '' = ∂n ke f f
• Momentum equation ρ μ ρβ v| (→ v · ∇)→ v = −∇ p − v→ − √ v→|→ ε2 K K
(4)
• Energy equation ε(ρc) f (→ v · ∇T ) = ∇ · (keff · ∇T ) where keff = εk f + (1 − ε)ks
(5)
In order to achieve the hydrodynamic and thermally developed flow conditions, the inlet–outlet boundary is provided with the translational periodic condition, with mass flow rate as m˙ = 4 × 10−5 kg/s (or Re = 10). All four walls of the square channel are provided with the no-slip boundary condition and the uniform heat flux as q '' = 10 KW/m2 . The details of boundary conditions are shown in Table 1.
3.3 Calculations of Hydrodynamic and Thermal Performance Parameters To compare the thermo-mechanical performance of the square channel inserted with the Primitive lattice for different values of micro-permeability, various hydrodynamic and heat transfer parameters are calculated. In the hydrodynamic study, macroscopic pressure drop across the channel length is calculated. In addition to this, overall permeability or its dimensionless form known as Darcy number is also calculated for all scenarios. In the Darcian regime, the dimensionless pressure drop and effective Darcy number are related as below:
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• Dimensionless pressure drop Δ P ∗ =
Δ P (μU/L)
(6)
1 Δ P ∗
(7)
• Darcy number Da =
In the heat transfer study, temperatures of the wall and bulk are necessary quantities for the calculations of different heat transfer coefficients. In all scenario, the wall and bulk temperatures are calculated as the area-weighted and mass-weighted average of temperatures for the corresponding regions. • Bulk mean temperature ∫ Tb =
Δ A
ρ f T |→ v |dA
∫
Δ A
(8)
v |dA ρ f |→
• Wall temperature 1 Tw = Δ A
∫ T dA
(9)
Δ A
Effective heat transfer coefficient is calculated for the channel for variation of the micro-permeability of the microporous zone, in order to calculate the augmentation in the heat transfer. Heat transfer coefficients are also individually calculated for void subdomain (fluid zone) and solid subdomain (microporous porous zone). • Effective heat transfer coefficient h eff =
qw'' h eff L , Nueff = Tw − Tb keff
(10)
• Solid phase heat transfer coefficient hs =
qw'' , Tws − Tbs
hs L keff
(11)
hfL qw'' , Nu f = Tw f − Tb f kf
(12)
N us =
• Fluid phase heat transfer coefficient hf =
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Table 2 Details of grids and results of grid independent test Mesh name No. of nodes Pressure % Change in Nusselt number % Change in gradient (Pa/m) pressure Nusselt number gradient Mesh 1
681
4.891
–
4.75
–
Mesh 2
3301
4.597
6.01
3.92
17.47
Mesh 3
47,540
4.572
0.54
3.32
15.30
Mesh 4
63,200
4.571
0.02
3.31
0.32
4 Results and Discussion 4.1 Grid Dependency Test For verification that the results of the numerical model should not depend on the grid employed in a non-porous channel, the sensitivity of the grid is tested using four distinct meshes. In order to resolve gradients with an appropriate level of precision, a non-uniform mesh is utilized in each case, with grids that are finer toward the edges than at the interior. All grids used here have maximum aspect ratio as 10 and minimum orthogonality as 0.4. The results of grid independent study are shown in the following table. It can be observed that while moving from Mesh 3 to Mesh 4, there is no significant improvement in the results, and therefore Mesh 3 is used for the study (Table 2).
4.2 Validation of the Computational Model The aforementioned solution methodology is validated for the flow and heat transfer in a mini-scale, non-porous, square channel with periodic boundary conditions between the inlet–outlet pair. For the hydrodynamic validation, the pressure gradients derived from the empirical results of Shah [11] as well as the present numerical result are displayed in Fig. 2a for various values of Reynolds number and mass flow rate, and they are determined to be an accurate match. The empirical results of the heat transfer of Shah and London [12] are used for validation of the present numerical result with an acceptable level of matching, as shown in Fig. 2b.
4.3 Hydrodynamic Results Figure 3 shows the flow trajectories through fluid and microporous zones. Out of ten values of micro-permeability used in the study, three values (Daμ =
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Fig. 2 Results of validation for a hydrodynamic study, and b heat transfer study, for different Reynolds number
10−10 , 10−3 , and 10−1 ) are picked up to illustrate their effect ( on flow trajecto) ries more clearly. For a very low value of micro-permeability Daμ = 10−10 , very small portion of flow is able to enter into the microporous region and flow is effectively passing through the void subdomain, creating a repetition of converging– diverging passage. A relatively high velocity is observed in this case at the throat (region. However, ) when the micro-permeability of the solid subdomain is increased Daμ = 10−3 , more portion of flow is able to enter into this region. This effect does also reduce the maximum velocity) inside the lattice. Finally, for the extremely high ( micro-permeability Daμ = 10−1 , , the flow gets to distributed uniformly in the void and solid subdomains, and creating an empty (non-porous) channel like flow field, where the lowest and highest velocities are observed at wall and core regions, respectively.
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Fig. 3 Flow trajectories inside the lattice and enlarged view in one unit-cell, for different values of micro-permeability shown for a Daμ = 10−10 , b Daμ = 10−3 , and c Daμ = 10−1
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Fig. 4 Effect of variation in the micro-permeability on effective Darcy number and dimensionless pressure drop of the channel
Figure 4 shows the effect of variation of micro-permeability of the solid subdomain (microporous region) on the overall pressure drop and the effective Darcy number of the channel. From Daμ = 10−10 to Daμ = 10−5 , there is no significant change in the pressure drop and it remain essentially high. However, after Daμ = 10−5 , a sharp reduction in the pressure drop value is observed, leading to 2-orders lower pressure drop in Daμ = 10−1 case. The reverse effect is observed in the effective Darcy number (Daeff ) of the channel, since in the Darcian regime, pressure drop is inversely related to the overall permeability of the channel. This effect is also clearly observable in the previous Fig. 3, where by the increase of Daμ , the flow trajectories are observed to becoming straighter and spreading throughout the lattice, contrary to low Daμ cases where the flow trajectories are constrained majorly inside the void subdomain.
4.4 Heat Transfer Results Figure 5 is showing the contours of temperature at the bottom wall and a plane parallel to the flow direction passing through the center. Similar to the pressure drop, from Daμ = 10−10 to Daμ = 10−3 , there is no significant change observed in the temperature distribution. In case of Daμ = 10−1 , however, a marginal change is seen.
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Fig. 5 Temperature contours on mid-plane and bottom wall of the channel for different values of micro-permeability shown for a Daμ = 10−10 , b Daμ = 10−3 , and c Daμ = 10−1
Figure 6 is showing the heat transfer performance using the Nusselt number for the solid and fluid phases, along with the effective heat transfer performance of the channel. Again, for the range of Daμ = 10−10 to Daμ = 10−5 , all three Nusselt numbers are invariant. After Daμ = 10−5 , the fluid and effective Nusselt numbers are observed to be increasing, due to the enhanced advection in the channel. However, the solid Nusselt number is found to be decreasing in this range which is due to the reduction in the dispersion effect in the microporous region by increment of its permeability.
4.5 Thermo-mechanical Performance The overall performance of the channel is measured by the Thermal Performance Factor (TPF) which calculates the relative increment to the heat transfer at the cost of relative decrement in the pressure drop. It is calculated using Eq. (13) shown
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Fig. 6 Effect of variation in the micro-permeability on fluid phase, solid phase, and effective Nusselt numbers
below. The reference values Δ P0∗ and Nu0 corresponds to the case, in which the solid subdomain is treated as solid zone itself (Daμ = 0). • Thermal Performance Factor, TPF =
Nu/Nu0 Δ P
∗
/Δ P0∗
1/ 3
(13)
Figure 7 is showing the pressure reduction factor, thermal augmentation factor, and TPF for different values of micro-permeability of solid subdomain. It is observed that effect of micro-permeability (or Daμ ) increases up to Daμ = 10−5 , , which has a very small effect on the pressure reduction as well as the thermal augmentation factors, since the pressure reduction factor (PRF) is near to 0.8 and thermal augmentation factor (TAF) is a little less than 1, in this range. However, a sharp change is observed in both PRF and TAF after the Daμ = 10−5 value, , as a result the TPF is observed to get increased up to 8 for extremely high micro-permeability (Daμ = 10−1 ). A large value of TPF is very desirable in case of mini-scale thermal management of electronic devices, therefore, present method of using microporous region in place of solid region turns out to be a beneficial approach.
5 Conclusions In the present work, parametric study is performed to evaluate the pressure drop and heat transfer performance of channel inserted with a TPMS-based Primitive lattice as the porous matrix. Steady and incompressible flow with constant thermophysical properties is considered for the fully developed situation. The solid subdomain of the
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Fig. 7 Effect of variation in the micro-permeability on the thermos-mechanical performance parameters
lattice is numerically treated as the microporous zone and its permeability is varied in the range of 10−10 < Daμ < 10−1 for a fixed value of micro-porosity. The results obtained in the parametric study are compared with the case of solid subdomain as solid zone (Daμ = 0). The major conclusions of this study are: (a) The drop in pressure is nearly invariant in the range of 10−10 < Daμ < 10−5 , thereafter it decreases at a sharp rate for higher micro-permeability scenario (10−5 < Daμ < 10−1 ). The reverse trend is observed in the overall permeability of the channel (measured using non-dimensional Daeff ). (b) The fluid phase and effective Nusselt numbers (Nu f andNueff ) are both invariant in the range 10−10 < Daμ < 10−5 and thereafter shows a marginal improvement in the range 10−5 < Daμ < 10−1 . . On the other hand, solid phase Nusselt number (Nus ) shows a marginal fall in its value at higher micro-permeability of the solid subdomain. (c) The thermo-mechanical performance results reveal that in the range 10−10 < Daμ < 10−5 , since the TPF is almost 1, there is no improvement in the performance by using the microporous zone in place of solid zone for the solid subdomain of the lattice. However, in the higher micro-permeability range 10−5 < Daμ < 10−1 , the TPF rises to a significantly higher values and using the porous zone for the solid subdomain in this range is highly beneficial.
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Nomenclature A cp Da h k K l L m˙ n Nu p q '' Re T u U v v→ w x y z
Area [m2 ] Specific heat capacity [J/kg-K] Darcy number (Da = K /L 2 ) Heat transfer coefficient [W/m2 -K] Thermal conductivity [W/m–K] Permeability [m2 ] Unit cell or pore size (l = L/4) [m] Channel width [m] Mass flow rate [kg/s] Normal distance from the surface [m] Nusselt number (h L/k) Pressure [Pa] Heat flux [W/m2 ] Channel size Reynolds no. (ρU L/μ) Temperature [K] Velocity in x direction [m/s] Average velocity in x direction [m/s] Velocity in y direction [m/s] Velocity vector [m/s] Velocity in z direction [m/s] X direction distance [m] Y direction distance [m] Z direction distance [m
Greek Letters β γ δ E μ ρ ∀
Inertial drag factor Periodic temperature gradient [K/m] Periodic pressure gradient [Pa/m] Porosity Dynamic viscosity of the fluid [kg/m-sec] Density of the fluid [kg/m3 ] Volume [m3
Subscripts b eff
Bulk quantity Effective property
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489
Fluid phase property Solid phase property Wall associated quantity Reference case
References 1. Moosavi R, Banihashemi M, Lin CX, Chuang PYA (2021) Combined effects of a microchannel with porous media and transverse vortex generators (TVG) on convective heat transfer performance. Int J Therm Sci 166:106961. https://doi.org/10.1016/j.ijthermalsci.2021.106961 2. Li Y, Gong L, Xu M, Joshi Y (2021) A review of thermo-hydraulic performance of metal foam and its application as heat sinks for electronics cooling. J Electron Packag 143(3). https://doi. org/10.1115/1.4048861 3. Ahmed HE, Fadhil OT, Salih WA (2019) Heat transfer and fluid flow characteristics of tubular channel partially filled with grooved metal foams. Int Commun Heat Mass Transfer 108:104336. https://doi.org/10.1016/j.icheatmasstransfer.2019.104336 4. Durgam S, Venkateshan SP, Sundararajan T (2018) A novel concept of discrete heat source array with dummy components cooled by forced convection in a vertical channel. Appl Therm Eng 129:979–994. https://doi.org/10.1016/j.applthermaleng.2017.10.061 5. Pastore N, Cherubini C, Rapti D, Giasi CI (2018) Experimental study of forced convection heat transport in porous media. Nonlinear Process Geophys 25(2):279–290. https://doi.org/10. 5194/npg-25-279-2018 6. Lu G, Zhao J, Lin L, Wang XD, Yan WM (2017) A new scheme for reducing pressure drop and thermal resistance simultaneously in microchannel heat sinks with wavy porous fins. Int J Heat Mass Transf 111:1071–1078. https://doi.org/10.1016/j.ijheatmasstransfer.2017.04.086 7. Kim SY, Koo JM, Kuznetsov AV (2001) Effect of anisotropy in permeability and effective thermal conductivity on thermal performance of an aluminum foam heat sink. Numer Heat Transfer Part A Appl 40(1):21–36. https://doi.org/10.1080/10407780121436 8. Khaled AR, Vafai K (2005) Heat transfer enhancement through control of thermal dispersion effects. Int J Heat Mass Transf 48(11):2172–2185. https://doi.org/10.1016/j.ijheatmasstransfer. 2004.12.035 9. Clarke DA, Dolamore F, Fee CJ, Galvosas P, Holland DJ (2021) Investigation of flow through triply periodic minimal surface-structured porous media using MRI and CFD. Chem Eng Sci 231:116264. https://doi.org/10.1016/j.ces.2020.116264 10. Rathore SS, Mehta B, Kumar P, Asfer M (2022) Flow characterization in triply periodic minimal surface (TPMS)-based porous geometries: Part 1—hydrodynamics. Transp Porous Media 1–33. https://doi.org/10.1007/s11242-022-01880-7 11. Shah RK (1978) A correlation for laminar hydrodynamic entry length solutions for circular and noncircular ducts. https://doi.org/10.1115/1.3448626 12. Shah RK, London AL (1978) Laminar flow forced convection in ducts, supplement 1 to advances in heat transfer. Academic, New York
Experimental and Numerical Studies on Liquid Bridge Stretching in Uni-port Lifted Hele-Shaw Cell for Spontaneous Fabrication of Well-Like Structures Makrand Rakshe, Sachin Kanhurkar, Amitabh Bhattacharya, and Prasanna Gandhi
Abstract The formation of liquid bridge between solid surfaces is of interest to fluid dynamics researchers due to the rich flow physics in the problem, as well as its various applications in engineering and biology. In this work, we propose a technique to spontaneously fabricate well-like structure by stretching the liquid bridge of highly viscous fluid in uni-port lifted Hele-Shaw cell (ULHSC). The numerical simulations are used to characterize optimal set of flow parameters for which wells are formed. Results from the simulations also yield insight into the flow physics governing micro-well formation. In the numerical simulation study, single hole is incorporated in the lower plate and upper plate is lifted with the uniform velocity. Parametric characterization of the system reveals that growth rate of the air bubble through the uni-port depends on the size of the hole diameter on a lower plate. Experiments are carried on ULHSC for two different holes diameters on bottom plate at same capillary number. Overall, the paper develops insights towards shaping of fluid in 3D well-like structures using ULHSC via numerical simulation and experimental studies. Keywords Liquid bridge · Lifted Hele-Shaw cell · Capillary number · 3D well-like structure · Spontaneous fabrication
M. Rakshe · S. Kanhurkar · P. Gandhi (B) Department of Mechanical Engineering, IIT Bombay, Powai 400076, India e-mail: [email protected] A. Bhattacharya Department of Applied Mechanics, IIT Delhi, New Delhi 110016, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_40
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1 Introduction Multiscale well-like structure and hollow tubes have a vast number of applications in engineering and biology. Micro-well systems are used for analysing individual cells by providing in vivo-like micro-environments. Analysis of a single cell with trapping is also possible with micro-well system [1, 2]. Various techniques have already been developed for micro-well fabrication like ice lithography [3], viscoelastic lithography [4]. Few researcher also used MEMS processes [5], 3D printing [6] approach for fabrication of well-like structures. Some mould replication techniques are also available for fabrication of well-like geometries. These available techniques have limitations in terms of scalability, cost effectiveness and time required for fabrication. In this paper, we study novel approaches to fabricate well-like structures where air fingers shape the liquid bridge into well-like structures. The formation of liquid bridge between solid surfaces is of interest to fluid dynamics researchers due to the rich flow physics in the problem, as well as its various applications in nature and industry. Certain reptiles and insects inject liquid and produce capillary liquid bridges, which enables them to stick to vertical walls via adhesive force. Sucking and injecting the fluid between toe pad and substrate eases walking over vertical objects for insects and animals [7, 8]. Detailed studies on static and dynamic bridge interface evolution have also been carried out to understand how to make printing of ink on paper more efficient. Depending on the flow parameters, liquid bridges can be analysed via onedimensional inviscid slice models as well as 1D viscous models [9–12]. Experimental results compare well with 1D viscous model for different Ohnesorge number. Here, μ and it characterizes the ratio of viscous Ohnesorge number is defined as Oh = (ρ Rσ )0.5 to the product of surface tension and inertial force. For low Oh number, the evolution of the bridge interface is mainly sensitive to the curvature of the meniscus [13]. At the start of stretching, a necking region with high curvature is formed towards the upper side of the liquid bridge. The capillary pressure is high in the necking region, due to its small radius of curvature. This leads to draining of liquid from the neck. For high Ohnesorge number, viscous forces become comparable to capillary forces, and provide resistance to the flow induced by gradients in capillary pressure. Therefore, at high Oh number, a long thin cylindrical film is formed with a high rupture length [14]. The primary motivation for doing these experiments and simulations is to develop a novel process of spontaneous fabrication of the well-like structure which has large scope in many fields in biology and engineering. In this paper, we presented numerical simulations and experimental results on stretching of liquid bridge between two surfaces, in which a hole is present on the bottom surface. The numerical simulations are used to characterize optimal set of flow parameters for which micro-wells are formed. Results from the simulations also yield insight into the flow physics governing well-like structure formation.
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Fig. 1 Schematic of liquid bridge stretching of radius R0 with hole of radius a0 in a bottom plate. The initial liquid bridge height is L 0 and the initial air bubble height is b0
2 Problem Specification Figure 1 shows a schematic of the flow set-up, in which a viscous liquid surrounded by air is being stretched between two plates. The top plate is moved upwards in axial direction with respect to the bottom fixed plate with a constant velocity. A hole is provided at the bottom plate to allow entry of air during plate separation. We consider the bridge to be axisymmetric with respect to the Z-axis. A cylindrical (R, Z) coordinate system is therefore used here, in which R is the radial coordinate. Initial dimensionless separation between plates is L 0 , dimensionless radius of the contact line of the bridge is R0 , and dimensionless radius of the hole in the bottom plate is a0 . In our numerical simulations, we initialize the curved elliptical interface of the bubble emerging from the hole with a non-dimensional height b0 with respect to the bottom plate. The instantaneous non-dimensional height of the liquid bridge and the non-dimensional bubble height along the Z-axis is h(τ ) and b(τ ), respectively, at a given non-dimensional time.
3 Numerical Methodology The liquid bridge is modelled via a diffuse interface level set method, in which the velocity field and level set function are evolved in time. Mass and momentum conservation equations are evolved for the velocity field, whereas the level set function is advected by the velocity field. A reinitialization equation is solved frequently to
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ensure that the level set function stays a normed distance function. All the equations have been solved in an axisymmetric domain. Surface tension force is applied at the interface via a discretized Dirac delta function supported over 2 grid cells around the interface. To simulate stretching of liquid bridge, the governing equations are mapped to a fixed domain via a timedependent stretching coordinate transformation along the vertical (Z) direction. The fluid properties are assumed to be a constant. A pinned contact line condition has been imposed at the three phase contact line.
4 Results and Discussion In this section, we discuss results for liquid bridge being stretched between two plates in the presence of a hole in the bottom plate. In the simulations, we fix capillary number, and we select liquid of viscosity μ = 1.129 Ns/m2 , density ρ = 1223 kg/m3 , liquid–gas surface tension is σ = 6.6 × 10−4 N/m, and gravity (g) = 0.001 m/s2 . Here, we are neglecting the effect of gravity, and the flow dynamics is governed by a balance between viscous, pressure and surface tension forces. We select air as the fluid surrounding the liquid bridge. In all our numerical experiments, the bottom plate is fixed and top plate is separated/lifted with velocity Wc = 0.6 cm/s. Initial bridge length is L 0 = 3.75 mm, radius of bridge is R0 = 6 mm and initial dimensionless bridge height is h ∗ = L 0 /R0 . The size of the hole a0 is changed systematically in the simulations.
4.1 Simulation Results In this section, we study growth of air bubble with respect to varying hole size, i.e. a0 ∈ {0.75, 1.5, 2.5}, keeping capillary number Ca = 10.26 fixed at a relatively high value. In Fig. 2, we plot the pressure contour, velocity vector field, streamlines and interface evolution for different values of a0 and at different time instances. We observe that, for any given time instance, with increasing a0 , the bubble size in Z-direction is larger (see Fig. 2 row-wise). As observed in Fig. 2, the range of pressure variation decreases with time. For the small and medium sized holes (a0 = 0.75, 1.5), a region with a large axial pressure gradient is present near the hole. The magnitude of pressure gradient reduces away from the hole. On the other hand, in the case of largest hole (a0 = 2.5), the bubble appears to influence the pressure gradient all the way to the top plate. The evolution of the outer interface is very similar for all the cases. In Fig. 3, we show the filled isocontours of vertical velocity field (W ), along with velocity vector field, streamlines and interface locations. Due to presence of the plates, W is almost always highest at the top plate and zero at the bottom plate (i.e. outside the hole). For smaller holes (a0 = 0.75), the vertical velocity almost
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Fig. 2 Numerically computed shape of the liquid bridge with the time (τ ) for the initial slenderness ratio h* = L 0 /R0 = 0.625, Ca = 10.26, dimensionless radius of the bubble/hole on the R-axis a0 ∈ ˜ = [0, 7.5] × [0, {0.75, 1.5, 2.5}, dimensionless height of bubble is 0.2, computational domain size 3.75], grid size is 54 × 54, contours shows the pressure distribution, red arrows show the velocity vectors and black lines shows the streamlines in the flow domain
increases monotonically with respect to height, except for the small regions near the bubble (see Fig. 3a–c); here the presence of hole does not appear to affect the velocity distribution away from the bubble very significantly. On the other hand, for larger hole sizes (a0 = 2.5), the air bubble influences the velocity field in the entire liquid (see Fig. 3g–i). It is also worth noting that the velocity of air entering the hole is higher for holes with larger radius. For the largest hole (a0 = 2.5), inlet velocity of air at τ = 0.02 is higher than the speed of the top plate. For the larger holes [Fig. 3(i)], the bubble shape appears to become conical at large time. The bubble interface also appears to be bounded by the streamline originating from the contact line (i.e. from R = a0 , Z = 0). The axial extensional strain (∂W / ∂Z > 0) induced by plate separation leads to convergence of radial flow (∂U/∂R < 0) above the bubble. The stream tube defined by the streamline originating from the contact line at the hole therefore becomes narrow at higher Z. Effectively, this leads to a more conical shape (rather than, say, a cylindrical shape) for the bubble, since it has to stay bounded within this stream tube.
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Fig. 3 Numerically computed shape of the liquid bridge with the time (τ ) for the initial slenderness ratio h∗ = L 0 /R0 = 0.625, Ca = 10.26, dimensionless radius of the bubble/hole on the R-axis a0 ˜ = ∈ {0.75, 1.5, 2.5}, dimensionless height of the bubble b0 is 0.2, computational domain size [0,7.5] × [0, 3.75], grid size is 54 × 54, contours shows the vertical axial velocity distribution, red arrows show the velocity vectors and black lines shows the streamlines in the flow domain
Based on Figs. 2 and 3, the bubbles exert their influence on the pressure and velocity field over a large domain in the liquid around the bubble for larger hole. On the other hand, the domain of influence around the bubble is much more localized for smaller hole. The dimensionless bubble height (b) has been plotted as a function of time (Fig. 4) for different a0 in Fig. 4, for Ca = 10.26 and h ∗ = 0.625. Consistent with the results discussed in Figs. 2 and 3, b increases with the time for all the hole radii, with faster growth observed for larger holes.
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Fig. 4 Dimensionless bubble height (b) on the Z-axis versus dimensionless time for a0 ∈ {1, 1.5, 1.75, 2.5}, b0 = 0.2, Ca = 10.26, h ∗ = 0.625
4.2 Experimental Results In this section, we present the experimental results of spontaneous formation of welllike structures in the stretched liquid bridge. Here, we performed experiments on the uni-port lifted Hele-Shaw cell (ULHSC) set-up, where liquid bridge between two acrylic plates stretched by moving one of the plates. Bottom acrylic plate (black colour plate) having single drilled hole at the middle of the plate. In this mechatronically controlled set-up, the two plates were separated by 4 mm/s velocity with help of motorized translational stage. Figures 4 and 5 capture the effect of the hole size on the growth of the air finger. As compared to 1 mm hole experiment (Fig. 5), growth of the air finger is more spontaneous in the 2 mm hole experiment (Fig. 6). In this way, experimental results of effect of the hole size on the penetration depth of the air finger in the liquid bridge shown in Figs. 5 and 6 are in-line with the numerical simulation results shown in Figs. 2 and 3, where at constant capillary number as hole size (a0 ) on the bottom plate increases then penetration depth of air finger in the liquid bridge increases.
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Fig. 5 Experimental result for stretching of liquid bridge for initial thickness of film (L 0 ) = 2 mm, separation velocity (Wc ) of plates = 4 mm/s, drill hole diameter in bottom plate is 1 mm, initial radius of liquid bridge (R0 ) = 6.25 mm, viscosity of liquid (μ) is 292.5 Ns/m2 liquid–air surface tension (σ ) is 0.02 N/m, Ca = μWc /σ = 58.5, h ∗ = L 0 /R0 = 0.32
5 Conclusions In this paper, we successfully demonstrated spontaneous formation of well-like structures in stretched liquid bridge on uni-port lifted Hele-Shaw cell. Parametric characterization of the system reveals that growth rate of the air bubble through the uni-port depends on the size of the hole diameter on a lower plate. Numerical simulation results clearly depict effect of the hole size on the growth rate of the air finger
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Fig. 6 Experimental result for stretching of liquid bridge for initial thickness of film (L 0 ) = 2 mm, separation velocity (Wc ) of plates = 4 mm/s, drill hole diameter in bottom plate is 2 mm, initial radius of liquid bridge (R0 ) = 6.25 mm, viscosity of liquid (μ) is 292.5 Ns/m2 , liquid–air surface tension (σ ) is 0.02 N/m, Ca = μWc /σ = 58.5, h ∗ = L 0 /R0 = 0.32
in the stretching liquid bridge. The same trend of air finger growth with different hole size is also captured experimentally. In this way, we developed the very elegant, time efficient way of well-like structure formation. Acknowledgements Authors like to acknowledge Mr. Raj Mashruwala for his generous donation for providing research facilities at ‘Suman Mashruwala Advanced Micro Engineering Laboratory’, Department of Mechanical Engineering, Indian Institute of Technology, Bombay. The work was supported by DST IMPRINT (Department of Science and Technology-IMPacting Research INnovation and Technology, Project Number 6722).
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Nomenclature Oh μ ρ σ h∗ τ a0 b0 L0 R0 Ca Wc
Ohnesorge number (–) Viscosity (Ns/m2 ) Density of fluid (kg/m3 ) Surface tension (N/m) Non-dimensional initial thickness of liquid film (–) Dimensionless time (–) Dimensionless bubble radius (–) Dimensionless bubble height (–) Dimensionless separation between plates (–) Dimensionless radius of the contact line of the bridge (–) Capillary number (–) Plate separation velocity (m/s)
References 1. Kim S-H, Lee GH, Park JY (2013) Microwell fabrication methods and applications for cellular studies. Biomed Eng Lett 3(3):131–137 2. Hosic S, Murthy SK, Koppes AN (2016) Microfluidic sample preparation for single cell analysis. Anal Chem 88(1):354–380 3. Park JY, Hwang CM, Lee S-H (2019) Ice-lithographic fabrication of concave microwells and a microfluidic network. Biomed Microdevices 11(1):129–133 4. Jeong GS, No DY, Lee J, Yoon J, Chung S, Lee SH (2016) Viscoelastic lithography for fabricating self-organizing soft micro-honeycomb structures with ultra-high aspect ratios. Nat Commun 7(1):1–9 5. Chen PC, Huang YY, Juang JL (2011) MEMS microwell and microcolumn arrays: novel methods for high-throughput cell-based assays. Lab Chip 11(21):3619–3625 6. Gregory C, Veeman M (2013) 3D-printed microwell arrays for Ciona microinjection and timelapse imaging. PLoS One 8(12):e82307 7. Federle W, Barnes WJP, Baumgartner W, Drechsler P, Smith JM (2006) Wet but not slippery: boundary friction in tree frog adhesive toe pads. J R Soc Interface 3(10):689–697 8. Persson BNJ (2007) Wet adhesion with application to tree frog adhesive toe pads and tires. J Phys Condens Matter 19(37):376110 9. Javier García F, Castellanos A (1994) One dimensional models for slender axisymmetric viscous liquid jets. Phys Fluids 6(8):2676–2689 10. Schulkes RMSM (1993) Nonlinear dynamics of liquid columns: a comparative study. Phys Fluids A 5(9):2121–2130 11. Javier García F, Castellanos A (1996) One-dimensional models for slender axisymmetric viscous liquid bridges. Phys Fluids 8(11):2837–2846 12. Mathues W, McIlroy C, Harlen OG, Clasen C (2015) Capillary breakup of suspensions near pinch-off. Phys Fluids 27(9):093301 13. Dodds S, Carvalho M, Kumar S (2011) Stretching liquid bridges with moving contact lines: the role of inertia. Phys Fluids 23(9):092101 14. Dodds S, da Silveira Carvalho M, Kumar S (2009) Stretching and slipping of liquid bridges near plates and cavities. Phys Fluids 21(9):092103
Numerical Investigation on Inertial Migration of Spherical Rigid Particle in the Entrance Region of a Microchannel K. K. Krishnaram and S. Kumar Ranjith
Abstract In this work, numerical investigation of inertial migration of neutrally buoyant rigid spherical particles in the entrance region of a microchannel is performed. Typically, inertial migration is examined for particles in a developed flow, in contrast, particles in developing flow are examined here. The entrance region having two-dimensional flow and fully developed region with the uni-directional flow is simulated with Re = 60 and a blockage ratio of 0.25. The particles are released from vertical positions of y/2H = 0.497 (center) and y/2H = 0.106 (near wall). It is observed that the particles released in the entrance region cover a longer longitudinal distance before reaching the equilibrium position. Indeed, the time taken for the particles to arrive at the zero-lift position in the developing flow is 23.8% and 7.8% more than that of a fully developed flow. Keywords Inertial migration · Entrance region · Poiseuille flow
1 Introduction Microfluidics has expanded rapidly throughout the past few years, which has empowered us to handle fluids and particles of size between ten to hundred micrometers [1]. Recently, microfluidics is widely used in the field of biomedical applications, owing to its capability to finely control the particle focusing and separation [2]. Inertial migration is one among the various methods for focusing particles in a microchannel, which solely depends on the size-dependent hydrodynamic effects [3].
K. K. Krishnaram (B) · S. Kumar Ranjith Micro/nanofluidics Research Laboratory, Department of Mechanical Engineering, College of Engineering Thiruvananthapuram, Thiruvananthapuram, Kerala 695016, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_41
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Firstly, Segre and Silberberg [4] discovered that randomly distributed particles in a tubular Poiseuille flow relocated to a stable equilibrium point near the wall due to the “tubular pinch effect”. They discovered that the particle migrates to a position at 0.6 times the radius of the tube, where the equilibrium position is reached when the net lift force equals zero. Eventually, theoretical studies revealed the existence of two competing forces, known as the wall-induced lift force and the shear gradientinduced lift force, behind this phenomenon [5–7]. The wall-induced lift force is generated by particle interaction with the wall and drives the particle toward the channel center, whereas the shear gradient-induced lift force is caused by the velocity gradient and it pushes the particle toward the wall [8]. Further, weak forces, like Saffman lift and Magnus lift forces exist on the particle. The slip velocity and shear create Saffman lift force and the direction of the force depends on whether the particle is lagging or leading the flow [9]. Magnus force is caused by particles spinning in viscous flows, and the direction is determined by the spin orientation [10]. The net lift force acting on the particle in plane Poiseuille flow is given by Asmolov ρU 2 D 4 [5], FL = Dm2 f c (Rec , xc ), where f c (Rec , xc ) is the coefficient of lift which is h a function of Reynolds number (ρUm Dh /μ) and particle position. Note that, for parallel walls, the hydraulic diameter Dh = 2H . Matas et al. [11] investigated the motion of rigid neutrally buoyant spherical particles in Poiseuille flow through the pipe at 67 ≤ Re ≤ 1700 and discovered that as Re increases, the narrow annular equilibrium position moved closer to the wall. Experimental results suggest that inertial migration in straight channels with rectangular and square cross sections differs significantly from circular tubes. Particles moved to two stable equilibrium locations along the lengthier side of the rectangular microchannel [12]. Liu et al. [13] observed that for high Reynolds numbers, two additional stable equilibrium positions have been identified near the shorter walls, causing particle migration to four equilibrium positions. Similarly, in square channels, eight equilibrium positions, including the channel face centers and corners, exist at higher Reynolds numbers [14]. However, at lower Reynolds numbers, particles in corner positions move toward the channel face centers [14]. Often, numerical techniques enable a thorough and simple understanding of the physics behind inertial migration. Numerous numerical techniques are employed, including the lattice Boltzmann method (LBM) [15], finite element method (FEM) [16], finite volume method (FVM) [17], and direct numerical simulation (DNS) [18] to examine the lateral shift of particles. All these investigations were focused on the particle placed in a developed flow. The lift force generated is due to the developed flow field. However, in the entrance region, the velocity profile is two-dimensional in a parallel microchannel. In this study, we employ a twodimensional numerical simulation using a finite element framework to investigate single particle migration in the entrance region of a flat Poiseuille flow. Here, the particle transport in the developing and developed flow is compared. The paper’s outline is provided below; Sect. 2 provides information on the numerical techniques employed. Section 3 explains the findings and related discussions. Finally, Sect. 4 summarizes the investigation and discusses the conclusions.
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2 Methodology Figure 1 shows the schematic representation of a particle placed in a 2D Poiseuille flow in the entrance zone. Both the developing and developed velocity profiles in a Poiseuille flow through infinite parallel plates are shown. The width (2H) and length (L) of the channel are chosen as 10 µm and 3000 µm, respectively. The Reynolds number for the flow is, Re = ρUmμ2H , where ρ, Um and μ are density, mean velocity, and viscosity of the fluid, respectively. The particle considered is rigid and neutrally buoyant, and has the same density as that of the fluid, ρ = 1178 kg/m3 , and has a blockage ratio of D/2H = 0.25, where D is the diameter of the particle. The fluid is assumed to be Newtonian, steady, incompressible, and two-dimensional. The governing equations solved are continuity and Navier–Stokes which are given by ∇ ·u =0
(1)
∇ · (ρuu) = ∇ P + μ ∇ 2 u
(2)
where ρ, u, and μ denote the density, velocity, and viscosity of the fluid, respectively. Meanwhile, particle tracing is governed by Newtonian formulation expressed by, d mp V = Ft dt
(3)
where m p , V and Ft are the mass of the particle, velocity, and total force. To capture the entrance hydrodynamics accurately, uniform flow is imposed at the channel inlet. The flow enters the channel section with a uniform velocity throughout the width. Due to the boundary layer development, the velocity profile eventually changes to a developed profile after a finite length. Next, the rigid particle is released in the
Fig. 1 Schematic diagram of a spherical particle placed in the entrance region of a hydrophilic microchannel
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Grid
Mesh elements
Central-line velocity (m/s)
G1
1256
1.178
G2
2180
1.179
G3
4280
1.8
G4
120,000
1.8
developed zone to understand the migration pattern. The walls are given a no-slip boundary condition. Finite element-based simulations are performed using Comsol Multiphysics™ 5.6 software. The fluid flow variables are separately calculated using the Eulerian grid method. With the aid of the Newtonian formulation, the particle trajectory is determined using this velocity and pressure field. The particle tracing and fluid flow in this case are connected through one-way coupling. The channel is given a structured rectangular mesh, and the results of the grid independence investigation are detailed in Table 1. The central-line velocity value converges with fewer meshes due to the simplicity of the channel design. For reliable findings, however, it is ensured that the mesh size is less than the particle size.
2.1 Validation To validate the numerical approach, the velocity profile of a fully developed flow through a hydrophilic parallel plate microchannel is compared with analytical results. The velocity profile of fully developed flow is given by [19] Eq. (4) 3 1 − η2 U (η) = 2
(4)
where U = u/U m , U m is the mean velocity and η = y/H is the non-dimensionalized vertical distance. Figure 2 shows the variation of velocity along the width of the microchannel with no-slip boundary conditions. The red dashed line represents the velocity profile obtained using Eq. (4). Due to no-slip boundary conditions, the velocity is zero at both walls and maximum velocity occurs at the channel center. The black colored dots represent the simulation result and it agrees well with the analytical result. Furthermore, the particle trajectory is obtained from simulations and is compared with the analytical model proposed by Ho and Leal [6]. Also, the result is compared with the experimental results of Tachibana [20] to confirm the accuracy of the numerical model. Also, the maximum error in the equilibrium position obtained in present study is 2.54% from the analytical [6] and experimental [20] results. The simulation was done with Re = 32.1 and a blockage ratio of 0.159. The numerical results have shown good agreement, with analytical results as shown in Fig. 3. Thus, the numerical scheme adopted is well capturing the Lagrangian–Eulerian problem of inertial migration.
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Fig. 2 Comparison of the fully developed velocity profile with analytical [19] results
Fig. 3 Comparison of particle trajectory with analytical [6] and experimental [20] results for Re = 32.1 and D/2H = 0.159
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3 Results and Discussion Using the Eulerian approach, the entrance hydrodynamics is obtained by solving mass and momentum equations. The fluid enters with a uniform velocity at the entrance and the boundary layer forms near the wall. As the flow proceeds, the boundary layer development completes and the flow is developed thereafter as shown in Fig. 4. Here, Fig. 4 velocity contours at the entrance region of microchannel at Re = 60. The maximum velocity occurs near the center and velocity gradually diminishes at the wall. The particles are inserted in the two-dimensional velocity field to study the migration. The particles are released from two different vertical locations y/2H = 0.106 and y/2H = 0.497 under-developed and developing flow conditions. Figure 5 compares particle trajectory during the inertial migration in the entrance and fully developed region for Re = 60 and a blockage ratio of 0.25. Figure 5a represents particle trajectory when released near the core of the channel, i.e., y/2H = 0.497 and Fig. 5b represents the trajectory when the particle is released near the wall of the channel, i.e., y/2H = 0.106. It is observed from Fig. 5 that, the particle travels a longer distance in the stream-wise direction in the entrance region than in the fully developed region prior to the lateral migration. The local velocity at the entrance zone depends on both the x and y directions, but it depends only on the y direction in the fully developed region. As a result, the velocity profiles in the entrance and fully formed regions are different, which in turn affects the shear gradient-induced lift force acting on the particle. The difference in the velocity profile causes a reduced shear gradient-induced lift force on the particle in the entrance region. Note that, at the entrance zone, the viscous forces are only present near the wall and the core moves with uniform velocity. Thereby a particle placed near the geometrical centerline experiences only the drag force in the flow direction. As a result, the particle gets dragged more along the longitudinal direction. However, for the fully developed region, the particle is placed in a velocity gradient, and subsequently greater shear gradient-induced lift is experienced on the particle. Thus, the particle in the fully developed region migrates to the lateral equilibrium position by traveling a shorter distance in the stream-wise direction. Figure 6 presents the particle migration trajectory as a function of nondimensionalized time (t*) for Re = 60 and (a) (b) blockage ratio 0.25. The trajectory of the particle released from an initial vertical position of y/2H = 0.497 and y/2H = 0.106 is shown in Fig. 6a and b, respectively. It is evident that the particle released in the entrance zone needs more time to migrate than the fully grown region, regardless of the initial positions. This is attributed to the difference in the velocity profile. The particle released at y/2H = 0.497 in the entrance zone migrates more slowly than the fully developed flow, adding 23.8% additional time. The particle released at y/2H = 0.106 in the entrance region also requires an additional 7.8% of the time. Furthermore, compared to the particle released close to the wall, the particle released at the core region requires more time. The particle travels a shorter distance to attain an equilibrium state when it is released close to the channel wall, and the lift force created by the wall promotes faster migration.
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Fig. 4 Velocity contours at the entrance region of microchannel at Re = 60
Next, we studied the effect of flow speed on inertial migration at the entrance zone. Here, Fig. 7 shows the effect of Reynolds number on the lateral migration of a particle in Poiseuille flow with a blockage ratio of 0.25. Simulations were conducted with three different Reynolds numbers and the particle trajectory is displayed in Fig. 7. It has been discovered that with higher Reynolds numbers, inertial migration occurs faster. This is because the velocity gradients are more at higher Reynolds numbers than at lower Reynolds numbers. Consequently, the particle encounters a greater shear gradient lift force and migrates more quickly, and moves to a shorter distance longitudinally to undergo migration.
4 Conclusions In this paper, the inertial migration of neutrally buoyant rigid spherical particles in the entrance zone of a microchannel is examined using the finite element approach. The numerical method is validated against benchmark problems in the literature. Numerical simulations are performed with Re = 60, blockage ratio = 0.25 and initial vertical position was at y/2H = 0.497 and y/2H = 0.106. Computations are performed for a two-dimensional entrance region and one dimensional fully developed region. The result shows that the entrance hydrodynamics significantly influences the lateral migration in the channel. Further, the particle in the entrance region has to travel a larger longitudinal distance to migrate in comparison with a fully developed scenario. Moreover, in comparison with the fully developed flow, the particle in the entrance region released at y/2H = 0.497 and y/2H = 0.106 requires an additional time of 23.8% and 7.8% respectively to migrate. Also, noted that the particle migration in the entrance region strongly depends on the Reynolds number. As Re increases, the particle migration becomes faster and travels a lesser longitudinal distance to reach the equilibrium position.
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Fig. 5 Particle trajectory for Re = 60 and D/2H = 0.25, released from a initial vertical position of y/2H = 0.497, b initial vertical position of y/2H = 0.106
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Fig. 6 Particle trajectory for Re = 60 and D/2H = 0.25 as a function of non-dimensionalized time, released from a initial vertical position of y/2H = 0.497, b initial vertical position of y/2H = 0.106
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Fig. 7 Particle trajectory as a function of Reynolds number for Re = 40, Re = 60, and Re = 80
Nomenclature Re D t* η Um Dh 2H L
Reynolds number (–) Particle diameter (µm) Non-dimensionalized time (–) Non-dimensionalized channel width (–) Mean velocity (m/s) Hydraulic diameter (µm) Channel width (µm) Channel length (µm)
References 1. Whitesides GM (2006) The origins and the future of microfluidics. Nature 442(7101):368–373 2. Di Carlo D (2009) Inertial microfluidics. Lab Chip 9(21):3038–3046 3. Bazaz SR, Mashhadian A, Ehsani A, Saha SC, Krüger T, Warkiani ME (2020) Computational inertial microfluidics: a review. Lab Chip 20(6):1023–1048 4. Segre G, Silberberg A (1962) Behaviour of macroscopic rigid spheres in Poiseuille flow part 2. Experimental results and interpretation. J Fluid Mech 14(1):136–157 5. Asmolov ES (1999) The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number. J Fluid Mech 381:63–87 6. Ho B, Leal L (1974) Inertial migration of rigid spheres in two-dimensional unidirectional flows. J Fluid Mech 65(2):365–400
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7. Schonberg JA, Hinch E (1989) Inertial migration of a sphere in Poiseuille flow. J Fluid Mech 203:517–524 8. Amini H, Lee W, di Carlo D (2014) Inertial microfluidic physics. Lab Chip 14(15):2739–2761 9. Saffman PG (1965) The lift on a small sphere in a slow shear flow. J Fluid Mech 22(2):385–400 10. Rubinow SI, Keller JB (1961) The transverse force on a spinning sphere moving in a viscous fluid. J Fluid Mech 11(3):447–459 11. Matas J-P, Morris JF, Guazzelli E (2004) Inertial migration of rigid spherical particles in Poiseuille flow. J Fluid Mech 515:171–195 12. Bhagat AAS, Kuntaegowdanahalli SS, Papautsky I (2008) Enhanced particle filtration in straight microchannels using shear-modulated inertial migration. Phys Fluids 20(10):101702 13. Liu C, Hu G, Jiang X, Sun J (2015) Inertial focusing of spherical particles in rectangular microchannels over a wide range of Reynolds numbers. Lab Chip 15(4):1168–1177 14. Miura K, Itano T, Sugihara-Seki M (2014) Inertial migration of neutrally Buoyant spheres in a pressure-driven flow through square channels. J Fluid Mech 749:320–330 15. Liu W, Wu C-Y (2019) Analysis of inertial migration of neutrally buoyant particle suspensions in a planar Poiseuille flow with a coupled lattice Boltzmann method-discrete element method. Phys Fluids 31(6):063301 16. Feng J, Hu HH, Joseph DD (1994) Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2. Couette and poiseuille flows. J Fluid Mech 277:271– 301 17. Wang Q, Yuan D, Li W (2017) Analysis of hydrodynamic mechanism on particles focusing in micro-channel flows. Micromachines 8(7):197 18. Zeng L, Balachandar S, Fischer P (2005) Wall-induced forces on a rigid sphere at finite Reynolds number. J Fluid Mech 536:1–25 19. Vijay S, Cletus J, Mg A, Kumar RS (2021) Analytical modelling of laminar developing flow between hydrophobic surfaces with different slip-velocities. J Fluids Eng 144(4):041301. https://doi.org/10.1115/1.4053251 20. Tachibana M (1973) On the behaviour of a sphere in the laminar tube flows. Rheol Acta 12(1):58–69
Dynamics of Electrically Actuated Carreau Fluid Flow in a Surface-Modulated Microchannel Subhajyoti Sahoo and Ameeya Kumar Nayak
Abstract The flow and mass transfer characteristics of an electroosmotic flow of Carreau fluid through a surface-modulated microchannel are presented. Flow is considered to be influenced by the coupling effect of both the externally applied electric field and heterogeneous zeta potential. The constitutive rheological behavior of the Carreau fluid is incorporated with the Nernst–Planck-Navier–Stokes-based flow governing equations to obtain the flow variation and species distribution depending on the behavior of the fluid properties. Moreover, the variation of electric doublelayer thickness and the zeta potential provides a complex flow phenomenon, which can be beneficial for mixing enhancement. The results indicate the practical application of electroosmotic flow over heterogeneous patched surfaces observed in the lab-on-a-chip device used in Bio-MEMS and can handle inhomogeneous surface conduction and geometric interfacial disorder. Keywords Electrokinetic flow · Carreau fluid · Micro-mixing · Heterogeneous zeta potential
1 Introduction In recent years, flow in micro- and nano-scale devices has been a new frontier area of research in science that includes physical chemistry, medicine, the manufacturing industry, electronics, and life science [1–4]. Numerous potential micro- and nanoscale fluid and ion transport research applications are observed in the biological and chemical industries [5]. Several factors, such as DNA hybridization for cytometry analysis, chemical treatment of biological species, emulsification of incompatible components, micro-reactions, and dispersion of gas phase bubbles or solid particles in liquid streams, make the flow mechanism a crucial problem in small-scale devices [6]. To execute tasks such as chemical synthesis, biological detection, singlecell analysis, and micro-scale processing, lab-on-a-chip (LOC) micro-devices have S. Sahoo (B) · A. K. Nayak Department of Mathematics, IIT Roorkee, Roorkee 247667, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_42
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been developed. These LOC micro-devices are based on micro- and nano-scale flow characteristics. LOC devices have shrunk over the last two decades, and the importance of microfluidic devices has grown in biomedical engineering [7]. In addition, microfluidic devices have arisen as essential uses of microfabrication technology in the biomedical and biochemical sciences due to their ability to handle micro-amounts of biological fluids fast and precisely [8]. Flow augmentation and species transfer in microfluidic devices are the most challenging task because the transport mechanism involves a slow diffusion process incorporating the weak inertial effect. Thus, mixing in such devices is primarily influenced by the diffusion process due to the low Reynolds number. This procedure is generally prolonged; hence, an adequate channel length is required to enhance the mixing. Moreover, the surface characteristics of the channel also significantly impact the mixing patterns of electroosmotic fluxes. Researchers considered microvortices forming due to surface heterogeneity governed by various patterned surfaces or complex surface-charge distribution to study non-uniform flow in micro/nanofluidic systems. Moreover, different channel geometry and the altered patch of surface charges are used to mix efficiently within the chosen channel length [9]. In such situations, electroosmotic flow (EOF) strongly correlates with surface roughness, mainly when the mean surface roughness is on the same scale as the thickness of the electric double layer. Experimental results showed that the geometric modulation of the microchannel wall could result in faster laminar mixing by increasing the interfacial area between the liquids to be mixed. The fluid in most of the microfluidic devices is non-Newtonian in nature. Researchers applied several models to describe the constitutive behavior of nonNewtonian fluids, including the power-law model [10, 11], the Carreau-Yasuda model, the Casson model, and the Mold flow second-order model [12, 13]. These non-Newtonian models are used to understand fluid rheology, especially in the blood flow phenomena, where an increment in the shear stress reduces the viscosity, demonstrating the interactive role of RBC [14]. Experimentally, it is shown that blood viscosity follows the Carreau fluid model [15]. This paper considers the nonNewtonian fluid as the transportation medium, which follows the Carreau model to describe the viscosity. The current study investigates the EOF transport impact of the Carreau fluid model in a modulated surfaced micro-channel by considering the effect of inertial forces due to surface heterogeneity. The current analysis addresses the variation of EOF parameters, such as the thickness of the Debye layer, externally applied electric field, and flow behavior index, which primarily influence the mixing.
2 Problem Formulation The EOF of a binary electrolyte is considered in a surface-modulated micro-channel of height H , length L, and width W . The channel is filled with an incompressible non-Newtonian electrolyte of constant permittivity ∈e, and the flow is laminar. It
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is assumed that O(H ) ≡ O(L) >> O(W ), i.e., the flow can be considered as two-dimensional. The modulated portion of the microchannel is positioned transversely along the wall, as represented in Fig. 1. The mathematical expression for the configuration of modulated walls is given by, yu = A + α[sin(2π x − δ) + cos(4π x − 2δ)]
(1)
yl = −A − α[sin(2π x − δ) + cos(4π x − 2δ)]
(2)
and
where α denotes the wave amplitude, δ denotes the phase shift. The external electric field E 0 is imposed along the axial direction of the micro-channel by setting the electrodes at both ends. The Carreau fluid model is employed to explain the rheological behavior of the non-Newtonian electrolyte. The constitutive relationship between the rate of the strain tensor and the shear stress tensor can be defined as: ( ) τ ∗ = 2μa∗ γ˙ ∗ γ˙ ∗ ,
(3)
√ 1 where τ ∗ and γ˙ = γ˙ ∗ : γ˙ ∗ ) denotes the stress tensor and strain tensor, 2( respectively. The apparent viscosity term )[ ( )2 ] n−1 ( 2 μa∗ (γ˙ ) = μ∗∞ + μ∗0 − μ∗∞ 1 + λ∗ γ˙ ∗ ,
(4)
where μ∗0 and μ∗∞ are the viscosities, zero and infinite shear rates, respectively. The fluid behavior index n indicates the shear thinning fluid for n < 1 and the shear thickening fluid for n > 1. The condition for Newtonian fluid is obtained for λ∗ = 0 or n = 1 or μ∗0 = μ∗∞ .
E0
H
L
Fig. 1 Schematic diagram of the microchannel
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3 Governing Equations Electroosmotic flow characteristics are obtained numerically by solving the Nernst– Planck, Poisson, and Navier–Stokes equations. A. Electric Potential Equation: The dimensional electric field potential Φ∗ is of the form, Φ∗ (x, y) = φ ∗ (x, y) + ψ ∗ (x, y)
(5)
where φ ∗ and ψ ∗ represent the induced and external electric potential, respectively. − → The electric field E ∗ satisfies Maxwell’s equation is given by, ( − ∗ →) → · ∇Φ → ∗ = ρe → · ∈e E ∗ = −∇ ∇ ∈e
(6)
∑ where ρe∗ = z i en i is the net charge density per unit volume and ∈e represents the permittivity of the medium which is defined as ∈e = ∈0 ∈r , where ∈0 is the dielectric constant of the solution and ∈r is the permittivity of the vacuum. The external and induced potentials are scaled by the same parameter φ0 (= RT /F), where R, T, and F, respectively, represent the gas constant, absolute temperature, and Faraday’s constant. Then, the dimensionless form of the potential equation is expressed as ∇2φ =
(κ H )2 ρe 2
(7)
and ∇ 2 ψ = 0.
(8)
B. Nernst–Planck Equation: The ion transport equation is governed by advection, diffusion, and electro-migration mechanisms is given by the Nernst–Plank equation as, ∂n i∗ → · Ni = 0 +∇ ∂t ∗
(9)
→ − → → i∗ + n i∗ ωi z i F − Here, Ni = −Di ∇n E ∗ + n i∗ q ∗ , where n i∗ represents the molar concentration of ith ionic species. Di , z i, and ωi are the diffusion coefficient, ionic valence, and mobility of ith species. To obtain the Nernst–Planck equation in dimensionless form, the time t ∗ is non-dimensionalized by the scale H/UHs , the ith species of ionic concentration n i∗ is scaled by n 0 (bulk ion concentration), the electric field E ∗ is scaled by E 0 . The generalized Helmholtz–Smoluchowski velocity UHS scales
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− → the velocity vector q ∗ . The non-dimensional form of the Nernst–Planck equation is given by, ρe (q · ∇n i ) = ∇ 2 n i + z i n i · ∇(φ + ψ) −
(κ H )2 z i n i ρe 2
(10)
C. Momentum Equation: The flow field, which is governed by the Cauchy momentum equation along with the continuity equation, is, [ − ] → )− − → →∗ →∗ ∂ q ∗ (− ρ + q · ∇ q = −∇ p ∗ + ∇ · τ ∗ + ρe E ∗ ∂t ∗ − → ∇ · q∗ = 0
(11) (12)
− → where q ∗ = (u ∗ , v ∗ ) is the fluid velocity vector, ρ is the fluid density, p ∗ is the pressure, and τ ∗ is the stress tensor. The governing equations in non-dimensional can be expressed as ] ∂q (κ H )n+! + (q · ∇q) = −∇ p + ∇ · τ + Re ρe ∇(φ + ψ) ∂t 2Δζ n n
(13)
∇ ·q =0
(14)
[
D. Boundary Conditions: A no-slip-based condition is assumed at the channel wall, i.e., u = 0 = v. Both the channel ends are considered as fully developed, i.e., ∂∂ux = 0 and ∂∂vx = 0. The walls are considered impermeable and adiabatic, i.e., (∇n i + z i n i ∇φ).nˆ = 0. At i =0 both ends of the channel domain, the boundary conditions are assumed to be ∂n ∂x (symmetrical). Here, nˆ denotes the unit normal vector indicating along the flow ( ) direction. The walls are imposed with positive zeta potential patches φ = ζ p in the modulated region to create vortices and backflow, where negative zeta (φ = ζ ) potential is used in the remaining portions. The boundary condition, ∂ψ = 0 along ∂x the channel inlet, outlet, and wall. E. Species Transport Equation The two different concentration streams are assumed to pass in the electrolyte solution to enhance the mixing activity in the micro-system. The controlled transportation equation includes both the convection and diffusion factors, whereas the numerical modeling approach excludes the species absorption and the chemical reaction of the eluted species. The dimensionless form of the transport equation (eluted species) is given by
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(q · ∇)C −
1 2 ∇ C =0 Pes
(15)
where C(= c/c R ), Pes (= HUHS /Ds ) and Ds represent the species concentration, the Peclet number, and the diffusion coefficient, respectively. At the channel wall, no mass flux condition is assumed, i.e., (∇C).n = 0. The boundary conditions at the problem domain’s inlet are given by, ( C=
1, in the lower half 0, in the upper half
(16)
The mixing efficiency at different positions of the channel is defined by a parameter σ (x) as, [ σ (x) = 1 −
∫ upper surface
|C
|dy
− C∞ lower surface ∫ upper surface |C0 − C∞ |dy lower surface
] × 100
(17)
Here C∞ and C0, respectively, represent the concentration at a completely mixed state at a fully unmixed state. C∞ = 0.5 C0, = 0 or 1. For the case of a fully mixed case, C = C∞, = 0.5, this indicates σ (x) = 100, and C = C0 indicates the case of a fully unmixed state, i.e., σ (x) = 0. The deflected geometry in (x y-domain) is transformed in a suitable way to a rectangular domain (X Y -domain) as X = x, Y =
y − yl y − yl = yu − yl F
(18)
Here F = yu − yl . The gradient operator ∇ and the Laplacian operator ∇ 2 in the transformed domain are defined as ( ) ) ( yl' ∂ 1 ∂ F ∂ − Y '+ , , (19) ∇≡ ∂X F F ∂Y F ∂Y [ ( ) ] ) 2 ( '' yl' yl F ' yl' ∂ ∂ F' 2 ∂2 F' F '' 2 + + 2Y − +2 2 ∇ ≡ −2 Y −Y ∂ X2 F F ∂ X ∂Y F F F F ∂Y ] [( ) 2 y2 ∂2 F' 1 Y + l + 2 . (20) + F F F ∂Y 2
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4 Numerical Methods A finite volume method-based staggered grid approach is used to solve the nonlinear coupled PDEs for the transport equation of flow field, ionic species transport, electric potential, and uncharged species transport. Integrating over respective control volumes yields the discretized form of the algebraic equations. The Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme is implemented to discretize the electroosmotic migration and the convective terms at each control volume. In contrast, a second-order central difference scheme is implemented to discretize the diffusion term. The transient terms are discretized using an implicit firstorder scheme. Newton’s linearization technique is used to deal with nonlinear terms. The resulting linear algebraic equations for the flow governing equation are then solved using the Semi-Implicit Method for the Pressure-Linked Equation (SIMPLE) method. The pressure link between continuity and momentum equations is achieved by converting the discretized continuity equation to the Poisson equation for correction of pressure term, which is solved in an iterative manner using the Successive over-relaxation (SOR) technique until the desired accuracy is obtained. The criteria for the convergence are given by, | | sup|Θ p+1 − Θ p | < Δ where Θ = (φ, ψ, f, g, u, v), Δ = 10−6 and p symbolize the number of iteration levels.
5 Model Validation We compared our numerically simulated results with the pre-existing analytical solution given by Zhao et al. [16] for different values of the Debye–Hückel parameter (κ H ). The present simulated results are in excellent agreement with the existing solution. Figure 2 represents a comparison of the velocity profile in the axial direction for flow behavior index n = 1, different Debye–Hückel parameters κ H = 3, 5, 10, 30 with the externally imposed electric field as 104 V/m for the case of fully developed electroosmotic flow. It has been shown that our results fully agreed with the results provided by Zhao et al. [16].
6 Result Discussion The physical parameters considered for the simulation are: the microchannel height is 10 µm, and the external electric field strength is 104 V/m. The temperature of the channel is chosen at 300 K. The dielectric constant of electrolyte is ∈e = 80∈0 ,
520 Fig. 2 Comparison between the present result (lines) with the analytical result (symbols) obtained by Zhao et al. [16], for κ H = 3, 5, 10, and 30 at the limiting case n = 1
S. Sahoo and A. K. Nayak 1
0.8
u
0.6
0.4
κΗ=3,5,10,30 n=1
0.2
0 0
0.2
0.4
y
0.6
0.8
1
where ∈0 = 8.85 × 10−12 F/m is the dielectric constant. The diffusion coefficient is considered equal, i.e., D1 = D2 = 2 × 10−9 m2 /s. Diffusivity of transport species is considered D S = 1 × 10−11 m2 /s. The variation of electroosmotic fluid flow rate with respect to EDL thickness for different fluid behavior index (n) is shown in Fig. 3. It is noticed that the fluid flow rate increases with the increment of the Debye Hückel parameter κ H for all the cases n, and the physics behind this may be described as for the case of overlapped EDL, the drag force which is responsible for the transport the fluid gets decreases thus for lower values of the Debye Hückel parameter, the fluid gets a longer retention time to flow in the microchannel; in other words, the micro-mixing will be better for the thick EDL case. Figure 4a, b represents the velocity vector profile for homogeneous wall potential and heterogeneous patches (ζ p = 1), respectively, where κ H = 10 and the fluid behavior index n = 1.4. For homogeneous wall potential, an ideal EOF profile is observed in both inlet and outlet of the channel, whereas the fluid profile gets slightly disturbed inside the modulated surface region. In contrast, the second case (Fig. 4b) velocity profile is impacted by the effect of a potential patch over the modulated region. Backward flow profile and vortices are observed closer to the modulated surface due to the overpotential patch region attached to the microchannel surface. Figure 5a–c shows the effect of the Carreau fluid index on concentration mixing when the external electric field is 104 V/m. These figures show that concentration streams are significantly larger in pseudoplastic fluids than in dilatant fluids. Further, Newtonian fluids mix faster in the modulated regions than dilatant fluids, whereas pseudoplastic fluids exhibit the opposite behavior. For higher fluid behavior indices,
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1
Uavg
0.8
0.6
n=0.6 n=0.8 n=1.0 n=1.2 n=1.4
0.4
10
20
κH
30
40
50
( ) Fig. 3 Average velocity profile U avg with respect to EDL thicknesses (κ H) for different flow behavior index (n)
Fig. 4 Velocity vector profile for a homogeneous zeta potential, b heterogeneous zeta potential where n = 1.4 and κ H = 10
the species concentrations are almost entirely mixed. The fluid behavior index significantly affects mixing because the thicker interface layer makes fluids interact slowly. The diffusion flux is more significant as the interfacial contact area is more prominent for n > 1, resulting in a more excellent mixing of species at the outlet.
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Fig. 5 Electroosmotic species mixing contour plot for a n = 0.6, b n = 1, c n = 1.4 when ζ p = 1.0 and κ H = 10
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100
n=1.4 80
n=1.2 n=1
60
n=0.8
40
n=0.6
20
0
0
2
x/H
4
Fig. 6 Distribution of mixing efficiency along the channel length for different flow behavior index (n), where κ H = 10, ζ = −1.0 and ζ p = 1.0
Figure 6 represents the mixing efficiency along the longitudinal direction of the microchannel. The mixing efficiency is proportional to the flow behavior index (n). We observed that apparent flow viscosity helps the flow by viscous forces and consequently increases the retention time, improving mixing within the microchannel. Figure 7 depicts the heterogeneous wall potential dependence on mixing efficiency. It is noticed that the downstream mixing efficiency curve follows a propor(tional ) relation with the flow behavior index (n) and the heterogeneous wall potential ζ p patch. For a significant potential heterogeneity, a bulk flow gets disturbed more; hence, the contact area between different streams and the holding time increases, resulting in efficient downstream mixing.
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Downstream mixing efficiency(%)
Fig. 7 Variation of downstream mixing efficiency as a function of flow behavior index (n) with κ H = 10.
80
60
40
ζp=2.0 ζp=1.5 ζp=1.0
20
0 0.6
0.8
1
1.2
1.4
1.6
Carreau fluid behaviour index(n)
7 Conclusion In this work, the mixing efficiency in a surface-modulated microchannel with variation of surface potential is analyzed for a two-dimensional electroosmotic flow. The wall potential heterogeneity changes due to the interfacial electrochemistry, resulting in several recirculation zones (vortices) near the channel wall resulting in more retention time and interfacial area. Moreover, the vortices formed by modulated surfaces due to increased interfacial contact zone between two streams improve the diffusion flux. The vortex size is expanded with an increase in the flow behavior index, causing a decrease in the flow rate to achieve adequate mixing. The concept described in this study can be helpful in the design of microfluidic devices to achieve excellent mixing results of biofluids at a decent flow rate. Acknowledgements The authors acknowledge the support from Science and Engineering Research Board (SERB) India, under project grant no. SER-1479-MTD during the preparation of this manuscript.
Nomenclature C Di Ds e
Eluted species (dimensionless) C ∗ /C R ith ionic species diffusivity [m2 s− 1 ] Eluted specie’s diffusivity [m2 s−1 ] Charge of an electron or proton C
Dynamics of Electrically Actuated Carreau Fluid Flow …
E0 F H L n n0 Pe Pes q Re Sc UHS u, v zi
525
External electric field [V m− 1 ] Faraday’s constant [C mol− 1 ] Channel height [m] Length of the channel [m] Fluid behavior index Ion density at the bulk [ions m− 1 ] Peclet number UHS H/Di Peclet number(uncharged species) UHS H/Ds Fluid velocity [ms− 1 ] Reynolds number HUHS /ν Schmidt number ν/Di Helmholtz-Smoluchowski velocity Components of fluid velocity ith ionic species valence
Greek ∈ γ˙ κ λ Δ μa φ ψ ρ ρe
Permittivity of the solution [CV−1 m−1 ] Rate of strain tensor γ˙ /(UHS /H ) Inverse of the Debye length [m− 1 ] Thickness of the EDL [m] Scaled applied electric field E 0 h/Φ0 Scaled apparent viscosity μa∗ /m(UHS H )n Induced electric potential V External electric potential V Density (fluid) [kg m− 3 ] Density (net charge) [kg m− 3 ]
Subscripts ∗
Dimensional
References 1. Wang X, Cheng C, Wang S, Liu S (2009) Electroosmotic pumps and their applications in microfluidic systems. Microfluid Nanofluid 6:145–162 2. Gorkin R, Park J, Siegrist J, Amasia M, Lee BS, Park JM, Cho YK (2010) Centrifugal microfluidics for biomedical applications. Lab Chip 10(14):1758–1773
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3. Meisel I, Ehrhard P (2006) Electrically-excited (electroosmotic) flows in microchannels for mixing applications. Eur J Mech-B/Fluids 25(4):491–504 4. Chen Y, Lv Z, Wei Y, Li J (2022) Mixing performance of the induced charge electro-osmosis micromixer with conductive chamber edges for viscoelastic fluid. Phys Fluids 34(8):083110 5. Villone MM, D’avino G, Hulsen MA, Greco F, Maffettone PL (2013) Particle motion in square channel flow of a viscoelastic liquid: migration vs. secondary flows. J Nonnewton Fluid Mech 195:1–8 6. Haque A, Nayak AK, Weigand B (2021) Bivariant species mixing and pressure drop within a hybrid periodic modulated microslit. Phys Fluids 33(10):102002 7. Stone HA, Stroock AD, Ajdari A (2004) Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu Rev Fluid Mech 36:381–411 8. Cavero I, Guillon JM, Holzgrefe HH (2019) Human organotypic bioconstructs from organ-onchip devices for human-predictive biological insights on drug candidates. Expert Opin Drug Saf 18(8):651–677 9. Wätzig H, Kaupp S, Graf M (2003) Inner surface properties of capillaries for electrophoresis. TrAC, Trends Anal Chem 22(9):588–604 10. Banerjee A, Nayak AK (2019) Influence of varying zeta potential on non-Newtonian flow mixing in a wavy patterned microchannel. J Nonnewton Fluid Mech 269:17–27 11. Haque A, Nayak AK, Bhattacharyya S (2021) Numerical study on ion transport and electroconvective mixing of power-law fluid in a heterogeneous micro-constrained channel. Phys Fluids 33(12):122014 12. Ghosh U, Chakraborty S (2015) Electroosmosis of viscoelastic fluids over charge modulated surfaces in narrow confinements. Phys Fluids 27(6):062004 13. Mehta SK, Pati S, Baranyi L. Heat transfer and fluid flow analysis for electroosmotic flow of Carreau fluid through a wavy microchannel considering steric effect 14. Rana A, Westein E, Niego BE, Hagemeyer CE (2019) Shear-dependent platelet aggregation: mechanisms and therapeutic opportunities. Front Cardiovasc Med 6:141 15. Johnston BM, Johnston PR, Corney S, Kilpatrick D (2004) Non-Newtonian blood flow in human right coronary arteries: steady state simulations. J Biomech 37(5):709–720 16. Zhao C, Zholkovskij E, Masliyah JH, Yang C (2008) Analysis of electroosmotic flow of powerlaw fluids in a slit microchannel. J Colloid Interface Sci 326(2):503–510
Heat Transfer Analysis of Peltier-Based Thermocycler for a Microfluidic-PCR Chip Nikhil Prasad, B. Indulakshmi, R. Rahul, and Ranjith S. Kumar
Abstract Polymeric chain reaction (PCR) is one of the commonly used technologies for the exponential amplification of a pathogen’s DNA or RNA for quick detection. A common PCR procedure includes three major steps: denaturation (90–97 °C), annealing (50–60 °C), and extension (68–72 °C). This process is often repeated in 25–30 cycles, and it may take up to 5–6 h for successful amplification of the target DNA. The implementation of PCR in a microfluidic platform would reduce the reaction time drastically as it may consume only a few microliters of samples and reagents. Also, it increases the sensitivity and capability of point-of-care applications. A thermal analysis of a microfluidic device fabricated using thermoplastic (PMMA) is presented in this paper. The heating and cooling of the sample are done using two thermoelectric modules. The numerical analyses of the system are carried out for comparing the performance of thermoplastic materials during the thermal ramping process. Keywords Microfluidics · Point-of-care · PCR · Peltier thermocycling · Heat transfer
1 Introduction Early pathogen detection is essential for managing and stopping the spread of a pandemic like COVID-19. There are various methods to detect a viral pathogen. One of the most prominent methods is the nucleic acid amplification test (NAAT). Out of the different NAAT methods available, the polymeric chain reaction (PCR) nucleic amplification process is the most popular technique for amplifying the pathogen DNA segments [1]. N. Prasad (B) · B. Indulakshmi · R. Rahul · R. S. Kumar Micro/nanofluidics Research Laboratory, Department of Mechanical Engineering, College of Engineering Thiruvananthapuram, Thiruvananthapuram, Kerala 695016, India e-mail: [email protected] N. Prasad APJ Abdul Kalam Technological University, Thiruvananthapuram, Kerala 695016, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_43
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A conventional PCR includes the following stages: (a) denaturation (95 °C), in which DNA that is double-stranded gets split into a single-strand (b) annealing (50 °C) involves the addition of primers, which are short pieces of template DNA whose activity is increased to produce single-strand DNA (c) extension stage (68 °C), where a full strand is extended to create two separate DNA strands. This cycle is repeated 25–30 times to accomplish the sufficient amplification of nucleic acid [1–3]. Due to the higher thermal capacity of the PCR materials, it usually takes 5–6 h to complete the process. With the introduction of the microfluidics system, the lab-on-a-chip concept has gained a lot of attraction in biomedical research [4]. The major highlight is the requirement of minimal sample volume which enables a high heat transfer rate due to the reduction in the thermal capacity of the material used. Hence, a microfluidicbased PCR device has much potential for more rapid amplification and detection [5, 6]. In the case of manufacturing a microfluidic device, thermoplastics have become an essential tool for creating microdevices at the industrial level due to their benefits such as low cost and quick bonding techniques. Poly methyl methacrylate (PMMA) is one of the best thermoplastics suitable for a microfluidic device, due to its low cost, biocompatibility, and optical transparency. But its low glass transition temperature (105 °C) tends other thermos plastics such as polycarbonates (PC) to be used for the same purpose. Sometimes, a combination of both is also used for better sealing and integrity at higher temperatures [7]. Generally, in a PCR process, the sample and reagents mixture are collected in the heating area and the thermal cycling procedure is carried out. Various methods [8] are available for heating the channel during denaturation and extension cycles. In this paper, a pair of Peltier elements are used to provide the necessary thermal ramping. The Peltiers are arranged on both sides of the chip to alternately heat and cool so that the thermal cycling process could be effectively carried out. Once the cycling process is completed, the samples are either taken out of the device for the detection or on-chip detection methods are employed. The schematic layout of the arrangement is shown in Fig. 1.
Fig. 1 Schematic of the PCR thermocycler assembly
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2 Literature Review Over the past two decades, significant progress has been achieved in the development of microfluidic PCR equipment. A summary of recent advancements in polymerase chain reaction (PCR) devices based on microfluidics for the detection of nucleic acid biomarkers was presented by Gorgannezhad et al. in their article [1]. They explore various microfluidic PCR device types, including spatial, continuous-flow, and transient PCR devices. This work considers a Peltier-based heating module, and the main concern is the thermal modelling of the module for numerical and analytical examination. Montecucco et al. [9] explored the one-dimensional transient heat conduction equation with internal heat generation. This concept can be used for several Peltier devices as well as thermoelectric generators (TEG). A novel, cost-effective PCR heating cycler method without the usual heating block with Peltier plates was discussed by Nasser et al. in [10]. Yang et al. [5] also studied polycarbonate-based Peltier-based PCR microreactors and verified their rapid performance, efficiency, detection sensitivity, and specificity in amplification. Considering the benefits and drawbacks of various materials, the choice of material is also very important. Sun et al. [7] introduce a new hybrid PMMA-PC microchip that contains a microchannel created by CO2 laser ablation and constructed by bonding a PC cover plate to a PMMA substrate. The thermal effectiveness of heat transfer is also very important because the PCR procedure requires accurate thermal cycling. In this paper, the microreactors are considered as rectangular channels in which the sample fluid undergoes thermal ramping and cyclising. In [11], Moharana and Lee [12] explored the heat transfer in microchannels and addressed the effect of aspect ratio in heat transmission. To the best of the authors’ knowledge, not much research has been done on thermocycler analysis of PMMA microchips for PCR applications.
3 Methodology In this section, the experimental determination of the heat flux from Peltier for numerical simulations and the computational methodology is discussed.
3.1 Determination of Peltier Heat Flux The heat flux from the Peltier device must be determined for the system’s thermal modelling. To make this determination, an experiment is performed. A 25 mm × 25 mm × 2 mm square block of the acrylic piece is kept on the top of the heating side of a Peltier (Fig. 2), and a voltage of 12 V and current of 3 A is provided at its terminals. A Peltier module (model TEC1-12706) is used in this experimental
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set-up. The technical specifications of the Peltier module are given in Table 1. The dimensions of the heater element are 40 mm × 40 mm × 3.8 mm. The temperature at the top surface of the PMMA is recorded for the first 130 s. According to Fourier’s law, heat conduction is mathematically stated as Q = −k AdT /dx
(1)
or Q/A = q = −k
dT dx
(2)
where the heat flux Q/A or q is a vector quantity which flows along the normal to the decreasing temperature. For a body which is assumed to be isotropic and homogeneous, the net heat flow Q will increase the internal energy of the volume element and the temperature increases from T1 to T2 in a given time (dt). The rate of internal energy accumulated within a volume element is given by: dE/dt = VρcdT /dt
(3)
where V is the volume of the solid, ρ is the density, c is the specific heat, dT is the temperature change, and dt is the time interval. By the energy balance on the volume
Fig. 2 Schematic of the experimental set-up for calculating the heat flux from the Peltier module
Table 1 TEC1-12706 technical specifications [13]
Hot side temperature (°C)
25
50
Qmax (W)
50
57
ΔT max (°C)
66
75
I max (A)
6.4
6.4
V max (V)
14.4
16.4
Module resistance (Ω)
1.98
2.30
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Fig. 3 Comparison of temperature development from experiment and numerical simulation
element, the rate of energy storage within the solid is equal to the difference between the rate of heat influx and the rate of heat efflux. The net heat flux required to raise the temperature (dT ) of a body in dt time is given by: q=
V dT ρc A dt
(4)
The heat flux q is estimated to be 2301.7 W/m2 based on Eq. (4). The cooling heat flux is assumed to have a value of −2301.7 W/m2 because the rate of heat transfer in a Peltier element is theoretically the same for both heating and cooling. A complementing numerical simulation is performed to validate the aspects of this experiment. A computational domain of a solid block of the same dimension (25 mm × 25 mm × 2 mm) with properties of PMMA is used for this purpose. The estimated heat flux is provided as a constant in the heat flux boundary condition. The transient temperature response at the top surface was estimated and compared with the experimental results (Fig. 3).
3.2 Numerical Model for Thermal Cycling The numerical study presented in this paper was done using ANSYS FLUENT version 2020 R1. The energy transport equation is ∂ v ρh) = ∇ · (k∇T ) (ρh) + ∇ · (→ ∂t
(5)
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where ρ is the density, h is the sensible enthalpy, k is the conductivity, and T is temperature. The numerical simulation is carried out with the assumption that the outer boundaries of this system are in an insulated condition and there is no convective heat transfer from the surface to the environment. In this simulation, the geometry represents a section of the microchip, in which the microchannel was engraved on a PMMA acrylic sheet (Body 2) and then sealed with another acrylic plate (Body 1) using thermal bonding [14] as shown in Fig. 4a. Body 1 and Body 2 are of dimensions 25 mm × 1 mm × 2 mm and the width and depth of the fluid domain are varied from 250 to1000 µm and 250 to 500 µm, respectively. Body 1 and Body 2 are considered as solid domains with the material properties listed in Table 2. A heat flux boundary condition required for the heating and cooling of the wall is implemented on either side of the body, which in turn heats up or cools the fluid in the channel. The system is heated until the fluid reaches an average temperature of 95 °C (denaturation stage). The temperature is then maintained at 95 °C for 30 s. After 30 s, the cooling side is provided with negative heat flux, bringing down the fluid’s average temperature to 50 °C (annealing stage). Following a 60 s of the cooling process, the fluid is heated once more until it reaches a temperature of 65 °C (extension stage). The fluid is then ultimately heated to 95 °C after 60 s and thus completes one cycle. The solver uses a user-defined function for inputting time-varying heat flux for the heating and cooling process. The volume average temperature at the fluid domain is monitored and plotted against time to achieve the required thermal ramping and cycling.
Fig. 4 a Schematic of the computational domain, b domain discretization
Heat Transfer Analysis of Peltier-Based Thermocycler … Table 2 Properties of materials
Table 3 Grid independency-the change of channel temperature with element number
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Property
PMMA
PC
Density (kg/m3 )
1190
1200
Thermal conductivity (W/m K)
0.17–0.19
0.19–0.22
Specific heat (J/kg K)
1400–1500
~ 1200
Elements
Nodes
4100 7500
Temperature (K)
5514
364.489
9597
429.606
11,774
14,576
429.608
17,688
21,339
429.612
The progress of heat transfer is indicated by the heat transfer index η=
Qt Q max
(6)
where Q t is the total energy transferred from the wall to the liquid from the initial time (t = 0) to the final time (t) and the term Q max represents the maximum amount of heat that can be transmitted.
3.3 Grid Independence Study An investigation of the grid’s independence from each element was done to investigate how element number impacts the numerical outcomes. For comparison, the temperature at the channel’s centre after 200 s of heating is used. Table 3 displays how the outcome changed depending on the grid’s element count. The table gives that the simulation began with an element number of 4100 and 5514 nodes and that there was not much variance observed after 7500 elements and 9597 nodes.
3.4 Validation A validation procedure using the benchmark results is carried out to guarantee that the numerical procedure accurately captures the heat transfer phenomena. A laminar convective heat transfer in the rectangular microchannel’s entry section [11] and [12] with constant heat flux at the walls is simulated. The variation of the Nusselt number along the channel length is observed compared with benchmark results.
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The Nusselt number is defined as Nu =
h DH kc
(7)
where DH is the hydraulic diameter and h is the heat transfer coefficient given as h=
(
Q
A Tw − T
)
(8)
where T is the fluid’s mean temperature specified by ∫ ρcu x T dV T = ∫ ρcu x dV
(9)
and Q is the heat flux on the walls. ρ and c indicate the fluid’s density and specific heat, respectively [15, 16]. Only the fluid domain with a constant fluid velocity (Fig. 5a) is taken into consideration to simplify the 3D conjugate heat transfer problem for the sake of validation. The aspect ratio (AR) of the channel is kept at 1. A continuous heat flux of 50 W/ cm2 is given over the four channel walls, while the Reynolds number is maintained at 100. The local Nusselt number is calculated as a function of channel length and plotted against z ∗ where z ∗ = z/(DH Re Pr ), where Pr is the Prandtl number which is given by Pr = v/ε. in which v is the momentum and ε is the thermal diffusivity. The results are compared with the results of Lee and Garimella [12] and Chandrupatla and Sastri [17] and are in good agreement as shown in Fig. 5b.
4 Results and Discuss In this section, results obtained from the simulations are discussed in detail.
4.1 Temperature–Time Response of PMMA Microchip in One Complete Cycle An investigation of the thermal response of PMMA materials during the thermal cycling process of a PCR application using two Peltier devices has been carried out. The response of temperature with change in time at various parts of the chip was discussed in the following sections:
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Fig. 5 a Computational domain of the microchannel in the current study for validation, b comparison of the current study with Lee and Garimella [12] and Chandrupatla and Sastri [17] for AR = 1
4.1.1
Temperature Response at Walls
To simulate the thermal ramping in a PCR process, the heat flux from both the Peltiers (heating and cooling) are systematically turned on and off resulting in time-varying pulses of heat fluxes at the heating side and cooling side of the microchip substrate. This is achieved by writing a user-defined code for the wall boundary conditions. The code observes the average fluid temperature in the channel and varies the heating and cooling fluxes accordingly as shown in Fig. 6a so that the fluid domain temperature
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follows the required thermal process in PCR. The variation of wall temperature on the heating and cooling side during a complete cycle is shown in Fig. 6b.
Fig. 6 a Variation of wall heat flux with time at heating and cooling side, b variation of temperature at the surface of heating and cooling side
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Fig. 7 Temperature–time response of PMMA material during the thermal ramping process
Table 4 Time taken for reaching PCR temperature zones
4.1.2
Temperature (C)
Time (s)
95
205
50
378
68
493
95
633
Temperature Response at Fluid Domain
Figure 7 shows the temporal variation of the temperature of the fluid with respect to the user-defined heat flux at the boundary wall. The time taken for the fluid in the microchannel to achieve several design thermal points in one cycle of a PCR amplification process from the initial time (t = 0) is observed and listed in Table 4.
4.2 Effect of Materials on Temperature Time Response in One Complete Cycle The effect of various materials used to manufacture microfluidic devices during the thermal cycling process is also investigated. Similar to PMMA material, the temperature time response of a microchip fabricated using polycarbonate (PC) is also carried out. Additionally, a chip with a combination of both materials is also analysed and compared. Figure 8 shows the comparison of the temperature response of PMMA, PC and the combination of PMMA and PC with respect to variation over time. The results show that PC is more thermally efficient and ramps through different temperature points more quickly than PMMA. However, using polycarbonate instead
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Fig. 8 Comparison of the temperature response of microchip materials (PMMA, PC, and combination of PMMA and PC)
of PMMA has other constraints such as high cost and difficulty in manufacturing. Therefore, combining PMMA and PC would provide the benefits of both materials.
4.3 Heat Transfer in Microchannel The process of heat transmission in a rectangular microchannel with continuous heat flux provided at the base of a solid domain is depicted in Fig. 9b. Only the heating phase of the PCR process is taken into account for the analysis and quantification of the heat transfer process. The bottom wall of the channel is primarily responsible for transferring heat to the fluid domain. Heat is quickly transported to the top layer of the chip because the water in the microchannel has a high thermal conductivity. Conduction from the side walls also affects the thermal variation in the fluid domain (Fig. 9b). The flow of heat from the solid material to the fluid domain is characterised by a dimensionless number, the heat transfer index (η) which is plotted in Fig. 9a and shows the variation of η for PMMA material with a channel aspect ratio of 1. The heat transfer index, which is zero initially, will gradually approach unity as the fluid temperature reaches the wall temperature of the material. The impact of heat transport on material change is also examined. As previously mentioned, PC has the highest rate of heat transfer as compared to PMMA. The heat transfer index variation for PMMA and PC is shown in Fig. 10. It is evident from the figure that PC has an efficient heat transfer index. To analyse the effect of aspect ratio in heat transfer, the width (W ) and the depth (D) of the channel are varied from the range of 250 µm to 1000 µm and 250 µm to 500 µm, respectively. The ratios considered are 0.25, 0.5, 1, and 2. There is no
Heat Transfer Analysis of Peltier-Based Thermocycler … Fig. 9 a Variation of η for PMMA with AR 1, b local temperature distribution across the channel and PMMA microchip with AR 1 channel cross-section after t = 175 s. Max temp is 357.348 K and min temp is 355.584 K
Fig. 10 Comparison of heat transfer index (η) obtained for PMMA and PC
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Fig. 11 Variation of heat transfer index with different aspect ratios
considerable change in the heat transfer index as shown in Fig. 11 but the maximum value for η is for AR equal to 2 minimum for AR equal to 1.
5 Conclusions In this study, a thermal analysis of a PCR-based microfluidic chip was numerically investigated. A thermal module using a pair of Peltier elements for nucleic acid amplification in a PCR process is numerically modelled and studied. The heat transfer rate through the Peltier systems was experimentally estimated and implemented in the numerical model. A user-defined functions were developed to systematically vary the heating and cooling effect on the chip so that the target thermal ramping process was achieved. The main purpose of this study was to estimate the time taken for reaching each step in a single cycle of the PCR process so that the total time for n number of cycles could be estimated. The effect of various materials and the dimensional characteristics of the channel such as accept ratio on the rate of heating and cooling was also investigated and compared. The effectiveness of the heat transfer was quantified using a dimensionless number called the heat transfer index. Based on the evaluation and results, the study indicates that PC material was thermally more efficient than PMMA. But due to other design limitations such as high cost and difficulty in fabrication, a combination of PMMA and PC was suggested. Acknowledgements The authors sincerely acknowledge the All India Council for Technical Education (AICTE), Government of India for providing funding under the RPS scheme (No. 864/FDC/RPS(Policy-1)/2021-22). The authors also thank the Kerala State Council for Science, Technology and Environment (KSCSTE) for supporting this research (No.33/WSD-BLP/2022-23/ KSCSTE).
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Nomenclature A c h k P V T Tw T ε u η Nu Dh Re Pr
Area [m2 ] Specific heat [kJ/kg K] Heat transfer coefficient [W/m2 K] Thermal conductivity [W/m K] Density [kg/m3 ] Volume [m3 ] Temperature [K] Wall temperature [K] Mean fluid temperature [K] Thermal diffusivity [m2 /s] Flow velocity [m/s] Heat transfer index [–] Nusselt number [–] Hydraulic diameter [m] Reynolds number [–] Prandtl number [–]
References 1. Gorgannezhad L, Stratton H, Nguyen N-T (2019) Microfluidic-based nucleic acid amplification systems in microbiology. Micromachines 10(6):408 2. Kulkarni MB, Goel S (2020) Advances in continuous-flow based microfluidic pcr devices—a review. Eng Res Express 2(4):042001 3. Li Z, Li Y, Sekine S, Xi H, Amano A, Zhang D, Yamaguchi Y (2019) Design and fabrication of portable continuous flow PCR microfluidic chip for DNA replication. Biomed Microdevices 22(1):5 4. Team TE (2021) Microfluidic PCR & QPCR. Elveflow 5. Yang J, Liu Y, Rauch CB, Stevens RL, Liu RH, Lenigk R, Grodzinski P (2002) High sensitivity PCR assay in plastic micro reactors. Lab Chip 2(4):179–187 6. Indulakshmi B, Prasad N, Kumar RS (2022) Performance analysis of a phase changing material based thermocycler for nucleic acid amplification. J Therm Sci Eng Appl 1–33 7. Sun Y, Satyanarayan MVD, Nguyen NT, Kwok YC (2008) Continuous flow polymerase chain reaction using a hybrid PMMA-PC microchip with improved heat tolerance. Sens Actuators B Chem 130(2):836–841 8. Miralles V, Huerre A, Malloggi F, Jullien M-C (2013) A review of heating and temperature control in microfluidic systems: techniques and applications. Diagnostics 3(1):33–67 9. Montecucco A, Buckle JR, Knox AR (2012) Solution to the 1-D unsteady heat conduction equation with internal joule heat generation for thermoelectric devices. Appl Therm Eng 35:177–184 10. Nasser GA, Abdel-Mawgood AL, Abouelsoud AA, Mohamed H, Umezu S, El-Bab AMF (2021) New cost effective design of PCR heating cycler system using Peltier plate without the conventional heating block. J Mech Sci Technol 35:3259–3268 11. Moharana MK, Khandekar S (2013) Effect of aspect ratio of rectangular microchannels on the axial back conduction in its solid substrate. Int J Microscale Nanoscale Therm Fluid Transp Phenom 4(3/4):211
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12. Lee P-S, Garimella SV (2006) Thermally developing flow and heat transfer in rectangular microchannels of different aspect ratios. Int J Heat Mass Transf 49(17–18):3060–3067 13. Al-Rubaye A, Al-Farhany K, Al-Chlaihawi K (2018) Performance of a portable thermoelectric water cooling system. Int J Mech Eng Technol 9(8):277–285 14. Arjun A, Ajith RR, Kumar Ranjith S (2020) Mixing characterization of binary-coalesced droplets in microchannels using deep neural network. Biomicrofluidics 14(3):034111 15. Arun MG, Dilip D, Ranjith SK (2021) Effect of interface curvature on isothermal heat transfer in a hydrophobic microchannel with transverse ribs and cavities. Int J Therm Sci 167:107014 16. Che Z, Wong TN, Nguyen N-T, Yang C (2015) Three dimensional features of convective heat transfer in droplet-based microchannel heat sinks. Int J Heat Mass Transf 86:455–464 17. Chandrupatla AR, Sastri VMK (1977) Laminar forced convection heat transfer of a nonNewtonian fluid in a square duct. Int J Heat Mass Transf 20(12):1315–1324
Effect of Viscosity on the Margination of White Blood Cells in an Inertial Flow Microfluidic Channel Dhiren Mohapatra, Rahul Purwar, and Amit Agrawal
Abstract White Blood Cells (WBCs) play a significant role in defending and eliminating infection-causing external elements and are indicators of several diseases, such as inflammatory diseases, cancer, and allergies. Here, microfluidic devices have an upper edge in the accurate and fast processing of the sample, which is needed for speedy disease prediction. In specific, passive microdevices are valuable as they are simple, cheap, and effective compared to the current inefficient conventional methods. Here we have analysed the effect of viscosity on the margination of WBCs in inertial flow in a rectangular channel with a bifurcation. We added known concentrations of viscoelastic fluid to vary the viscosity of the sample. WBC margination was analysed from the values of WBC enrichment with respect to viscosities. Keywords WBC · Margination · Viscoelastic · Microdevice · Passive · Hydrodynamic
1 Introduction Conventional methods for cell separation from human blood, like density gradient centrifugation, FACS, and MACS are inefficient due to significant numbers of cell loss, the requirement of skilled human resources, and more working space. So, a passive microfluidic device is suitable for these processes as it can reduce effort and cost. In a passive device, the focusing line or plane of cells can be manipulated by the biophysical properties of human blood components and flow properties. Out of those, viscosity, hydraulic diameter, and velocity play a significant role. Lift forces responsible for the cell margination and focusing depend on these parameters significantly. D. Mohapatra (B) · A. Agrawal Department of Mechanical Engineering, IIT Bombay, Powai, Mumbai 400076, India e-mail: [email protected] R. Purwar Department of Biosciences and Bioengineering, IIT Bombay, Powai, Mumbai 400076, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_44
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Inertial focusing uses inertial forces to focus specific cells to a single streamline based on their size. Irrespective of the channel cross-section of microdevices, the inertial effect on the particles can’t be ignored, as it used to be earlier. Assuming the flow as Stokes flow may give the result, but it is far from reality as the inertial effect and viscosity play a significant role in particle/cell movements. There are two types of migration: Lateral migration (normal to the flow direction) caused due to the inertial effects and migration in the direction of flow caused due to viscous drag [1]. Due to the parabolic nature of Poiseuille flow inside the channel, flow velocity at the centre is maximum and reduces symmetrically towards the walls. This difference in velocity produces a shear gradient, the source of lift force (shear-induced lift force) acting on the particles, which drives them toward the walls. But when particles reach the wall, they are repelled by another lift force, wall-induced lift force, caused due to the wake formation of the rotating particles moving towards the wall. This wallinduced lift force is dominant near the walls, whereas the shear-induced lift force is dominant at the centre. So, the particles move to an equilibrium position where these forces are balanced [2]. The net lift force can be calculated by [3] FL = CL (Re, x/ h).
ρU 2 a 4 DH2
(1)
where U is the maximum velocity, DH is the hydraulic diameter, a is the particle diameter, and C L is a non-dimensional lift coefficient whose value depends on the Reynolds number of the channel and the specific position of the particle in the cross-section. In addition to this, using a viscoelastic fluid can enhance the viscosity and exert an extra lift force, Elastic force. This force contributes to an increase in the further margination of the WBCs, which can be assumed as [4] Fel = −2Cel d 3 ηp λ∇ γ˙ 2
(2)
Here in this study, we analysed the effect of viscosity by keeping the other parameters responsible for WBC margination constant.
2 Materials and Methods This section presents the design, fabrication, and detailed sample preparation procedures. The channel considered for this study is of dimension 40 µm height × 80 µm width. A bifurcation is at the end of the focusing channel to separate the WBCs from the main streamline. The height of the microchannels remains the same throughout. The detailed fabrication and experimental procedures are described in the subsections of this section.
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Fig. 1 Fabrication process schematics
2.1 Microfluidic Device Design and Fabrication The channel mould consists of the microchannel required for the microfluidic device, which is replicated from the photomask. The channel is designed using AutoCAD. First, the geometry is transferred from the photomask to the silicon wafer. Following this, RCA cleaning is performed for the removal of oxides and ionic and organic contaminations. Then, a thin-layer silicon oxide layer is generated to force the oxidizing agent to diffuse into the silicon wafer for better bonding between the photoresist SU-8 and the wafer. For the fabrication of SU-8 patterns, the standard photolithography process was followed. For the channel on the PDMS device, the curing agent was added to PDMS with a ratio of 10:1 (PDMS: Curing Agent). The mixture was vigorously steered, ensuring uniform mixing. Then, vacuum degassing was done to remove the bubbles, followed by pouring the mixture onto the mould. It was kept in a furnace at 65 °C temperature for 40 min for baking. Peeling off the PDMS from the mould and punching can be done now on the PDMS device to make inlet and outlet ports. We used Plasma Cleaner and bonder to activate the surfaces of PDMS-glass and bond them. A graphical view of the whole process is shown in Fig. 1.
2.2 Experimental Set-up Our experimental set-up comprises tubes, connections, and a syringe pump (D101963, Cole-Parmer) fitted with a 10-ml syringe (BD Discardit II) to aid flow in the microdevice. We used an inverted microscope (ZEISS Axio Vert.A1) with a digital microscopy camera (Axiocam 305 colour) for the experimental images and
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videos. The detailed experimental setup is shown in Fig. 2, and the layout is shown in Fig. 3. All investigations using human blood samples were carried out in accordance with protocols and guidelines recommended by the Institute Bio-Safety Committee, IIT Bombay. Fresh blood samples were taken from healthy, willing donors with informed consent.
Fig. 2 Experimental set-up
Fig. 3 Layout of experimental set-up
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2.3 Sample Preparation The blood sample was collected in a tube coated with anticoagulant ethylenediaminetetraacetic acid (EDTA). We processed the blood sample on the same day of collection. The suspension medium used here for elastic-dominant was polyvinylpyrrolidone (PVP). This aqueous solution is bio-compatible and has low shear thinning properties [5]. The viscous solution was prepared by dissolving PVP powder (Mw = 360,000, Sigma Aldrich, USA) in PBS at various concentrations (1, 3.125, 5, 10, and 20 wt%). This PVP solution was used to dilute the whole blood sample. Only whole human blood was used for all the experiments instead of washed cells.
3 Results and Discussion As lift force depends on the parameters like hydraulic diameter, viscosity, and flow rate, the margination of WBCs can be manipulated with these. Here, the output sample analysis was done based on the WBCs enrichment factor. And this enrichment factor can be defined as the ratio of the number of WBCs in the sample per unit volume to the number of WBCs in actual condition per unit volume.
3.1 Margination Using Whole Blood and Diluted Blood First, we compared the WBC enrichment values with whole blood and diluted blood. As shown in Fig. 4, with the whole blood sample, we got ~ 3.75-fold enrichment. Whereas, PBS diluted sample, we achieved only around onefold enrichment. In this case, we used the same device and flow rate range. So, every other parameter was the same as before except viscosity. In the following case, we achieved WBC enrichment nearly the same as the whole blood sample using a PVP diluted sample. The PVP diluted sample’s viscosity was the same as the blood’s by maintaining the PVP solution viscosity. Using viscoelastic fluid results in an increase in lift force which enhances WBC margination. The dilution factor was 1:2 for both PBS and PVP diluted samples.
3.2 Margination Using Different PVP Concentrations To find out the effect of higher concentrations of PVP on WBC margination, we went for several concentrations from 1 to 20 wt%. For the same dilution factor of
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Fig. 4 Enrichment of WBCs using whole blood, PBS-diluted blood, and PVP-diluted blood
Fig. 5 Enrichment of WBCs with various PVP concentrations with a dilution factor of 1:2
1:2, WBC enrichment decreased with the increase in PVP concentrations, as shown in Fig. 5. It almost went down to 0.5 with a PVP concentration of 20 wt%.
3.3 Margination Using Different Dilution Factors For further analysis, we considered higher diluted PVP diluted samples (1 wt%). WBCs enrichment was found to be relatively higher at a dilution factor of 1:2 in comparison to the higher diluted samples. Though the dilution varied from 1:2 to
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Fig. 6 Enrichment of WBCs with various dilution factors at a PVP concentration 1 wt%
1:20, the viscosity remained constant as both whole blood and the PVP solution had nearly the same viscosity, as shown in Fig. 6.
3.4 Discussion All the experiments show that viscoelastic fluid enhances the WBC margination compared to a PBS-diluted sample. But using a higher PVP concentration, more viscous than blood, decreases the margination of WBCs. Keeping the viscosity constant as of blood and using a higher dilution factor also results in less margination of WBCs. This decrease in margination is related to the less presence of RBC aggregation in the channel’s mid-region, which pushes the bigger cells like WBCs towards the outer wall. In contrast, the smaller cells are focused on the mid-region of the channel [6]. This creates a cell-free layer zone close to the wall. So, as covered in the previous section, the net lift force, RBC aggregation, and geometrical factors decide the cell-free layer’s width. This technique was used by several blood-plasma separation devices to yield plasma with improved purity [7, 8]. For cell focusing, the width of the cell-free layers can be optimized, as the margination tends to be less at very low or high widths. Adding parameters like viscosity, discussed here, to these studies can lead to a higher WBC enrichment of WBCs after separation. This will help the WBCs and various cells of the same size, like EPCs and CECs, due to size dependency focusing.
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4 Conclusions It can be concluded that the WBC margination mainly depends on cell–cell interaction in the focusing zone. The margination tends to be barely affected at lower and higher viscosity. Only maintaining the blood sample’s viscosity and ensuring more cell– cell interaction, for more RBC aggregation, in the focusing zone of the channel can enhance the WBC margination. Acknowledgements We are grateful to the Centre for Excellence in Nanotechnology (CEN), IIT Bombay, for providing fabrication facility.
Nomenclature U DH a CL C el ρ Re FL Fe ηp γ˙ λ
Maximum velocity [m/s] Hydraulic diameter [m] Particle diameter [m] Lift coefficient [–] Elastic lift coefficient [–] Density [kg/m3 ] Reynolds number [–] Net lift force [N] Elastic force [N] Polymeric contribution to the solution viscosity [Pa s] Average shear rate [s− 1 ] Relaxation time [s]
References 1. Martel JM, Toner M (2014) Inertial focusing in microfluidics. Annu Rev Biomed Eng 16(1):371– 396. https://doi.org/10.1146/annurev-bioeng-121813-120704 2. Ho BP, Leal LG (1974) Inertial migration of rigid spheres in two-dimensional unidirectional flows. J Fluid Mech 65(2):365–400. https://doi.org/10.1017/S0022112074001431 3. Bhagat AAS, Kuntaegowdanahalli SS, Papautsky I (2008) Continuous particle separation in spiral microchannels using dean flows and differential migration. Lab Chip 8(11):1906. https:// doi.org/10.1039/b807107a 4. Lu X, Liu C, Hu G, Xuan X (2017) Particle manipulations in non-Newtonian microfluidics: a review. J Colloid Interface Sci 500:182–201. https://doi.org/10.1016/j.jcis.2017.04.019 5. Ravin HA, Seligman AM, Fine J (1952) Polyvinyl pyrrolidone as a plasma expander. N Engl J Med 247(24):921–929. https://doi.org/10.1056/NEJM195212112472403 6. Laxmi V, Tripathi S, Joshi SS, Agrawal A (2020) Separation and enrichment of platelets from whole blood using a PDMS-based passive microdevice. Ind Eng Chem Res 59(10):4792–4801. https://doi.org/10.1021/acs.iecr.0c00502
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7. Prabhakar A, Kumar YVBV, Tripathi S, Agrawal A (2015) A novel, compact and efficient microchannel arrangement with multiple hydrodynamic effects for blood plasma separation. Microfluid Nanofluidics 18(5–6):995–1006. https://doi.org/10.1007/s10404-014-1488-6 8. Tripathi S, Kumar YVB, Agrawal A, Prabhakar A, Joshi SS (2016) Microdevice for plasma separation from whole human blood using bio-physical and geometrical effects. Sci Rep 6(1):26749. https://doi.org/10.1038/srep26749
Experimental Investigation of Two-Phase Immiscible Liquid Flow Through a Microchannel Rohit Kumar, Chandan Nashine, Arman Mohaddin Nadaf, Harish Kumar Tomar, and Manmohan Pandey
Abstract Two-phase flow is getting more and more applicable as component sizes in mechanical systems are miniaturized. The applications of two-phase microfluidic devices include energy conversions, chemical synthesis, and thermal management. Hence, the two-phase flow largely determines the functionality and performance of these devices. Immiscible two-phase liquid flow in microchannels has a wide area of applications such as extraction processes, emulsion production and other biochemical applications. Therefore, modern researchers are trying to study the flow dynamics of two-phase flow and its effect on microscale devices. In this work, experimental studies are done for flow visualization and determining flow patterns of the two immiscible liquids flowing through the microchannel using the inverted microscope, and the images are captured with a high-speed camera. The microchannel has a square cross-section with an area of 100 × 100 µm2 with two inlets and one common outlet. The liquids are passed through the T-shaped microchannel through their respective inlets. De-ionized water and silicone oil are selected as the working fluids for the experiment. Experiments are conducted for the different combinations of the flow rate of the fluids. The images are captured along the flow length at various locations to analyze the flow patterns inside the channel. The background noise of the captured images is removed by the postprocessing method. The processed images are analyzed using the Phantom Camera Application software to obtain the stratified length, bubble length, and bubble velocity of the flow regimes. Keywords Two-phase flow · Immiscible liquids · µ-PIV · High-speed camera · Microchannel · Flow patterns
R. Kumar (B) · C. Nashine · A. M. Nadaf · H. K. Tomar · M. Pandey Department of Mechanical Engineering, IIT Guwahati, Guwahati 781039, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_46
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1 Introduction Two-phase flow in mini and microchannels comprises a dynamic and rapidly developing area. Two-phase microfluidic devices have been widely employed in various applications like thermal management, energy conversions, and chemical synthesis and hence the two-phase flow largely determines the functionality and performances of these devices. Also, the advancements in science and technology have forced modern researchers to study the flow dynamics and their effect on microscale devices. Fluorescence microscope with a high-speed camera, micro-particle image velocimetry (µ-PIV) are such techniques that can be utilized to visualize and then analyze the flow dynamics inside the microscale devices. Experimental work in the field of microchannels is continually developed over the last three-four decades. µ-PIV, fluorescence microscope, and high-speed camera are good experimental techniques that are still progressing due to their parts like camera, laser, and highperformance computer. So experimentally, we can analyze the fluid characteristics and different flow patterns or regimes with the help of these techniques.
2 Literature Review and Objective Zhao et al. [1] observed the liquid–liquid two-phase flow patterns in a rectangular PMMA microchannel at the T-junction. Five flow patterns are obtained in the microchannel: monodispersed droplets flow, droplet population flow, slug flow, parallel flow, and annular flow. Liquid–liquid two-phase flow patterns transition maps at the T-junction and in the microchannel were constructed. The experiment was done by injecting one fluid at a constant flow rate and the second fluid at a variable flow rate. Thus, different flow patterns were identified, and the corresponding two-phase pressure drops were measured. When oil was injected first, the patterns observed in both quartz and glass microchannels were droplet, slug, and annular flows with the water as the dispersed phase [2]. Sarkar et al. [3] used a serpentine microchannel made of glass for the experiments on two-phase liquid–liquid flow in which water and butanol were used as the working fluids. The main flow patterns observed were slug flow, slug and droplet flow, droplet flow, unstable annular flow, annular flow, annular dispersed flow, and fully dispersed flow. Kashid and Kiwiminsker [4] experimented on two-phase liquid–liquid flow using a T-junction microchannel that were studied for square and trapezoidal cross-sections. Water was used as the aqueous phase, while toluene was used as the organic phase. For both the channels, slug flow, slug-drop flow, and annular/parallel flow were observed. Wehking et al. [5] observed that the transition between formation regimes was so vital during experimentation, specific observable characteristics for droplets produced in each of the three primary regimes: droplet in T-junction (DTC), droplet in channel (DC), and parallel flow (PF) were reported. Three different combinations of fluids were used: n-butanol-water, n-butyl acetate-water, and toluene-water. The
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main flow patterns observed were slug flow, droplet flow, slug and droplet flow, and parallel flow [6]. The visualization of the two-phase flow was performed at the capillary tube using a high-speed camera just at the exit of the T-junction. The three flow regimes (drop, plug, and annular) were observed, and mappings were plotted for the three configurations [7]. Cao et al. [8] experimentally reported that the aqueous phase was the continuous phase due to the hydrophilic microchannel walls, and the organic phase was the dispersed phase. Three main flow patterns were observed, i.e., annular flow, slug flow, and droplet flow. Additionally, slug velocities and slug length were investigated. The current work is focused on the investigation of flow patterns observed in T-shaped microchannel using two immiscible liquids. Effect of varying flow rate on the flow patterns is also studied and the average size and bubble velocity is calculated by postprocessing of the images.
3 Experimental Setup The experimental setup is shown in Fig. 1. The setup consists of two syringe pumps, T-junction microchannel, microscope, and a high-speed camera attached with a computer system. Since the microchannel has two inlets, so both fluids will be entered with the help of two different syringe pumps into the channel. Combination of aq-glycerol and Si-oil, de-ionized water and Si-oil was used in the experiments.
3.1 Experiment with Aq-Glycerol and Si-Oil (50 cSt) The first experiment is carried out with the fluids aqueous-glycerol and Si-oil with a viscosity of cSt. The concentration of aq-glycerol is 55% water and 45% glycerol to make the solution’s refractive index 1.401, which is close to the refractive index of Sioil. The solution of aq-glycerol is also seeded with fluorescent polystyrene particles of size 1 µm for better flow visualization. According to the manufacturer specification, 1 ml of the supplied solution of polystyrene contains 1 × 1010 microparticles,
Fig. 1 Schematic diagram of the setup
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which gives us a volume concentration (v/v) of 0.52%. For our experiment, 90 µl of these seeding particles are mixed with 50 ml of an aqueous-glycerol solution, which gives the concentration of 9.414 × 1010. Now the concentration of seeding particles remains fixed, while the flow rate of both fluids varies. Figure 2 shows that the Si-oil becomes the continuous phase, while the aq-glycerol becomes the discontinuous phase. The Taylor bubbles of aq-glycerol are formed having a length of 0.276 mm and moving with a velocity of 18.55 mm/s. The dark boundary around the Taylor bubbles is due to seeding particles. Since there is no laser light for illumination, the particles do not emit any light and appear black. Figure 3 shows a similar result as that of case-1, i.e., the Si-oil becomes the continuous phase, while the aq-glycerol becomes the discontinuous phase. The Taylor bubbles of aq-glycerol are formed having the length of 0.251 mm and moving with the velocity of 29.78 mm/s. The increase in the flow rate of aq-glycerol results in the reduction in the size of the Taylor bubbles, but there is an increase of about 60% in the bubble velocity.
Fig. 2 Flow visualization in T-section with aq-glycerol and Si-oil with flow rates 0.2 ml/h and 0.02 ml/h, respectively
Fig. 3 Flow visualization in T-section with aq-glycerol and Si-oil with flow rates 2 ml/h and 0.02 ml/ h, respectively
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3.2 Experiment with De-ionized Water and Si-Oil (20 cSt) Figure 4 shows that the Si-oil is in the continuous phase, while the DIW becomes the discrete phase. Near the junction, there is a formation of slug up to 1.325 mm, and after that, Taylor bubbles of DIW start to form, which are length of 0.69 mm and move with a velocity of 35.33 mm/s. Figure 5 shows that the Si-oil is in the continuous phase, while the DIW becomes the discrete phase. Near the junction, there is a formation of slug up to 0.8552 mm, and after that, Taylor bubbles of DIW start to form, which are length of 0.63 mm and move with the velocity of 32.12 mm/s. As the flow rate of DIW decreases, the slug length, bubble size, and bubble velocity also reduce. Due to the presence of oil and particles at certain locations, the bubble gets distorted at that location. Figure 6 shows that the Si-oil is in the continuous phase, while the DIW becomes the discrete phase. Near the junction, there is a formation of slug up to 0.7618 mm, and after that, Taylor bubbles of DIW start to form, which are length of 0.39 mm and move with a velocity of 29.83 mm/s. As the flow rate of DIW decreases, the slug length, bubble size, and bubble velocity also reduce.
Fig. 4 Flow visualization in T-section with DIW and Si-oil with flow rates of 1 ml/h and 1 ml/h, respectively
Fig. 5 Flow visualization in T-section with DIW and Si-oil with flow rates of 0.9 ml/h and 1 ml/ h, respectively
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Fig. 6 Flow visualization in T-section with DIW and Si-oil with flow rates 0.8 ml/h and 1 ml/h, respectively
Figure 7 shows that the Si-oil is in the continuous phase, while the DIW becomes the discrete phase. Near the junction, there is a formation of slug up to 0.691 mm, and after that, Taylor bubbles of DIW start to form, which are in length of 28.57 mm and move with a velocity of 25.45 mm/s. With this flow rate of DIW, Taylor bubbles reduce significantly, indicating a transition from Taylor bubbles to bubbly flow. Figure 8 shows that the Si-oil is in the continuous phase, while the DIW becomes the discrete phase. Near the junction, there is a formation of slug up to 0.643 mm, and after that, Taylor bubbles of DIW start to form, which are length of 20.54 mm and move with a velocity of 22.89 mm/s. Again, the size of bubbles gets reduced as the flow rate of DIW is reduced to 0.6 ml/h. With the flow rate of 0.5 ml/h of the de-ionized water, the slug length reduces to 0.53 mm, and after that, bubbly flow occurs, in which the average size of bubbles is 0.1945 mm, moving with an average velocity of 32.52 mm/s. Since there is a transition from Taylor bubbles to bubbly flow, there is an increase in the bubble velocity. This is because the bubbles are smaller in size, hence reducing the area of contact with the walls, which allows them to move faster than Taylor bubbles (Fig. 9).
Fig. 7 Flow visualization in T-section with DIW and Si-oil with flow rates of 0.7 ml/h and 1 ml/ h, respectively
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Fig. 8 Flow visualization in T-section with DIW and Si-oil with flow rates of 0.6 ml/h and 1 ml/ h, respectively
Fig. 9 Flow visualization in T-section with DIW and Si-oil with flow rates of 0.5 ml/h and 1 ml/ h, respectively
With the flow rate of 0.4 ml/h of the DIW, the slug length reduces to 0.46 mm, and after that, bubbly flow occurs, in which the average size of bubbles is 0.1605 mm moving with an average velocity of 30.50 mm/s. The average velocity of bubbles, in this case, is less than that of the earlier case, but the velocity is still greater than that of Taylor bubbles (Fig. 10). The formation of bubbles happens in seven stages noted in Fig. 11a–g. There is a small gap at the junction in the first stage, denoted as the lag phase. Later with time, the fluid starts to rise in the channel, which vanishes the lag phase. Then the slug formation starts to happen, which goes up to the maximum length of 0.3288 mm, and after that, the slug breaks into the tiny bubbles of size 0.13 mm moving with an average speed of 19.94 mm/s. This is the least speed of bubbles we get in these series of experiments done. This might be because, with a flow rate of 0.3 ml/h, the flow occurs with difficulty.
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Fig. 10 Flow visualization in T-section with DIW and Si-oil with flow rates of 0.4 ml/h and 1 ml/ h, respectively
4 Summary The present work investigates the visualization of the flow patterns formed by two immiscible liquids inside a T-shaped microchannel made of PDMS. Initially, the flow visualization was tried using the micro-PIV setup, but that could not happen due to technical issues. Then the flow visualization is done using the inverted microscope, and the videos are taken out with the help of a high-speed camera. Later, videos are postprocessed using the Phantom Camera Control (PCC) application to determine the slug length, bubble size, and average velocity of the bubble. For different flow conditions, different results are obtained. Two different sets of experiments are carried out. For the first experiment, a mixture of water and glycerol in the ratio of 55% and 45% by volume, respectively, while the other fluid is Si-oil of viscosity 50 cSt is chosen. In this experiment, the flow rate is set to be 0.2 ml/h for aqueous-glycerol and 0.02 ml/h for Si-oil. For the other part of this experiment, the flow rate of aq-glycerol is increased to 2 ml/h, while for Si-oil, it remains the same. For the second set of experiments, de-ionized water and Si-oil of 20 cSt are taken as the working fluids. The flow rate of de-ionized water is initially set to 1 ml/ h, and with the reduction of 0.1 ml/h, the final flow rate is set to 0.3 ml/h, while the flow rate of Si-oil is kept constant as 1 ml/h. It is seen that the flow inside the microchannel is not happening when the flow rate of DIW is kept below 0.3 ml/h. The bubble formations for the flow rate of 0.3 ml/h have happened in several stages discussed earlier.
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Fig. 11 Flow visualization with a flow rate of DIW 0.3 ml/h and Si-oil 1 ml/h
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Acknowledgements This work was partially funded through Grant No. EMR/2016/003513 of SERB, DST, Govt. of India.
Nomenclature DIW µ-PIV PMMA
De-Ionized Water (–) Micro Particle Image Velocimetry (–) Polymethyl Methacrylate (–)
References 1. Zhao Y, Chen G, Yuan Q (2006) Liquid-liquid two phase flow patterns in a rectangular microchannel. Am Inst Chem Eng J 52:4052–4060 2. Salim A, Fourar M, Pironon J, Sausse J (2008) Oil-water two-phase flow in microchannels: flow patterns and pressure drop measurements. Can J Chem Eng 86:978–988 3. Sarkar PS, Singh KK, Shenoy KT, Sinha A, Rao H, Ghosh SK (2012) Liquid-liquid two-phase flow patterns in a serpentine microchannel. Ind Eng Chem Res 51:5056–5066 4. Kashid MN, Kiwiminsker L (2011) Quantitative prediction of flow patterns in liquid-liquid flow in micro-capillaries. Chem Eng Process 50:972–978 5. Wehking JD, Gabany M, Chew L, Kumar R (2014) Effects of viscosity, interfacial tension, and flow geometry on droplet formation in a microfluidic T-junction. Microfluid Nanofluid 16:441–453 6. Darekar M, Singh K, Mukhopadhyay S, Shenoy K (2017) Liquid-liquid two-phase flow patterns in Y-junction microchannels. Ind Eng Chem Res 56:12215–12226 7. Mahdi Y, Daoud K, Tadrist L (2017) Two-phase flow patterns and size distribution of droplets in a microfluidic T-junction: experimental observations in the squeezing regime. C R Mecanique 345:259–270 8. Cao Z, Wu Z, Sundén B (2018) Dimensionless analysis on liquid-liquid flow patterns and scaling law on slug hydrodynamics in cross-junction microchannels. Chem Eng J 344:604–615
Elastohydrodynamics of Electromagnetically Actuated Deformable Microfluidic Systems Apurba Roy and Purbarun Dhar
Abstract Fluid structure interaction (FSI) resulting out of flow of Newtonian electrolytic fluid through a parallel plate microchannel having elastic walls has been semi-analytically investigated in this article. The fluid is exposed to external electric fields in the axial and transverse direction, and a magnetic field in the transverse direction. Taking in to account all the participating forces, a comprehensive examination of the two-way interrelationship between the flow hydrodynamics and channel deformation is presented in this article. Considering the case of constant flow rate within the channel, the variation of pressure distribution in the channel has been demonstrated for different flow rates and their effect in channel wall deformation. The velocity profile for different flow rates at different channel sections has also been depicted. For the case of constant pressure gradient in the channel, the channel wall deformation along the channel length has been depicted. The effect of magnetic field to bring about flow enhancement has also been explored in our study. The effect of the elasticity of channel walls on flow dynamics has also been investigated in this article. Keywords Microfluidics · Elastic channel · Lubrication theory · Electroosmotic flow · Electro-magneto hydrodynamics
1 Introduction Understanding and modelling of human physiological microscale phenomenon has been a subject of great interest among researchers since a long time [1, 2]. With the development of microfabrication in the late twentieth century, several in vitro studies have been performed to model arterial flow by constructing lab-on-chip (LOC) devices [3, 4]. The microchannels are usually made with soft polymeric materials, such as crosslinked polydimethylsiloxane (PDMS), gelatin-methacryloyl, etc., A. Roy (B) · P. Dhar Hydrodynamics and Thermal Multiphysics Lab (HTML), Department of Mechanical Engineering, IIT Kharagpur, Kharagpur-721302, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_48
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to model the deformed nature of arteries when fluid flows through it. The deformability of the channel walls leads to the key feature of fluid structure interaction. The hydrodynamics within the channel introduces deformation in the wall, and the deformation in the wall also alters channel hydrodynamics, leading to two-way coupling between the fluid and solid domain. Additionally, the pressure drop in deformable channels are seen to be much lower than that of rigid channel resulting in low flow rates [5, 6]. Since, the channel materials are made of soft polymers, excessive pressure gradients cannot be applied without risking physical disintegration of channel material. One of the popular ways to overcome the bottleneck of slow flow rates is by applying external body forces to the fluid of the channel. Flow of an electrolytic solution through these soft polymeric channels leads to setting up of surface charges and consequent setting up of electrical double layer (EDL). Applying an electric field in the axial direction sets up electroosmotic flow leading to flow enhancement in the channel [7, 8]. Several recent studies [9–11] have investigated the phenomenon of fluid flow through deformed channel driven by combined pressure and electroosmotic flow. However, enhanced strength of axial electric field leads to high Joule heating, and subsequent changes to both the fluid and channel wall medium [12–14]. Another way to aid flow through the channel is by applying transverse electric and magnetic fields. The Lorentz force due to coupling of the external fields creates a body force in the direction of flow [15, 16]. By controlling the strength of the electric and magnetic fields, the flow rate can be precisely fine-tuned and the flow can even be reversed. From the literature reported above, we observe that a comprehensive work on flow of an ionic electrolytic solution through a deformed elastic channel, driven by the combined action of pressure force, electroosmotic force, and Lorentz force due to application of transverse electric and magnetic fields has not yet been done. A systematic investigation of such flow physics can provide deep insights to help understand the interlink between the participating forces and help in the development and improvement of microfluidic lab-on-chip devices. Advancing the above view, we semi-analytically examine the induced pressure and pressure gradient in the channel in presence of external pressure gradient, and axial and transverse electric and magnetic fields in this article. We also study the wall deformation arising due to pressure-induced fluid structure interactions in the channel. Additionally, we investigate the extent of flow acceleration possible by increasing the strength of electric and magnetic fields. This study is expected to provide deeper insights into electroosmotically and electromagnetically actuated elastohydrodynamics of compliant channels, which can help in designing and developing microfluidic systems composed of soft polymers.
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2 Problem Formulation The physical model of our problem is represented by the schematic diagram as shown in Fig. 1. We consider parallel plates initially separated by a distance 2h 0 . To mimic the behaviour of biological tissues, the plates are assumed to be made of elastic material, initially of thickness ts , which are prone to deformation under the influence of pressure force created when fluid flows between the plates. The outer boundary of the plates is rigid and does not undergo any deformation. The width of the plates is taken to be infinitely large so that infinitesimally small gradients are developed in that direction. The plates are assumed to be made of soft polymers (e.g. hydrogels), which assume a surface charge when it comes in contact with a monovalent binary electrolytic solution, thereby leading to setting up of electric double layer (EDL) in the channel. The fluid is also acted upon by a constant electric field in the axial direction, constant electric and magnetic fields in the transverse directions, and a pressure gradient force along the channel.
2.1 Electrostatics The surface charge developed in the channel walls disturb the ionic distribution in the fluid. Counter-ions are attracted to the walls, and their gradient decreases as one moves towards the bulk solution. If n + and n − are the number of cations
Fig. 1 Schematic diagram of a deformable microchannel filled with a Newtonian fluid driven by combined pressure force, electroosmosis and electromagnetic forces
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and anions in the ionic solution and n 0 is the bulk ionic concentration, then their ( ) distribution is governed by the Boltzmann distribution, n ± = n 0 ∓zeψ ' /kB T . Here, k B is the Boltzmann constant, z is the valency of ions, e is the ionic charge, and T is the absolute temperature. The charge density, ρq' in such scenario would ( ) be, ρq' = (n + − n − )ze = −2n 0 ze sinh zeψ ' /kB T , where, ψ ' is the electrostatic potential. This non-uniform distribution of charges in the fluid leads to non-uniform potential distribution, given by the Poisson-Boltzmann distribution ) ( ρq' 2n 0 ze ∂ 2ψ ' zeψ ' = , = − sinh ∂ y '2 ∈ ∈r ∈0 kB T
(1)
where ∈ is the permittivity of the fluid. For low surface potential, we can conveniently employ Debye–Huckel linearization to the electrostatic potential distribution. Employing the boundary conditions of constant zeta potential (ζ ) at the walls and zero flux at the centreline, Eq. (1) yields the following distribution of electrostatic potential, ) ( cosh κ ' y ' , ψ =ζ cosh(κ ' h ' ) '
where κ ' is the Debye–Huckel parameter, given as κ ' =
(2) √
2z 2 e2 n 0 /∈kB T .
2.2 Hydrodynamics (Fluid Domain) For creeping flow, the momentum conservation equation in the x—direction for flow of a Newtonian fluid inside the channel, can be given as, 0=−
d p' d2 u 'x + μ + Fx' . dx ' dy '2
(3)
In Eq. (3), p ' is the pressure. Due to the applied electric fields in the axial and transverse direction, E = E x' eˆ x − E z' eˆ z and magnetic field in transverse direction, B = B y' eˆ y , the current density can be given as, J = γe (E + u × B), where γe is the average conductivity of the electrolytic solution, and u is the velocity field. The ' Lorentz force can be calculated as F = ρq E + J × B. The momentum equation in the x—direction yields, ( ' ') d p' d2 u 'x 2 ' '2 ' cosh κ y . 0 = − ' + μ '2 + −γe Bz u x + σe B y E z − ∈κ E x ζ dx dy cosh(κ ' h ' )
(4)
We use the reference parameters, xref = L, yref = h 0 , u ref = U , pref = λ + 2G, κref = 1/ h 0 , E x,ref = ζ / h 0 and h ref = h 0 , and using non-dimensional
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√ parameters, Hartmann number, Ha = γe /μB y h 0 , elastoviscous number, β = ) ( ( λ+2G ) h 20 , electroviscous number, Nlv = ∈ζ 2 /γe h 0 , electric Reynolds’ number, L μU √ ( ) ∈ E z2 h 0 Ree = ∈U/γe h 0 , and S = Re1 e μU , the nondimensionalized Eq. (4) yields, 0 = −β
cosh(κ y) d p d2 u x . B y + HaS − Ha2 u x − Nlv κ 2 E x + 2 dx dy cosh(κh)
(5)
Note that in the following discussions, the nondimensionalized parameters are devoid of any overbars for the sake of clarity. Equation (5) can be solved by utilizing the boundary conditions, u x (y = h) = 0 (no slip at the walls) and du x /dy(y = 0) = 0, to obtain the velocity profile inside the channel. [ ][ ] dp cosh(Hay) 1 β − HaS −1 ux = dx cosh(Hah) Ha2 [ ] Nlv κ 2 E x cosh(Hay) cosh(κ y) − + Ha2 − κ 2 cosh(Hah) cosh(κh)
(6)
It must be noted that due to the pressure of fluid in the channel, the channel height gets deformed. Thus, the velocity profile is not only a function of the channel height, but also of the channel length. The discussion on variation of wall deformation with pressure is given in the next subsection.
2.3 Deformation (Solid Domain) The stress developed in the channel, for no net body force in the wall material, is given by the Navier equation, ∇ · τ = 0. For linear, homogenous and isotropic material, the stress developed can be expressed in terms of the strain ] rates with the] help of the two Lame’s constants, λ and G, as τ = λ(∇ · δ)I + G (∇δ) + (∇δ)T . Here, I is ] ]T the identity tensor, and δ is the displacement vector, given as δ = δx' , δ 'y1 . From scaling estimates (ts 0 ⎪ Δ2 ) ⎨ ( U −U min i ψi = min min 1, if Δ2 > 0 Δ2 ⎪ ⎪ ⎩ 1 if Δ2 = 0
(11)
) ( ) ) ( ( Δ2 = 1/2 ∇Ui ∗ ri→ Umax = max Ui , max j , U j Umin = max Ui , min j , U j j and max j and min j refer to the maximum and minimum values of the neighboring cells for the central cell i. After the gradient and the limiter calculation, the solution for the central cell is finally evaluated from the Taylor series expansion. Viscous flux calculation is done using a central differencing scheme.
4 Results and Discussion Model Validation: In the present study, our solver was studied for a geometry of 45% stenosis. The inlet waveform is sinusoidal. The density and viscosity of the fluid were 755 kg/m3 and 0.00143 Ns/m2 . The sinusoidal flow was 4.3 ± 2.6 ml. Flow was laminar with a mean Reynolds number of 590. The Womersley number was 7.75.
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Fig. 2 Validation with experimental results
Results were validated with the experimental results of Ojha et al. [3]. Figure 2 compares our results of centerline velocities at the inlet with the experimental results in one cardiac cycle. Our results with the experimental data matched quite accurately. The small differences might be due to the different time steps and grid densities taken into consideration. The two cases considered were (a) less than 50% occlusion and (b) 80% occlusion. After the geometry extraction, the meshing was done after a grid-independent study. The image with less than 50% blockage was axis-symmetric, whereas the whole geometry was considered for the artery with 80% stenosis. Figure 3 shows the geometry taken from the Internet (https://myheart.net/articles/heart-blockage-explai ned-with-pictures/) as well as the constructed meshed geometry.
4.1 Study of Velocity Profile Along the Cardiac Cycle for 50% Stenosis With the change in the inlet velocity sinusoidally, the velocity profile was studied at four distinct positions, i.e., at the start of the cycle, at maximum velocity, mid-phase, and minimum velocity of the cycle just beyond the stenosis. With the changes in the velocity of the sinusoidal inlet, the velocity peak also varied accordingly. Negative velocity at the walls is observed, which indicates a region of flow separation. This region of flow separation changes with changes at different phases of the cycle. t/ tp = 1.0 refers to the region of the fully accelerated flow domain. In this region, no flow separation was observed. In all the other three time frames, flow recirculation and separation were seen. Hence we could conclude that flow separation is observed in almost all phases of the cycle in the region just beyond the stenosis. These trends also matched with Lee et al. results [27]. The details of the same are highlighted in Fig. 4.
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(a)
(b) Fig. 3 Extracted geometry and meshing for a 50% occlusion b 70% occlusion
Fig. 4 Velocity profiles at four different time frames for 50% stenosis
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4.2 Wall Shear Stress Calculation for 50% Stenosis Wall shear stress is one of the basic parameters for the growth and initiation of plague. Figure 5 shows the wall shear stress at the same four points of the cycle. Results show that the value of shear stress in the walls rose directly with the flow rate. Negative wall shear stress indicates flow recirculation, and a dip just after the stenosis is observed throughout, indicating flow recirculation. The negative value of wall shear stress after the stenosis is observed in the decelerating phases of the cycle, i.e, from t/tp = 0.25 to 0.75. This negative wall shear stress indicates disturbed flow instead of the unidirectional flow in healthy arteries. Also, the sudden rise in the value of WSS indicates the area of stenosis in the arteries.
4.3 Study of Velocity Profile Along the Cardiac Cycle for 80% Stenosis The geometry extracted from the image for 80% stenosis was not axis-symmetric, and hence the simulation was run for the whole geometry. The peak value of the velocity profile rose drastically when the stenosis was increased to 80%. A rise in peak velocity is an indication of an increase in stenosis area. Other than that, the patterns of the velocity profile obtained were similar, but the region of flow separation was more. Negative velocity at the walls was obtained for all the regions except in the region of fully accelerating flow, at t/tp = 1.0. Details of the resultant velocity profile are shown in Fig. 6. The region of disturbed flow and recirculation also increased as the stenosis area increased to 80%.
4.4 Wall Shear Stress Calculation for 80% Stenosis Under normal conditions, the wall shear stress varies within a narrow range from 0.3 Pa to around 1.5 Pa in arteries, but with stenosis, there is a rise in the value depending upon the severity. This rise impacts the endothelial cells of the arteries. Results for 80% stenosis showed a tremendous increase in wall shear stress in the stenosis region. A greater negative value after the stenosis indicates larger recirculation, as shown in Fig. 7.
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Fig. 5 WSS at four different time frames for 50% stenosis
t/tp = 0.25
t/tp = 0.5
t/tp = 0.75
t/tp = 1.0
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Fig. 6 Velocity profiles at four different time frames for 80% stenosis
5 Conclusions Extracted geometry from real images showed similar trends and patterns as that of the constructed geometry. Velocity peak increases greatly when the stenotic area increases to 80%. Healthy arteries account for unidirectional flow in arteries, but the presence of stenosis will lead to conditions like flow separation and flow recirculation. The negative values of velocity near the walls are indications of such disturbed flows. Results also show that the negative wall shear stress value increases as the stenosis increases, thus indicating recirculation.
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Fig. 7 WSS at four time frames for 80% stenosis
t/tp = 0.25
t/tp = 0.50
t/tp = 0.75
t/tp = 1.0
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Nomenclature Ec Gc Ev Gv τ HLLC-AC β Θ R* W
Convective flux in z-direction Convective flux in r-direction Viscous flux in z-direction Viscous flux in r-direction Pseudo time Harten Lax and van Leer with contact Artificial compressibility factor Matrix Residue term Flow variables
References 1. Wang N, Feng J, Wang Y (2018) The effect of left coronary bifurcation angle on vascular fluid hemodynamics as well as plaque formation and distribution. J Biomed Eng Res 2. Kelidis P, Konstantinidis E (2018) Pulsatile flow through a constricted tube: effect of stenosis morphology on hemodynamic parameters. Comput Methods Biomech Biomed Engin 21(7):479–487. https://doi.org/10.1080/10255842.2018.1481505 3. Ojha M, Cobbold RSC, Johnston KW, Hummel RL (1989) Pulsatile flow through constricted tubes: an experimental investigation using photochromic tracer methods. J Fluid Mech 203(173):173–197. https://doi.org/10.1017/S0022112089001424 4. Bauer A et al (2020) Analysis of the wall shear stress in a generic aneurysm under pulsating and transitional flow conditions. Exp Fluids 61(2). https://doi.org/10.1007/s00348-020-2901-4 5. Gharib M, Beizaif M (2003) Correlation between negative near-wall shear stress in human aorta and various stages of congestive heart failure. Ann Biomed Eng 31(6):678–685. https:// doi.org/10.1114/1.1574025 6. Henriques HAM et al (2016) Blood flow in the common carotid artery with stenosis. In: ECCOMAS Congress 2016—Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering, vol 1, pp 98–104. https://doi.org/10.7712/100 016.1795.9043 7. Zhang JM et al (2015) Hemodynamic analysis of patient-specific coronary artery tree. Int J Numer Method Biomed Eng 31(4):e02708. https://doi.org/10.1002/cnm.2708 8. Seo J-H, Eslami P, Caplan J, Tamargo RJ, Mittal R (2018) A highly automated computational method for modeling of intracranial aneurysm hemodynamics. Front Physiol 9. https://doi.org/ 10.3389/fphys.2018.00681 9. Praharaj P, Sonawane C, Ingalhalikar M (2021) Bibliometric survey on image processing techniques using Lattice Boltzmann method for CFD simulations. Libr Philos Pract 2021(March):1– 22 10. Cr S (2018) High order accurate numerical simulation of flow past an oscillating circular cylinder. Fluid Mech Res Int J 2(5):230–232. https://doi.org/10.15406/fmrij.2018.02.00042 11. Zhao Y, Forhad A (2003) A general method for simulation of fluid flows with moving and compliant boundaries on unstructured grids. Comput Methods Appl Mech Eng 192(39– 40):4439–4466. https://doi.org/10.1016/S0045-7825(03)00424-9 12. Mhamed S, Essam AB, Thamer AB, Mojtaba M (2017) Investigation of blood flow modeling in artery using ALE formulation. Int J Comput Methods 14(1):1–13. https://doi.org/10.1142/ S0219876217500013
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13. Sonawane CR, More YB, Pandey A (2019) Numerical simulation of unsteady channel flow with a moving indentation. In: E3S Web conference, vol 128, pp 3–5. https://doi.org/10.1051/ e3sconf/201912810010 14. Sonawane CR, Mandal JC, Rao S (2019) High Resolution Incompressible Flow Computations over Unstructured Mesh using SDWLS Gradients. J Inst Eng Ser C 100(1):83–96. https://doi. org/10.1007/s40032-017-0390-x 15. Mandal JC, Sonawane CR, Iyer AS, GosaviInamdar SJ (2011) Incompressible flow computations over moving boundary using a novel upwind method. Comput Fluids 46(1):348–352. https://doi.org/10.1016/j.compfluid.2010.08.011 16. Ünsal B et al (2021) Impact of inflow boundary conditions on the calculation of CT-based FFR. Thermophys Aeromech 1(5):2682–2689. https://doi.org/10.1007/s10439-016-1625-3 17. Xu P et al (2021) The hemodynamics of patent ductus arteriosus in patients after central shunt operation. Comput Math Methods Med 2021. https://doi.org/10.1155/2021/6675613 18. Roy M, Sikarwar BS, Bhandwal M, Ranjan P (2017) Modelling of blood flow in stenosed arteries. Procedia Comput Sci 115:821–830. https://doi.org/10.1016/j.procs.2017.09.164 19. Carvalho V, Pinho D, Lima RA, Teixeira JC, Teixeira S (2021) Blood flow modeling in coronary arteries: a review. Fluids 6(2):1–13. https://doi.org/10.3390/fluids6020053 20. Campinho P, Vilfan A, Vermot J (2020) Blood flow forces in shaping the vascular system: a focus on endothelial cell behavior. Front Physiol 11(June):1–12. https://doi.org/10.3389/fphys. 2020.00552 21. Ai L et al (2009) Optimization of intravascular shear stress assessment in vivo. J Biomech 42(10):1429–1437. https://doi.org/10.1016/j.jbiomech.2009.04.021 22. Banerjee MK, Ganguly R, Datta A (2012) Effect of pulsatile flow waveform and Womersley number on the flow in stenosed arterial geometry. ISRN Biomath 2012:1–17. https://doi.org/ 10.5402/2012/853056 23. Sonawane C, Praharaj P, Kulkarni A, Pandey A, Panchal H (2022) Numerical simulation of heat transfer characteristics of circular cylinder forced to oscillate elliptically in an incompressible fluid flow. J Therm Anal Calorim. https://doi.org/10.1007/s10973-022-11621-z 24. Sonawane C, Praharaj P, Pandey A, Kulkarni A, Kotecha K, Panchal H (2022) Case studies on simulations of flow-induced vibrations of a cooled circular cylinder: Incompressible flow solver for moving mesh problem. Case Stud Therm Eng 34:102030. https://doi.org/10.1016/j. csite.2022.102030 25. Sonawane CR, More YB, Pandey AK (2021) Numerical simulation of unsteady channel flow with a moving indentation using solution dependent weighted least squares based gradients calculations over unstructured mesh. Heat Transf Eng 1–16. https://doi.org/10.1080/01457632. 2021.1874661 26. Sonawane C, Praharaj P, Pandey A, Kulkarni A (2021) High order accurate numerical simulation of vortex-induced vibrations of a cooled circular cylinder case using solution dependent weighted least square gradient calculations. In: E3S Web Conference, vol 321, p 04005. https:// doi.org/10.1051/e3sconf/202132104005 27. Lee KW, Xu XY (2002) Modelling of flow and wall behaviour in a mildly stenosed tube. Med Eng Phys 24(9):575–586. https://doi.org/10.1016/S1350-4533(02)00048-6
Highlighting the Importance of Nasal Air Conditioning in Septoplasty Using Virtual Correction Tools: A Numerical Study Kartika Chandra Tripathy and Ajay Bhandari
Abstract Nasal air conditioning of the inspired air is one of the primary functions of the upper airway. However, it can get significantly altered due to abnormal changes in the nasal anatomy. Understanding the alteration in nasal air conditioning in a noninvasive manner is essential for effective septoplasty planning. Therefore, the current study aims to investigate this alteration due to nasal septum deviation with the help of computational fluid dynamics (CFD). Additionally, its relevance in septoplasty planning has been demonstrated using the virtual septoplasty technique. A patientspecific nasal cavity model having S-shaped deviation has been reconstructed using the CT scan images, which are virtually corrected to mimic septoplasty. The computational model is employed to investigate the alteration in airflow velocity, pressure drop, wall shear stress, relative humidity, and water mass fractions in deviated and virtually corrected cases. The insilico results demonstrate that deviated nasal septum (DNS) leads to asymmetric distribution of airflow velocity and water mass fractions, which becomes symmetric in both the cavities after the virtual septoplasty procedure. Additionally, there is an increase and decrease of relative humidity and pressure drop, respectively, in virtually corrected case that leads to delivery of saturated air to the lungs and ease in breathing. Keywords Nasal air conditioning · Nasal septum deviation · Virtual septoplasty · CFD · Patient-specific model
K. C. Tripathy · A. Bhandari (B) Biofluids Research Lab, Department of Mechanical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_52
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1 Introduction Nasal breathing problems are one of the most frequent health conditions, affecting 40% of the world’s population. In India, around 20–30% of the population suffers from some nasal obstruction problems such as congestion, stuffed nose, and allergic rhinitis and about 15% of them, later on, tend to develop asthma [1]. Nasal septum deviation (NSD) is one of the recurrent causes of nasal obstruction that changes the airflow mechanics and leads to several nasal problems. There are many treatment options to relieve the nasal symptoms posed by NSD, such as decongestants, nasal sprays, and over-the-counter medications. However, all these treatment options provide temporary relief, leaving septoplasty the only successful treatment option for NSD where the twisted bone and cartilage are straightened using surgery. However, despite the effective surgical correction, approximately 50% of the patients are not appeased with their postoperative outcomes [2]. To avoid this postoperative discontent, surgeons have developed several methods, but still predicting postoperative outcomes is a challenging task. To this end, computational fluid dynamics (CFD) is a promising tool to evaluate the complex nasal airflow in detail non-invasively. When combined with modern imaging acquisition techniques, CFD can help surgeons develop objective indicators that can help them optimize their surgical procedures suitable for a particular patient and alleviate the patient’s quality of treatment and diagnosis.
2 Literature Review and Objective Many studies talk about the prediction of nasal airflow using CFD [3–5] and studied flow parameters that govern the physiological functions of the nasal cavity. In a similar direction, some researchers have attempted to model the nasal humidification process inside the healthy nasal anatomies [6] and highlighted the importance of nasal air conditioning inside the nose. However, the alteration of nasal humidification due to the deviated nasal septum (DNS) remains unexplored. A dearth of studies explains the importance of nasal air conditioning in septoplasty procedures. In this regard, the current research aims to develop an image-based CFD model to analyse the effect of DNS on nasal airflow and air conditioning. A detailed investigation of changes in nasal humidification has been performed with importance given to relative humidity and water mass fractions. Additionally, the importance of nasal humidification in septoplasty procedures has been highlighted by virtually correcting the deviation of a patient’s septum using the virtual septoplasty method [7].
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3 Materials and Methods 3.1 CT Scan Acquisition and 3D Reconstruction of the Nasal Cavity The CT scan images of a patient having deviated septum (“S-shaped” septal deviation) were acquired from the Fortis Memorial Research Institute Gurugram, INDIA. The images were taken with a slice thickness of 0.45 mm and a resolution of 512×512 pixels with a pixel size of 0.4296×0.4296 mm in all sagittal, coronal, and axial planes as illustrated in Fig. 1a. After acquiring the images, the airway segmentation of the nasal cavities was done by using the open-source segmentation software 3D slicer. Manual segmentation of the nasal airways was done by using segmentation tools such as thresholding, masking, level tracing, and smoothing operations that are available in the software. Thresholding ranges were selected from − 1024HU to − 659.5HU to mask only the airways (please refer to Fig. 1b). After masking, the paranasal sinuses were manually confiscated as they do not affect the nasal airflow. Repeating the same procedure on all the slices, a 3D model of the deviated nasal cavity was developed (please refer to Fig. 1c), which was further imported in STL format to another open-source software meshmixer to eliminate holes and unwanted artefacts. Figure 1 shows the schematic of the procedure for nasal airway segmentation. Virtual septoplasty technique was adopted to develop the 3D model of the corrected nasal cavity. Prior virtual corrections were performed on the deviated septum using tissue deformation techniques in the 3D slicer. Non-deformable points were added on the septum and the nasal cavity central line, and wrap transformation Thresholding, masking and level tracing
(b) Masking of nasal airways along with sinus
(a) Load of CT scan data in 3D slicer
Sinus removal and smoothing (c) Final 3D model of the nasal cavity Fig. 1 Steps for 3D model reconstructions of the nasal cavity
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Fig. 2 Nasal cavity a before and b after correction
(a)
(b)
methodology was applied to provide deformation. The dimensions of the deviated septum were compared with the average dimensions of 46 healthy nasal anatomies obtained from the literature [7]. The thickness of abnormal areas was reduced, and the deviated regions were corrected, as illustrated in Fig. 2.
3.2 Grid Generation Final STL geometry obtained from the segmentation process was imported to ICEM CFD for finite volume grid generation. Unstructured tetrahedral mesh elements were used to mesh the geometry. The sharp velocity gradients inside the boundary layer were captured by providing ten prism layers near the wall region with a growth rate of 1.1. The final computational mesh comprised 3.1 million cells obtained after the mesh independence test.
3.3 Mathematical Model In the current study, resting breathing condition (inhalation rate = 15 L/min) has been assumed for which the flow inside the nasal cavity is considered laminar. Assuming air to be an incompressible and Newtonian fluid, its flow inside the nasal cavity is governed by the mass and momentum conservation equations that can be expressed as ∂ (ρu i ) = 0 ∂ xi
(1)
∂σi j ∂ ∂p + + ρg j ρu i u j = − ∂x j ∂x j ∂x j
(2)
In addition to nasal airflow, nasal air conditioning is also considered that is governed by energy and species transport equations.
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∂T ∂ ∂ k u i ρCp T = ∂ xi ∂ xi ∂ xi ∂ ∂ ∂F D (u i F) = ∂ xi ∂ xi ∂ xi
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(3) (4)
The humidification inside the nasal cavity is accomplished by a highly vascularized mucus membrane coated over the nasal wall. This biological phenomenon has been captured by mathematically modelling the heat and mass transfer through the mucus membrane instead of physically incorporating it in the 3D model because of the segmentation constraints. Simplified models for heat and mass transfer were utilized (please refer to Fig. 3a, b) by assuming that the boundary layer does not store any water flux coming from the mucus membrane for mass transfer. These can be mathematically expressed as Q memb = kmemb
T0 − TS δmemb
Wmemb = Wbl Wmemb = ρv × Dmemb Wbl = ρv × Dbl
∂F FO − FS = ρv × Dmemb ∂η δmemb
∂F Fs − F = ρv × Dbl ∂η δbl
(5) (6) (7) (8)
Equating Eqs. (7) and (8) gives the mass fraction of water vapour at the wall that can be expressed as Fs =
FO + Dδblbl F Dmemb + Dδblbl δmemb
Dmemb δmemb
(9)
The values of the parameters mentioned in Eqs. (1)–(9) have been listed in Table 1.
3.4 Numerical Methodology For pressure velocity coupling, the coupled approach is used [8] in which both mass and momentum conservation equations were solved simultaneously. The pseudo transient approach was used to simulate the steady flow governing equations with a time step of 0.0001 s for better convergence and stability. The least-square cellbased technique was used for discretization of the gradient terms and a second-order scheme was used for pressure. The QUICK scheme was used for the convective term
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(a)
(b) Fig. 3 Models for a heat and b mass transfer
Table 1 List of parameters Name of the parameter
Values
(Dmemb )
2.5 × 10−5 m2 /s
References [6]
(Dbl )
3 × 10−5
[6]
(δmemb )
2 mm
[6]
(kmemb )
0.6 W/m K
[6]
δbl
1 mm
[6]
Mass fraction at organ side (FO )
0.0334
[6]
T0
310 K
[8]
m2 /s
appearing in momentum, energy, and species transport equations as it is third-order accurate. A central difference scheme was applied for diffusion terms as they are direction-independent. For airflow, mass flow rate and no-slip boundary conditions were specified at the inlet (nostril) and the walls, respectively. For the temperature,
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Fig. 4 3D model of the nasal cavity showing all the boundaries
INLET (Boundary conditions: Mass flow inlet, T=298K, F=0.00684)
WALL (Boundary Conditions: No slip, Equation-5, Equation-9)
OUTLET (Boundary conditions: Outflow)
fixed temperature (298 K) and heat flux boundary conditions (Eq. 5) were specified at the inlet (nostril) and the walls, respectively. For the species transport, the mass fraction of water vapour for inlet air was taken to be 0.00684 (35% relative humidity) with a specified mass fraction (Eq. 9) defined at the walls. The outlet was prescribed as the outflow boundary condition (please refer to Fig. 4).
4 Results and Discussion 4.1 Effect of Deviation and Virtual Correction on Nasal Airflow After virtual correction of the septal deviation, the airflow in the nasal airways changes significantly. Figure 5 shows the velocity contours in different coronal sections of the deviated and virtually corrected nasal cavity. It can be observed that increased values of velocity, which were found in some regions of the deviated nasal cavity are significantly reduced after virtual correction. This has a profound effect on the reduction of wall shear stress values (please refer to Fig. 6), which reduces the chances of bleeding from the nose and other agonizing effects such as damage to the nasal wall in those areas. A closer analysis of Fig. 5a1 shows the non-uniform velocity distribution in the nasal valve region inside the deviated case, which becomes uniform after virtual correction (Fig. 5b1). A similar type of behaviour is observed in the nasopharynx region (Fig. 5a2–b2), resulting in a significant decrease in the
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pressure drop from the nostril to the nasopharynx from section 1 to 19 (please refer to Fig. 7). Pressure drop inside the nasal cavity is directly related to the nasal resistance ) whose value decreases from 52,800 to 42,000 Pa s/m3 after (Nasal resistance = p Q virtual correction. This decrease in pressure drop and nasal resistance values can be
(a1)
(a2)
(b1)
(b2)
Fig. 5 Velocity contours for deviated (a1, a2) and virtually corrected (b1, b2) case Fig. 6 Shear stress distribution in deviated and virtually corrected case
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Fig. 7 Pressure distribution in deviated and virtually corrected case
attributed to the reduced velocity and the net convective acceleration that happens because of virtual correction, which further decreases the breathing difficulties experienced by the patient.
4.2 Effect of Deviation and Virtual Correction on Nasal Humidification In addition to nasal airflow, nasal humidification is significantly altered due to virtual correction of the septum. Two essential parameters were analysed for quantifying nasal humidification: relative humidity and water mass fractions. It can be observed from Fig. 8 that a significant increase in relative humidity occurs in the nasopharynx region after virtual correction. However, the increase was not found to be substantial in the other coronal sections before the nasopharynx. This can be attributed to the lesser and higher temperature difference in the coronal sections lying before (Sections 1–15) and after (Sections 15–20) the nasopharynx region, respectively (please refer to Fig. 9). In the sections lying after the starting of the nasopharynx region (Sections 15–20) of the virtually corrected case, mixing of relatively less hot and cold air jets coming from both the cavities takes place. This mixing of jets and heat interaction with the nasal walls collectively results in a higher temperature drop than in the deviated case. On the contrary, for the sections lying before the nasopharynx regions (Sections 1–15), because of no mixing, the temperature difference between the deviated and the corrected case remains on the lower side. Due to higher temperature drop in the nasopharynx region, the air becomes nearly saturated, which further ensures saturated air delivery to the lungs. Figure 10a, b show the local mass fractions of water vapour inside the left and right nasal cavities in different coronal sections for deviated and virtually corrected cases, respectively. A closer analysis of Fig. 10a reveals a non-uniform distribution of the mass fractions in the
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left and right nasal cavity. Further, the regions on the deviated side seem to have more mass fractions of water vapour than the non-deviated side, which can be attributed to the accelerating flow and diffusion of more water vapours in the deviated region. In the current study, S-shaped deviation has been analysed where the deviations are significant in the posterior regions. This results in more mass fractions of water vapour (> 0.030) in the left nasal cavity as compared to the right ones in the coronal sections (Sections 7–11). This also results in the local drying of the mucus membrane, which further leads to the problems like epistaxis. After virtual correction, it was found that the non-uniformities in the mass fractions are minimized in most of the coronal sections (Sections 5–11), decreasing the chances of mucosal damage (Fig. 10b). However, for some of the coronal sections (Sections 1–4), nonuniform distribution remains. This is because the septoplasty has been performed Fig. 8 Humidification in deviated and virtually corrected case
Fig. 9 Temperature distribution in deviated and virtually corrected case
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(a)
(b)
Fig. 10 Mass fractions of water vapour in deviated (a) and corrected (b) case
virtually by correcting the deviated septum according to the average dimensions of 46 healthy nasal anatomies available in the literature. This can result in some discrepancies. Nevertheless, variations in the mass fractions found in Sections 1–4 may not cause much damage to the mucus membrane as the quantitative values are not as high as in Sections 5–11. The current study investigates the alteration in nasal air conditioning due to DNS and highlights its importance in septoplasty. The analysis demonstrated in this study is only limited to one type of deviation (S-shaped) because of the data limitation. However, in the future, the analysis will be extended to a large number of deviations frequently encountered in nasal patients to produce more generic conclusions.
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5 Conclusions In the current study, the effect of DNS on alteration in nasal air conditioning has been numerically investigated. A patient-specific model of S-shaped nasal deviation has been developed, which is virtually corrected by comparing it with healthy nasal anatomy dimensions available in the literature. The results demonstrate that the septal deviation significantly alters the nasal air flow and humidification with increased airflow velocities and water mass fractions in the deviated regions. This leads to damage of the nasal walls due to high wall shear stresses and local mucosal drying. The study also demonstrates the need to include nasal humidification as an essential parameter while performing septoplasty procedures using the virtual septoplasty method. The results presented in this study may be used as a necessary framework by ENT surgeons for effective septoplasty planning. Acknowledgements The authors would like to thank Dr. R. K. Gupta and Dr. Anup Singh for providing CT scan data sets for this research. Further, the authors acknowledge the support received by a grant from the Science and Engineering Research Board (Grant Number: SRG/2021/000053) and Faculty research scheme of IIT (ISM) Dhanbad (Grant Number: FRS (147)/2020-2021/MECH).
Nomenclature Dmemb Dbl δmemb kmemb T0 δbl F FS F0 Wmemb Wbl σi j gj T TS u i, j ρ ρv
Mass diffusion coefficient of the membrane (m2 /s) Mass diffusion coefficient of the boundary layer (m2 /s) Mucus layer thickness (mm) Thermal conductivity of mucus membrane (W/m K) Temperature at organ side (K) Thickness of boundary layer (mm) Mass fraction of water vapour in air (–) Mass fraction of water vapour on nasal surface (–) Mass fraction of water vapour on organ side (–) Water flux from membrane (kg/m2 s) Water flux from boundary layer (kg/m2 s) Stress tensor (N/m2 ) Acceleration due to gravity (m/s2 ) Temperature of the air (K) Temperature of the nasal wall (K) Velocity of the air (m/s) Density of air (kg/m3 ) Density of water vapour (kg/m3 )
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References 1. Varshney J, Varshney H (2015) Allergic rhinitis: an overview. Indian J Otolaryngol Head Neck Surg 67(2):143–149. https://doi.org/10.1007/s12070-015-0828-5 2. Sundh C, Sunnergren O (2015) Long-term symptom relief after septoplasty. Eur Arch Otorhinolaryngol 272(10):2871–2875. https://doi.org/10.1007/s00405-014-3406-7 3. Wen J, Inthavong K, Tu J, Wang S (2008) Numerical simulations for detailed airflow dynamics in a human nasal cavity. Respir Physiol Neurobiol 161(2):125–135. https://doi.org/10.1016/j. resp.2008.01.012 4. Chen XB, Lee HP, Chong VFH, de Wang Y (2009) Assessment of septal deviation effects on nasal air flow: a computational fluid dynamics model. Laryngoscope 119(9):1730–1736. https:// doi.org/10.1002/lary.20585 5. Doorly DJ, Taylor DJ, Schroter RC (2008) Mechanics of airflow in the human nasal airways. Respir Physiol Neurobiol 163(1–3):100–110. https://doi.org/10.1016/j.resp.2008.07.027 6. Kumahata K, Mori F, Ishikawa S, Matsuzawa T (2010) Nasal flow simulation using heat and humidity models. J Biomech Sci Eng 5(5):565–577. https://doi.org/10.1299/jbse.5.565 7. Moghaddam MG, Garcia GJM, Frank-Ito DO, Kimbell JS, Rhee JS (2020) Virtual septoplasty: a method to predict surgical outcomes for patients with nasal airway obstruction. Int J Comput Assist Radiol Surg 15(4):725–735. https://doi.org/10.1007/s11548-020-02124-z 8. Inthavong K, Fletcher DF, Kamooshi M, Vahaji S, Salati H (2022) Wet surface wall model for latent heat exchange during evaporation. Int J Numer Methods Biomed Eng. https://doi.org/10. 1002/cnm.3581
Thrombosis Modelling in a Stenosed Artery Prateek Gupta, Rakesh Kumar, Sibasish Panda, and Mohammad Riyan
Abstract This chapter presents a discussion on blood clotting in an idealized stenosed artery and its effect on the local flow dynamics. We make use of a simple residence-time-based model to mimic the clot growth. The blood flow is modelled using Ansys Fluent, while the clotting process is supervised using a UDF subroutine. Blood flow is considered to be Newtonian and non-pulsatile. We believe that the model presented in this chapter can be extended to study clotting patterns in geometries much more complex than the one presented here. Keywords Stenosed artery · Clotting · Residence-time modelling · Blood flow · Newtonian
1 Introduction Cardiovascular Diseases (CVDs) are one of the foremost reasons for early deaths. This was recently highlighted during the (ongoing) Coronavirus pandemic, which was linked to a higher mortality rate in people with cardiovascular comorbidities. One of the most popular and frequent CVD is ‘atherosclerosis’, which occurs due to the build-up of fats and cholesterol (commonly called as ‘plaque’) on the inner wall of the arteries. This results in a narrowing of the arterial lumen, thereby causing ischemic heart problems. Besides reducing the flow of the blood, atherosclerotic regions are welcome sites for thrombosis or blood clotting. It is believed that any damage to the atheromatous plaque can initiate thrombosis [1]. This results in a production of blood clots near the damaged site, which increases the lumen blockage even further. In some cases, these clots may detach from the vessel wall and give rise to embolisms [2]. It is therefore important to have a better understanding of the degree to which the stenosis has progressed and its role as a causal factor in atherothrombosis. There are several methods for modelling thrombosis, and a brief review of each method is given in [3]. Complexitywise, all models can be categorized into three P. Gupta · R. Kumar (B) · S. Panda · M. Riyan Department of Aerospace Engineering, IIT Kanpur, Kanpur, 208016, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_53
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types: Pure biochemistry models, biochemistry-plus-flow models, and biochemistryplus-flow-plus-structure models. The third category models can properly capture all the elements of Virchow’s triad, which are: intravascular vessel wall damage, stasis of flow, and the presence of a hypercoagulable state. Enormous efforts have been made in the past to come up with highly sophisticated models of blood clotting in the vasculature. However, none of these models give a complete description of all the elements of Virchow’s triad at a continuum level, which cements the complexity of this task. In the present work, we use the concept of residence-time to model clot initiation and growth. This approach avoids the modelling of the complex coagulation cascade that is responsible for clot formation. However, by using this approach, we can still get a good idea of the pattern in which a clot grows and its effect on the blood flow. Several researchers have used residence-time modelling to portray thrombosis. Bernsdorf et al. [4] implemented the residence-time model within the Lattice–Boltzmann (LBM) framework to show clotting patterns in a 3-D idealized rectangular stenosis. They further extended their work to explain clotting patterns near a venous valve [5]. Narracott et al. [6, 7] used a clotting model based on the residence-time of the fluid and combined it with a variable viscosity model to study clotting patterns in idealized stenosis and cerebral aneurysms. They used the commercial CFD software package Ansys CFX to perform the calculations. Friedrich and Reininger [8] used a residence-time model to study clotting patterns on indwelling catheters. As per their findings, fluid residence time is the single most important parameter in determining clotting patterns. Herein, we present a study of the clotting patterns in an idealized Gaussian stenosis with varying degrees of severity. The Reynolds number (Re) of the flow based on the vessel diameter is taken to be 100, and the flow is predicted to fall in the laminar regime. Section 2 describes the complete mathematical modelling of the problem and its implementation in Ansys Fluent. Section 3 describes the results, while Sect. 4 presents the conclusions and ends with a brief discussion on future prospects.
2 Mathematical Modelling 2.1 Geometry and Boundary Conditions The shape of the flow domain is assumed to be ideal. An axis-symmetric bulge characterizing the stenosis is given by the following Gaussian distribution function: r (z) = 1 − λ ∗ exp −ξ ∗
z 2 R
(1)
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Fig. 1 Schematic of the stenosed blood vessel
where r (z) is the bell-shaped Gaussian distribution profile determining the flow domain; λ is the stenosis severity, ξ denotes the degree of curvature, and R is the maximum radius of the flow domain. An analogy of λ can be mentioned as (Di -Ds )/ Do . For the current study, we have chosen λ = 0.3 and 0.6, which is tantamount to 30% and 60% severity, respectively. A higher value of the curvature parameter ξ denotes a smooth curvature. Here, ξ = 5 ensures the absence of sudden flow disturbances (Fig. 1). At the domain boundaries and blood vessel wall, the boundary conditions for the present problem are written as follows: • At the inlet: A parabolic flow profile in the x- direction and zero velocity in the y-direction can be written mathematically as y2 u x = u ∞ 1 − 2 and u y = 0 Di
(2)
• On blood vessel wall: No-slip boundary condition is applied throughout. It can be written as ux = u y = 0
(3)
• At the outlet: The pressure-outlet boundary condition is used throughout. It can be written as ∂u y ∂u x = 0; =0 ∂x ∂x
(4)
2.2 Clotting Model As mentioned earlier, the clotting process is modelled using the concept of fluid residence time. In general, the residence time of a fluid particle is defined as the total time that it has spent inside a given control volume (which in our case is the entire vessel domain). This definition of residence time is applied to the fluid that is flowing through the stenosed artery. It is obvious that the fluid near the inlet has a
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lower residence time as it has spent lesser time inside the domain. We call it ‘new fluid’. Similarly, the fluid closest to the outlet will have the highest residence time and is termed ‘old fluid’. Now, since the residence time (or the ‘age’ of the fluid) cannot be modelled directly in Ansys Fluent, so we introduce a massless passive scalar in our domain. The concentration of this passive scalar species is a measurable quantity. Moreover, the local concentration of this passive scalar is directly correlated to the fluid residence time in that area. Thus, a higher concentration implies a higher residence time. It is important to note that we are talking about a ‘passive’ scalar species, which means that its presence will have no effect on the flow field. A constant amount of this scalar species will be injected inside every element (or cell) of the mesh at each time-step. Fluent will then solve the governing equations and compute the velocity and concentration fields at each time step. Parts of the domain where the concentration exceeds a threshold limit will be clotted (or solidified). Of particular interest is the region that is immediately downstream of the stenosis. Several studies [9–12] have shown that recirculation zones are formed downstream of the stenosis, which increases the chance of blood clotting. This is because fluid parcels trapped inside the recirculation zone remain restricted to that region. With time, the fluid in that region gets ‘old’. In other words, the concentration of the passive scalar will rise inside the recirculation zone because there is no convective mass-transfer between the Eddy and the rest of the fluid. Only diffusion can take place.
2.3 Governing Equations There are three main governing equations to solve. The continuity equation, momentum equation, and the scalar transport equation for the passive scalar [6]. Continuity equation: ∇. v=0
(5)
Momentum equation: ρ
∂ v + ( v .∇) v = −∇ p + μ∇ 2 v ∂t
(6)
Scalar Transport Equation: ∂c = ∇.(D∇c) − ∇.( v c) + S ∂t where D = 1.26e − 09, and S = 1.
(7)
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2.4 Implementing the Clotting Model in Ansys Fluent Ansys fluent is a Finite-Volume Method (FVM) based software with a cell-centred formulation. It means that the flow variables are computed and stored at the cell centroid. This information is important for writing correct UDFs as node-based values are not directly available in Fluent. The first thing to take care of is the species transport of the passive scalar. This is modelled by switching ON the User-Defined Scalar (UDS) transport equation and setting the diffusivity value in the ‘materials’ pane. By default, fluent chooses the unit of diffusivity as kg/m-sec. Therefore, the original diffusivity in m2 /sec needs to be multiplied by the fluid density. We also need to model the solidification of those regions where the species concentration exceeds the threshold. Within the LBM framework, Bernsdorf et al. [4] implemented a bounce-back condition on those nodes which exceeded the threshold concentration. Within Ansys Fluent, solidification can be achieved by deactivating those cells in which the concentration threshold is violated. This can be done by marking the cells using the ‘cell register’ utility in Fluent. The marked cells are then separated and deactivated from the cell zone, and are no longer taken into the solver loop. This is equivalent to having a no-slip wall boundary condition at the deactivated cells. In this manner, we are able to capture the feedback of the growing clot on the local flow dynamics. The UDF that helps with the clotting process is written using a DEFINE_ADJUST macro. This DEFINE_ADJUST subroutine is called at the start of every iteration. Its purpose is to report back the location of that first cell where the concentration threshold is exceeded. This cell then acts as a nucleation point for the clot growth. All subsequent cells that will be solidified will either lie in the neighbourhood of this cell or in the vicinity of the vessel wall. This is important because physiologically speaking, a clot needs a thrombogenic surface to grow. The UDF further takes care that no isolated nucleation points are formed in the bloodstream as they will have no surface to adhere to. Care should also be taken about the outlet region where the fluid residence time will be highest. As per Eq. (7), a unity source term is being added in all the cells at each time step. Given sufficient time, all cells near the outlet region will exceed the threshold and the outlet region will be completely solidified, which is unrealistic. To avoid this, another subroutine is written that adjusts the local concentration of the outlet cells every time they exceed the threshold. The value of the concentration threshold (C T ) is chosen such that it is small enough to allow sufficient cells to clot within a reasonable simulation time, and large enough to avoid the solidification of entire domain within a few time-steps.
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3 Results and Discussion Clotting simulation results have been presented for two degrees of stenosis: 30 and 60%. As mentioned earlier, a concentration threshold is defined, and all cells that exceed this threshold are solidified with no further mass transfer. Simulations are initially run without the injection of the passive scalar, and injecting was done only after the velocity field became steady in time. For the 30% stenosis, the velocity field became steady at around a flow-time of 0.2 s with a time-step size (t) of 0.001 s. This is equivalent to a wall time of 12.9 min on a single compute node. Another 12,620 iterations are performed to capture the progression of the clot (Fig. 3). It can be observed that the clot begins to grow from the wall, and thickens as more and more cells are deposited. The region distal to the stenosis throat shows a small patch of marked cells formed due to the presence of very tiny recirculation zones. Similarly, for the 60% stenosis severity, an initial run of 14,000 iterations with a t of 0.0001 s is made, which is equivalent to a wall time of 23.8 min. The time-step size is reduced by a factor of 10 in order to stabilize the solution as the velocity gradients in the stenosis region are higher. A further clock time of 93 min is registered for showing the clot progression (Fig. 4). It is immediately noticeable that the size of the final clot is much larger for this case, which can be attributed to the presence of much bigger recirculation zones immediately downstream of the stenosis throat (Fig. 2). A mild asymmetry in the clot growth can be seen in Fig. 4b, which stems from the asymmetric flow field for this case. Figure 5 shows how the flow field has adapted to the final clot. The recirculation zones at the upper and lower side have been replaced by a mass of solid clot, which restricts the movement of any fluid inside it. Moreover, the maximum velocity magnitude at the centreline is observed to increase from 15.1 to 16.1 m/sec. As predicted, there is no turbulence in the flow, and no vortex shedding takes place for the chosen value of the Reynolds number.
4 Conclusions and Future Work A residence-time based approach was used to model the development of a blood clot in an idealized stenosed blood vessel with a varying degree of stenosis severity. The complex biochemical kinetics was replaced by a single convection–diffusion equation, which governed the flow of the aging species (the passive scalar). Mesh cells that exceeded the concentration threshold were solidified to mimic clot growth. It was observed that 60% stenosis resulted in the formation of a much bigger clot accompanied by some noticeable changes in the flow field. This work does not take into account the shear-stress activation thresholds for clot formation. In a future work, the shear stress thresholds will be incorporated into the code to account for more realistic clotting patterns. Also, the geometry used in
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Fig. 2 Recirculation zones formed at the distal end of stenosis for a 30% and b 60% severity
Fig. 3 Clot progression at 30% stenosis for flow-time a 0.2 b 0.687 and c 0.831 (in seconds)
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Fig. 4 Clot progression at 60% stenosis for flow-time a 0.281 b 0.293 and c 0.311 (in seconds)
Fig. 5 Flow field adapts to the growing clot (60% stenosis severity.)
this study was 2D and ideal. Future works will extend the concepts presented in this paper to more realistic and 3D geometries.
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Nomenclature c Re u∞ p R R μ D S Ld Di Ds t
Concentration of passive scalar Reynolds number of flow Free stream velocity [m/s] Pressure [Pa] Maximum radius of flow domain [m] Variable radius of the flow domain [m] Blood viscosity [Pa.s] Diffusivity [m2 /s] Source term Length of arterial stenosis [m] Internal diameter of ideal geometry [m] Internal diameter of stenosed geometry [m] Time-step size in seconds [s
Greek Symbols λ ξ ρ
Stenosis severity Curvature parameter Density of blood [kg/m3
References 1. Lendon C, Born GV, Davies MJ, Richardson PD (1992) Plaque fissure: the link between atherosclerosis and thrombosis. Nouv Rev Fr Hematol (1978) 34(1):27–29. PMID: 1523097 2. Goodman P, Barlow E, Crapo P, Mohammad S, Solen K (2005) Computational model of device-induced thrombosis and thromboembolism. Ann Biomed Eng 33(6):780–797 3. Bodnár T, Galdi GP, Neˇcasová Š (eds) (2014) Fluid-structure interaction and biomedical applications. Springer Basel 4. Bernsdorf J et al. (2006) Concurrent numerical simulation of flow and blood clotting using the lattice Boltzmann technique. Int J Bioinf Res Appl 2(4):371–380 5. Harrison SE et al (2007) Development of a lattice Boltzmann framework for numerical simulation of thrombosis. Int J Mod Phys C 18(04):483–491 6. Narracott A et al. (2003) Development of a model for the investigation of blood clotting in cerebral aneurysms following coiling. Proc Int Congr Comput Bioeng, Zaragosa 7. Narracott A et al. Development and validation of models for the investigation of blood clotting in idealized stenoses and cerebral aneurysms. J Artif Organs 8(1):56–62 (2005) 8. Friedrich P, Reininger AJ (1995) Occlusive thrombus formation on indwelling catheters: in vitro investigation and computational analysis. Thromb Haemost 73(01):066–072 9. Razavi A, Shirani E, Sadeghi MR (2011) Numerical simulation of blood pulsatile flow in a stenosed carotid artery using different rheological models. J Biomech 44(11):2021–2030
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10. Bluestein D et al. (1997) Fluid mechanics of arterial stenosis: relationship to the development of mural thrombus. Ann Biomed Eng 25(2):344–356 11. Schoephoerster RT et al. (1993) Effects of local geometry and fluid dynamics on regional platelet deposition on artificial surfaces. Arterioscler Thromb J Vasc Biol 13(12):1806–1813 12. Bark J, David L, Ku DN (2010) Wall shear over high degree stenoses pertinent to atherothrombosis. J Biomech 43(15):2970–2977
Gold Nanoparticle-Antibody Bio-Probe Analysis: Synthesis, Conjugation, Characterization and Dot Blot Assay on Paper Prateechee Padma Behera, Shubham Kumar, Monika Kumari, Pranab Kumar Mondal, and Ravi Kumar Arun
Abstract Gold Nanoparticle (AuNP) surface functionalized with antibodies is critical for the optimization and development of AuNP-enabled biosensing technologies. The optical properties of AuNPs and the binding specificity of antibody–antigen interactions help in the amplification of the assay signals in point-of-care technologies. Obtaining the stable AuNP–Antibody conjugate poses a major challenge as there is a chance of aggregation of the particles when the pH shifts. In this work, 37.7-nm sized spherical citrate-capped AuNPs are synthesized and are conjugated with Anti-25-OH vitamin D3 antibodies using a direct physical absorption method which requires no chemical functionalization of the AuNP or the antibody surface. The UV–vis spectroscopy and DLS outputs are used to characterize and prove the efficiency of the system. The UV–vis analysis is used to indicate the adsorption of antibodies onto the AuNP surface with the change in the absorbance peak. It was observed that the negatively charged AuNPs binded to the positive charge of the antibodies to form effective bioprobes for further detection of antigen in a short time as well as visible to the naked eye. The AuNP-Ab bioprobe’s stability and efficiency were checked with repeatability under 24 h observation. Keywords Gold nanoparticles · Antibodies · Conjugation · Bioprobes · UV–vis spectroscopy analysis
P. P. Behera (B) · P. K. Mondal Department of Mechanical Engineering, IIT Guwahati, Guwahati 781039, India e-mail: [email protected] S. Kumar Department of Physics, IISER Tirupati, Tirupati 517507, India M. Kumari · R. K. Arun Department of Chemical Engineering, IIT Jammu, Jammu 181221, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_54
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1 Introduction Antibody-labelled Gold Nanoparticles (AuNPs) recently have gained much attention to biological interactions which can be used for the detection of antigens in Lateral Flow Assays (LFA), drug delivery [1], gene transfer, bio-probes in cell and tissue analysis. Biosensors designed with high efficiency depend on the development of novel materials to improve the transduction and recognition steps. These nanomaterials deliver excellent practicality in biomedical applications owing to their size scale and their high surface-to-volume ratios. Exceedingly, one of the most remarkable features of AuNPs is the high extinction coefficients in the visible spectral range which is higher than conventional dyes [2]. Direct physical adsorption of antibodies onto the surface of AuNPs (citrate-capped) is the most straightforward approach in forming bio-probes for the detection of antigens [3]. This physisorption strategy relies on electrostatic, hydrophobic, and hydrophilic interactions between AuNPs and antibodies surface to form the conjugate. Additionally, dried papers coated with AuNP aggregates can be stored for at least several weeks without loss of biosensing function [4]. It has been well reported that the antibodies have been shown to interact with negatively charged AuNP to create stable conjugates. Protein attachment occurs on the metal surface of the gold nanoparticle, provided that the pH is optimized for the isoelectric point of the protein. These days, optimal use of AuNP-labelled antibodies can be found in LFA strips. LFA strips use filter papers/nitrocellulose charged with AuNP-Ab bioprobes for detecting targeted analyte/antigen. This serves as the very base of a lateral flow assay. Filter papers/nitrocellulose are affordable, have different pore sizes and their surface can be modified according to the need of the antigen earmarked. The powerfree flow of fluid by capillary action of fluid inside the porous channel of paper is made to most use in LFAs, dipsticks, etc. The paper’s porous network maintains protein stability and ensures a consistent flow inside the membrane. The capillary force regulates the flow rate inside the membrane which is also dependent on the surface tension, contact angle, and viscosity of the fluid sample [5].
2 Literature Review and Objective The simplicity and practicality of colorimetric detection have been widely accepted for POCT devices. Gold nanoparticle-based colorimetric assays have received significant attention due to their unique optical properties, low cost, and simplicity [6]. AuNPs modified with goat anti-mouse IgG and Horseradish Peroxidase (HRP) were synthesized for the detection of Pb(II) in an enhanced ELISA method developed by Zhou et al. [7]. Liu et al. [8] coupled the optical properties of AuNPs and the monoclonal anti-HA antibody to generate AuNP-mAb conjugates for the colorimetric detection of H3N2 IAV. The virus was quantified with absorption spectral measurements and detected by the naked eye as the AuNP-mAb bio-probes colour
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changed from red to purple. AuNP-based immunoassay devices present an easy-touse rapid diagnostic kit. Although, for the quantitative assessment of any nutritional deficiency [9, 10] or to check the spoilage of any food material [11], assays can be accompanied by a smartphone-assisted portable imaging device that autonomously performs the necessary image processing. From previous research, AuNP sizes of around 20–40 nm are the most desired ones in the application of a gold nanoparticlebased immunochromatographic assay. It was found that if the diameter of AuNPs was too small (40 nm) are unstable, and self-coagulation occurs [12]. In the present work, we have coupled the excellent colorimetric properties of AuNPs with the monoclonal anti-25-OH vitamin D3 antibody-based specific recognition to create AuNP-Ab bioprobes which will be further used to detect vitamin D3 deficiencies in serum samples. AuNP-Ab bioprobes can be prepared by physical adsorption [13] or chemisorption [14]. This work presents direct physisorption of antibodies on nanoparticle surface considering: (a) the pH of the AuNP solution, (b) the isoelectric point (pI) of the antibody, and (c) the added quantity of the antibody. The prepared bioprobes are characterized and are further used for the detection of vitamin D3 on a nitrocellulose membrane platform.
3 Materials and Methods 3.1 Materials and Reagents Sodium tetra chloroaurate (III) dehydrate (AuCl4 Na.2H2 O, 99%, Sigma-Aldrich), Trisodium citrate dehydrate (TSC, HOC(COONa)(CH2 COONa)2 · 2H2 O, SigmaAldrich), Polyoxyethylenesorbitan monolaurate (Tween-20, C58 H114 O26 , SigmaAldrich), Bovine serum albumin (BSA, C123 H193 N35 O37 , Assay > = 98%), Phosphate-buffered saline (0.01 M, pH 7.4, containing 0.138 M NaCl), Cholecalciferol (C27 H44 O; CMS334-1G), Anti-25-OH vitamin D3 antibody Rabbit monoclonal, clone RM3 recombinant, purified immunoglobulin.(Sigma-Aldrich). All solutions are made in autoclaved Milli-Q water sourced from the Centre for Environment, IITG. The antibody was reconstituted in Phosphate-buffered Saline (PBS). Membrane strips of 0.22 µm pore size (MF-Millipore Membrane Filter) are used as a detection platform. The prepared solutions required for the assay have been examined in triplicate and characterized using UV–Vis.
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3.2 Apparatus Transmission Electron Microscopy (TEM) analysis, zeta potential measurements, Dynamic Light Scattering (DLS), and UV–Vis spectroscopy were used to obtain information about the morphology, structure, and size of the AuNPs synthesized in laboratory and AuNP/Antibody conjugates. The absorption spectra were obtained using a UV Visible Spectrophotometer SHIMADZU CORP 81,191 UV-1900. The TEM images were obtained with Transmission Electron Microscope (TEM), Make: JEOL, Model: JEM 2100 (CIF Facility, IITG). To measure the particle size and to measure the charge of the particles, Anton Paar Litesizer 500 (Particle size analyzer) is used.
3.3 Preparation of AuNPs and AuNP-Ab Bioprobes Standard citrate reduction procedures [15] were used for AuNP synthesis with slight modifications. AuNPs of 37.7 nm were obtained. Briefly, all the glassware used in the conjugation experiment was thoroughly washed and rinsed in deionized water, oven-dried, and autoclaved before use. 5 ml milli-Q water is boiled on a hot plate up to its boiling point. Then, an aqueous solution of HAuCl4 (0.01 M) was prepared and added to the boiling milli-Q water with continuous stirring on a magnetic hot plate. In 1 ml of milli-Q water, 5 mg of TSC is added and stirred properly. Then, the 1 ml sodium citrate solution was quickly added to the boiling milli-Q water. The solution is then stirred continuously on the magnetic stirrer. The solution colour changed from grey to black, to purple, and finally to wine red during this period. Then, the heating source was removed, the suspension was stirred for another 15 min, and allowed to cool to room temperature. It was then stored in an autoclaved glass reagent bottle. For further use, the AuNP solution was stored in dark at 4 °C. The direct physical adsorption to construct AuNP/Ab conjugates is a straightforward method needing minimal expertise in synthesis. It is therefore an easy method to screen antibodies. The Anti-25-OH vitamin D3 antibody was first diluted in PBS (0.01 M, 7.4 pH) for further use. The conjugates of AuNP/Ab were prepared by the addition of 0.01 µg/mL antibody to a suspension of AuNPs (pH 7) followed by incubation for 2 h at 27 °C with a gentle rocking, during which the antibodies were adsorbed onto the AuNPs through a combination of ionic and hydrophobic interactions. After being blocked by 30 µL of 0.05% (w/v) BSA, the conjugates were left at RT for 20 min. The AuNP/Ab conjugates were centrifuged at 13 000 rpm at 4 °C for 20 min. The supernatant was then discarded and the bottom sediment was rinsed with 0.05% (v/v) Tween 20 in PBS. It is crucial to remove the free antibodies and the unbound BSA, so the washing step was repeated three times. After another centrifugation for 20 min at 13 000 rpm at 4 °C, the conjugate was resuspended in 500 µL PBS. For further use finally, the conjugates were stored at 4 °C.
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4 Results and Discussion 4.1 Characterization of AuNPs The prepared gold nanoparticles are characterized by using UV–vis spectroscopy, TEM, DLS, and Zeta-potential. The UV–vis absorbance spectra showed a strong absorption around 517 nm, called plasmonic absorption [16], as shown in Fig. 1. The confinement of the valence electron in a reduced space gives this effect, which allows the electron to oscillate in resonance with the electromagnetic radiation. The particle size analyser instrument measures the average hydrodynamic diameter of the AuNPs which came out to be 37.7 nm. However, the DLS method gives only the hydrodynamic diameter which also includes the effect of the double layer of the liquid on the nanomaterial surface. Further, SEM and TEM images are acquired to confirm the spherical shape of the AuNPs and the actual diameter came out to be 35.7 nm. The potential of the nanoparticles was measured using zeta-potential and negatively charged particles are confirmed with − 38.8 mV. The resulting gold colloids are stored at 4 °C in the dark for a long duration without precipitation. The negative charge around the AuNPs repels the other particles and keeps the particles from aggregating. So, the nanoparticles were well stabilized because the change in colour was not observed. The Derjaguin–Landau–Verwey–Overbeck (DLVO) theory explains the stabilization of nanoparticles. The balance between the repulsive electrostatic force and attractive van der Waals force is given by the DLVO theory which explains the stability in colloidal solution, thereby represented by the below equation; G = GVan der Waals + Gelectrostatic
(1)
Note that, DLVO theory accounts for the potential energy variations that occur when two particles approach each other with the resultant net attraction and repulsion forces as a function of interparticle distance [17]. The results across different experimental settings are not entirely explained by the classical DLVO theory.
4.2 Characterization of AuNP-Ab Bioprobes The stability of the antibody-AuNP complex is an important concern in the biological environment with ionic strength. The direct absorption of the antibody onto the gold surface requires the isoelectric point of the IgG antibody to be around 7. In this pH, the antibody can preserve its configuration and bioactivity and can bind to the AuNP for protein adsorption on the surface [18]. In the present work, we are conjugating 37.7 nm AuNPs with Anti-25-OH-vitamin D3 monoclonal antibodies at pH 7.4. The synthesized AuNPs exhibited similar sizes, spherical shape, and good
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Fig. 1 a SEM (scanning electron microscopy) image of the synthesized AuNPs; b TEM (transmission electron microscopy) image of the AuNPs; and c absorbance spectra of AuNP
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Gold Nanoparticle-Antibody Bio-Probe Analysis: Synthesis … Fig. 2 a 37.7 nm sized AuNPs (1.5 ml); b Antibodies and BSA 0.5%(w/v) is added to the AuNPs and the total volume is kept 1.5 ml; c antibodies, tween 20 0.5% (v/v) and BSA 0.5%(w/v) is mixed is with AuNPs and total volume is kept 1.5 ml
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monodispersity. The unbound AuNPs had a maximum absorbance at a wavelength of 517 nm (Fig. 1, curve (c)). As shown in Fig. 2, three vials are taken of 1.5 ml capacity to understand the stability and efficiency of the AuNP-Ab conjugates under two different conditions. The first vial (Fig. 2a) contains the synthesized AuNPs and is taken as a control for the experiment. The second vial (Fig. 2b) contains the AuNPs, to which antibodies are added along with 30 µL BSA 0.5% (w/v). The third vial (Fig. 2c) constitutes of the AuNPs, the Anti-25-OH-Vitamin D3 monoclonal antibodies, 30 µL BSA 0.5% (w/v), and also tween 20 0.5% (v/v) is added to see the effect of it on the conjugates. The vials are agitated on a rocker for 20 min and are kept at room temperature for 3 h. After centrifugation and resuspension, the samples are taken for UV–vis spectroscopy and DLS analysis. Figure 3 shows the UV–vis data of the three samples three hours after the conjugation procedure. A modest red shift can be seen in sample B and sample C peaks. The surface plasmon band shifted from 517 nm (AuNP, sample A) to 524 nm in sample B and from 517 to 520 nm in sample C. The Spectral shift was due to the change in the local refractive index resulting from the antibody adsorption onto the metal surface and is consistent with the previous reports [3]. There is an increase in the hydrodynamic diameter of the AuNPs from 37.7 to 53.59 nm after the direct conjugation of antibodies to the AuNPs. It is a characteristic of a single-layer IgG adlayer [19]. The samples were again analysed under UV–vis for a stability test after 24 h. Figure 4 gives the absorbance spectra of the samples on day 2 of the conjugation experiment. The Spectra of Sample B shifted 1 nm after 24 h whereas sample C shows no change in wavelength. This shows that the AuNP/Ab bioprobes are stable even after 24 h when stored at 4 °C.
4.3 Immunoblotting Analysis Before going for dry storing the bioprobes on a conjugate pad for lateral flow assay, the AuNP/Ab conjugates are tested on dot-blot assay as shown in Fig. 5. A dot blot
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Fig. 3 Absorbance spectra of the samples taking AuNP as control. Absorbance is taken after 3 h of conjugation
Fig. 4 Absorbance spectra of the samples taking AuNP as control. Absorbance is taken after 24 h of conjugation
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Fig. 5 a AuNP blotted on the paper; b AuNP-Ab conjugate assayed with vitamin D3 ; c AuNP-Ab bio-probes on the paper
assay was used to test whether the antibodies retain their functionality (i.e., antigen binding) after adsorption onto the AuNPs. Nitrocellulose membranes are used here by us for the detection area. Square paper strips are cut from the nitrocellulose membrane and put on a petri dish. On the paper strip (b), 10 µL of vitamin D3 (10 µM) is dropped from a micropipette and the area is air dried for 20 min. The spot is then washed with PBS to discard any unbound vitamin D3 on the porous substrate. Then 20 µL of BSA solution is dropped to the paper strip to block the sites to stop unspecific binding. Again after 20 min of air drying, the paper strip is washed with PBS to free the paper from unbound BSA molecules. Further 10 µL of AuNP-labelled anti-Vitamin D3 is added to the spot. As can be seen in Figure (c) only AuNP-labelled anti-Vitamin D3 is dropped on the paper strip to show the colorimetric difference. Vitamin D3 analyte when makes immunocomplex with the antibody, the colour of AuNP fades to a certain extent, whereas the AuNP labelled anti-Vitamin D3 on the adjacent paper shows bright red colour on the periphery. In the next Figure (a), the same strip is now compared with only the AuNP solution to show the visual difference between the protocols followed, the AuNP-labelled anti-Vitamin D3 and the colloidal AuNP solution. It can be seen that among the three-dot blots, the one with AuNP shows the brightest colour, the AuNP-labelled anti-Vitamin D3 shows a faint red colour and the protocol strip with vitamin D3 shows an even fainter red colour because of the immunoreaction on the porous substrate.
5 Conclusions Rich surface chemistry, low toxicity, strong optical absorption, and high electron density give gold nanoparticles a unique role in future medicine and sensors. In this work, the bio-probes are successfully prepared by the direct physical absorption method. The stability of the conjugates was intact as was seen from the UV–vis analysis after 24 h. The DLS for the AuNP-Ab showed an increase in diameter due
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to the bioconjugation of the antibody on the metal nano-surface and also surface passivation. This method of conjugation is highly anticipated in the LFAs because of the broad applications and easy-to-prepare procedure. The working of the bio-probes was also tested in a later phase to validate and the properties were found intact while antigen detection. The present work uses 37.7 nm AuNPs which is the ideal size for protein/Antibody conjugation onto the metal surface. The conjugates can be further dry stored on paper for lateral flow assembly for detection of any antigen/analyte. Acknowledgements We would like to thank New Generation Innovation and Entrepreneurship Development Centre, the branch of Department of Science and Technology and Ministry of Education, Govt. of India for the support, and for funding this project. We would also like to thank Dr. Sachin Kumar (Professor, Virology Lab), IITG for lending their laboratory space for conducting various experiments.
Nomenclature POCT LFA Ab μM μL DLS
Point-of-care Technology Lateral Flow Assay Antibodies Micro Molar concentration Micro Litre Dynamic Light Scattering
References 1. Yafout M, Ousaid A, Khayati Y, El Otmani IS (2021) Gold nanoparticles as a drug delivery system for standard chemotherapeutics: a new lead for targeted pharmacological cancer treatments. Sci African 11:e00685 2. Chang C-C, Chen C-P, Wu T-H, Yang C-H, Lin C-W, Chen C-Y (2019) Gold nanoparticlebased colorimetric strategies for chemical and biological sensing applications. Nanomaterials 9:861 3. Tripathi K, Driskell JD (2018) Quantifying bound and active antibodies conjugated to gold nanoparticles: a comprehensive and robust approach to evaluate immobilization chemistry. ACS Omega 3:8253 4. Zhao W, Brook MA, Li Y (2008) Design of gold nanoparticle-based colorimetric biosensing assays. ChemBioChem 9:2363 5. Kasetsirikul S, Shiddiky MJA, Nguyen NT (2020) Challenges and perspectives in the development of paper-based lateral flow assays. Microfluid Nanofluidics 24:1 6. Song Y, Wei W, Qu X (2011) Colorimetric biosensing using smart materials. Adv Mater 23:4215 7. Zhou Y et al (2011) An enhanced Elisa based on modified colloidal gold nanoparticles for the detection of Pb(II). Biosens Bioelectron 26:3700 8. Liu Y, Zhang L, Wei W, Zhao H, Zhou Z, Zhang Y, Liu S (2015) Colorimetric detection of influenza a virus using antibody-functionalized gold nanoparticles. Analyst 140:3989 9. Vemulapati S, Rey E, Dell DO, Mehta S, Erickson D (2017) OPEN a quantitative point-of-need assay for the assessment of vitamin D 3 deficiency 1
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A Computational Analysis of the Impact of Blood’s Viscoelastic Properties on the Hemodynamics of a Stenosed Artery Sourabh Dhawan, Pawan Kumar Pandey, Malay Kumar Das, and Pradipta Kumar Panigrahi
Abstract Treatment of atherosclerotic diseases, including stenosis and arterial aneurysms, requires a more accurate prediction of hemodynamic flow features. In addition to the degree of stenosis and the body’s physiological state, blood rheology substantially impacts the haemodynamics of a stenosed artery. The current study shows how the assessment of hemodynamic wall indicators is affected when blood viscoelasticity is taken into account as opposed to when it is ignored. For this, multi-mode Giesekus and Simplified Phan-Thein/Tanner (sPTT) models were used to mimic the rheology of real and whole blood. The finite volume-based solver rheoFOAM, part of the rheoTOOL package, was used to run numerical simulations in a planar 75% stenosed artery. The Newtonian and Carreau-Yasuda model (for purely shear thinning fluid) were also used to carry out numerical simulations, together with the non-linear viscoelastic models. The post-stenotic zone’s temporal streamwise velocity evolution at various planes demonstrates how the flow separation zone’s size and symmetry rely on the blood’s rheology. When the blood’s elasticity is taken into account, there are much fewer reattachment points, which is a sign of the number of recirculation zones along the artery wall during one cardiac cycle. Additionally, non-linear viscoelastic models predict higher values of hemodynamic wall indicators than the Newtonian and Carreau–Yasuda models. The current data demonstrate that the blood’s rheology cannot be discarded when computational fluid dynamics simulations are employed as a tool in the diagnosis, prevention, and treatment of severely stenosed arteries. Keywords Haemodynamics · Shear-thinning/non-newtonian · Viscoelasticity · Stenosis · Pulsatile flow
S. Dhawan (B) · P. K. Pandey · M. K. Das · P. K. Panigrahi Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_55
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1 Introduction About seven million fatalities yearly are caused by coronary artery diseases [1], such as atherosclerosis, resulting from the gradual deposition of lipoproteins and other fat molecules in the arterial walls. The stenosis that results from atherosclerotic plaque can alter the hemodynamic behaviour of blood flow. Because it is difficult to obtain precise data through experiments or measurements, a tool is needed to prevent, diagnose, and treat CAD [2]. This was achieved by using Computational Fluid Dynamic (CFD) simulations; however, the models’ effectiveness directly affected the CFD simulations’ outcomes. Previous studies have ignored the blood’s viscoelastic properties and modelled the blood as either a Newtonian fluid or a solely shear-thinning fluid while examining the effects of various factors on the hemodynamic wall indicators, such as the degree of stenosis, the type of the pulsatile waveform, and arterial wall deformation. The purpose of the present study is to do CFD simulations using a model that takes into account both the shear-thinning and elastic characteristics of blood and then investigate how taking into account blood’s viscoelasticity affects the prediction of hemodynamic wall indicators.
2 Literature Review and Objective According to a review of the literature, significant progress has been made in the development and understanding of the flow field in the stenosed artery under the assumption that blood is a Newtonian fluid and a non-Newtonian (Generalized Newtonian) fluid whose viscosity dependency on the shear rate is controlled by the Carreau-Yasuda model [3]. Although the significance of blood viscoelasticity— which results from the elastic behavior of RBC’s suspended in plasma as they stretch and deform under shear—is widely recognized in the literature [4, 5], only a few authors, including Bodnár et al. [6, 7], Good et al. [8], and Javadzadegen et al. [9], have taken this attribute into account in numerical simulations. Javadzadegen et al. [9, 10] used a generalized Oldroyd-B model to simulate the unsteady blood flow in single and multiple stenosed arteries. However, most of these earlier numerical investigations that treated blood as a viscoelastic fluid used hypothetical combinations of model parameters. In other words, most of this research did not consider the model parameters created by fitting the rheological responses of whole and actual blood to various viscometric flows. Deano et al. [5] conducted a detailed and thorough analysis based on passive micro-rheology experiments to assess the actual and whole blood viscoelasticity. They used the multi-mode Giesekus and multi-mode sPTT viscoelastic models to fit their experimental data, which can be used for computational rheology. In this regard, a handful of researchers, including Chauhan et al. [11], evaluated a basic sinusoidal waveform while simulating the blood flow in a symmetrically stenosed artery under
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steady and pulsatile flow circumstances. The main goal of the current study is to use a patient-specific waveform [12] to numerically investigate how blood’s viscoelastic properties affect the flow pattern and hemodynamic indicators in comparison to the case where blood is assumed to be Newtonian as well as Generalized Newtonian Fluid during its flow through a planar stenosed artery. The blood flow patterns, wall pressure and shear stress distribution, pressure drop, velocity field, and reattachment points in stenosed arteries were all presented here.
3 Methodology The current study investigates the flow characteristics of an analogous blood fluid through the 75% planar stenosed artery (Fig. 1) under both steady and pulsatile flow conditions. Even though the artery wall is elastic, we perceive it as rigid. Using the multi-mode Giesekus and multi-mode sPTT models, the whole blood is modeled as an incompressible, shear-thinning, and viscoelastic fluid. Simulations are also carried out assuming that blood is a Newtonian fluid and a non-Newtonian shearthinning fluid to examine how the blood elasticity solely affects the flow properties of blood in the stenotic and post stenotic zone. Figure 2 depicts the rheology of human blood as predicted by the Carreau–Yasuda Model, multi-mode sPTT, and multi-mode Giesekus. The typical incompressible mass and momentum conservation equation is solved to determine the blood flow characteristics. When blood is viewed as a merely shearthinning fluid, Eq. (3) governs viscosity; however, when blood is considered a shearthinning plus viscoelastic fluid, Eq. (4) breaks down the stress term into the solvent and elastic contributions.
Fig. 1 Schematic diagram of the 75% planar stenosed artery
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Fig. 2 Comparison of the 4-modes sPTT and 4-modes Giesekus model with the Carreau–Yasuda models’ predictions for the apparent viscosity of human blood
Mass conservation ∂u i =0 ∂ xi Momentum conservation ∂τi j ∂p ∂u i ∂u i =− + uj ρ + ∂t ∂x j ∂ xi ∂x j
(1)
(2)
Carreau–Yasuda model τi j = 2η(γ˙ )Di j n−1 η(γ˙ ) = η∞ + (η0 − η∞ ) 1 + (k)a a
(3)
Multi-mode Non-Linear viscoelastic model τi j = τiSj + τiPj τiSj
1 ∂u j ∂u i = 2η S Di j ; Di j = + 2 ∂ xi ∂x j τiPj =
N
τikj
(4) (5)
(6)
k=1 ∇
k k λk τi j + λk τikj −αk k τink · τnkj = − 2ηkP Dikj f τnn ηP
(7)
k k λk εk τnn =1+ f τnn ηkP
(8)
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Table 1 Parameters of the multi-mode Giesekus and sPTT model obtained by fitting the rheological data of the real and whole blood [5] Modes (k)
Giesekus model
sPTT model
ηkP (Pa.s)
λk (s)
αk (−)
ηkP (Pa.s)
λk (s)
εk (−)
1
0.05
7
0.06
0.05
7
0.2
2
0.001
0.4
0.001
0.001
0.4
0.5
3
0.001
0.04
0.001
0.001
0.04
0.5
4
0.0016
0.006
0.001
0.0016
0.006
0.5
5 (solvent)
η S = 0.0012
η S = 0.0012
The Giesekus and sPTT constitutive models differ depending on whether the extensibility coefficient, εk and mobility factor, αk , are present or absent for each k mode of Eqs. (7) and (8). In the Giesekus model, the extensibility coefficient equals 0, whereas the mobility factor equals to αk . However, the extensibility coefficient for the sPTT model is equal to εk , whereas the mobility factor is equal to 0. Table 1 lists the parameter values for each mode of the Giesekus and sPTT models.
3.1 Model Parameters for Blood Blood is assumed to be an isotropic, homogenous fluid with a constant density, (ρ) of 1050 kg/m3 . The following constitutive models for blood, along with the corresponding parameters, are considered in this study. • The Newtonian model, with a constant viscosity, μ = 0.00359 Pa.s. • The Carreau-Yasuda model, a purely shear thinning GNM with a time constant, λ, of 8.2 s, a transition parameter, a, of 1.23, zero-shear viscosity, η0 , of 0.16 Pa.s, and infinity shear viscosity, η∞ , of 0.00359 Pa.s [13]. • The multi-mode Giesekus and sPTT viscoelastic models, along with the related fits by Campo-Deno [5], are shown in Table 1.
3.2 Dimensionless Numbers and Boundary Conditions The current flow is governed by the following dimensionless numbers: Reynolds number (W i = λm Um /H ) and Womernumber (Re = ρW P Um /η∞ ) Weissenberg 1/ sley number W o = H (ρω/η) 2 , where W P is the width of the unstenosed artery, Um is the time-averaged mean velocity of flow, ω is the frequency of the pulsatile waveform and λm is the mean relaxation time defined by λm =
N
ηkP λk k . k=1 η P
k=1 N
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Table 2 Boundary conditions at various boundaries of the computational domain Boundary
Velocity
Pressure
Extra-stress tensor
Inlet
u = Uin , v = 0
τi j = 0
Outlet
∂u ∂n
∂p ∂n
Artery wall
=0
u = 0, v = 0
=0
p=0
n · ∇τi j = 0
∂p ∂n
τi j, f w = τi j,P + ∇τi j,P · d P f w
=0
Additionally, Table 2 provides the boundary conditions needed to complete the formulation of the problem.
3.3 Numerical Details The open source CFD software OpenFOAM, based on finite volumes, has been used to solve the governing equations, namely the mass and momentum conservation equations. The viscoelastic constitutive stress equations for blood are additionally solved using the rheoFOAM solver included in the rheoTool toolbox [14]. The time derivatives were discretized using the Euler scheme, while the diffusion terms in the momentum and constitutive equations were discretized using the Gauss linear corrected scheme. The high-resolution CUBISTA (Convergent and Universally Bounded Interpolation Scheme for Treatment of Advection) scheme was used to discretize the advective terms in the momentum and constitutive equations because of its improved convergence. Preconditioned Conjugate Solver (PCG) with DIC (Diagonal-based Incomplete Cholesky) preconditioner was used to solve the linear systems of the pressure and velocity fields. Preconditioned Bi-conjugate Gradient Solver (PBiCG) with DILU (Diagonal-based Incomplete LU) preconditioner was used to solve the linear systems of the stress fields. The log-conformation tensor technique [15] is employed for stabilization, and the SIMPLE algorithm is used to establish the pressure velocity coupling. The computational domain and mesh for the current investigation were made using the ‘Ansys Workbench,’ and the grid design was made using a hexahedral mesh element. A relatively fine mesh is constructed and employed close to the artery wall to capture the steep gradients of velocity and stress components. The mesh’s specifics are as follows: N y = 80, Ns = 150, N x = 700 and N T = 56000 are the number of elements along the artery’s width, along the stenosis, along the axial direction, and overall inside the computational domain, respectively. Additionally, the simulations are performed for both steady and pulsating flows. Under steady flow conditions, the time-averaged velocity at the inlet is Um = 0.4241 m/s, whereas under pulsatile flow conditions, a pulsing waveform with a cycle duration of T = 0.425 s is imposed at the inlet. The nature of the pulsatile waveform employed in this study and the time planes considered for analysis are shown in Fig. 3a.
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Fig. 3 a Information on the critical time planes and the pulsatile waveform that were taken into account, b schematic of the computational mesh employed
4 Results and Discussion Under steady and pulsatile flow conditions, the blood flow through a 75% planar stenosed artery is simulated using the validated numerical code. This article presents the haemodynamic parameters and flows pattern at different times in the cardiac cycle. The Womersley number (W o = 4.212) and the mean Reynolds number (Rem = 496.16) are also constant throughout this work.
4.1 Flow Pattern Prediction Due to its potential pathogenic implications, the research of the temporal flow pattern in the stenosed artery is essential, especially in the post-stenotic zone. To understand how the elasticity consideration leads to variation in Flow Separation Zones (FSZ), Fig. 4 shows the axial velocity profiles at various X-locations for all four models taken into consideration at the time instant, t/T = 0.2. Results shown in Fig. 2 for an artery with a 75% stenosis level demonstrate that the size and symmetry of the flow separation zone depend on whether or not the blood’s elasticity is considered. Additionally, as shown in Fig. 5, reattachment points throughout one cardiac cycle are used to demonstrate the number of FSZ adjacent to the arterial wall. The distance between the two points reflects the size of the flow separation zone. Figure 5 shows that the size and number of flow separation zones next to the arterial wall depend on the blood rheology. These flow separation zones indirectly dictate the blood cell residence duration, which has physiological consequences that could result in additional complications.
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Fig. 4 Axial velocity profiles for each model at various axial planes at a specific time instant, (t/T = 0.2)
4.2 Haemodynamic Descriptors Based on Wall Shear Stress Our objective is to assess the changes in the artery wall’s hydrodynamic indicators when the elastic energy contribution from RBC’s is considered. Or, to put it another way, specific hemodynamic parameters that serve as indicators for the emergence and progression of arterial disorders like atherosclerosis and thrombosis are evaluated to explain the effect of blood’s viscoelasticity clearly. To better understand the regions more susceptible to developing plaque, we have used WSS-based descriptors like time-averaged wall shear stress (TAWSS), and Relative Residence Time (RRT) as vital hemodynamic parameters. The Relative Residence Time (RRT) descriptor shows how long particles stay close to the artery wall. The TAWSS and Oscillating Shear Index (OSI) effects are included in RRT,
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Fig. 5 Reattachment points for each of the four models under consideration throughout various time instants during one cardiac cycle
which has a direct correlation to the OSI and an inverse correlation to the TAWSS. Since it depends on the other two descriptors, RRT is the best metric for measuring blood flow disturbances. The following are the mathematical definitions of TAWSS, OSI, and RRT: 1 TAWSS(s) = T
T |WSS(s, t)|dt
(9)
0
⎤ T 0 WSS(s, t)dt ⎦ ⎣ OSI(s) = 0.5 1 − T |WSS(s, t)|dt 0 ⎡
(10)
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RRT(s) =
1 (1 − 2 × OSI) × TAWSS
(11)
Figure 6 shows the TAWSS distribution along the top and bottom walls for each model being considered. Furthermore, the TAWSS is presented separately in the stenotic and post-stenotic zones for clarity’s sake. In the stenotic zone, the 4-modes sPTT model predicts a maximum TAWSS value of 69.4 and 57.46% higher than those predicted by the Newtonian and Carreau–Yasuda models for the top wall. However, compared to Newtonian and Carreau–Yasuda models, the Giesekus model predicts 816 and 758.54% more values of maximum TAWSS. Compared to Newtonian and Carreau–Yasuda models, the 4-modes sPTT model predicts maximal TAWSS values of 54.65 and 61.83% higher for the bottom wall in the stenotic zone. In contrast, the Giesekus model predicts maximum TAWSS values of 799.3 and 836.4% higher than those of the Newtonian and Carreau–Yasuda models.
Fig. 6 TAWSS distribution in the stenotic zone along a the top wall, b the bottom wall, and in the post-stenotic zone along c the top wall and d the bottom wall
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Figure 7 also depicts the variation in the maximum WSS value over one cardiac cycle in the stenotic and post-stenotic zones along the top and bottom. The point of maximum stress can be found anywhere along the wall in the stenotic and poststenotic zones. Figure 7 depicts the maximum wall shear stress value variation over one cardiac cycle. The maximum stress in both zones occurs during the systolic phase of the waveform, roughly at the instant t/T = 0.2. The distribution of RRT along the top and bottom walls in the stenotic and poststenotic zones is shown in Fig. 8 for each model taken into consideration. Zones vulnerable to atherosclerotic plaque development are those with RRT levels above the threshold of 8 Pa−1 [16]. Blood’s viscoelastic properties in the stenotic zone do not support Newtonian or Carreau–Yasuda model predictions, as only these models predict RRT values higher than the threshold value. The Carreau–Yasuda model’s predictions for residence times over the threshold value are supported by taking into account the blood’s viscoelastic properties in the post-stenotic zone.
Fig. 7 Variation in the maximum value of WSS in the stenotic zone along a the top wall, b the bottom wall, and in the post-stenotic zone along c the top wall and d the bottom wall. The maximum shear stress location along the wall is not fixed
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Fig. 8 RRT predictions from all models taken into account along the top and bottom walls in the a stenotic zone and b the post-stenotic zone
4.3 Wall Pressure Distribution The pressure distribution along the artery wall is another significant hemodynamic indicator. Figure 9 depicts the axial distribution of the wall pressure along the top and bottom walls at two time instants, namely t/T = 0.2 and 0.753 s in one cardiac cycle. It can be seen that the arterial wall pressure distribution is dramatically affected by consideration of blood’s viscoelasticity. All the results discussed above demonstrate that blood rheology is crucial to consider before claiming anything about the zones sensitive to plaque formation and propagation.
4.4 Steady Versus Pulsatile Flows Figure 10a shows, for each of the four models taken into account, the time-averaged pressure drop P between the inlet and exit of a stenosed artery for both steady and pulsatile flow. The contribution of the blood’s elasticity results in a larger time-averaged pressure drop regardless of the inlet flow conditions. For steady and pulsating flow conditions, the multi-mode sPTT model’s percentage increase in pressure drop over the Carreau–Yasuda model is 2.03% and 5.43% respectively. Contrarily, for steady and pulsating flow conditions, the percentage increase in pressure drop over the Carreau–Yasuda model for the multi-mode Giesekus model is 6.82% and 10.27% respectively. Figure 10b demonstrates that when blood is seen as either purely shear thinning or viscoelastic fluid, the pressure drop under steady flow conditions falls between the maximum and minimum pressure drop under pulsatile flow conditions.
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Fig. 9 Distribution of wall pressure along the artery wall a, b at t/T = 0.2 along the top and the bottom walls, respectively, and c, d at t/T = 0.753 along the top wall and the bottom walls
Fig. 10 a Time-averaged pressure difference for all models taken into account, b maximum and minimum pressure drop in pulsatile flow conditions versus steady flow conditions
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5 Conclusions To better understand the impact of blood’s viscoelasticity on the hemodynamic indicators, simulations are run under both steady and pulsatile flow conditions imposed at the inlet of the stenosed artery. Important inferences from the study include the following: (1) The size, number, and symmetry of flow separation zones are considerably affected by considering blood elasticity. (2) Compared to the post-stenotic zone, the stenotic zone is less susceptible to plaque formation when blood elasticity is considered. (3) Multimode Gesekus model predicts 816 and 758.54% greater TAWSS than the Newtonian and Carreau–Yasuda model on the top artery wall in the stenotic zone. In contrast, the multi-mode sPTT model predicts 69.4 and 57.46%. (4) For steady and pulsating flow conditions, the multi-mode sPTT model’s percentage increase in pressure drop over the Carreau–Yasuda model is 2.03% and 5.43% respectively. For the multi-mode Giesekus model, it is 6.82% and 10.27%. Acknowledgements The HPC facility offered by the Indian Institute of Technology Kanpur is gratefully acknowledged by the authors.
Nomenclature α ε ηs ηp η η0 , η∞ γ˙ λ τ τ S, τ P D ρ ω k u H Wt L1 L0
Mobility factor Extensibility coefficient Solvent viscosity [Pa s] Elastic viscosity [Pa s] Total dynamic viscosity [Pa s] Zero and infinity shear viscosity [Pa s] Shear rate [s− 1 ] Relaxation time [s] Extra stress tensor [Pa] Solvent and Elastic contribution of Extra stress tensor, respectively [Pa] Strain rate tensor [s− 1 ] Density of air [kg/m3 ] Cardiac frequency [rad/s] Number of modes Axial velocity component [m/s] Half-width of unstenosed artery [m] Width of the throat of the stenosis [m] Location of stenosis [m] Half-width of the stenosis [m]
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Height of the stenosis [m] Cardiac cycle duration [s]
References 1. Tortora GJ, Derrickson BH (2014) Principles of anatomy and physiology. 14th Editi. https:// doi.org/10.1007/978-3-540-75863-1_1 2. Yilmaz F, Gundogdu MY (2008) A critical review on blood flow in large arteries; relevance to blood rheology, viscosity models and physiologic conditions 197–211 3. Pandey PK, Das MK (2021) Effect of foam insertion in aneurysm sac on flow structures in parent lumen: relating vortex structures with disturbed shear. Phys Eng Sci Med 44:1231–1248. https://doi.org/10.1007/s13246-021-01058-3 4. Thurston GB, Henderson NM (2006) Effects of flow geometry on blood viscoelasticity. Biorheology 43:729–746 5. Campo-Deaño L, Dullens RPA, Aarts DGAL, Pinho FT, Oliveira MSN (2013) Viscoelasticity of blood and viscoelastic blood analogs for use in polydimethylsiloxane in vitro models of the circulatory system. Biomicrofluidics 7:1–11. https://doi.org/10.1063/1.4804649 6. Bodnár T, Sequeira A, Prosi M (2011) On the shear-thinning and viscoelastic effects of blood flow under various flow rates. Appl Math Comput 217:5055–5067. https://doi.org/10.1016/j. amc.2010.07.054 7. Sequeira TBA, Pirkl L (2009) Numerical simulations of blood flow in a stenosed vessel under different flow rates using a generalized oldroyd-B model 2:645–649 8. Good BC, Deutsch S, Manning KB (2016) Hemodynamics in a pediatric ascending aorta using a viscoelastic pediatric blood model. Ann Biomed Eng 44:1019–1035. https://doi.org/10.1007/ s10439-015-1370-z 9. Javadzadegan A, Esmaeili M, Majidi S, Fakhimghanbarzadeh B (2009) Pulsatile flow of viscous and viscoelastic fluids in constricted tubes. J Mech Sci Technol 23:2456–2467. https://doi.org/ 10.1007/s12206-009-0713-9 10. Javadzadegan A, Fakhimghanbarzadeh B (2010) The pulsatile flow of Oldroyd-B fluid in a multi-stenosis artery with a time-dependent wall. Proc Inst Mech Eng Part CJ Mech Eng Sci 224: 915–923. https://doi.org/10.1243/09544062JMES1810 11. Chauhan A, Sasmal C (2021) Effect of real and whole blood rheology on flow through an axisymmetric stenosed artery. Int J Eng Sci 169:103565. https://doi.org/10.1016/j.ijengsci. 2021.103565 12. Pandey PK, Agrawal R, Mukul P, Das MK (2021) Study on pulsatile blood flow in cerebral stenosed artery. Lect Notes Mech Eng 581–588. https://doi.org/10.1007/978-981-16-0698-4_ 63 13. Abraham F, Behr M, Heinkenschloss M (2005) Shape optimization in steady blood flow: a numerical study of non-newtonian effects. Comput Methods Biomech Biomed Engin 8:127– 137. https://doi.org/10.1080/10255840500180799 14. User Guide/RehoTool/version3.0 (2018) 15. Pimenta F, Alves MA (2017) Stabilization of an open-source finite-volume solver for viscoelastic fluid flows. J Nonnewton Fluid Mech 239:85–104. https://doi.org/10.1016/j.jnnfm. 2016.12.002 16. Doorly D, Sherwin S (2009) Geometry and flow. Model Simul Appl 1:177–209. https://doi. org/10.1007/978-88-470-1152-6_5
Effect of Induced Helicity on the Hemodynamics of Carotid Artery Passage L. Rakesh, Arun Kadali, K. Prakashini, and S. Anish
Abstract Abrupt narrowing of the carotid artery known as atherosclerosis is a common cardiovascular disease, increasing the risk of stroke which is one of the leading causes of death. Helicity in the arterial passage is found to be one of the effective ways to minimize plaque formation. Using Autodesk Meshmixer, an opensource software, the stenosed portion of the diseased artery is removed to obtain what is referred to in this study as the base case. The helicity and hemodynamic characteristics of a patient-specific geometry with and without stent in repaired instance are examined. The current study found that when novel stent design is placed there is a reduction in recirculation zone size and Relative Residence Time (RRT), but also resulted in increased pressure drop across the artery. Keywords Carotid artery · Atherosclerosis · Relative residence time · Helicity · Pressure drop
1 Introduction Elderly people are most affected by the chronic, progressive vascular disease atherosclerosis. The innermost layer of the arteries, the tunica intima, which is in direct contact with the blood, becomes inflamed. This inflammation causes lipoproteins to oxidise and build up in the artery walls, which over time can result in a considerable stenosis (narrowing) of the blood arteries. A clot forms at the stenosis location as a result of narrowing artery walls and atherosclerotic plaque rupture, resulting in thrombotic stroke. A clot breaks off and lodges in a smaller artery near the brain, resulting in an embolic stroke. The carotid artery bifurcation region is the most prevalent site of ischemic stroke. The Common Carotid Artery (CCA), the L. Rakesh · A. Kadali · S. Anish (B) Department of Mechanical Engineering, NIT Surathkal, Karnataka 575025, India e-mail: [email protected] K. Prakashini Department of Radiodiagnosis and Imaging, KMC, Manipal, Udupi, Karnataka 576104, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_56
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Internal Carotid Artery (ICA), and the External Carotid Artery (ECA) make up this area. Atherosclerosis is characterized by the patchy thickening and hardening of the arterial wall due to fatty material deposits. Carotid Angioplasty and Stenting (CAS) and Carotid Endarterectomy (CEA) are two procedures used to treat or prevent strokes. Although CEA is currently the gold standard in the treatment of carotid stenoses, carotid stent implantation is a beneficial option in the prevention of strokes since it is less intrusive and avoids numerous surgical problems [10].
1.1 Scope of Present Work The focus of present work is to perform numerical simulations to investigate the role of haemodynamics in the atherogenesis for patient-specific carotid artery. Clinicians continue to struggle with stenting failure owing to restenosis and late thrombosis. To address this, a novel stent design that increases helicity in the bifurcated artery passage has been proposed and is placed in Common Carotid Artery (CCA) instead of traditionally placing normal stent at the stenosis site. This suggested innovative stent design, would produce helicity in the blood flow, washing away atherogenic particles that may collect at low wall shear stress locations. The proposed conceptual design minimizes the fluid layer’s relative residence time near the walls.
2 Literature Review and Objectives 2.1 Role of Hemodynamic Factors A number of illnesses, including diabetes, hypercholesterolemia, and hypertension, have been connected to artery intimal layer thickening. According to studies, some specific artery locations are more likely than others to develop lesions. Additionally, these locations typically exhibit oscillating Wall Shear Stress (WSS) and reduced WSS [4, 9]. The frictional force generated by blood flow in a vessel’s inner lining that resists its motion is known as WSS. Initially, it was thought that the development of high WSS caused these lesions to progress [8]. In contrast, more recent and sophisticated studies have unequivocally demonstrated that lesion accumulation occurs in areas of poor WSS, flow separation, and deviance from axially oriented unidirectional flow [2]. The oscillatory shear index (OSI) was proposed to represent the shear stress operating in directions other than the direction of mean shear stress and to provide a numerical value for the shear stress exerted on the artery wall in pulsatile flow [7].
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2.2 Effect of Helical Flow In vivo research has shown that the heart’s rotation around its own axis causes the blood to flow through the arteries in a helical spiral pattern. The human circulatory system is affected both favourably and unfavourably by the spiralling blood flow [10]. In order to replicate pulsatile spiral blood flow through a conduit with a 75% crosssectional reduction stenosis, the k-turbulence model was employed [11]. At the inlet boundaries of their investigation, spiral components were utilized. They discovered that the flow properties of this spiral component have various observed consequences, both positive and negative, on the human circulatory system. Using the PIV velocity field measurement approach the impact of pulsatile swirling inlet flow on the flow structure in the post-stenosis zone was investigated [11]. They demonstrated that at the post-stenosis, the whirling flow increases WSS and decreases OSI. These findings demonstrate that the whirling flow is what creates the athero-protective flow fields at the post-stenosis. Weakly whirling flow characteristics under steady flow were experimentally and numerically investigated in the asymmetrically stenosed straight tube model [6]. They developed a technique for measuring the reverse flow region’s length using the ultrasonic Doppler method. The swirling action resulted in a 5–10% reduction in the length of the reverse-flow zone in the stenosis’ other plane (symmetry plane).
2.3 Objectives 1. To design a novel stent design which induces helicity in the artery model. 2. To comprehend how induced helicity affects the carotid artery passage haemodynamics.
3 Materials and Methods Computational analysis on actual artery geometries is carried out in this study. Table 1 displays information about the patients investigated in this study. Table 1 Details of patient Age (years)
Gender
Right carotid artery % stenosis
Left carotid artery % stenosis
67
M
70
–
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Fig. 1 a Carotid artery after segmentation b carotid artery after revascularization
3.1 Computational Model Generation 3.1.1
Artery Segmentation and Revascularization
DICOM (Digital Imaging and Communications in Medicine) is a medical image format that may be used to share data while maintaining a high level of quality that meets clinical requirements. 3D Slicer, a free and open-source software is used for carotid artery segmentation. Various effects available in segment editor module are used to perform segmentation. In this work Threshold, Islands and Scissors effects are used to get desired carotid artery as shown in Fig. 1a. This is stored in STL format. Due to a lack of healthy (stent treated) patient data, the stenosed segment of the carotid artery was repaired and restored manually using Autodesk Meshmixer software an open-source 3D modelling application for generating, analysing, and optimising 3D models. This step is equivalent to carotid angioplasty. The stenosed area was removed using Autodesk Meshmixer’s 3D Sculpting brush tools. The resulting repaired carotid artery is shown in Fig. 1b.
3.1.2
Mesh Generation
The carotid bifurcation CFD models were generated with unstructured tetrahedral mesh using ANSYS Fluent. To resolve steep gradients of mean velocity present in boundary layer inflation layers are provided at the arterial wall. Figure 2 shows the inflation layers as ICA ad ECA outlet.
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Fig. 2 Inflation layers at ICA and ECA outlet
3.2 Flow Physics ANSYS Fluent is a programme that uses the Finite Volume method to solve the incompressible Navier-Stokes equation. Computational meshes are imported into it. The simulations which included SIMPLE pressure velocity coupling scheme, a second order spatial discretization of pressure and momentum and a second order implicit transient formulation were run using fluent’s pressure based solver. Blood is simulated as a Newtonian fluid with density and dynamic viscosity values of 1060 Kg/m3 and 0.003 Pa-s, because the current study concentrates on haemodynamics through big arteries. The continuity and velocity residuals were subjected to 10−4 CFD convergence conditions. Three cardiac cycles were simulated for each patient to reduce transient model effects, with the last cardiac cycle data being utilised for analysis. A time step value of 0.001 s is employed. The carotid arteries mean Re is around 300, hence the assumption of laminar flow is fair for healthy models of this artery [3]. The blood flow in carotid arteries is governed by Continuity Eq. (1) and Navier Stokes’ Eq. (2). Continuity equation ρ f ∇ · V = 0
(1)
Navier Stokes equation ρf
3.2.1
∂ V + ρ f V · V V = −∇ P + μ∇ 2 V ∂t
(2)
Boundary Conditions
The use of realistic Boundary Conditions (BCs), together with the acquisition of geometry, are essential for the validity of the simulations. The Common Carotid
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Fig. 3 Transient velocity profile at inlet [17]
Artery (CCA) serves as the entrance for the carotid artery, the blood channel upstream of the bifurcation. The Internal Carotid Artery (ICA) and External Carotid Artery (ECA), the two vessels that emerge from the bifurcation, are the two exits. In this study, transient velocity profile is applied at CCA as shown in Fig. 3 and steady pressure condition of zero-gauge pressure is applied at the 2 outlets.
3.3 Grid Independence Study By increasing the number of mesh elements by one million between each succeeding mesh, three different meshes namely coarse, medium, and fine mesh were produced for patient 1. 2 million (coarse), 3 million (medium), and 4 million (fine) mesh elements in total were taken into consideration. As shown in Table 2, the peak velocity across the whole domain and the time-averaged WSS averaged over the wall were used to compare the various meshes. It can be seen that there is not much of a difference between medium mesh and fine mesh in terms of maximum velocity or wall averaged TAWSS. For further computational models, medium mesh with 3 million elements is used since it has a shorter computational time than fine mesh. Table 2 Grid independence study Number of grid elements
2 million (coarse) 3 million (medium) 4 million (fine)
Maximum velocity over the domain 0.2492 (m/s)
0.2522
0.2524
Wall averaged TAWSS (Pa)
2.238
2.246
2.248
Computational time (hr)
11
14
22
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Fig. 4 Validation study a Re = 290 b Re = 700 [1]
3.4 Validation Validation is the process of comparing numerically anticipated model outputs to actual physical model findings. In the present study for validation purpose we have considered experimental analysis on an idealised carotid artery model using Particle Image Velocimetry (PIV) [1]. Simulations were run at Re = 290 and Re = 700, which are typical of the mean and peak systolic flow, with the assumption that the flow is stable for the sake of validation. The ICA and ECA flow division ratio was set at 7/3. Laminar flow modelling is used, as the blood flow stays laminar at the given Reynolds Number. Figure 4 shows the comparison of numerically estimated WSS along the outer wall of the ICA of idealized carotid artery models with PIV observations [1].
3.5 Novel Stent Design The concept design for a novel stent that can create a helical flow structure in bifurcated arteries in presented here. To begin, a normal carotid artery stent is built similarly to Boston Scientific Carotid Wallstent in SOLIDWORKS software. The rib height, number of ribs, rib pitch and rib revolution are the geometric parameters of triangular rib used in this study. We have examined eight different stent cases with various designs but because of certain constraints contours of other stent designs are not included here. To reduce the relative residence duration of the fluid layer closer to the wall, avoid a large pressure drop, and provide little fluid resistance in the artery channel, a helical flow pattern must be incorporated in the innovative stent design. Based on these specifications, triangular ribs are inserted on the inside wall of the stent. Figure 5 shows the front, side, and isometric view of the novel stent design used in this study.
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Fig. 5 Front, side and isometric view of novel stent design used in this study
Fluid flows across this novel stent design, but it cannot pass through the solid walls of the stent and triangle rib. So, over the novel stent design, a cylinder is constructed, and a Boolean subtraction operation is used to remove the novel stent design, yielding fluid domain. This fluid area is implanted at the CCA passage of the patient specific carotid artery at a suitable distance from the bifurcation location to enhance hemodynamics at the bifurcation region. Figure 6a depicts the computational domain of a patient’s carotid artery prior to the placement of novel stent design, Figure 6b depicts the computational domain of a patient’s carotid artery after the placement of novel stent design.
Fig. 6 a Patient-specific carotid artery (base case) b patient specific carotid artery fitted with novel stent design
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4 Results and Discussion 4.1 Helicity Indicators The helicity of the blood flow is required in order to assess the magnitude and extent of the helical flow inside the geometries. The kinetic helicity density is defined as the dot product of velocity and the vorticity vector Hk = v · (∇ × v) = v · ω
(3)
where v and ω are the velocity and the vorticity vector respectively. This variable can be normalised with regard to the vorticity and velocity magnitudes, producing the local normalised helicity (LNH), as stated [5]. LNH =
v·ω = cos θ |v||ω|
(4)
where θ denotes the angle formed by the vectors of vorticity and velocity. The LNH has a range of − 1 ≤ LNH ≤ 1. Helicoidal structures with positive (negative) values of LNH are considered to be right- (left-) handed and rotate in the clockwise direction when viewed in the direction of forward motion (counter clockwise). The local velocity and vorticity vectors’ alignment or misalignment is measured by the LNH. Figure 7 shows the LNH contours at the axial plane for the base case and the artery fitted with novel stent design. Another descriptor for swirl flow used in this study is helical intensity [5] Helical intensity, h =
1 TV
|Hk |dV dt T
(5)
V
where ‘V ’ stands for the volume of fluid in the domain, where ‘T ’ represents the heart pulse cycle. Integration is done over both volume and time (T ) to define the bulk flow parameter (V ). Take note that the temporal integration is completed for the most recent cardiac cycle. This parameter defines total amount of helical flow in the fluid domain, irrespective of direction. Table 3 summarizes helical intensity values for base case and artery fitted with novel stent design. Left-handed helical structures being induced to the arterial passage can be visualized by observing LNH contours (Fig. 7), by placing novel stent design 24% of helical content is increased in the arterial passage as seen in Table 3.
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Fig. 7 LNH contours
Base Case
Novel stent design
Table 3 Helical intensity Helical intensity Base case
12.09
Novel stent design
14.98
% Deviation from base case – 23.96
4.2 Flow Field Analysis Velocity contours at various cross-sectional planes at diastolic time step is shown in Fig. 8. One plane prior to bifurcation zone, one plane at bifurcation zone and one plane after bifurcation plane is taken into consideration. It can be seen that at all these three planes there is a reduction in size of recirculation zones when novel stent design is placed. The helical movement has caused a redistribution of kinetic energy from the arterial passage’s centre to its periphery. This helps in reducing the recirculation zones. Vorticity (Spiral Flow) is nothing but translational and rotational flow combined together. Clinically its effect is to reduce the recirculation zones and minimize the RRT. Figure 9 shows the velocity contours at an axial plane during diastolic time step. Similar observations made from velocity contours at cross-sectional plane can be made from velocity contours at axial plane. We can observe from Fig. 9 that recirculation zones are formed behind the helical rib as they are obstructing the flow path. The objective of this work is to understand how helicity will effect haemodynamics
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Fig. 8 Velocity contours at cross sectional plane
Base Case
Novel Stent design
Fig. 9 Velocity contours at axial plane
Base case
Novel stent design
of patient specific carotid artery. It can be seen that as helicity is induced into the flow there is a reduction in size of recirculation zones and RRT which are favourable,
4.3 Relative Residence Time (RRT) Analysis Figure 10 shows the RRT vs vertical distance Z plot. RRT is evaluated on the outer wall of ICA. The peak in RRT value in base case is at the location of bifurcation zone. By placing novel stent design peak value of RRT at bifurcation zone is reduced. But a zone with erratic fluctuations of RRT is present when novel stent design is placed, this zone is identified as the region where the stent is placed. Particles are residing at the walls of novel stent design which is leading to this sudden increase in RRT value. Overall, the maximum value of RRT recorded is less when novel stent design
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Fig. 10 Relative residence time (RRT) on the outer wall of ICA
is placed. In the case where stent is not deployed there is a substantial increase in RRT. When the novel stent is placed, the RRT value is significantly reduced. Hence any detrimental effects like stenosis or plaque build up will be minimised because of low RRT.
4.4 Pressure Drop An increase in the pressure drop signals an increase in artery flow resistance. The area-averaged pressure differential between the computational model’s inlet and the bifurcation section is used to determine static pressure drop. Next, a time average for the previous cardiac cycle is calculated. In the various stents which we have designed and examined this particular configuration gives the least pressure drop In future more simulations can be done by varying geometric features of the proposed stent design which will definitely provide a compromise for pressure loss. A swirl generator should not significantly reduce static pressure over time. Table 4 summarizes the pressure drop calculations. Pressure drop when novel stent design is placed is 37% more than base case. Table 4 Pressure drop Pressure drop (Pa) Base case Novel stent design
77.22 106.21
% Deviation from base case – 37.55
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5 Conclusions The present work details the numerical investigation on patient specific carotid artery with and without novel stent design to understand helicity and haemodynamic characteristics. From LNH contours and helical intensity, it can be seen that helical content in the arterial passage is increased when novel stent design is placed. Twentyfour percent of helical content is improved when novel stent design is placed when compared to base case. This resulted in the reduction of recirculation zones size and RRT. But 37% pressure drop is increased when novel stent design is placed when compared to base case. The objective behind this work is to develop an understanding of helical flow and its effects. Hence, we have investigated only on certain important parameters like helicity index, RRT etc. In the near future, investigations are to be carried out using the optimum stent design for other parameters like WSS. Acknowledgements We are grateful to the funding agency (DST-SERB) for their generous and continuous support for this study. We also thank Department of Radiodiagnosis and Imaging, Kasturba Medical College (KMC) Manipal, Manipal Academy of Higher Education (MAHE) and NIT Karnataka for the support rendered for this investigation.
Nomenclature CB CCA CTA DICOM ECA ICA IGES LNH OSI ROI RRT STL TAWSS WSS ρ μ τw Re
Carotid Bifurcation Common Carotid Artery Computer Tomography Angiography Digital Imaging and Communication in medicine External Carotid Artery Internal Carotid Artery Initial Graphics Exchange Specification Local Normalized Helicity Oscillatory Shear Index Region of Interest Relative Residence Time Stereolithography Time averaged wall shear stress Wall shear stress Density (kg/m3 ) Dynamic Viscosity (Pa-s) All shear stress (N/m2 ) Reynold’s Number
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References 1. Buchmann Nicolas A, Miharu Y, Mark J, Tim D (2010) Particle image velocimetry (PIV) and computational fluid dynamics (CFD) modelling of carotid artery haemodynamics under steady flow: a validation study. J Biomech Sci Eng 5(4):421–436 2. White CR, Frangos JA (2007) The shear stress of it all: the cell membrane and mechanochemical transduction. Phil Trans R Soc B 1459–1467 3. Lopes D, Puga a H, Teixeira J, Lima R (2020) Blood flow simulations in patient-specific geometries of the carotid artery: a systematic review. J Biomech 111 4. DiCarlo AL, Holdsworth DW, Poepping TL (2019) Study of the effect of stenosis severity and non-newtonian viscosity on multidirectional wall shear stress and flow disturbances in the carotid artery using particle image velocimetry. Med Eng Phys 65:8–23 5. Gallo D, Steinman DA, Bijari PB, Morbiducci U (2012) Helical flow in carotid bifurcation as surrogate marker of exposure to disturbed shear. J Biomech 45(14):2398-2404 6. Gataulin YA, Zaitsev DK, Smirnov EM, Fedorova EA, Yukhnev AD (2015) Weakly swirling flow in a model of blood vessel with stenosis: numerical and experimental study. St. Petersburg Polytechnical Univ J Phys Math 1(4):364–371 7. Ku DN, Giddens DP, Zarins CK, Glagov S (1985) Pulsatile flow and atherosclerosis in the human carotid bifurcation. Positive correlation between plaque location and low oscillating shear stress. Arterioscler Thromb Vasc Biol 5(3):293–302 8. Lutz RJ, Cannon JN, Bischoff KB, Dedrick RL, Stiles RK, Fry DL (1977) Wall shear stress distribution in a model canine artery during steady flow. Circul Res 41(3):391–399 9. Joel ME, Anburajan M (2013) 3D Modelling of stenotic internal carotid artery treated with stent: A CFD analysis of blood. Int Conf Comput Netw Commun Eng 10. Wissgott C, Schmidt W, Behren P, Brandt C, Schmitz KP, Andresen R (2014) Experimental investigation of modern and established carotid stents. Fortschr Röntgenstr 186:157–216 11. Linge F, Hyde MA, Paul MC (2014) Pulsatile spiral blood flow through arterial stenosis. Comput Methods Biomech Biomed Engin 17(15):1727–1737
Numerical Simulation of Flow in an Idealized Intracranial Aneurysm Model to Study the Effect of Non-newtonian Blood Flow Rheology Suraj Raj, S. Anil Lal, and Anjan R. Nair
Abstract Rupture risk assessment of intracranial aneurysms has gained relevance in the recent years. Serious health issues such as Subarachnoid Haemorrhage (SAH), damage to the brain and death can occur if an aneurysm ruptures. Rupture risk prediction based on size cannot be considered accurate. There is no clinical method to predict rupture risk of aneurysm so far. Computational modelling is highly relevant in the study of aneurysms since the experimental approach is difficult. Thus, the rupture risk can be predicted using Computational Fluid Dynamics (CFD). Although blood has non-Newtonian rheology, most CFD research to date have assumed Newtonian behaviour. Comparative study of various available non-Newtonian models and scope of developing a reliable model for intracranial blood flow simulations still remains. In the present work, numerical simulations have been carried out on idealized 2D half spherical aneurysm geometry to study the basic flow patterns and the impact of blood rheology on wall shear stress (WSS). Keywords Intracranial aneurysm · Risk assessment · Non-Newtonian blood rheology · Wall shear stress
1 Introduction Cerebral aneurysm is a disease affecting the cerebral artery or vein. It is characterised by local dilation of the vasculature wall when it weakens. In many cases, it is asymptomatic. The likelihood of cerebral aneurysms rupturing is influenced by both S. Raj (B) · A. R. Nair Department of Mechanical Engineering, College Of Engineering Trivandrum, Trivandrum, Kerala 695016, India e-mail: [email protected] APJ Abdul Kalam Technological University, Trivandrum, Kerala 695016, India S. A. Lal School of Engineering, Amrita Vishwa Vidyapeetham, Amritapuri Campus, Karunagapally, Kollam, Kerala 690525, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_57
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non-modifiable and modifiable factors [1]. The modifiable factors include hypertension, smoking habits, alcohol abuse etc. The non-modifiable factors mainly include family history and genetic disorders. There are three types of cerebral aneurysms which occur. They are (i) saccular type (ii) fusiform (iii) dissecting type. The most common type of aneurysm of all occurring intracranial aneurysms is the saccular type aneurysm. They resemble berry or sac and occur at the locations where branching occurs for the arteries [2]. Rupture of aneurysm can cause subarachnoid haemorrhage, brain damage, and even death. Although clinical care for subarachnoid haemorrhage patients has significantly improved, one quarter of the survivors still die, while approximately half live with persistent neurological disorders [3]. There are mainly three methods to treat aneurysms to prevent rupture. They are the endovascular coiling, flow diversion using stent and surgical clipping by craniotomy. Recent progress and greater use of neurovascular imaging has improved the identification of asymptomatic intracranial unruptured aneurysms. To decide which unruptured aneurysm to treat first has become more difficult as a result of this. The decision making also involves whether to resort to endovascular treatment (coiling or flow diversion) or clipping. Both endovascular and microsurgical procedures have a risk of related morbidity and mortality. Therefore, an objective method for aneurysm rupture risk assessments is really important, which can reliably predict the aneurysms with highest risk. Understanding the implications of hemodynamic stresses on the origin, development, and rupture of cerebral aneurysms has been the focus of recent research. Mechanical engineering and molecular biology study the impact of haemodynamics on the pathophysiology of intracranial aneurysms. The most frequent factor utilised in clinical practise in the past to determine the likelihood of rupture was the size of the aneurysm. But depending solely on aneurysm size actually does not address the complexity of the actual problem. Hemodynamic as well as morphological parameters related to aneurysm must be considered. These factors include the aneurysm’s location, size, shape, presence of daughter sacs, blood flow direction through the aneurysm, aneurysm neck diameter, blood flow angle with respect to the parent artery, impingement size, residence time of blood inside the aneurysm, Wall Shear Stress (WSS), normal stress, and impingement size. Anyone with enough knowledge about cerebral aneurysms may draw the main conclusion that the interaction of the aforementioned variables is what causes aneurysms to grow and eventually rupture. Aneurysm rupture risk has been more accurately predicted, and efforts have been undertaken to comprehend the mechanisms underlying their genesis, growth, and rupture. Since the experimental technique is challenging in many research, computational modelling is extremely pertinent. The assessment of rupture risk using computational fluid dynamics (CFD) has been given some thought [4–6]. The outcomes of CFD simulations can yield thorough diagnostic data that is helpful for a better comprehension of cerebral aneurysms and the danger of rupture that goes along with them [7, 8].
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2 Literature Review and Objective There have been numerous experimental and numerical investigations on cerebral aneurysm reported in the literature. Using idealized geometric models moulded in glass and flow being visualised using dye injection method, the physical factors related to intracranial aneurysm were analysed and the following hypothesis are made. The initiation of aneurysm is caused by local damage to the internal elastic membrane due to forces resulting from bloodstream axial impingement on the aneurysm walls. Aneurysm growth is due to the wall vibrations resulting from the turbulence in aneurysm sac. The rupture of aneurysm is attributed to the decrease in wall strength as a result of the increase in aneurysm size, decrease in wall thickness and rise in blood pressure [9–11]. A significant central zone of recirculation (zone of low shear stresses) was observed in all types of aneurysms in one of the earliest numerical studies, which was conducted in three idealised aneurysm shapes (half-spherical, spherical, and pearshaped) representing different stages of growth of cerebral aneurysm [12]. Predictions of rupture risk based on morphology had an impact. It is widely accepted that bigger aneurysms have a significant likelihood of rupturing. The size of an aneurysm is defined by the mean diameter of the aneurysm. There are exceptions to this theory, however, as small aneurysms also rupture whereas larger aneurysms remain un-ruptured [13–15]. In addition to size and shape, intra-aneurysmal flow structures are also influenced by the topology of the parent vessel, which directly affects how blood flows from the parent arterial to the aneurysm sac [16]. The flow field through ideal models of saccular aneurysms on curved arteries was numerically simulated, and the results showed that the already weakened wall of the aneurysm becomes more susceptible to dilation in conjunction with stretching from the distal side of neck brought on by increased pulsatile pressure. Additionally, the aneurysm grows to include healthier sections of the parent artery as the wall deterioration spreads from the distal neck. The aneurysm thus grows, continuously suffers flow impingement, and further degrades and stretches As a result, higher arterial curvature and aneurysm neck size enhance the probability of aneurysm growth [17]. Wall Shear Stress (WSS) has been used as an important parameter for the rupture risk assessment of cerebral aneurysms. The interpretation based on this parameter is controversial, though. According to certain findings [18], aneurysm rupture is caused by increased WSS. On the other hand, some publications claim that low WSS causes the thinning and degeneration of artery walls, which in turn causes rupture [19]. Another numerical study’s findings revealed that abnormal haemodynamics in both low and high WSS can cause cerebral aneurysms to form and rupture through multiple biological mechanisms [20]. The origin and evolution of vascular disorders are significantly influenced by hemodynamic flow characteristics such recirculations, jet impingement, and secondary flows, according to a recent numerical research on intracranial aneurysm [21]. Blood has been regarded as a Newtonian fluid in the majority of CFD investigations [14, 22, 23].
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The present study is based on an idealized 2D half spherical geometry of intracranial aneurysm which represents one of the early stage of intracranial aneurysm. By assuming that blood is a Newtonian fluid, the fundamental flow patterns within the half-spherical saccular type aneurysm are examined. Blood is a shear-thinning non-Newtonian fluid. Thus, the Newtonian fluid assumption for blood with constant viscosity deviates from actual rheological behaviour. Therefore, the impact of nonNewtonian blood flow rheology on WSS acting on the parent artery wall containing the aneurysm is also examined.
3 Model Geometry and Computational Methodology A schematic diagram of idealized 2D geometry of a half spherical aneurysm model selected for validating the numerical method is shown in Fig. 1. The aneurysm taken into consideration in this study had no neck, and the angles at which it connected to the parent artery were extremely acute. The diameter of the aneurysm ostium (base) is taken as 2R, where R is the radius of the parent artery. In the numerical simulations, three Reynolds numbers 100, 400, and 700 were employed. Here, the Reynolds number is defined as Re = UD/ϑ, where U is the input velocity, d is the parent artery’s diameter, and ϑ is the blood’s kinematic viscosity. These Reynolds number values indicate the flow variability seen in various people under various hemodynamic situations [12].
3.1 Computational Domain and Governing Equations The geometry was generated in ANSYS spaceclaim 2019 R2. A straight tube with a diameter of 4 mm (the average size of the internal carotid artery) represents the parent artery or blood vessel. The proximal end of the aneurysm is 10 mm from the inlet and on the downstream side the outlet is at 10 mm from the distal end of
Fig. 1 Schematic diagram of half spherical aneurysm model
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aneurysm [12]. A semi-circular portion of 2 mm radius in the upper wall of parent artery represents the saccular type aneurysm in the present study. Thus, the aneurysm size (height of dome from the base) in the present study is 2 mm which is considered as a small sized aneurysm as in literature [27]. The flow was assumed to be steady and laminar in nature for the Reynolds numbers investigated. Flow is fully developed at the inlet to the parent artery. This is incorporated using User-defined Function (UDF) written for 2D fully developed parabolic velocity profile. The UDF is compiled and then integrated in ANSYS FLUENT 2019 R2. The aneurysm and parent artery walls are subjected to a no-slip boundary condition for velocity components. Using ANSYS Fluent 19.0 R2, steady state Navier–Stokes equations are solved for the present 2D numerical analysis. The governing equations that the solver uses are: Equation of continuity ∇ · (ρu) = 0
(1)
∇ · (ρuu) = −∇ p + ∇.(μ∇.u)
(2)
Equation for momentum
where ρ is the density of blood, u represents the velocity vector, p is pressure and μ is dynamic viscosity of blood. Carreau viscosity model, that has been proven to be suitable for modelling actual blood, is used to represent the shear thinning behaviour of blood for non-Newtonian blood flow simulations. [25, 26]. n−1/2 μeff (γ˙ ) = μinf + (μ0 − μinf ) 1 + (λγ˙ )2
(3)
Here μ0 , μinf represents viscosity at zero shear rate and viscosity at infinite shear rate respectively, n represents a power index and λ the relaxation time. These represent material coefficients and their values for actual human blood are λ = 3.313 s, n = 0.3568, μ0 = 0.056 Pa.s and μinf = 0.003450 Pa.s.
3.2 Computational Mesh and Details of Grid Convergence Study ANSYS ICEM CFD was used to create the computational mesh (Fig. 2). The Roache Method is used to conduct the grid convergence study in a methodical manner [24]. The details are provided in Table 1 for the coarse, medium, and fine quadrilateral cell meshes, which were numbered 1, 2, and 3 accordingly. The Roache Method states that GCICoarse = 3|ε|r o /(r O -1), GCIFine = 3|ε|/(r O -1) and grid independence is attained when GCICoarse ≈ r 0 (GCIFine ). As a result, Mesh 2 is chosen as the final mesh. GCI
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Table1 Details of the meshes and maximum WSS Sl. No
Number of nodes
Maximum WSS in aneurysm dome (Pa)
Mesh 1
175,626
1.0517642
Mesh 2
213,456
1.0510541
Mesh 3
261,264
1.0503811
Table 2 Roache grid convergence study details
r 32
1.223
r 21
1.2154
ε32
0.00673
ε21
0.007
GCICoarse = 3|ε|r o /(r O -1)
6.5006
GCIFine = 3|ε|/(r O −1)
4.0727
r o *(GCIFine )
6.1916
stands for Grid Convergence Index, r is the ratio of generated grid’s element counts, o is the assumed order of accuracy (here taken as 2) and ε is the difference between a property value determined at a given grid position in two separate meshes. GCIcoarse and GCIFine reflect the grid convergence index between grids 2 and 1 and grids 3 and 2, respectively. Table 2 provide information of the various grid convergence parameters. The maximum wall shear stress (WSS) in the aneurysm dome was chosen as the parameter for the grid independence investigation. The WSS metric was chosen since it has been noted to be a significant parameter impacting the assessment of the risk of cerebral aneurysm rupture.
3.3 Numerical Modelling Blood flow simulations were performed inside the half-spherical aneurysm model using ANSYS FLUENT 2019 R2 (ANSYS, Inc.). Using the SIMPLE pressure– velocity coupling method and a laminar solver, the governing incompressible Navier– Stokes equations were solved. Upwind central and second order upwind were used for the spacial discretization of the pressure and velocity components. The threshold for residual error convergence was chosen as 10−4 . The numerical validation is conducted under the assumption that blood is a Newtonian fluid with kinematic viscosity (ϑ) of blood set at 0.0035 Pa.s and density ρ = 1060 kg/m3 respectively. The Carreau viscosity model is utilised for non-Newtonian simulations.
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Fig. 2 Computational mesh with an enlarged view in the vicinity of aneurysm dome
4 Results and Discussion 4.1 Numerical Validation The values of wall shear stress for a half -spherical geometry along the upper wall for the entire length has been obtained for Reynolds numbers 100, 400, and 700 and the results have been validated against those reported in [12] as shown in Fig. 3. Table 3 shows the comparison of the maximum value of wall shear stress in the walls of the parent vessel and the aneurysm. The comparison shows reasonable agreement with the results presented in [12].
4.2 Flow Features In all the three cases of Reynolds numbers studied, three basic flow patterns were noted. (i) For the distal end (downstream side), there is volume flow into aneurysm from the parent artery
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Fig. 3 Validation
Table 3 Maximum WSS (in dynes/cm2 ) in the aneurysm and parent artery Region of wall
Ostium width
Re
Maximum WSS (present study)(Pa)
Maximum WSS [12] (Pa)
Half spherical aneurysm
2R
100
1.05
1.03
400
3.5
3.46
700
5.98
5.92
100
0.58
0.57
400
2.27
2.27
700
4.01
3.97
Parent artery
2R
(ii) The aneurysm sac represents a low velocity recirculation zone (iii) For the Proximal end (upstream side) of the aneurysm, there is outflow from the aneurysm sac into the parent artery. Change in Reynolds number influences the flow pattern as shown in the streamline plots in Figs. 4, 5, and 6.
4.3 Newtonian vs Non-Newtonian Blood Rheology Numerous numerical studies have relied on the Newtonian assumption rather than taking into account the non-Newtonian character of blood. However, in order to accurately predict haemodynamics, along with associated flow patterns and fluid mechanical forces, the real blood flow rheology must be taken into consideration.
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Fig. 4 Streamline plot (Re = 100)
Fig. 5 Streamline plot (Re = 400)
Simulations were performed in the present work by taking into consideration the non-Newtonian rheology of blood, and changes in WSS were compared to those under Newtonian consideration for the half-spherical geometry. The comparison of Newtonian and non-Newtonian blood rheology is shown in Figs. 7 and 8 by taking into account the WSS fluctuation along the top wall containing the aneurysm at Reynolds numbers 100 and 400. The axial distance along the wall (x) containing the aneurysm is non-dimensionalized by the diameter of the parent artery (D). At both the Reynolds numbers studied (100, 400), the Newtonian consideration leads to underestimation of WSS. Given its significance as a hemodynamic parameter, WSS should provide quantitatively reliable predictions of the impact on the aneurysm
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Fig. 6 Streamline plot (Re = 700) Fig. 7 WSS versus x/D at Re = 100
wall. Therefore, for correct qualitative analysis in hemodynamic simulations, nonNewtonian models should be taken into account. In assessing the risk of cerebral aneurysm rupture, this would in fact produce more accurate results.
5 Conclusions The WSS values predicted for Reynolds numbers 100, 400, 700 shows similar variations as in [12] with a minimum percentage error. When compared to the parent artery, the aneurysm dome’s WSS values are lower, indicating lower velocity areas inside
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Fig. 8 WSS versus x/D at Re = 400
the aneurysm. This indeed may lead to stagnation zones within the aneurysm and affecting the normal functioning of the endothelial cells of aneurysm wall compared to the parent artery. The growth and rupture mechanism can be attributed to lower WSS values and the stagnation of blood within the aneurysm sac. Thus it can concluded that the low wall shear stress initiates an inflammatory cell mediated mechanism of intracranial aneurysm growth and rupture. When compared to the Newtonian fluid assumption for blood, the non-Newtonian consideration results in higher levels of WSS within the aneurysm. The observed trend as well as aneurysm representing stagnation zones signifies the relevance of considering non-Newtonian models for blood flow simulations within the aneurysm to take into account the property of blood to thin under shear, as well as to make precise numerical predictions regarding the evaluation of risk of aneurysm rupture.
Nomenclature Re ρ ν r o WSS
Reynolds number Density of blood [kg/m3 ] Kinematic viscosity [Pa.s] Grid refinement index Order of accuracy Wall shear stress [Pa]
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References 1. Aoki T, Nozaki K (2016) Preemptive medicine for cerebral aneurysms. Neurol Med Chir 56(9):552–568 2. Weir B (2006) Unruptured intracranial aneurysms: a review. J Neurosurg 96(1):3–42 3. Gao BL, Hao H, Hao W, Ren CF, Yang L, Han Y (2022) Cerebral aneurysms at major arterial bifurcations are associated with the arterial branch forming a smaller angle with the parent artery. Sci Rep 12(1):1–10 4. Jou LD, Quick CM, Young WL, Lawton MT, Higashida R, Martin A, Saloner D (2003) Computational approach to quantifying hemodynamic forces in giant cerebral aneurysms. Am J Neuroradiol 24(9):1804–1810 5. Rayz VL, Boussel L, Acevedo-Bolton G, Martin AJ, Young WL, Lawton MT, Higashida R, Saloner D (2008) Numerical simulations of ow in cerebral aneurysms: comparison of CFD results and in vivo MRI measurements. J Biomech Eng 130 6. Berg P, Vob S, Becker M, Serowy S, Oliver (2016) Bringing hemodynamic simulations closer to the clinics: a CFD prototype study for intracranial aneurysms. In: 2016 38th Annual international conference of the IEEE engineering in medicine and biology society (EMBC), pp 3302–3305 7. Qian Y, Takao H, Umezu M, Murayama Y (2011) Risk analysis of unruptured aneurysms using computational uid dynamics technology: preliminary results. Am J Neuroradiol 32(10):1948– 1955 8. Valen-Sendstad KristianMardal, Mortensen K-A, Reif M, Langtangen BAP, Petter H (2011) Direct numerical simulation of transitional ow in a patient-specific intracranial aneurysm. J Biomech 44(16):2826–2832 9. Ferguson GG (1972) Physical factors in the initiation, growth, and rupture of human intracranial saccular aneurysms. J Neurosurg 37(6):666–677 10. Ferguson GG (1970) Turbulence in human intracranial saccular aneurysms. J Neurosurg 33(5):485–497 11. Roach MR, Scott S, Ferguson GG (1972) The hemodynamic importance of the geometry of bifurcations in the circle of Willis (glass model studies). Stroke 3(3):255–267 12. Burleson AC, Strother CM, Turitto VT (1995) Computer modeling of intracranial saccular and lateral aneurysms for the study of their hemodynamics. J Neurosurg 37(4):774–784 13. Ohashi Y, Horikoshi T, Sugita M, Yagishita T, Hideaki N (2004) Size of cerebral aneurysms and related factors in patients with subarachnoid haemorrhage. Surg Neurol 61(3):239–245 14. Kyriacou SK, Humphrey JD (1996) Influence of size, shape and properties on the mechanics of axisymmetric saccular aneurysms. J Biomech 29(8):1015–1022 15. Russell SM, Lin K, Hahn SA, Jafar JJ (2003) Smaller cerebral aneurysms producing more extensive subarachnoid hemorrhage following rupture: a radiological investigation and discussion of theoretical determinants. J Neurosurgery 99(2):248–253 16. Sforza DM, Putman CM, Cebral JR (2009) Hemodynamics of cerebral aneurysms. Ann Rev Fluid Mech 41:91–107 17. Hoi Y, Meng H, Bendok SH, Hanel BR, Guterman RA, Hopkins LR, Nelson L (2004) Effects of arterial geometry on aneurysm growth: three-dimensional computational fluid dynamics study. J Neurosurgery 101(4):676–681 18. Cebral JR, Mut F, Weir J, Putman C (2011) Quantitative characterization of the hemodynamic environment in ruptured and unruptured brain aneurysms. Am J Neuroradiol 32(1):145–151 19. Kono K, Fujimoto T, Shintani A, Terada T (2012) Hemodynamic characteristics at the rupture site of cerebral aneurysms: a case study. Neurosurgery 71(6):E1202–E1209 20. Meng H, Tutino VM, Xiang J, Siddiqui A (2014) High WSS or low WSS? complex interactions of hemodynamics with intracranial aneurysm initiation, growth, and rupture: toward a unifying hypothesis. Am J Neuroradiol 35(7):1254–1262 21. Jeong W, Rhee K (2012) Hemodynamics of cerebral aneurysms: computational analyses of aneurysm progress and treatment. Comput Math Methods Med 2012
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22. Zhang Z, Xu L, Liu R, Liu X, Zhao B, Liang F (2019) Importance of incorporating systemic cerebroarterial hemodynamics into computational modeling of blood flow in intracranial aneurysm. J Hydrodyn 1–14 23. Feletti A, Wang X, Talari S, Mewada T, Mamadaliev D, Tanaka R, Yamada Y, Kei Y, Suyama D, Kawase T et al. (2018) Computational fluid dynamics analysis and correlation with intraoperative aneurysm features. Trends Manage Cerebrovasc Dis 3–9 24. Roache PJ (1994) Perspective: a method for uniform reporting of grid refinement studies. J Fluids Eng 116(3):405–413 25. Cho YI, Kensey KR (1991) Effects of the non-newtonian viscosity of blood on flows in a diseased arterial vessel. Part 1: Steady Flows, Biorheol 28:241–262 26. Jabir E, Lal SA (2016) Numerical analysis of blood flow through an elliptic stenosis using large eddy simulation. Proc Inst Mech Eng 230(8):709–726 27. Merritt WC, Berns HF, Ducruet AF, Becker TA (2021) Definitions of intracranial aneurysm size and morphology: a call for standardization. Surg Neuro Int 12
On the Replication of Human Skin Texture and Hydration on a PDMS-Based Artificial Human Skin Model Aditya Ranjan, Vijay S. Duryodhan, and Nagesh D. Patil
Abstract In the current study, we have developed an artificial skin sample and replicated the textures and hydration of the human skin on it. Polydimethylsiloxane (PDMS), Polyvinyl alcohol (PVA), and Glutaraldehyde (GA) are used to fabricate an Epidermal Skin Equivalent (ESE) of human skin. Additionally, a PVA-based hydrogel is synthesized, and its hydration property is examined at varying concentrations of PVA. The values of roughness (6.2–10.8 µm), wettability (72–122°), and hydration (10–40% after 24 h) of artificial skin samples fabricated in the present study are found to be very close to that of human skin. The hydration of PVA hydrogel is observed to increase (17–96%) with an increase in PVA concentration (10–30%). Our results highlight that the fabricated artificial skin model closely resembles the surface and hydration characteristics of human skin. Keywords Epidermal skin equivalent · Artificial skin sample · Skin texture · Hydrogel · Hydration
1 Introduction Human skin is one of the vital organs in the body. It comprises mainly three different layers, namely the epidermis, dermis, and hypodermis [1]. Each layer has a distinct composition, thickness, and functional properties. It acts as a protective layer for the inner body parts and has various functions, such as providing pathways for drug delivery, protection from pathogens, UV radiation, water loss, and injury [2]. So, it becomes very important to develop a scientific understanding of skin by experimentally analyzing it. This will be helpful in the development of cosmetic products, delivering a drug through the skin, or improving skin comfort. However, due to A. Ranjan · V. S. Duryodhan · N. D. Patil (B) Department of Mechanical Engineering, IIT Bhilai, Raipur 492015, India e-mail: [email protected] N. D. Patil Department of Bioscience & Biomedical Engineering, IIT Bhilai, Raipur 492015, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_58
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various ethical issues, human skin is not used for analysis. The need for a human skin alternative arises mainly to evaluate the permeation of various drugs or compounds (in vitro study). Animal skin, cultured skin, and physical skin models are widely used as its alternative. Porcine skin and rat skin are found to closely resemble human skin [3, 4]. Artificial skin is also prepared using cell culture; however, they are expensive, difficult to store, and could not exactly replicate the actual human skin. However, the physical skin model has certain advantages over cultured skin, such as it is less expensive, easy to store and handle, and helps to tune the properties [5].
2 Literature Review and Objective In vivo study of drug diffusion via human skin is not possible due to several risks associated with drug contact with human skin, as well as ethical concerns. However, as shown in Table 1, some animal skin, such as a rat or porcine skin, is discovered to be quite similar to human skin, making it an excellent alternative for conducting numerous experiments [6]. Human skin is heterogeneous and anisotropic [7] having multiple physical properties such as thermal, surface, mechanical, electrical, optical, etc. [5]. To replicate all these properties in a single artificial sample is very difficult. Thus, different materials are used to replicate the specific properties of human skin. For example, polyurethanebased skin models are used to replicate the surface characteristics of human skin, or liquid suspension is used to replicate the optical properties [5]. Morales-Hurtado et al. [2] fabricated a PDMS-based epidermal skin equivalent and successfully replicated the roughness, elastic modulus, and hydration of human skin. Various research suggests that the skin is slightly hydrophobic in nature, with roughness lying in the range of microns. Ohtsuki et al. [8] report the roughness value of human skin to lie in the range of 7–16 µm. Elkhyat et al. [9] studied the wettability of human skin and confirmed that the skin sites poor in lipids (having a lesser number of sebaceous glands) are hydrophobic with contact angles ranging from 91º–102º. However, according to Rabost-Garcia et al. [10], the skin surface is slightly hydrophobic, with the contact angle oscillating between 80º and 110º. In contrast, the hydration of the stratum corneum of a normal human lies in the range of 15–40% [2]. Various skin models have successfully replicated the skin texture. Nachman and Franklin [11] fabricated an Elastosil-based skin model in which Alja safes breeze Table 1 Skin thickness comparison for various species at different anatomical locations Species
Stratum corneum (µm)
Epidermis (µm)
Whole skin (µm)
Human [3]
18.2 ± 3.3
51.2 ± 12.2
2.58 ± 0.07
Pig [3]
17.5 ± 2.4
50.7 ± 11.4
1.74 ± 0.18
Rat [4]
18
32
2.09
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is used to replicate the skin texture. Conventional silicone elastomers like Elastosil are hydrophobic and are not able to absorb water. So, it is modified with hydrophilic alpha olefin sulfonate, which helped to create a hydrophilic water-absorbing skin model. This skin model closely resembles the surface and mechanical properties of actual human skin [11]. Lir et al. [12] fabricated a Gelatin-based skin model in which Silflo is used to replicate the skin texture. In the present work, a PDMS-based ESE is fabricated because its fabrication process is not complex, and previous research findings [2] suggest that it closely mimics the surface and mechanical properties of the skin. PDMS is mixed with PVA and GA to produce the skin equivalent. Human skin is viscoelastic and has an elastic modulus in the range of a few MPa. PDMS has an elastic modulus in the same range, but it is hydrophobic and hence cannot absorb water. This is mixed with PVA to increase its ability to retain water [2]. In the present work, PVA hydrogel has also been fabricated. Hydrogels are threedimensional networks of cross-linked polymers [13] that are employed in a variety of biological applications such as biosensors, drug delivery vectors, and carriers or matrices for cells in tissue engineering. When PVA molecules are frozen and thawed again, they cross-link physically. Also, PVA gels are widely cross-linked with Glutaraldehyde or Formaldehyde to produce insoluble networks for biological application [14]. The presence of hydroxyl groups in the structure of PVA allows for Glutaraldehyde to be used in the reaction. The hydroxyl groups of PVA react with Glutaraldehyde at 90 °C to generate a cross-linked network. This reaction requires an acidic medium, which is already present in the solution [15]. As an acidic medium, H2 SO4 (96–98%) can be used. Based on the literature review, it is found that the replication of skin texture on a PDMS-based skin model is not reported. Also, the effect of PVA concentration on the hydration of PDMS-based skin equivalents is rarely found. Therefore, the objectives of the present work are as follows: • Replication of human skin texture on PDMS- based skin equivalent. • Effect of PVA concentration on the hydration of skin equivalent and hydrogel.
3 Materials and Methods For the fabrication of PDMS-based epidermal skin equivalent, the materials used are Polyvinyl Alcohol (PVA) (Mw 31,000–50,000, 87–89% hydrolyzed) from SigmaAldrich, Glutaraldehyde (GA) (50 wt.% in H2 O) from Sigma-Aldrich, H2 SO4 (98% solution in water), PDMS with curing agent, Teflon sheet, Fevicol SH, and Sodium Chloride (NaCl) [2].
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3.1 Fabrication Method The fabrication of the artificial skin sample starts with preparing a 20 wt.% (w/ v) solution of PVA in distilled water (DW) and a 2 wt.% (v/v) solution of GA in distilled water in a separate round-bottom (RB) flask. The magnetic stir bar is dipped in both flasks, and the flask is fixed to a reflux unit. The solution is stirred for around 12 h at 90 °C and around 250 rpm. After 12 h, both solutions are taken out of the reflux unit. 10 wt.% H2 SO4 solution is added to the prepared PVA solution in the RB flask and stirred for around two minutes. This is to be done to initiate cross-linking between PVA and GA. PDMS is prepared in a 20:1 ratio of PDMS and curing agent. The prepared PDMS is desiccated to remove air bubbles. A Teflon sheet is used to prepare the mold to which the final solution will be transferred. Fevicol (adhesive) is poured on the forearm of a 25-year-old male volunteer to replicate the skin texture. Volunteer skin is properly cleaned before pouring. After pouring an adequate amount, the adhesive is properly spread and allowed to dry. After drying, the adhesive is carefully peeled off, and a section is cut and pasted on the Teflon mold using double-sided tape. The separately prepared PVA and GA solutions are mixed in a 10:1 ratio and stirred for around 1 min. PDMS is now mixed with prepared PVA-GA solution in the ratio of 7:3. Further, 1 wt.% of NaCl is added to the mixture and stirred manually for 1 min. The final mixture is then transferred to the Teflon mold having a negative replica of skin glued to the bottom of the mold. The mold is then covered and placed in the oven for 24 h at 40 °C. After 24 h, the mold is taken out, and the final sample is peeled off carefully so that the transferred texture does not get affected [2]. The final mixture in the Teflon mold and a negative replica of the skin is shown in Figs. 1a, b, respectively. PVA hydrogels are prepared by mixing the above-prepared PVA and GA solution in a 10:1 ratio. However, to understand the role of PVA concentration on hydration, 10, 15, 20, 25, and 30% (w/v) PVA solutions in distilled water are used to prepare hydrogels. Fig. 1 a Final PDMS, PVA-GA mixture in teflon mold, and b negative skin replica as observed under optical profilometer
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3.2 Characterization The peeled-off sample is placed properly in the petri dish with the textured surface facing upward, and the dish is covered properly to prevent any dust contact with it. For surface texture and hydration analyses, the sample is gently cut in half. Using an optical profilometer, the surface is characterized. The fabricated hydrogel and skin sample are weighed first before being immersed in distilled water in order to examine the hydration of artificial skin samples and hydrogels with various PVA concentrations. After that, the samples’ weights are recorded each day to determine the sample’s hydration level.
4 Results and Discussion The surface and hydration characterization of the fabricated artificial human skin sample and hydrogel is performed, and its results have been discussed in the subsequent section.
4.1 Surface Characterization Skin provides an alternative route for drug delivery. When a drug or compound is placed on the skin (considering passive diffusion), it becomes important to characterize the surface because the larger the spreading, the greater the permeation area of the drug. In the present work, surface and hydration characteristics of fabricated skin samples have been analyzed. To characterize the fabricated artificial skin sample for its surface roughness, an arithmetic mean height (Ra value) is obtained using an Optical profilometer. The contact angle of the sessile water droplet is also measured using the Kruss Advance Droplet Shape Analyzer (as shown in Fig. 2). The contact angle shown in Fig. 2a is 92º, and Fig. 2b is 107º which describes the hydrophobic nature of the fabricated substrates. Table 2 illustrates the contact angle and roughness related to various fabricated samples. The range of the measured skin Fig. 2 Sessile droplet contact angle measurement using droplet shape analyzer
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Table 2 Average roughness value and water droplet contact angle for the fabricated skin sample Sample No.
Arithmetic roughness value (Ra) in µm
Droplet contact angle
a
6.20
93º–109º
b
6.36
72º–104º
c
7.44
109º–126º
d
10.15
89º–113º
e
10.84
91º–103º
Fig. 3 Optical profilometer images of four skin samples
roughness Ra value, which is almost comparable to actual human skin, is determined to be 6–11 microns. In Fig. 3, images of four skin samples are displayed. Images of the fabricated skin sample are acquired using an Optical profilometer. These results allow us to draw the conclusion that the fabricated skin sample has human skin’s surface features because its contact angle and Ra value are within the same range.
4.2 Hydration Characterization Three hydrogel samples corresponding to PVA concentration varied from 10 to 30% are fabricated. Each sample is dipped in distilled water for 24 h and their hydration is determined using Eq. 1 shown below [2].
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Hydration % =
705
Swollen weight − Dry Weight ∗ 100 Dry weight
(1)
The curve of hydration of PVA hydrogels (in percent) versus PVA concentration is shown in Fig. 4. The average hydration of 10%, 15%, 20%, 25%, and 30% PVA concentration hydrogels measured after one-day span are found to be 17.3%, 48.7%, 47.9%, 65.4%, and 95.8%, respectively. It can be well understood from the curve that the hydration content of PVA hydrogel increases upon increasing the PVA concentration. This happens because PVA consists of hydroxyl groups, which increase upon increasing the PVA concentration. Therefore, more hydroxyl groups are now available for interaction with water molecules through hydrogen bonds. It is to be noted that the solubility of PVA in water is inversely proportional to its degree of hydrolysis and molecular weight [16]. In the present work, partially hydrolyzed PVA (87–89%) has been used instead of fully hydrolyzed PVA (>99%). Fully hydrolyzed PVA has a strong intramolecular H-bond that results in its near insolubility in water at lower temperatures and is only soluble in hot water (above 50ºC). The solubility of partially hydrolyzed PVA is nearly independent of its molecular weight while increasing with a decrease in molecular weight for fully hydrolyzed PVA (due to lesser intramolecular interaction) [17]. Figure 5 represents the variation of hydration of artificial skin samples versus the day for which samples were dipped into the water. Two different ESE models are fabricated using 20 and 25% PVA concentrations. Three different samples for each concentration of PVA are fabricated, and their hydration is calculated using Eq. 1. The hydration is monitored for seven days, and a standard deviation error bar is plotted as shown in Fig. 5. It is evident that the hydration of both ESE samples increases with time. It can also be understood that the hydration of the skin sample increases upon increasing the PVA concentration. The average hydration of the 20% and 25% PVA Fig. 4 Mean hydration of PVA hydrogel with standard deviation error bars
120
Hydration (%)
100
80
60
40
20
0
5
10
15
20
25
PVA concentration (%)
30
35
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Fig. 5 Mean hydration of ESE with standard deviation error bars
80
20% PVA conc. 25% PVA conc
70
Hydration (%)
60 50 40 30 20 10 1
2
3
4
5
6
7
Day
concentration-based ESE sample is found to increase from 10.34% and 21.98% on the first day to 35.14% and 52.24% on the seventh day, respectively. The hydration of the stratum corneum of human skin lies in the range of 15–40%. Hence, based on the above data, it can be concluded that ESE fabricated using 20% PVA is closer to providing the hydration characteristics, same as that of skin compared to 25% PVA concentration. The hydration is associated with two factors: (a) the presence of hydroxyl groups which allows interaction with water molecules through hydrogen bonds [16]; (b) the presence of porosity in the sample. Due to the material’s porosity, it allows water to permeate and then diffuse through its interior structure. From Fig. 5, we can infer that the larger the time for which the sample is dipped in distilled water, the larger its hydration.
5 Conclusions An artificial skin sample is fabricated using PDMS, PVA, and GA. The objective is to replicate the skin texture and hydration characteristics. The hydrogel of different PVA concentrations is also fabricated to understand its role in hydration. Various skin samples are fabricated having different PVA concentrations, and their hydration is investigated. The arithmetic mean height, Ra value of fabricated ESE is found to lie in the range of 6.2–10.8 µm while that of actual human skin lies in the range of 7–16 µm. The sessile water droplet contact angle on the ESE is found to lie in the range of 72°–122°. The droplet contact angle on human skin oscillates between 80° and 110°, which indicates that it is hydrophobic. The hydration of PVA hydrogel is found to increase upon increasing the PVA concentration. The average hydration
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of PVA hydrogel (after 24 h) is found to lie in the range of 17% for 10% PVA concentration to 96% for 30% PVA concentration. Similarly, skin equivalents with 20 and 25% PVA concentrations are also fabricated, and it is found that hydration of ESE increases upon increasing PVA concentration. Average hydration of 20 and 25% PVA concentration-based ESE is found to be 35 and 52% after 7 days. Therefore, it can be concluded that the fabricated skin sample closely resembles human skin in terms of its surface properties, wetting behavior, and hydration. Such artificial skin samples are very much useful to carry out experimental studies on passive drug diffusion through the skin.
References 1. Alkilani AZ, McCrudden MTC, Donnelly RF (2015) Transdermal drug delivery: Innovative pharmaceutical developments based on disruption of the barrier properties of the stratum corneum. Pharmaceutics 7(4):438–470. https://doi.org/10.3390/pharmaceutics7040438 2. Morales-Hurtado M, Zeng X, Gonzalez-Rodriguez P, ten Elshof JE, van der Heide E (2015) A new water absorbable mechanical Epidermal skin equivalent: the combination of hydrophobic PDMS and hydrophilic PVA hydrogel. J Mech Behav Biomed Mater 46:305–317. https://doi. org/10.1016/j.jmbbm.2015.02.014 3. Sato K, Sugibayashi K, Morimoto Y (1991) Species differences in percutaneous absorption of nicorandil. J Pharm Sci 80(2):104–107. https://doi.org/10.1002/jps.2600800203 4. Jung EC, Maibach HI (2015) Animal models for percutaneous absorption. J Appl Toxicol 35(1):1–10. https://doi.org/10.1002/jat.3004 5. Dabrowska AK et al (2016) Materials used to simulate physical properties of human skin. Skin Res Technol 22(1):3–14. https://doi.org/10.1111/srt.12235 6. Neupane R, Boddu SHS, Renukuntla J, Babu RJ, Tiwari AK (2020) Alternatives to biological skin in permeation studies: Current trends and possibilities. Pharmaceutics 12(2). https://doi. org/10.3390/pharmaceutics12020152 7. Bischoff JE, Arruda EM, Grosh K (2000) Finite element modeling of human skin using an isotropic, nonlinear elastic constitutive model. J Biomech 33(6):645–652. https://doi.org/10. 1016/S0021-9290(00)00018-X 8. Ohtsuki R, Sakamaki T, Tominaga S (2013) Analysis of skin surface roughness by visual assessment and surface measurement. Opt Rev 20(2):94–101. https://doi.org/10.1007/s10043013-0014-5 9. Elkhyat A, Mac-Mary S, Humbert P (2009) Skin wettability and friction. Handbook Cosmet Sci Technol 427–436 Third Edition January 2013. https://doi.org/10.1201/b15273-41 10. Rabost-Garcia G, Farré-Lladós J, Casals-Terré J (2021) Recent impact of microfluidics on skin models for perspiration simulation. Membranes (Basel) 11(2):1–13. https://doi.org/10.3390/ membranes11020150 11. Nachman M, Franklin SE (2016) Artificial skin model simulating dry and moist in vivo human skin friction and deformation behaviour. Tribol Int 97:431–439. https://doi.org/10.1016/j.tri boint.2016.01.043 12. Lir I, Haber M, Dodiuk-Kenig H (2007) Skin surface model material as a substrate for adhesionto-skin testing. J Adhes Sci Technol 21(15):1497–1512. https://doi.org/10.1163/156856107782 844783 13. Ren T, Gan J, Zhou L, Chen H (2020) Physically crosslinked hydrogels based on poly (vinyl alcohol) and fish gelatin for wound dressing application: fabrication and characterization. Polymers (Basel) 12(8). https://doi.org/10.3390/POLYM12081729 14. Amjad Z (2002) Water soluble polymers: solution properties and applications May 2007. https:// doi.org/10.1007/0-306-46915-4
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15. Gadhave RV, Mahanwar PA, Gadekar PT (2019) Effect of glutaraldehyde on thermal and mechanical properties of starch and polyvinyl alcohol blends. Des Monomers Polym 22(1):164– 170. https://doi.org/10.1080/15685551.2019.1678222 16. Harpaz D, Axelrod T, Yitian AL, Eltzov E, Marks RS, Tok AIY (2019) Dissolvable polyvinylalcohol film, a time-barrier to modulate sample flow in a 3D-printed holder for capillary flow paper diagnostics. Materials 12(3):1–11. https://doi.org/10.3390/ma12030343 17. Chan LW, Hao JS, Heng PWS (1999) Evaluation of permeability and mechanical properties of composite polyvinyl alcohol films. Chem Pharm Bull (Tokyo) 47(10):1412–1416
Simulation of Lateral Migration of Red Blood Cell in Poiseuille Flow Using Smoothed Particle Hydrodynamics Justin Antony and Ranjith Maniyeri
Abstract Cell separation is a process of isolating one or more specific cell populations from a heterogeneous mixture of cells. Understanding the dynamics of cells in different flow conditions is necessary to develop and improve the cell separation methods based on mechanical properties of cells. The present work numerically investigates the lateral migration of a deformable red blood cell in Poiseuille flow and the effect of the initial position of the cell on migration time and final shape of RBC. A meshless particle-based method known as smoothed particle hydrodynamics (SPH) is used in the simulations, which has several advantages over conventional gridbased methods in simulating fluid–structure interactions problems involving large deformation. A numerical model has been developed using a modified SPH that implements various improvements reported in the literature. The numerical simulations are parallelized on GPU using CUDA Fortran. The developed numerical model has been validated with existing results in the literature and it captures the deformation and migration of the deformable cell very well. It is observed that the RBC migrates towards the centre and attains similar shape at steady state irrespective of the initial position. Keywords Lateral migration · Meshless particle method · Poiseuille flow · Red blood cell · Smoothed particle hydrodynamics
Nomenclature x1 , x2 i, j m ρ W
Position vector [m] Indices representing particles Mass [kg] Density [kg/m3 ] Smoothing kernel function
J. Antony · R. Maniyeri (B) Biophysics Laboratory, Department of Mechanical Engineering, National Institute of Technology Karnataka (NITK), Surathkal, Mangalore 575025, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_59
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ν p μ F ES ED EB κ h υ
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Velocity [m/s] Pressure [N/m2 ] Viscosity [Pa s] Acceleration due to external force [m/s2 ] Shear modulus [N/m] Dilation modulus [N/m] Bending modulus [Nm] Local curvature [1/m] Smoothing length [m] Kinematic viscosity [m2 /s]
1 Introduction Cancer is a disease in which some of the body cells grow uncontrollably and spread to other parts of the body. Cancer is one of the major cause of death in the world. When a healthy cell becomes cancer cell, the mechanical properties of the cell changes drastically such as its deformability increases several folds that enables them to penetrate through surrounding tissue and blood vessels into the blood circulation system and spread to other parts of the body. Hence the name circulating tumour cells (CTCs). Separation of the CTCs from blood sample is necessary both for detection of the disease and for research purposes. The blood contains 45% of red blood cells (RBC), 1% of white blood cells and platelets, and 55% plasma. Removing RBC from blood samples will improve the efficiency of the cancer cell separation. Hence, it is important to study the hydrodynamic interaction of RBC under different flow conditions to develop and improve the RBC separation techniques.
2 Literature Review and Objective Smoothed particle hydrodynamics (SPH) is a meshless particle-based Lagrangian method that was invented for solving astrophysical problems [1] and later its application extended for solving computational fluid dynamic (CFD) problems. Even though the SPH has several advantages over conventional grid-based methods in simulating problems involving large deformations, moving interfaces etc., they have very low accuracy due to several inconsistencies arising from non-uniform particle distribution and SPH approximations near boundaries. Decoupled finite particle method (DFPM) proposed by Zhang and Liu [2] overcomes these errors by introducing new terms in the denominator of the SPH approximations that are derived from Taylor series expansions. The inaccuracies in the SPH approximations arises from the nonuniform particle distribution cause the formation of voids or clumps in the particle
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distribution. An artificial particle displacement has been proposed by Shadloo et al. [3] in which the particles at region of higher concentration are moved towards region of lower concentration, thus maintaining a uniform distribution of particles in the domain. Density diffusive term has been introduced by Marrone et al. [4] to remove the spurious oscillations in pressure arising from the use of artificial equation state. RBC dynamics have been investigated by several researchers in the literature. Hosseini and Feng [5] investigated the deformation of RBC in shear flow, Poiseuille flow and in sudden constriction using SPH. They reproduced the tank-treading motions and axisymmetric motions of the RBC. Polwaththw-Gallage et al. [6] investigated motion and deformation of a single RBC in stenosed capillary using SPH and showed that the membrane stiffness and geometrical parameters of the capillary have significant impact on the RBC’s motion. Shi et al. [7] investigated the lateral migration of cells in Poiseuille flows using immersed boundary method and shown that the rate of migration towards centre of the channel depends on the swelling ratio and the deformability of the cells. The lateral migration of deformable bodies using SPH is not reported much in the literature. The present work is a preliminary study to investigate the lateral migration of a deformable cell in Poiseuille flow using SPH and to analyse the effect of initial position of the cell in the migration time and final position. More complex fluid– structure interaction problems of RBC under various flow conditions as well as its interaction with other cells will be investigated in the future to develop or improve efficiency of cell separation methods.
3 Methodology 3.1 Discretization Using SPH A function f of position vector x1 can be written in the integral form as a function of another position vector x2 with the help of Dirac delta function as in Eq. (1) where the Dirac delta function is given by Eq. (2). ∫ f (x2 )δ(x1 − x2 )dx2
f (x1 ) = Ω
δ(x1 − x2 ) =
(
1, x1 = x2 0, x1 /= x2
(1)
(2)
The SPH particle approximation for a function and its derivative can be written as in Eqs. (3) and (4) by replacing the Delta function in Eq. (1) with smoothing kernel function W (x1 − x2 , h) (or W12 ) and replacing the integral with summation over neighbouring particles in the support domain.
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(3)
N ∑ mj ( ) f x j · ∇i Wi j ρj j=1
(4)
f (xi ) ≥
∇ · f (xi ) ≥
The angled bracket is used to indicate the kernel approximation operator. In DFPM, the above particle approximations are modified using Taylor series expansion as given in Eqs. (5) and (6). The derivation of these approximations can be found in Zhang and Liu [2]. mj ( ) j=1 ρ j f x j Wi j ∑N m j j=1 ρ j Wi j
∑N f (xi ) ≥
(5)
mj ( ) j=1 ρ j f x j · ∇i Wi j ) ∑N m j ( j=1 ρ j x j − x i · ∇i Wi j
∑N
∇ · f (xi ) ≥
(6)
The denominators in the Eqs. (5) and (6) help in satisfying the normalization condition of the kernel near the boundaries of the domain and thus overcomes the kernel inconsistency and particle inconsistency problems. Finding the Laplacian of a function in DFPM requires nested approximation in which higher-order derivatives are obtained from lower-order derivatives. This nested approximation increases the computational time. Huang et al. [8] proposed a novel discrete scheme for Laplacian operator given in Eq. (7) that requires less computational cost. ∑N ∇ 2 f (xi ) > 2
mj j=1 ρ j
( ( ) ) f x j − f (xi ) Wi j
(7)
χψ
where χ =15/7πh2 and ψ=31h4 π/210. From the above approximations, the SPH formulations of continuity and momentum equations can be written as given in Eqs. (8) and (9), respectively. ∑N
( ) ∂W β β v j − vi · βi j ∂ xi ) mj ( x j − xi · ∇i Wi j ρj
mj j=1 ρ j
Dρi = −ρi ∑ N Dt j=1 Dviα Dt
∑N
=−
1 ρi ∑ N
j=1
Mj j=1 ρ j
∂W
( p j− pi ) ∂ x αi j μi ( ) i δ αβ + 2 Mj β β ρi x j − x i · ∇i Wi j ρj
∑N
Mj j=1 ρ j
(8) (
) uαj − uiα Wi j
χψ
+ fi (9)
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3.2 Spring Network Model of RBC Membrane The RBC membrane has high modulus against areal contraction and expansion due to the lipid-bilayer structure. Hence, the surface area of the RBC membrane is conserved. However, the membrane is highly flexible due to bending and shear deformation which enables them to pass through narrow capillaries. In the present study, the spring network model used in Hosseini and Feng [5] is adopted. For RBC membrane particles, additional terms are added to right-hand side of the SPH equations to represent the forces due to stretching and bending elasticity as explained by Hosseini and Feng [5]. The stress due to stretching of the membrane is given by Eq. (10), and the bending moment is given by Eq. (11). √ [ )] ( ) 1 + Cλ41 C 1 + Cλ21 ES ( 2 λ1 λ1 − 1 T = × 1+ ( )2 2 1 + Cλ21 1 + Cλ41
(10)
m = E B (κ − κ0 )
(11)
where C = E D /E s , the ratio of dilatation to shear moduli and λ1 is the ratio of deformed length to initial length, E B is the bending modulus, κ0 and κ are local curvature of the resting and deformed states. The bending moment is converted into nodal forces by replacing the moments with a pair of equal and opposite forces acting on the spring segments.
3.3 GPU Acceleration Using NVIDIA FORTRAN The hardware acceleration for CFD calculations can be achieved in two ways. One is using high performance computing (HPC) on super computers consisting of large number of CPU cores. Second is using novel computing architectures such as graphics processing units (GPU). Compared to large multicore HPC systems, GPU has the advantage of low cost and ease of maintenance while having comparable performance. The basic computational unit on GPU is a thread processor (core). The thread processors are grouped into multiprocessors, which contain a limited amount of resources used by the resident threads namely registers and shared memory. Multiple multiprocessors can reside on a single GPU, but the number of multiprocessors that can run are simultaneously limited by the resources required by each multiprocessor. Figure 1 shows the computational units in GPU. The capability of GPUs to simulate SPH has been demonstrated by Crespo et al. [10]. The data transfer between the GPU and CPU is computationally expensive, and thus the memory transfer is done only at the initial step and whenever the output is required to export to file. All SPH calculations are done in the GPU similar to Crespo et al. [10].
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Fig. 1 Computational units in GPU [9]
3.4 Numerical Simulation To investigate the lateral migration of RBC, a channel of dimension 24 μm × 10 μm is considered. The RBC is assumed to be circular shape with a diameter of 5.5 μm. The blood plasma and cytoplasm of RBC are considered to be same fluid with density 1000 kg/m3 and viscosity 1.2 × 10−3 Pa s. Acceleration is given to the fluid in the x-direction to initiate the flow. No-slip boundary conditions are applied on top and bottom walls, and periodic boundary conditions are applied on the inlet and outlet boundaries. The initial position of the cell is varied to understand the effect of position on migration time. The domain is discretized with uniformly distributed particles as shown in Fig. 2, and all the properties are initialized and stored in CPU. The initial values are transferred to GPU for SPH calculations. The first step in GPU is to identify the neighbours of every particle. The fast algorithm proposed by Rhoades [11] is modified and parallelized in GPU to identify and create a list of neighbours to each particle. The properties of virtual particles are evaluated secondly. The virtual particles are the particles created outside the boundary for implementing boundary conditions at top and bottom walls and are created initially at domain discretization step. The properties of these particles are evaluated using SPH approximations at their mirrored node in fluid domain. For no-slip boundary, the velocity of virtual particle is taken as negative of velocity components of its mirrored node. Then, the artificial equation of state [2] is solved to obtain pressure of each particle. The forces on RBC membranes are evaluated next, and finally the momentum equation is solved and the position of particles are updated. The density of each particle is also updated using continuity equation adding density diffusive term to remove unwanted spurious oscillation in density. The artificial particle displacement is used to maintain the uniform distribution of particles. This procedure is continued till steady state is reached. The results at intermediate time intervals are saved by transferring the required data back to the CPU. All the simulations are carried out on a laptop with hexa-core AMD Ryzen 5 4000 processor and Nvidia GeForce GTX 1650Ti GPU with 4 GB dedicated memory.
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Fig. 2 Initial particle distribution of the domain
4 Results and Discussion The motion and deformation of single cell in a channel is investigated to understand the RBC dynamics and dependence of initial position on migration time. An acceleration is given in x-direction to get an average velocity 0.1 m/s without the cell that corresponds to Re of 0.5556 as in Shi et al. [7]. The time step and particle independency study has been conducted to remove the dependency of the results on them, and the validation of the numerical model is done. The effect of initial position of the RBC is investigated by considering three different initial locations by varying the y-coordinate of the centre of RBC (yc ).
4.1 Time Step Independency Study Time step independency study has been conducted to remove dependency of the result on the time step size. The maximum permissible time step size is determined as the time step that satisfies the CFL condition (Eq. 12) and viscous diffusion condition (Eq. 13). Δt ≤ 0.25
h c
Δt ≤ 0.125h 2 /ϑ
(12) (13)
In present work, the viscous diffusion condition gives lower time step size values. The optimum time step size is determined by varying the constant (0.125) in the right-hand side of the Eq. (13). The particles are distributed uniformly with an initial particle spacing of 0.4 μm. Simulations are carried out for a time period of 0.001 s. Figure 3 shows the comparison of the final shape of RBC at time step 0.001 s for different values of the constant. Table 1 shows the comparison of time step size and the time taken for the simulations. The constant 0.01 is found to be optimum in terms
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Fig. 3 Comparison of final shape at 0.001 s for different constants in the viscous diffusion condition
Table 1 Comparison of time step size and time taken for different constants
Constant
Δt (s)
Time taken (s)
0.125
7.9 × 10−9
325
0.060
5.0 × 10−9
549
0.010
1.16 × 10−9
2220
0.008
4.63 × 10−10
6013
0.005
2.01 × 10−10
13,392
of both accuracy and computational time and is used for all the following numerical simulations.
4.2 Particle Independency Study The particle independence (initial particle spacing independence) study is conducted to determine the optimum number of particles that is both accurate and computationally efficient. Figure 4 shows the comparison of final shapes at 0.001 s for different number of particles, and Table 2 shows the comparison of time taken for the simulations. About 96 × 40 is taken as the optimum number of particles considering both accuracy and computational time.
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Fig. 4 Comparison of final shape at 0.001 s for different number of particles
Table 2 Comparison of time taken for different number of particles Particle distribution
Initial particle spacing (μm)
Time taken (s)
60 × 25
0.4
2220
96 × 40
0.25
15,694
120 × 50
0.2
44,403
4.3 Validation of the Numerical Model The developed numerical model has been validated for Poiseuille flow [12] and for the motion and deformation of a single RBC in shear flow [13] in previous work. The deformation of RBC in a channel placed near the wall at a distance of 2.0 × 10−6 m from the centreline is simulated with optimum time step and number of particles. The obtained results are validated with that of Shi et al. [7] as shown in Fig. 5. It can be observed that the results of present work are in good agreement with the results in the literature [7].
4.4 Effect of Initial Position on Shape of RBC The RBC has been released from four different locations by varying yc as − 2.0 μm, − 1.0 μm and 0.0 μm. The comparison of their shape deformation is shown in Fig. 6. A particle at the front edge of the RBC (marked in blue) is tracked to determine the motion of the RBC membrane. The RBC released near the wall has undergone maximum deformation and the deformation reduces as the initial location is moved
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Fig. 5 Comparison of shape of RBC at different instances obtained from the developed model (red) with that of Shi et al. [7] (blue) at yc = 2.0 × 10−6 m
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Fig. 6 Lateral migration of RBC at different initial positions
towards the centre of channel. The tracking particle revolves in anticlockwise direction as the cell moves. This motion of RBC membrane is known as tank-treading motion. However, the RBC placed initially at the centre of the channel remains in the centre and the tracking particle remains at the front edge of the cell. At steady state, the final shape of the RBCs are similar irrespective of their initial location. The non-linear velocity profile resulting from Poiseuille flow causes asymmetric forces on the RBC when RBC is in asymmetric position. This results in the lateral migration of the RBC towards centre of the channel and tank-treading motion of the cell membrane. The tracking particle on RBC remains at the same location when the RBC released at the centre of the channel as there is no asymmetry in the forces acting on RBC since the velocity profile is symmetric about the centreline of RBC.
4.5 Effect of Initial Position of Migration Time The initial centre of the RBC (black particle) is tracked to determine the migration length of RBC. Figure 7 shows the path traced by the initial centre of RBC when released from different locations. It should be noted that the black particle in Fig. 6
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Fig. 7 Path traced by the initial centre of RBC for different initial locations
Table 3 Effect of initial position on migration time
yc
Migration time (s)
−2.0 μm
0.00195
−1.0 μm
0.00125
0.0 μm
0
is not the mass centre of RBC, but the particle close to initial centre point of RBC. The approximate migration time is tabulated in Table 3. Kaoui et al. [14] presented that due to non-linear velocity profile the cell migrates to the centre of flow. However, in the present work, the path traced by the particle as shown in Fig. 7 does not coincide with the centre of flow exactly. This is due to the reason that the particle being tracked is not the mass centre of RBC, but a particle close to the mass centre at the initial position of RBC. However, the approximate migration time can be determined from that particle. From Table 3, it is clear that as the initial position becomes closer to the centreline of channel, the migration time reduces.
5 Conclusions The lateral migration of single RBC in Poiseuille flow is investigated numerically using the developed numerical model based on modified smoothed particle hydrodynamics. The developed code is parallelized in GPU using CUDA Fortran. The results obtained using developed model are validated with existing results in the literature. It is observed that irrespective of the initial location of RBC, the final position and shape of RBC is similar. The non-uniform velocity profile causes asymmetric forces
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on RBC when RBC is not at the centre of the channel, resulting in the lateral migration and tank-treading motion of the RBC membrane. It is also observed that the migration time will increase as the initial position moves away from the centreline of the channel. The developed numerical model has successfully simulated the fluid– structure interaction of a single cell in Poiseuille flow. We believe that the model can be extended to investigate more complex cell interactions and thus to develop an efficient cell separation techniques.
References 1. Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices R Astronom Soc (Oxford University Press Oxford, UK) 181(3):375–389 2. Zhang ZL, Liu MB (2018) A Decoupled finite particle method for modeling incompressible flows with free surfaces. Appl Math Model 60:606–633 3. Shadloo MS, Zainali A, Yildiz M, Suleman A (2012) A robust weakly compressible SPH method and its comparison with an incompressible SPH. Int J Numer Meth Eng 89:939–956 4. Marrone S, Colagrossi A, Antuono M, Colicchio G, Graziani G (2013) An accurate SPH modeling of viscous flows around bodies at low and moderate Reynolds numbers. J Comput Phys 245:456–475 5. Hosseini SM, Feng JJ (2009) A Particle-based Model for the Transport of Erythrocytes in Capillaries. Chem Eng Sci 64:4488–4497 6. Polwaththe-Gallage HN, Saha SC, Sauret E, Flower R, Gu Y (2015) Numerical investigation of motion and deformation of a single red blood cell in a stenosed capillary. Int J Comput Methods 12(4):1540003 7. Shi L, Pan T, Glowinski R (2012) Numerical simulation of lateral migration of red blood cells in Poiseuille flows. Int J Numer Meth Fluids 68:1393–1408 8. Huang C, Lei JM, Liu MB, Peng XY (2016) An improved KGF-SPH with a novel discrete scheme of Laplacian operator for viscous incompressible fluid flows. Int J Numer Meth Fluids 81:377–396 9. Ruetsch G, Fatica M (2014) CUDA Fortan for scientists and engineers best practices for efficient CUDA Fortran programming. Elsevier, Amsterdam 10. Crespo AJC, Dominguez JM, Barreiro A, Rogers BD (2011) GPUs a new tool of acceleration in CFD: efficiency and reliability on smoothed particle hydrodynamics methods. Public Lib Sci One 6(6):e20685 11. Rhoades CE Jr (1992) A fast algorithm for calculating particle interactions in smooth particle hydrodynamic simulations. Comput Phys Commun 70:478–482 12. Antony J, Maniyeri R (2020) Numerical simulation of fluid flow in a channel using smoothed particle hydrodynamics. In: Proceedings of 65th Congress of ISTAM 13. Antony J, Maniyeri R (2022) Numerical simulation of RBC in shear flow using smoothed particle hydrodynamics. In: Paper No. 4144, 20th ISME conference on advances in mechanical engineering, IIT Ropar, India, 19–21 May 2022 14. Kaoui B, Ristow GH, Cantat I, Misbah C, Zimmermann W (2008) Lateral migration of twodimensional vesicle in unbounded Poiseuille fow. Phys Rev E 77:111702
Effect of Stenosis Severity on the Hemodynamics of an Idealized Straight Arterial Tube Pawan Kumar, Somnath Roy, and Prasanta Kumar Das
Abstract The present study examines the effect of stenosis severity on arterial hemodynamics using the immersed boundary method-based in-house numerical code. A realistic physiological velocity waveform varying in time and parabolic in space is adopted as an inlet boundary condition in a straight arterial tube with single asymmetric stenosis. The flow is considered to be laminar and incompressible, assuming blood as a Newtonian fluid. The degree of stenosis (DS) is defined by the obstructed cross-sectional area. The velocity and vorticity contours demonstrate that increment in DS results in further disturbances in the flow downstream of the stenosis. The areas with low Time-Averaged Wall Shear Stress (TAWSS) and high Oscillatory Shear Index (OSI) are more prone to further plaque deposition, and areas with high TAWSS, such as throat of the stenosis, are potential sites for thrombus formation. The magnitude of the peaks in the cyclic pressure drop across the stenosis rises with its severity. However, in the present study, a greater pressure drop is noted at 45% stenosis (DS45) case as compared to 65% stenosis (DS65) case as some pressure recovery occurs in the latter case. Keywords Stenosis · Pressure drop · Immersed boundary method, wall shear stress · Oscillatory shear index · Thrombus
P. Kumar (B) Advanced Technology Development Centre, IIT Kharagpur, Kharagpur 721302, India e-mail: [email protected]; [email protected] S. Roy · P. K. Das Department of Mechanical Engineering, IIT Kharagpur, Kharagpur 721302, India e-mail: [email protected] P. K. Das e-mail: [email protected] S. Roy Centre for Computational and Data Sciences, IIT Kharagpur, Kharagpur 721302, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_60
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1 Introduction Atherosclerosis is a chronic disease that generally occurs in the systemic arteries like the carotid artery, coronary arteries, abdominal aorta, and superficial femoral arteries [1]. The arterial lumen becomes locally constricted due to the deposition of plaque, which is made up of deposits of fatty substances, cholesterol, cellular waste products, calcium, and fibrin. This constriction/narrowing of the artery is termed stenosis. The extent of constriction is defined based on the degree of stenosis (DS), which is defined as the ratio of cross-sectional area obstructed due to plaque deposition to the normal cross-sectional area. The dynamics of blood flow are very useful to understand the onset and progression of atherosclerosis. The flowing blood exerts a tangential force on the inner linings of the walls of the artery, made of endothelial cells, that results in wall shear stress (WSS). Downstream of the stenosis in arteries, the mean WSS is very low and oscillates between positive and negative directions during the cardiac cycle [1], which is quantified using the mechanical parameter Oscillatory Shear Index (OSI) [2]. The blood flows in the reverse direction of the streamlined flow, which gives rise to the recirculation zone. This can result in decreased blood flow to tissues downstream, weaken the wall locally, and change the endothelial function, vascular tone, cell adhesion molecules, and platelet adhesion (onset of thrombosis) [3]. Atherosclerosis affects arterial compliance. This situation remains asymptotic unless the stenosis gets severe and the person shows symptoms like chest pain, shortness of breath, and pain in the jaw, arms, and shoulders [4]. The pressure force gets reduced because of the loss of energy in the recirculation zone due to the vortex formation. This leads to an increase in the left ventricular load as the heart has to do additional work for pumping the blood. This leads to impairment of heart function [3]. Therefore, it is important to understand the effect of varying DS on the functional characteristics of the heart.
2 Literature Review and Objective Many numerical and experimental simulations have been carried out, using either steady state or physiological pulsatile velocity as the inlet boundary condition, to understand the effect of stenosis on the hemodynamics of the arteries. However, the researchers considered the geometry of the stenosis as axisymmetric, whereas it has been found that the stenosis present in the diseased arteries is irregular and asymmetric [2, 5, 6]. Some of the mechanical parameters used to quantify the stenosis severity are Time-Averaged Wall Shear Stress (TAWSS), OSI, pressure drop, etc. [2, 7]. TAWSS is obtained by averaging the wall shear stress over the entire pulsation cycle [8] as:
Effect of Stenosis Severity on the Hemodynamics of an Idealized …
1 TAWSS = T
725
T |τw dt|
(1)
0
Here, T is the time period over which average WSS is calculated and τw is the wall shear stress. OSI is a mechanical parameter that represents the extent of fluctuations in the wall shear stress [8]. The deflection of the shear force vector from the predominant direction of the flow can be calculated through the values of OSI during a pulsation cycle. The mathematical expression for the OSI is given below: ⎤ T τ dt ⎥ w ⎢ 0 ⎥ 1⎢ ⎥ 1− T OSI = ⎢ ⎥ ⎢ 2⎣ ⎦ |τw |dt ⎡
(2)
0
For the same value of DS, the fluid characteristics observed downstream of the asymmetric stenosis may be different from that of axisymmetric stenosis. This may have serious implications on the hemodynamics of the stenosed artery and can be determined using the above-mentioned mechanical parameters. Pressure drop across stenosis is also a useful parameter to quantify the hemodynamics of the diseased arteries [9]. Cyclical pressure drop across the stenosis, for varying degrees of stenosis, in an artery can be a useful parameter to identify the functionally significant stenosis, which may be asymptotic (mild or moderate) as per the anatomical classification of the arteries based on the degree of stenosis. The present study aims to quantify the effect of the stenosis severity on the hemodynamics of the arteries for the physiological pulsatile flow using various mechanical parameters such as streamline velocity, vorticity, TAWSS, OSI, and pressure drop across the stenosis. Blood is assumed as a Newtonian fluid, and flow is considered to be incompressible and laminar. The arteries are assumed to be rigid, straight, having a circular cross-sectional area.
3 Methodology 3.1 Geometry and Grid Configuration We considered the three-dimensional straight arterial tube with single asymmetric stenosis of the following degree of stenosis, expressed in percentage as 0%, 45%, 65%, and 75%, and are given the names, DS0, DS45, DS65, and DS75, respectively in the present paper. The normalized length of the tube is 33D. The length of the stenosis is 2D, and the length of the tube upstream and downstream of the stenosis is 6D and 25D, respectively, as shown in Fig. 1. Here, D is the diameter of the normal tube.
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Fig. 1 Three-dimensional geometry of the stenosed artery; a isometric view, b front view, c crosssectional view (gray color shows the stenosed portion), d healthy artery
The grid size of D/80 is taken for the simulation. The three-dimensional geometry of asymmetric stenosis is made using SOLIDWORKS, a 3D CAD software, as per the geometrical specifications used in the work of Young and Tsai [10].
3.2 Governing Equations The basic equations which govern the fluid flow are the continuity equation and Navier–Stokes equations, which are given in the normalized form by ∂u j = 0, and ∂x j
(3)
∂u i u j ∂u i ∂p 1 ∂ 2ui + =− + ∂t ∂x j ∂ xi Re ∂ x j ∂ x j
(4)
Here, u, t, and x represent the normalized velocity, time, and position (dimension of which is length), respectively, and i and j are the index notation for three directions (x, y, and z). Re is the Reynolds number.
3.3 Boundary Conditions At the inlet, we are imposing a parabolic physiological pulsatile velocity, used by Chabi et al. [6], as given in Fig. 2. The waveform repeats its cyclic nature after every 0.8 s (75 beats per minute). The Reynolds number (ratio of inertial force to viscous force) for the simulation is considered to be 142 based on average velocity (V ) over a pulsation cycle and inlet tube diameter (D). The Womersley number (ratio of unsteady
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Fig. 2 Normalized pulsatile velocity waveform at the inlet [6]
inertial force to viscous force) for the present work is 2.31, which is based on the tube diameter (D), time period (T ) of the pulsatile cycle, dynamic viscosity (μ), and density of the blood (ρ). At the outlet, Orlanski boundary condition is applied [11] and for the pipe wall, a no-slip boundary condition is given.
3.4 Numerical Scheme Kumar and Roy [12] have demonstrated a sharp interface immersed boundary (IB) method for solving flow in complex arterial channels. In their proposed methodology, the IB method is implemented over a 3-D full N-S solver where the solution of the governing equations is obtained by the MAC algorithm [2]. The IB method used by Kumar and Roy [11] has been made faster using GPU, which substantially reduces the computational time [13, 14]. For simulating the cases in the present study, the GPUaccelerated IB method of Raj et al. [13] is used. The flow solver uses a Cartesian mesh framework and has an overall second-order accuracy. The finite difference method (FDM) approach has been used to discretize the governing equations and the discrete forcing approach is considered while implementing the IB method. Time integration is obtained using a second-order accurate Adams–Bashforth method. Non-dimensionalization has been done with appropriate scaling.
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Fig. 3 Validation of the in-house code at Re = 800. Axial velocity profiles for steady flow (shown with orange color) compared with the experimental result (shown with dots) of Usmani and Muralidhar [7]
4 Results and Discussion The focus of this study is to analyze CFD results of flow patterns, pressure drops, and WSS distributions, which are related to the diagnosis and progression of arterial stenosis in clinical settings. For the general applicability of the various findings of the present study, the results which are discussed in the subsequent sections are shown in the normalized form using relevant scale and parameters.
4.1 Validation The numerical results obtained using in-house code are validated with the experimental results of Usmani and Muralidhar [7] for steady flow through a stenosed artery at 75% area reduction, at the Reynolds number (Re) of 800. The velocity profiles at three different axial positions (X1 = 5, X2 = 7, X3 = 9) of the stenosed artery show satisfactory matching with those obtained through the PIV-based experiment, as shown in Fig. 3.
4.2 Flow Patterns The physiological waveform, shown in Fig. 2, can be divided into two parts: systole and diastole. The diastole has its peak at t/T = 0.525, and the increasing and decreasing curves show diastolic acceleration and deceleration phases, respectively. It becomes important to understand the influence of stenosis on the hemodynamics during the deceleration phase of the diastole, as the velocity decreases (flow retards) during this phase (between t/T = 0.525 and 1). Therefore, results obtained through the numerical simulations shown here focusses on the flow behavior during the deceleration phase. The velocity contours for different degrees of stenosis at the diastolic deceleration phase (t/T = 0.6) are shown in Fig. 4. The magnitude of the velocity is positive and maximum at the throat of the stenosis, which increases with an increase
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Fig. 4 Velocity contours for a DS45, b DS65, and c DS75, at the deceleration phase (t/T = 0.6)
Fig. 5 Vorticity contours for a DS45, b DS65, and c DS75, at the deceleration phase (t/T = 0.6)
in the stenosis severity. The axial velocity gets disturbed and recirculation zones are formed downstream of the stenosis. The disturbance leads to the vortex formation, and with the increase in the value of the degree of stenosis, the effect of vortex fluctuations can be seen to travel further downstream region of the flow, as shown in Fig. 5. The reason for showing the vorticity contours for the deceleration phase (at t/T = 0.6) is that the effect is predominant during the deceleration phase of the pulsatile waveform. As we can see in Fig. 6, in the case of the healthy artery, DS0, no flow disturbance takes place as there is no stenosis; therefore, no vortex formation takes place.
4.3 Pressure Drop Across the Stenosis The normalized pressure drop across the stenosis is calculated between the axial distance of X = 4 and 10. The cyclic variation of the normalized or non-dimensional
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Fig. 6 For a healthy artery at the deceleration phase (t/T = 0.6): a velocity contour, and b vorticity contour
pressure drop is shown in Fig. 7 for two cycles. The positive pressure drop represents favorable pressure gradient, whereas the negative pressure drop (below 0 mm Hg pressure drop line) corresponds to adverse pressure gradient. It can be seen that the maximum fluctuation in the pressure drop is for DS75, and the minimum fluctuation is observed in the case of DS0 (healthy artery). The pressure fluctuation should increase with the increasing stenosis severity. But in the present study, more pressure drop is noticed for DS45 case as compared to DS65 case due to some pressure recovery taking place in the latter case. Also, in case of DS65, the flow is taking place in such a way that both favorable and adverse pressure gradient have almost similar area under the curves which results in the lesser value of mean pressure drop in the case.
Fig. 7 Cyclic pressure drop across stenosis (between X = 4 and 10) for two cycles. (Plots are shown with different colors)
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4.4 Time-Averaged Wall Shear Stress (TAWSS) and Oscillatory Shear Index (OSI) The non-dimensional TAWSS and OSI for the three cases are shown in Figs. 8 and 9, respectively. If we compare the three cases, then for the DS75, we can observe that low TAWSS is spread over a large area as compared to DS45 and DS65. The maximum TAWSS is at the throat of the stenosis in all three cases. It is also observed that for the cases of DS65 and DS75, at the post-stenotic zone (between X = 8 and 14), the OSI is very high as compared to the case of DS45. Fig. 8 Non-dimensional TAWSS a DS45. b DS65. c DS75
Fig. 9 Oscillatory shear index (OSI) a DS45. b DS65. c DS75
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5 Conclusions In this present work, simulations are done in idealized straight tubes with 0, 45, 65, and 75% as the degree of stenosis (based on the obstructed cross-sectional area) using physiological pulsatile inlet velocity. These four degrees of stenosis show the case of healthy, mild, moderate, and severe stenosis. From the results obtained for different parameters in the present study, we can find that for 75% stenosis (DS75), TAWSS is the lowest and OSI is very high in the post-stenotic zone, as compared to the other cases (DS0, DS45, and DS65). The regions of low TAWSS and high OSI zone are prone to plaque accumulation. Also, at the throat of the stenosis, the TAWSS is maximum for all the cases except DS0. This is the site prone to thrombus formation. This can cause serious arterial diseases because the phenomenon of thrombus formation and plaque deposition can also lead to thrombotic or embolic stroke. The plots of cyclic pressure drop across stenosis show that the pressure fluctuations increase with the increasing degree of stenosis. In the case of DS45, the cyclic pressure drop is found to be more than the DS65. This is an unusual behavior that needs further investigation to find the reason behind this deviation from the trend. This numericalbased study can help take the critical decision to assess the stenosis severity not just based on its anatomical appearance, but also based on its impact on the functionality of the stenosed artery. The effect of stenosis severity on hemodynamics can be studied non-invasively using the parameters like the pressure drop across stenosis along with the other well-defined mechanical parameters which are discussed in the present work. This study facilitates to identify the clinically significant stenosis based on its functional characteristics, which would not be possible just on the basis of the data obtained from computed tomography and magnetic resonance imaging techniques alone. Acknowledgements We acknowledge the high-performance computing facility, PARAM Shakti (IIT Kharagpur)—a National Supercomputing Mission, Government of India, for providing their computational resources. We are thankful to the Ministry of Education, Government of India, for providing a scholarship to carry out the research work.
Nomenclature D DS WSS TAWSS OSI Re ρ μ T
Diameter of the tube [m] Degree of stenosis Wall shear stress [kg/m-s2 ] Time-averaged wall shear stress [kg/m-s2 ] Oscillatory Shear Index Reynolds number Density of blood [kg/m3 ] Dynamic viscosity of blood [kg/m-s] Time period of the pulsatile cycle [s]
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References 1. Ku DN (1997) Blood flow in arteries. Annu Rev Fluid Mech 29:399–434. https://doi.org/10. 1146/annurev.fluid.29.1.399 2. Sood T, Roy S, Pathak M (2018) Effect of pulse rate variation on blood flow through axisymmetric and asymmetric stenotic artery models. Math Biosci 298(6):1–18. https://doi.org/10. 1016/j.mbs.2018.01.008 3. Vlachopoulos C, O’Rourke M, Nichols WW (2011) McDonald’s blood flow in arteries: theoretical, experimental and clinical principles. McDonald’s Blood Flow Arter. https://doi.org/10. 1201/B13568 4. Heart Attack Symptoms, Risk, and Recovery | cdc.gov 5. Dodds SR (2002) The haemodynamics of asymmetric stenoses. Eur J Vasc Endovasc Surg 24(4):332–337. https://doi.org/10.1053/ejvs.2002.1729 6. Song J, Kouidri S, Bakir F (2021) Numerical study on flow topology and hemodynamics in tortuous coronary artery with symmetrical and asymmetrical stenosis. Biocybern Biomed Eng 41(1):142–155. https://doi.org/10.1016/j.bbe.2020.12.006 7. Usmani AY, Muralidhar K (2016) Pulsatile flow in a compliant stenosed asymmetric model. Exp Fluids 57(12):1–24. https://doi.org/10.1007/s00348-016-2274-x 8. Uttam S, Khan PM, Alam MI, Roy S (2020) Behavior of wall shear stress near carotid artery bifurcation at elevated pulse rates. J Flow Vis Image Process 27(3):249–267. https://doi.org/ 10.1615/JFlowVisImageProc.2020031021 9. Freidoonimehr N, Chin R, Zander A, Arjomandi M (2020) An experimental model for pressure drop evaluation in a stenosed coronary artery. Phys Fluids 32(2). https://doi.org/10.1063/1.513 9701 10. Young FYTDF (1973) Flow characteristics in models of arterial stenoses. I steady flow. J Biomech 6:395–410 11. Alvarado-Rodríguez CE, Klapp J, Sigalotti LDG, Domínguez JM, de la Cruz Sánchez E (2017) Nonreflecting outlet boundary conditions for incompressible flows using SPH. Comput Fluids 159:177–188. https://doi.org/10.1016/j.compfluid.2017.09.020 12. Kumar M, Roy S (2016) A sharp interface immersed boundary method for moving geometries with mass conservation and smooth pressure variation. Comput Fluids 137:15–35. https://doi. org/10.1016/j.compfluid.2016.07.008 13. Harlow FH, Welch JE (1965) Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys Fluids 8(12):2182–2189. https://doi.org/10.1063/1.176 1178 14. Mohan P, Raj A, Alam I, Chakraborty S (2023) Prediction of vortex structures in pulsatile flow through S-bend arterial geometry with different stenosis levels 43:298–312
Microdevice for Plasma Separation and in Vitro Quantification of Plasma Proteins Tony Thomas, Neha Mishra, and Amit Agrawal
Abstract Recent advances in engineering have demonstrated the use of microfluidic devices as a platform for bioanalytical applications. Over the last few decades, microdevices for blood plasma separation are globally recognized due to their numerous benefits as compared to conventional techniques. One of the recently reported work from our research group has successfully demonstrated a simple and efficient passive microfluidic device capable of plasma separation from the whole blood. The design utilized bifurcated microfluidic channel dimensions of hundred microns for separating plasma with almost 100% separation efficiency. In the present research, we report blood plasma separation using this microfluidic chip and in vitro quantification of albumin (plasma protein) using colorimetric techniques. Undiluted blood (hematocrit up to 45%) at a flow rate of 0.3–0.6 ml/min was used for performing experiments. The separated plasma was further utilized for the quantification of albumin. Albumin present in the plasma binds with a reagent Bromocresol green (BCG) to form a blue-green colored complex (Albumin-BCG). The intensity of the resultant color formed after mixing is proportional to the concentration of albumin. The absorbance of the colored complex at 628 nm was measured using a spectrophotometer for quantitative analysis of albumin. The concentration of albumin obtained from microfluidic-separated plasma was compared with that of albumin detected from centrifuged plasma. The comparative study shows that the results are within the reasonable limits of agreement (error of ± 3%). The potential outcomes of this research build confidence toward the design and development of microfluidic plasma separation and detection devices at large. Keywords Plasma separation · Microfluidic chip · Hematocrit · In vitro quantification · BCG
T. Thomas (B) · N. Mishra · A. Agrawal Department of Mechanical Engineering, IIT Bombay, Mumbai 40006, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_61
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1 Introduction Blood plasma plays a very vital role in regulating and maintaining various metabolic activities of the body [1]. In addition to water, mineral salts and proteins constitute the major components of plasma. Blood plasma plays a very crucial role in the transportation of nutrients and waste products, regulating the body temperature, maintaining the pH, etc. In addition to that, the plasma proteins serve many significant functions, which include blood clotting (fibrinogen), providing immunity (immunoglobulin), and maintaining oncotic pressure (albumin). The biomarkers, minerals, analytes, and proteins present in the plasma serve as vital indicators for clinical disease diagnosis. The separation of plasma from whole blood is inevitable for in vitro quantification of these parameters of interest [1, 2]. Clinical facilities rely on the efficient separation of plasma from blood cells for the quantification of these disease biomarkers. Plasma separation minimizes the interference of blood cells thereby improving the accuracy of diagnosis [1, 3]. Conventional techniques of plasma separation utilize the process of centrifugation, which has several drawbacks that include a tedious process, low throughput, requires more power and time, etc. [1, 4]. Microfluidic chip-based blood plasma separation devices have gained widespread recognition due to their simple and cost-effective fabrication, higher separation efficiency and throughput, faster process time, etc. [3, 4]. The potential capabilities of integrating these microfluidic chips into biosensors can realize a complete lab-on-a-chip device for disease diagnosis. Active methods of microfluidic separation techniques utilize an external source of energy, which makes the system bulky, and complex that may require more power for its operation [5–10]. Passive techniques are comparatively simpler and costeffective where various inertial, biophysical effects, and channel geometries are being explored in a smart and intelligent way [11–13]. Fahraeus effect and ZweifachFung bifurcation laws are the two different biophysical effects that are being utilized by researchers for microfluidic blood plasma separation. Fahraeus effect is more significant in microfluidic channels with dimensions of 300 µm or less where the blood cells tend to migrate toward the center of the channel forming a cell-free region along the channel walls [14–16]. Similarly, the Zweifach-Fung bifurcation law explains the behavior of cells at channel bifurcations. Whenever a microchannel splits into two channels, the blood cells tend to flow into the channel with a higher flow rate provided certain specific conditions are satisfied [17]. Bifurcation laws are applicable when the critical velocity ratio (branch with higher velocity to that of the branch with lower velocity) should be greater than or equal to 2.5. Similarly, the size ratio (ratio of the cell to microchannel dimensions) should be of the order of one [18]. Researchers have employed the combined effects of channel geometry such as bends, curvatures, and constriction–expansion along with the biophysical phenomena to improve the plasma separation efficiency. One of the earlier reported works utilized T-shape microchannel bifurcations for plasma separation [19]. Later Blattert et al. demonstrated the use of curved microchannels for increased cell-free layer formation due to the centrifugal effect
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[20]. Yet another research group has employed constriction–expansion of the channels for cell focusing followed by plasma skimming [21]. Further, Kerhoas et al. were successful in employing a hybrid design, which combines bifurcation law, Fahraeus effect, constriction–expansion, and high flow rates [22]. Lee et al. explored the possibilities of a constriction expansion array for developing inertial lift force and Dean’s drag force for plasma separation [23]. Yet another group reported plasma separation with curved microchannels using inertial forces for focusing [24]. Tripathi et al. employed T-microchannels with elevated dimensions and biophysical effects for clog-free operations [25]. Prabhakar et al. proposed a tau-shaped hybrid microdevice where several biophysical effects together with centrifugal effects and constriction–expansion were used to achieve plasma separation with higher efficiency [26]. Many of these devices reported in the literature used diluted blood and separation efficiency decreased with high hematocrit or whole blood [4, 27]. Though some of the hybrid design has shown better separation efficiency close to 99% with whole blood, other drawbacks such as clogging due to low microchannel dimensions, very low plasma yield, and complexity in design and fabrication limit the application of these devices [3]. The poor quality of plasma extracted from the diluted blood results in uncertainties in analyte/biomarker detection and its quantification [3]. Tripathi et al. from our research group in their recently reported work has addressed many of these challenges by modifying their earlier design of T and Tau microchannels [3, 4]. They were successful in fabricating a simple, passive hydrodynamic microfluidic device capable of providing complete separation of cells from the blood. Almost 100% separation efficiency was achieved with whole blood (low to high hematocrit values up to 42%) at a controlled flow rate of 0.3–0.5 ml/min [3]. The higher dimensions of the channels (hundred microns) ensure clog-free operations and comparatively higher yield (up to 6%) [3] of good quality plasma are some of the significant features of this device that builds confidence in its usage for real-time bioanalytical applications. We employed this design for plasma separation, and as a further extension, we have performed experiments on the extracted plasma for the quantification of albumin (plasma protein). Clinical biochemistry laboratories have adopted the dye-binding method of albumin estimation for a few decades [28]. This method offers several advantages such as faster analysis, precise estimation, fairly specific reactions, low cost, simple operation, etc. [28]. We used BCG dye reagent for the detection of albumin. The color change of the test sample was compared with that of a reference (standard) solution for estimating the concentration of albumin using the photometric method. The quality of plasma extracted after separation is a critical parameter that ensures the accuracy of any quantitative or qualitative analysis. The comparative study of the present method of plasma separation with that of the conventional method (centrifuge) reveals that this microdevice has a potential scope of application toward the point of care (POC) diagnostic devices. The concentration of albumin detected from the plasma extracted by two different methods of separation is in good agreement with each other. Integration of biosensors to this device can deliver a complete standalone on-chip plasma separation and detection device.
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Fig. 1 Design of the microdevice for blood plasma separation
2 Materials and Methodes In this section, we discuss the design of the microfluidic device for blood plasma separation, fabrication protocols, and the experimental setup used for this study.
2.1 Design of Blood Plasma Separation Microdevice The device constitutes a main input channel (200 µm) to which the blood is pumped using a syringe pump connected to the input port (I) at a controlled flow rate (0.3– 0.6 ml/min). The main channel gets bifurcated into two output channels of different dimensions as shown in Fig. 1. The narrow channel (60 µm) was used to collect the separated plasma at the second output port (P), whereas the broad channel (300 µm) was used to collect the blood from the first output port (O). This design utilizes biophysical effects such as the Zweifach-Fung bifurcation laws and the Fahraeus effect to realize the plasma separation. Besides the above biophysical effects, some geometrical effects such as constriction and expansion of the channels, and bends were introduced to enhance the separation efficiency [3]. More detailed descriptions such as the theoretical aspects of the design and the dimensions of the channels can be found in previous literature reports [3, 4].
2.2 Fabrication of Microchannels We employed conventional techniques of photolithography for fabricating the micro mold on a 2-inch silicon wafer. Photoresist SU8-2050 was spin-coated on the silicon wafer to achieve a channel depth of 60 µm [3, 4]. Further, PDMS (Poly dimethyl siloxane) replicas of the channels were fabricated using soft lithography. PDMS was
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developed at a 10:1 mixing ratio (base to curing agent ratio) which was further cured at a temperature of 70 °C for 90 min. The PDMS replicas were bonded to glass substrates using the oxygen plasma bonding technique. The fabricated microfluidic chip was later connected with connectors and tubes to facilitate the blood flow through the microchannels formed between the glass and PDMS.
2.3 Experimental Setup Human blood with a hematocrit content of less than 45% was used for our experiments. Fresh blood samples were collected in EDTA (anticoagulant)-coated vacutainer. A syringe pump (Cole-Parmer) was utilized to feed the blood into the microdevice connected via microfluidic connectors and tubes. The blood was pumped into the channel at a controlled flow rate of 0.6 ml/min. The separated plasma was collected in a 1 ml Eppendorf tube using micropipettes. The extracted plasma was later used for detecting the concentration of albumin. The colorimetric reagent BCG (Bromocresol green) used for albumin quantification was procured from Erba diagnostics GmbH. The CCD camera connected to the microscope (Olympus CH 20i) was used to capture the experimental images. The optical absorbance of the resultant combination of plasma and BCG reagent at 628 nm was measured using a plate reader and spectrophotometer (Thermofischer scientific) interfaced with a PC. The centrifuge (Thermofischer scientific) used to separate plasma from blood was operated at 3000 rpm for 20 min. The schematic of the experimental setup used for this study is shown in Fig. 2.
3 Results and Discussion Plasma separated from whole blood using the microdevice with a yield of 2–3%. We used 3 ml of blood from three adult male donors for our experiments. The extracted plasma was used for albumin detection using BCG reagent.
3.1 Detection of Albumin Albumin, a major plasma protein, plays a significant role in the regulation and distribution of extracellular fluid, and the transportation of various hormones, vitamins, and trace metals [28]. The levels of albumin in the blood remain as a figure of merit for normal metabolic operations in the body. For a healthy human adult, albumin concentration in the blood shall be in the range of 3.5–5.2 g/dl. Increased levels are observed in case of (a) dehydration due to reduced plasma water content. (b) Also
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Fig. 2 Experimental setup used for plasma separation and quantification of albumin. The separated plasma is extracted using a micropipette which was mixed with BCG reagent in an Eppendorf tube. The resultant solution was transferred to a well plate for measuring the absorbance using a spectrophotometer interfaced with a PC. The absorbance at 628 nm was used to calculate the concentration of albumin. The experimental images were captured using a microscope
during stasis venipuncture causes fluid to escape into the extravascular compartment. Decreased levels are observed in cases where (a) excessive protein loss from the kidney, skin, or intestine. (b) Decreased synthesis due to dietary, hepatic diseases, or malabsorption. (c) Increased catabolism in fever, untreated diabetes mellitus, and hypertension.
3.2 Methodology for Detection of Albumin Albumin present in the plasma binds with a reagent BCG to form a blue-greencolored complex (Albumin-BCG) [29]. The intensity of the resultant color formed after mixing is proportional to the concentration of albumin [29, 30]. The resultant BCG–albumin complex shows an absorbance peak at 628 nm. Albumin present in the plasma was quantified by comparing the absorbance (628 nm) of the test sample with that of the absorbance of the standard solution of albumin. The standard solution contains a known value of albumin concentration (3.6 g/dl) supplied along with the diagnostic reagent kit. The concentration of albumin was calculated from Eq. 1. C=
AT ×S AS
(1)
where ‘C’ represents the concentration of albumin (g/dl). ‘S’ represents the concentration of the standard solution (g/dl); A T represents the absorbance value of the
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Fig. 3 The UV–visible absorbance spectrum of BCG–albumin complex over a range of wavelength from 600 to 650 nm. The spectrum shows a peak absorbance at 628 nm
test (BCG + plasma), and A S denotes the absorbance value of the standard solution (BCG + standard) The absorbance spectrum of the albumin–BCG complex is plotted in Fig. 3. The absorbance spectrum shows a shift toward the blue-green region when albumin binds with BCG. The efficiency of plasma separation using this microfluidic device was compared with that of plasma separated using conventional techniques of centrifugation. Quantification of albumin was performed on the plasma separated from both techniques for comparative study. The plasma separated from three different blood samples was used for experiments. A total of 500 µL of BCG reagent was mixed with 20 µL of plasma in the Eppendorf tube. About 100 µL of the resultant solution was transferred to a well plate for absorbance measurement. The experimental results of albumin detection are tabulated in Table 1. The results are within the normal range of albumin concentration for adults. Analysis of the result reveals that the device can provide excellent separation efficiency, which is a very vital parameter for analytical experiments. The results are comparable with the centrifugation techniques that are widely used in clinical diagnosis. The comparative study depicts that a relative error of ± 3% is within the reasonable Table 1 Concentration of albumin calculated from plasma separated using centrifugation and microdevice Sl. no Sample
Concentration of albumin (g/dl) Plasma separated using microdevice Plasma separated using centrifugation
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Sample 1 4.5
4.3
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Sample 2 3.8
3.7
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Fig. 4 Concentration of albumin detected for whole blood samples with hematocrit < 45%
limits of agreement. The concentration of albumin (g/dl) detected from plasma separated using this microdevice and centrifuge is plotted in Fig. 4. This result reaffirms the efficiency of the microdevice in delivering clean and pure plasma that can be directly used for further steps of detection protocols. However, at higher hematocrit (> 45%), the separation efficiency decreases that in turn deteriorates the accuracy of quantitative detection.
4 Conclusions In this study, we performed experiments on plasma separation using a microfluidic device for whole blood samples with a hematocrit of less than 45%. The blood was fed into the channel using a syringe pump at a controlled flow rate of 0.6 ml/min. Though the yield of the plasma was on the lower side, excellent separation efficiency (100%) was achieved. Clean and pure plasma was extracted from the microdevice which is highly desirable for bioanalytical experiments. The plasma was used for photometric detection and quantification of albumin. The quality of plasma obtained through microfluidic separation was compared with that of the centrifuge. The concentration of albumin detected from both plasma samples reveals that the results are within the reasonable limits of agreements. The experimental study confirms that this technique of microfluidic separation has the potential to replace the conventional techniques of plasma separation in clinical diagnosis. Low plasma yield and poor separation efficiency for higher hematocrit (> 45%) remain a major challenge that needs to be addressed. Despite these drawbacks, faster process time, simple construction, and ease of operation make this device a promising alternative in the domain of point-of-care diagnostics. This work can be further extended toward the design of a complete standalone on-chip plasma
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separation and detection device. The integration of biosensors to the output port of this microdevice can avoid many intermediate steps involved in the current detection protocols. Acknowledgements The authors are thankful to the Department of Science and Technology (DST), Govt. of India, for providing financial support to this project. The authors would like to extend thanks to the Centre of Excellence in Nanotechnology (CEN), Department of Electrical Engineering, IIT, Bombay, for providing the micro-nano fabrication facilities. The authors would like to acknowledge the support from Prof. Siddhartha Tripathi, Faculty at BITS Pilani, Goa campus.
References 1. Kerhoas MK, Sollier E (2013) Micro-scale blood plasma separation: from acoustophoresis to eggbeaters. Lab Chip 13:3323–3346 2. Sollier E, Rostaing H, Pouteau P, Fouillet Y, Achard JL (2009) Passive microfluidic devices for plasma extraction from whole human blood. Sens Actuat B 141:617–624 3. Tripathi S, Bala Varun Kumar YV, Agrawal A, Prabhakar A, Joshi SS (2016) Microdevice for plasma separation from whole human blood using biophysical and geometrical effects. Sci Rep 6:267499 4. Tripathi S, Kumar YVBV, Prabhakar A, Joshi SS, Agrawal A (2015) Passive blood plasma separation at the microscale: a review of design principles and microdevices. J Micromech Microeng 25:083001 5. Nakashima Y, Hata S, Yasuda T (2010) Blood plasma separation and extraction from a minute amount of blood using dielectrophoretic and capillary forces. Sens Actuat B 145:561–569 6. Laurell T, Petersson F, Nilsson A (2007) Chip integrated strategies for acoustic separation and manipulation of cells and particles. Chem Soc Rev 36:492–506 7. MacDonald MP, Spalding GC, Dholakia K (2003) Microfluidic sorting in an optical lattice. Nature 426:421–424 8. Huh D, Bahng JH, Ling YB, Wei HH, Kripfgans OD, Fowlkes JB, Grotberg JB, Takayama S (2007) Gravity-driven microfluidic particle sorting device with hydrodynamic separation amplification. Anal Chem 79:1369–1376 9. Lee BS, Lee JN, Park JM, Lee JG, Kim S, Cho YK, Ko C (2009) A fully automated immunoassay from whole blood on a disc. Lab Chip 9:1548–1555 10. Jung J, Han K-H (2008) Lateral-driven continuous magnetophoretic separation of blood cells. Appl Phys Lett 93:223902 11. Pamme N (2007) Continuous flow separations in microfluidic devices. Lab Chip 7:1644–1659 12. Sajeesh P, Sen AK (2014) Particle separation and sorting in microfluidic devices: a review. Microfluid Nanofluid 17:1–52 13. Sollier E, Rostaing H, Pouteau P, Fouillet Y, Achard L (2009) Passive microfluidics devices for plasma extraction from whole human blood. Sens Actuat B 141:617–624 14. Fahraeus R (1929) The suspension stability of the blood. Physiol Rev 9:241–274 15. Fahraeus R, Lindqvist T (1931) The viscosity of the blood in narrow capillary tubes. Am J Physiol 96:562–568 16. Barbee JH, Cokelet GR (1971) The fahraeus effect. Microvas Res 3:6–16 17. Fung YC (1973) Stochastic flow in capillary blood vessels. Microvasc Res 5:34–48 18. Fung YC (1981) Biomechanics—Mechanical Properties of Living Tissues. Springer), New York 19. Yang S, Undar A, Zahn JD (2006) A-microfluidic device for continuous, real-time blood plasma separation. Lab Chip 6:871–80
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20. Blattert C, Jurischka R, Tahhan I, Schoth A, Kerth P, Menz W (2004) Separation of blood in microchannel bends In: Conference proceedings IEEE engineering in medicine and biology society, vol 4, pp 2627–2630 21. Faivre M, Abkarial M (2006) Geometrical focusing of cells in a microfluidic device: an approach to separate blood plasma. J Biorheol 43:147–159 22. Kerhoas MK, Kavanagh DM, Dhariwal RS, Campbell CJ, Desmulliez MPY (2010) Validation of a blood plasma separation system by biomarker detection. Lab Chip 10:1587–1595 23. Lee MG, Choi S, Kim HJ, Lim HK, Kim JH, Huh N, Park JK (2011) Inertial blood plasma separation in a contraction–expansion array microchannel. Appl Phys Lett 98:253702 24. Di Carlo D, Edd JF, Irimia D, Tompkins RG, Toner M (2008) Equilibrium separation and filtration of particles using differential inertial focusing. Anal Chem 80:2204–2211 25. Tripathi S, Prabhakar A, Kumar N, Singh SG, Agrawal A (2013) Blood plasma separation in elevated dimension T-shaped microchannel. Biomed Microdev 15:415–425 26. Prabhakar A, Bala Varun Kumar YV, Tripathi S, Agrawal A (2015) A novel, compact and efficient microchannel arrangement with multiple hydrodynamic effects for blood plasma separation Microfluid. Nanofluid 18:995–1006 27. Goldsmith L, Marlow JC (1979) Flow behavior of erythrocytes. II. Particle motions in concentrated suspensions of ghost cells. J Colloid Interface Sci 71:383–407 28. Kumar D, Banerjee D (2017) Methods of albumin estimation in clinical biochemistry: Past, present, and future. Clin Chim Acta 469:150–160 29. Doumas BT, Watson WA, Biggs HG (1971) Albumin standards and the measurement of serum albumin with bromcresol green. Clin Chim Acta 31:87–96 30. Tietz NW (ed) (1995) Clinical guide to laboratory test, 3rd edition, wb Saunders, vol 22
White Blood Cell Separation and Blood Typing Using a Spiral Microdevice Sanjay Mane, Vadiraj Hemadri, Sunil Bhand, and Siddhartha Tripathi
ABSTRACT Our immune system is shielded from various pathogens by white blood cells (WBCs), which work as soldiers. WBCs reach the infectious site and kill the invading pathogens. The assessment of WBC activity and function is important in various diseases such as cancer, HIV, and autoimmune disorders; hence their separation is essential. The present study describes a method for WBC separation using the inertial microfluidic technique. A simple spiral microfluidic device with one inlet and three outlets is constructed here for WBC separation. The device functions on a diluted blood sample. Only 2 finger-pricked droplets of blood (~20 µl) are required to prepare the minute volume of a diluted blood sample. A syringe pump is used to infuse the sample into the channel reservoir. The microdevice takes less than 22 s for WBC separation. We report WBC separation efficiency of nearly 90% with 92% RBC rejection ratio. In addition to WBC separation, the spiral microdevice is capable of blood group testing. Keywords Spiral microdevice · Inertial forces · Hematocrit · Blood group
1 Introduction The immune system depends on how well white blood cells (WBCs) function. WBCs are immune cells that protect us from various infectious diseases. Apart from WBCs, red blood cells, platelets, and plasma are other blood constituents. The major portion of blood is occupied by plasma (i.e., 55% v/v) and RBCs (i.e., 45% v/v). WBCs and platelets occupy less than 1% of total blood volume [1]. WBCs are very few in numbers (5000–11,000 /µl) compared to RBCs (45,00,000–55,00,000 /µl) and platelets (1,50,000–4,50,000 /µl). WBCs are spherical, less deformable than RBCs, S. Mane · V. Hemadri · S. Tripathi (B) Department of Mechanical Engineering, BITS-Pilani, K K Birla Goa Campus, Mormugao, Goa 403726, India e-mail: [email protected] S. Bhand Department of Chemistry, BITS-Pilani, K K Birla Goa Campus, Mormugao, Goa 403726, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_62
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and nucleated cells have a diameter ranging from 7–22 µm [2]. WBCs are further classified into granulocytes and agranulocytes. RBCs are deformable, biconcave disk-shaped cells with a diameter of 5–8 µm. Platelets are the smallest cells (dia. 2–4 µm) among all cells having discoid in shape [3]. WBCs play a crucial role in engulfing various pathogens like bacteria, viruses, and fungi. The activity of WBCs engulfing pathogens is known as phagocytosis [4]. To study phagocytosis activity and other immune-related disorders, their effective separation from other blood cells is important. The conventional techniques used for WBC separations are centrifugation, RBC lysis, and flow cytometry [5–7]. Centrifugation is the “gold standard” technique in clinical pathology laboratories. Although centrifugation is essential for WBC separation, there are substantial constraints, such as the need for processing steps, costly chemicals, and skilled technicians [8]. RBC lysis requires additional lysing reagents that may change WBC viability, activation, and light scattering properties [9, 10]. Flow cytometry provides highly accurate results, but it is costly [11]. Recent advances in microfluidic technology can be a viable option to overcome the drawbacks associated with conventional techniques. Microfluidic devices can study blood cell dynamics and their separation. In microfluidics, cell separation can be achieved using active and passive methods [12]. The active methods use external forces like acoustic, magnetic, and electric with complex geometry for cell separation [12, 13]. On the other side, the passive method utilizes channel geometry, biophysical laws, and the properties of cells for cell separation [14]. The passive methods are simple in geometry, cost-effective, require a small blood sample volume, and can be integrated into a point-of-care device for immediate analysis. Some of these passive methods include microfiltration, deterministic lateral displacement (DLD), hydrodynamic methods, and inertial techniques [15, 16]. The microfiltration technique is based on differences in cell properties (i.e., cell size and deformability) for cell separation. Cheng et al. [17] used clogging-free polycarbonate microporous membranes with an integrated bidirectional micro-pump. They reported a 72.1% WBC separation efficiency using undiluted blood. DLD approach uses micro-post arrays to separate cells. Here, the cells are compelled to take different pathways based on their diameter. Zheng et al. [18] used the DLD concept and achieved 91% WBC separation efficiency using diluted blood. Yamada and Seki [19] introduced the hydrodynamic filtration technique, which depends on the flow rate and size of channel widths. Using this technique, they were able to enhance the ratio of WBCs to RBCs by 29 times. Another technique is inertial separation, in which the position of particles/cells within the channel is governed by the balance of inertial lift force and Dean Drag force [20–24]. Wu et al. [25] employed a spiral channel with a trapezoidal cross-section to isolate WBCs. The trapezoidal cross-section alters the velocity profile and results in strong Dean vortices on smaller particles means on RBCs. Therefore, RBCs shift toward the outer wall and WBCs toward the inner wall making their efficient separation. Using diluted blood with a hematocrit (volume percentage of RBCs) of 1–2%, they obtained a WBC separation efficiency of > 80%. Mane et al. [26] utilized the hypertonic buffer to separate WBCs using a wavy-type microfluidic device. The hypertonic buffer causes RBCs to shrink, exposing them to lower inertial and centrifugal forces, which helps in cell placement within the
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channel. Using a wavy channel, we achieved 88% separation efficiency with a purity of 83%. Our survey reveals that there is enough scope to enhance the WBC separation process.
2 Objective This research aims to separate WBCs in a simple spiral microchannel using inertial microfluidics approach. The separation involves a syringe pump to infuse the blood sample into the channel. The study also includes the use of the same microdevice to identify the person’s blood group. This novel approach helps in isolation of WBCs for further disease diagnostics.
3 Methods and Materials 3.1 Microchannel Design and Fabrication A spiral microchannel was designed having one inlet and three outlets, as shown in Fig. 1. The main channel width is kept at 200 µm, and the depth is 50 µm throughout the channel. The microdevice fabrication was carried out in two steps. Initially, a master mold was created using a CNC milling machine. Later, the soft lithography technique was employed. The detailed fabrication process is given in our previous study [26]. The final fabricated PDMS (polydimethylsiloxane) device is shown in Fig. 1, and the channel is highlighted by passing the blue color dye through it. Fig. 1 Fabricated PDMS device with working principle
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3.2 Working of the Microdevice As depicted in Fig. 1, the channel consists of three outlets. Outlet 3 is provided to receive target cells, i.e., WBCs, whereas outlet 1 and outlet 2 are provided for removing the majority of RBCs. Removing RBCs in outlets 1 and 2 will reduce the number of RBCs getting in outlet 3, thereby increasing the purity of WBCs in outlet 3. As diluted blood passes into the channel, the RBCs and WBCs acquire different positions in the channel. Cells flowing in a spiral geometry experience two types of forces; inertial lift and Dean Drag force [20–22]. Two prominent inertial lift forces are present within the channel: shear-induced and wall-induced lift forces. The shearinduced lift force is due to the interaction between cells, which causes the migration of cells from the channel center to the channel walls. On the contrary, the interaction between cell and wall causes wall-induced lift force acting toward the channel center. The Dean Drag force is due to the difference in velocity fields at the inner and outer walls of the channel. As a result, RBCs get shifted toward the outer wall, and WBCs shift toward the inner wall of the channel. The smaller particles (i.e., RBCs) quickly respond to Dean force compared to larger particles (i.e., WBCs) and they shift toward the outer wall of the channel [23]. Larger particles (i.e., WBCs) experience a strong lift force on them and they shift toward the inner wall of the channel [24]. The net inertial lift force (F L ) and Dean Drag force (F D ) can be calculated as [22, 23]: FL =
ρu 2 d 4p dh2
(1)
where ρ is the density of the fluid, u is the velocity of the fluid, d p is the diameter of the particle/cell, and d h is the hydraulic diameter of the microdevice. FD = 3π μU D d p
(2)
where μ is the dynamic viscosity of the fluid, and U D is the Dean velocity of the fluid. Dean velocity can be calculated as, U D = 1.8 × 10−4 × De1.63 in which De is the Dean number, and it is estimated as De = Rech × (dh /2R)0.5 , Rech is the channel Reynolds number, and R is the mean radius of curvature. Using Eqs. (1) and (2), we calculated the ratio of net lift force to Dean Drag force (Rf ), and their values are 0.48 and 2.26 for RBC and WBC, respectively. This shows that a strong net lift force acts on WBC, whereas a strong Dean Drag force acts on RBC. Due to high Dean Drag force, RBCs move toward outlet 1 and outlet 2, whereas WBCs reach outlet 3 by experiencing a strong net lift force.
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3.3 Experimental Setup and Procedure The experimental setup includes an inverted microscope (Olympus-IX73) with a high-speed camera (Phantom Miro-C210). The blood sample was continuously infused into the microdevice using a syringe pump (Cole Parmer). Nearly 20 µl of blood sample was obtained using the finger prick method. Later, the blood sample was diluted using normal saline (0.9% NaCl) to form nearly 1% Hct. The diluted blood sample was fed into the channel inlet via a silicon tube using a syringe pump. All the experiments were conducted at room temperature within 24 h after blood collection. The cells available at the inlet of the channel and cells after separation (i.e., at outlet 3) were counted using an auto cell counter (Countess-3L, Thermo Fisher Scientific). Results were also confirmed using flow cytometry. Acridine orange (AO) was added to the sample to differentiate the WBCs from other blood constituents. AO stains the WBCs; they excite at 489 nm and emit green fluorescence at 493 nm. Flow cytometry differentiates the stained WBCs on the basis of AO excitation–emission wavelengths. The performance of the microdevice is based on WBC separation efficiency (ηsep ) and RBC rejection ratio (Φ). The separation efficiency and RBC rejection ratio are calculated according to Eqs. (3) and (4), respectively [7, 10]. no. of separated WBCs present at outlet 3/μl total no. of WBCs available at inlet/μl
(3)
no. of RBCs present at outlet 1 and outlet 2/μl total no. of RBCs available at inlet/μl
(4)
ηsep = =
4 Results and Discussion This section discusses the WBC separation in a spiral microfluidic device using a syringe pump and the identification of an individual’s blood group.
4.1 WBC Separation The microdevice shown in Fig. 1 was tested for WBC separation. The results are shown in Fig. 2. The diluted blood samples were infused into the inlet reservoir of the channel using a syringe pump. The experiments were carried out at various hematocrits. The best WBCs separation is obtained at nearly 1% Hct with an inlet flow rate calculated as ~ 45 µl/s. Figure 2a shows the image of the microdevice captured at the separation branch, whereas Fig. 2b is the same image presented in a zoomed view. As flow occurs, the first branch removes approximately half the number of RBCs with very few WBCs into outlet 1. This is because RBCs are present near the outer
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Fig. 2 a Microscopic bright field images of WBC Separation at second branch b zoomed view of second branch c Identification of person’s blood group using spiral microdevice
wall, and WBCs are present near the inner wall of the channel due to strong inertial forces acting on them. The presence of RBCs near the outer wall of the channel makes it easy to remove RBCs into outlet 1. As the remaining cells arrive in second branch, Fig. 2a and b, the majority of RBCs with few WBCs moves into outlet 2 of the microdevice. Finally, the remaining cells contain the majority of WBCs, with very few RBCs received into outlet 3. We obtained 90 ± 5% WBC separation efficiency with a 92 ± 4% RBC rejection ratio using this spiral microdevice.
4.2 Blood Group Test We also utilized the same spiral channel for blood typing. Blood typing is quite common and important. The test is essential during a blood transfusion or transplant. Conventional laboratory techniques require a significant volume of blood samples. However, the microfluidic device requires fewer blood samples [27]. The microfluidic blood group testing method can lead to the simplification of medical care in emergencies. It has the potential to considerably reduce costs and essential manpower. The microfluidic blood group typing can also be used to determine the expression of the antigen on the RBCs. Antigen expression assists our body in identifying and eliminating hazardous invaders [28]. Figure 2c shows the blood group test with the use of a spiral device. The inlet, outlet 1, and outlet 2 reservoirs are used for blood grouping tests once WBCs are separated in outlet 3. The blood group test works on the antigen–antibody principle. Three types of antibodies are used in blood grouping tests, i.e., Anti-A, Anti-B, and Anti-D. Anti-A and Anti-B identify the blood group type, whereas Anti-D indicates whether the blood group is positive or negative. When an antigen on the surface of RBCs binds to a specific antibody, RBCs form a cluster known as agglutination, and the test is positive [27]. If there is no visible agglutination, the test is negative. The criteria for identifying the person’s blood group are shown in Table 1. The blood group test was conducted after the separation of WBCs. As shown in Fig. 2c, a small drop of Anti-A, B, and D was placed on the samples present at the inlet, outlet 1, and outlet 2 of the channel. Later on, the sample and antibody solution were thoroughly mixed. The mixing was carried out
White Blood Cell Separation and Blood Typing Using a Spiral Microdevice Table 1 Criteria to identify patient’s blood group
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Anti-A
Anti-B
Anti-D
Blood group
+ ve
− ve
+ ve
A+
+ ve
− ve
− ve
A−
− ve
+ ve
+ ve
B+
− ve
+ ve
− ve
B−
+ ve
+ ve
+ ve
AB+
+ ve
+ ve
− ve
AB−
− ve
− ve
+ ve
O+
− ve
− ve
− ve
O−
using a micropipette, and the solution was observed after 2 min. As an example, the experimental observations reveal the individuals blood group as “B + ”.
4.3 Cell Counting and Quantification Using a flow cytometer and an auto cell counter, cells were counted. RBC and WBC initial concentrations were determined to be 80,000 /µl and 90 /µl, respectively. Upon separation, the concentrations of RBCs and WBCs were found to be 6500 / µl and 81 /µl, respectively. The separation efficiency and RBC rejection ratio were calculated, as discussed in Sect. 3.3. The separation efficiency is 90 ± 5%, whereas the RBC rejection ratio is 92 ± 4%. Figure 3 shows the flow cytometry results on the samples present at the inlet (Fig. 3a) and outlet 3 (Fig. 3b) of the microdevice. Acridine orange was added to the sample to differentiate the WBCs from other blood constituents. The cell differentiation is based on forward scatter height (FSC-H) and side scatters height (SSC-H). The FSC shows the cell size, whereas SSC shows the granularity of cells. The RBCs, WBCs, and platelets are shown in Fig. 3 using their initial letters. Figure 3b confirms that WBCs (granulocytes and agranulocytes) are enriched in outlet 3 with few RBCs and platelets compared to their concentrations present at the inlet (Fig. 3a). This demonstrates that the spiral microdevice can separate WBCs with high purity.
5 Conclusions This work demonstrated a simple spiral microfluidic device for WBC separation. Experiments are conducted using diluted blood samples prepared from two fingerpricked blood drops. A syringe pump is used to inject the sample into the microdevice. WBC separation is carried out using an inertial technique in which RBCs shifted toward the outer wall and WBCs toward the inner wall of the channel. Using a spiral
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Fig. 3 Quantification of cells using flow cytometry a cells present at inlet of the channel before separation b cells present at outlet 3 of the channel after separation. The top ellipse represents granulocytes, whereas the lower ellipse indicates agranulocytes
channel, the WBC separation efficiency obtained is 90%, with 92% RBC rejection ratio at nearly 1% hematocrit (Hct). In addition, the reported microdevice can be utilized to identify the person’s blood group. Also, this novel method takes less than 22 s for WBC separation and can be used for point-of-care diagnostic. Acknowledgements The authors acknowledge funding support under strategic research projects from BITS BioCyTiH Foundation (a Section 8 not for profit company) hosted by BITS Pilani supported under the National Mission of Interdisciplinary Cyber Physical Systems (NM-ICPS), Department of Science & Technology (DST), Government of India. We want to thank Mr. Tushar, BITS Pilani, K. K. Birla Goa campus, for assisting with the experiments related to flow cytometry. Ethical Clearance Every experiment was carried out in accordance with the guidelines and standards established by the Human ethical committee at BITS Pilani, K K Birla Goa campus. All blood samples were screened for infectious illnesses before being used.
Nomenclature ρ μ dp dh Rc Vf UD De R
Density of fluid [kg/m3 ] Dynamic viscosity of fluid [Pa.s] Particle/cell diameter [m] Hydraulic diameter [m] Radius of curvature of a microchannel [m] Velocity of fluid [m/s] Dean velocity [m/s] Dean number Mean radius of curvature [m]
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Rech FD FL Rf ηsep Φ
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Channel Reynolds number Dean drag force [N] Net lift force [N] Ratio of inertial lift to Dean Drag force Separation efficiency [%] RBC rejection ratio [%]
References 1. Fung YC (2013) Biomechanics: mechanical properties of living tissues. Springer Science Business Media 2 2. Caro CG (2011) The mechanics of the circulation. 2nd Edn. Cambridge University Press 3. Viallat A, Abkarian M (2020) Dynamics of blood cell suspensions in microflows. CRC Press, New York 4. Li X, Chen W, Liu G, Lu W, Fu J (2014) Continuous-flow microfluidic blood cell sorting for unprocessed whole blood using surface-micromachined microfiltration membranes. Lab Chip 14:2565–2575 5. Choi HS, Kim JW, Cha YN, Kim C (2006) A quantitative nitroblue tetrazolium assay for determining intracellular superoxide anion production in phagocytic cells. J Immunoassay Immunochem 27:31–44 6. Cheng X, Irimia D, Dixon M, Sekine K, Demirci U, Zamir L, Tompkins RG, Rodriguez W, Toner M (2007) A microfluidic device for practical label-free CD4+ T cell counting of HIV-infected subjects. Lab Chip 7:170–178 7. Tan JKS, Park SY, Leo HL, Kim S (2017) Continuous separation of white blood cells from whole blood using viscoelastic effects. IEEE Trans Biomed Circuits Syst 11:1431–1437 8. Bruil A, Aken WGV, Beugeling T, Feijen J (1991) Asymmetric membrane filters for the removal of leukocytes from blood. J Biomed Mater Res 25:1459–1480 9. Sethu P, Anahtar M, Moldawer LL, Tompkins RG, Toner M (2004) Continuous flow microfluidic device for rapid erythrocyte lysis. Anal Chem 76:6247–6253 10. Zhang J, Yuan D, Sluyter R, Yan S, Zhao Q, Xia H, Tan SH, Nguyen NT, Li W (2017) Highthroughput separation of white blood cells from whole blood using inertial microfluidics. IEEE Trans Biomed Circuits Syst 11. Sethu P, Moldawer LL, Mindrinos MN, Scumpia PO, Tannahill CL, Wilhelmy J, Efron PA, Brownstein BH, Tompkins RG, Toner M (2006) Microfluidic isolation of leukocytes from whole blood for phenotype and gene expression analysis. Anal Chem 78:5453–5461 12. Kim M, Jung SM, Lee KH, Kang YJ, Yang S (2010) A microfluidic device for continuous white blood cell separation and lysis from whole blood. Artif Organs 34:996–1002 13. Han KH, Frazier AB (2006) Paramagnetic capture mode magnetophoretic microseparator for high efficiency blood cell separations. Lab Chip 6:65–273 14. Urbansky A, Olm F, Scheding S, Laurell T, Lenshof A (2019) Label-free separation of leukocyte subpopulations using high throughput multiplex acoustophoresis. Lab Chip 19:1406–1416 15. Vandelinder V, Groisman A (2007) Perfusion in microfluidic cross-flow: separation of white blood cells from whole blood and exchange of medium in a continuous flow. Anal Chem 79:2023–2030 16. Kuan DH, Wu CC, Su WY, Huang NT (2018) A microfluidic device for simultaneous extraction of plasma, red blood cells and on-chip white blood cell trapping. Sci Rep 8:1–9 17. Mane S, Hemadri V, Tripathi S (2021) Isolation of white blood cells from human blood in a microfluidic device, FMFP2021–08-180. In: 48th National conference on fluid mechanics and fluid power (FMFP), BITS Pilani, Pilani Campus, Rajasthan, India, December 27–29
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18. Cheng Y, Ye X, Ma Z, Xie S, Wang W (2016) High-throughput and clogging-free microfluidic filtration platform for on-chip cell separation from undiluted whole blood. Biomicrofluidics 10:014118 19. Zheng S, Tai YC, Kasdan H (2006) A micro device for separation of erythrocytes and leukocytes in human blood. In: Proceedings engineering medicine and biology, pp 1024–1027 20. Yamada M, Seki M (2005) Hydrodynamic filtration for on-chip particle concentration and classification utilizing microfluidics. Lab Chip 5:1233–1239 21. Carlo DD, Irimia D, Tompkins RG, Toner M (2007) Continuous inertial focusing, ordering, and separation of particles in microchannels. Proc Natl Acad Sci USA 104:18892–18897 22. Tripathi S, Kumar YVBV, Agrawal A, Prabhakar A, Joshi SS (2019) Microdevice for plasma separation from whole human blood using biophysical and geometrical effects. Sci Rep 6:26749 23. Nivedita N, Papautsky I (2013) Continuous separation of blood cells in spiral microfluidic devices. Biomicrofluidics 7:1–14 24. Chiu PL, Chang CH, Lin YL, Tsou PH, Li BR (2019) Rapid and safe isolation of human peripheral blood B and T lymphocytes through spiral microfluidic channels. Sci Rep 9 25. Tay HM, Petchacup C, Dalan R, Hou HW (2015) Leukocyte fractionation using inertial microfluidics. In: Conference of miniaturized system chemistry life science, pp 367–369 26. Wu L, Guan G, Hou HW, Bhagat AAS, Han J (2012) Separation of leukocytes from blood using spiral channel with trapezoid cross-section. Anal Chem 84:9324–9331 27. Mane S, Hemadri V, Tripathi S (2022) Separation of white blood cells in a wavy type microfluidic device using blood diluted in a hypertonic saline solution. BioChip J 16:291–304 28. Chang YJ, Ho CY, Zhou XM, Yen HR (2018) Determination of degree of RBC agglutination for blood typing using a small quantity of blood sample in a microfluidic system. Biosens Bioelectron 102:234–241 29. Kline TR, Runyon MK, Pothiawala M, Ismagilov RF (2008) ABO, D blood typing and subtyping using plug-based microfluidics. Anal Chem 80:6190–6197
Effect of Arterial Flow on Heat Transfer During Magnetic Hyperthermia Application Subeg Singh and Neeraj Kumar
Abstract This paper investigates the effects of location arterial flow on heat transfer during hyperthermia applications. A three-dimensional tumor model consisting of a blood vessel (artery) is modeled and simulated for magnetic hyperthermia physics. The location of the blood vessel (artery) with respect to the tumor is altered in the tumor models and it passes through different positions from the tumor center. The size (diameter) of the artery is 4 mm, and the tumor size is 10 mm. The different artery positions x = 3, 6, 9, and 12 mm from the tumor center are considered in the physical models. Results show that the position of the artery plays a crucial role in heat dissipation from the tumor volume during magnetic hyperthermia. The therapeutic temperature in the tumor tissue decreases when an artery is closer to the tumor center. This is due to the higher heat transfer effect induced by arterial blood flow. However, when the artery is located at the periphery or away from the tumor, higher temperature is induced in the tumor volume during magnetic hyperthermia in comparison to the cases when the artery is located deep inside the tumor volume. Keywords Magnetic hyperthermia therapy · Effects of artery on hyperthermia · Magnetic nanoparticles (MNPs)
1 Introduction Hyperthermia is a treatment methodology that refers to the use of heat to cure a disease [1]. Heat treatment as a therapy is not new, but it is becoming more popular for tumor treatment due to the negligible adverse effects associated with it. Various studies have shown that when hyperthermia, coupled with other cancer treatment methodologies like radiation or chemotherapy, enhances the overall treatment efficacy [2, 3]. Furthermore, hyperthermia has negligible side effects compared to radiation or chemotherapy [3]. The range of temperature rise during hyperthermia is usually 4– 8 ° above the human body temperature, i.e., 41–45 ° [4]. Temperature-induced above S. Singh · N. Kumar (B) Thapar Institute of Engineering & Technology, Patiala 147001, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_63
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this range is used to thermally ablate the tumor tissue. Magnetic nanoparticle hyperthermia (MNPH) is one of the emerging modalities for hyperthermia therapy for the cancer treatment [3, 5]. In this therapy, magnetic nano-particles (MNPs) are delivered to the tumor tissue. These particles encapsulated in the tumor tissue produced heat under the action of the oscillating magnetic field [3]. Thus, this localized heat produced by the MNPs is controlled by the externally applied magnetic field. Various parameters like MNP distribution, magnetic field parameters, and tumor shape, size and its locations affect the overall temperature elevation in the tumor tissue. Vasculature embedded in the tissue plays a vital role in the transfer of heat from the tissue to thermo-regulate the tissue temperature. Many tumors, like carotid body tumors, grow near the major artery. The hyperthermia applications of these tumors will be affected by the flow of blood through the artery. In the present work, the effects position of the artery with respect to the tumor for magnetic hyperthermia are analyzed. The position of blood-carrying artery is changed with respect to the tumor to measure its effects on the temperature elevation in the tumor zone during MNPH. The MNPH simulations are done on a three-dimensional tumor model with an artery at different positions. The computations are performed using COMSOL multiphysics 5.1. Results show that the position of the artery plays a key role in heat dissipation from the tumor during magnetic hyperthermia. Results suggested that therapeutic temperature decreases when arteries are closer to the tumor due to the convective heat transfer effect, and temperature in the tumor volume increases during magnetic hyperthermia when the artery is away from the tumor.
2 Literature Review and Objective The objective of this study is to compare the heat dissipation and temperature profile due to the presence of an artery at different locations with respect to the tumor. This is analyzed by varying the position (distance) of the artery from the center of the tumor. The temperature profile in the tumor as well as surrounding tissue is computed by the solving Pennes bioheat transfer model during MNPH. Various studies (both experimental and computational) have been reported in the last few decades, covering different aspects of magnetic nano-particle hyperthermia therapy. Johannsen et al. [5] reported the outcomes of the first clinical trial of MNPH on patients with prostate cancer. Salloum et al. [6] have reported in vivo experimental study on mice inducing magnetic hyperthermia and reported the temperature elevation in the tissue during MNPH. Through the experimental study, Salloum et al. [6] have demonstrated that MNP distribution in the tissue depends upon the injection parameters tissue structure. The MNP distribution in the tissue is one of the key parameters that affect the temperature elevation in the disuse domain during MNPH. Attaluri et. al. [7] have experimentally investigated the effects of MNP distribution on temperature elevation in prostate tumors of mice during MNP. Various computational studies by different groups have investigated various parameters and its effects on MNPH. Like Singh et al. [8] have demonstrated the effects of different
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injection strategies on MNPH. Singh and Kumar [9] have investigated the effects of tumor shape on this therapy. Kandala et al. [10] have tried to optimize this therapy through power modulation of the magnetic field. Salloum et al. [11] also presented an optimization procedure for enhancement in treatment planning for MNPH. Nain et al. [12] have investigated the effects of tumor position and ambient conditions on magnetic nano-particle thermo therapy. Even though MNPH has been investigated extensively in the last few decades, a few researchers [13] have investigated the effects of blood vessels (BVs) on heat dissipation and temperature elevation during MNPH. However, physiologically tumors usually appear around the BVs [14]. The presence of BVs around the tumor greatly influences the heat dissipation process during MNPH. The temperature elevation during thermotherapy is drastically affected by the presence of BVs. Thus, the current work investigates the effects of the presence of BVs and their location with respect to the tumor mass during MNPH.
3 Methodology The MNPH therapy is computationally investigated on the 3D tumor model embedded with a blood vessel (as shown in Fig. 1). The following section contains a detailed description of the physical model, boundary conditions, and governing equations used to simulate the MNPH on the tumor models. Fig. 1 Schematic diagram of the physical model
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3.1 Physical Modeling Methods Tumors can have any arbitrary shape; however, many computational studies [9, 10] have considered tumors as spherical or ellipsoidal in shape. In this work, 3D spherical tumor surrounded by healthy tissue (Fig. 1) is considered a physical model to investigate MNPH physics. It should be noted from the figure that a blood-carrying vessel is also passing nearby to the tumor tissue. The physical model shown in the Fig. 1 consists of spherical tumor, surrounded by the spherical healthy tissue domain along with a cylindrical artery passing through the tumor. The healthy tissue domain is 60 mm in diameter, the tumor diameter is 10 mm [9], and the cylindrical blood vessel (artery) is 4 mm in diameter. The blood vessel passes through the complete tissue domain. For MNPH, MNPs are injected into the tumor tissue. The external high-frequency magnetic field is applied. Due to excitation induced by an external magnetic field, the MNP embedded in the tissue produces heat due to Neel, Brownian relaxations, and hysteresis loss mechanisms [8, 9, 12]. The heat dissipation in the tissue volume depends upon the distribution of MNP as well as magnetic field parameters [9]. However, for the sake of simplification, the heat-generating MNPs are considered as uniformly distributed in the tumor tissue. The uniform heat generation from MNP in terms of specific loss of power (SLP) during MNPH is considered 106 W/ (m3 of tissue) [9]. The effect of artery position on MNPH is investigated by choosing four different artery positions with respect to the tumor center (as shown in Fig. 2). These artery positions (distance between tumor center and axis of the blood vessel) are at a distance of 3 cm, 6 cm, 9 cm, and 12 cm from the tumor center. These artery positions are represented as positions 1, 2, 3, and 4 respectively. It should be noted form these positions that the artery is embedded inside the tumor for positions 1 and 2. The artery is touching the tumor tissue surface for position 3; however, for 4th position, artery is outside the tumor tissue. In all the tumor models, MNPH is simulated using COMSOL multiphysics software. All the physical models are created in this software. To numerically simulate the MNPH, physics discretization (mesh generation) of the physical modes are done using COMSOL multiphysics. Tetrahedral mesh as shown in Fig. 3 is used to discretize the tumor model. The tetrahedral element used for meshing has a minimum element size of 0.504 mm. The distribution of tetrahedral mesh in the critical zone, i.e., at the interface of tumor and artery, is shown in Fig. 3b.
3.2 Mathematical Modeling The governing mathematical model used to simulate heat transfer during MNPH is Pennes bioheat transfer equation (PBHTE) [reference for PBHTE]. The mathematical description of this equation is given below:
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Fig. 2 Artery position with respect to tumor position
Fig. 3 Meshing model
k∇ 2 T = ρc
∂T ∂t
− wb cb ρb (Tb − T ) − (Q met + Q ext )
(1)
where ρ, c, and k are the density, specific heat, and thermal conductivity of the tissue, respectively. cb , wb , ρ b , and Tb denote the specific heat of the blood, blood perfusion rate, blood density, and blood temperature, respectively. T is the absolute temperature. Qmet denotes the metabolic heat generated by the chemical reaction inside the tissue, and it is considered constant, and the external heat source is given by Qext = Qext (t). Here, external heat source is heat generation by MNPs under the influence of high-frequency alternate magnetic field. The term wb cb ρb (Tb − T ) heat transfer due to the blood perfusion’. . The thermophysical properties of the tumor model are given in Table 1. The blood flow through the artery is assumed to be laminar, incompressible, Newtonian, and inviscid in nature. The flow of blood in arteries is usually in the lamina
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Table 1 Material properties
Material
w (1/s)
ρ (kg/m3 )
C p (J/(kg·K)
Tumor
0.025
1060
3500
Tissue
0.0064
1060
3500
Blood
–
1000
4800
flow regime [15]. Thus, the blood flow in the artery is mathematically modelled with Navier–Stokes equation as expressed by Eq. 2, ρ(u.∇)u = ∇. − p I + μ(∇u + (∇u)T + F
(2)
where u is the velocity, p is pressure, and μ is the dynamic viscosity of blood. The heat convected away in the artery due to blood flow is modeled by the convection–diffusion transport equation as by Eq. 3. ρc
∂T + ρcu.∇T = ∇.(k∇T ) ∂t
(3)
3.3 Boundary Conditions The physically relevant interface boundary conditions (BCs) are imposed at the tumor–healthy tissue interface, on the outer surface of healthy tissue, and on the artery wall. These boundary conditions are imposed in the different interfaces as shown in Fig. 4. For PBHTE the continuity of the temperature as well as heat flux is imposed at the tumor–healthy tissue interface. Mathematically this condition is imposed by the following equation. Ttumour |interface =THealthy tissue |interface − ktumour
∂ THealthy tissue ∂ Ttumour = − k Healthytissue ∂η interface ∂η interface (4)
A detailed description of these boundary conditions is given by Singh et al. [8]. The outer boundary of the healthy tissue is far away from the heating zone; thus, temperature at this outer surface is assumed to be the core body temperature, i.e., 37 ◦ C. Similarly, at the tissue artery interface, wall heat flux conducted through the wall is convected away by flowing fluid. Heat flux at the wall is computed with the following equation, −n.q = kt n x Tx + n y Ty + n z Tz
(4)
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Fig. 4 Computational model for numerical simulation
For blood flow through the artery, the inlet condition is prescribed in terms of the uniform bold flow rate [15, 16]. The uniform blood flow rate at the inlet is taken as 3.53 × 10−7 m3 /s [17]. A fully developed condition is imposed at the outlet of the BV, and on the inner surface, a no-slip boundary condition is applied. At the inlet, blood temperature is at core body temperature, i.e., 37 ◦ C.
4 Results and Discussion As described in the physical modeling section, the MNPH is simulated with four different positions of the artery. The MNPH is induced for 1200 s in all the tumor models. The temperature of the tissue in the heating zone as well as its surrounding transiently increases as the therapy starts. This behavior is captured in Fig. 5. In this figure, transient temperature elevation at different locations of the tissue in a radially outward direction (y = 3, 6, 9, 12 mm) from the tumor center is depicted for all four artery positions.
4.1 Temperature–Time Graph for Different Artery Positions The transient temperature distribution in the tumor tissue and outside it are shown in Fig. 5. The temperature time plots are for the varying artery distance from the center of the tumor. It can be noted from this figure (Fig. 5) that the maximum steady-state temperature successively decreases as the distance of the probe point increases from the tumor center. Furthermore, the effect of artery position on the temperature is also visible in these figures (Fig. 5). Like when artery position is at x = 3 mm, the maximum temperature in the tissue at y = 3 mm is less than the maximum temperature attained
762 Fig. 5 Temperature–time graph for artery at a x = 3 mm b x = 6 mm c x = 9 mm d x = 12 mm
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by the tissue at y = 6 mm. It shows that when the artery is inside the tumor, the temperature elevation in the tissue nearing the artery is reduced during MNPH. This is due to higher heat dissipation by convection produced by flowing blood in the blood vessel. Similarly, in Fig. 5b, it can be seen that the maximum temperature for y = 3 mm rises above y = 6 mm due to the fact that the artery moves away from the tumor. This trend is similar for Fig. 5c and d.
4.2 Temperature Contour for Different Artery Positions The effect of heat dissipation by varying the distance of artery from the center of the tumor is illustrated in the Fig. 6 using temperature contours. It can be noted from this figure the presence of an artery reduces the temperature in the nearby tissue region.
Fig. 6 Temperature contours for artery at a x = 3 mm b x = 6 mm c x = 9 mm d x = 12 mm
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Fig. 7 Maximum temperature with varying artery distance
4.3 Maximum Temperature Graph for Different Artery Positions Below given plot in Fig. 7 shows the maximum temperature attained in the tumor region with varying distances of the artery from the center of the tumor. It can be clearly seen that when the artery is moving away from the tumor center, the maximum temperature of the tumor region is increased. From this, it can be concluded that the presence of an artery increases heat dissipation by its heat transfer through blood flow.
5 Conclusions In this study, the effects of different artery position with respect to tumor tissue on magnetic hyperthermia are investigated. The results show that the presence of an artery substantially reduced the temperature of the tissue around it. The position of the artery also affects the maximum temperature induced during MNPH. The temperature distribution is more uniform when the artery is away from the tumor mass. Acknowledgements I would like to thank Ph.D. Scholar Amritpal Singh for his valuable input for this work. I would like to Virginia Tech- TIET CEEMS(Center of Excellence in Emerging, Materials) for providing me computational facility for this work.
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Abbreviations Nomenclature kt wbti wbtu Ta cpb cpt qart u n T Q q Fi Pi gi Qext Qmet
Tumor conductivity [W/m·K] Perfusion rate healthy tissue [1/s] Perfusion rate tumor [1/s] Artery temperature (K) Blood heat capacity [J/kg·K] Tissue heat capacity [J/kg·K] Flow rate in hepatic artery [m3 /s] Velocity space vector [m/s] Unit vector Temperature vector Heat source [W/ m2 ] Heat flux [W/m2 ] Body force per unit volume [N/m3 ] Surface force per unit volume [N/m3 ] Gravitational acceleration [m2 /s] External heat source [W/m3 ] Metabolic heat source [W/m3
Greek Letters ρ μ ∇ ∇2
Density of the fluid [kg/ m3 ] Dynamic viscosity of blood Del operator Laplace operator
Subscripts B Ti tu
Blood Tissue Tumor
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References 1. Habash RW, Krewski D, Bansal R, Alhafid HT (2011) Principles, applications, risks and benefits of therapeutic hyperthermia. Front Biosci-Elite 3(3):1169–1181 2. Adam A, Kenny LM (2015) Interventional oncology in multidisciplinary cancer treatment in the 21st century. Nat Rev Clin Oncol 12(2):105–113 3. Thiesen B, Jordan A (2008) Clinical applications of magnetic nanoparticles for hyperthermia. Int J Hyperth 24(6):467–474 4. Zee JVD (2002) Heating the patient: a promising approach? Ann Oncol 13(8):1173–1184 5. Johannsen M, Thiesen B, Wust P, Jordan A (2010) Magnetic nanoparticle hyperthermia for prostate cancer. Int J Hyperth 26(8):790–795 6. Salloum M, Ma RH, Weeks D, Zhu L (2008) Controlling nanoparticle delivery in magnetic nanoparticle hyperthermia for cancer treatment: experimental study in agarose gel. Int J Hyperth 24(4):337–345 7. Attaluri A, Ma R, Qiu Y, Li W, Zhu L (2011) Nanoparticle distribution and temperature elevations in prostatic tumours in mice during magnetic nanoparticle hyperthermia. Int J Hyperth 27(5):491–502 8. Singh G, Kumar N, Avti PK (2020) Computational evaluation of effectiveness for intratumoral injection strategies in magnetic nanoparticle assisted thermotherapy. Int J Heat Mass Transf 148:119–129 9. Singh A, Kumar N (2022) Parameterizing the effects of tumor shape in magnetic nanoparticle thermotherapy through a computational approach. ASME J Heat Transfer 144 10. Kandala SK, Liapi E, Whitcomb LL, Attaluri A, Ivkov R (2019) Temperature-controlled power modulation compensates for heterogeneous nanoparticle distributions: a computational optimization analysis for magnetic hyperthermia. Int J Hyperth 36:115–129 11. Salloum M, Ma R, Zhu L (2009) Enhancement in treatment planning for magnetic nanoparticle hyperthermia: optimization of the heat absorption pattern. Int J Hyperth 25(4):309–321 12. Nain S, Kumar N, Kumar Avti P (2022) Computational investigation of the tumor position and ambient conditions on magnetic nanoparticle thermo-therapy. Thermal Sci Eng Prog 34:101396 13. Tang Y, Jin T, Flesch RC (2017) Numerical temperature analysis of magnetic hyperthermia considering nanoparticle clustering and blood vessels. IEEE Trans Magn 53(10):1–6 14. Cao Z, Ding BS, Guo P, Lee SB, Butler JM, Casey SC, Simons M, Tam W, Felsher DW, Shido K, Rafii A (2014) Angiocrine factors deployed by tumor vascular niche induce B cell lymphoma invasiveness and chemoresistance. Cancer Cell 25(3):350–365 15. Daly BJ (1976) A numerical study of pulsatile flow through stenosed canine femoral arteries. J Biomech 9:465–475 16. Kandpal A, Kumar N (2018) Computational hemodynamic study of healthy and pathological abdominal aorta, paper no. FMFP2018–483. In: 7th International and 45th national conference on fluid mechanics and fluid power (FMFP) December 10–12, IIT Bombay, Mumbai, India 17. Paulsen AW, Klintmalm GB (1992) Direct measurement of hepatic blood flow in native and transplanted organs, with accompanying systemic hemodynamics, hepatology 16(1):100–111
Flow Separation and Pressure Drop Analysis for Blood Flow in Symmetric Stenosed Arteries of Various Shapes Anamika Maurya, Janani Srree Murallidharan, and Atul Sharma
Abstract The past two decades witnessed that the symmetric/asymmetric stenosed arteries are an active area of research as it causes numerous arterial diseases such as thrombosis and atherosclerosis, which lead to human death; hence, they need therapeutic attention. Much research has been reported concerning the shape of the stenosis as the elliptical one, but several medical data available in the literature suggest that the stenotic form is of no particular kind. Thus, the present work aims to numerically investigate the severity of a symmetric stenotic region for four different stenotic shapes (plug, triangular, trapezoidal, and elliptical) in terms of the leading pressure drop noted in the narrowed artery. The present work has been carried out for a wide range of Reynolds numbers (30–650) and % area stenosis (60.9–93.8) at stenotic length, β = 2. The new extensive results have been reported regarding the flow characteristics, separated flow zones, Euler number, etc. The separated zones, leading to the propagation of the static blood flow region, are a vital function of Reynolds number and % area stenosis for a fixed stenotic length. Based on the pressure drop analysis and separation zones, it has been noted that the plug shape is the most severe compared to other investigated ones. The triangular shape is seen to be the least severe in terms of causing the pronounced separated zones and pressure drop in the vicinity of the stenosis. Keywords Blood flow · Stenosed artery · Stenotic shapes · Pressure drop · Separation zone
A. Maurya (B) · J. S. Murallidharan · A. Sharma Department of Mechanical Engineering, IIT Bombay, Mumbai Maharashtra - 400076, India e-mail: [email protected] A. Maurya School of Engineering and Applied Science, Ahmedabad University, Navrangpura, Ahmedabad, Gujrat 380009, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_64
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1 Introduction Every year, worldwide, atherosclerosis in human arteries leads to atherosclerosis stroke, cerebral stroke, etc., causing approximately 31% of deaths [1]. A recent survey by American Heart Association also predicted that 21% of human deaths globally are only caused by atherosclerosis stroke [2]. Atherosclerosis occurs in human arteries due to the deposition of cholesterol, fatty acids, low-density lipoprotein, and other substances. Such accumulation of blood substances leads to the narrowing of the lumen and its hardening, resulting in the blood flow impedance and hence several ailments that require careful treatment. Thus, the blood flow in the stenosed artery remains an active area of research. Several medical data are available in the literature, suggesting that the stenosis shape does not have a particular form [3]. Much research has been expended to understand the blood flow behaviour in the stenosed artery but is limited to elliptical stenotic shapes [4, 5]; still, information related to the comparison of several kinds of stenotic shapes on the blood flow is scarce. Predicting the physiological severity of localized stenosis in a single- or multi-branched network is challenging using a standard angiogram. Thus, the pressure drop analysis can evaluate the actual functional seriousness of the stenosis [6]. The pressure drop analysis is still considered an essential haemodynamic parameter in developing countries [6]. Therefore, the present work aims to understand the effect of different stenotic shapes in predicting flow fields in the symmetric artery’s vicinity, separated zones, and pressure drops across the stenotic region. The predicted pressure drop in this work can be further used as a precursor to training the machine learning models to predict the pressure drop for real-life physiological conditions.
2 Literature Review and Objective In recent years, several researchers expanded their work to understand the blood flow behaviour in the stenosed artery through analytical, experimental, and computational work. A few have been reviewed herein, directly relevant to the present work. Young and Tsai [7] did experiments to analyse the flow characteristics in terms of the pressure drop for the elliptical shape stenosed artery only. They studied the blood flow, considering both the steady and unsteady flow through a tube with curved stenosis. They used both the axisymmetric and non-symmetric models for a wide range of Reynolds numbers (100–5000), % area stenosis (56–89), and aspect ratio (2–4). The results showed that the non-symmetric models’ pressure drop was significantly higher than the axisymmetric models. They also mentioned that flow separation is a vital function of the size of the stenotic region. They noted the presence of turbulence in the flow domain for a high Reynolds number, i.e. Reo > 2000. Also, they stated slightly different results for the unsteady flows. They also discussed the viscous and inertial effects on the pressure drop.
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Subsequently, Seeley and Young have also done similar discussions regarding predicting viscous and inertial effects on the flow characteristics [8]. They did in vitro steady flow experiments for the blood flow through a stenotic region (i.e. plug) over a wide range of flow and geometric parameters. They observed an increase in the pressure drop with geometric parameters such as stenotic length for plug shape. In contrast, the present work numerically investigates the blood flow behaviour not only limited to plug shape but also discusses the elliptical, triangular, and trapezoidal shape stenosis geometries. Further, Dash et al. [9] analytically modelled the blood flow as a Newtonian fluid. They worked with steady, laminar, and incompressible flow assumptions for a slight curvature and mild stenosis. They used the perturbation method to predict the hemodynamic flow characteristics for Reynolds number from 25 to 100 and aspect ratio as 0.1. They discussed correcting the pressure drop values using catheter size as a passive parameter. They also reported the modification in the secondary streamlines in a cross-sectional plane due to the combined influence of catheterization, curvature, and stenosis. Further, Neofytou and Drikakis [10] carried out a numerical investigation on the unsteady periodic blood flow in a 2-D channel along with elliptical stenosis. They used the Casson, power-law, and Quemada models to model the shear-thinning behaviour of the blood for a wide range of conditions. They reported the formation of many vortices in the flow domain for such models. They demonstrated the difference in the results whilst treating the blood flow as Newtonian and shear-thinning fluids. Moreover, Kamangar et al. [6] numerically investigated the steady and transient flow of blood through a tube along with three stenotic shapes, triangular, trapezoidal, and elliptical, over a wide range of conditions as % area stenosis 70 ≤ %A ≤ 90, flow rate, 175 ≤ Q (mL/min) ≤ 115, and stenotic length, β = 3.33. They used Carreau model to treat blood as a shear-thinning fluid. They reported that the various shapes of the stenosis significantly influence the intraluminal flow, which substantially affects the pressure drop. They said that in the case of an intermediate stenosis, there is a risk of misinterpretation of the diagnosis on the seriousness of the presence of the stenosis. Hence, they suggested a region of uncertainty between an area ratio of 76.5–82.7% in a single constraint vessel. In contrast, the present work predicts flow patterns, flow velocity and pressure profiles, flow separation zones, pressure drop analysis, etc., comparing four stenotic shapes reported above and more severe stenosis as %A = 93.8. Most recently, Chen et al. [11] have used the Carreau rheological model to numerically investigate the shear-thinning behaviour of the blood vessel with elliptical shape stenosis. They worked with axisymmetric, 2-D, steady, laminar, and incompressible assumptions spanning various conditions. This work reported an augmentation in the separation zones with Reynolds number and stenosis rate. They noted an increase in the resistance coefficient with % area stenosis for a fixed Re. Thus, a voluminous body of knowledge is currently available on the blood flow in a stenosed artery using various shapes, lengths, heights, luminal areas, etc., for both symmetric and non-symmetric stenotic regions. However, very little information is available in the literature regarding a comparative analysis of four different stenotic shapes to predict their severity through separation zones and pressure drop analysis.
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Thus, the present work aims to fill the gap between the body of knowledge through a systematic study of the effect of different shapes on the blood flow behaviour from clinical aspects; it is well known that the stenosis shape is arbitrary. In particular, the present work numerically investigates the steady laminar and incompressible flow of blood in a 2-D stenosed micro-channel with four different geometric shapes of symmetric stenosis such as triangular (GTri ), elliptical (GElli ), plug (GP ), and trapezium (GTrap ) on the laminar flow of blood. The blood has been modelled as a shear-thinning fluid using the Bird–Carreau model over a wide range of conditions: Reynolds number, 30 ≤ Reo (based on unobstructed channel width) ≤ 650, % area stenosis, 60.9 ≤ %A ≤ 93.8, and for a fixed aspect ratio, β = 2. To the best of our knowledge, this is the first attempt to analyse the combined effects of four stenotic shapes and the shear-thinning behaviour of blood over such a wide range of conditions stated above.
3 Problem Description and Numerical Strategy In this work, the blood, i.e. treated as shear-thinning and single-phase homogenous fluid, enters into the flow domain through the inlet plane with a fully developed velocity profile with an average velocity, U o , and get influenced by four stenotic shapes as elliptical (GElli ), triangular (GTri ), plug (GP ), and trapezium (GTrap ). Based on the reviewed literature, the flow is assumed to be laminar, steady, incompressible, and two dimensional in this work. Figure 1 shows the schematic representation of the computational domain, the description of pre-stenotic, stenotic, and post-stenotic regions, and the characterization of separation zones. The pre-stenotic and poststenotic channel length, i.e. R and 2R, respectively, used in the present simulation ensures the fully developed flow in the specified regions. The unobstructed channel artery width and length have been embraced as W o and R, respectively, for each case [4]. In this work, all the governing equations and field variables have been solved numerically only in the half domain to economize the computational efforts, as the present work explores the symmetric stenosed artery. The governing equation solved in this work is as follows: Fig. 1 Schematic representation of the computational set-up
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• Mass conservation equation: ∇·U =0
(1)
• Momentum conservation equation: [U · ∇U ] = −∇ P +
1 ∇ ·τ Re
(2)
The length and velocity scales used to non-dimensionalize the governing equations are the height of the channel artery (H) and the average velocity (U o ) of the blood, respectively. Thus, the pressure is non-dimensionalized by using ρUo2 , and the viscous stress components by using η(Uo /H ). The deviatoric stress tensor, τ , for the blood flow is given as follows, considering blood as an incompressible fluid [11, 12]: τ = 2η(γ˙ ) D and
D=
1 ∇U + ∇U T 2
(3)
Here, γ˙ is the rate of the strain tensor, and related to the second invariant of the rate of deformation tensor D, i.e. coupled with the velocity gradient of the flow field. Here, η is the blood viscosity which depends on the local shear rate of the blood in the artery. In the literature, numerous non-Newtonian models have been proposed to treat blood as a shear-thinning fluid [10, 12]. The present work used the Bird–Carreau model, as the literature suggests that it is the most accurate model to predict the blood viscosity of the actual physiological blood in the arteries [13]: (n−1)/ 2 η(γ˙ ) = η∞ + (ηo − η∞ ) 1 + (λγ˙ )a
(4)
Here, ηo (= 0.056 Pa .s), η∞ (= 0.00345 Pa .s), λ (= 3.313 s), a (= 2), and n (= 0.3568) are the zero shear viscosity, infinite shear viscosity, time constant, Carreau parameter, and power-law index, respectively, which has been taken from Mendieta et al. [14]. In this work, the Reynolds number based on the unobstructed region width has been varied from 30 to 650 and calculated using blood viscosity and density as 1060 kg/m3 and viscosity as 2.78 mPa .s [15]. The continuity and momentum equations, along with appropriate boundary conditions, have been solved in conjunction with the Bird–Carreau model. A zero gauge pressure, i.e. P = 0 at the outlet along with the no-slip conditions at the artery channel walls, has been used herein. Further, the non-dimensionless parameters of interest of the present work have been defined as follows: • Reynolds number, Re Re =
ρUo Wo η
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• Euler number, Eu Eu =
(PProximal − PDistal ) ρUo2
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L Wo
(7)
• Aspect ratio, β β=
Herein, PProximal is the inlet channel pressure, and PDistal is the outlet channel pressure. This work has been carried out numerically using an open-source CFD code OpenFOAM (v7.0) [16]. This is free software based on the finite volume method (FVM) written in C++ object-oriented code. The “blockMesh” utility of the OpenFOAM has been used to create the geometry and meshing. The solver named as “nonNewtonianIcoFoam” has been chosen herein to work with the Carreau model, a transient solver for the incompressible, laminar flow of non-Newtonian Fluids. The validation of the non-Newtonian model has been done recently by Kowalczyk [17]. The default pressure–velocity coupling algorithm PISO (pressure-implicit with splitting of operators) has been used here. Here, “nCorrectors,” i.e. pressure corrector loops, have been chosen as 2. For both pressure (p) and velocity (U), the tolerance has been set as 10–6 , which ensures the stabilizations of variable up to at least five decimal places. To ensure the accuracy and reliability of the present numerical scheme, the current numerical model has been validated with the available experimental results of Seeley and Young [8] for the plug shape, GP stenosed artery (the figure is not being shown herein for the sake of conciseness). The present and experimental results are in perfect agreement with each other for low Reynolds numbers up to Reo = 200. A slight deviation at a high Reynolds number, i.e. Reo > 200, has been observed. Such deviations are not unfamiliar in numerical studies and may be attributed to the grid structure and/or the assumptions [18].
4 Results and Discussion 4.1 Local Velocity and Pressure Profiles Figure 2 shows the flow visualization in terms of centreline velocity profile, lateral velocity profile, and centreline pressure profile for multiple values of %A at Reo = 300 and β = 2, only for plug shape stenosis, GP . Similar analysis has also been carried out for all other shapes, along with a comparative analysis between each shape (for the conciseness of the paper, the figure has not been shown herein). The influence of stenosis on the blood flow for GP can be noted in Fig. 2a at fixed stenotic length, β =
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2 and Reynolds number, Reo = 300 for four values of %A as 60.9, 75, 85.9, and 93.8, which suggest continuous narrowed lumen area. As expected, no significant change in the centreline velocity profile has been noted in the pre-stenotic and post-stenotic regions, regardless of %A. In the stenotic region, the maximum velocity has been noted for the case of higher %A, i.e. 93.8. For instance, ~ 64% enhancement in the flow acceleration has been observed for %A = 93.8 compared to %A = 60.9, see Fig. 2a. This is because of the reduced flow area for blood flow. A similar trend of enhancement in the velocity profile has also been noted by Kamangar et al. [6], but not for the plug shape, GP stenosis. Further, Fig. 2b demonstrates the flow characteristics in terms of separation zones, elucidating the lateral velocity in the post-stenotic region, close to stenosis, i.e. at X = 73.77. Herein, in Fig. 2b, the negative lateral velocity represents the presence of the flow separation region that has been noted for either of %A at Reo = 300 and β = 2 in this work. The cross-sectional length up to which the negative velocities are present in the flow domain demonstrates the width of the separation zones, i.e. W R . Such width is seen to bear a positive dependence on %A, which indicates the severity introduced in the stenosed artery along with the present stenosis. The blood flow can experience such separation zones as a local obstruction to the flow in the post-stenotic region. In such regions, the flow recirculates continuously, which may lead to the rupture of the artery wall for a high shearing level. Figure 2c represents the centreline pressure profile along the axial distance (in the flow direction, at Y = 0) of the computational domain for various %A (60.9– 93.8) at β = 2 and Reo = 300. It has been plotted bearing in mind that in developing countries, doctors treat %A and pressure drop as important hemodynamic parameters to understand the severity of the stenosed artery instead of investigating a particular stenotic shape [6]. As the blood flows in the downstream section, it sees an obstruction to flow introduced by stenosis itself, hence the flow acceleration in the stenotic region, as discussed above. The pressure drops to a specific value in the flow acceleration region due to the Bernoulli effect. It achieves a local minimum (i.e. close to the flow separation point from the wall). After that, as the flow moves in the downstream section, a recovery in the pressure has been observed to maintain the fluid momentum in the vicinity of the stenotic region. The fluid achieves a local maximum (i.e. close to the point of reattachment of fluid to the walls) before dropping linearly in the fully developed region far away from the stenosis. However, Fig. 2c shows that a more extended recovery length of the channel is required for the artery, which is stenosed with a higher %A. Thus, the blood flow experiences a substantial pressure drop in the stenotic region for high %A. Moreover, from Fig. 2c, it can also be seen that the maximum inlet pressure is found for higher %A due to a significant drop in the pressure for the same case in the stenotic reason. In general terms, in the pre-stenotic region, the pressure is seen to be increasing function of %A due to the decreased available flow area for blood to flow. No significant change in the post-stenotic region pressure profile has been observed; herein, far downstream region for various %A as the flow has been developed, and the blood has no memory of the stenotic region.
774 Fig. 2 Variation of a centreline velocity profile, U c b lateral velocity profile, U x c centreline pressure profile, Pc with %A at Reo = 300 and β = 2
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4.2 Separated Zones The separation zone length (i.e. L R normalized with the width of the channel, W o ) has been calculated herein as the difference between the reattachment point, L r (i.e. point of reattachment of blood to the wall) to the separation point, L s (i.e. the point of detachment of the blood from the wall) [19, 20]. The separation zone length, L R , has been monitored herein because of two particular reasons (1): It indicates the length in the post-stenotic region up to which the blood flow experiences a local obstruction to flow along with stenosis. Such zones indicate the local growth of the plaque formation in the post-stenotic region, hence needing attention in the diagnostic and therapeutic. (2) It represents the local region in which the blood recirculates continuously at a relatively more minor velocity than the available flow area; still, it can lead to a rupture and may promote arterial diseases. Figure 3a shows the effect of % area stenosis and fluid inertia, Reo on the separation zone length, L R for the plug shape, GP at β = 2. The zero values of the separation length in Fig. 3a demonstrate no observation of flow separation zones for such cases. From Fig. 3a, it is seen that the separation zone length bears a positive relationship with the inertial force, i.e. Reo . This is because higher fluid inertia suppresses the blood flow resistance [21]. Thus, blood gets squeezed out for a longer downstream distance in the post-stenotic region under the action of the dominance of inertial force. The maximum separation length that has been observed herein is ~ 38 for Reo = 650, β = 2 and %A = 93.8 for GP . Further, Fig. 3a also demonstrates that the separation zone length is significantly higher for high %A for a fixed Reynolds number; for instance, at β = 2, Reo = 650, ~ 88% enlargements in the separation zone have been noted for %A = 93.8 than %A = 60.9, which suggests that the growth of the plaque formation may lead such a long distance. Furthermore, Fig. 3b compares different stenotic shapes in terms of the noted separation zone length for a fixed %A = 75 and β = 2. A longer separation zone has been noted for plug shape, GP , whilst the shortest one has been observed for triangular shape, GTri . This is because of the geometric effect, as the maximum loss in the kinetic energy has been noted due to the sharp edge (or sudden change in the flow direction) present in the plug shape than the triangular one. Thus, to recover the loss in kinetic energy, the fluid extends for a longer distance in the post-stenotic region and experiences a longer separation zone. From Fig. 3b, a geometric sequence in terms of causing longer separation zone for a given Reo for a fixed %A = 75 and β = 2 is as follows: GP > GTrap > GElli > GTri . From such observations, one can qualitatively say that the most severe stenotic shape is a plug, GP . In contrast, the least severe one is the triangular shape, GTri , as ~ 38% enlargement in the separation zone has been noted in the case of plug shape than the triangular shape.
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Fig. 3 a Variation of separation zone length, L R with Reo for plug shape, GP at β = 2 b comparison of shapes in terms of separation zone at %A = 75, β = 2
4.3 Pressure Drop Analysis Figure 4a elucidates the effect of the reduced flow area in the stenosis on the pressure drop for plug shape, GP at β = 2 for 30 ≤ Reo ≤ 650. As expected, the maximum pressure drops in the constricted channel, i.e. Eu, have been observed for higher %A as also discussed above, see Fig. 2c. The more pronounced effect of the percentage area stenosis on the pressure drop is seen at the lower values of the Reynolds numbers, Reo . For instance, ~ 30% enhancement in the pressure drop is seen for %A = 93.8 compared to %A = 60.9. This is due to the dominance of the viscous forces over the inertial force leading to higher frictional resistance for the blood to flow through the artery; hence, high pressure drops. Also, at a fixed stenotic length and for a particular shape, it has been observed that as the Reynolds number increases, the influence of %A gets suppressed; for instance, Reo > ~ 400, see, Fig. 4. Figure 4b compares different stenotic shapes in terms of Euler number, for a fixed stenotic length as β = 2 and percentage area stenosis as %A = 93.8. In broad terms, the Euler number decreases with the Reynolds number regardless of the choice of the stenotic shapes. However, the plug shape noted a higher pressure drop. On the other hand, a lower pressure drop is seen for the triangular shape stenotic region. Notably,
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Fig. 4 a Variation of pressure drop, Eu with % area stenosis, b influence of studied stenotic shapes on the pressure drop
~31% increment in the pressure drop of GP has been noted compared to GTri. Thus, the triangular shape stenotic region may be the least severe in the studied geometries. The sequence of the severity of the different stenotic shapes, which requires more attention to diagnose and therapeutic, as predicted from the noted pressure drop values from Fig. 4b, is as GP > GTrap > GElli > GTri .
5 Conclusions The present work numerically investigates the steady laminar and incompressible flow of blood in a 2-D obstructed micro-channel with four different stenotic shapes as triangular (GTri ), elliptical (GElli ), trapezium (GTrap ), and plug (GP ). Herein, the blood has been treated as a homogenous shear-thinning fluid over a wide range of conditions as: Reynolds number, 30 ≤ Reo (based on unobstructed channel width) ≤ 650 and % area stenosis, 60.9 ≤ %A ≤ 93.8 at fixed stenotic length, i.e. β = 2. As far as the authors know, this is the first systematic study comparing the four different stenotic shapes in terms of separation zone caused by the stenosis and pressure drop across it. The major thrust of the current work is to investigate the most severe stenotic shape through flow visualization and pressure drop analysis and
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its comparative analysis. The key results of the present work have been summarized as follows: • Significant modulation in the flow characteristics has been observed by the governing parameters along with stenotic shapes. A substantial enhancement in the flow acceleration has been noted with %A in the stenotic region. • The separation zone is seen to be a strong function of the governing parameters and stenotic shapes. A more extended separation zone is seen for high %A and high Reo, for a fixed stenotic length and shape. • The maximum pressure drop is caused by GP across the stenosis, whilst the lowest Eu has been noted for GTri . • The sequence of the geometry noted in terms of severity based on the caused separation zones and pressure drop analysis is as follows: GP > GTrap > GElli > GTri . Thus, the plug shape stenotic region requires significant medical attention to diagnose and cure.
Nomenclature %A β Eu L Lr Ls LR η Pc R Reo ρ Ux Uc Wo WR W1 X, Y
Percentage area stenosis Stenotic length Euler number Stenotic length [mm] Reattachment point Separation point Separation zone length Fluid viscosity [Pa .s] Central pressure profile Pre-stenotic length [m] Reynolds number (based on unobstructed region) Fluid density [kg/m3 ] Axial velocity profile Central line velocity profile Width of unobstructed region [mm] Width of separation zones Width of obstructed region [mm] Dimensionless coordinate system
Acronyms GP GTrap
Plug shape Trapezium shape
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Elliptical shape Triangular shape
References 1. World Health Organization (WHO) (2017) Cardiovasc. Dis. Fact Sheet No. 317 2. Virani SS et al (2021) Heart disease and stroke statistics—2021 update, a report from the American Heart Association. Circulation 143(8):254–743 3. Berglund H, Luo H, Nishioka T, Fishbein MC, Eigler NL, Tabak SW, Siegel RJ (1997) Highly localized arterial remodeling in patients with coronary atherosclerosis. Circulation 96:1470– 1476 4. Pijls NHJ, Bruyne BD, Peels K et al (1996) Measurement of fractional flow reserve to assess the functional severity of coronary artery stenoses. New Engl J Med 334:1703–1708 5. Kabir MA, Alam MF, Uddin MA (2021) Numerical simulation of pulsatile blood flow: a study with normal artery, and arteries with single and multiple stenosis. J Eng Appl Sci 68:1–25 6. Kamangar S, Badruddin IA, Ahmad NA, Govindaraju K, Nik-Ghazali N, Ahmed NJS, Badarudin A, Khan TMY (2017) The influence of geometrical shapes of stenosis on the blood flow in stenosed artery. Sains Malaysiana 46:1923–1933 7. Young DF, Tsai FY (1973) Flow characteristics in models of arterial stenoses—I. Steady flow. J Biomech 6:395–410 8. Seeley BD, Young DF (1976) Effect of geometry on pressure losses across models of arterial stenoses. J Biomech 9:439–448 9. Dash RK, Jayaraman G, Mehta KN (1999) Flow in a catheterized curved artery with stenosis. J Biomech 32:49–61 10. Neofytou P, Drikakis D (2013) Effects of blood models on flows through a stenosis. Int J Numer Meth Fluids 43:597–635 11. Chen X, Zhan Y, Fu Y, Lin J, Ji Y, Zhao C, Fang Y, Wu J (2021) The effect of stenosis rate and Reynolds number on local flow characteristics and plaque formation around the atherosclerotic stenosis. Acta Bioeng Biomech 23:135–147 12. Konala BC, Das A, Banerjee RK (2011) Influence of arterial wall-stenosis compliance on the coronary diagnostic parameters. J Biomech 44:842–847 13. Albadawi M, Abuouf Y, Elsagheer S, Ookawara S (2021) Predicting the onset of consequent stenotic regions in carotid arteries using computational fluid dynamics. Phys Fluids 33:123106 14. Mendieta JB, Fontanarosa D, Wang J et al (2020) The importance of blood rheology in patientspecific computational fluid dynamics simulation of stenotic carotid arteries. Biomech Model Mechanobiol 1477–1490 15. Thurston GB (1979) Rheological parameters for the viscosity viscoelasticity and thixotropy of blood. Biorheology 16:149–162 16. The OpenFOAM Foundation (2019) OpenFOAM User Guide 7.0. https://cfd.direct/openfoam/ user-guide-v7 17. Westermaier S, Kowalczyk W (2020) Implementation of non-Newtonian fluid properties for compressible multiphase flows in OpenFOAM. Open J Fluid Dynam 10:135–150 18. Roache PJ (1997) Quantification of uncertainty in computational fluid dynamics. Ann Rev Fluid Mech 29:123–160 19. Maurya A, Tiwari N, Chhabra RP (2019) Effect of inclination angle on the forced convective flow of a power-law fluid in a 2-D planar branching channel. Int J Heat Mass Transfer 134:768– 783
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20. Maurya A, Mishra L, Chhabra RP (2021) Forced convection from a sphere to power-law fluids in a tapered tube. Int Commun Heat Mass Transf 126:105447 21. Maurya A, Tiwari N, Chhabra RP (2021) Controlling the flow and heat transfer characteristics of power-law fluids in T-junctions using a rotating cylinder. Int J Therm Sci 163:106854
Comparative Study of Uniform and Pulsatile Blood Flow Through Single Stenosed Carotid Artery Swapnil Rajmane and Shaligram Tiwari
Abstract Symmetric occlusion with the range of 20–70% reduction in diameter of carotid artery is numerically investigated. Computations have been performed for both steady and pulsatile inlet velocities with the average Reynolds number of 400. Comparison has been made between steady and unsteady flow based on the timeaveraged quantities like WSS, reattachment point, etc. The effect of pulsatile inlet velocity is observed in terms of WSS transient behaviour. OSI is monitored to look for the regions favourable in the new plaque formation. Study shows that the result outcomes are realistic and superior when the inlet is employed with a pulsatile flow instead of a steady inlet. Flow behaviour found to be very sensitive to inlet pulse, and associated rapid changes in the WSS might rupture plaque and cause blood clots. OSI distribution shows that the region downstream of stenosis is more prone to new plaque formation. Keywords Atherosclerosis · Blood flow · Pulsatile flow · Wall shear stress · Oscillatory shear index
1 Introduction Atherosclerosis has become a fatal disease amongst all cardiovascular diseases. It occurs due to the deposition of low-density lipoprotein (LDL) also known as bad cholesterol present in the bloodstream. Permeable blood arteries allow the transport of excessive LDL from the lumen to intimal layer which leads to the formation of plaque within the intima. The carotid artery is a blood vessel that supplies the oxygenated blood to the parts of the body above the neck. The formation of plaque in the carotid arteries might lead to neurological trouble or stroke. It becomes necessary to get familiar with the flow pattern behaviour associated with the stenosed artery for which CFD can be utilized as a reliable tool. S. Rajmane (B) · S. Tiwari Department of Mechanical Engineering, IIT Madras, Chennai 600036, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_65
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2 Literature Review and Objective In the last few decades, several investigations have been made into arterial stenosis generation and proliferation. The study of flow characteristics remained at the core of these investigations. Lee [1] has done a detailed flow pattern analysis of a double constricted artery having a severity of 50% for Re in the range of 5–200. It observed that circulations are well established in the spacing between two constrictions, and their elongation along the post-stenotic length is significantly affected by Re and spacing between two constrictions. Deshpande et al. [2] have done the numerical analysis for laminar flow through stenosed artery having severity of 75 and 90%, and it reported that turbulence incorporation is essential for the severe stenosed cases. In the above-mentioned literature, the pulsatile nature of inlet velocity and nonNewtonian behaviour was not taken into account. Cho and Kensey [3] reported the effect of shear rate on the shear thinning of blood using several non-Newtonian constitutive models. Liao et al. [4] performed numerical studies for physiological pulsatile flow and concluded that flow characteristics for time-averaged pulsatile flow and steady flow have analogous variations concerning change in Reynold’s number. Similarly, Long et al. [5] have numerically investigated the pulsatile flow through arterial stenosis having severity between 25 and 75%. It concluded that the post-stenotic flow is sensitive to inlet waveform and stenosis severity. Similarly, Deplano and Siouffi [6] showed the influence of waveform on the velocity and wall shear stress (WSS) downstream of the blockage. OSI is the haemodynamic parameter associated with the pulsatile inlet velocity where the flow has oscillatory nature in the lumen. Ku et al. [7] suggested that the downstream region of stenosis is more favourable in plaque progression where low mean WSS and oscillatory flow usually occur. In the present study, three-dimensional, non-Newtonian, pulsatile, and laminar flow through the stenosed carotid artery is numerically investigated. Symmetric constriction within the range of 20–70% has been chosen. The average Re of 400 is considered for steady flow computations, and its characteristics are distinguished with that of the pulsatile flow aiming to observe the effect of pulsatile nature. WSS and OSI are quantitatively compared for various severities.
3 Methodology 3.1 Governing Equations In this work, laminar and incompressible flow is considered with blood having nonNewtonian behaviour. For the simulation, continuity and Navier–Stokes equations are used as the governing equations of mass and momentum conservation (Eqs. 1 and 2, respectively). Non-Newtonian behaviour is taken into account by using the Carreau model [3].
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∇ · u→ = 0
(1)
∂(ρ u→) + ρ(→ u · ∇)→ u = −∇ P + ∇ · (μ∇ u→) ∂t
(2)
Here, ρ is taken as 1060 kg/m3 . The parameter μ is defined by the Carreau model using Eq. (3). It explicitly depends on the shear rate (γ˙ ) only. ] (n−1) [ μ = μ∞ + (μ0 − μ∞ ) 1 + (λγ˙ )2 2
(3)
Here, μ∞ = 0.00345 Pa s, μ0 = 0.056 Pa s, λ = 3.313 s, and n = 0.3568. The relation between shear stress and shear rate can be given by Eq. (4). τ = μγ˙
(4)
3.2 Geometry and Boundary Conditions The Gaussian profile has been used for the symmetric occlusion specified in Eq. (5). The severity implies the percentage reduction in the diameter of an artery as taken by Liao et al. [4]. In Eq. (5), c can be given as (D − d 0 )/D, and η is taken as 8. Figure 1 shows the geometry of the stenosed artery. The total length of an artery is 25D with L i = 10D and L o = 15D. r (Z ) 2 = 1 − ce−η(Z /D) R
(5)
A realistic physiological pulse as shown in Fig. 2 is provided at the inlet. The waveform has a period of 0.86 s for a cycle with 70 beats/s. In the present work, the non-dimensional period of 1 is considered. A parabolic velocity profile for fully developed flow can be obtained from Eq. (6). No-slip boundary condition is provided at the impermeable artery wall. Zero gauge pressure is imposed at the outlet. [ ( r )2 ] u(r, t) = 2U (t) 1 − R
Fig. 1 The geometry of stenosed artery
(6)
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Fig. 2 Inlet physiological pulse [8]
3.3 Numerical Technique and Grid Independence Study In the present study, computations have been performed using ANSYS Fluent 20.1. Finite volume method-based solvers are used for solving the governing equations. A SIMPLE scheme is implemented for resolving the pressure–velocity coupling. For the transient formulation, a second-order implicit scheme is used. Preferred structured grids are generated using ICEM CFD 20.1. Grid refinement study is carried out on the 60% severe artery model. Maintaining the better grid quality in such severe cases ensures the quality for other cases too. Four refinements have been studied with the hexahedral cells by varying the first cell height and grid density in the domain of interest. From Fig. 3, it can be seen that the velocity profile does not show significant variation for the last three grid structures, and the relative error for maximum WSS is less than 1% in latter two cases (Table 1). Based on this, grid 3 has been chosen for all the computations in the present study. Figure 4 represents a schematic of the grid. Non-dimensional time step size of 2 × 10−5 found to be accurate for the present study.
3.4 Validation For the validation, the computational results are validated against the results reported by Long et al. [5]. The work [5] has done a detailed analysis of the pulsatile flow through the stenosed artery. Velocity profiles are compared for 50% of the severe case at the axial distance of 1D from the throat and at time instant t = 0.2. The results are matching appreciably except near the centreline (Fig. 5).
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Fig. 3 Velocity profile at Z = 0 for different grids
Table 1 Grid independence study S. No.
Number of grids
First cell height (mm)
Maximum WSS (Pa)
Relative error (%)
1
313,548
0.075
181.65
–
2
494,361
0.05
193.34
6.43
3
692,622
0.03
198.01
2.41
4
875,864
0.02
199.95
0.92
Fig. 4 a Isometric view of the grid, b cross-sectional view at Z = 0
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Fig. 5 Comparison of the velocity profile at z = 1D and t = 0.2
4 Results and Discussion 4.1 Effect of Severity on the Steady and Unsteady Flow Flow behaviour can be effectively rendered by using the streamlines. Figure 6 represents the streamlines for different severities having steady inlet velocity corresponding to an average Re of 400. The flow manifests steady behaviour irrespective of the severity. The recirculation zone is found to be well-established downstream of the blockage, and its length is found to increase as the severity of blockage (stenosis) increases (Table 2). Referring to that, mild stenosis of 20% severity does not form the circulation zone behind the stenosis (Fig. 6a). Flow direction perturbs slightly but remains along the artery wall and does not separate. In cases of 60 and 70% severity, flow downstream of the throat becomes plug type jet which elongates the recirculation zone beyond the proposed length of the artery model (Fig. 6e, f). Figure 7a–f constitutes a vector representation of the axial velocity at the various intervals shortly after the throat. When it comes to mild stenosis of 20%, the profile
Fig. 6 Steady flow streamlines for a 20%, b 30%, c 40%, d 50%, e 60%, f 70% stenosis
6.0374
11.6200
21.8409
42.4574
88.4876
231.7890
30
40
50
60
70
0 0.00719 0.0185 0.0457 – –
0
− 0.5790
− 1.0544
− 2.1477
− 4.4876
− 14.5128 184.271
70.2660
33.7580
17.4205
9.3175
4.8793
− 11.1364
− 3.6900
− 1.6057
− 0.7778
− 0.4054
0
Minimum WSS (Pa)
Maximum WSS (Pa)
Reattachment length (m)
Maximum WSS (Pa)
Minimum WSS (Pa)
Unsteady flow (Time-averaged)
Steady flow (at mean Re = 400)
20
Severity (%)
Table 2 Quantitative comparison of steady flow and pulsatile flow
–
–
0.0465
0.0195
0.00752
0
Reattachment length (m)
–
–
1.75
5.40
4.58
0
% difference in reattachment length
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does not found to be changing along the axial direction. It can be seen that axial velocity increases as the severity of stenosis increases. Along the centreline of the artery, velocity is found to be almost equal at the interval of 1D, 2D, and 3D. Near the wall of the artery where circulation region is present, reversal flow velocity diminishes along the axial direction for the severity of 30 and 40%. Whereas Fig. 7 Axial velocity vectors for a 20%, b 30%, c 40%, d 50%, e 60%, f 70% stenosis
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in the severe cases of 50, 60, and 70% stenosis, the reversal flow velocity is found to be increasing from 1 to 3D. It suggests that the high momentum of the fluid in the severe stenosed cases (Fig. 7d–f) restricts flow to attain a parabolic velocity profile. Another interesting thing happening with the velocity profile at the throat section can be visualized in Fig. 7a–f. Maximum velocity at the throat section occurs at the centreline for the severity of 20, 30, and 40% (Fig. 7a–c), but it exists near artery wall in cases of severe stenosis (Fig. 7d–f). In other words, the steepness of velocity gradient near the wall is found to increase as the severity increases. In fact, due to this reason, the severe stenosed cases experience higher WSS near the throat as compared to mild and moderate ones (Table 2). Maximum and minimum WSS for all severities have been arranged in Table 2. It is appropriate to compare steady flow characteristics with corresponding timeaveraged characteristics of pulsatile flow. Comparison has been made accordingly between the maximum and minimum WSS. The magnitude of maximum and minimum WSS is found to be slightly higher in the steady case than that in the case of time-averaged pulsatile flow. Reattachment length is also found to be in good agreement within 1–5% range. Figure 8 shows the transient behaviour of streamlines for the severity of 50%. It can be seen that streamlines show significant changes compared to that of steady flow (Fig. 6d). The squeezing and elongation of circulatory zones and their associated directional changes in WSS, that plays a vital role in the new plaque formation, remain unnoticed under the assumption of steady flow. Similarly, in Fig. 9, which represents the WSS transient behaviour, WSS is severely undermined in the case of steady flow (Table 2). The above discussion suggests that though the steady flow assumption gives a rough picture of flow characteristics of time-averaged pulsatile flow, it oversights the transient behaviour in the pulsatile flow.
Fig. 8 Streamlines for 50% stenosis at various time instants
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Fig. 9 WSS distribution for 50% stenosis at various time instants
4.2 Effect of Pulsatile Inlet Velocity As seen earlier, steady flow assumption leads to significant lapses in result outcomes. Analysis of flow patterns and parameters like WSS for pulsatile inlet velocity could be of much interest in contrast to steady inlet flow. Figure 10a, b depicts the transient behaviour of streamlines for the case of 60 and 70% severity. These severities are chosen due to the occurrence of fascinating flow patterns. Asymmetric vortices are observed at some instants. In the case of 60% severe stenosis, it can be seen from Fig. 10a that unsteadiness onsets in the flow at t = 0.3 and starts generating vortices. Though asymmetric vortices are formed at one instant, axisymmetric circulatory zones are observed at the next instant of t = 0.4. Similarly, for the case of 70% severity, the vortices appear at both the time instants of t = 0.3 and t = 0.4 (Fig. 10b). It is accounted for the corresponding velocity changes at the inlet. The flow remains aligned with the axial direction at other instants of time, which may be attributed to mere squeezing and elongation of the recirculation zone. Figure 11a, b represents the WSS distribution for the severity of 60 and 70% over a cardiac cycle. It can be seen that WSS is maximum slightly upstream of the throat. It is because the velocity gradient is very steep at this location compared to other locations. At the time instant t = 0.2, WSS is maximum compared to other instants because the inlet pulse reaches its peak at the same instant. At the next instant of t = 0.3, maximum WSS falls drastically and remains approximately half of the previous one. Similarly, at other instants also, WSS distribution changes accordingly with the inlet pulse. These sudden changes in WSS within the short instants might lead to plaque rupture which has a severe consequence on blood clot formation. Intermittent changes in WSS may become the reason for the activation of platelets and consequently thrombus generation. From the above discussion, it can be concluded that the inlet velocity pulse steers well the flow behaviour and WSS distribution throughout the cycle.
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Fig. 10 Streamlines for a 60%, b 70% stenosis at various time instants
4.3 Oscillatory Shear Index OSI is the factor associated with the haemodynamic study which deals with the foreseeing of regions favourable in the formation of new plaque. It is predominantly governed by the WSS magnitude and its associated direction. It can be defined using Eq. (7), and its value lies between 0 and 0.5 [9]. The maximum value of OSI infers more oscillating flow, whereas the peak value of 0.5 ensures the fully oscillating flow such that net forward flow does not appear. [ 1 OSI = 1− 2
|] | t |∫ τw dt | 0 ∫t0 |τw |dt
(7)
In the pulsatile flow, flow separation and attachment points are highly sensitive to the inlet velocity. The shift in the separation and attachment point due to varying inlet velocity leads to the directional changes in the WSS vector. It leads to the oscillating flow and inherently amplifies the OSI value. Figure 12 represents the streamlines for the time-averaged mean velocities. Observing Figs. 12 and 13 altogether concludes
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Fig. 11 WSS distribution for a 60%, b 70% stenosis at various time instants
that the maximum OSI location is consistent with the reattachment point. It is interesting to observe that OSI distribution is symmetric in the case of severities between 20 and 50%, and it is mainly because of the symmetric nature of the flow. In contrast, OSI is asymmetric for severe stenosis of 60 and 70%. The severity of the stenosis affects the OSI distribution significantly. It can be seen from Fig. 13a that the OSI factor is minimum for 20% severe stenosis. This is a consequence of no flow separation happening for such mild severity. As the severity increases, the region associated with a peak value of OSI shifts further downstream with the widening of the peak OSI region (Fig. 13b–d). This means that the region favourable for the formation of new plaque also moves downstream with the increment in the severity of stenosis. For the case of 60 and 70% severity, the proposed length of the artery model is not found to be sufficient to capture the peak OSI region (Fig. 13e–f). In fact, detailed results confirm that the region downstream of stenosis is more prone to the formation of new plaque.
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Fig. 12 Time-averaged streamlines for a 20%, b 30%, c 40%, d 50%, e 60%, f 70% stenosis
Fig. 13 OSI distribution for a 20%, b 30%, c 40%, d 50%, e 60%, f 70% stenosis
5 Conclusions In the present work, numerical computations are performed on the stenosed carotid artery with the severity in the range of 20% to 70%. Inlet is provided with both steady and physiological pulsatile flows with assuming blood as a non-Newtonian fluid. The effect of severity is represented using the streamlines and velocity vectors. It is observed that at the throat location, the steepness of the velocity gradient near the wall rises as the severity increases which inherently causes enhancement in WSS also. The comparison of flow characteristics for steady and pulsatile flows concludes that the time-averaged quantities are found to be in good agreement. Noticing the transient characteristics and WSS distribution for the pulsatile flow, it can be stated that the steady flow assumption is inaccurate in the haemodynamic study. It is also observed that the pulsatile inlet velocity superiorly governs the flow behaviour and WSS distribution throughout the cardiac cycle. Along with the inlet pulse, the artery wall in the stenosed region experiences intermittent changes in WSS which may lead to undesirable incidents like plaque rupture and platelet activation. OSI distribution is found to be symmetric for the severity of 20, 30, and 40%, whereas it appeared asymmetric for 60 and 70% severity because of the compliant behaviour of OSI with the flow separation and reattachment. The peak OSI region is observed to be shifting
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downstream as the severity increases. OSI distribution signifies that the region downstream of the stenosis is more susceptible to the formation of new atherosclerotic plaque, which is also supported by a literature [10]. Acknowledgements The authors acknowledges the support of the High Performance Computing facility at Indian Institute of Technology Madras, Chennai, India. All the computations have been performed using “AQUA super cluster”.
Nomenclature WSS OSI P u→ ρ μ μ∞ μ0 γ˙ τ D d0 r R c η Z U Li Lo t T Tp
Wall shear stress [Pa] Oscillatory shear index Pressure [Pa] Velocity vector [m/s] Density of blood [kg/m3 ] Dynamic viscosity of blood [kg/m s] Infinite shear viscosity [kg/m s] Zero shear viscosity [kg/m s] Shear rate [1/s] Shear stress [Pa] Normal diameter of artery (0.006 m) [m] Diameter at throat of the artery [m] Radius of artery [m] Normal radius of artery [m] Constriction ratio Coefficient of shape of stenosis Axial distance [m] Axial velocity [m/s] Entrance length [m] Post-stenotic length [m] Non-dimensional time instant Dimensional time instant [s] Cardiac cycle period [s]
References 1. Lee TS (1990) Numerical studies of fluid flow through tubes with double constrictions. Int J Numer Meth Fluids 11:1113–1126 2. Deshpande MD, Giddens DP, Mabon RF (1976) Steady laminar flow through modelled vascular stenosis. J Biomech 9:165–174 3. Cho YI, Kensey KR (1991) Effects of the non-Newtonian viscosity of blood on hemodynamic of diseased arterial flows: part 1, steady flows. Biorheology 28:241–262
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4. Liao W, Lee TS, Low HT (2004) Numerical studies of physiological pulsatile flow through constricted tube. Int J Numer Meth Heat Fluid Flow 14:689–713 5. Long Q, Xu XY, Ramnarine KV, Hoskins P (2001) Numerical investigation of physiologically realistic pulsatile flow through arterial stenosis. J Biomech 34:1229–1242 6. Deplano V, Siouffi M (1999) Experimental and numerical study of pulsatile flow through stenosis: wall shear stress analysis. J Biomech 32:1081–1090 7. Ku DN, Giddens DP, Zarins CK, Glagov S (1985) Pulsatile flow and atherosclerosis in the human carotid bifurcation. AJ Arterioscler 5:293–302 8. Perktold K, Rappitsch G (1995) Computer simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model. J Biomech 28(7):845–856 9. He X, Ku DN (1996) Pulsatile flow in the human left coronary artery bifurcation. J Biomech Eng 118:74–82 10. Jahangiri M, Saghafian M, Sadeghi MR (2015) Numerical study of turbulent pulsatile blood flow through stenosed artery using fluid-solid interaction. Comput Math Methods Med 1–10. https://doi.org/10.1155/2015/515613
Image-Based Retinal Haemodynamics Simulation of Healthy and Pathological Retinal Vasculature Shivam Gupta and Ajay Bhandari
Abstract The current study aims to develop a comprehensive computational framework for quantitatively investigating haemodynamics in retinal arterial and venous networks. Two different pathological conditions, diabetic retinopathy and hypertension, have been considered, and the retinal haemodynamics change due to these pathologies has been investigated. Further, the comparison has been made with the healthy retinal vasculature. The retinal vascular tree has been extracted from the fundus images. The blood flow through the vasculature has been simulated, keeping into account the Fåhraeus–Lindqvist effect to incorporate the viscosity changes. Simulated results reveal that the blood flow velocity in arteries and veins decreases with the increase in the radial distance for all the cases. Further, higher average blood velocity and wall shear stress (WSS) are observed in the retinal network of diabetic and hypertension cases compared to the healthy ones. The current numerical model will help delineate the effect of structural changes in the retinal vasculature due to different pathological conditions on the retinal haemodynamics, which may be used as a helpful prognosis tool by ophthalmologists. Keywords Fundus image · Retinal haemodynamics · Diabetic retinopathy · Hypertension · And WSS
1 Introduction The human eye is a paired sense organ and one of the primary senses’ gateways. It can primarily be disseminated into two parts, the anterior and posterior chamber, with the retina being one of the essential components of the eye’s posterior chamber. It is highly vascular, making it prone to vascular diseases such as arteriolar narrowing, leaking, and arteriovenous nicking [1]. All these vascular complications in the retina are mainly caused by diabetes and hypertension, which, if left unchecked, can lead S. Gupta · A. Bhandari (B) Biofluids Research Lab, Department of Mechanical Engineering, IIT (ISM) Dhanbad, Dhanbad 826004, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_66
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to severe alterations in retinal microvascular structure. According to WHO in India, around 74 million people and 32.8% of the Indian population suffer from diabetes and hypertension, respectively, and 21.7% of these people suffer from retinal diseases such as haemorrhage, microaneurysms, and retinopathy [2]. The alteration in vascular structure caused by these diseases severely impairs the haemodynamic parameters inside the retinal vessels, such as blood flow velocity, pressure drop, and wall shear stresses (WSS). In the long term, this leads to abnormal outpouchings of the retinal capillary walls (microaneurysms) and blockage of minute blood vessels inside the retina. Currently, no such objective tool or precursor is available that can help ophthalmologists in the early prognosis of such retinal vascular complications. Nevertheless, computational fluid dynamics (CFD), with the help of medical imaging, can help the ophthalmologist study variation in retinal haemodynamic parameters for the early identification of retinal disorders non-invasively.
2 Literature Review and Objective Work in the field of retinal haemodynamics has continued since the 1980s. However, in the last decade, image-based numerical models have substantially helped understand retinal architecture’s fundamental haemodynamics. This is achieved by extracting the actual geometry of the retinal vasculature from the medical images. In this regard, a 2D numerical model was developed to predict the blood flow and oxygen distribution in a healthy retinal artery of the human eye extracted from the fundus images [3]. Taking this analysis forward, Malek et al. [4] simulated the blood flow in healthy retinal arteries and veins separately, focusing more on the quantitative evaluation of venous circulation. A computational model combining two approaches was developed to examine the effect of different ocular shapes on retinal haemodynamics [5]. Rehban et al. developed a computational framework to investigate retinal tissue stresses by analysing different viscosity models, blood flow (pulsatile/steady), and retinal tissue’s outer architecture [6]. It is evident from the literature that some studies focus on investigating retinal haemodynamics, yet it is unclear how different pathological conditions alter the retinal geometry and further haemodynamics. Additionally, there is a dearth of studies that thoroughly analyse the retinal haemodynamics for various pathological conditions and compare them with the healthy ones. To this end, the current research aims to numerically investigate the retinal haemodynamics in arteries and veins for different pathological conditions. The retinal architecture has been extracted from the fundus images of diabetic and hypertensive cases, and the haemodynamic distribution has been compared with healthy retinal vasculature.
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3 Methods 3.1 Fundus Image Acquisition and Processing The coloured retinal fundus images having a resolution of 4288×2488 of healthy and two pathological subjects, consisting of proliferative diabetic retinopathy (PDR) and hypertension, were acquired from the Harpreet eye care centre, Ludhiana, Punjab, using Kowa VX 10 alpha with 50/30-degree field of view (Fig. 1a–c). These digital RGB fundus images were then decomposed into the green channel to get higher contrast between the background and retinal vessels [7]. Additionally, some preprocessing steps like histogram stretching and bilateral filtering were applied to increase the contrast further and eliminate noise. A vessel enhancement filtering algorithm [8] was developed to extract the retinal vessel networks from preprocessed images. A novel MATLAB script is written to generate copies of input images by reproducing the image at different resolutions. The higher dimension images help filter out thick vessels more accurately, whilst lower dimensions images help capture finer details [7]. The filtered images were rescaled to original dimensions and binarized through the hysteresis thresholding before superimposing onto one final binary image (Fig. 2a–c). Also, the authors have tried to validate these binary images with their respective fundus images to ensure complete accuracy.
Fig. 1 Fundus images of a healthy, b PDR, and c hypertension cases
Fig. 2 Binary images of a healthy, b PDR, and c hypertension cases
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3.2 CAD Geometry Construction The final binary fundus image is subdivided into two individual images containing arterioles and venules, respectively. The separation of arterioles and venules was performed manually based on the topological differences. In the fundus image, the arteries appear thinner and brighter in colour due to high levels of oxygen concentration than the veins. Additionally, arterioles have a wider central/light reflex than venules [6], which helps its segmentation. The segmented binary images were imported into SolidWorks (Dassault Systèmes SOLIDWORKS Corp.) and scaled to realistic dimensions using the standard optic disc diameter of 1.85 mm [9]. Both geometries for arteries and venous networks were generated in each case. The spline feature in SolidWorks was used to manually trace the segmented vessels to construct a smooth vessel network, which was then exported as DFX files containing a plane geometry.
3.3 Mathematical Modelling In the current study, the mass and momentum conservation equations govern blood flow in the retinal arteries and veins. Additionally, the following assumptions have been considered. • Blood has been assumed to be an incompressible fluid, and its flow inside the arteries and veins has been considered to be steady [4]. • The blood vessel wall is assumed to be inelastic or rigid, and gravity forces are neglected [4]. Based on these assumptions, the following equations can express the blood flow inside the retinal vasculature. ∇ ·v =0
(1)
ρ(v · ∇v) = −∇ p + μ∇ 2 v
(2)
Although the blood is considered as a non-Newtonian fluid with two major haemodynamic constituents: blood cells and plasma. However, in this study, we have modelled blood as Newtonian fluid with viscosity as a function of haematocrit and vessel diameter, according to the Fåhraeus–Lindqvist effect [10]. This effect is applicable in the current scenario as the retinal blood vessel’s diameter is less than 150 µm [11]. To account for this effect, the following empirical viscosity model has been adopted for the current study [11]: μ = 6e−0.085D + 3.2 − 2.44e−0.06D
0.645
(3)
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3.4 Numerical Methodology The numerical simulations were performed using commercial software COMSOL Multiphysics version 5.5. The simulations were performed using an Intel(R) Core (TM) i7-8750H CPU @ 2.20 GHz personal computer. The Navier–Stokes equations have been discretized and solved numerically using the linear algebraic multigrid (AMG) method available in the COMSOL Multiphysics package. For the current study, the continuum approach is valid as the blood vessel’s diameter is way more as compared to the molecular diameter (~ 10 nm). A constant velocity boundary condition is prescribed for blood flow at the inlet (the root of the arterial tree), where the blood flow enters the vascular tree for all three cases. Due to the unavailability of subject-specific flow measurements, clinical data available for the healthy eye from the previous literature are used [6]. We scaled the healthy flow data based on the inlet diameter to create waveforms for other pathological subjects, so the inlet velocity in each case remains the same. The mean blood flow in the central retinal artery (CRA) has been reported as 43.2 µL/min. A no-slip boundary condition was applied on all the walls. For all the outlets, an impedance boundary condition is applied in all the cases. Asymmetrically structured fractal trees were generated at the end of each visible network outlet because of the difficulty in extracting smaller arterioles and venules from the fundus image [12]. In this fractal tree network, the radii of the successive daughter vessels (rd1 and rd2 ) were obtained by introducing the scaling parameters α and β for the radius of the root vessel (r p ) such that [4]: rd1 = α · r p ; rd2 = β · r p
(4)
ri, j = α i · β j−i · r p
(5)
− ξ1 ξ √ α = 1+γ2 ; β=α γ
(6)
The fractal tree has been extended until the mean radius of any daughter’s vessel becomes less than rmin [3]. For any vessel, the resistance to the blood flow can be expressed as [12]: Ro =
8μlrr πro3
(7)
The total resistance R O of each fractal tree has been computed iteratively, starting from the terminal branches. This resistance has been used to evaluate the pressure drop between outflow pressure PO and terminal of the virtual extended networks Pend is given by [12]: P = PO − Pend = R O · Q O
(8)
802 Table 1 Values of parameters
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Parameters
Value
Power-law coefficient (ξ )
3
Minimum radius (rmin )
15 µm
Asymmetry index (γ ) for (r ≤ 22 µm)
0.62
Asymmetry index (γ ) for (r > 22 µm)
0.4
Length-to-radius ratio (lrr ) for (r ≤ 22 µm)
27
Length-to-radius ratio (lrr ) for (r > 22 µm)
29
Terminal pressure (Pend )
0
Q O of central retinal artery (CRA)
43.2 µL/min
Central retinal vein pressure (CRVP)
13 mmHg
Blood density (ρ)
1050 kg/m3
The PO obtained from the above equation was prescribed at the respective outlet vessel. For the venous network, unique constant inlet velocity boundary conditions are applied at the corresponding inlets of the model in each case. The polynomial relation developed by [13] was adopted to prescribe venous flow velocity ranging approximately from 5 to 20 mm/s depending on the inlet diameter. A central retinal vein pressure (CRVP) of 13 mmHg has been defined at the final outlet in each case (Table 1). Triangular elements throughout the arterial and venous networks and the boundary layer elements in all the near-wall regions were used for meshing. Further, the corner refinement method decreased the element size at sharp corners in the geometry whilst generating the final mesh for each case. Mesh convergence has been achieved by systematically increasing the number of mesh elements from 9000 to 0.5 million. The final computational mesh consisted of 0.25 million mesh elements obtained after the mesh independence test.
4 Results and discussion 4.1 The Geometry of the Retinal Arterioles The healthy and pathological cases observed a significant difference in arterial tree diameter (Fig. 3). The healthy geometry had smaller mean ± standard deviation (SD) diameters (64 ± 43 µm) than the PDR case (81 ± 47 µm). In comparison, the mean ± SD diameters of the hypertension case (56 ± 28 µm) were smaller than the healthy ones. Additionally, it was observed that diabetic arterioles terminated at larger diameters as compared to others. This is because of the nature of the pathological condition, which leads to thinning of smaller arterioles making them less visible due to focal constraints [6].
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Fig. 3 Average diameters of healthy and pathological cases
4.2 Retinal Haemodynamics Figure 4 shows blood flow velocity distribution in the arterial and venous trees for the healthy, diabetic, and hypertensive cases. The distribution indicates that the blood flow velocity in arteries and veins decreases with the increase in the radial distance, having maximum and minimum values at the centre and peripheral of the vessel, respectively. Further, it can be observed that in the arterial network (Fig. 4a–c), the blood velocity decreases as it travels from larger arteries to smaller ones and subsequently to the capillaries in all the cases [14]. This low blood flow velocity in the arterioles provides sufficient time to carry out exchanges with tissue cells. However, in the
Fig. 4 Blood flow velocity distribution in a healthy, b PDR, c hypertension artery network and in d healthy, e PDR, and f hypertension venous network
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pathological conditions, the decreased blood flow velocity was significantly larger than the respective blood velocity in healthy cases. On the contrary, in the case of the venous network (Fig. 4d–e), the blood flow velocity increases as it flows from smaller venules to larger veins [14]. This is because the capillary beds facilitate the exchange of gas and nutrients before travelling in the venules and further into the veins. This qualitative observation was found to be similar in all the cases. Figure 5 shows the spatially averaged blood flow velocity distribution in the case of arteries and veins for all the cases. It can be observed that the averaged blood flow velocity in the case of arteries in PDR and hypertension cases was found to be 69% and 41% higher than the healthy ones, respectively. Subsequently, the average pressure drops in pathological cases were higher than in healthy ones. This increased blood pressure can cause the breakdown of the blood–retina barrier or obliteration of capillaries, resulting in intraretinal bleeding, retinal oedema, or swelling of the optic disc [15]. The change in average arterial blood velocity can be attributed to the altered topology of the retinal architecture in case of pathological conditions such as tortuosity, which is consistent with the experimental findings [16]. On the other hand, the averaged blood flow velocity in the case of veins in PDR and hypertension cases was found to be 41% and 36% lower than the healthy ones, respectively, which is consistent with the experimental findings [14]. Subsequently, average pressure drops in pathological cases were lower than the healthy ones causing an insufficient momentum for the return blood flow and resulting in low mean velocity. This low-velocity blood in case of veins also leads to inadequate return of blood flow back to the veins.
Fig. 5 Average blood velocity of healthy and pathological cases
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4.3 Wall Shear Stress (WSS) Figure 6 shows the WSS distribution in the arterial and venous trees for healthy, diabetic, and hypertensive cases. In all cases, WSS is higher in the blood vessels that are getting thin with the downstream flow. This is because as the diameter decreases, blood flow velocity increases, leading to a higher shear rate, which results in higher WSS. Figure 7 shows the spatial-averaged WSS distribution in the case of arteries and veins for all the cases. It can be observed that the average WSS in the retinal arteries was higher than the corresponding venule network for each case. These results are consistent with the non-invasive WSS measurements in retinal microcirculation [17]. Further, it can be concluded that the average arterial WSS in the PDR case was highest, followed by hypertension and healthy. In the venous network, we observed a relatively low WSS in the pathological conditions compared to healthy cases. This
Fig. 6 WSS distribution in a healthy, b PDR, c hypertension artery network and in d healthy, e PDR, and f hypertension venous network Fig. 7 Average WSS of healthy and pathological cases
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difference is mainly due to the low blood mean velocity which is consistent with the experimental findings [14], resulting in a lesser overall WSS in veins. Recent in vitro research reports that approximately 2 pascals of shear stress can stimulate the retinal endothelial cells to release nitric oxide (NO) in pathological conditions [18], which is also in line with our simulated results (Fig. 7). This release plays a vital role in vasoregulation, especially where the retinal arterioles are adjacent to the paired venules leading to the diffusion and dilation of NO produced by venules into nearby paired arterioles. This results in some characteristic changes observed mainly in diabetic retinopathy patients like venous bedding, looping, and duplication.
5 Conclusions In this study, an analysis and comparison of the retinal circulation in healthy and two pathological cases have been carried out using realistic fundus images. A detailed quantitative comparison of retinal haemodynamics between healthy and pathological retinal vasculatures has been performed. It has been inferred that the average blood flow velocity for the arterial network of PDR and hypertension is higher than the healthy case. This further results in higher mean WSS in pathological conditions compared to healthy ones. On the contrary, the average blood flow velocity for the venous network of PDR and hypertension is less than the healthy ones. The current numerical framework can be further extended to investigate retinal haemodynamics in various pathological conditions. Acknowledgements The authors thank Dr. Harpreet Singh for providing the fundus images of healthy and pathological eyes. Further, the authors would like to acknowledge the support received by a grant from the Science and Engineering Research Board (Grant Number: SRG/2021/000053) and the Faculty research scheme of IIT(ISM) Dhanbad (Grant Number: FRS (147)/2020-2021/ MECH).
Nomenclature v ρ p μ r D α, β γ l rr QO
Blood velocity vector [m/s] Blood density [kg/m3 ] Blood pressure [kg/m s2 ] Blood viscosity [kg/m s] Vessel radius [m] Vessel diameter [µm] Scaling parameters Asymmetry index Length-to-radius ratio Volumetric flow rate [m3 /s]
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PO Pend
807
Outflow pressure [kg/m s2 ] Terminal pressure [kg/m s2
References 1. MacGillivray TJ, Trucco E, Cameron JR, Dhillon B, Houston JG, van Beek EJR (2014) Retinal imaging as a source of biomarkers for diagnosis, characterization and prognosis of chronic illness or long-term conditions. Br J Radiol 87(1040):20130832. https://doi.org/10.1259/bjr. 20130832 2. Kulkarni S et al (2019) Estimating the magnitude of diabetes mellitus and diabetic retinopathy in an older age urban population in Pune, western India. BMJ Open Ophthalmol 4(1):e000201. https://doi.org/10.1136/bmjophth-2018-000201 3. Liu D, Wood NB, Witt N, Hughes AD, Thom SA, Xu XY (2009) Computational analysis of oxygen transport in the retinal arterial network. Curr Eye Res 34(11):945–956. https://doi.org/ 10.3109/02713680903230079 4. Malek J, Azar AT, Nasralli B, Tekari M, Kamoun H, Tourki R (2015) Computational analysis of blood flow in the retinal arteries and veins using fundus image. Comput Math with Appl 69(2):101–116. https://doi.org/10.1016/j.camwa.2014.11.017 5. Dziubek A, Guidoboni G, Harris A, Hirani AN, Rusjan E, Thistleton W (2016) Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model. Biomech Model Mechanobiol 15(4):893–907. https://doi.org/10.1007/s10237-015-0731-8 6. Rebhan J, Parker LP, Kelsey LJ, Chen FK, Doyle BJ (2019) A computational framework to investigate retinal haemodynamics and tissue stress. Biomech Model Mechanobiol 18(6):1745– 1757. https://doi.org/10.1007/s10237-019-01172-y 7. Budai A, Bock R, Maier A, Hornegger J, Michelson G (2013) Robust vessel segmentation in fundus images. Int J Biomed Imaging 2013. https://doi.org/10.1155/2013/154860 8. Frangi AF, Niessen WJ, Vincken KL, Viergever MA (2013) Multiscale vessel enhancement filtering. Lect Notes Comput Sci (including Subser. Lect Notes Artif Intell., Lect. Notes Bioinform) 1496(2013):130–137. https://doi.org/10.1007/BFb0056195 9. Kristen A, Kelsey L, Wintermantel E, Doyle B (2016) Fundus image based blood flow simulation of the retinal arteries. In: Computational biomechanics for medicine. Springer International Publishing, Cham, pp 143–154 10. Fåhræus R, Lindqvist T (1931) The viscosity of the blood in narrow capillary tubes. Am J Physiol Content 96(3):562–568. https://doi.org/10.1152/ajplegacy.1931.96.3.562 11. Pries AR, Secomb TW (2011) Microcirculatory blood flow: functional implications of a complex fluid. In: 3rd Micro Nano flows conference, Aug 2011, pp 22–24 [Online]. Available: http://bura.brunel.ac.uk/handle/2438/6848 12. Olufsen MS, Peskin CS, Kim WY, Pedersen EM, Nadim A, Larsen J (2000) Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann Biomed Eng 28(11):1281–1299. https://doi.org/10.1114/1.1326031 13. Riva CE, Grunwald JE, Sinclair SH, Petrig BL (1985) Blood velocity and volumetric flow rate in human retinal vessels. Investig Ophthalmol Vis Sci 26(8):1124–1132 14. Burgansky-eliash Z, Nelson DA, Bar-tal OP, Lowenstein A, Grinvald A, Barak A (2010) Reduced retinal blood flow velocity in diabetic retinopathy. Retina 30(5):765–773. https:// doi.org/10.1097/IAE.0b013e3181c596c6 15. G. K. Lang, D. Recker, C. W. Spraul, and K. Gerhard, Ophthalmology, Third. Mosby/Elsevier, 2009. 16. Malek J, Azar AT, Tourki R (2014) Impact of retinal vascular tortuosity on retinal circulation. Neural Comput Appl 26(1):25–40. https://doi.org/10.1007/s00521-014-1657-2
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17. Nagaoka T, Yoshida A (2006) Noninvasive Evaluation of Wall Shear Stress on Retinal Microcirculation in Humans. Invest Opthalmol Vis Sci 47(3):1113. https://doi.org/10.1167/iovs.050218 18. Lakshminarayanan S, Gardner TW, Tarbell JM (2000) Effect of shear stress on the hydraulic conductivity of cultured bovine retinal microvascular endothelial cell monolayers. Curr Eye Res 21(6):944–951. https://doi.org/10.1076/ceyr.21.6.944.6985
Numerical Study on the Effect of Exercise on Various Configurations of Stenosis in Coronary Artery Siddharth D. Sharma, Piru Mohan Khan, Suman Chakraborty, and Somnath Roy
Abstract One of the primary factors in adult fatalities is coronary artery disease. Hence, the study on the effect of stenosis position and number of stenoses becomes necessary to predict the susceptibility of formation of another stenosis and predict atherosclerotic plaque rupture. The probability of plaque rupture increases under exertion; hence, the current study uses two different pulse rates (75 BPM for the rest condition and 120 BPM for the exercise condition). Higher time-averaged wall shear stress (TAWSS) values are observed in case of double stenoses even at rest conditions. A substantial increase in peak TAWSS value is observed under exercise conditions compared to rest conditions. A downstream shift in oscillatory shear index (OSI) region is observed under exercise conditions for all cases, whereas a significant reduction in OSI region is observed in the case of proximal stenosis. Keywords Coronary artery · Computational fluid dynamics · Atherosclerotic plaque rupture
S. D. Sharma (B) · S. Roy Center of Computational and Data Sciences, IIT Kharagpur, Kharagpur 721302, India e-mail: [email protected] P. M. Khan · S. Chakraborty · S. Roy Department of Mechanical Engineering, IIT Kharagpur, Kharagpur 721302, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_67
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1 Introduction In India, one of the leading causes of death for adults is coronary artery disease. According to data released by The Registrar General of India, more than 30% of deaths in the period 2016–2020 were due to circulatory system diseases [1]. Moreover, the occurrences of cardiac arrest during exercise have increased in recent times. Heart attack is a common cause leading to cardiac arrest. A heart attack occurs when coronary arteries providing blood to heart muscles get entirely blocked due to the rupture of atherosclerotic plaques. Hence, the study of haemodynamics in atherosclerotic coronary arteries under exercise conditions is necessary to predict atherosclerotic plaque rupture. There have been many studies on the effect of haemodynamics in the presence of stenosis [2–5]. Under exercise conditions, the pulse rate gets elevated. Sood et al. [6] reported that elevated pulse rates cause disturbances in flow past the stenotic region, leading to higher wall shear forces fluctuations. Coronary arteries are generally twisty in nature. The inner walls of bends are susceptible to plaque development [7], and research indicates that these regions have lower wall shear stresses [8]. Though there are many studies on the effect of radius of curvature on the flow as well as considering different grades of stenosis [9–11], the effect of exercise on the haemodynamics of coronary artery with consecutive bends having atherosclerosis has not been explored much.
2 Materials and Methods 2.1 Geometry To study the effect of multiple bends found in coronary arteries, an idealized artery with an S-bend is considered in the current study. Because of the curvature, recirculation zones form near the bend’s inner walls, making them prone to atherosclerosis. Moreover, stenosis shape plays an essential role in haemodynamics regardless of the severity of blockage. Eccentric stenosis shows more serious haemodynamic complications than concentric stenosis [2]. Hence, in the current study, eccentric stenosis is considered at the inner walls of the bends. As the occurrence of coronary atherosclerotic plaque rupture is found to be high in ≤ 50% diameter stenosis [12], 50% blockage is considered for stenosis. The diameter of coronary artery is taken to be 3 mm. The radius of curvature taken is 2D. Length of 10D is taken before the S-bend, and 20D is taken after the S-bend. Based on the position and number of stenosis, three configurations are explored. The first configuration considers stenosis on the interior wall of the artery’s first bend, whilst the second configuration considers stenosis on the interior wall of the second bend. Stenoses are considered at the interior wall of both bends in the third
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Fig. 1 Geometry of S-bend for different configurations
arrangement. From here on, the first, second, and third configurations are referred to as proximal, distal, and double stenoses, respectively (Fig. 1).
2.2 Mathematical Model and Boundary Conditions The fluid flow simulations are carried out using the incompressible Navier–Stokes equations: ∇·V =0 )
∂V ρ + V · ∇V ∂t
(1)
) = −∇ · σ
(2)
The stress tensor consists of deviatoric and hydrostatic stress tensors as shown in Eq. (3). σ = −pI + τ
(3)
Here, I is the identity tensor. As shown in Eq. (4), τ is a function of the shear strain rate tensor, S. The dynamic viscosity μ in the present work is a function of strain rate γ˙ , considering the non-Newtonian Carreau model. τ = μ(γ˙ )S
(4)
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Fig. 2 Pulsatile velocity profile at inlet for rest (75 BPM) and exercise (120 BPM) condition
( ) Here, S is given by S = 21 ∇ V + ∇ V T , and the strain rate is given by γ˙ = / ∑∑ 1 Si j Si j . The apparent viscosity is calculated for the Carreau model, as shown 2 i
j
in Eq. (5). μ − μ∞ n−1 = (1 + (λγ˙ )) 2 μo − μ∞
(5)
Here, μ∞ represents dynamic viscosity at an infinite strain rate (0.0035Pa s), μo denotes dynamic viscosity at zero strain rate (0.056Pa s), λ is relaxation time (3.313s), and n denotes power index (3.568). The density of blood is taken as 1050 kg/ m3 . To solve the aforementioned equations, a sharp interface immersed boundary method-based solver accelerated on GPU using OpenACC is used [13]. The solver uses marker and cell (MAC) algorithm to solve the Navier–Stokes equation. The Poisson equation is solved using red–black successive over-relaxation (RBSOR) method. The solver is extended for non-Newtonian flows by Irshad et al. [14] Realistic pulsatile inlet velocity used in the current study is taken from Song et al. [15] for two different pulse rates. For the rest condition, a pulse rate of 75 BPM is considered, whilst a pulse rate of 125 BPM is considered for the exercise condition as shown in Fig. 2. At the outlet, convective boundary condition is used for velocity.
2.3 Validation and Grid Independence Study The current code is validated against an experimental study carried out by Taylor et al. [16] on an S-shaped duct at Re = 790. As shown in the Fig. 3, velocity profiles obtained at three sections are compared with the experimental data and show good
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agreement at grid size of D/48. In the current study, three different configurations based on position and number of stenoses (proximal, distal, and double stenoses) are considered at rest (75 BPM) and exercise (120 BPM) conditions. Grid independence study is carried out for double stenoses case under exercise condition. To check grid independence, time-averaged axial velocity at x/D = 5 is compared for three different grid sizes, and it can be observed in Fig. 4 that velocity profiles converse for D/48 and D/64 grid sizes. Hence, in the current study, D/48 grid size is used.
Fig. 3 Comparison of velocity profiles with Taylor et al. [16] at sections AA, BB, and CC
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Fig. 4 Time-averaged axial velocity at x/D = 5 for three different grid spacings
3 Results and Discussion 3.1 Variation in Velocity Profiles In Fig. 5, streamwise velocity is shown for different configurations of stenoses at rest condition. In the case of single stenosis (Fig. 5) at either bend, only a single recirculation zone attached to stenosis is obtained at rest condition. Whilst under exercise condition, a second recirculation zone is also observed near bend 2 in case of proximal stenosis due to increased axial velocity.
3.2 Variation in Vortex Structures at Peak Velocity As shown in Fig. 7, there is substantial increase in the length of vortex structures at peak flow rate in all the cases. The dean vortices generated after the S-bend are sustained for a longer duration under exercise conditions as compared to that of rest conditions, although the length of elongated vortices are lower in case of double stenosis as compared to single stenosis cases (Fig. 7).
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Fig. 5 Velocity contours at peak inlet velocity during rest condition for different configurations of stenoses Fig. 6 Velocity contours at peak inlet velocity during exercise condition for different configurations of stenoses
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Fig. 7 Vortex structures at peak flow rate
3.3 Tawss TAWSS is an important parameter to predict susceptibility of formation of stenoses as well as atherosclerotic plaque rupture. TAWSS represents the total wall shear stress exerted throughout the cardiac cycle. TAWSS =
1 T ∫|τw |dt T 0
(6)
A low value of TAWSS coupled with a high value of OSI shows susceptibility of formation of stenosis, whilst high values of TAWSS can result in plaque rupture. Figure 8 shows that peak TAWSS values for single stenosis cases are pretty comparable, whilst that of double stenoses are significantly higher. As compared to the rest condition, peak TAWSS values are remarkably higher under exercise condition, as shown in Fig. 9. Furthermore, peak TAWSS values are at dangerous levels even in case of single stenosis under exercise conditions.
3.4 OSI OSI represents the zones where wall shear stresses show directional changes over the cardiac cycle. High values of OSI denote susceptibility of formation of stenosis. ( |) | T |∫ τw dt | 1 0 OSI = 1− T 2 ∫0 |τw |dt
(7)
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Fig. 8 TAWSS under rest condition (75 BPM)
In the case of proximal stenosis shown in Fig. 10a, b, OSI downstream of the stenosis undergoes significant reduction under exercise condition and shifts further downstream. This shift is due to higher axial jet velocity occurring during exertion, which can be seen in Fig. 6. For distal stenosis shown in Fig. 10c, d, the OSI region downstream of the stenosis shifts further downstream under exercise condition. Whilst in the case of double stenosis, no significant OSI is found post the first stenosis, the behaviour of OSI post the second stenosis is similar to that of distal stenosis.
3.5 Pressure Drop Pressure drop is recorded across the S-bend with time for both rest and exercise conditions. In Fig. 11, it can be observed that the pressure drop is similar for both proximal and distal stenoses. However, pressure drop across double stenosis is remarkably higher as compared to single stenosis cases, i.e. proximal and distal stenosis. Further, it can be observed in that the pressure drop is slightly higher in case of distal stenosis as compared to that of proximal stenosis. Also, pressure drop in case of exercise condition (120 BPM) is significantly higher than rest condition (75 BPM).
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Fig. 9 TAWSS at exercise condition (120 BPM)
4 Conclusions Across a stenosis, pressure drops significantly more in cases of double stenosis than in cases of single stenosis. More so, in all instances, the pressure drop significantly increases when subjected to exercise. One can also observe that there is a sharp increase in TAWSS under exercise condition, even for single stenosis configurations, which can lead to an increased probability of atherosclerotic plaque rupture. OSI region shifts further downstream under exertion for all cases, but in case of proximal stenosis, a significant reduction in OSI region is observed.
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Fig. 10 OSI contour for different configurations a and b show proximal stenosis, c and d show distal stenosis, and e and f show double stenoses; a, c, and e are at rest condition whilst (b), (d), and (f) are at exercise condition]
Fig. 11 Pressure drop across S-bend under different conditions
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Acknowledgements We have utilized Paramshakti, a supercomputing facility built by the National Supercomputing Mission of the Government of India at IIT Kharagpur. This study is supported by the SERB project: CRG/2019/00265.
Nomenclature V ρ σ p τ γ˙ μ S
Velocity [m/s] Density [Kg/m3 ] Stress tensor [Pa] Pressure [N/m2 ] Shear stress tensor [Pa] Strain rate [s− 1 ] Dynamic viscosity [Pa s] Strain rate tensor [s− 1 ]
References 1. Registrar General of India (2020) Report on medical certification of report on medical certification of cause of death 2020 [Online]. Available: https://censusindia.gov.in/nada/index.php/ catalog/42681 2. Freidoonimehr N, Chin R, Zander A, Arjomandi M (2021) Effect of shape of the stenosis on the hemodynamics of a stenosed coronary artery. Phys Fluids 33(8). https://doi.org/10.1063/5. 0058765 3. Freidoonimehr N, Chin R, Zander A, Arjomandi M (2020) An experimental model for pressure drop evaluation in a stenosed coronary artery. Phys Fluids 32(2). https://doi.org/10.1063/1.513 9701 4. Timofeeva M, Ooi A, Poon EKW, Barlis P (2022) Numerical simulation of the blood flow through the coronary artery stenosis: effects of varying eccentricity. Comput Biol Med 146:105672. https://doi.org/10.1016/j.compbiomed.2022.105672 5. Khan PM, Raj A, Alam MI, Chakraborty S, Roy S (2023) Prediction of vortex structures in pulsatile flow through S-bend arterial geometry with different stenosis levels. Biocybern Biomed Eng 43(1):298–312. https://doi.org/10.1016/j.bbe.2023.01.003 6. Sood T, Roy S, Pathak M (2018) Effect of pulse rate variation on blood flow through axisymmetric and asymmetric stenotic artery models. Math Biosci 298(February):1–18. https://doi. org/10.1016/j.mbs.2018.01.008 7. Liepsch D (2002) An introduction to biofluid mechanics—basic models and applications. J Biomech 35(4):415–435. https://doi.org/10.1016/S0021-9290(01)00185-3 8. Malek AM (1999) Hemodynamic shear stress and its role in atherosclerosis. JAMA 282(21):2035. https://doi.org/10.1001/jama.282.21.2035 9. Xie X, Wang Y, Zhou H (2013) Impact of coronary tortuosity on the coronary blood flow: a 3D computational study. J Biomech 46(11):1833–1841. https://doi.org/10.1016/j.jbiomech.2013. 05.005 10. Xie X, Li Y, Xie S (2018) Computation of hemodynamics in eccentric coronary stenosis: a morphological parametric study. Technol Heal Care 26(2):229–238. https://doi.org/10.3233/ THC-160529
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11. Wong KKL, Wu J, Liu G, Huang W, Ghista DN (2020) Coronary arteries hemodynamics: effect of arterial geometry on hemodynamic parameters causing atherosclerosis. Med Biol Eng Comput 58(8):1831–1843. https://doi.org/10.1007/s11517-020-02185-x 12. Fishbein MC, Siegel RJ (1996) How big are coronary atherosclerotic plaques that rupture? Circulation 94(10):2662–2666. https://doi.org/10.1161/01.CIR.94.10.2662 13. Raj A, Khan PM, Alam MI, Prakash A, Roy S (2022) A Gpu-accelerated sharp interface immersed boundary method for versatile geometries. J Comput Phys 478:111985. https://doi. org/10.2139/ssrn.4086449 14. Alam MI, Raj A, Khan PM, Kumar S, Roy S (2021) Numerical simulation of flow of a shearthinning Carreau fluid over a transversely oscillating cylinder. J Fluid Mech 921:1–33. https:// doi.org/10.1017/jfm.2021.485 15. Song J, Kouidri S, Bakir F (2021) Numerical study on flow topology and hemodynamics in tortuous coronary artery with symmetrical and asymmetrical stenosis. Biocybern Biomed Eng 41(1):142–155. https://doi.org/10.1016/j.bbe.2020.12.006 16. Taylor AMKP, Whitelaw JH, Yianneskis M (1984) Developing flow in S-shaped ducts, vol 2: circular cross-section duct
Effect of Aging on Passive Drug Diffusion Through Human Skin Aditya Ranjan, Vijay S. Duryodhan, and Nagesh D. Patil
Abstract In the present work, a comparative study of the passive diffusion of a drug through the intercellular (ICR) and sweat duct (SDR) routes of human skin has been performed. We have tested trans-cinnamic acid and caffeine as drugs, as those are the most commonly available compounds. The effect of aging on transdermal drug diffusion has been considered by performing the above analysis for young age (< 40 years) and old age (> 60 years). A mathematical model based on Fick’s law is adopted to understand drug diffusion through various layers. Each of the routes is described by a compartment model along with the donor and receiver compartments at the top and bottom, respectively. Code is validated by comparing present results with the published experimental findings. For both tested drugs, it is found that the intercellular route provides a faster route of drug delivery as compared to the sweat duct route. It is also found that the amount of drug diffusion increases upon the aging of human skin. Keywords Transdermal · Drug diffusion · Intercellular · Sweat duct · Aging
1 Introduction Transdermal drug delivery is a noninvasive mode of delivering a drug that avoids first pass and gastrointestinal absorption [1]. Hence, this mode is advantageous among other modes of drug delivery such as oral, inhalation, and injection. However, all drugs are not found to have a high rate of diffusion through the skin. The drug can diffuse via intercellular, sweat duct, hair follicle, and transcellular route. The intercellular route refers to the pathway through the spaces between the cells. Sweat duct and hair follicles are appendages present in the outer layer of skin A. Ranjan · V. S. Duryodhan · N. D. Patil (B) Department of Mechanical Engineering, IIT Bhilai, Raipur 492015, India e-mail: [email protected] N. D. Patil Department of Bioscience & Biomedical Engineering, IIT Bhilai, Raipur 492015, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_68
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which provides a route of permeation [2]. The sweat duct route comprises the upper coil duct, dermal duct, and secretory coil duct [3]. The topmost layer of human skin known as the stratum corneum shows the highest resistance to drug diffusion [4]. Underneath of the stratum corneum lies the epidermis and dermis.
2 Literature Review and Objective Even though skin acts as a barrier against permeating drugs or compounds, Michaels et al. [5] showed that some drugs are significantly permeable through the skin. This led to the development of transdermal patches in the year 1979. The drug delivered via human skin must have low molecular weight, be lipophilic, and be effective at low dosages. To enhance the drug delivery or to deliver larger molecules, active methods such as iontophoresis, cavitational ultrasound, or thermal ablation can be opted [6]. As per Cygan et al. [2], possible pathways for passive diffusion of chemicals through the skin to the circulation network are as follows: (a) transcellular route, (b) intercellular route, (c) sweat duct, (d) hair follicle. According to Trommer and Neubert [7], drug permeating through the transcellular route is offered maximum hindrance. Therefore, the intercellular route, hair follicle route, and sweat duct route are more suitable pathways for drug permeation. Various mathematical models have been presented in the literature to understand passive drug diffusion through human skin using numerical methods [8–11]. Grassi and Colombo [8] employed both numerical and analytical approaches to study the passive diffusion of drugs through a swollen membrane. Later, Coceani et al. [9] used a similar mathematical model to describe the acyclovir diffusion through the rat skin. They have validated the results from the mathematical model with that of in vitro experiments, which were found to be in good agreement. In the present work, the same mathematical model is utilized to understand drug diffusion through human skin diffusing through both ICR and SDR. Upon conducting the literature review, it is found that an extensive study has been performed on drug diffusion through the hair follicular route and overall skin. However, very few works of literature have reported passive drug diffusion through the SDR [2]. Moreover, studies on the effect of aging on transdermal drug delivery are also rarely found. This work presents a comparative study of passive diffusion through intercellular and sweat duct routes considering trans-cinnamic acid and caffeine as drugs. Skin properties of two age groups (< 40 years and > 60 years) have been considered to understand the effect of age on passive drug diffusion.
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3 Materials and Methods In the present work, one-dimensional Fick’s law has been used to describe diffusion through ICR and SDR. Here, it is assumed that the concentration gradient in the other directions is negligible. ∂ 2C ∂C =D 2 ∂t ∂x
(1)
∂C ∂ 2C = D∗ 2 ∂t ∂x
(2)
Equation 1 represents Fick’s second law that describes the variation of drug concentration C as a function of space (x) and time (t). Here, D represents the drug diffusion coefficients in various skin layers. This equation describes the drug diffusion through the ICR. Equation 2 represents the drug diffusion through SDR, and D* represents the effective diffusion coefficient which considers the effect of porosity and tortuosity associated with the sweat duct. Figure 1 represents the compartment model [9] adopted to visualize various skin routes mentioned. In the present study, diffusion analysis of trans-cinnamic acid and caffeine through skin routes has been performed. Trans-cinnamic acid is well known for its antimicrobial, antitumor, antioxidant, and antimycobacterial properties [12] while caffeine is mainly effective in neurodegenerative diseases [13]. Here, the drugs are assumed to be present in powder form, and it mixes with solvent to form a solution that passively diffuses through skin routes. Initially, powder drugs having an undissolved mass and particle radius M o and Ro , respectively, are assumed to be present in the donor compartment. The variation of mass M in the donor compartment is calculated by Eq. 3. Drug dissolution constant K t (Eq. 5) describes the dissolution of the drug. As a result, the drug size will vary and is given by Eq. 4. Further, V d represents the volume of the donor compartment. The variation of drug concentration in donor compartment C d with time is obtained from Eq. 6. Here, Dsc and C sc represent the diffusion coefficient and drug concentration in the stratum corneum. The variation of drug concentration in stratum corneum (sc), viable epidermis (ve), and dermis (de) is governed by Eq. 7, 9, and 11, respectively. C r represents the drug just entering the receiver compartment. The continuity of drug concentration between different layers is ensured by interface conditions such as Eq. 8 between stratum corneum and viable epidermis and Eq. 10 between viable epidermis and dermis. However, drug partitioning varies at each interface along ICR. These partitions at the interface are governed by partition coefficients K 1 , K 2 , and K 3 described by. Equation 13 (whose values are mentioned in Table 1). Finally, the variation of drug concentration in the receiver compartment is given by Eq. 12. Initial condition assumes that the C do amount of the drug is present in the donor compartment only, as shown in Eq. 14. The below partial differential equations (PDEs) are made non-dimensional using a set of characteristic variables shown in Eq. 15 and 16 and then discretized using the
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Fig. 1 A three-layer compartment model for skin routes
Table 1 Values of coefficients for drugs used in present study [16–18] Drug
Partition coefficient
Diffusion coefficient (cm2 /s)
Diffusion coefficient in water (cm2 /s)
Trans-cinnamic acid
K 1 = 7.26 K 2 = 2.43 K 3 = 3.47
D1 = 9.81 × 10–5 D2 = 5.4 × 10–5 D3 = 9.82 × 10–3
9.9 × 10–6
Caffeine
K 1 = 7.84 K 2 = 0.94 K 3 = 1.12
D1 = 9.83 × 10–5 D2 = 4.23 × 10–5 D3 = 7.96 × 10–3
9 × 10–6
second-order central difference and backward difference schemes. These discretized equations are solved using the finite difference method (FDM) in explicit form using an in-house built-in MATLAB code. Temporal and spatial variations of drug concentration diffused through intercellular and sweat duct routes and for both young and old age are plotted and discussed in subsequent sections. h sc
M = M0 + Vd (Cd0 − Cd ) − Vr Cr − ∫ Csc Sdx 0
−
h sc +h ve
∫
h sc
Cve Sdx −
h sc +h ve +h de
∫
h sc +h ve
Cde Sdx
(3)
√ R = R0
3
M M0
K t = 4π R 2 Vd
Kd Vd
dCd ∂Csc = Vd K t (Cs − Cd ) + Dsc S |x=0 dt ∂x
(4) (5) (6)
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( ) ∂ ∂Csc ∂Csc = Dsc ∂t ∂x ∂x Dsc
Dve
Cd+ =
∂Csc ∂Cve |x=h sc = Dve |x=h sc ∂x ∂x ( ) ∂Cve ∂Cve ∂ Dve = ∂t ∂x ∂x
(8) (9)
∂Cve ∂Cde |x=h sc +h ve = Dde |x=h sc +h ve ∂x ∂x ( ) ∂ ∂Cde ∂Cde = Dde ∂t ∂x ∂x
Vr k1 =
(7)
∂Cr ∂Cde = Dde S |x=h sc +h ve +h de ∂t ∂x
(10) (11) (12)
Csc Cve Cve Cde ; k2 = ; k3 = ; kr = =1 Cd Csc Cde Cr
(13)
Cd = Cd0 ; Csc = Cve = Cde = Cr = 0
(14)
Cd Cr Dve Vd ; Cr+ = ; t + = t 2 ; Vd+ = Cdo Cdo h ve sh ve
(15)
Vr+ =
Vr x ; x+ = 1 + ; sh ve h ve
K t+ = K t
h 2ve Dve
(16)
A similar set of PDEs (Eq. 6–12) is used to obtain the variation of drug concentration in the sweat duct route. However, to account for the porosity and tortuosity of the sweat duct, the diffusion coefficient is modified as the effective diffusion coefficient, De , and is given by Eq. 17 [14]. De ∂ 2 C ∂C = ∂t ε ∂x2
(17)
De is the product of porosity, apparent tortuosity, and diffusion coefficient. To consider some amount of drug being carried away by the bloodstream for circulation, the receiver sampling technique is adopted similar to Ref. [9]. Around 10% of the drug is assumed to enter the bloodstream periodically. It is mathematically modeled as Eq. 18. Cra =
Crb (Vr − ΔVr ) Vr
(18)
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C r b and C r a represent the drug concentration in the receiver compartment before and after sampling, while ΔVr represents the volume being carried away for circulation. An initial amount of drug in the donor compartment (C do ) is assumed to be 2.62 mg. The initial powder drug radius (Ro ) is taken as 5.7 µm, and an initial amount of undissolved drug (M o ) is assumed as 1 mg. The permeation area (S) is taken as 1 cm2 . The volume of the donor and receiver compartment is taken as 3 cm3 . The value of powder dissolution coefficient (K d ) is taken as 5.01 × 10–5 cm/s [9]. Table 1 describes the diffusion and partition coefficient of the drug in three different layers 1, 2, and 3 representing stratum corneum (sc), viable epidermis (ve), and dermis (de) respectively. The partition coefficient is defined as the ratio of drug or solute concentration on either side of an interface at equilibrium conditions [15]. It also shows the drug diffusion coefficient in water because 99% of the medium inside the sweat duct is aqueous. The effect of aging has also been considered in the present work. It has been found that the thickness of the stratum corneum almost remains unchanged with age. The thickness of viable epidermis reduces slightly with age, while the dermis layer is found to significantly shrink upon aging [19]. The variation of thickness of various layers has been shown in Table 2. With age, the pore size is found to increase [20]. Hence, the diameter of the duct also increases with age which leads to an increase in porosity. Porosity is defined as the ratio of the area of all ducts per unit area of skin. Tortuosity is defined as the ratio of the actual curvilinear path followed by a drug to the straight line path [2]. The variation of porosity and tortuosity [19, 21] of sweat ducts with age has been shown in Table 3. Table 2 Thickness variation of skin layers upon aging Layers in ICR
Young age (< 40 years) (µm)
Old age (> 60 years) (µm)
Stratum corneum
15
17
Viable epidermis
62
52
Dermis
982
616
Table 3 Porosity and tortuosity variation of sweat duct upon aging Layers in SDR
Pore diameter (µm)
Tortuosity
Young
Old
Young
Old
Upper coil duct
20
60
1.22
1.39
Dermal duct
10
20
2.00
2.72
Secretory coil duct
30
40
5.06
3.82
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Fig. 2 Grid independence test for ICR for young age
3.1 Grid and Time Independence Test Before taking up detailed simulations, the grid and time independence tests are performed by considering the diffusion of trans-cinnamic acid through ICR and SDR for a young age. For the sake of brevity, in Fig. 2, the grid independence test is shown only for ICR. For the intercellular route, the grid points in each layer are varied, respectively, from N = 6–31. Figure 2 shows the temporal variation of trans-cinnamic acid concentration in the receiver compartment at different number of grids used to discretize the skin layers. It shows that the results remain unchanged with the number of grid points. Similar to the grid independence, the non-dimensional time step dt + is varied for ICR, and the curve is plotted. Figure 3 represents the time independence test. It can be observed that for ICR, time independence behavior is observed from dt + = 10–5 . Hence, based on the above test, the number of grid points chosen in each layer for ICR is 6, while the non-dimensional time step for ICR is taken as 10–5 .
3.2 Code Validation Figure 4 illustrates the code validation of the present work with the published result for acyclovir diffusion through the rat skin [9]. Coceani et al. [9] performed permeation experiments over various rat skin samples using the Franz diffusion cell apparatus. The experimental result for rat skin (sample 9 of Ref. [9]) has been compared with the numerical results obtained using the in-house MATLAB code (as shown in Fig. 4), and the results are found to be in good agreement.
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Fig. 3 Time independence test for ICR for young age
Fig. 4 Code validation with acyclovir permeation through the rat skin
4 Results and Discussion The temporal and spatial variations of drug concentration permeating through the two routes are obtained based on the numerical solution of the above equations. The drug concentration is found to increase with time and vary with spatial location in each of the routes. Results have been discussed for both tested drugs in the subsequent sections.
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4.1 Temporal and Spatial Variations of Trans-cinnamic Acid Figure 5a shows the C r versus time plot of trans-cinnamic acid. A comparison of trans-cinnamic acid diffusing through the intercellular route for young and old age has been shown. It is found that amount of trans-cinnamic acid reaching the receiver compartment after 60 min is 0.136 mg and 0.1361 mg through ICR and 0.05 µg and 0.22 µg through SDR for young and old age, respectively. It can be observed that the drug diffusion rate is relatively higher for old age as compared to a young age for both ICR (Fig. 5a) and SDR (Fig. 5d), but there is a drastic increment in the rate of permeation through SDR upon aging. Upon aging, an increment of 0.05% and 376%, respectively, in ICR and SDR is observed for the trans-cinnamic acid. Since the change in thickness of skin layers upon aging is insignificant, this leads to a very small change in drug concentration through ICR. On the contrary, upon aging, the pore size of the sweat duct changes. The diameter of the upper coiled duct changes from 20 to 60 µm upon aging (Table 3). Due to the increase in pore size, the hindrance offered to drug diffusion reduces; hence, the amount of drug diffusion increases through SDR in old age. However, it should be noted here that the amount of drug diffusing through the intercellular route is approximately 1000 times more than the sweat duct route. This is because the sweat duct has a very small pore size and a tortuous pathway that resists drug permeation. Figure 5 shows the spatial variation of trans-cinnamic acid diffusing through ICR (Fig. 5b, c) and SDR (Fig. 5e, f) for young (Fig. 5b, e) and old age (Fig. 5c, f). The spatial variation represents the variation of non-dimensional drug concentration (C + ) along the non-dimensional length (X + ). It is observed that a sharp drop in drug
Fig. 5 Temporal (a and d) and spatial variations (b, c, e, f) of trans-cinnamic acid diffusion through ICR (a–c) and SDR (d–f). (b, e) and (c, f) represent spatial variation for young and old age, respectively
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concentration exists in layer 1 of ICR (i.e., stratum corneum) irrespective of age (Fig. 5b, c). This is because of the higher partition coefficient of trans-cinnamic acid for layer 1 compared to that for other layers of ICR. Note that the partition coefficient of a drug at a particular interface describes the ratio of solute at either side of the interface at equilibrium. For example, the partition coefficients of trans-cinnamic acid in stratum corneum, viable epidermis, and dermis are 7.26, 2.83, and 3.47, respectively. It means the drug received in the stratum corneum is 7.26 times lower than that available in the donor compartment and likewise. The spatial variation curves have been plotted at t = 0.05, 5, and 60 min in each case. In the inset of Figs. 5b, c, a jump in drug concentration can be observed. In Fig. 5b, at the interface of sc–ve, the drug concentration jumps from 0.135 to 0.048 mg while from 0.048 to 0.014 mg at the interface of ve–de (at t = 60 min). This is because of the partition coefficient’s role at each layer’s interface. However, such shifting in drug concentration is not observed in the case of SDR (Figs. 5e, f), as there is no such interface present in the sweat duct. Hence, there is no role of partition coefficient in case of SDR. The effective diffusion coefficient describes the drug diffusion through the sweat duct route. So, as porosity increases upon aging (Table 3), the effective diffusion coefficient increases. Because of this, the slope of the C + versus X + curve in Fig. 5f is more than that can be seen in Fig. 5e.
4.2 Temporal and Spatial Variations of Caffeine A similar analysis is performed for studying the passive permeation of caffeine through ICR and SDR as well. The results for temporal and spatial variations of caffeine diffusing through the ICR and SDR for young and old age are obtained. The trend in variation is almost similar to that in the case of trans-cinnamic acid. However, there is a certain difference in the amount and rate of drug permeated. It is observed that after 60 min, the amount of caffeine reaching the receiver compartment is 1.061 mg and 1.0624 mg, respectively, for young and old age, while the amount of trans-cinnamic acid reaching the receiver compartment is 0.1360 mg and 0.1361 mg, respectively, for young and old age. Hence, the amount of caffeine that has diffused is more than that of trans-cinnamic acid because the diffusion coefficient of caffeine is slightly higher than that of trans-cinnamic acid (Table 1). Also, the partition coefficient of caffeine is relatively lower; hence, more caffeine is available for diffusion. However, for the sweat duct route, the amount of caffeine reaching the receiver compartment after 60 min is 0.0405 µg and 0.195 µg for young and old age, respectively. While the amount of trans-cinnamic acid reaching the receiver compartment is 0.0462 µg and 0.22 µg for young and old age, respectively. This is because of the reason that the sweat duct route is mainly aqueous, and the diffusion coefficient of trans-cinnamic acid in water (Dw = 9.9 × 10–6 cm2 /s) is slightly more than that of caffeine in water (Dw = 9 × 10–6 cm2 /s) (Table 1). The trend in the spatial variation
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of caffeine is similar to trans-cinnamic acid. But, due to variation in the partition coefficient, the jump in drug concentration is different.
5 Conclusions To analyze the drug diffusion for human skin through the intercellular and sweat duct route, a mathematical model based on Fick’s law has been adopted. The model has been solved using explicit FDM. The present study determines which pathway is more effective for drug transport. This study also helps to understand the effect of aging on drug permeation. For instance, the trans-cinnamic acid and caffeine permeation increased by 0.05%, 0.12% and, 376%, 381% upon aging for ICR and SDR, respectively. Drug diffusion is found to increase with age with a significant increment through the sweat duct. This can be attributed to an increase in pore size upon aging. It is found that the intercellular route provides a faster route of delivering medicines via human skin as compared to the sweat duct route. For example, if compared to young age, trans-cinnamic acid concentration in the receiver compartment after 60 min is found to be 0.14 mg and 0.05 µg for ICR and SDR, respectively, while that for caffeine is found to be 1.06 mg and 0.04 µg for ICR and SDR, respectively. This is due to the lower value of the effective diffusion coefficient of the drug while diffusing through the sweat duct route. Therefore, it can be concluded that the intercellular route is a more effective route for delivering a drug through human skin. The desirable parameter for a drug to be delivered through the skin is a high diffusion coefficient and a low partition coefficient. Acknowledgements N.D.P. gratefully acknowledges the financial support under startup-researchgrant (SRG) scheme from SERB-DST, Govt. of India, New Delhi, by grant number SRG/2020/ 001947, for the computational workstation used in the present work’s simulation.
NOMENCLATURE D C C+ K Vd ε S
Diffusion coefficient [cm2 /s] Drug concentration [mg] Non-dimensional drug concentration Drug partition coefficient Volume of donor compartment (cm3 ) Porosity Permeation area (cm2 )
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References 1. Dhote V, Bhatnagar P, Mishra PK, Mahajan SC, Mishra DK (2012) Iontophoresis: a potential emergence of a transdermal drug delivery system. Sci Pharm 80(1):1–28. https://doi.org/10. 3797/scipharm.1108-20 2. Cygan RT, Ho CK, Weiss CJ (2002) Linking the geosciences to emerging bio-engineering technologies 3. Rabost-Garcia G, Farré-Lladós J, Casals-Terré J (2021) Recent impact of microfluidics on skin models for perspiration simulation. Membranes (Basel) 11(2):1–13. https://doi.org/10.3390/ membranes11020150 4. Andrews SN, Jeong E, Prausnitz MR (2013) Transdermal delivery of molecules is limited by full epidermis, not just stratum corneum. Pharm Res 30(4):1099–1109. https://doi.org/10.1007/ s11095-012-0946-7 5. Michaels AS, Chandrasekaran SK, Shaw JE (1975) Drug permeation through human skin: theory and invitro experimental measurement. AIChE J 21(5):985–996. https://doi.org/10. 1002/aic.690210522 6. Alkilani AZ, McCrudden MTC, Donnelly RF (2015) Transdermal drug delivery: Innovative pharmaceutical developments based on disruption of the barrier properties of the stratum corneum. Pharmaceutics 7(4):438–470. https://doi.org/10.3390/pharmaceutics7040438 7. Trommer H, Neubert RHH (2006) Overcoming the stratum corneum: the modulation of skin penetration. a review. Skin Pharmacol Physiol 19(2):106–121. https://doi.org/10.1159/000 091978 8. Grassi M, Colombo I (1999) Mathematical modelling of drug permeation through a swollen membrane. J Control Release 59(3):343–359. https://doi.org/10.1016/S0168-3659(98)00198-9 9. Coceani N, Colombo I, Grassi M (2003) Acyclovir permeation through rat skin: mathematical modelling and in vitro experiments. Int J Pharm 254(2):197–210. https://doi.org/10.1016/ S0378-5173(03)00028-0 10. Sugibayashi K, Todo H, Oshizaka T, Owada Y (2010) Mathematical model to predict skin concentration of drugs: toward utilization of silicone membrane to predict skin concentration of drugs as an animal testing alternative. Pharm Res 27(1):134–142. https://doi.org/10.1007/ s11095-009-9987-y 11. George K, Kubota K, Twizell EH (2004) A two-dimensional mathematical model of percutaneous drug absorption. Biomed Eng Online 3:1–13. https://doi.org/10.1186/1475-925X3-18 12. Pontiki E, Hadjipavlou-Litina D, Litinas K, Geromichalos G (2014) Novel cinnamic acid derivatives as antioxidant and anticancer agents: design, synthesis and modeling studies. Molecules 19(7):9655–9674. https://doi.org/10.3390/molecules19079655 13. Cappelletti S, Daria P, Sani G, Aromatario M (2015) Send orders for reprints to reprints@ benthamscience.ae Caffeine: cognitive and physical performance enhancer or psychoactive drug? 14. Shackelford CD, Moore SM (2013) Fickian diffusion of radionuclides for engineered containment barriers: diffusion coefficients, porosities, and complicating issues. Eng Geol 152(1):133–147. https://doi.org/10.1016/j.enggeo.2012.10.014 15. Schlosser PM, Asgharian BA, Medinsky M (2010) Inhalation exposure and absorption of toxicants 16. Ellison CA et al (2020) Partition coefficient and diffusion coefficient determinations of 50 compounds in human intact skin, isolated skin layers and isolated stratum corneum lipids. Toxicol In Vitro 69:104990. https://doi.org/10.1016/j.tiv.2020.104990 17. di Cagno MP et al (2018) Experimental determination of drug diffusion coefficients in unstirred aqueous environments by temporally resolved concentration measurements. Mol Pharm 15(4):1488–1494. https://doi.org/10.1021/acs.molpharmaceut.7b01053 18. Delgado JMPQ (2007) Molecular diffusion coefficients of organic compounds in water at different temperatures. J Phase Equilib Diffus 28(5):427–432. https://doi.org/10.1007/s11669007-9160-4
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19. Ezure T, Amano S, Matsuzaki K (2021) Aging-related shift of eccrine sweat glands toward the skin surface due to tangling and rotation of the secretory ducts revealed by digital 3D skin reconstruction. Skin Res Technol 27(4):569–575. https://doi.org/10.1111/srt.12985 20. Flament F et al (2015) Facial skin pores: a multiethnic study. Clin Cosmet Investig Dermatol 8:85–93. https://doi.org/10.2147/CCID.S74401 21. Sonner Z et al (2015) The microfluidics of the eccrine sweat gland, including biomarker partitioning, transport, and biosensing implications. Biomicrofluidics 9(3). https://doi.org/10.1063/ 1.4921039
Computational Investigation on the Empirical Relation of Murray’s Law Mudrika Singhal and Raghvendra Gupta
Abstract The human circulatory system is complex, and about one-sixth of the resting metabolic rate is consumed for keeping the blood flowing through the system. It consists of geometrical complications such as tapering, branching, channel bifurcations and curvatures. For minimizing the biological work accounting for the blood flow at the bifurcations, Murray’s law was formulated by Cecil D. Murray. The work done in overcoming the viscous drag and the maintenance of the vessel is considered for deriving the empirical relation of Murray’s law. The empirical relation states that the volumetric flow rate is proportional to the cube of the vessel radius. Here, in this study, we have computationally investigated the above stated empirical relation. Keywords Bifurcations · Flow rate · Friction factor · Murray’s law · Pressure
1 Introduction Most of the biological networks such as arteries, lungs and trees have a number of bifurcations. Literature studies have shown that in most of such systems, a cubic law is followed, according to which the cube of the radius of the mother tube is equal to the summation of the cube of the radii of all the daughter tubes. Mathematically, it can be expressed by Eq. (1), 3 Rm =
∑
Rd3i
(1)
i=1,n
M. Singhal (B) · R. Gupta Department of Chemical Engineering, IIT Guwahati, Guwahati 781039, India e-mail: [email protected] R. Gupta Jyoti and Bhupat Mehta School of Health Sciences and Technology, IIT Guwahati, Guwahati 781039, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_69
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Here, Rm is the radius of the mother tube and Rd is the radius of the daughter tube. To explain this, Murray [1] proposed an explanation based on cost minimisation principle.
2 Literature Review and Objective Murray assumed that two major energy terms contribute to the cost: (1) the energy required to drive the flow which is equal to Wf and (2) the energy required to maintain the fluid and the involved vessel tissues (Wm ) [2]. The two major energy requirement terms have a contrary relationship with variation in radius, and thus, it becomes important to have an optimal radius so that the energy requirement is minimized. Assuming that the flow follows Hagen–Poiseuille’s law, i.e. the flow is steady, laminar, Newtonian and fully developed in nature, the power required for overcoming the viscous drag (Wf ) experienced by the fluid flow can be given by Eq. (2). Wf = QΔP
(2)
Here, Q is the volumetric flow rate of blood in the vessel and ΔP is the pressure drop. For Poiseuille flow, ΔP can be given by Eq. (3). ΔP =
8μQ L π R4
(3)
Here, μ is the dynamic viscosity, L is the length of the vessel, and R is the radius of the vessel. It may be noted that friction factor for Hagen–Poiseuille flow is given by Eq. (4). f =
16 Re
(4)
ρu D μ
(5)
Here, Re is the Reynolds number. Re =
Here, ρ is the density of the fluid, u is the flow velocity, D is the diameter of the pipe, and μ is the dynamic viscosity. Murray suggested that the maintenance cost is proportional to the volume of the blood and can be given by the Eq. (6). Wm = kπ R 2 L
(6)
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Here, k is a coefficient accounting for the chemical cost required for keeping the blood constituents fresh and functional. Hence, the total power requirement can be given by the sum of both the terms Wf and Wm . Wtotal = Wf + Wm Wtotal =
8μQ 2 L + kπ R 2 L π R4
(7)
(8)
For a constant volumetric flow rate in the vessel of a given length, Wtotal can be expressed as; Wtotal =
a Q2 + b R2 R4
(9)
Here, a = 8μL and b = kπ L π On minimizing the above expression with respect to the radius, we obtain a relation, stating that Q ∝ R 3 . Hence, Murray’s law states that the volumetric flow rate of blood through the channel∑is proportional to the cube of the of the channel. ∑radius 3 Q i, j which results in Rm = Ri,3 j . However, flow At the bifurcation, Q m = separation occurs at the bifurcation, and the length of the daughter tubes can be very short resulting in non-validity of Hagen–Poiseuille relation. Further, a different relation will be obtained when the flow is turbulent. In this work, we computationally model laminar flow at a bifurcation and obtain a relation between ΔP and Q. In the non-dimensional form, this relation between ΔP and Q can be represented as f versus Re. Further, we have also modelled the turbulent flow regime in a similar manner, and the friction factor for turbulent flow regime is given by Blasius equation.
3 Computational Fluid Dynamics (CFD) Methodology 3.1 Computational Domain The geometrical domain consists of a bifurcation, where the mother tube bifurcates into the daughter tubes. The diameter of the mother tube is taken as 3.5 mm, as seen in Fig. 1, which is in accordance with the diameter of a coronary artery, as in the literature [3]. The length of both the mother and the daughter tubes is taken as 30 mm, which is sufficient for the flow to become fully developed. The bifurcation angle is ◦ assumed to be symmetric and α = β = 30 . The daughter tubes are assumed to be equal in diameter, and by application of the continuity equation at the bifurcation, the diameter of the daughter tubes is calculated as 2.77 mm each, following Murray’s law.
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Fig. 1 Bifurcation geometry following dimensions according to Murray’s law
Following these dimensions, a 3D geometry was created in Design Modeler, ANSYS Fluent, the commercial CFD software. Meshing with tetrahedron elements was carried out, and an inflation boundary layer consisting of 5 prism layers near the wall was generated to account for the gradients near the wall.
3.2 Governing Equations and Boundary Conditions The blood is assumed to be Newtonian and incompressible with the value of density equal to 1060 kg/m3 and the dynamic viscosity as 0.003 Pa s. The mass and momentum conservation equations can be written as follows; Mass conservation equation: ∇ ·V=0
(10)
Momentum conservation equation: ( ρ
DV Dt
) = −∇ p + (μ + μT )∇ 2 V
(11)
Steady state simulations are carried out with a parabolic velocity profile applied at the inlet and zero-gauge pressure at the outlet. The walls are considered to be rigid. Simulations are carried out for both the laminar and turbulent regimes. k-epsilon model is used for turbulence modelling and the governing equations are as follows: ∂(ρk) ∂(ρku i ) ∂ + = Sk ∂x ∂ xi ∂x j
[( ) ] μt ∂k μ+ + Gk σk ∂ x j
+ G b − ρε − YM + Sk ∂(ρε) ∂(ρεu i ) ∂ = + ∂x ∂ xi ∂x j
[( ) ] μt ∂ε ε μ+ + C1ε σε ∂ x j k
(12)
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Fig. 2 Mesh independence study: comparison of x-velocity on a vertical line at the bifurcation for three grid densities: 0.135 million, 0.245 million and 0.4 million
+ (G k + C3ε G b ) − C2ε ρ
ε2 + Sε k
(13)
Here, G k is the generation of turbulence kinetic energy due to mean velocity gradients and G b is the generation of turbulence kinetic energy due to buoyancy. YM represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. C1ε , C2ε and C3ε are constants, σk and σε are the turbulent Prandtl numbers for k and ε, respectively. Sk and Sε are user-defined source terms.
3.3 Mesh Independence Study The simulation results must be independent of the grid size to have accurate results. Hence, simulations were carried out for three grid sizes, 0.135, 0.245 and 0.4 million elements. The variation of the x-velocity along a vertical line on the wall with y is studied to choose the size of the grid. From the Fig. 2, it can be observed that the velocity profile does not vary much for 0.245 million and 0.4 million elements, and hence, we have chosen the grid density as 0.245 million elements for our simulation studies.
4 Results and Discussion Laminar Regime For the laminar flow regime, the volumetric flow rate is varied from 0.06 LPM to 1 LPM, and the inlet boundary condition user-defined function (UDF) is altered
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in accordance with it. Laminar model is used, and coupled scheme is used for the simulations with a solution convergence criterion of 10–5 for all the variables. The pressure drop in a pipe of length L can be calculated by the Darcy–Weisbach equation, given by: ΔP =
f Lρu 2 2D
(14)
Substituting the equation for friction factor (Eq. 4) in the expression for pressure drop and after rearranging the terms, we obtain that; ΔP ∝ Q
(15)
Q is the volumetric flow rate of the blood. Simulations were carried out for varying flow rates, and the pressure drop was calculated in the daughter tube from the simulation data by calculating area-weighted average pressure at two planes shown in Fig. 1. Pressure drop in the daughter tube (ΔP) can be given by the following expression, ΔP = Pbifurcation − Poutlet
(16)
By taking the above value of pressure drop, the friction factor is calculated for each flow rate from the Darcy–Weisbach equation. A curve is plotted for friction factor versus Reynolds number, and curve fitting is carried out to obtain a relation between the two parameters, as seen in Fig. 3. From the above-obtained curve, the following relation is obtained between friction factor and Reynolds number: f = 82.721Re−1.036
(17)
Substituting this relation into the Darcy–Weisbach equation and after rearranging the terms, we obtain that: Fig. 3 Relation between friction factor and Reynolds number for laminar flow regime in daughter tube
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Fig. 4 Isosurface of zero x-component of velocity superimposed with the velocity vector
ΔP ∝ Q 0.964
(18)
Thus, the variation of the pressure drop with the volumetric flow rate for a given diameter of the vessel can be given by the above expression, and it is slightly different from the empirical relation of Murray’s law. Further, when the above-obtained relation is substituted into the expression for cost function (eq. no. 8) and the equation is minimized, it is obtained that: Q ∝ R 3.05
(19)
From Fig. 4, it can be observed that recirculations form near the bifurcation region. Though flow separation happens near the bifurcation, the CFD results reveal that Murray’s law is not affected greatly by the formation of these recirculations. As the volumetric flow rate is proportional to the cube of the radius and the same relation holds true for the whole system, it leads to the conclusion that same velocity profile will characterize each vessel. Further, the velocity gradient and consequently the wall shear stress shall remain same throughout the system. To examine this, wall shear stress has been calculated, as seen in Fig. 5. It is observed from Fig. 5, that there is a sharp decrease in the value of wall shear stress in the regions near to bifurcations, and further, it becomes constant. These regions are characterized by flow separation which can be seen from Fig. 4. Hence, it becomes important to examine the empirical relation in this region. To fulfil this, the relation has been examined by calculating a pressure drop for half of the length of the daughter tube. It might be beneficial in those cases where the length of the daughter tubes is short. From Fig. 6, a relation between friction factor and Re is obtained, f = 201.64Re−1.126
(20)
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Fig. 5 Variation of wall shear stress along a line on the inner wall of the daughter tube Fig. 6 Relation between friction factor and Reynolds number for laminar flow regime in daughter tube for half of the length
Further, on minimizing this relation for the cost minimization principle, we obtain following relation between volumetric flow rate and radius: Q ∝ R 3.13
(21)
Hence, it can be said that the empirical relation of Murray’s law will be slightly changed when the daughter tubes are short in length, such that the recirculation zones persist towards their end. Turbulent Regime Further, we extend our studies for turbulent flow regime. For the turbulent flow regime, the volumetric flow rate is varied from 1.1 to 2.5 LPM, and the inlet boundary condition user-defined function (UDF) is altered in accordance with it. k-epsilon model and coupled scheme are used for the simulations with a solution convergence criterion of 10–5 for all the variables. For some cases, as the flow bifurcates, it laminarizes in the daughter tubes; hence, for our analysis, we have chosen flow rates varying from 1.7 to 2.5 LPM.
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Friction factor
Fig. 7 Relation between friction factor and Reynolds number for turbulent flow regime in daughter tube
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0.8
y = 214.62x-0.69
0.6 0.4 0.2 0 2300
2500
2700
2900
Re
3100
3300
3500
We know, friction factor ( f ) for turbulent flow in a circular pipe can be given by Blasius equation: f =
0.316 Re0.25
(22)
Here, Re is the Reynolds number, as given by Eq. 5. If the flow is assumed to be turbulent and we derive Murray’s law with this assumption, following relation is obtained for cost minimization: Q ∝ R 2.45
(23)
Hence, it can be said that Eq. 23 states the empirical relation of the cost minimization principle. Analysis similar to laminar regime is performed, and the relation between ΔP and Q is plotted in the non-dimensional form which consists of f v/s Re, as shown in Fig. 7. Minimizing the relation obtained from the trend in Fig. 7, following expression is obtained, Q ∝ R 2.73
(24)
Hence, it can be said that the relation obtained from CFD simulations varies from the empirical relation.
5 Conclusions The flow phenomenon at bifurcations is governed by the Murray’s law. According to it, for the minimization of the energy requirement, the vessels are arranged in such a way that the volumetric flow rate of the blood is proportional to the cube of the radius of the vessel. Murray’s law holds good for the assumption of laminar,
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steady and fully developed flow. The regions near to bifurcation are characterized by recirculation formation which can lead to non-validity of the Hagen–Poiseuille relation. On computational investigation, it is revealed that Murray’s empirical relation is affected by the recirculation formation when modelled for a laminar flow, if the daughter tubes are short in length. For the daughter tubes which are long enough for the flow to become flow developed in nature, the empirical relation remains unaffected. Similarly, if the relation is derived for a turbulent flow and is investigated computationally, the relation is affected.
Nomenclature f L ΔP Q Re V ρ μ
Friction factor Length of the vessel [m] Pressure drop [Pa] Volumetric flow rate [m3 /s] Reynolds number Velocity [m/s] Density of blood [kg/m3 ] Dynamic density of blood [Pa s]
References 1. Sherman TF (1981) On connecting large vessels to small: the meaning of Murray’s law. J Gen Physiol 78(4):431–453 2. Stephenson D, Patronis A, Holland DM, Lockerby DA (2015) Generalizing Murray’s law: an optimization principle for fluidic networks of arbitrary shape and scale. J Appl Physiol 118 3. Taylor CA, Fonte TA, Min JK (2013) Computational fluid dynamics applied to cardiac computed tomography for non-invasive quantification of fractional flow reserve. J Am Coll Cardiol 61(22):2233–2241
Investigation of Impulse Jet Dispersion Mechanism of Needle-Free Drug Delivery Device Priyanka Hankare, Sanjeev Manjhi, and Viren Menezes
Abstract Impulse jet injector creates a high-speed liquid microjet through a small orifice that is used for drug injection into the skin. A requisite amount of dosage is administered at a disseminable depth into the skin sample. One of the key aspects of dispersion investigations is to study the flow mechanics of jet propagation inside the target. The injection of the drug into soft materials like polyacrylamide gel slabs is examined in order to understand the process for drug dispersion and uptake. Highspeed imaging is used to capture the rapid dynamics of the fluorescent liquid injected into the ballistic polyacrylamide gel slab. A 2D numerical simulation is performed to confirm the jet impact pressure and velocity, and it is found to be in good agreement with experimental results. Understanding physics of jet penetration through polyacrylamide gels allows one to deduce jet injection into skin targets. The findings demonstrate that drug distribution into the target was effective. Keywords Needle-free · Shockwave · Jet dispersion · Polyacrylamide gel · Finite volume method
1 Introduction Needle-free drug delivery devices are in the limelight because of their ability to inject the drug subcutaneously, intradermally, or intramuscularly without actually piercing the skin of the target. This phenomenon of the jet injection lasts for microseconds, making the procedure minimally invasive and painless [1]. A fine stream of liquid microjet is generated that promptly deposits the drug up to the dermal level. How the drug is transferred into the skin sample is one of the research topics that can help us better understand the mechanism of drug delivery [2]. Given the widespread popularity of needle-free drug delivery, it is intriguing to learn about the interaction of the high-speed jet with the skin and the peculiarities P. Hankare (B) · S. Manjhi · V. Menezes Department of Aerospace Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_70
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of the jet penetration and dispersion into the target skin [3]. Here, synthetic skin samples made of homogenous, isotropic materials that mimic the viscoelastic properties of skin are tested. Since skin is opaque, polyacrylamide gels are utilized as such target materials for clear visualization of the liquid jet flow inside the target. Penetration occurs in a variety of shapes and sizes within the target sample. Physics of jet penetration is studied as the liquid jet advances into the target. The jet in the skin infiltrates in the stages; first, the skin is breached with the impact of the jet, and later the jet propels into the skin through the breached hole and attains maximum depth that disperses into the skin in a controlled manner. The imaging of the procedure included varied elliptical or spherical patterns of drug dispersion depending on various injection frameworks [4]. The delivery of the jet is impulsive and quickly gets dispersed in the skin. This dispersion pattern states the reach of the jet and its interaction with the skin sample. The impact pressure and jet velocity that controls the depth of jet penetration are also discussed in this paper.
2 Literature Review and Objective Schramm-Baxter [2] used a commercial jet injector to inject the liquid jet onto polyacrylamide gels and anticipated that jet penetration can be split into three discrete events: erosion, stagnation, and dispersion. With increasing Young’s modulus of skin sample, the jet penetration depth and erosion rate decreased. Mohizin [3] stated that the injected fluid interacts with numerous skin tissue layers and interfaces, which suggests that the corresponding injection profile is reliant upon their mechanical properties. Zeng [5] aims to present the penetration characteristics of larger volume injections, including dynamic properties, dispersion patterns, and percent delivery. This study uses high-speed photography and impact experiments of the liquid jet into porcine skin tissues to capture the dynamic properties of the injection. Understanding the mechanism of the drug uptake necessarily requires research into the pattern of drug dispersion into a target skin. The goal of this research is to understand the links that exist between the jet dispersion profile of fluid and injection parameters which will aid in optimization of needle-free drug delivery. To aid this experimental study and gain a deeper understanding of flow mechanics, numerical simulations of impact pressure and jet velocity are also performed.
3 Materials and Methods The shockwave principle is used to design and build the jet injector for the present study [1]. The needle in a typical syringe is replaced by a microjet. Given that polyacrylamide gels are translucent, they are chosen as skin targets.
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Fig. 1 Schematic of microjet injector setup
3.1 Jet Injector Device The device is built as a micro-shock tube with high-pressure driver and low-pressure driven sections separated by a thin primary diaphragm. Figure 1 shows the dimensions of the syringe. The driver section is filled with nitrogen gas until the primary diaphragm made up of Mylar bursts. A miniature drug compartment is followed by driven section and is separated by a thin, flexible membrane made of elastic rubber. The rubber membrane reflects the incident shock wave from the driven section, and the high pressure left behind that results from the shock reflection is transferred to the liquid in the drug compartment. This causes the generation of a micro-liquid jet that serves as a needle for penetration into the target. In this case, liquid medicine is water mixed with rhodamine dye for fluorescence. The drug compartment can hold liquid up to 1.5 ml. A schematic of the injector is presented in Fig. 1 [6].
3.2 Polyacrylamide Gel Ballistic acrylamide-based models are among the best for simulating human soft tissues [7]. Ballistic polyacrylamide gel models are made in-house with a 10% weight ratio. A transparent acrylamide-water solution was produced by continuously stirring the mixture while it was heated in the hot oil bath at 60–70 °C. This mixture was later put into a transparent mold and cooled for 2 h in a freezer at 10 °C. The bloom value of the 10% w/w ballistic polyacrylamide gel model is approximately 220–240, and it gets lower over time. The dispersion pattern is visualized by impacting the jet into 20 mm thick transparent and homogeneous slab of the polyacrylamide gel. Mechanical properties of slab vary by changing % w/w of ballistic acrylamide.
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Fig. 2 Numerical model (a) with boundary conditions (b) and (c) with FVM structured mesh
3.3 Numerical Simulation for Microjet Numerical simulations are performed to determine the velocity and pressure of the impacting jet using ANSYS workbench 16.2. A 2D fluid domain with jet inlet is created with a structured grid as shown in Fig. 2. To keep the error inbound during simulations, the mesh size of 200,000 is set as the optimal outcome of the gridindependent test. Finite volume simulations are conducted for a density-based absolute transient planer mode. Using the multiphase-Eulerian mode, the fluid is taken as an air–water mixture. In this case, the k-omega SST (air–water) mixture model is activated with pressure-inlet, pressure-outlet, and adiabatic (Q = 0) wall with no-slip boundary (u = v = 0) conditions. The inlet pressure is set to 40 bar (diaphragm bursting pressure), and the exit pressure is kept constant at atmospheric pressure while maintaining a temperature uniformity of 300 K. The fluid (air) is assumed to be ideal. The problem is initiated using hybrid initialization with convergence criteria of 10e−06 and step size of 7e−06 and a time step count of 10,000.
4 Results and Discussion 4.1 Impact Jet Pressure Measurement For the desired penetration into the target body for drug delivery, the impact pressure is one of the crucial parameters. The impact pressure of the jet is measured by using a µ-pressure sensor (FlexiForce-A 301, Tekscan, MA, USA). This sensor is capable to capture impact force in a time span of 5 µs and pressure range up to 5 ± 2 bar. The sensitivity of the sensor is 390.5 mV/N with a sensing diameter of 0.345'' . The circuit assembly is as shown in Fig. 3. The sensor measures the impact pressure of 2.5 ± 0.80 bar as shown in Fig. 4.
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Fig. 3 Impact pressure measurement assembly
Fig. 4 Jet impact pressure measured with pressure sensor
The jet velocity is determined by the flow dynamics that are recorded in conjugate images of the jet and the corresponding time required for the jet to traverse, which resulted a value of 72–80 m/s.
4.2 Numerical Simulation The simulation results of the microjet impacting the target body are discussed in this section. Post-processing yields impact pressure and jet velocity. Figure 5a depicts the velocity and contour velocity vectors (c). A close match between the measured jet velocity of 76.2 m/s (Fig. 5b) and the experimental value of 72–80 m/s was found. As illustrated in the velocity contour plot, a stagnation point is formed at the impact region, where the velocity of the fluid becomes zero (Fig. 5). The pressure will be highest at the impact location due to this flow phenomenon. The impact pressure was calculated to be 2.77 bar (Fig. 6), which is close to the experimental value of 2.5 bar. Figure 6 depicts the pressure contour plot. This impact
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Fig. 5 a Velocity contour b velocity distribution along the flow direction
Fig. 6 a Pressure contour b pressure distribution along the flow direction
pressure is adequate to allow the drug to enter the target body. This simulation will also aid in the development and analysis of high-impact jet-type devices.
4.3 Jet Dispersion Study in Polyacrylamide Gel Transparency of polyacrylamide gel offers us the opportunity for using high-speed imaging to record the pattern and penetration depth of jet. The shape and size of the dispersion depict how the drug interacts with the liquid jet. Figure 7 illustrates timed images of the jet dispersion into 10% polyacrylamide gel at a jet volume of 0.4 ml. A small hole of circular nature, whose diameter is nearly twice that of the jet, is formed as the jet impacts the target’s surface, channeling the flow inside to the desired depth. As this jet advances into the target, it is seen that the drug appears to be disseminated in an ellipsoid pattern after reaching a particular depth. Until the drug
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Fig. 7 Time-lapsed images of the jet dispersion pattern in polyacrylamide gel slab
is fully utilized, this elliptical lump of the drug continues to move. A 2D dispersion pattern can be seen in Fig. 7. Since the target is homogeneous, the jet enters it by creating a fracture plane on the surface. Flow mechanics were recorded using high-speed photography with the Phantom v 710 (v12.2) from vision research (Wayne, NJ, USA). A liquid jet measuring 125 µm was used to strike the polyacrylamide gels, and a halogen light source was used to illuminate them. The images were taken at a rate of 13,000 frames per second. For the instantaneous behavior of the jet, each image is processed using the ImageJ tool. Drug dispersion is shown to begin right at the beginning, followed by a microchannel that can be seen clearly during high-speed imaging and demonstrates the maximal penetration depth. This jet injection procedure is extremely rapid, lasting only 3.87 ms. Jet then settles at that depth. The flow begins to move quickly, becomes static, and eventually disperses within the skin, and this transition takes place in a few milliseconds. The rate of depth increase is 15.7 m/s, which is nearly five times slower than the jet impact velocity, indicating that the majority of KE energy is used for the breach of the surface.
5 Conclusions The governing parameters for a shock wave needle-free microjet drug delivery are computed numerically and experimentally. A numerical simulation for the experimental parameters is run using the finite volume technique to examine the flow profile, impact pressure, and jet velocity. The outcomes of the experiments and the numerical analysis show good agreement. The average errors between experimental and numerical results for impact pressure and jet velocity are found to be 7% and 3.5%, respectively. High-speed imaging is used to investigate the jet dispersion pattern inside the target. This study showed that, in contrast to syringe drug administration,
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a good spread of the drug is seen, which can result in adequate drug dispersion into the target. The drug dispersion in actual skin samples can be predicted with the aid of this investigation. Acknowledgements This work was financially supported by HDFC-ERGO—IIT Bombay Innovation Lab with WBS number DO/2021-HDRD002-002.
Nomenclature FVM KE
Finite volume method (–) Kinetic energy (–)
References 1. Battula N, Menezes V, Bhalekar S, Bhalekar H, Nejad SM, Hosseini H (2017) Impulse-powered needle-free syringe for vaccine/drug injection. Technol Health Care 25(6):1131–1138 2. Schramm-Baxter J, Katrencik J, Mitragotri S (2004) Jet injection into polyacrylamide gels: investigation of jet injection mechanics. J Biomech 37(8):1181–1188 3. Mohizin A, Kim JK (2022) Dispersion profile of a needle-free jet injection depends on the interfacial property of the medium. Drug Deliv Transl Res 12(2):384–394 4. Grant TM, Stockwell KD, Morrison JB, Mann DD (2015) Effect of injection pressure and fluid volume and density on the jet dispersion pattern of needle-free injection devices. Biosys Eng 138:59–64 5. Zeng D, Wu N, Qian L, Shi H, Kang Y (2020) Experimental investigation on penetration performance of larger volume needle-free injection device. J Mech Sci Technol 34(9):3897–3909 6. Hankare P, Agrawala A, Menezes V (2022) High-speed jet injector for pharmaceutical applications. J Med Devices 16(3):034502 7. D˛abrowska AK, Rotaru GM, Derler S, Spano F, Camenzind M, Annaheim S, Stämpfli R, Schmid M, Rossi RM (2016) Materials used to simulate physical properties of human skin. Skin Res Technol 22:3–14
Analysis of 2D Human Airway in Laminar and Turbulent Flow Model Vivek Kumar Srivastava and Aman Raj Anand
Abstract In this article, an airflow study of a two-dimensional human airway model is performed using computational fluid dynamics (CFD). The human respiratory system is one of the important parts of the human body where CFD is used to understand the physics of flow, diagnosis, prognosis, and treatment of respiratory diseases. The function of the human respiratory system is to deliver oxygen between the atmosphere and the lungs and remove carbon dioxide. Airflow and particle deposition in the human respiratory tract provide important information for clinical purposes (regional ventilation) and inhalation treatment. In this article, a sixth-generation twodimensional airflow characteristic of the human respiratory tract is simulated using commercial CFD software. Three different inspiratory flow rates of 10 L/min (normal respiration), 45 L/min (moderately rapid respiration), and 60 L/min (rapid respiration) are considered to see the effects of flow rates. Velocity and pressure contours are calculated to understand the flow physics of the human respiratory system. Keywords Human airway model · Computational fluid dynamics · Laminar and turbulent flow
1 Introduction Luo and Liu [1] studied inspiratory airflow in a fifth-generation computed tomography model of the human lung. Computer simulations have been run for the Reynolds number in the range 900–2100. It is found that as the Reynolds number increases, airflow in the center increases and is diverted into the left main bronchus. Wang et al. [2] studied airflow patterns in three models of the human airway. V. K. Srivastava (B) School of Advanced Sciences, Vellore Institute of Technology, Amaravati, Andhra Pradesh, India e-mail: [email protected] A. R. Anand Department of Mechanical Engineering, Motihari College of Engineering, Motihari, Bihar, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_71
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The model was built in two parts. The upper part was constructed with a computed tomography model, the lower part with the Weibel lung model. The simulated result shows high shear stress on the wall of the epiglottis and bronchial bifurcations. Kang et al. investigated an effect of geometric variations on the pressure loss of the human lung model [3]. The three-dimensional model is taken from the Weibel lung model. They have achieved a very low pressure drop for different branch angles in the smooth pipe. Srivastav et al. [4] compared the flow physics between the computed tomography model and the simplified model. It was found that the flow physics in the computed tomography model differs significantly from the simplified model. Cartilaginous rings have also been found to have significant effects on the physics of airflow and particle deposition. In another study, they examined the effects of the tumor in a computed tomography model [5]. Huang et al. [6] studied obstructive sleep apnea in the upper airways using the fluid structure interaction (FSI) method. They compared the results obtained with the two methods, namely FSI and CFD [7]. Local aerodynamic characteristics, such as velocity profiles of airflow at various cross sections of the respiratory tract, depend mainly on the structures of the airway segment under consideration and upstream airways [8]. The simulated results show that the FSI method is more realistic and closed than the CFD method. The present work focuses on the two-dimensional investigation of the sixthgeneration model of the human airway. It has been found that two-dimensional studies provide faster and more meaningful results than three-dimensional studies. The doctor needs fast patient results, which can be obtained by two-dimensional imaging. Some respiratory diseases occur mainly in the terminal airways because fine particles can reach the end of the deeper airways. The residence time of the pollutant particles can cause damage to the airway epithelium, inflammation, and also tumor formation travel. The results will be useful to the doctor for a quick review of the patient. It is seen that over the last one decade, the airflow studies through the human airways has been studied widely, both experimentally and via computer simulations. Studies utilizing only the lower airways typically use a laminar profile at the trachea, which is the point of demarcation between the upper and the lower airways [9].
2 Geometry of the Respiratory Tract The 2D reconstruction of the airway model is based on a three-dimensional model that was created with computed tomography data. CT scan is made from the entrance of the trachea to the lower bronchi. A total of 519 slices with a slice spacing of 6.25 × 4 m and a pixel size of 512 × 512 were used for the full reconstruction of the sixth-generation model. After the 3D model was created, it was converted into a 2D model using the image from 3D picture. The six-generation model is shown in Fig. 1.
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Fig. 1 Two-dimensional sixth-generation human respiratory tract
3 Grid Generation The mesh generation of the two-dimensional human airway model was performed using the triangular elements. The triangular element was found to fit curved surfaces with minimal distortion, so it is used in the present study (Fig. 2). Fig. 2 Grid generation of sixth-generation model
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Table 1 Grid independency test Number of elements
Max. velocity (m/s)
Case-1
2569
3.36
Case-2
5914
3.56
Case-3
9328
3.62
Case-4
19,618
3.6
Case-5
40,402
3.6069
Error deviation (%) 5.617978 1.657459 − 0.55556 0.1913
4 Grid Independency Test The network independence test is performed for different cases. A network independence test is performed to verify the computational accuracy of the airway model. The grid is refined by adjustment techniques in the Ansys-Fluent software. First, 2569 items were generated, which were refined three times to 5914, 9328, 19,618, and 40,402 items. The grid stands alone for case-4 at element number 19,618. The generated mesh model is given in Table 1.
5 Governing Equations The airflow was assumed to be steady and incompressible; therefore, continuity equation: ∂u j =0 ∂x j
(1)
∂ ui u j 1 ∂p μ ∂ 2ui =− + ∂x j ρ ∂ xi ρ ∂ xi ∂ x j
(2)
Momentum equation:
μ = viscosity coefficient, ui , uj (i, j = 1, 2, 3) is the velocity component in x, y and z direction. p = pressure, ρ = density of fluid.
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Fig. 3 Velocity a U max = 2.38 m/s, b U max = 2.28 m/s
6 Computational Validation The present validation was done with the results reported by Luo et al. [7]. The velocity contours for the two results is shown in Fig. 3.
7 Results and Discussion The 2D airflow in the human airway is performed for two different flow rates, namely normal and severe breathing conditions. Velocity and pressure contours are calculated in the next section. Velocity Contours Velocity contours were calculated for two flow rates of 10 and 45 L/min, shown in Figs. 4 and 5. The plotted velocity contours show the maximum velocity in the oral cavity due to the sudden airway contraction route. The sudden contraction reduces the cross-sectional areas, further increasing the magnitude of the velocity. The maximum speed corresponding to the flow rates 10 L/min and 45 L/min is 1.2 m/s and 5.51 m/ s, respectively. It is found that most of the flow disturbance occurs at the trachea and bifurcation junctions. Pressure Contours Pressure is another important parameter useful in predicting flow physics in the human respiratory system. Figures 6 and 7 represents pressure contours corresponding to the flow rates 10 L/min and 45 L/min, respectively. Static pressure is found at the maximum closure of the oral cavity entrance and bifurcation junction due to the direct influence of the main airflow on these places. It is well known that particles are transported by air currents. So, when toxic particles travel through the air, they are likely to be deposited at spurs and in the oral cavity. This leads to an increase in high-risk diseases such as oral cancer.
860 Fig. 4 Velocity contours of sixth-generation model (10 L/min)
Fig. 5 Velocity contours of sixth-generation model (45 L/min)
V. K. Srivastava and A. R. Anand
Analysis of 2D Human Airway in Laminar and Turbulent Flow Model Fig. 6 Static pressure contour (10 L/min)
Fig. 7 Static pressure contour (45 L/min)
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Fig. 8 Turbulence kinetic energy (45 L/min)
Turbulence Turbulence is one of the most important aspects that help to analyze the flow in the human respiratory system. Turbulence kinetic energy contour of human respiratory model is shown in Fig. 8. As turbulent flow, type of fluid flow (gas or liquid) in which the fluid is subject to erratic fluctuations or mixing, in contrast to laminar flow, where the fluid moves in smooth trajectories or strata under forced inspiration, the fluid’s velocity at a point is subject to constant changes in magnitude and direction. The maximum turbulent kinetic energy at 45 L/min flow rate is 0.4.
8 Conclusion The CFD simulation is performed on a sixth-generation 2D model. Bifurcation joints have been found to be the most likely site in high breathing conditions [turbulent versus lower breathing conditions (normal)].
References 1. Luo HY, Liu Y (2008) Modeling the bifurcating flow in a CT-scanned human lung airway. J Biomech 41:2681–2688 2. Wang Y, Liu Y, Sun X, Yu S, Gao F (2009) Numerical analysis of respiratory flow patterns within human upper airway. Acta Mech Sin 25:737–746
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3. Kang M-Y, Hwang J, Lee J-W (2011) Effect of geometric variations on pressure loss for a model bifurcation of the human lung airway. J Biomech 44:1196–1199 4. Srivastav VK, Paul AR, Jain A (2013) Effects of cartilaginous rings on airflow and particle transport through simplified and realistic models of human upper respiratory tracts. Acta Mech Sin 29(6):883–892 5. Srivastav VK, Kumar A, Shukla SK, Paul AR, Bhatt AD, Jain A (2014) Airflow and aerosol-drug delivery in a CT scan based human respiratory tract with tumor using CFD. J Appl Fluid Mech 7(2):345–356 6. Huang R, Li X, Rong Q (2013) Control mechanism for the upper airway collapse in patients with obstructive sleep apnea syndrome: a finite element study. Sci China Life Sci 56(4):366–372 7. Luo XY, Hinton JS, Liew TT et al (2004) LES modelling of flow in a simple airway model. Med Eng Phys 26:403–413 8. Wang C (2005) Airflow in the respiratory system. In: Interface science and technology, vol 5. Elsevier (Chapter 3) 9. Azarnoosh J, Sreenivas K, Arabshahi A (2016) Computational fluid dynamics simulation of the airflow through the human respiratory tract, vol 6, pp 688–692
Effects of Stenosis Profile on Hemodynamic and Mass Transport in Axisymmetric Geometries: A Numerical Study Ankani Sunil Varma and K. Arul Prakash
Abstract In this study, irregular stenosis profile is considered, with its throat area being varied into three different configurations. Ideal geometries are viable for numerically analyzing hemodynamics under various stenosis severities. Numerical work has been carried out on these stenosis profiles under pulsatile flow conditions using open-source software OpenFOAM. The irregular stenosis profile highly influences wall shear stress and its variations. Significant variations in the amplitude of the WSS profile are observed. These stenosis profiles are liable for higher chances of rupture. Mild stenosis further promotes the aggregation of plaque deposits, which the magnitude of WSS fluctuations can predict. The severity of the stenosis has strongly affected mass transfer near the arterial wall by dominating concentration transport by convection into the lumen region. Convection dominated region is strongly limited to upstream of throat section, predominantly for severe stenosis configurations. Hence, actual representation of the stenosis profile with surface irregularity elucidates hemodynamic and mass transfer phenomenon for a better understanding of diseased condition. Keywords Irregular stenosis · Hemodynamics · Mass transfer
1 Introduction Atherosclerosis is a progressive disease formed by the aggregation of plaque deposits inside arterial walls. Severe occlusion of arteries leads to inadequate supply of blood, vital elements such as nutrients, and oxygen to the organs. Ideal stenosis geometries are viable for numerically analyzing hemodynamics and for studying severity of the occlusion by considering different variations. However, very few research has been carried out on effects of stenosis profile on hemodynamics and mass transfer over the arterial wall. Ideal geometry emphasizes to study effects of various stenosis A. S. Varma (B) · K. A. Prakash Department of Applied Mechanics, IIT Madras, Chennai 600036, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_72
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profiles on hemodynamics. In reality, accumulation of plaque on the arterial wall out-turns shape of the lesion surface in the lumen region with several irregularities. Significance of irregular stenosis profile is required for realistic modeling of diseased arteries and its effects on hemodynamics. Arteries are the blood vessels which carry blood from the heart to various organs for their functioning and are also responsible for transporting vital elements such as oxygen and nutrients. Inner lining of arteries is deposited by plaque which progressively develops and occludes the arteries [1]. Flow dynamics of blood analogous fluid within an axisymmetric stenosed artery under steady and pulsatile conditions are numerically investigated by Chauhan and Sasmal [2]. For accurate and realistic modeling of blood, a multi-mode sPTT (simplified Phan–Thein–Tanner) model which accounts for both the shearthinning and viscoelastic rheological properties of blood is considered and observed that fluid flow becomes more chaotic than Newtonian fluid [2]. Sakthivel et al. [3] studied effects of surface irregularities for a mild stenosis configuration using offlattice Boltzmann solver. Parameters such as amplitude and frequency factors of the irregularity profile are varied, and their significance on hemodynamics is analyzed. Shear thinning effect of blood has reduced magnitude of wall shear stress than that of Newtonian [3]. The significance of mass transport in the progression of atherosclerosis is investigated numerically by Kaazempur et al. [4] on two different stenosis models. They reported that asymmetric stenosis representation is more realistic than compared with those of axisymmetric for analyzing flow, mass transfer patterns, and also quantified well [4]. Gerhard et al. [5] conducted simulation by considering concentration distribution which describes transport of oxygen in an axisymmetric stenosis geometry and focused on how it gets influenced by wall shear stress and recirculation zone. Solute flux at the arterial wall was modeled by shear-dependent permeability and with constant wall permeability. They found that vascular permeability and concentration boundary layer are strongly affected by the shear-dependent permeability and wall flux got strongly affected [5]. Hemodynamics and mass transport phenomenon on different patient-specific deformed aorta are investigated and concluded that for analyzing vascular diseases parameters such as wall shear stress, luminal surface concentration, and oxygen flux are inclusively important by Jie et al. [6]. Modeling of arterial wall transport and its aspects such as governing equations and boundary conditions are reviewed well by Mehrzad et al. [7]. Challenges are to really prescribe the arterial anatomy and transport process which can actually represent arterial wall tissue. Major models described are wall-free, homogenous-wall, and multi-layer which have been analyzed and propounded multi-layer wall with a porous medium that serves as accurate arterial wall anatomy [7]. Oxygen mass transport in asymmetrical stenosed artery with homogenous wall model is numerically modeled by Moore and Ethier [8] and proposed that for modeling transport of oxygen must inclusively consider transport of oxygen from lumen to wall tissue and also consumption of oxygen within the wall. Significance of wall shear stress on luminal surface concentration of low-density lipoprotein (LDL) on the walls of ideal aortic aneurysm geometry was studied, and a correlation was proposed which could identify higher LDL concentrations with direct WSS plots by Satyajit et al. [9].
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2 Methodology The geometry of the artery with constricted lumen analogous to represent stenosis is generated by the following Eq. 1 [3] with the length of 5D from throat toward upstream and of 15D toward downstream. Smooth profile is defined by cosine function and the surface irregularity by sine function. Figure 1 shows stenosis in asymmetric geometry with constriction and the controlled parameters for varying the construction have listed in Table 1. Different stenosis profiles showing variation between smooth and irregular characteristics are shown in Fig. 2 with their respective configuration. Mild stenosis configuration is mentioned by M 1 with occlusion percentage of 56%, in the similar way severe stenosis (M 0 ) with 75%, and high-risk stenosis (M 2 ) is considered by the blockage of 89%, respectively. R(z)|irregular = R −
[ ( )[ βs R 1 + cos πS0z 2
( + As sin
FS π z 2DS0
) − S0 ≤ z ≤ S0
(1)
where R represents the nominal radius of the axisymmetric geometry, β S = 1 − (r stenosis /r nominal ) reduction in diameter of the artery, S 0 for the stenosis span and F S = 0.025D, AS = 12D the frequency and amplitude factors for the irregularities on the stenosis surface. Blood is modeled as incompressible and as the size of the arteries considered here are large where non-Newtonian effects are not predominant, so Newtonian behavior of fluid is considered. Dynamics of fluid flow is governed by Reynolds Number of 200, Re = 2Ruxρ/η where ux is average velocity, ρ is the density of fluid and η is the constant viscosity. For modeling mass transfer of vital elements such as adenosine triphosphate (ATP) and free oxygen in blood proportionally small species are being considered by non-dimensional Schmidt number of 3000 defined by Sc = ν/Dc . These species are incorporated by uniform scalar concentration C = 1 is imposed at inlet and walls are treated as rigid and impermeable to concentration, imposed with Dirichlet boundary condition of C = 0 and zero gradient at the
Fig. 1 Axisymmetric stenosed artery
Table 1 Details of different stenosis configuration
Stenosis configuration
β S (R)
S 0 (R)
Area reduction (%)
M0
0.5
4
75
M1
0.33
4
56
M2
0.667
4
89
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Fig. 2 Axisymmetric stenosis profiles
outlet [4]. To describe psychological importance of blood in axisymmetric geometry, pulsatile flow condition at the inlet of artery with an unsteady velocity waveform u x = u x + A sin(ωt) is employed and governed by two non-dimensional numbers, 1 namely Womersley number of 10, defined by W o = R(ω/ν) 2 and unsteady ratio of 10 defined by φ = A/u x . At the outlet plane, Neumann boundary condition for all the variables are used except for the pressure a zero value is assigned. No-slip boundary condition is used on the walls [2]. Blood flow is described by the following continuity and momentum Eqs. 2 and 3, respectively. Concentration field is being described and solved by advection and diffusion Eq. 4. These governing Eqs. 2–4 are discretized and solved numerically by finite volume method based open-source software OpenFOAM. Advection terms in the momentum equation were treated with high-resolution CUBISTA (Convergent and Universally Bounded Interpolation Scheme for Transport of Advection) scheme and diffusion terms by using Gauss linear orthogonal interpolation scheme and the unsteady is by Euler scheme. The linear system of algebraic equations formed by discretizing schemes were solved by simple corrected algorithm method, and the same solver was also modified to incorporate advection–diffusion equation. Schematic diagram of computational domain with grids of stenosis configuration M 0 is shown in Fig. 3. ∇ ·u =0
(2)
( ) p ∂u + (u · ∇)u = −∇ + ν∇ 2 u ∂t ρ
(3)
∇ · (−Dc ∇c + cu) = 0
(4)
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Fig. 3 Schematic grid density of M 1 stenosis configuration
Fig. 4 Comparison of the axial velocity component between the present study ‘ − ’ and Chauhan and Sasmal ‘ ’ [2] for M 1 stenosis configuration at Re = 200
2.1 Validation Numerical solver is validated by considering above mentioned conditions on an ideal stenosis geometry of M 1 configuration with smooth profile which is 56% occluded. Axial velocities for different time intervals along the mid-plane of the stenosis throat of the current study are compared with that of similar profile [2]. Velocity variation is shown in Fig. 4 which shows good agreement with the current numerical setup.
3 Results and Discussion Simulations are performed to understand the flow characteristics and mass transfer phenomenon over axisymmetric geometry by modeling blood as a Newtonian fluid. A constricted section is introduced at this axisymmetric geometry which resembles an ideal stenosed artery, and this stenosis profile is obtained by Eq. 1 [3]. Furthermore, to represent its profile accurately, irregularities are considered on its wall surface. These profiles are classified into three major stenosis configurations based on the constant parameters listed in Table 1. Fluid phenomenon around the irregular stenosis is given much attention for analyzing the hemodynamics and mass transfer patterns. Results are presented here
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by considering simulations on mainly three stenosis configurations which can physiologically treated as mild (M 1 ), severe (M 0 ), and high-risk (M 2 ) stenosis. Irregular stenosis profile causes fluid flow to be much more complex than the smooth profile. Figure 5 illustrates streamlines of stenosis configuration M 1 to analyze effects of irregular wall profile and demonstrates recirculation zones across the stenosis region. Effect of irregular stenosis profile on fluid flow and mass transfer patterns are illustrated in terms of time averaged quantities for a period of one cardiac cycle (T ). Wall shear stress (τ w ) is one of the significant hemodynamic parameters and is being calculated from the velocity field shown in Eq. 5, and negative sign describes about the reverse flow. Cycle averaged wall shear stress (WSS) and its magnitude are defined by Eqs. 6 and 7, respectively, and they are normalized by considering viscosity, velocity, and diameter. The variation of WSS on smooth and irregular stenosis of M 1 configuration is shown in Fig. 6. Global maximum value of WSS is attained just before the stenosis throat and minimum after the throat with a negative value, shown in Fig. 7. Unlike on the smooth stenosis, there are several peaks and valleys of the WSS profile, and these fluctuations are liable for higher chances of rupture. Magnitude of velocity does not seem to be high not only at the throat also toward the region of downstream till T = 0.5, i.e., accelerating stage of the periodic cycle. τw (x, t) = −μ
∂u x |r =wall ∂r
(5)
T
WSS = ∫ τw (x, t)dt
(6)
0
T
|WSS| = ∫|τw (x, t)|dt
(7)
0
OSI =
) ( WSS 1 1− |WSS| 2
(8)
which can be seen in Fig. 5. High frequency of the WSS profile is due to the formation of recirculation zones at peaks of irregular surface, but their initiation is found to be in the downstream first peak adjacent to throat. During the deceleration stage (T ≥ 0.5) vortices are found to formed at the peaks in upstream region. In contrast, similar small vortices at downstream region have been reported in axisymmetric geometry with smooth stenosis profile by considering viscoelastic fluid model, which quantified these vortices that are due to shear thinning behavior [2]. Thus, irregular profiles are one among the factors for actual modeling of the stenosis. As with increase in severity of the stenosis, same trend of WSS is observed among all the stenosis configurations. Magnitude of WSS is increased radically for the M 2 stenosis with its peak value at upstream to stenosis throat, but there is not much variation between the global and local minimum values, which are with very finite difference. The change in local maximum and minimum of WSS from positive to negative value at the downstream of the stenosis throat is due to the formation
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Fig. 5 Streamline plots of irregular profile M 1 stenosis configuration
Fig. 6 Comparison of cycle averaged WSS between smooth and irregular profiles M 1 stenosis configuration
Fig. 7 Axial variation of WSS on various irregular stenosis configuration
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of the primary recirculation zone, and emphasizing strength of this zone is much higher for M 2 stenosis. At the instance of T = 0.25, during accelerating stage, the maximum velocity has advancing further downstream from the throat into the post stenosis region with increase in the stenosis severity. During the decelerating stage , i.e., T = 0.5, recirculation zone spreads into the post stenotic region while vortices start forming at the irregularities in the upstream of the stenosis profile near to the wall surface. A prominent parameter OSI (oscillatory shear index) given in Eq. 8 is used to represent variations of WSS from large magnitude of positive to negative values. Figure 8 shows variations of OSI along the walls of all the stenosis configuration. OSI with values larger than 0.5 indicates complete flow reversal region and with values equals zero indicates forward stream flow. Fluctuations in OSI with higher amplitudes are found in the region downstream of throat section, and in the upstream, values are less than 0.5 indicating less variations of WSS. Frequency of these oscillations are comparatively higher in downstream of the stenosis throat than in the upstream which acquaints region of flow reversal. Particularly in the regions of constriction which are subjected to OSI ≥ 0.5, advances aggregation of flat deposits like plaque in turns lead to progression of occlusion. From Fig. 8, it can be concluded that mild stenosis further extremely stimulates fat deposits at its downstream region than with severe stenotic sections. Mass transfer phenomenon takes the fluid flow pattern over the stenosis configuration domain. This phenomenon is quantified by non-dimensional Sherwood number, defined by Sh D = n Dc /(Cw − C0 ), where n = −∂C/∂ y is the concentration gradient along the wall. ShD exhibits several peaks and valleys all the way along the stenosis wall. Variations of ShD are shown in Fig. 9 for all the stenosis configurations. Amplitudes of the Sherwood number increase from the upstream till the throat section of the stenosis, and ShD exhibits higher magnitudes for M 2 configuration among all the stenosis configuration considered. Convective transport of the concentration is significantly higher in the M 2 stenosis due to the spreading of high velocity region along the stenosis section axially, in turn inhibits transport of vital elements
Fig. 8 Axial variation of OSI for various irregular stenosis configuration
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Fig. 9 Axial variation of Sherwood Number for various irregular stenosis configuration
into the arterial wall near by the stenosis profile. Downstream within the stenosis frequency and amplitude of the mass transfer pattern is substantially reduced due to flow reversal which indicates residual time of concentration that is much higher at this region. Sherwood number variations almost remained same in terms of magnitude along the stenosis profile for M 1 configuration. Hence, in the region of recirculation, mass transfer distribution plays an prominent role.
4 Conclusions Numerical work has been carried out using open-source software OpenFOAM for simulating blood flow on ideal axisymmetric stenosis geometry. Irregular stenosis profile of the geometry is being considered with three different configurations. Effects of irregularities have strongly influenced cycle averaged wall shear stress profile than with smooth stenosis profile. Within the same occlusion when compared smooth profile, irregular stenosis has higher magnitudes of WSS over the stenotic region. Upstream of stenosis throat have reported larger amplitudes of WSS and rapid increase is observed for M 2 configuration. Frequency of the WSS profile pattern is almost constant with irrespective of stenosis configuration. Mild stenosis further promotes aggregation of plaque deposits much faster than severe stenosis, as they are subjected to higher variations from positive to negative WSS values which are quantified by OSI ≥ 0.5. Upstream of stenosis throat section is prominently influenced by the recirculation zone than compared with downstream section during deceleration stage of cardiac cycle. Dominance of mass transport is by convection for severe stenosis configuration which predominantly effects the mass transfer to the arterial wall tissues near the stenosis region. This emphasizes the actual representation of stenosis profile with surface irregularity elucidates hemodynamic and mass transfer phenomenon for better understanding of diseased condition.
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Nomenclature A Dc ω ν
Amplitude of velocity waveform [–] Concentration diffusivity in lumen region [m2 /s] Frequency [s− 1 ] Kinematic viscosity [m2 /s]
Subscripts w
Wall
References 1. Lusis AJ (2000) Atherosclerosis. Nature 407:233–241 (Google Scholar There is no corresponding record for this reference) 2. Chauhan A, Sasmal C (2021) Effect of real and whole blood rheology on flow through an axisymmetric stenosed artery. Int J Eng Sci 169:103565. https://doi.org/10.1016/j.ijengsci.2021. 103565 3. Sakthivel M, Anupindi K (2021) An off-lattice Boltzmann method for blood flow simulation through a model irregular arterial stenosis: the effects of amplitude and frequency of the irregularity. Phys Fluids 33(3). https://doi.org/10.1063/5.0044948 4. Kaazempur-Mofrad MR, Wada S, Myers JG, Ethier CR (2005) Mass transport and fluid flow in stenotic arteries: axisymmetric and asymmetric models. Int J Heat Mass Transf 48(21–22):4510– 4517. https://doi.org/10.1016/j.ijheatmasstransfer.2005.05.004 5. Rappitsch G, Perktold K (1996) Computer simulation of convective diffusion processes in large arteries. J Biomech 29(2):207–215. https://doi.org/10.1016/0021-9290(95)00045-3 6. Chen J, Gutmark E, Mylavarapu G, Backeljauw PF, Gutmark-Little I (2014) Numerical investigation of mass transport through patient-specific deformed aortae. J Biomech 47(2):544–552. https://doi.org/10.1016/j.jbiomech.2013.10.031 7. Khakpour M, Vafai K (2008) Critical assessment of arterial transport models. Int J Heat Mass Transf 51(3–4):807–822. https://doi.org/10.1016/j.ijheatmasstransfer.2007.04.021 8. Moore JA, Ethier CR (1997) Oxygen mass transfer calculations in large arteries. J Biomech Eng 119(4):469–475. https://doi.org/10.1115/1.2798295 9. Choudhury S, Anupindi K, Patnaik BSV (2019) Influence of wall shear stress and geometry on the lumen surface concentration of low density lipoprotein in a model abdominal aortic aneurysm. Phys Fluids 31(1). https://doi.org/10.1063/1.5074125
Experimental and Numerical Study of Flow Through Ventilator Splitter Aniruddh Mukunth, Raj Shree Rajagopalan, and Naren Rajan Parlikkad
Abstract Mechanical ventilation is a technique in which the air and oxygen mixture is pumped into the lungs of patients who have low levels of oxygen and are unable to breathe on their own. During emergencies such as an outbreak like COVID-19, hospitals run short of ventilator devices to meet the growing number of patients who require ventilator support. In such cases, a single ventilator device shall be used to ventilate multiple patients using a flow splitter. This study aims to analyse the hydrodynamics of flow through various splitters for two patients having the same compliance and requiring equal flow rates. For the analysis a 3D Y-Splitter having split angle over a range of 30–180° is designed with an inlet diameter of 21.8 mm splitting into two exit arms each of 19 mm based on the ventilator dimensions available at Meenakshi Hospital, Thanjavur. A transient 3D models are simulated for a range of inlet mass flow rates in ANSYS 2019 R1, and the best splitter with equal outlet mass flow rate is identified. The results are validated experimentally under similar flow conditions. Keywords CFD · Mechanical ventilation · Ventilator splitter · Split angle · Flow distribution
1 Introduction A mechanical ventilator is a device used for patients who are unable to maintain the required level of oxygen and carbon dioxide in their body on their own due to internal blockages in the lungs or caused by diseases like acute respiratory distress syndrome (ARDS). Based on the patient’s compliance and the airway resistance in the lungs, the tidal volume of air is appropriately supplied. The tidal volume is the volume of air that is exhaled or inhaled during the respiratory cycle. The mechanical ventilator provides a positive pressure to supply the required tidal volume of air to the patient. Based on the patients’ condition, the ventilator can be kept in appropriate A. Mukunth · R. S. Rajagopalan · N. R. Parlikkad (B) School of Chemical and Biotechnology, Sastra Deemed to be University, Thanjavur 631 401, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_73
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modes such as assist-control, synchronized intermittent mechanical ventilation, and pressure support ventilation. The tidal volume, respiratory rate, and the positive end-expiratory pressure are set to facilitate breathing. The SARS-CoV-2 is part of the coronavirus family which uses healthy cells to multiply and evade other cells [1]. This type of coronavirus infects the lower and upper part of the trachea and damages the alveoli where oxygen and carbon dioxide are exchanged with the blood. As ARDS progresses, shortness of breath is observed in patients; therefore, they require external support for breathing to supply oxygen needed to the body and for the lungs to recover. During the surge in the second wave of coronavirus cases in India, the infrastructure was devastated and an increase in the mortality rate was observed due to a shortage of medical equipment like a mechanical ventilator [2]. As an emergency resort, multiple patients were connected to a single ventilator using a ventilator splitter to distribute the flow among two patients. The major concern of using a splitter is the uneven flow distribution among the patients as medically higher-level care is required for ARDS patients. Hence, the ventilator splitter design must be carefully chosen. The requirements of the ventilator splitter are to provide equal and sufficient tidal volume to two or more patients and have minimum pressure drop. In the present work, a preliminary approach was used to study the hydrodynamics by understanding the behaviour and distribution of the flow for patients having equal compliance. The analysis was carried out by varying the continuous flow of inlet air of the patients and maintaining the inspiratory pressure to be constant.
2 Literature Review and Objective 2.1 Previous Study on Ventilator Splitter Experimental and simulation reports on mechanical ventilation for multiple patients were consolidated as part of the current work. Neyman and Irvin [3] reported the use of a ventilator splitter with a 180° angle of split at the junction for mechanical ventilation for multiple patients. They reported that 180° angle split was acceptable based on their experimental study with simulated lung models. They reported the acceptance of the design based on sensors in mechanical ventilator that ensures set values of pressure, and tidal volume is supplied without difficulty. Similarly, Ayyildiz et al. [4] performed experiments with a custom two port 3D printed splitter with approximately 90° split angle with healthy humans. They also reported the acceptance of the design based on sensors in a mechanical ventilator. However, the actual flow distribution ratio at outlet arms was not reported in Neyman and Irvin [3] and Ayyildiz et al. [4]. Advincula et al. [5] reported that the use of a ventilator splitter would not guarantee adequate amount of airflow to the patients. In all the studies, the designs used for the ventilation were not conclusive enough in terms of equal flow distribution. The authors suggested performing simulation studies through CFD
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to understand the presence of dead zones or any other flow parameters involved in the flow distribution. Duke et al. [6] have used transient state time marching method of solution to model the breathing cycle using OpenFOAM for various restrictors to analyse the efficacy of using ventilator splitters for patients with different lung compliance. The pressure drop between the inlet and outlet plane was analysed, but the equality of flow distribution was not discussed.
2.2 Previous Study on General Flow Splitter The experimental and simulation results of various flow splitters used in other applications such as heat exchangers were also studied as part of the current work. Wang et al. [7] performed experiments with water for compact parallel-flow heat exchangers for a nine number of bundle tubes with diameter 2–3 mm and various inlet headers. They reported flow maldistribution as non-uniformity (standard deviation) measured using β i and β. Flow ratio, β i , is measured as ratio of volumetric flow rate in ‘i’th tube to total volumetric flow rate and β, average flow ratio for the total tubes. The volumetric flow rate in the range 0.5–3.0 LPM was used in their experiments. They reported non-uniformity ranged from 0.0013 to 0.0096 for various configuration of the inlet header. Peng et al. [8] performed CFD studies with air for plate-fin heat exchanger on various number and configuration of splitter plates. The mass flow rate was varied from 0.09 to 0.24 kg/s in their study. The flow maldistribution was measured as standard deviation calculated using outlet channel mass flow rates. The flow maldistribution for different inlet headers was found to be from 0.05867 to 0.00498 kg/s. These studies indicate that for any configuration or design of the geometry there exists flow disturbance or uneven flow distribution at the outlet caused due to junction. Therefore, a detailed study on the flow splitter is required to understand the hydrodynamics of splitting and to optimize the design and geometry to get equal flow distribution.
2.3 Medical Data for Mechanical Ventilation The medical data to be used for a single patient were collected directly from the Meenakshi hospital, Thanjavur. The medical data were collected from two critical care patients and a test lung ventilator. The range of pressure, tidal volume, and respiratory rate data collected for a single patient are presented in Table 1.
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Variable Pressure (gauge)
Range
Unit
15–30
mbar
Tidal volume
350–800
mL
Inspiration time
1–1.2
s
3 Materials and Methods 3.1 Ventilator Splitter CAD Models Design In this current work, ventilator design type or shape, inlet and outlet diameters, inlet and outlet pipe lengths were taken as fixed variables. The nominal values for the fixed variables were determined from the literature survey and ventilator port connector dimensions from Meenakshi hospital, Thanjavur. The ventilator splitter inlet dimensions were designed to get directly installed at the oxygen port of the mechanical ventilator. The outlet dimension was designed to connect with connecting tubes to the masks of patients. Junction geometry dimensions and angle of split were taken as design variables in the current work. The values of these variable were changed, and the effect of these variables on flow distribution ratio is to be optimized in this study. Fixed variable values are presented in Table 2. The diameters of the inlet and outlet arms were taken as constant for the length of pipe in this present study. The values of inlet and outlet pipe lengths were taken to be λ times inlet and outlet diameter, respectively. For the present study, λ was assumed to be 2.
4 Experimental Setup Experiments were setup to ascertain the flow distribution in a Y-Splitter to measure the outlet flow rates. The flow rates were measured at each outlet arm and the results were compared. Three Y-Splitters were designed and 3D printed to be used for the experiments. The CAD models were designed with AUTODESK FUSION 360 with a wall thickness of 1.4 mm. The Y-Splitter was printed using polylactic acid (PLA) Table 2 Fixed variables for geometry
Variable
Dimension ( mm)
Inlet diameter (internal diameter)
21.8
Outlet diameter (internal diameter)
19
Inlet pipe length
43.6
Outlet pipe length
38
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Fig. 1 3D printed Y-Splitter
with 100% strength and 0.2 mm layer height for the angles 45, 60, and 90°. The image of the 3D printed splitter is shown in Fig. 1. The schematic diagram and experimental setup are shown in Figs. 2 and 3, respectively. An air compressor with a capacity of up to 7 kg/cm2 was used as a source of air for the experiment. To maintain constant pressure, a pressure regulator with a least count of 0.05 kg/cm2 was used and was set to 1.25 kg/cm2 . The inlet air flow rate was measured and controlled using a rotameter (SS304 1/4 in. panel mount) with a least count of 2.5 LPM. The Y-Splitter was setup horizontally and levelled using a spirit level to avoid the effect of gravity. To avoid the disturbances caused by the bending of the hose, a constant length pipe of diameter 3/4 inch was used before the inlet of the Y-Splitter. All the components were connected with a 10 mm pressure hose.
Fig. 2 Schematic diagram for experiments with air
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Fig. 3 Experimental setup for air
4.1 Measurement Technique Measurement of velocity at the outlet arms was recorded using a digital anemometer. The anemometer AVM-03 (Make: Work Zone) had a least count of 0.001 m/s with 2% error in measurement as reported by the manufacturer. The anemometer was directly kept at the outlet of the splitters separately. Due to the fluctuations in the air flow time, the outlet velocity was time averaged. A sampling study was performed to determine the sampling duration. For the sampling study, velocity values with 90° Y-Splitter were collected for 10, 15, and 20 s with a frequency of 1 s, and the average values of velocity were compared. Comparing the results, for the sampling times of 15 and 20 s, the variation was less. Therefore, 15 s was used as sampling time for the analysis. Experimental run was performed by flipping the splitter to ascertain whether there was any bias with respect to the outlet arm. The analysis was conducted for 45° split angle for the inlet flow rate of 55 LPM. The results showed that irrespective of the geometry flipping, the flow rate of fluid was higher in left outlet arm (before and after flipping). Thus, this indicates that there is no bias in outlet arms. Similarly, experiment was carried out for a long duration of time (1 h) to ensure the outlet flow rates does not change with time for the same angle and inlet flow rate. The outlet velocities were recorded with a time interval of 15 min. No significant change was observed in the flow distribution ratio; therefore, the outlet flow rates can be termed to be constant over the entire period of the experiment.
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5 Computational Model Computational model was developed for simulating flow through Y-Splitter. The computational domain consists of an inlet pipe and two outlet pipes joined smoothly at the junction. The two-dimensional images of the CAD models created using ANSYS 2019 R1-SpaceClaim are shown in Fig. 4 for 90° split angle. As the flow was transient and turbulent in nature, standard k-epsilon model with standard wall function was used to model the flow behaviour. The fluid was assumed to be incompressible. The effect of gravity was not considered in the simulation. In a mechanical ventilator, air and oxygen is mixed and used. However, in the simulation, air was used as fluid material with a density of 1.225 kg/m3 and a viscosity of 1.7894e-05 kg/m.s.
5.1 Boundary Conditions The tidal volume requirement of two patients ranges from 700 to 1600 mL, and the inspiration time is kept constant at 1.2 s. The inlet velocity was computed as 1.56–3.57 m/s from tidal volume and inspiration time. The Reynolds number was calculated based on inlet diameter of 21.8 mm and is given in Table 3. The inlet and outlet boundary conditions were taken as velocity inlet and pressure outlet with a gauge pressure of 2000 Pa, respectively. No-slip boundary condition were used for the wall.
Fig. 4 CAD design of 90° Y-Splitter with dimensions
882 Table 3 Reynolds number for inlet flow rates
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Superficial velocity (m/s)
Reynolds number
1.56
2315
2.07
3071
2.57
3813
3.07
4555
3.57
5297
5.2 Mesh Independence Study and Sampling Studies Mesh independence study was carried out to establish that the simulated solution is independent of the mesh used in the numerical simulation. Polyhedral mesh with three inflation layers was used to carry out the mesh independent study. Mesh-independent study indicated that 0.9 mm polyhedral mesh can be used for the simulation. Sampling times denote the time period of averaging performed in unsteady state simulation resulting in time averaged flow profiles. The sampling time chosen must capture the values to represent the flow behaviour for the entire duration. The outlet parameters with various sampling times were recorded to find the sampling time. For the Y-Splitter with a 90° split angle, the values were recorded for sample times of 1, 2, 4, and 6 s with a frequency of 1 time step. On the outlet surface, 50 radial bands were created and the circumferential average velocity in the axial direction at the outlet arms was measured for various sample times. Comparing the outlet mass velocity for Y-Splitter sampling times of 1, 2, 4, and 6 s, there was no variation among them. Hence, a sample time of 1 s was used in further simulations for the Y-Splitter.
6 Results and Discussion 6.1 Comparison of Simulated Results with Experimental Data The Y-Splitter with split angles of 45, 60, 90, and 120° was simulated over a range of air flow rates between 35 and 80 LPM. The experimental and simulation flow distribution ratios of 45°, 60°, and 90° Y-Splitters were compared, and a variation of less than 2% was observed between the simulated and experimental results. The uncertainty analysis indicated that the average percentage error in flow distribution ratio was 0.24% using Eq. 1. σ = φ
/(
∂φ ∂u 1
)2
( σu21 +
∂φ ∂u 2
)2 σu22
(1)
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The error bars plotted in Figs. 5, 6 and 7 have an average variation of 1.78%. The flow distribution ratio obtained from the experiments was compared with the simulation for 45°, 60° and 90° and is shown in the plots (Figs. 5, 6 and 7).
Fig. 5 Experimental results for 45° Y-Splitter
Fig. 6 Experimental results for 60° Y-Splitter
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Fig. 7 Experimental results for 90° Y-Splitter
6.2 Flow Distribution Ratio of Y-Splitter for Various Split Angles The simulation study was extended to simulate flow through splitter over a wide range of split angles from 30 to 180° to ascertain optimum split angle. The flow distribution ratio (Φ) was calculated as the ratio of flow rates as given in Eq. 2. φ=
Mass Flow Rate at Outlet 1 Mass Flow Rate at Outlet 2
(2)
It is evident from the simulated results that the Y-splitter with a 150° split angle showed the least deviation (0.0296%), and 102° has the highest deviation (0.1057%) from the ideal ratio (φ = 1). The variation of the flow distribution ratio for Y-Splitter with respect to the angle for a particular flow rate is shown in Fig. 8. It was observed that the flow distribution ratio depends on both the angle of split and the flow rate. At certain angles of split between the ranges 55–56° and 105–106°, the flow distribution ratio is independent of the flow rate.
6.3 Pressure and Velocity Contours for Flow Through Y-Splitter Pressure and velocity contours were plotted and studied for Y-Splitter to understand the maldistribution of flow. The splitter designs with the least and highest deviation from the ideal flow distribution ratios were recorded and compared. Unequal pressure distribution at the junction and low-pressure region at the sharp bends were observed from Figs. 9 and 11. At certain sections of the low-pressure region, recirculation
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Fig. 8 Y-Splitter simulation results plot
of flow was observed. The recirculation region for the Y-Splitter for 150–102° at 3.57 m/s is presented in Figs. 10 and 12. It was observed that the recirculation is less in 150° split angle when compared to the 102° split angle. This indicates that the recirculation must be minimized in order to achieve equal distribution of flow. Velocity contour plotted for Y-Splitter showed that the location of maximum velocity is skewed away from the centre at the outlet arms. Due to the skewness, there is uneven flow rate in the cross-sectional area of the outlet arms and can be observed from Figs. 10 and 12.
Fig. 9 Pressure contour of 150° Y-Splitter at 3.57 m/s
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Fig. 10 Recirculation region at point A and B for 150° Y-Splitter at 3.57 m/s
Fig. 11 Pressure contour of 102° Y-Splitter at 3.57 m/s
At the junction of the Y-Splitter geometries, two lines were plotted as shown in Fig. 13 to ascertain the pressure distribution between the points on the lines. The lines drawn, namely Line 1 and Line 2, were at the point of split in junction parallel to Y-axis. The point pressure values at 15 equidistant points from the X-axis from both the lines were recorded. The pressure distribution must be symmetrical, or the
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Fig. 12 Recirculation region at point A and B for 102° Y-Splitter at 3.57 m/s
pressure difference between the points on Line 1 and Line 2 must be zero for the flow distribution to be equal. The average pressure difference at the junction is presented in Table 4 for the best (150°) and the worst splitting angle (102°) for various flow rates. It is evident from Table 4 that the lesser the pressure difference at the junction tends to provide more equal flow distribution.
Fig. 13 At Z = 0 plane with line at junction Y-Splitter 102°
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Table 4 Average pressure difference at the junction of the best and worst Y-Splitter geometry at various flow rates Inlet velocity (m/s)
Angle (θ)
Average junction pressure difference × 10–2 (Pa)
Deviation from ideal ratio × 10–2 (%)
1.56
30°
0.42
10.02
60°
0.40
0.28
102°
1.25
9.70
55°
0.84
0.30
102°
1.85
10.46
150°
0.91
0.89
102°
2.60
11.59
150°
1.28
0.26
102°
3.11
12.56
150°
2.33
0.99
2.07 2.57 3.07 3.57
7 Summary In the present study, hydrodynamics of the ventilator splitter was studied to optimize the geometry for equal distribution of flow. The simulation and experimental data were compared, and less than 2% variation was observed for the split angles 45, 60 and 90°. The difference in the flow distribution ratio was of the same magnitude as the uncertainty of the measurement device. Hence, a more precise measuring techniques are required to ascertain the trend. The simulation results of 14 different split angles were simulated and studied. The flow distribution ratio was dependent on the flow rate as well as the split angle. The least deviation from ideal ratio was observed at 150° split angle, and 102° split angle had the highest deviation from ideal ratio. It was observed that at a certain range of split angles for Y-Splitter; 55–56° and 105–106°, the flow distribution ratio was independent of the flow rates. The average percentage deviation from ideal ratio of Y-Splitter ranges from 0.0148 to 0.1057%. The pressure and velocity contours were constructed to understand the flow distribution. The use of symmetric planes to mesh a section of the symmetric geometry was done in many studies. This is also a commonly suggested meshing techniques in CFD studies [9]. This method of symmetric meshing was used by Longest and Vinchurkar [10] to study the effect of particle deposition in bifurcating airway models where only one-quarter of the geometry was meshed. Bao et al. [11] meshed half of their geometry because of symmetry in the geometry for the Pico propeller hydro-turbine applied in fish farms. In the current work, the complete domain of the geometry was simulated instead of simulating only a particular section to understand the flow maldistribution of the outlet arms. It was evident from the results of present study that there exists flow maldistribution in spite of symmetric geometry. In other words, the flow is asymmetric in nature. Hence, simulating flow by assuming symmetric domains are not
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appropriate to capture the hydrodynamics in such systems. The method of reducing the computational domain to a particular section based on geometric similarity may not be appropriate. Limitations The variables involved in the complicated clinical settings cannot be easily replicated in a numerical and experimental setup. These include, but not limited to: 1.
The patient’s lung response (change in compliance) during the course of ventilation; 2. Flow reversal or cross-contamination; 3. The 3D printed model cross-contamination due to exposure to humidity and environment; 4. Leaks in the ventilation line and loss of pressure due to faulty connections. Future Work The ventilator splitter can be modelled to simulate the actual breathing pattern using a user-defined function in ANSYS. This can be done by periodically varying the inlet pressure and velocity based on the mode which is being studied. Acknowledgements The authors thankfully acknowledge the financial support provided by Indian Institute of Chemical Engineers (IIChE) for carrying out B.E./B.Tech project work in this subject. We thank Meenakshi Hospital, Thanjavur, for helping us collect all the medically and technically relevant data. Aniruddh and Raj Shree extend thanks to SASTRA Deemed University for providing the remote server access to perform the simulation with ANSYS 2019 R1.
Nomenclature σΦ Φ
Standard error in flow distribution ratio [–] Flow distribution ratio [–]
References 1. COVID-19 and the lungs: how does COVID-19 affect the lungs? https://www.nhlbi.nih.gov/ coronavirus/lungs. Accessed 06 Aug 2022 2. Ghosh DD, Sarkar A, Chouhan DP (2020) COVID-19 second wave: district level study of concentration of confirmed cases and fatality in India. Environ Challenges 5:100221 3. Neyman G, Irvin CB (2006) A single ventilator for multiple simulated patients to meet disaster surge. Acad Emerg Med 13:1246–1249 4. Ayyıldız S, Dursun AM, Yıldırım V et al (2020) 3D-printed splitter for use of a single ventilator on multiple patients during covid-19. 3D Printing Additive Manufact 7:181–185
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5. Advincula RC, Dizon JRC, Chen Q et al (2020) Additive manufacturing for COVID-19: devices, materials, prospects, and challenges. MRS Commun 10:413–427 6. Duke DJ, Clarke AL, Stephens AL (2022) A computational fluid dynamics assessment of 3D printed ventilator splitters and restrictors for differential multi-patient ventilation. 3D Printing Med 8:2 7. Wang C-C, Yang K-S, Tsai J-S, Chen IY (2011) Characteristics of flow distribution in compact parallel flow heat exchangers, part I: typical inlet header. Appl Therm Eng 31:3226–3234 8. Peng X, Li D, Li J, Jiang S, Gao Q (2020) Improvement of flow distribution by new inlet header configuration with splitter plates for plate-fin heat exchanger. Energies 13:1323 9. Using symmetry to reduce model size. https://www.comsol.com/support/learning-center/art icle/Using-Symmetry-to-Reduce-Model-Size-35921. Accessed 06 Aug 2022 10. Longest PW, Vinchurkar S (2007) Effects of mesh style and grid convergence on particle deposition in bifurcating airway models with comparisons to experimental data. Med Eng Phys 29:350–366 11. Bao TN, Jun-Ho K (2019) Design and analysis of a pico propeller hydro turbine applied in fish farms using CFD and experimental method. J Korean Soc Mar Environ Saf 25:373–380
Bioconvective MHD Flow of Micropolar Nanofluid Over a Stretching Sheet Due to Gyrotactic Microorganisms with Internal Heat Generation/ Absorption and Chemical Reaction P. Vimala and R. Dhivyalakshmi
Abstract The present work deals with the study of bioconvective MHD flow of micropolar nanofluid containing gyrotactic microorganisms along with chemical reaction over a stretching sheet. Nonuniform heat source/sink and viscous dissipation effects are also considered. The chemical reaction phenomenon reduces the nanoparticle concentration and enhances the mass diffusion rate on the sheet. The nonlinear coupled system of PDEs is converted into nonlinear system of ODEs by using a similarity transformation, and the transformed equations are solved by using MATLAB bvp4c solver. The skin friction coefficient, the microrotation parameter, the Nusselt number, the Sherwood number, and the motile density number are numerically computed and presented. The effects of the pertinent physical parameters on the velocity, angular velocity, temperature, nanoparticle concentration, and motile density of microorganisms are inspected. The present results are relevant in improving the performance of microbial fuel cells and heat transfer devices. Keywords Bioconvection · Micropolar nanofluid · Nonuniform heat source/sink · Chemical reaction · Motile microorganisms
1 Introduction Enhancement of heat transfer is one of the major concerns in chemical industries. Suspension of nanoparticles (< 100 nm) in the traditional base fluid such as water, oil, and ethylene glycol are termed as nanofluid. These suspended nanoparticles in any base fluid can improve the thermophysical properties of the fluid. Nanoparticles can easily pass through the capillaries and microchannels. They do not create any blockage in the flow. Thus, scientists used nanofluids in different industries such as P. Vimala (B) · R. Dhivyalakshmi Department of Mathematics, Anna University, Chennai 600025, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_74
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solar collectors, heat exchangers, dielectric characteristics, bio-sensors, fuel production, automobile engine cooling, and food industry. The interaction of nanoparticles is highly useful in the biomedical industries, i.e., cancer treatment, chemotherapy, diagnosis of brain tumor, and microsurgical technology. Choi and Eastman [1] first presented the idea of nanofluid and analyzed the impact of heat transfer experimentally. Buongiorno [2] explained the convective transport of nanoparticles by considering only Brownian diffusion and thermophoresis features among the seven slip conditions. Wang and Fan [3] determined that the nanoparticles are of four scales: the microscale, the molecular scale, the macro-scale, and the mega scale. Micropolar fluid consists of suspensions of randomly oriented rigid particles in a viscous medium where the microdeformation is neglected. They possess their own microrotations and microspins. The behavior of the micropolar fluid is identified by microstructure of the particles, spin inertia and couple stress. The modeling of the micropolar fluid theory was initially proposed by Eringen [4]. Later, many researchers shifted their attention toward micropolar theory, because of its wide applications in various fields such as electrical and magnetic fields. Examples of micropolar fluids are concrete with sand, muddy fluids, polymeric suspensions, magnetic fluids, liquid crystals, cervical flows, and animal blood with rigid cells. Sandeep and Sulochana [5] investigated the unsteady mixed convection flow of micropolar fluid with the impact of heat source and sink and concluded that the velocity boundary layer was increased by micropolar parameter. To enhance the thermophysical properties, researchers used nanoparticles in bioconvection. Kuznetsov [6] first initiated the idea of suspension of nanoparticles along with the motile microorganisms in the nanofluid bioconvection. The phenomenon of bioconvection is the upward movement of the microorganisms in a water-based fluid. These motile microorganisms increase the density of the base fluid which move in a specific direction in response to various stimuli such as oxytaxis, gravitaxis, gyrotaxis, chemotaxis, and phototaxis. But the movement of nanoparticles depend on Brownian motion and thermophoresis. Thus, the motion of the microorganisms and the nanoparticles both are independent. In addition, bioconvection phenomenon helps to improve the stability of nanoparticles suspension. The application of bioconvection in many fields include electronics, mechanical, chemical, petroleum engineering, environmental system, bio-microsystem and biotechnology. Chemical reaction plays a vital role in nanofluid bioconvection. Study of chemical reaction has significance in chemical industry and bio-engineering applications such as manufacturing of ceramics, polymer production, chemical processing of materials, and food processing. Rout et al. [7] studied MHD flow of micropolar fluid over a vertical plate with destructive chemical reaction effects on velocity field. Motivated by the above articles, the present paper studies the MHD micropolar nanofluid flow of gyrotactic microorganisms including the effects of heat generation/ absorption and chemical reaction over a stretching sheet. The governing equations of the flow problem are transformed into a system of nonlinear ordinary differential equations using a similarity transformation and solved by MATLAB bvp4c solver. To validate the present work, the results are compared with the existing ones in literature, and they are shown to agree well. The main purpose of the study is to
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examine the impact of pertinent physical parameters on the flow, heat transfer, and mass transfer characteristics. These are presented numerically and graphically.
2 Mathematical Formulation A steady, two-dimensional, incompressible, laminar, MHD flow of micropolar nanofluid in the presence of gyrotactic microorganisms is considered. Nonuniform heat source/sink, viscous dissipation, and chemical reaction effects are also incorporated. Water-based nanofluid is taken into account. It is assumed that the temperature of the solute is less than 42 °C to ensure the living condition of the microorganisms. The geometry of the problem is shown in Fig. 1. Using these assumptions, the boundary layer approximation of the problem can be expressed as ∂v ∂u + =0 ∂x ∂y
(1)
) ( k f ∂ 2u kf ∂N ∂u ∂u σ B02 u u + +v = ϑf + − ∂x ∂y ρ f ∂ y2 ρ f ∂y ρf ) ( 1 [ + (1 − C∞ )ρ f βg(T − T∞ ) − g(C − C∞ ) ρ p − ρ f ρf )[ ( − gϑ ∗ (n − n ∞ ) ρm − ρ f (2) ] [ kf γ f ∂2 N ∂u ∂N ∂N 2N + u − +v = ∂x ∂y ∂y (ρ j ) f ∂ y 2 (ρ j) f ( ) (μ + k) f ∂u 2 σ B02 u 2 ∂T ∂2T ∂T δ ) ) +( ) +v =α 2 + ( u +( ∂x ∂y ∂y ∂ y ρc p f ρc p f ρc p f Fig. 1 Schematic representation of the problem
(3)
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]
( ) ] DT ∂ T 2 ∂C ∂ T + + τ DB ∂y ∂y T∞ ∂ y
(4)
∂ 2C ∂C DT ∂ 2 T ∂C − k0 (C − C∞ ) +v = DB 2 + ∂x ∂y ∂y T∞ ∂ y 2 ( ) ∂n b1 Wc ∂C ∂ 2n ∂ ∂n +v + n = Dm 2 u ∂x ∂y ∂y ∂y (Cw − C∞ ) ∂ y
u
(5)
(6)
The boundary conditions are, u = u w = bx, v = 0, N = 0, T = Tw , C = Cw , n = n w at y = 0, u → 0, N → 0, T → T∞ , C → C∞ , n → n ∞ as y → ∞ (7) In Eq. (3), γ f is the spin gradient viscosity which is defined as ) ( kf j γf = μf + 2
(8)
ϑ
where j = bf is the micro-inertia density and k f is the vortex viscosity. The internal heat generation/absorption parameter δ in Eq. (4) is defined as δ=
[ kb [ ∗ A (Tw − T∞ )e−η + B ∗ (T − T∞ ) ϑf
(9)
where A∗ and B ∗ denote space and temperature-dependent internal heat generation/ absorption coefficients. It is assumed that the coefficients A∗ > 0 and B ∗ > 0 represent the heat generation internally and A∗ < 0 and B ∗ < 0 represent the internal heat absorption. Equations (2)–(6) are reduced into the system of ODEs, by using the following similarity transformations: η=
√
√ blϑ f y, u = bx f ' (η), v = − bϑ f f (η),
θ (η) =
N=
√ b3 lϑ f xg(η),
T − T∞ C − C∞ n − n∞ , ϕ(η) = , ξ (η) = Tw − T∞ Cw − C∞ nw − n∞
(10)
The transformed equations are ( )2 (1 + K ) f ''' − f ' + f f '' + K g ' − M f ' + λ[θ − Nrϕ − Rbξ ] = 0 ( ) [ [ K '' 1+ g − f ' g + f g ' − K 2g + f '' = 0 2
(11) (12)
Bioconvective MHD Flow of Micropolar Nanofluid Over a Stretching …
( )2 ( )2 θ '' + Pr f θ ' + Pr Ec(1 + K ) f '' + Pr EcM f ' + A∗ e−η + B ∗ θ ( )2 + Pr Nbθ ' ϕ ' + Pr Nt θ ' = 0 ( ) Nt '' ϕ '' + Sc f ϕ ' − kc ϕ + θ =0 Nb [ [ ξ '' + Lb f ξ ' − Pe (ξ + Ω)ϕ '' + ξ ' ϕ ' = 0
895
(13) (14) (15)
The transformed boundary conditions are f (0) = 0,
f ' (0) = 1, g(0) = 0, θ (0) = 1, ϕ(0) = 1,
ξ (0) = 1, at η = 0, f ' → 0, g → 0, θ → 0, ϕ(∞) → 0, ξ (∞) → 0, as η → ∞.
(16)
The dimensionless numbers are ϑf kf ϑf b1 Wc σ B02 , Pe = , K = , M= , Lb = , α Dm μf bρ f Dm ϑf gβ(1 − C∞ )(Tw − T∞ ) n∞ , Sc = λ= , Ω= , 2 b x DB nw − n∞ ( ) ρC p p D B (C W − C∞ ) u 2w ( ) , Nb = , Ec = (cp) f (Tw − T∞ ) ρC p f ϑ f ) ( ( ) ρC p p DT (Tw − T∞ ) ρ p − ρ f (Cw − C∞ ) k0 ( ) , , kc = , Nr = Nt = b βρ f (1 − C∞ )(Tw − T∞ ) ρC p f T∞ ϑ f ) ( ϑ ∗ (n w − n ∞ ) ρm − ρ f Rb = . βρ f (1 − C∞ )(Tw − T∞ ) Pr =
To describe the flow characteristics, the physical quantities used are the skin friction coefficient C f , local Nusselt number Nux , local Sherwood number Shx , and the motile microorganism density number Nnx expressed, respectively, by τw xqw xqm , Shx = , , Nux = ρ f u 2w k(Tw − T∞ ) D B (Cw − C∞ ) xqn Nnx = Dn (n w − n ∞ )
Cf =
(17)
where τw , qw , qm , and qn represent the wall shear stress, surface heat flux, surface mass flux, and the surface motile microorganism flux given by [ ) ] ( ∂T ∂u τw = μ (1 + K ) qw = −k ∂ y y=0 ∂ y y=0
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( qm = −D B Taking Rex =
bx 2 ϑf
∂C ∂y
(
) qn = −Dn y=0
∂ξ ∂y
) (18) y=0
and substituting Eqs. (10) and (18) in Eq. (17), we get
'' −1/2 C f Re1/2 = −θ ' (0) x = (1 + K ) f (0), Nux Rex
= −ϕ ' (0), Nnx Re−1/2 = −ξ ' (0) Shx Re−1/2 x x
(19)
3 Solution Methodology The system of coupled nonlinear ordinary differential equations (11)–(15) subject to the boundary conditions (16) is solved by using MATLAB bvp4c solver. Equations (11)–(15) are converted into the following system of first-order ordinary differential equations by introducing the new variables f = y1 , f ' = y2 , f '' = y3 , g = y4 , g ' = y5 , θ = y6 , θ ' = y7 , ϕ = y8 , ϕ ' = y9 , ξ = y10 , ξ ' = y11 , and the system is given by y1' = y2 , y2' = y3 , [ 1 [ 2 y − y1 y3 − K y5 + M y2 − λ(y6 − Nry8 − Rby10 ) , y3' = 1+K 2 y4' = y5 , 2 y5' = [y2 y4 − y1 y5 + K (2y4 + y3 )], 2+K y6' = y7 , [ [ y7' = (− Pr) y1 y7 + Ec(1 + K )y32 + EcM y22 + Nby7 y9 + Nt(y7 )2 − A∗ e−η − B ∗ y6 , y8' = y9 , Nt ' y, y9' = Sc[kc ϕ − y1 y9 ] − Nb 7 ' y10 = y11 , [ [ ' y11 = −Lby1 y11 + Pe y9' (y10 + Ω) + y9 y11 The boundary conditions in (16) reduce to y1 = 0, y2 = 1, y4 = 0, y6 = 1, y8 = 1, y10 = 1 at η = 0, y2 → 0, y4 → 0, y6 → 0, y8 → 0, y10 → 0 as η → ∞.
Bioconvective MHD Flow of Micropolar Nanofluid Over a Stretching …
897
4 Results and Discussion In all the results, unless otherwise specified the parameter values used are: K = 0.2, M = 0.05, λ = 0.1, Nr = 0.1, Rb = 0.1, Pr = 1.2, Ec = 0.02, Nb = 0.1, Nt = 0.1, A∗ = 0.01, B ∗ = 0.01, Sc = 0.2, kc = 0.4, Lb = 1.2, Pe = 1.2, and Ω = 0.2. Table 1 presents a comparison of present results of skin friction coefficient, microrotation, heat transfer rate, and mass transfer rate with those of Eldabe [8]. It is seen that the results agree well with each other with four decimal places of accuracy. Table 2 presents the effects of various parameters on the skin friction coefficient and the microrotation parameter. It is noted that for increasing the values of the micropolar constant K , magnetic number M, buoyancy ratio parameter Nr and bioconvection Rayleigh number Rb, both skin friction coefficient and microrotation are enhanced. On the other hand, for an increase in mixed convection parameter λ, skin friction coefficient and microrotation decrease. Table 3 analyzes the effects of different parameter values on the heat transfer rate. It is observed that for the growing values of micropolar constant K and Prandtl number Pr, the Nusselt number increases. But reverse trend is seen for increasing each magnetic number M, Eckert number Ec, thermophoresis parameter Nt, Brownian motion parameter Nb, space-dependent heat generation/ absorption parameter A∗ , and temperaturedependent heat generation/absorption parameter B ∗ . Table 4 portrays the effect of different parameters on the local Sherwood number. The results show that gradually escalating values of the Schmidt number Sc, Brownian motion parameter Nb, chemical reaction parameter kc , space-dependent heat generation/absorption parameter A∗ , and temperature-dependent heat generation/absorption parameter B ∗ cause an increase in the rate of mass transfer. On the other hand, Sherwood number is reduced for mounting values of thermophoresis parameter Nt. Table 5 describes the impact of various parameters on the microorganism density. It is noticed that for increasing values of the bioconvection Rayleigh number Rb, Peclet number Pe, and concentration difference parameter Ω, the density of microorganism is reduced, whereas the opposite effect is observed for an increasing value of bioconvection Lewis number Lb. Table 1 Skin friction, microrotation, heat transfer rate, and mass transfer rate—a comparison M 0
K 0.2
− f '' (0)
g ' (0)
−θ ' (0)
−ϕ ' (0)
[8]
Presentynnt
[8]
Present
[8]
Present
[8]
Present
0.9098
0.9098
0.0950
0.0950
0.4688
0.4688
0.2149
0.2149
0.5
1.1144
1.1144
0.1051
0.1051
0.4250
0.4250
0.1972
0.1972
1.0
1.2871
1.2871
0.1121
0.1121
0.3914
0.3913
0.1857
0.1857
0
1.4142
1.4142
0
0
0.3734
0.3734
0.1790
0.1790
0.5
1.1407
1.1408
0.2112
0.2112
0.4119
0.4119
0.1938
0.1938
2.0
0.7696
0.7697
0.3586
0.3586
0.4675
0.4675
0.2204
0.2204
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Table 2 Skin friction coefficient and microrotation parameter—effects of various parameters K
M
λ
Nr
Rb
−C f Rex
g ' (0)
0.2
0.05
0.1
0.1
0.1
1/2
1.0736
0.0941
0.5
1.1922
0.1746
2
1.6150
0.2864
0
1.0466
0.0929
0.5
1.2931
0.1030
1
1.5020
0.1101
0.1
1.0736
0.0941
0.2
1.0303
0.0923
0.3
0.9882
0.0906
0.2
1.0858
0.0953
0.4
1.1108
0.0979
0.6
1.1373
0.1010
0.2
1.0788
0.0943
0.4
1.0891
0.0946
0.6
1.0995
0.0950
0.2
0.05
0.1
0.1
In this analysis, the influence of various physical parameters on velocity, angular velocity, temperature, nanoparticle concentration, and motile density are studied. Figure 2 demonstrates the effect of magnetic number M on the x-direction velocity f ' (η). It is seen that f ' (η) is reduced for an increasing magnetic number M. The physical behavior of the magnetic effect induces the Lorentz force which slows down the flow velocity. Figure 3 shows the effects of micropolar constant K on the velocity f ' (η). It is observed that the boosting values of the micropolar constant enhances the velocity profile. Large values of K cause change in viscosity resistively low as a result of which the velocity f ' (η) increases. Figure 4 investigates the effect of magnetic number M on the angular velocity field g(η). When M increases, the angular velocity g(η) decreases. An increase in magnetic number M means an increase in the drag force which is a resistive force causing a decrease in g(η). Figure 5 portrays the impact of micropolar constant K on the angular velocity profile. It is indicated that for mounting values of K , the angular velocity g(η) is increased. Figure 6 produces the effect of space-dependent internal heat generation/ absorption coefficient A∗ on the temperature profile θ (η). It is observed that higher values of A∗ enhances the temperature distribution. The presence of heat source results in the generation of energy, and hence, the thermal energy rises.
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Table 3 Local Nusselt number—effects of various parameters −1/2
K
M
Pr
Ec
Nt
Nb
A∗
B∗
Nux Rex
0.2
0.05
1.2
0.02
0.1
0.1
0.01
0.01
0.5684
0.5
0.5871
2
0.6354
0.2
0
0.5744
0.5
0.5205
1
0.4769
0.05
2
0.7898
5
1.3174
7
1.5563
1.2
0.2
0.4724
0.3
0.4193
0.4
0.3665
0.02
0.2
0.5504
0.4
0.5152
0.6
0.4807
0.1
0.2
0.5483
0.4
0.5075
0.6
0.4681
0.1
0.02
0.5234
0.03
0.4797
0.04
0.4369
0.01
0.03
0.5493
0.05
0.5294
0.07
0.5087
Figure 7 visualizes the effect of temperature-dependent internal heat generation/ absorption coefficient B ∗ on the thermal velocity profile. Growing values of B ∗ > 0 increases the temperature field. Figure 8 explains the impact of Schmidt number Sc on the nanoparticle concentration profile ϕ(η). The concentration profile is decreased for the increasing values of Schmidt number Sc. Mass diffusivity creates the opposite effect on Schmidt number. The larger values of Sc tend to lower the mass diffusivity. Therefore, concentration ϕ(η) is declined. Figure 9 exhibits the effect of chemical reaction parameter kc on nanoparticle concentration distribution. It is noticed that the increasing values of chemical reaction kc decrease the concentration ϕ(η). In the case of destructive reaction, the chemical reaction slows down the concentration of the fluid. The fluid concentration gradually
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Table 4 Local Sherwood number—effects of various parameters −1/2
Sc
Nt
Nb
Kc
A∗
B∗
Shx Rex
0.2
0.1
0.1
0.4
0.01
0.01
− 0.0075
0.3
0.1176
0.4
0.2227
0.5 0.2
0.3144 0.3
− 0.6474
0.5
− 1.1676 − 1.5737
0.7 0.1
1
0.3393
2
0.3571
3
0.3621
0.1
0.7
0.1174
0.9
0.1854
1.2 0.4
0.2731 0.1
0.3067
0.2
0.6182
0.3
0.9116
0.01
0.06
0.0373
0.07
0.0468
0.08
0.0565
Table 5 Microorganism density—effects of various parameters −1/2
Rb
Lb
Pe
Ω
Nnx Rex
0.1
1.2
1.2
0.2
0.6583
0.2
0.6582
0.4
0.6578
0.6 0.1
0.6575 1
0.5807
2
0.9174
3 1.2
1.1748 2
0.6454
3
0.6299
4 1.2
0.6150 0.4
0.6481
0.6
0.6377
0.8
0.6273
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Fig. 2 Effect of magnetic number M on velocity f ' (η)
Fig. 3 Effect of micropolar constant K on velocity f ' (η)
changes from the higher to lower value when the strength of the chemical reaction rate is higher than the kinematic viscosity of the fluid. Figure 10 examines the effect of Peclet number on motile density ξ (η). It is seen that with the growing values of Peclet number Pe, motile density profile is declined. An increment in the Peclet number enhances the motion of fluid particles that cause depletion in thickness of motile microorganisms.
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Fig. 4 Effect of magnetic number M on angular velocity g(η)
Fig. 5 Effect of micropolar constant K on angular velocity g(η)
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Fig. 6 Effect of space-dependent heat generation coefficient A∗ on temperature distribution θ (η)
Fig. 7 Effect of temperature-dependent heat generation/absorption coefficient B ∗ on temperature distribution θ (η)
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Fig. 8 Effect of Schmidt number Sc on nanoparticle concentration ϕ(η)
Fig. 9 Effect of chemical reaction kc on nanoparticle concentration ϕ(η)
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Fig. 10 Effect of Peclet number Pe on motile microorganism density ξ (η)
5 Conclusion In this paper, MHD flow of micropolar nanofluid containing gyrotactic microorganisms in the presence of heat generation/absorption and chemical reaction effects over a stretching sheet has been studied. The following conclusions are arrived at: • Both velocity and angular velocity are increased for the growing values of the micropolar constant K , but a decrease is noted for gradually increasing values of the magnetic number M. • The temperature distribution is enhanced when increasing both the space and the temperature-dependent heat generation parameters A∗ and B ∗ . • The nanoparticle concentration is reduced when increasing the chemical reaction rate parameter and Schmidt number. • A decreasing behavior of motile density is seen for increasing values of the bioconvection Peclet number.
References 1. Choi SUS, Eastman JA (1995) Enhancing thermal conductivity of fluids with nanoparticles. ASME Fluids Eng 231:99–105 2. Buongiorno J (2006) Convective transport in nanofluids. J Heat Transfer 3(128):240–250 3. Wang L, Fan J (2010) Nanofluids research: key issues. Nanoscale Res Lett 8(15):1241 4. Eringen AC (1966) Theory of micropolar fluids. Int J Math Mech 16:1–18 5. Sandeep N, Sulochana C (2015) Dual solutions for unsteady mixed convection flow of MHD micropolar fluid over a stretching/shrinking sheet with non-uniform heat source/sink. Eng Sci Technol Int J 18:738–745 6. Kuznetsov AV (2010) The onset of nanofluid bioconvection in a suspension containing both nanoparticles and gyrotactic microorganisms. Int Commun Heat Mass Transfer 37:1421–1425
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7. Rout PK, Sahoo SN, Dash GC, Mishra SR (2016) Chemical reaction effect on MHD free convection flow in a micropolar fluid. Alex Eng J 55:2967–2973 8. Eldabe N, Ouaf MEM (2006) Chebyshev finite difference method for heat and mass transfer in a hydromagnetic flow of a micropolar fluid past a stretching surface with ohmic heating and viscous dissipation. Appl Math Comput 177(2):561–571
Machine Learning in Fluid Mechanics
Application of Machine Learning for Forced Plume in Linearly Stratified Medium Manthan Mahajan, Nitin Kumar, Deep Shikha, Vamsi K. Chalamalla, and Sawan S. Sinha
Abstract Direct numerical simulation (DNS) is very accurate; however, the computational cost increases significantly with the increase in Reynolds number. On the other hand, we have the Reynolds-averaged Navier–Stokes (RANS) method for simulating turbulent flows, which needs less computational power. Turbulence models based on linear eddy viscosity models (LEVM) in the RANS method, which use a linear stress–strain rate relationship for modeling the Reynolds stress tensor, do not perform well for complex flows (Shih et al. in Comput Methods Appl Mech Eng 125:287–302, 1995). In this work, we intend to study the performance of nonlinear eddy viscosity model (NLEVM) hypothesis for turbulent forced plumes in a linearly stratified environment and modify the standard RANS model coefficients obtained from machine learning. The general eddy viscosity hypothesis supported by the closure coefficients generated from the tensor basis neural network (TBNN) is used to develop TBNN-based K-∈ model. The aforementioned model is used to evaluate the plume’s mean velocity profile, and maximum height reached. The comparison between standard LEVM, NLEVM, and the experimental results indicates a significant improvement in the maximum height achieved, and a good improvement in the mean velocity profile. Keywords Forced plume · Modified RANS · General eddy viscosity hypothesis · TBNN-based K-∈
1 Introduction There are many real-life scenarios where we can see forced plumes, e.g., plume formation after a volcanic eruption, fountains, pollutants flows in the atmosphere, and chimney smoke from the industries. Hence, it becomes an important aspect of engineering to simulate the flow. Also, stratified fluid studies are commonly used in geophysical fluid mechanics. M. Mahajan (B) · N. Kumar · D. Shikha · V. K. Chalamalla · S. S. Sinha Department of Applied Mechanics, Indian Institute of Technology, Delhi 110016, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_75
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Morton et al. [2] suggested in his study that plume is simply used for the flow in which there is an effect of buoyancy only, and jets are used for the flow with continuous momentum supply. Hence, “forced plume” or “buoyant jets” terms are used for the flow with both properties, simple plume, and jet. Fox [3] in their study investigated the turbulent buoyant jet in a linearly stratified environment and figured out that the entrainment is dependent on the Reynolds stress, the form of similarity profiles, and local mean Froude number. Bloomfield et al. [4] in their work developed a theoretical model for an axisymmetric turbulent fountain in both homogeneous and stratified mediums. The model also incorporates the entrainment of the ambient fluid into the initial fountain up flow, and entrainment of ambient, and up flow into the subsequently formed downflow. The entrainment of the denser fluid (fountain) with the ambient fluid gives two prominent effects, an increase of overall volume flux with the entry of ambient fluid and the velocity of the up flow decreases to zero at a certain maximum height. After that, up flow falls on the subsequent up flow as a turbulent plume and surrounds the central up flow. The developed model was approximately 95% accurate in calculating flow parameters in linearly stratified medium. Mirajkar et al. [5] in their research presented the study of the effects of varying ambient stratification strength N∞ for the turbulent plume flow field. Parameters like maximum height, spreading height, and radial propagation of the plume were characterized to study the behavior of the plume. The radial spread of the plume was found to be independent of the buoyant frequency. The entrainment coefficient was found to be larger for the higher values of N∞ . The maximum height of the plume and spread of the plume decreases with an increase in N∞ . Mirajkar et al. [6] studied the turbulent forced plume experimentally and compared different parameters with the jet flow field. The center line velocity declined linearly with height until suddenly decreased to zero, while the jet’s velocity dropped off endlessly. Although the two had different mean velocities, there was a good agreement. In their study, Kumar et al., [7] analyzed and compared the RANS simulation results for mean centerline velocity and maximum height reached by the turbulent forced plume in a linearly stratified environment with the experimental results by Mirajkar et al. [6]. The mean velocity profile gives accurate results in the vicinity of the source, but away from the source velocity profile deviates away from the experimentally determined profile. This also leads to significant errors in the maximum height prediction by the RANS method. Various variants of the K-∈ were analyzed, and standard K-∈ was found to be most accurate among linear eddy viscosity models. Tsang-Hsing Shih et al. [1] in their work analyzed that the linear eddy viscosity models are unable to do well in flow with separation and curvatures. LEVM like standard K-ε defines Reynolds stress tensor as proportional to the strain rate tensor alone. It has been observed that these models partially follow realizability conditions. The realizable algebraic equation for Reynolds stress tensor was derived in which linear to quadratic function of mean strain rate and mean rotation rate tensor were considered for defining Reynolds stress tensor. The model coefficients were derived considering all realizability conditions.
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Pope [8] aims to suggest a better formulation for the effective viscosity approach for Reynolds stress tensor closure. In their research, a hypothesis was proposed in which anisotropic part of Reynolds stress tensor was present as a linear summation of ten tensors. These ten tensors include higher-order terms of mean strain rate and rotation rate normalized with the k/ε ratio. All tensors are multiplied with the coefficients, which are the function of a finite number of known invariants. Bruntonv et al. [9] presented an article on the history, recent developments, and new opportunities in the field of machine learning when combined with the fluid mechanics problems. Gholami et al. [10] had compared computational fluid dynamics (CFD) and artificial neural network (ANN) methods against experimental observations studying over open channel sharp bend flow characteristics. Results indicate better performance of ANN with less R.M.S error as compared to CFD. Ling et al. [11] justified the significant need for modification in the conventional RANS method to find Reynolds stress anisotropic tensor to define the richer set of turbulence physics. In their work, they aim to solve the Reynolds stress tensor with the help of an artificial neural network called tensor-based neural network (TBNN). The equation used for the neural network was the same as given by Pope [8]. The neural network was trained with accurate than RANS, but they could not match the level of DNS. The previous RANS studies on the turbulent forced plume in the stratified environment were done, using simple RANS with a LEVM. No work has been recorded regarding modifying the simple RANS method and use of NLEVM for the forced plume. In the present work, we modified the RANS model with the help of a machine learning algorithm. We used the LES data set for the neural network training and applied the solution of TBNN to the realizable algebraic Reynolds stress equation model [1] to compile a new TBNN-based K-∈ model. In this sense, we present a new cost-efficient RANS method incorporating a machine learning based turbulence model of the turbulent forced plume. Aforementioned tasks are achieved by working on the following objectives: • Use of ML tools and high-fidelity LES database to arrive at improved closure coefficients for general eddy viscosity closure paradigms which will help in developing TBNN-based K-ε model. • Evaluation of ML enhanced RANS methodology in simulating and predicting flow statistics in plume flow. Specifically, we look at the following quantities: • 1. Mean velocity profile. • 2. The maximum height reached by the plume. All evaluations are performed by comparing our ML enhanced RANS predictions against available experimental results by Mirajkar et al. [6] and RANS results by Kumar et al. [7].
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2 Methodology This section presents the governing equations and problem formulation using RANS and ML methodology used for simulation of forced plume in stratified environment. Section 2.1presents the governing equation based on RANS and concise description of realizable algebraic Reynolds stress model [1] and the standard K-∈ model. Section 2.2 presents the theory for used ML methodology, and Sect. 2.3 presents the computational setup for simulations.
2.1 Governing Equations The unsteady Reynolds-averaged Navier–Stokes equations (URANS, Eq. (2)) and energy equations (Eq. 3) under the Boussinesq approximation are solved numerically using an open-source code OpenFOAM-5.0. These set of equations are given as:
ρ
∂u i =0 ∂ xi
(1)
∂u i ∂ρ ∂u i ∂ pd ∂ ∂u i + uj =− [v − u i' u 'j ] − gk xk ∂t ∂x j ∂ xi ∂ x j ∂ x j ∂ xi
(2)
[ ] ∂u j Td v ∂ Td ∂ Td ∂ ∂ Ta (z) ' ' + = − T uj − w ∂t ∂x j ∂ x j Pr ∂ x j ∂z
(3)
ρ = 1 − β(T − Tb )
(4)
where ρu i' u 'j represents the Reynolds stress tensor which is solved with the help of turbulence model. pd is defined as p = pd + ρg3 x3 . ρ is Reynolds averaged total density normalized with reference density taken at the bottom (ρb ). The relation between temperature and density is give in Eq. 4. T d is the deviation temperature which can be represented as the difference between Reynolds averaged temperature and the background temperature. Evolution equation of T d is found by using Eq. (5) in the evolution equation of T Td = T − Ta (z)
(5)
where T is Reynolds averaged temperature and Ta (z) is the background temperature varying linearly with height. A similar set of equations and the numerical setup were used by Kumar et al. [7] in their work to evaluate different variants of K-∈ model for the forced plume flow in a stratified environment. The nonlinear eddy viscosity-based realizable algebraic Reynolds stress model (Shih’s quadratic model [1]) is considered to extend the study over the turbulent
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forced plume. In this model, Reynolds stress depends on a combination of mean strain rate (S) and rotation rate (W ) tensors up to the second order. Comparing the results for this model with the standard K-∈ [7] and experimental [6] is essential because the TBNN-based K-∈ model is compiled in OpenFOAM-5.0 after editing this model. The realizable algebraic Reynolds stress model directly takes the modeling equation for k (turbulent kinetic energy) and ∈ (dissipation rate) from the standard K-∈ model, which are ) ] [ ( νt k, j k,t + U j k − ν + = P + B− ∈ (6) σk ,j )] [ ( ε ε2 νt ε (7) ε,t + U j ε − ν + = C1 P − C2 + C1 (1 − C3 ) B σε , j k k k where νt = Cmu
K K2 2/3 , Cmu = ,ζ = W ε A1 + η + αζ ε
(8)
νt is turbulent viscosity. (m),t represents derivative of (m) with respect to time, and (m),i represents spatial derivative of (m) with respect to x i . Here, ‘m’ represents any arbitrary variable. P is the turbulence production term due to shear, while B is the turbulence production term due to buoyancy. Standard and machine learning based models are modified to account for this aspect. B is modeled using gradient diffusion hypothesis as in [7], according to which it is dependent on the total Reynolds stress B = (β)
( )( )( ) ) ∂T 2 k ( Cmu ui u j gj 3 σt ε ∂x j
(9)
Other than the order of mean strain rate and rotation rate tensor, another significant difference between algebraic stress and the standard K-∈ model is the factor C mu . This factor is constant in the standard K-∈ model and variable in the algebraic stress model to follow the realizability criteria defined by Shih et al. [1].
2.2 Machine Learning Methodology Integration of the standard RANS method with machine learning has been done with the help of a tensor basis neural network (TBNN). TBNN was formulated by Ling et al. [11], which is an artificial neural network. The purpose of this network is to predict the Reynolds stress anisotropic tensor by taking highly accurate data for training. The Galilean invariance is embedded into the predicted Reynolds stress anisotropic tensor. The purpose for embedding the Galilean invariance is to ensure that the anisotropic tensor is rotated by the same amount when the coordinate frame is
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Fig. 1 TBNN architecture
rotated. Without invariance property, the machine learning model will make different predictions of the same flow if its direction is changed. Figure 1 shows the basic structure of TBNN. There are two input layers. One is for the five tensor invariants, and the other is used to input the ten tensor basis. The first input layer is followed by some optimized number of hidden layers, which apply the activation function to the input parameters. The final hidden layer has ten nodes representing ten values of g(n) which will be merged with ten tensor input layer T (n) to give the output as. b=
10 ∑
g (n) (λ1 , λ2 , . . . . . . , λ5 )T (n)
(10)
n=1
where b is the normalized anisotropic Reynolds stress tensor. Equation 10 for Reynolds stress anisotropic tensor was given by Pope [8]. Five tensor invariants (λ1 , λ2 , . . . , λ5 ) are the scalar functions of the normalized mean strain rate tensor (Sn ) and rotation rate tensor (Wn ). The tensors of T (n) (n = 1 to 10) are represented in terms of Sn and Wn as T (1) = Sn T (2) = Sn Wn − Sn Wn T (3) = Sn2 − 1/3 I. T r (Sn2 ) T (4) = Wn2 − 1/3 I. T r (Wn2 ) T (5) = Wn Sn2 − Sn2 Wn T (6) = Wn2 Sn + Sn Wn2 − 1/3 I. T r (Sn2 Wn ) T (7) = Wn Sn Wn2 − Wn2 Sn Wn
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T (8) = Sn Wn Sn2 − Sn2 Wn Sn T (9) = Wn2 Sn2 + Sn2 Wn2 − 1/3 I. T r (Sn2 Wn2 ) T (10)= Wn Sn2 Wn2 − Wn2 Sn2 Wn
(11)
Various scalar invariants functions can be given as λ1 = T r (Sn2 ) λ2 = T r (wn2 ) λ3 = T r (Sn2 ) λ4 = T r (wn2 Sn ) λ5 = T r (wn2 Sn2 )
(12)
Calculating the values of the scalar coefficients g(n) (λ1 , λ2 , . . . , λ5 ) is the main purpose of the neural network. These coefficients are used in the nonlinear stress equation to make a new turbulence model in OpenFOAM-5.0.
2.3 Computational Methodology The computational domain under the study is a three-dimensional box with dimensions as L x = 0.91 m, L y = 0.91 m, and L z = 0.6 m with a source of plume at the center indicated by a patch as shown in Fig. 2 To compare the results with the experimental data of Mirajkar et al. [6] and RANS-based data of standard K-∈ for the turbulent forced plume, the same flow parameters are used in the study. wo = 0.22 ms−1 is the initial velocity in the vertical direction from the source, which corresponds to Re = 3100. All other components of velocity are zero. The stratification strength of the ambient fluid (N ∞ = 0.4 s−1 ) is used. A constant temperature T = 316 K is used at the source. The diameter (d) of the source is 0.0127 m. At the source, a fixed velocity is used as the boundary condition, and the no-slip boundary is applied at the bottom boundary. The domain’s lateral and top boundaries are treated with a zero-gradient boundary condition. At the source, a constant temperature is used, while the zero gradient is used at all other boundaries. LES data for turbulent plume is used for the training of the TBNN. The overall domain consists of approximately 7 million grid points. This large number of data points can slow the training of the neural network and may also lead to overfitting and generation of bias, as most of the data have negligible information about the flow characteristics. We choose a subdomain to pick the data points that can significantly describe the turbulent plume’s behavior. Finally, the data set is reduced to 1.7 million. We tested different configurations of the neural network and its parameters for the sample data set, which is a proportional extraction of the complete data set. The change in learning rate does not affect the loss versus epoch curves. The larger batch
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Fig. 2 Computational domain
sizes are used to change the higher learning rate curve to a good learning rate curve. It is also observed that increasing the number of hidden layers and the number of neurons per hidden layer helps decrease the R.M.S error value between the predicted and true value of the Reynolds stress anisotropic tensor. The comparison for the selection of the neural network configuration is made based on the root mean square (RMS) error value as in Table 1 (numerically) and with the prediction plot over the test data (visually) in Fig. 4. Network 1 shown in Table 1 is 24 hidden layers with 88 nodes per hidden layer with 55,000 batch size, Network 2 is 24 hidden layers with 104 nodes with 65,000 batch size, and Network 3 is 32 hidden layers with 128 nodes in each hidden layer with 80,000 batch size. Network 2 is selected as it has less R.M.S error for the diagonal components and approximately similar error for the diagonal elements as compared to Network 1. For the network with 24 hidden layers and 104 nodes per hidden layer, the loss versus epoch curve and output prediction plot for test data are shown in Figs. 3 and 4, respectively. Figure 3 shows the convergence of training and validation loss, representing a good fit curve. Figure 4 shows the alignment of true and predicted data to a diagonal line with a slope as one. Over this line, the true and predicted values Table 1 R.M.S error value for different networks
Stress components
Network 1
Network 2
Network 3
A00
0.0592
0.0583
0.0606
A01
0.0514
0.0518
0.0520
A02
0.0571
0.0575
0.0579
A10
0.0514
0.0518
0.0520
A11
0.0765
0.0752
0.0797
A12
0.0474
0.0478
0.0483
A20
0.0571
0.0575
0.0579
A21
0.0474
0.0478
0.0483
A22
0.0568
0.0559
0.0584
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Fig. 3 Loss versus epochs curve
are equal. The important hyper parameters which are used to optimize the TBNN algorithm for forced plume data set are in Table 2.
Fig. 4 Output prediction plot over test data
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Number of hidden layers
24
Number of nodes per hidden layer
104
Loss function
Mean square error
Total data set
17.5 × 105
Split fraction
0.7
Optimizer
ADAM
Batch size
65,000
Learning rate
0.01
3 Results and Discussions A grid-independent mesh of 1.34 million cells used by Kumar et al. [7] previously in their work for analyzing different variants of K-∈ model has been utilized. The stratification strength N ∞ used is 0.4 s−1 . The data for same value of N ∞ is used for the training of TBNN algorithm. We modified the standard Shih’s quadratic model to develop a new turbulence model for the turbulent forced plume referred to as TBNN-based K-ε model. The nonlinear stress equation is coded in reference to the Pope [8] in this model. The ML coefficients, generated from the TBNN code after training the neural network with the generated LES data, are used as the closure coefficients for solving the Reynolds stress equation. The simulations are performed in OpenFOAM-5.0. The modified solver named as buoyant Boussinesq Pimple Foam is used for the simulations. The solver is modified by Kumar et al. [7] to capture ambient stratification’s effects. Simulations are performed for a time of 160 s. Results are time-averaged from the span of 140 s to 160 s. The comparisons of the results were made obtained from the Shih’s quadratic model, the standard K-ε model by Kumar et al. [7], the experimental results by Mirajkar et al. [6] and the TBNN-based K-ε. In the comparison, we have considered the mean velocity profile and the maximum height reached by the plume. The maximum height of the plume is the height where the mean velocity becomes zero. All plots drawn represent the normalized (⟨mean ⟩) velocity versus the normalized height. The vertical center line mean velocity U y is normalized using the initial source velocity (wo ), and the height (Z) is normalized using the source diameter (d). When released from the source at a higher temperature and lesser density than the ambient environment, the plume rises due to both momentum and buoyancy effects. The ambient environment has varying density. Near the source, momentum effects are dominant, while away from the source, buoyancy effects are dominant. As the plume begins to rise, there is constant entrainment of the cold fluid into the plume from the ambient. This process makes the plume denser while rising. The height where the plume’s density becomes equal to the ambient density is known as neutral height. The maximum height reached by the plume is greater than the neutral height. The inertia in plume surpasses the neutral height and then falls back.
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Fig. 5 Mean vertical velocity contour with normalized height (Z/d) a Experimental [6] b K-ε model [7]
Fig. 6 Shih’s quadratic model mean vertical velocity contour plot
Figures 5 and 6 represent the mean vertical velocity contour plots of the turbulent forced plume obtained from the experimental data of Mirajkar et al. [6], standard K-∈ model, and Shih’s quadratic model. The mean vertical velocity from the two models is compared with that of experimental data. It can be observed that the red color portion in mean vertical velocity contour plot of Shih’s quadratic model is present up to higher height than that of the experimental and standard K-ε model. Hence, the mean velocity near the source is over predicted for the quadratic model.
3.1 Validation of the Model The TBNN algorithm is trained with LES data for the turbulent forced plume with a stratification strength (N ∞ ) of 0.4 s−1 . For this N ∞ value, there was an overall
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Fig. 7 Normalized mean center line velocity profile comparison between standard K-ε, TBNNbased K-ε model, and experimental results for N ∞ = 0
improvement in predicting the results for mean velocity profile and maximum height reached by plume with the TBNN-based K-ε model. To validate our model, we used the same TBNN-based K-ε model, which we formulated for flow with N ∞ = 0.4 s−1 to the turbulent forced plume with N∞ = 0 (Fig. 7). The observations made from the contour plots in Figs. 5 and 6 can also be verified from the normalized center line velocity plot shown in Fig. 8. The standard Shih’s quadratic model predicts higher values for the mean velocity in the vicinity of the source. The center line mean velocity contour plot for TBNN-based K-∈ model is shown in Fig. 8. This contour plot matches the experimental data contour more than any standard turbulence model. The TBNN-based K-∈ model has significantly improved overall results for the mean velocity profile and maximum height reached by the plume. The maximum height predicted by the linear and quadratic models given in Table 3 has an error up to 14% when compared with the experimental maximum height. However, the maximum height prediction by TBNN-based K-∈ is very close to the experimental results with an error of 0.26% as given in Table 3.
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(⟨ ⟩ ) Fig. 8 Normalized mean center line velocity U y /wo versus normalized height (Z/d) comparison for standard K-ε, Shih’s quadratic model, TBNN-based K-ε, and experimental for N ∞ = 0.4 s−1 . The error bar represents that values can vary in this range
Table 3 Maximum height (in cm) attained by plume at 0.4 stratification strengths
Turbulence models
Maximum height attained
% Error over experimental
Standard K-ε
26.87
13.99
Shih’s quadratic model
27.36
12.42
Experimental [6]
31.24
–
Theoretical height [12]
30.60
2.05
TBNN-based K-ε
31.32
0.26
4 Conclusion In this paper, we present the modified RANS model for the turbulent forced plume using the TBNN-based turbulence model. The evaluation of different neural network configurations was done based on the R.M.S error between true and the predicted value of the Reynolds stress tensor for the test data. The LES data of forced plume stratification strength of 0.4 s−1 was used for training and testing the neural network. The coefficients generated from the TBNN were implemented in the K-∈ model to compile a new TBNN-based K-∈ model. The evaluation of TBNN-based K-∈ model significantly improved the normalized mean velocity profile and maximum height prediction of the forced plume at N∞ = 0.4 S-1 . The TBNN-based K-∈ was tested on the flow with N∞ = 0, and the agreement was very good with the experimental data as well as the standard K-∈ model results.
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References 1. Shih TH, Zhu J, Lumley JL (1995) A new reynolds stress algebraic equation model. Comput Methods Appl Mech Eng 125(1):287–302 2. Morton BR (1959) Forced plumes. J Fluid Mech 5(1):151–163 3. Fox DG (1970) Forced plume in a stratified fluid. J Geophys Res 75(33):6818–6835 4. Bloomfield LJ, Kerr RC (2000) A theoretical model of a turbulent fountain. J Fluid Mech 424:197–216 5. Mirajkar HN, Balasubramanian S (2017) Effects of varying ambient stratification strengths on the dynamics of a turbulent buoyant plume. J Hydraulic Eng 143(7):04017013 6. Mirajkar HN, Mukherjee P, Balasubramanian S (2020) PIV study of the dynamics of a forced plume in a stratified ambient. J Flow Vis Image Proces 27(1) 7. Kumar N, Mukherjee P, Chalamalla VK, Dewan A, Balasubramanian S (2022) Assessment of rans-based turbulence model for forced plume dynamics in a linearly stratified environment. Comput Fluids 235:105281 8. Pope SB (1975) A more general effective-viscosity hypothesis. J Fluid Mech 72(2):331–340 9. Brunton SL, Noack BR, Koumoutsakos P (2020) Machine learning for fluid mechanics. Annu Rev Fluid Mech 52:477–508 10. Gholami A, Bonakdari H, Zaji AH, Akhtari AA (2015) Simulation of open channel bend characteristics using computational fluid dynamics and artificial neural networks. Eng Appl Comput Fluid Mech 9(1):355–369 11. Ling J, Kurzawski A, Templeton J (2016) Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J Fluid Mech 807:155–166 12. Morton BR, Taylor GI, Turner JS (1956) Turbulent gravitational convection maintained and instantaneous sources. Proc Roy Soc Lond Series Math Phys Sci 234(1196):1–23
Comparative Study of Future State Predictions of Unsteady Multiphase Flows Using DMD and Deep Learning Neil Ashwin Raj, Danesh Tafti, Nikhil Muralidhar, and Anuj Karpatne
Abstract Flow across an array of solid obstructions is a common phenomenon observed in many applications such as multiphase flows, heat exchangers, and environmental flows. In this work, we aim to train deep learning models and to predict the time evolution of unsteady flow fields in a domain of randomly arranged 2D cylinders at Reynolds number 50. Two different approaches are used and compared in this paper, dynamic mode decomposition (DMD) which is a dimensionality-reduction algorithm based on singular value decomposition (SVD) and long short-term memory (LSTM) neural networks. In both cases, the model is trained on the first 165 time steps and then is tested on predicting the next 300 time steps. Two flow fields with different spectral characteristics are used to compare the performance of the two techniques. The LSTM architecture owing to its ability to learn nonlinear dynamics performs better than the DMD algorithm in the case with more temporal time scales present. Keywords Dynamic mode decomposition (DMD) · Convolutional auto-encoder (CAE) · Long short-term memory (LSTM) · Multiphase flow · Unsteady flow · Deep learning
N. A. Raj · D. Tafti (B) Department of Mechanical Engineering, Virginia Tech, Blacksburg, USA e-mail: [email protected] N. Muralidhar Assistant Professor, Department of Computer Science, Stevens Institute of Technology, Hoboken, USA A. Karpatne Department of Computer Science, Virginia Tech, Blacksburg, USA © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_76
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1 Introduction Fluid flow through an arrangement of structures is a common phenomenon observed in various industrial processes, biological phenomena, and flow in urban environments. Flow past protrusions such as cylinders or cuboids are very common in heat exchangers and have thus been extensively studied using various techniques since the 1980s [1]. Flow over various structures in an urban environment is necessary to study processes such as contaminant dispersion and breathability [2]. Flow through a random array of structures can lead to complex interactions of wakes and resulting flow fields and therefore making it worthwhile to explore methods that can be used to calculate these unsteady flow fields, other than solving the full Navier–Stokes equations or through experimental measurements. In this work, we describe and compare two different techniques to predict the time evolution of unsteady flow— dynamic mode decomposition (DMD) and a deep learning architecture consisting of long short-term memory (LSTM) and convolutional auto-encoder (CAE) neural networks. Dynamic mode decomposition was first introduced by Schmid [3]. It is a dimensionality reduction algorithm, which approximates a nonlinear evolving dynamical system as a solution to a linear ODE. DMDs can be used to analyze dynamical systems by extracting relevant spatially coherent structures “modes” and the temporal evolution of these modes over time. DMD has been used to study many unsteady fluid flow problems [4]. Liu et al. [5] have used proper orthogonal decomposition (POD) and DMD in a rotary multiphase flow pump. DMD has also been used in studies to predict the future state of fluid mechanical processes. Gryzlov et al. [6] compared LSTMs and DMDs for virtual flow metering (VFM, a petroleum industry term referring to a method to calculate flow rate, without measuring flow rates directly). They used two time series inputs, the pressure, and temperature, to predict the time evolution of liquid and fluid flow rate and predict their future states, and concluded that the DMD was more accurate. Huang et al. [7] use three methods DMD, Conv-LSTMs, and a combination of both to learn the time evolution of unsteady flow in a T-junction and found that the model combining both methods performed the best. Deep learning refers to a class of techniques in artificial intelligence that are based on the biological neuron. Deep learning models are good at discovering intricate features in high-dimensional and nonlinear datasets and have thus found use in many spheres such as financial predictions, speech and image recognition, and natural language processing. One of the earliest descriptions of recurrent neural networks can be found in the work by Rumelhart et al. [8]. Unlike standard feed-forward networks, RNNs retain a hidden state which can retain some ‘memory’ from the input sequence. Another feature of recurrent neural networks is their ability to deal with inputs or generate outputs of varying lengths, which is not the case with other standard feed-forward networks. This has made these networks very popular in times series predictions, particularly for tasks such as translation, statistical language modeling, sentimental analysis, stock predictions, weather predictions, image captioning, and video predictions [9–14]. Long short-term memory (LSTM) was first introduced
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by Hochreiter [15]. LSTMs are a type of neural network architecture that contain an additional state along with the hidden state called the cell state. The cell state can retain long-term information. LSTMs have been used successfully for many problems involving fluid mechanics. Hasegawa et al. [16] have used convolutionalauto-encoders (CAE) and long short-term memory (LSTM) networks to train on unsteady vortex shedding of 80 different randomly shaped bluff bodies. The CAE was used to get lower dimensional representations of flow field snapshots which were then utilized by the LSTM to learn the time evolution of the flow field. Hasegawa et al. [17] studied a similar problem, where a similar CAE-LSTM model was used to train over the snapshots of vortex shedding behind a cylinder at different Reynolds numbers. They observed that the model did not perform well in situations where the flow field structure changed compared to the training flow structure. The combination of CAE and LSTMs has also been used by Romit et al. [18] to study two different problems, the Burger’s equation and shallow water equation. Similar to the previous work, the CAE was used to get a lower-dimensional representation of the time snapshots, and the LSTM was employed for predicting future states. Further, for the Burger’s equation, they did a parametric interpolation, where they appended the viscosity associated with the Burger’s equation solution to the latent space vector before using it in the LSTM. The trained network performed well on unseen Reynolds numbers. This work also compared the predictions of the CAE-LSTM with a proper orthogonal decomposition (POD) with Galerkin projection (GP) method and concluded that the CAE-LSTM model performed better for the same compression ratio from real space to the reduced space. Mohan et al. [19] used a combination of reduced order modeling and deep learning. They used POD and GP to project temporal data to a reduced space consisting of spatial modes and time coefficients. A linear combination of the spatial modes and time coefficients are then used to reconstruct the temporal data in the original space. They used LSTMs to learn the temporal variation of the time coefficients and discussed the performance of the trained model on two different datasets, isotropic turbulence, and magnetohydrodynamic turbulence. The objective of this paper is to compare the performance of the DMD and CAELSTM to learn the dynamics of two flow fields of different spectral content. One of them is close to periodic, while the other is much more chaotic.
2 Methods In this section, a brief description of the particle resolved simulations (PRS) used to generate the x-velocity field data, and the two different time projection methods used in this study, DMD and CAE-LSTM.
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Fig. 1 Calculation domain with random arrangement of cylinders
2.1 PRS Simulations Two different random arrangements of 2D cylinders are simulated with packing/solid fraction ratio of 0.1 and Reynolds number (based on the diameter of the cylinder and the mean velocity) of 50. The packing fraction is defined as the ratio of the area of the cross section of all the cylinders to the total area of the computational domain. The domain is of size 10 × 10, and each cylinder is of diameter 1. The computational domain is shown in Fig. 1. A 401 × 401 structured grid is used in the simulation. All the simulations are performed using an in-house code GENIDLEST (generalized incompressible direct and large eddy simulation of turbulence) [20]. The immersed boundary method (IBM) is used for modeling the cylinders. All four boundaries are set as periodic. A constant pressure gradient is imposed in the flow direction along the x-axis to drive the flow, and the reference velocity is adjusted to set the Reynold number to 50. Once the nominal Reynolds number is reached, 465 snapshots of the flow field are saved every 20 time steps at a non-dimensional period of 0.06.
2.2 Dynamic Mode Decomposition The DMD method provides a spatiotemporal decomposition of data into a set of dynamic modes that are derived from snapshots or measurements of a given system in time [21]. The schematic of the DMD algorithm is shown in Fig. 2. First, the data is flattened and stacked as shown in the figure, to format the snapshot matrix X = {x1 , x2 , . . . , x M } where every column of size N is a flattened snapshot at
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time step M. Similarly, the snapshot matrix starting from the next time step is also constructed X = {x2 , x3 , . . . , x M+1 }. It is to be noted that these snapshot matrices are usually very tall matrices where N > M. The aim of DMD is calculating the matrixA, such that X = AX . The A matrix can be calculated by taking the pseudo (Moore–Penrose inverse, shown here as any tensor with superscript +) of X to get A = X X + . However, this is computationally expensive, considering that N = 4012 (PRS spatial resolution) and M = 165. In this case, the size of matrix A will be 4012 × 4012 . Thus, we project A to a reduced space to get A˜ using the relation A˜ = U + AU , where U is the matrix housing the orthogonal modes obtained from the singular value decomposition (SVD) of the first snapshot matrix, X = U V + . The SVD results in a product of three different matrices, the first is an orthogonal matrix U with each column representing an orthogonal mode of the dynamical system. The is a diagonal matrix of singular values arranged in a hierarchical/descending order like σ1 ≥ σ2 , · · · , σr , where r is any value lesser than M based on the number modes of modes chosen, these singular values are representative of the energy distribution of the system. The final matrix V + is also orthogonal like U , every column of V + representing the time dynamics/evolution of a mode. ˜ given by A˜ = U + AU , is modified, so that A is not used The definition of A, directly. Using X = U V + and X = AU V + , A = X V −1 U + . Using this ˜ we get A˜ = U + X V −1 . Performing relationship for A and the definition of A, ˜ ˜ ˜ the DMD modes are obtained by an Eigen decomposition on A to get AW = λ A, ∼
using the relation = X V −1 W . Finally, the eigenvalues of A, λ and can be combined to get, X = eλt b, where b is found at t = 0, X t=0 = eλ0 b thus giving b = + X t=0 .
Fig. 2 Schematic of DMD algorithm
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2.3 CAE-LSTM The first step in the CAE-LSTM architecture is the CAE or convolutional autoencoder. In a most basic sense, a CAE is a deep learning model which learns the identity function. An auto-encoder with only linear layers solves an identical minimization problem as principle component analysis (PCA) without the orthogonality constraint [22]. A basic auto-encoder has the same number of input nodes as output nodes and its task is to reconstruct the input. A convolutional auto-encoder as used here has convolution layers which retain spatial structure unlike fully connected layers as in a traditional auto-encoder. The CAE is used to reduce the full-scale flow field snapshot (S) of size 4012 to a latent vector (l) of size 128. Figure 3 shows the CAE architecture used, and the part encoding the input to a lower dimensional space is referred to as the CAE-encoder and the part reconstructing the latent space back to the full-scale images is the CAE-decoder. The loss function used here is mean squared error loss b MSECAE =
i=1 (SGround Truth
− SPredicted )2
b
where b is the batch size of 16. The optimizer used is Adam [23], with a learning rate of 5 × 10−5 . A different CAE model is trained for each of the two different particle arrangements using the first 165 snapshots of the x-velocity field. A LSTM cell as shown in Fig. 4 consists of three gates, the forget gate, input gate, and the output gate. At any time instant, an input to an LSTM is the cell state from the previous time step (Ct−1 ), the input from the current time step (lt ), and the hidden state from the previous time step (h t−1 ). The forget gate is represented by f t = σ (W f · [h t−1 , lt ] + b f )
Fig. 3 Schematic of CAE encoder and decoder
(1)
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Fig. 4 Schematic of CAE-LSTM architecture
where W f is a weight matrix, operating on both the hidden state from the previous time step and the input from the current time step; the output of this gate is multiplied with the previous cell state and added to the outputs of the input gate which is described next. σ is the sigmoid activation function which is defined as σ =
1 1 + e−x
(2)
The sigmoid function returns values between zero and one. The cell state contains the long-term memory of the LSTM, and the forget gate’s weight matrix learns to retain only necessary information, which it does by deleting or ‘forgetting’ information in the cell state. The next gate is the input gate which consists of two different operations, shown in the equations. i t = σ (Wi · [h t−1 , lt ] + bi )
(3)
C˜ t = tanh WC · h t−1 , lt + bC
(4)
where tanh is the hyperbolic tangent function. The product of the above two operations i t · C˜ t is then added to f t ∗ Ct−1 , creating a new cell state. Finally, the output state is calculated using the new cell state, the previous hidden states, and the current input in the following manner. ot = σ Wo · h t−1 , lt + bo
(5)
h t = ot ∗ tanh(Ct )
(6)
In our implementation, the input tensor at every time step are the latent vectors obtained from the CAE, specifically we input the latent spaces of five consecutive
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time-steps to predict the next five time-steps, i.e., the LSTM learns to map lt=n...n+5 to lt=n+6...n+10 . This dataset of five input to five output time steps is generated from the first 165 time steps of the PRS simulations. After the network is trained, the LSTM is just provided the first five time-steps (1–5), it uses them to predict the next five time steps, which are then used as input to predict the next five time-steps, and so on till the 465th time step, thus projecting 460 time steps into the future as we will be shown in the results section. The LSTM architecture by itself can be divided into an encoder and decoder part, the encoder receives lt=n,...,n+5 and learns how to efficiently encode this time series data in the form of hidden and cell states at time tn+5 , these states are then used as inputs to the decoder part of the LSTM. The decoder LSTM then predicts the latent vectors for the next five-time steps, which are compared to the ground truth latent vectors to calculate the loss function. The LSTM architecture used here is three layers deep and has a hidden and cell state size of 256, and once again the optimizer used is Adam [23], with a learning rate of 5 × 10−5 for ~ 2000 epochs. Similar to the CAE, the mean squared error loss function is used with a batch size of 4. b (lGround Truth − lPredicted )2 MSELSTM = i=1 b A different LSTM model is trained for the two different flow fields using the latent spaces obtained from the corresponding CAE model.
3 Results and Discussion 3.1 Flow Field Characterization Based on the random arrangement of the cylinders, the flow field can exhibit a range of behaviors from cyclic to chaotic. To quantify the state of the flow, the number of dominant frequencies is calculated for each of the two flow fields. To calculate the number of dominant frequencies, n points are randomly selected in the flow field and the x-velocity variations over time are recorded. A fast Fourier transform is performed on these n signals, and the n power spectrums obtained are then summed and scaled with the value of the most powerful frequency. The number of dominant frequencies above a cut off value of 0.01 value is then counted. The larger the number of frequencies above this cut off value, the larger the deviation from cyclic flow. The result of this exercise is presented in Fig. 5 for the two flow fields together with the corresponding temporal evolution of x-velocity. Clearly, the flow field in Fig. 5b (Case B) is more chaotic and has a wider range of time scales compared to the flow field in Fig. 5a (Case A). The number of dominant modes associated with the first and second flow fields are ~ 5 and ~ 18, respectively (the entire spectrum has not been plotted as there are not many dominant modes at higher frequencies).
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(a)
(b) Fig. 5 X-velocity signals and power spectrums of two different flow fields a Case A; b Case B
These two data sets are used to test the performance of the DMD and CAE-LSTM to predict the future dynamic states of the system.
3.2 Prediction Accuracy The accuracy of prediction is estimated by defining the instantaneous mean error as a difference between the predicted field and the ground truth normalized by the mean x-velocity of the flow and is defined as 1 u i,ground truth − u i,predicted MRE = N u where the summation is over all cells (N ) in the domain and u is the mean x-directional velocity.
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Fig. 6 MRE versus time using DMD and CAE-LSTM for low with few dominant modes (Case A)
3.3 Future State Predictions for Case A Figure 6 shows the MRE versus time for both the DMD and LSTM predictions for Case A. As expected, both the models perform very well in predicting future states in this flow field—the errors are less than 2% at each time instant, noting that the errors are approximately of the same magnitude during training and prediction and they grow very gradually during the prediction stage from time step 166–465. The MRE is much lower for DMD compared to LSTM predictions, which are of the order ~ 10–4 and ~ 10–2 , respectively. As mentioned earlier while the latent space vector size for the LSTM is 128, the DMD uses the first 25 modes. The reason that the LSTM MRE is much larger than DMD is due the reconstruction error of the trained CAE. While training the CAE to reduce a high-resolution flow field to a latent space vector and reconstruct it back to original space, the CAE can only reconstruct the image with an MRE ~ 10–3 ; thus, no matter how well the LSTM is trained it will not be able to predict flow fields with accuracies better than ~ 10–3 . The DMD on the other hand does not have this constraint.
3.4 Future State Predictions for Case B Figure 7 compares the prediction accuracy of the two methods for Case B which has ~ 18 dominant frequencies. Here, the prediction accuracy of the number of modes including in the DMD is also evaluated—25 and 75 modes are included for evaluation. Firstly, there is a marked difference in accuracy between the training stage and prediction stage, and the prediction accuracy decreases with time. Note that during prediction only, the first five time steps (1–5) use the ground truth fields, after which the predicted fields are used to project forward in time (time steps 6–465), unlike other studies which periodically interject the prediction stage with ground truth values
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Fig. 7 MRE versus time using DMD and CAE-LSTM for chaotic flow (Case B)
during the prediction stage, thus steering the prediction toward the ground truth. Counter to expectations, it is found that while DMD-75 has much better accuracy than DMD-25 during the training phase, and does better during the early stages of prediction, it starts deviating much more from the ground truth during the later prediction stages. This could be because during training, DMD-75 is exposed to more high frequency content which could be overshadowing the low frequency modes which play a critical role in long-term predictions, it could also be due to the presence of modes with diverging temporal coefficients. Between the three methods, CAELSTM performs the best keeping mean errors within 15%. Figure 8 presents the x-velocity signal at one location in the flow field to compare predictions to the ground truth x-velocity. All three models are quite accurate in predicting the training data, but start to deviate from each other and from the ground truth after 300 time steps. As reflected in the mean errors, CAE-LSTM does the best in long-term prediction, followed by DMD-25. DMD-75 exhibits considerable deviation from the ground truth during the later stages of projection during which the mean errors exceed 20%.
4 Conclusions Two different techniques DMD and CAE-LSTM were used to learn the temporal evolution of two statistically different 2D unsteady flow fields in a random arrangement of infinitely long cylinders. The DMD and CAE-LSTM were then used to project ahead in time to predict the future state of the system and compared to the ground truth values. It was found that both the LSTM and DMD performed well on flow fields with low spectral content with prediction errors confined to very low
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Fig. 8 X-velocity signal versus time for the three methods and ground truth data
values (within 2%). When it came to the flow with a wide range of temporal scales, the LSTM performed better than DMD with 25 modes, and much better than DMD using 75 modes. The accuracy of the CAE-LSTM was within 15% of ground truth, whereas DMD with 75 modes exceed 20% toward the end of the prediction stage. It is suspected that DMD with 75 nodes overfits the high-frequency modes in the training set and does not generalize well over long-term predictions which would be more dependent on learning low frequency content. Therefore, it is concluded that to project accurately to future states, it is important to select the optimal number of modes so as not to overfit the training data time frames or include erroneous modes with diverging temporal coefficients. Similarly, the size of the latent space vector in CAE-LSTM could also have an impact which was not investigated in this work. Acknowledgements We would like to thank the Virginia Tech ARC (Advanced Research Computing) for providing the computational resources required for this work.
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Deep Learning Approach to Predict Remaining Useful Life of Axial Piston Pump Md Adil and Pratik Punj
Abstract In the hydraulic machines like hydraulic pump, while in use, many faults start to appear in these machines resulting in the undesirable output. One of such parameters is the leakage fault, which is very common in pumps and motors. In pumps, it has been observed that over a period of time with increase in wear, leakage also increases. This relationship between the wear and leakage value is utilized here using the NARX NN to estimate the RUL of the pump. In this work, the different training algorithm has been used in order to train the NARX NN, and the models that has been trained are used to estimate the remaining useful life (RUL) of an axial pump that is being used to control the hydraulic system of sheet metal casting process. Results of the model are promising. Keywords Hydraulic pump · Failure · Wear · Life prediction · Preventive maintenance · Forecasting · Leakage · Deep learning · ANN · NARX
1 Introduction Axial pumps are among the most vital component the hydraulic systems [1]. It is a positive displacement pump [2] which needs to be compact and lightweight in order to fulfill the need of various industries. Among all the available methods, the most effective and widely used one to reduce the weight/power ratio is by increasing the working pressure. But the force experienced by the piston cylinder pair is very high when working under extremely high pressure which increases the possibility of the wear by significant amount [3, 4]. Excessive wear could lead to the detreating condition for the oil suction and discharge which could ultimately result in failure of the pump. In order to take precautionary measures and avoid the machine breakdown, M. Adil (B) Department of Mechanical Engineering, Jadavpur University, Kolkata, India e-mail: [email protected] P. Punj Department of Mining Machinery Engineering, IIT (ISM) Dhanbad, Dhanbad, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_77
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we need to be aware of the wear condition of the machine components. For scenario like this, preventive maintenance comes handy. Some principal activities like condition monitoring, diagnosis, and prognosis of faults are being performed in the field of the maintenance engineering of mechanical system [5, 6]. Nowadays, prognosis and fault diagnosis are attracting a lot of researcher’s attention. Most of these efforts are going into increasing the performance and the reliability of the model to predict the fault in the machine before its actual breakdown. Bedotti et al. [7, 8] established a model for the axial pump on the basis of previous wear and friction data for fault diagnosis. These models are nonparametric type of model which uses the previous data in order to estimate the RUL of the component. Baptista et al. [9] established a model to forecast the fault in the component to ensure the proper maintenance on time. They compared the result of various data driven strategies using an ARMA model technique and computing. Fung et al. [10] predicted the roundness error of the cutting tools. RUL forecast accuracy could help in avoiding the unexpected breakdown of the machines. Even defect like wear, rupture, and other flaws can have a very big effect on the performance of the mechanical equipment and can even result in the breakdown [11, 12]. Enrico et al. [13] developed a data-driven fuzzy model to estimate the RUL of nuclear system due to dynamic failure. This paper focuses on applying NARX NN model for estimating the remaining useful life of the axial pump using the limited data that is measured at the industrial site. Many researchers have been done to do the fault diagnosis of the pump using different methods. Du et al. [14] have developed a layer clustering algorithm to diagnose the aircraft axial piston pump. Rivera et al. [15] established the technique for estimating the maintenance of the pump. Abd Kadir et al. used the basic feed-forward neural network for predicting the RUL [16]. They used the vibration as the criteria of failure and used it to predict the point of failure or in other words calculated the remaining useful life of the rotating machines. Deep learning is now one of the most popular methods for big data process and its analysis as it gives useful data which can be utilized for better prediction. Its multilayer structure further enhances its data processing capability. Great advance has been made in the building the architecture for deep neural network for applications in the domain of image, speech, and natural language recognition and processing. Now, its application also includes fault diagnosis on extraction of raw signal as well as time domain feature. Even though much of the success in the field of deep learning has been observed in the field of classification problem, deep learning has been found to be effective in predicting car traffic, weather, wind speed, tide, and Internet speeds. In this paper, deep learning-based method has been employed for axial piston pump’s remaining useful life prediction using its leakage data has been done. This approach is tested and validated using data collected from test. Results shows the promising RUL estimation performance.
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2 Dataset It has been observed that the amount of leakage can be a good indicator of the amount of degradation that the pump has experienced. As the wear increases, the volume of leakage also increases which can be used to predict the degree of the degradation in the pump. Data of the leakage flow rate of six axial pump is measured at the TATA steel (Jamshedpur, India). Average leakage value has been used for RUL prediction. Data set cover a length of 2-year 9 months, with an observation period between 2016 and 2019 at the interval of 15 days.
3 NARX NN Nonlinear autoregressive model with exogenous inputs (NARX) is a type of ANN which is composed of interconnected nodes which is inspired from the neural system. Each individual node receives one or more input, and it sums them to give output, which passes through an activation function, which is nonlinear for the case of NARX. These nodes represent artificial neurons. Based on the direction of information processing, different categories of ANN can be defined. When information flows only in the forward direction, such network are generally called as the feedforward neural network while, in network like NARX, information flows in both direction, which allow the connection between the neurons of the same layer as well as between the current and previous layer. Faster convergence to optimal connection weight between input and the neurons as well as the lower number of ladders to calibrate makes this model effective. The basic NARX model can be defined with the following equation: t(n + 1) = f [t(n), . . . , t(n − dt + 1); u(n), u(n − 1), u(n − du + 1)] Or in compact form, we can write this as: t(n + 1) = f [T (n); U (n)] where t(n) ∈ R and u(n) ∈ R represent the output and input of the model, respectively. Term dt > 1 and du > 1 represent the represent the output memory and input memory orders. Vector T (n) and U (n) represent the input and output regressor, respectively. NARX is a very powerful class of dynamic model which has been proven to be computational equivalent of the Turing machines.
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4 Training Algorithm 4.1 Levenberg–Marquardt (LM) LM algorithm is among the most widely accepted algorithm for ANN-based timeseries prediction. It gives the benefits of both the Gaussian–Newton and the steepest descent method. LM algorithm, search for the minima of the function F(z) and optimizes the solution. F(z) is the sum of square of nonlinear function. 1∑ F(z) = [ f i (z)]2 2 i m
Hessian matrix can be approximated by the LM method as: [ ]−1 T Δ w = J T (w)J (w) + ρ I J (w)e(w) Use of the Jacobian matrix J with that of the identity matrix term ρ I can ensure that the Hessian matrix is always positive. It results in the significant reduction of the computational cost.
4.2 Scaled Conjugate Gradient (SCG) SCG is generally used for feed-forward neural. SCG follows the concept of general optimization with some differences, like it chooses the step size and the search direction more effectively than the general optimization strategy using the secondorder approximation, the equation of which is represented as: 1 E(w + v) ≈ E(w) + E ' (w)T + v T E '' (w)v 2 In scaled conjugate gradient, each iteration gives the optimal distance. The line search then determines the optimum distance that need to be moved in the search direction as the given equation. wk+1 = wk + bk ∗ pk After this, next search direction is performed which is conjugate to the previous search instructions.
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4.3 Bayesian Regularization Bayesian regularization is an ANN algorithm that correct the value of weight and that of refraction based on Levenberg–Marquardt optimization. The algorithm first reduces the value of square of error and weight, then finds the optimal combination so as to give the optimal network. It introduces the network weight into the objective function of the training which can be represented as: F(ω) = αεω + βε D Following is applied to the Bayesian rule to optimize the value of the objective parameter α and β. P(α, β|D, M) =
P(D|α, β, M)P(α, β|M) P(D|M)
5 Performance Parameter In this paper, following error matrices are used to compare the accuracy of each model as well as to compare different deep learning architecture. Mean average error (MAE): MAE =
n 1∑ (error) n t=1
MSE =
n 1∑ (error)2 n t=1
Mean square error (MSE):
Root mean square error (RMSE): [ | n |1 ∑ RMSE = √ (error)2 n t=1 Mean absolute percentage error (MAPE): ( MAPE =
n 1∑ error n t=1 actual value
) × 100%
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where error = actual value−estimated value
6 Results The data measured between the year 2016 and 2019 were used for testing, training, and validation of NARX NN. First of all a network is trained using these data after this it is tested and then used to predict the future leakage. The proposed network is trained using the following training algorithms: Levenberg–Marquardt (LM), scaled conjugate gradient (SCG), and Bayesian regularization (BR). The training of the network is done on the criteria of MSE in which the network calculates the gradient and updates the weight to ensure that the network converges to the point of minimum error. Autocorrelation as shown in Fig. 1 is used to validate the network performance. It was getting influenced by the value of delays. So, the delays have been selected in such a way to ensure the values remain within the confidence level. In this study as can be seen in Fig. 1, auto-correlation remains within the 95% confidence level in the case of Levenberg–Marquardt and Bayesian regularization, while it is slightly off target in the case of scaled conjugate gradient but is still within a good degree of confidence level. Figure 2 shows the correlation between input and the errors. It is within the confidence limit at all the lags. This leads to the fact that the model has captured all the feature of the system, and input and output have been modeled accurately. Different algorithm has been chosen to train the network, and the predicted result of all the algorithm has been compared with the actual measured value. The failure is detected by defining the upper limit of the leakage. The main objective of the study is to evaluate the remaining useful life of the pump, so that timely action can be taken to increase its life as well as breakdown of the mechanical system can be avoided. Figure 3 shows the response of the network for the input data. All three algorithms have given pretty accurate result with error ranging between + 0.5 and −0.5 for some values which is quite good when compared with the value of the entire time series. Apart from the three algorithm, i.e., LM, BR, and SCG, some other algorithm was also used to forecast the data using the NARX network and compared with the actual result. Table 1 presents the parameter to evaluate the performance of the different algorithm. All the parameter have been calculated using the measured and forecasted data using the closed-loop NARX NN. As can be observed that, all the three algorithms that have been explained in detail in this paper have good accuracy in forecasting the time-series data of the axial piston pump. Among these, Bayesian regularization was most accurate followed by the Levenberg–Marquardt and then scaled conjugate gradient. The ranking based on the MAPE as this is the only performance parameter that is dimensionless which make it an ideal choice
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Fig. 1 Autocorrelation errors a LM b BR c SCG
to compare all the algorithm. In the term of other performance parameter also, the accuracy of forecasting is pretty good but the ranking changes to scaled conjugate gradient being most accurate followed by Bayesian regularization and Levenberg– Marquardt. It can also be observed that in term of the magnitude of the error, resilient backpropagation is more effective than one of the most widely used Levenberg– Marquardt algorithm. Table 2 contains the performance data of the open-loop NARX network which is used to train, test, and validate the data. Performance parameter for different algorithm is not consistent with the one that has been calculated using the forecasted value of the leakage which simply means that training performance do not guarantees a good forecasted value. But if the training performance is not good, then it will be tough for the network to predict a good result.
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Fig. 2 Correlation between input and errors a LM, b BR, and c SCG
Figure 4 shows the plot of predicted data of closed-loop NARX NN using different algorithm and its comparison with the measured data. Data predicted by all the algorithm are pretty off the target in the initial steps but goes on to predict pretty accurately later on, and this happens because in the initial steps, network fail to converge, but once it converges the prediction is pretty accurate as compared to the
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Fig. 3 Time-series prediction a LM, b BR, and c SCG
measured data. From the graph, it can be observed that the Levenberg–Marquardt take the most amount of time to converge followed by Bayesian regularization and scaled conjugate gradient, which is same as the sequence for the maximum RMSE value.
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Table 1 Different performance parameter for different training algorithm Algorithm
RMSE
MSE
MAE
MAPE
Bayesian regularization
0.31947
0.1020
−0.0521
−0.338
Levenberg–Marquardt
1.52368
2.3216
−0.0407
−1.536
Scaled conjugate gradient
0.07424
0.0055
0.0041
−3.16
Resilient backpropagation
0.24949
0.0622
0.0177
−1.684
Conjugate gradient with powell/beale restarts Fletcher–powell conjugate gradient
12.7041 5.11757
161.39 26.189
12.045 −4.3751
54.270 −31.19
Table 2 Different training, testing, and validation performance parameter for different training algorithm Algorithm
MSE
R
Bayesian regularization
0.0032134
Training: 0.9998 Testing: 0.99981
Levenberg–Marquardt
0.0041012
Training: 0.99872 Testing: 0.99897 Validation: 0.99927
Scaled conjugate gradient
0.031208
Training: 0.99979 Testing: 0.99988 Validation: 0.99985
Resilient backpropagation
0.10205
Training: 0.9983 Testing: 0.99766 Validation: 0.9973
Conjugate gradient with powell/beale restarts
0.003282
Training: 0.99965 Testing: 0.9993 Validation: 0.99956
Fletcher–powell conjugate gradient
0.005282
Training: 0.99971 Testing: 0.99947 Validation: 0.99985
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Fig. 4 Comparison of the forecasted and measured value a LM, b BR, and c SCG
7 Conclusion Wear pattern is dependent on the condition of the component and its running time which make it possible to use the leakage which always increase because of increase in wear with time. Using NARX NN, leakage behavior of axial piston pump with time has been analyzed. Different training algorithm was used to train the NARX NN, and their results have been compared; then, the trained network is used to predict the future leakage value. Network parameters like hidden nodes and time delay were varied to get the most optimum network. It has been observed that the scaled conjugate gradient is better in predicting the leakage as compared to Levenberg–Marquardt and Bayesian
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regularization. Scaled conjugate gradient also gave the least number wild forecasting as compare to the others. Result shows the good accuracy of the NARX NN in predicting the leakage of the axial piston pump which depends on the wear. However, as every ANN is heavily dependent on the amount of training data available, prediction accuracy can further be enhanced by increasing the size of training data set.
References 1. Yang H-Y, Pan M (2015) Engineering research in fluid power: a review. J Zhejiang Univ Sci 16:427–442 2. Manring N (2013) Fluid power pumps and motors: analysis, design and control, McGraw Hill Professional 3. Xia S, Zhang J, Ye S, Xu B, Huang W, Xiang J (2019) A spare support vector machine based fault detection strategy on key lubricating interfaces of axial piston pumps. IEEE Access 7:178177–178186 4. Gels S (2011) Einsatz konturierter und beschichteter Kolben-Buchse-Paare in Axialkolbenmaschinen in Schragscheibenbauweise 5. Li H, Zhang Y, Zheng H (2009) Hilbert-Huang transform and marginal spectrum for detection and diagnosis of localized defects in roller bearings. J Mech Sci Technol 23(2):291–301 6. Zhanqiang X et al (2017) Gear fault diagnosis under variable conditions with intrinsic timescale decomposition-singular value decomposition and support vector machine. J Mech Sci Technol 31(2):545–553 7. Bedotti A et al (2018) Dynamic modelling of the swash plate of a hydraulic axial piston pump for condition monitoring applications. Energy Procedia 148:266–273 8. Borutzky W (2020) A hybrid bond graph model-based-data driven method for failure prognostic. Procedia Manuf 42:188–196 9. Baptista M et al (2018) Forecasting fault events for predictive maintenance using data-driven techniques and ARMA modeling. Comput Ind Eng 115:41–53 10. Fung EHK, Chung APL (1999) Using ARMA models to forecast workpiece roundness error in a turning operation. Appl Math Model 23(7):567–585 11. Huang HZ et al (2015) Support vector machine based estimation of remaining useful life: current research status and future trends. J Mech Sci Technol 29(1):151–163 12. Choi Y-S, Lee K-H (2010) Investigation of blade failure in a gas turbine. J Mech Sci Technol 24(10):1969–1974 13. Zio E, Di Maio F (2010) A data-driven fuzzy approach for predicting the remaining useful life in dynamic failure scenarios of a nuclear system. Reliab Eng Syst Saf 95(1):49–57 14. Du J, Wang S, Zhang H (2013) Layered clustering multi-fault diagnosis for hydraulic piston pump. Mech Syst Signal Process 36(2):487–504 15. Rivera DL et al (2018) Towards a predictive maintenance system of a hydraulic pump. IFACPapersOnLine 51(11):447–452 16. Mahamad AK et al (2010) Predicting remaining useful life of rotating machinery based artificial neural network. Comput Math Appl 60:1078–1087
Machine Learning-Assisted Modeling of Pressure Hessian Tensor Deep Shikha and Sawan S. Sinha
Abstract Velocity gradient dynamics play a pivotal role in understanding various nonlinear phenomena in turbulent flows. In the evolution of velocity gradient dynamics, the pressure Hessian and the viscous Laplacian are two mathematically unclosed terms which need separate modeling. The current study models the pressure Hessian term using the tensor basis neural network (TBNN). The network is trained on direct numerical simulation (DNS) data of stationary incompressible turbulence conditioned on local flow topologies. We compare the topology-based TBNN model performance with the DNS results as well as with the unconditioned (raw) TBNN model. The model results are evaluated in terms of the strain rate and the pressure Hessian eigenvector alignments. The model captures some of the essential alignment features of the DNS results. Keywords Velocity gradient · Pressure Hessian · Machine learning
1 Introduction The study of velocity gradient dynamics evolution is important to understand the physics of various nonlinear processes like energy cascading [25], intermittency [11, 12], scalar mixing [6, 17, 23], and material element deformation [1, 8]. Indeed, a direct numerical solution (DNS) is one of the most comprehensive ways to access the pressure Hessian and the velocity gradient tensor using the gradient operator as a postprocessing step. However, to get temporal evolution of the velocity gradient tensor needs Lagrange particle tracker [16, 18]. Another approach to getting the velocity gradient tensor is the experimental approach which is also very challenging as we do not have any device to measure the gradients directly. Although the measurements of strain rate and vorticity vector are reported by Xu et al. [28], there are no such studies reported for pressure Hessian so far. D. Shikha (B) · S. S. Sinha Department of Applied Mechanics, IIT Delhi, Delhi 110016, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 K. M. Singh et al. (eds.), Fluid Mechanics and Fluid Power, Volume 4, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-7177-0_78
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Developing a simple dynamical model for the velocity gradient dynamics is one simpler approach which has been developed over subsequent years. These dynamical models are set of ordinary differential equations which are computationally easier to solve and more intuitive than solving partial differential equations like Navier– Stokes. Moreover, such velocity gradient dynamic models can be used directly as closure models for the so-called Lagrangian probability density function (PDF) [21] methods of turbulence computations. In this line of research, the first model for incompressible flows was proposed by Vieillefosse [27]. Subsequently, there have been various attempts made by many researchers [2, 3, 7, 10, 13, 24, 25] to improve the model. The two main challenges faced by the velocity gradient dynamics models, pressure Hessian and the viscous Laplacian terms, are nonlocal and mathematically unclosed in nature. Hence, these two terms need separate modeling. In Vieillefosse [27] model, the isotropic part of the pressure Hessian term is taken from the pressure Poisson’s equation of incompressible flow field, whereas the anisotropic part of the pressure Hessian tensor and the viscous Laplacian tensor are ignored. The resultant model is called the restricted Euler equation (REE) model. The first model for the anisotropic pressure Hessian term was given by Chevillard and Meneveau [4] using the recent fluid deformation approach, named as recent fluid deformation (RFD) model. The first model of the diffusion process was proposed by Martin et al. [15] which was further modified by Jeong and Girimaji [10] and called the Linear Lagrangian diffusion (LLD) model. In compressible flows, the first model was proposed by Suman and Girimaji [24] which was further modified by Danish et al. and Suman and Girimaji [7, 25]. In recent years, machine learning has been used by various researchers to model turbulence processes. Recently, Ling et al. [14] used machine learning to improve the predictions of the RANS model. The author developed a tensor-based neural network architecture (TBNN) which predicts the Reynolds stress anisotropy tensor in incompressible flows. The same architecture was also used by Parashar et al. [19] to predict the pressure Hessian tensor in incompressible flows. Encouraged by this trend in this work, we also used a machine learning approach to model the pressure Hessian tensor. We pursue this study in two stages. These two stages differ in terms of the training process of the machine learning (ML)-based modeling strategy. In the first stage, we employ a tensor-based neural network (TBNN) [14] to model the pressure Hessian process using a raw (unconditioned) DNS dataset of isotropic turbulence for the training purpose. In the second stage of this work, we explore improving the pressure Hessian model by forcing the ML tools to evolve models conditioned on local flow field topology [5]. Indeed, even with the traditional modeling approaches, it has been demonstrated that velocity gradient and pressure Hessian statistics, when conditioned on local flow field topology, provide insightful and distinguishable features of various nonlinear phenomena [25]. Subsequently, the ML-assisted models developed in this study are extensively evaluated by comparing their performance against available DNS results.
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This paper is organized into five sections. Section 2 presents the governing equations of the velocity gradients in an incompressible flow field. In Sect. 3, we present the details of the ML-assisted modeling strategy employed in this work. In Sect. 4, we present extensive evaluations of the trained networks against the DNS behavior. Section 5 concludes the paper with a brief summary.
2 Governing Equations We start with the continuity and the momentum equations for an incompressible flow field: ∂ Vk =0 ∂ xk
(1)
∂ Vi ∂ Vi 1 ∂p μ ∂ 2 Vi + Vk =− + ∂t ∂ xk ρ ∂ xi ρ ∂ xk ∂ xk
(2)
Here, V i , x i , ρ, and μ represent the velocity, spatial coordinate, density, and dynamic viscosity, respectively. The evolution equation of velocity gradient tensor (Ai j = ∂∂ xVij ) can be derived by taking spatial derivative of Eq. (2): D Ai j 1 ∂2 p μ ∂ 2 Ai j = − Aik Ak j − + Dt ρ ∂ xi ∂ x j ρ ∂ xk ∂ xk
(3)
Here, Pij (term II in RHS of Eq. 3) and γ ij (term III in RHS of Eq. 3) represent the pressure Hessian and the viscous Laplacian tensors influencing the evolution of the velocity gradient tensor. The second and third terms (pressure Hessian and viscous Laplacian tensors) given on the right-hand side (RHS) of Eq. (3) are nonlocal and mathematically unclosed. The pressure Hessian tensor can be decomposed into two parts [27]: the isotropic (Zδ ij ) and the anisotropic (Qij ) parts: Pi j =
Z δi j + Qi j 3
(4)
Using Poisson’s equation of incompressible flows, the isotropic part Zδ ij can be written in terms of the second invariant (Alm Aml ) of velocity gradient tensor as follows: Z = − Alm Aml
(5)
The anisotropic part of the pressure Hessian tensor (Qij ) is modeled using the machine learning technique (more details are given in Sect. 3). Flow topology represents the streamlined pattern around a fluid element as observed by a nonrotating observer who is translating with the fluid element. Chong
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Table 1 Conditions to find different local flow topologies
SFS
UFC
SNSS
(q > 0 and r < 0) or (q > 0 and r > 0) or q < 0 (q < 0 and r 1a > r ) (q < 0 and r 1b < r ) r 1a < r
UNSS q β p > γ p and (α s > β s > γ s ) respectively [3]. Their corresponding pressure Hessian eigenvectors are defined as (ˆeα p , eˆ β p , eˆ γ p ). The strain rate eigenvectors defined as (ˆeαs , eˆ β s , eˆ γ s ).
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In Fig. 2, we show the alignments of the eigenvectors of the pressure Hessian tensor with the local vorticity vector (ω) (taken directly from the DNS velocity field). From the DNS alignments, we observe that the intermediate eigendirection (ˆeβ p ) of pressure Hessian tensor tends to align along the vorticity vector. The TBNN predicts the tendency of the vorticity vector to align with the eˆ αp eigendirection. The topology-based conditioned TBNN also captured the similar behavior as that of unconditioned TBNN. Figure 3 presents the alignments of the smallest eigendirection (ˆeγ s ) of the strain rate tensor with the largest pressure Hessian eigendirection (ˆeαp ). From the figure, we observe that the DNS curve has a shallow peak in between 0.6 and 0.8 which is captured by the TBNN predictions with the higher peak values than the DNS peak value. Conditioned TBNN predictions show sensitivity toward the alignment statistics as the alignments of different topologies show different features which need to be compared with the conditioned DNS results. Figure 4 presents the alignments of the intermediate eigendirection (ˆeβ s ) of the strain rate tensor with the largest pressure Hessian eigendirection (ˆeαp ). From the figure, we observe that the DNS results show that (ˆeβ s ) eigendirection tends to align along with the (ˆeαp ) eigendirection which is captured by the TBNN predictions with higher magnitude. The topology-based TBNN alignments are also similar to the unconditioned TBNN alignments. Figure 5 presents the alignments of the largest eigendirection (ˆeαs ) of the strain rate tensor with the largest pressure Hessian eigendirection (ˆeαp ). From the figure, we observe that the DNS results show orthogonal alignment tendency of (ˆeβ s ) eigendirection with the (ˆeαp ) eigendirection as the PDF has a peak at 0. The TBNN predictions also show peak at 0, but it also shows a false peak near 0.75. The SNSS and UNSS topology TBNN do not show any false peak, but the magnitude of the peak at 0 is much higher than the DNS results. In the above results, the topology-based conditioned TBNN predictions are compared against the unconditioned DNS results which shows significant difference in the alignments due to the sensitivity of individual flow topology toward their alignments statistics. Hence, further comparison of the topology-based conditioned TBNN against the corresponding topology-based condition DNS results is needed. The detailed comparison of the conditioned TBNN will be given in the conference presentation.
5 Conclusions The study of various nonlinear turbulent processes hinges on the velocity gradient evolution process. The pressure Hessian being one of the unclosed terms in the velocity gradient evolution equation needs closure modeling. The current study aims to model the anisotropic part of the pressure Hessian tensor using a machine learning approach. We used a tensor-based neural network (TBNN) to predict the pressure Hessian tensor using the velocity gradient tensor as input to the neural network. In this
Machine Learning-Assisted Modeling of Pressure Hessian Tensor Fig. 2 PDFs of the cosine of the angle between the vorticity vector and the different eigendirections of pressure Hessian tensor: a eˆ γ p ω, b eˆ β p ω, c eˆ αp ω
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958 Fig. 3 PDFs of the dot product of the angle between the strain rate eigendirection (ˆeγ s ) and the pressure Hessian eigendirection (ˆeα p )
Fig. 4 PDFs of the dot product of the angle between the strain rate eigendirection (ˆeβ s ) and the pressure Hessian eigendirection (ˆeα p )
Fig. 5 PDFs of the dot product of the angle between the strain rate eigendirection (ˆeαs ) and the pressure Hessian eigendirection (ˆeα p )
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work, we trained the model using the isotropic incompressible stationary turbulence data for the neural network training. We trained five different neural networks using the unconditioned DNS data and the topology-based conditioned data. We evaluate the trained networks by comparing their results with the DNS behavior. The comparison is made based on the alignment tendencies of the pressure Hessian eigendirections with the strain rate eigendirections and the vorticity vector. With the available alignment plots, we conclude that the trained TBNN can predict some essential features of the DNS results but cannot capture the exact DNS behavior. While comparing the performance of conditioned TBNN with the unconditioned DNS, we conclude that the different flow topologies have different alignment tendencies. The comparisons presented in this paper are given with the unconditioned DNS results which are significantly different than the topologies-based conditioned TBNN predictions; hence, further comparison of the topology-based conditioned TBNN against the corresponding topology-based conditioned DNS results is needed. The detailed comparison of the conditioned TBNN will be given in the conference presentation. Acknowledgements The authors acknowledge the computational support provided by the HighPerformance Computing (HPC) center of the Indian Institute of Technology Delhi, India.
References 1. Batchelor GK (1952) The effect of homogeneous turbulence on material lines and surfaces. Proc R Soc Lond Ser A Math Phys Sci 213(1114):349–366 2. Cantwell BJ (1992) Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys Fluids A Fluid Dyn 4(4):782–793 3. Chevillard L, Meneveau C, Biferale L, Toschi F (2008) Modeling the pressure Hessian and viscous Laplacian in turbulence: comparisons with direct numerical simulation and implications on velocity gradient dynamics. Phys Fluids 20(10):101504 4. Chevillard L, Meneveau C (2006) Lagrangian dynamics and statistical geometric structure of turbulence. Phys Rev Lett 97(17):174501 5. Chong MS, Perry AE, Cantwell BJ (1990) A general classification of three-dimensional flow fields. Phys Fluids A Fluid Dyn 2(5):765–777 6. Danish M, Sinha SS, Srinivasan B (2016) Influence of compressibility on the Lagrangian statistics of vorticity–strain-rate interactions. Phys Rev E 94(1):013101 7. Danish M, Suman S, Srinivasan B (2014) A direct numerical simulation-based investigation and modeling of pressure Hessian effects on compressible velocity gradient dynamics. Phys Fluids 26(12):126103 8. Girimaji SS, Pope SB (1990) Material-element deformation in isotropic turbulence. J Fluid Mech 220:427–458 9. http://turbulence.pha.jhu.edu 10. Jeong E, Girimaji SS (2003) Velocity-gradient dynamics in turbulence: effect of viscosity and forcing. Theor Comput Fluid Dyn 16(6):421–432 11. Li Y, Meneveau C (2005) Origin of non-Gaussian statistics in hydrodynamic turbulence. Phys Rev Lett 95(16):164502 12. Li Y, Meneveau C (2006) Intermittency trends and Lagrangian evolution of non-Gaussian statistics in turbulent flow and scalar transport. J Fluid Mech 558:133–142
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