Fluid Mechanics and Fluid Power, Volume 5: Select Proceedings of FMFP 2022 (Lecture Notes in Mechanical Engineering) [2024 ed.] 9819960738, 9789819960736
This book comprises select peer-reviewed proceedings of the 9th International and 49th National Conference on Fluid Mech
114 79 37MB
English Pages 864 [824] Year 2024
Contents
About the Editors
Multiphase Flow
Comparative CmFD Study on Geometric and Algebraic Coupled Level Set and Volume of Fluid Methods
1 Introduction
2 Governing Equations
3 Numerical Methodology
3.1 CLSVOF Advection Equations
3.2 Navier–Stokes Equations
3.3 Solution Algorithm
4 Results and Discussion
4.1 Static Droplet
4.2 Dam Break
4.3 Buoyant Rise of a Bubble
5 Conclusions
References
Numerical Modelling of a Binary Droplet on Solid Surface
1 Introduction
2 Numerical Modelling
2.1 Geometry Description
2.2 Governing Equation
2.3 Boundary Conditions for the Binary Liquid Sessile Droplet
2.4 Algorithm
3 Code Validation
4 Conclusion
References
Characterization of the Spray Flow Field at the Exit of a Pintle Injector
1 Introduction
2 Literature Review and Objectives
3 Materials and Methods
3.1 Design of the Pintle Injector
3.2 Test Conditions
3.3 Diagnostic Techniques
4 Results and Discussion
4.1 Structure of the Pintle Injector Spray
4.2 Variation of Cone Angle and Penetration Length
4.3 Droplet Velocity Field at the Exit of the Injector
5 Conclusions
References
Experimental Investigation of Vapour Bubble Condensation in Subcooled Water Using Different Nozzles
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Image Processing Method
4 Results and Discussion
4.1 Single Bubble Condensation
4.2 Bubble Formation Frequency
4.3 Maximum Bubble Radius
4.4 Effect of Nozzle Diameter and Pool Temperature on Bubble Collapsing Duration
5 Conclusions
References
VOF Simulations of Evaporation and Condensation Phenomenon Inside a Closed-Loop Thermosyphon
1 Introduction
2 Literature Review and Objective
3 Experimental
3.1 Experimental Setup
3.2 Experimental Procedure
4 Numerical Methods
4.1 Governing Equations
4.2 Geometry and Boundary Conditions
5 Results and Discussion
5.1 Validation
5.2 Temperature Distributions
5.3 Evaporation
5.4 Condensation of Water Film
5.5 Conductivity Calculations
6 Conclusions
References
Hydrodynamics of Droplet Generation Under Squeezing Regime in a T-junction Cylindrical Microfluidic System
1 Introduction
2 Mathematical Modeling
2.1 Problem Statement
2.2 Governing Equations
2.3 Computational Approach
3 Results and Discussion
3.1 Phase Profiles
3.2 Droplet Length
3.3 Instantaneous Pressure Profiles
3.4 Velocity Profiles and Recirculation Zones
4 Conclusions
References
Coalescence of Disc-Shaped Falling Droplets Inside Quiescent Liquid Media
1 Introduction
2 Objective
3 Materials and Methods
4 Results and Discussion
5 Conclusions
References
Effects of Power Law Fluid Characteristics on Core-Annular Flow in a Horizontal Pipe
1 Introduction
2 Flow Equations
2.1 Core Fluid
2.2 Annular Fluid
2.3 Boundary Conditions
3 Results and Discussion
3.1 Increasing Viscosity of Core Fluid, Both Fluids Being Newtonian
3.2 Varying Flow-Behaviour Index of Low Viscosity Core Fluid
3.3 Varying Flow-Behaviour Index of Highly-Viscous Core Fluid
3.4 Varying Flow-Behaviour Index of Low-Viscous Annular Fluid
4 Conclusions
References
Evaporation Dynamics of Bidispersed Colloidal Suspension Droplets on Hydrophilic Substrates Under Different Relative Humidity and Ambient Temperature
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Preparation of Colloidal Solution and Droplet Generation:
3.2 Experimental Set-Up
4 Results and Discussion
5 Conclusions
References
Comparative CFD Analysis of Heat Transfer and Melting Characteristics of the PCM in Enclosures with Different Fin Configurations
1 Introduction
2 Literature Review and Objective
3 Geometrical Description
3.1 Numerical Methodology and Governing Equation
3.2 Initial and Boundary Conditions
3.3 Grid Independency Test and Model Validation
4 Results and Discussion
4.1 Melting Characteristics of the PCM
4.2 Enhancement Ratio
4.3 Evolution of Stream Lines
5 Conclusions
References
Influence of Contact Line Velocity Implementation in Dynamic Contact Angle Models for Droplet Bouncing and Non-bouncing Dynamics on a Solid Substrate
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Numerical Methodology
3.2 Dynamic Contact Angle Modelling and Contact Line Velocity Approximation
4 Results and Discussion
4.1 Comparison of Experimental and Numerically Obtained Droplet Wetted Diameters with Non-Bouncing Mood
4.2 Comparison of Experimental and Numerical Maximum Droplet Heights with Non-Bouncing Mood
4.3 Comparison of Experimental and Numerically Calculated Droplet Wetted Diameters with Bouncing Mood
4.4 A Comparison of Experimental and Numerically Calculated Droplet Maximum Height with Bouncing Mood
5 Conclusions
References
Droplet Impact and Wetting on a Micropillared Surface
1 Introduction
2 Methodology
2.1 Fabrication Process for Making Micropillared Surfaces
2.2 Generation and High-Speed Visualization of a Droplet
3 Results and Discussion
3.1 Wetting Characteristics
3.2 Effect of Weber Number for Non-bouncing Condition
4 Conclusions
References
Multiphase Modelling of Thin Film Flow Over the Vertical Plate Under Gravity in Pseudo-Laminar Region
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Simulation Modelling
3.2 Analytical Modelling
4 Results and Discussion
4.1 Grid Independency Test
4.2 Validation with Kapitsa Equation
4.3 Variation of Film Thickness Along the Plate Length
4.4 Variation of Film Reynolds Number Along the Plate Length
4.5 Velocity Profile of Film
4.6 Variation of Wall Shear Stress Along the Plate Length
4.7 Variation of Film Thickness with Time
5 Conclusions
References
Evolution of Marangoni Thermo-Hydrodynamics Within Evaporating Sessile Droplets
1 Introduction
2 Literature Review and Objective
3 Mathematical Formulation
3.1 Governing Equations
3.2 Initial and Boundary Conditions
3.3 Numerical Scheme
4 Results and Discussion
4.1 Temporal Variation of Internal Velocity Field and Temperature Distribution
4.2 Influence of Contact Angle on the Internal Circulation Velocity and Temperature
5 Conclusions
References
Investigation of Liquid Vaporization Characteristics at Low–Pressure Conditions
1 Introduction
2 Experimental Setup and Procedure
3 Performance Characteristics
4 Results and Discussion
4.1 Temperature Characteristics
4.2 Mass Evaporated (mw,v)
4.3 Nonequilibrium Fraction (NEF)
4.4 Heat Transfer Coefficient (h)
5 Conclusions
References
Effect of Surface Structures on Droplet Impact Over Flat and Cylindrical Surfaces
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Mathematical Modeling
3.2 Geometric Modeling
3.3 Validation of Numerical Model
4 Results and Discussion
4.1 Impact of Droplet on Structured Flat Surfaces
4.2 Impact of Droplet on Structured Cylindrical Surfaces
5 Conclusion
References
Numerical Investigation on Bubble Dynamics Using DOE Approach for Cavitation Machining Process
1 Introduction
2 Literature Review and Objective
3 Numerical Model and Methodology
3.1 Numerical Model
3.2 Parameters Selection
3.3 Factorial Design
4 Results and Discussion
4.1 Significant Parameters
4.2 Confirmation Study
5 Conclusions
References
Effect of Surface Tension on the Thermal Performance of Pulsating Heat Pipe with and Without Surfactant
1 Introduction
2 Literature Review and Objective
3 Experimental Setup
3.1 Experimental Procedure
4 Results and Discussion
4.1 Thermodynamic Behavior of PHP
5 Conclusions
References
Study of Physical Characteristics of a Bi-porous Composite Capillary Wick for a Flat Miniature Loop Heat Pipe
1 Introduction
2 Literature Review and Objective
3 Methodology
4 Characterization of Wick Parameters
4.1 Capillary Rise
4.2 Wettability
4.3 Porosity
4.4 Permeability
4.5 Pore Size
4.6 XRD Analysis
5 Conclusions
References
Influence of Air Injection on Cavitation in a Convergent–Divergent Nozzle
1 Introduction
2 Numerical Scheme
2.1 (8)
3 Results and Discussion
4 Conclusions
References
Investigation of the Droplet Impingement on a Hydrophobic Surface with a Fixed Particle
1 Introduction
2 Methodology
3 Results and Discussion
4 Conclusions
References
Effect of Impact Velocity on Spreading and Evaporation of a Volatile Droplet on a Non-porous Substrate
1 Introduction
2 Materials and Methods
2.1 Experimental Setup
2.2 Image Processing
3 Results and Discussion
3.1 Regimes of Droplet Impact Outcome
3.2 Influence of Droplet Height/Velocity on Impact Outcome
4 Conclusions
References
Study of Liquid–Vapor Oscillating Nature in a U-Shaped Tube for a Pulsating Heat Pipe
1 Introduction
2 Theoretical Model
2.1 Governing Equations
2.2 Non-dimensionalization of Governing Equations
2.3 Numerical Solution of Non-dimensionalize Equations
3 Results and Discussion
4 Conclusions
References
Hydrodynamics of Two-Phase Immiscible Flow in T-Junction Microchannel
1 Introduction
2 Problem Description and Methodology
3 Results and Discussion
3.1 Phase Profiles of the Flow Regimes
3.2 Pressure Evolutions of the Flow Regimes
4 Conclusions
References
Experimental Study of Onset of Nucleate Boiling from Submerged Ribbon Heaters of Varying Width
1 Introduction
2 Literature Review and Objective
3 Methodology
4 Results and Discussion
4.1 Heat Flux Variation with Subcooling and Heater Width
4.2 Wall Superheat Variation with Subcooling and Heater Width
5 Conclusions
References
Numerical Study of Bubble Growth on a Hydrophilic Surface
1 Introduction
2 Literature Review and Objective
3 Methodology
3.1 Governing Equations
3.2 Computation Domain and Boundary Conditions
3.3 Numerical Technique and Grid Independent Study
3.4 Validation
4 Results and Discussion
4.1 Bubble Morphology
4.2 Bubble Equivalent Radius
4.3 Bubble Base Radius (Br)
5 Conclusions
References
Frequency Analysis of Direct Contact Condensation Using the Wavelet Transform During the Vertical Steam Injection on the Subcooled Water Pool
1 Introduction
2 Literature Review and Objective
3 Experimental Setup and Methodology
3.1 Method of Analysis
4 Results and Discussion
4.1 Continuous Wavelet Transformation (CWT) Analysis of Pressure Signal
4.2 Analysis of Acoustic Sound Signal
5 Conclusions
References
A Numerical Analysis of Flat-Fan Spray Injection into Coflow of Air
1 Introduction
2 Numerical Methodology
2.1 Computational Domain and Mesh
2.2 Governing Equations
2.3 Operating and Boundary Conditions
2.4 Grid Independence Study
2.5 Validation
3 Results and Discussion
3.1 Spray Structure
3.2 Axial Evolution of Spray Parameters
3.3 Effect of Liquid Preheating
4 Conclusions
References
Capillary Rise in the Interstices of Tubes
1 Introduction
2 Experimental Setup
3 Methodology
4 Results and Discussion
5 Conclusions
References
A Study of Flow Patterns Near Moving Contact Lines Over Hydrophobic Surfaces
1 Introduction
2 Methodology
3 Results
4 Conclusions
References
An Experimental Investigation of an Effect of Swirl Flow Field and the Aerodynamic Force on the Droplet Breakup Morphology
1 Introduction
2 Methodology
2.1 Experimental Set-Up
3 Results and Discussion
4 Conclusions
References
Droplet Growth and Drop Size Distribution Model for Dropwise Condensation on Hydrophobic Tubular Surfaces
1 Introduction
2 Literature Review and Objectives
3 Mathematical Model Framework
3.1 Estimation of the Apparent Contact Angle
3.2 Single Droplet Heat Transfer
3.3 Drop Size Distribution
3.4 Overall Condensation Heat Transfer Rate
4 Results and Discussion
4.1 Validation
4.2 Curvature Effect of the Tube
4.3 Effect of Degree of Sub-Cooling
5 Conclusions
References
Analysis of Deformation Effects on Falling Spray Droplet Motion Under Postulated Sodium Spray Fire Scenario in SFR
1 Introduction
2 Literature Survey and Objective
3 Numerical Model for Droplet Motion
3.1 Equation of Motion for Rigid Spheres Falling in the Gas Medium
3.2 Model for Liquid Droplet Deformation
4 Results and Discussion
4.1 Evaluation of Settling Velocity of Falling Water Droplets
4.2 Evaluation of Settling Velocity of Sodium Droplets
5 Conclusions
References
Particle Filtration in Suspension Droplet Breakup
1 Introduction
2 Methodology
3 Results and Discussion
4 Conclusions
References
Dynamic Characteristics of Submicron Particle Deposited on the Charged Spherical Collector
1 Introduction
2 Literature Review and Objective
3 Methodology
3.1 Model Geometry
3.2 Basic Governing Equations
4 Results and Discussion
4.1 Validation of Results
4.2 Electric Potential and Electric Field Norm Distribution Within the Domain
4.3 Trajectories of Particle Drifting Towards Charged Collector
4.4 Analysis of Forces Acting on Particle Drifting Towards the Charged Collector
5 Conclusions
References
Accommodating Volume Expansion Effects During Solid–Liquid Phase Change—A Comparative Study
1 Introduction and Literature Review
2 Problem Description and Methodology
2.1 Physical Model
2.2 Numerical Methodology
2.3 Solution Procedure
2.4 Numerical Model Validation
3 Results and Discussion
4 Conclusions
References
Self-Similar Velocity Profiles in Granular Flow in a Silo with Two Asymmetrically Located Exits
1 Introduction
2 Methodology
3 Results and Discussion
4 Conclusions
References
Droplet Impact and Spreading Around the Right Circular Cone: A Numerical Approach
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 The Problem as a Physical Description
3.2 Governing Equations
3.3 Boundary Conditions
3.4 Numerical Method
3.5 Grid Independence Test
3.6 Validation of Current Computational Scheme
4 Results and Discussion
4.1 Profile of Droplets Over Time: A Temporal Analysis
4.2 Weber Number and Contact Angle are Affected by the Deformation Factor
5 Conclusions
References
Effect of Direct Current Electrowetting on Dielectric on Droplet Impingement Dynamics
1 Introduction
2 Methodology
3 Results and Discussion
3.1 Dynamics of Droplet Impingement on EWOD Chip Without Voltage
3.2 Dynamics of Droplet Impingement on EWOD Chips with Voltage
3.3 Effect of Voltage on Droplet Height and Diameter on Droplet Impact Dynamics.
4 Conclusions
References
Evolution of Ferrofluid Droplet Deformation Under Magnetic Field in a Uniaxial Flow
1 Introduction
2 Problem Formulation
2.1 Governing Equations
2.2 Nondimensionalization
3 Numerical Method
3.1 Grid Independence and Cahn Number Independence Test
3.2 Model Validation
4 Results and Discussion
5 Conclusions
References
Time-Dependent Droplet Detachment Behaviour from Wettability-Engineered Fibers during Fog Harvesting
1 Introduction
2 Materials and Methods
3 Results and Discussion
4 Conclusions
References
Stability Analysis from Fourth-Order Nonlinear Multiphase Deep Water Wavetrains
1 Introduction
2 Governing Equations
3 Evolution Equations for Deep Water
4 Modulational Instability Analysis and Results
5 Discussion and Conclusions
Appendix
References
Drop Size and Velocity Distributions of Bio-Oil Spray Produced by Airblast Atomizer
1 Introduction
2 Experimental Setup
2.1 Measurement Techniques
3 Results and Discussion
3.1 Simultaneous Droplet Size and Velocity Correlation
3.2 Secondary Atomization
4 Conclusions
References
Numerical Investigation of Oil–Water Two-Phase Flow Through Sudden Contraction Tube
1 Introduction
2 Methodology
2.1 Physical Problem Description
3 Mathematical Modeling
4 Boundary Conditions
5 Result and Discussion
6 Conclusion
References
Validation of the Time Model in Gas–Liquid Horizontal Pipe Flow
1 Introduction
2 Literature Review and Objectives of Present Work
3 The Model for Transition to Slug (Time Model)
3.1 The Vertical Growth Time
3.2 The Horizontal Growth Time
4 Experimental Procedure
5 Results and Discussion
5.1 Image Processing for the Average Liquid Height
5.2 Transition Zone Determination
5.3 Validation of the Time Model
5.4 Wave Crest Tracking and its Characteristics
6 Conclusions
References
Experimental Investigation on the Influence of Bed Height and Bed Particle Size on Bed Expansion for a Bubbling Fluidized Bed
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
4 Results and Discussion
5 Conclusions
References
Study on Escapes Probability of Gas Bubble in Surge Tank Using Water Model Experiment
1 Introduction
2 Experimental Model Details
3 Experimental Procedure
4 Results and Discussion
4.1 Effect of Water Flow Rate in Surge Tank
4.2 Effect of Air Injection Rate to the Surge Tank
5 Conclusions
References
Mathematical Modelling and Optimization of Cylindrical Heat Pipe
1 Introduction
2 Literature Review and Objective
3 Mathematical Modelling
3.1 Cylindrical Heat Pipe
3.2 Thermo-Physical Properties of Water
3.3 Heat Transfer Limitations of Cylindrical Heat Pipe with Sintered Capillary Structure
3.4 Parameters of Cylindrical Heat Pipe
3.5 Parameters of Capillary Structure
4 Results and Discussion
4.1 Heat Transfer Limitations of Cylindrical Heat Pipe with Sintered Capillary Structure for 180° Axial Orientation
4.2 Heat Transfer Limitations of Cylindrical Heat Pipe with Sintered Capillary Structure for 90° Axial Orientation
4.3 Comparison of Capillary Limit of Cylindrical Heat Pipe on the Basis of Axial Orientation
4.4 Comparison of Heat Transfer Limitations of Cylindrical Heat Pipe on the Basis of Axial Orientations
4.5 Comparison of Capillary Limit of Cylindrical Heat Pipe on the Basis of Spherical Diameter of Copper Powder for 180º Axial Orientation
4.6 Comparison of Cylindrical Heat Pipe on the Basis of Spherical Diameter of Copper Powder (56 μm) and Width of Wick Structure for 180º Axial Orientation
5 Conclusion
References
Anomalous Motion of a Sphere upon Impacting a Quiescent Liquid: Influence of Surface Wettability
1 Introduction
2 Experimental Setup and Methodology
3 Results and Discussion
4 Conclusions
References
Analysis of Fission Gas-Fuel Particle Dispersion in a Voided Triangular Channel Under Sustained PCM Conditions
1 Introduction
2 Equation of Motion for Gas-Particle Mixture Flow
3 Model Parameters
4 Results and discussion
5 Conclusions
References
Experimental Investigation of Droplet Spreading on Porous Media
1 Introduction
2 Experimental Set-Up
2.1 Droplet Creation
2.2 Image Recording and Data Processing
3 Results and Discussion
4 Conclusions
References
Investigation of the Liquid Sheet Breakup Dynamics in Like-On-Unlike Impinging Injectors
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Experimental Arrangements
3.2 Test Conditions
3.3 Diagnostic Techniques
3.4 Data Processing Techniques
4 Results and Discussion
4.1 Morphology of the Liquid Sheet
4.2 Dynamics of the Liquid Sheet Breakup
4.3 Ligament Shedding Process
5 Conclusions
References
Numerical Investigation of Mist Flow Characteristics in a Hexagonal Fuel Rod Bundle
1 Introduction
2 Literature Review and Objective
3 Methodology
3.1 Geometric Details
3.2 Computational Scheme and Boundary Condition
3.3 Validation
3.4 Grid Independence Test
4 Results and Discussion
4.1 Effect of Heat Flux
4.2 Effect of Mass Flux
4.3 Effect of Quality
5 Conclusions
References
Droplet Impact on a Superheated Concave Surface Having a Curvature Ratio of Unity
1 Introduction
2 Methodology
3 Results and Discussion
3.1 Dynamics of Droplet Contact on Superheated Flat and Concave Surfaces
4 Conclusions
References
Numerical Study on the Jet Breakup of Molten Nuclear Fuel in the Coolant in Dripping Regime
1 Introduction
2 Governing Equation
2.1 Initial and Boundary Condition
2.2 Calculation of Interfacial Tension
2.3 Crust Formation Due to Solidification
3 Numerical Methodology
4 Results and Discussion
4.1 Effect of Interfacial Tension on Jet Breakup
4.2 Effect of Crust Formation on Jet Breakup
5 Conclusion
References
Electromagnetohydrodynamic (EMHD) Flow Actuation with Patterned Wettability
1 Introduction
2 Methodology
2.1 Problem Description
2.2 Mathematical Model
3 Results and Discussion
3.1 Validation
3.2 Hydrodynamic Analysis
4 Conclusions
References
Mixing in a Size Segregated Fluidized Bed: Simulations and Experiments
1 Introduction
2 Literature Review and Objective
3 Methods
3.1 Experiments
3.2 Simulations
4 Results and Discussion
5 Conclusions
References
Drop Impact on a Deep Pool: A Revisit to the Large Bubble Entrapment Regime
1 Introduction
2 Methodology
3 Results and Discussion
4 Conclusions
References
2D Numerical Simulation of the Electrospraying Process of a Viscoelastic Liquid in an Ambient, Highly Viscous Liquid
1 Introduction
2 Problem Formulation and Governing Equations
2.1 Fluid Flow and Viscoelastic Constitutive Equation
2.2 Log Conformation Method
2.3 Electrostatic Equations
2.4 Two-Phase Flow Equations
2.5 Grid Independence Test
3 Results and Discussion
3.1 Viscoelastic Jet Analysis
4 Conclusions
References
Deformation Dynamics During Complete Rebounding During Impact of a Falling Droplet of Varied Surface Tension on a Sessile Drop
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
4 Results and Discussion
4.1 Droplet Deformation During Full Rebounding
4.2 Changes in the Dynamic Contact Angle During Rebounding
5 Conclusions
References
Real-Time Strengthening of Natural Convection and Dendrite Fragmentation During Binary Mixture Freezing
1 Introduction
2 Materials and Methods
3 Results and Discussion
4 Conclusions
References
Study of Bubble Growth on a Heated Vertical Surface: Influence of Axial Flow Vibration
1 Introduction
2 Methodology
2.1 Nucleating Bubble
2.2 Wall Temperature
2.3 Force Balance Model
3 Results and Discussion
3.1 Validation
3.2 Influence of Subcooling and Vibration Frequency
4 Conclusions
References
Comparative Study of Droplet Impact Characteristics with Various Viscous Liquids: A Study of Both Miscible and Immiscible Droplet Impacts
1 Introduction
2 Literature Review and Objective
3 Experimentation
4 Results and Discussion
4.1 Miscible Droplet Impact
4.2 Comparison of Miscible and Immiscible Droplet Impact
5 Conclusions
References
Parametric Study on Marangoni Instability in Two-Layer Creeping Flow
1 Introduction
2 Formulation
2.1 Base State
2.2 Linearized Eigenvalue Problem for Stability Analysis
2.3 Numerical Technique
3 Results
3.1 System Stability Under Peclet Number Variation
3.2 System Stability Under Diffusivity Ratio Variation
4 Conclusions
References
Measurement of Force in Granular Flow Past Cylindrical Models for Various Inclination Angle
1 Introduction
2 Objective
3 Experimental Setup
3.1 Experiment for Force Measurement on Cylinder
4 Results and Discussion
4.1 Drag Force Calculation
4.2 Effect of Inclination Angle on Force
4.3 Effect of Diameter on Drag Force
5 Conclusions
References
Experimental Interfacial Reconstruction and Mass Transfer Modelling of a Slug Bubble During Co-current Flow in a Millimetric Tube
1 Introduction
2 Literature Review and Objective
3 Materials and Methods
3.1 Co-current Experimental Apparatus
3.2 Principle of Colorimetric Technique
3.3 Image Processing and Concentration Contours
4 Results and Discussion
4.1 Superficial Velocity of Gas and Unit Cell Length
4.2 Estimation of Liquid Side Mass Transfer Coefficient
4.3 Radial Oxygen Concentration Profile in Liquid Slug
5 Conclusions
References
Novel and Efficient Superhydrophilic Surface for Improved Critical Heat Flux in Heat Pipe Applications
1 Introduction
2 Materials and Methods
2.1 Contact Angle Determination
2.2 Surface Fabrication
3 Results and Discussion
3.1 Structural Growth of Microstructure
3.2 Surface Morphology
3.3 Thermal Performance
4 Conclusions
References
Effects of Wettability on the Flow Boiling Heat Transfer Enhancement
1 Introduction
2 Literature Review and Objective
3 Model Description
3.1 Computational Domain
3.2 Governing Equations
3.3 Grid Independence Study
4 Results and Discussion
4.1 Validation of Numerical Model
4.2 Effect of Bubble Dynamics on Pressure, Temperature, and Velocity Distribution of Flow
4.3 Effect of Surface Wettability on Bubble Dynamics During Flow Boiling Heat Transfer
4.4 Effect on Local Heat Flux and HTC Along the Channel Length
4.5 Effect on Local Removed Heat Flux and HTC with Time
5 Conclusions
References
Fluid-Structure Interaction
Computational Study to Assess the Usage of Asymmetric Canard as Yaw Control Device for a Generic Fighter Aircraft
1 Introduction
2 Details of Grid and CFD Computations
3 Results and Discussion
3.1 Pressure Distribution
3.2 Side Force
3.3 Yawing Moment
4 Conclusions
References
Physiological FSI Study for Phonoangiography-Based Rupture Risk Prediction in Abdominal Aortic Aneurysms
1 Introduction
2 Methodology
2.1 Computational Setup and Parametric Details
2.2 Numerical Model
2.3 Boundary Conditions and Flow Parameters
2.4 Validation: Wave Propagation in a Flexible Tube
3 Results and Discussion
3.1 Integrated Pressure Force Rate (IPFR)
3.2 Rupture Potential Index (RPI)
3.3 Correlation of RPI with Frequency
4 Conclusions
References
Impact of Building Configurations on Fluid Flow in an Urban Street Canyon
1 Introduction
2 Literature Review and Objective
3 Computational Model and Methodology
3.1 Computational Domain and Grid Setting
3.2 Boundary Conditions
3.3 Solver Settings
4 Results and Discussion
4.1 Effect of Building Height
4.2 Effect of Building Configuration
4.3 Effect of Wind Direction
5 Conclusions
References
Experimental and Theoretical Analysis of Flow-Induced Vibration of Cantilevered Flexible Plate
1 Introduction
2 Methodology
2.1 Experimental Setup
2.2 Wake Oscillator Model
3 Results and Discussion
4 Conclusions
References
Flow-Induced Vibration of an Elastically Mounted Cylinder Under the Influence of Downstream Stationary Cylinder
1 Introduction
2 Computational Model
2.1 Problem Description
2.2 Governing Equation and Numerical Method
2.3 Grid Generation and Independence Study
2.4 Validation
3 Results and Discussion
3.1 Vibration Characteristics
3.2 Flow States
4 Conclusions
References
Recommend Papers
![Fluid Mechanics and Fluid Power, Volume 4: Select Proceedings of FMFP 2022 (Lecture Notes in Mechanical Engineering) [2024 ed.]
9819971764, 9789819971763](https://ebin.pub/img/200x200/fluid-mechanics-and-fluid-power-volume-4-select-proceedings-of-fmfp-2022-lecture-notes-in-mechanical-engineering-2024nbsped-9819971764-9789819971763.jpg)
![Fluid Mechanics and Fluid Power, Volume 5: Select Proceedings of FMFP 2022 (Lecture Notes in Mechanical Engineering) [2024 ed.]
9819960738, 9789819960736](https://ebin.pub/img/200x200/fluid-mechanics-and-fluid-power-volume-5-select-proceedings-of-fmfp-2022-lecture-notes-in-mechanical-engineering-2024nbsped-9819960738-9789819960736.jpg)
File loading please wait...
Citation preview
Lecture Notes in Mechanical Engineering
Krishna Mohan Singh Sushanta Dutta Sudhakar Subudhi Nikhil Kumar Singh Editors
Fluid Mechanics and Fluid Power, Volume 5 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
Lecture Notes in Mechanical Engineering (LNME) publishes the latest developments in Mechanical Engineering—quickly, informally and with high quality. Original research or contributions reported in proceedings and post-proceedings represents the core of LNME. Volumes published in LNME embrace all aspects, subfields and new challenges of mechanical engineering. To submit a proposal or request further information, please contact the Springer Editor of your location: Europe, USA, Africa: Leontina Di Cecco at [email protected] China: Ella Zhang at [email protected] India: Priya Vyas at [email protected] Rest of Asia, Australia, New Zealand: Swati Meherishi at [email protected] Topics in the series include: ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Engineering Design Machinery and Machine Elements Mechanical Structures and Stress Analysis Automotive Engineering Engine Technology Aerospace Technology and Astronautics Nanotechnology and Microengineering Control, Robotics, Mechatronics MEMS Theoretical and Applied Mechanics Dynamical Systems, Control Fluid Mechanics Engineering Thermodynamics, Heat and Mass Transfer Manufacturing Engineering and Smart Manufacturing Precision Engineering, Instrumentation, Measurement Materials Engineering Tribology and Surface Technology
Indexed by SCOPUS, EI Compendex, and INSPEC. All books published in the series are evaluated by Web of Science for the Conference Proceedings Citation Index (CPCI). To submit a proposal for a monograph, please check our Springer Tracts in Mechanical Engineering at https://link.springer.com/bookseries/11693.
Krishna Mohan Singh · Sushanta Dutta · Sudhakar Subudhi · Nikhil Kumar Singh Editors
Fluid Mechanics and Fluid Power, Volume 5 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 978-981-99-6073-6 ISBN 978-981-99-6074-3 (eBook) https://doi.org/10.1007/978-981-99-6074-3 © 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
Multiphase Flow Comparative CmFD Study on Geometric and Algebraic Coupled Level Set and Volume of Fluid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orkodip Mookherjee, Shantanu Pramanik, and Atul Sharma Numerical Modelling of a Binary Droplet on Solid Surface . . . . . . . . . . . . Mradul Ojha, Lalit Kumar, and Rajneesh Bhardwaj Characterization of the Spray Flow Field at the Exit of a Pintle Injector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sanjay Kumar Gupta, Aman Bakshi, Rachit Bundiwal, and K. P. Shanmugadas Experimental Investigation of Vapour Bubble Condensation in Subcooled Water Using Different Nozzles . . . . . . . . . . . . . . . . . . . . . . . . . . Samarendu Biswas, Aranyak Chakravarty, and Mithun Das VOF Simulations of Evaporation and Condensation Phenomenon Inside a Closed-Loop Thermosyphon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vivek K. Mishra, Saroj K. Panda, Biswanath Sen, M. P. Maiya, and Dipti Samantaray
3 17
29
41
49
Hydrodynamics of Droplet Generation Under Squeezing Regime in a T-junction Cylindrical Microfluidic System . . . . . . . . . . . . . . . . . . . . . . Pratibha Dogra and Ram Prakash Bharti
59
Coalescence of Disc-Shaped Falling Droplets Inside Quiescent Liquid Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deepak Kumar Mishra, Raghvendra Gupta, and Anugrah Singh
71
Effects of Power Law Fluid Characteristics on Core-Annular Flow in a Horizontal Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sumit Tripathi
77
v
vi
Contents
Evaporation Dynamics of Bidispersed Colloidal Suspension Droplets on Hydrophilic Substrates Under Different Relative Humidity and Ambient Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mahesh R. Thombare, Suryansh Gupta, and Nagesh D. Patil Comparative CFD Analysis of Heat Transfer and Melting Characteristics of the PCM in Enclosures with Different Fin Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arun Uniyal and Yogesh K. Prajapati
85
97
Influence of Contact Line Velocity Implementation in Dynamic Contact Angle Models for Droplet Bouncing and Non-bouncing Dynamics on a Solid Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Priyaranjan Sahoo, Javed Shaikh, Nagesh D. Patil, and Purnendu Das Droplet Impact and Wetting on a Micropillared Surface . . . . . . . . . . . . . . 121 Yagya Narayan and Rajneesh Bhardwaj Multiphase Modelling of Thin Film Flow Over the Vertical Plate Under Gravity in Pseudo-Laminar Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 N. Shiva, Nilojendu Banerjee, and Satyanarayanan Seshadri Evolution of Marangoni Thermo-Hydrodynamics Within Evaporating Sessile Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Arnov Paul and Purbarun Dhar Investigation of Liquid Vaporization Characteristics at Low–Pressure Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Sarvjeet Singh, Jaydip Basak, Prodyut Chakraborty, and Hardik Kothadia Effect of Surface Structures on Droplet Impact Over Flat and Cylindrical Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Saptaparna Patra, Avik Saha, and Arup Kumar Das Numerical Investigation on Bubble Dynamics Using DOE Approach for Cavitation Machining Process . . . . . . . . . . . . . . . . . . . . . . . . . 187 Amresh Kumar, Tufan Chandra Bera, and B. K. Rout Effect of Surface Tension on the Thermal Performance of Pulsating Heat Pipe with and Without Surfactant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Vaishnavi K. Wasankar and Pramod R. Pachghare Study of Physical Characteristics of a Bi-porous Composite Capillary Wick for a Flat Miniature Loop Heat Pipe . . . . . . . . . . . . . . . . . . 211 Toni Kumari and Manmohan Pandey Influence of Air Injection on Cavitation in a Convergent–Divergent Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Pankaj Kumar, Santosh Kumar Singh, Jaisreekar Reddy, and Mihir Shirke
Contents
vii
Investigation of the Droplet Impingement on a Hydrophobic Surface with a Fixed Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 K. Niju Mohammed, P. S. Tide, Franklin R. John, A. Praveen, and Ranjith S. Kumar Effect of Impact Velocity on Spreading and Evaporation of a Volatile Droplet on a Non-porous Substrate . . . . . . . . . . . . . . . . . . . . . . 241 Amit Yadav, Allu Sai Nandan, and Srikrishna Sahu Study of Liquid–Vapor Oscillating Nature in a U-Shaped Tube for a Pulsating Heat Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Anoop Kumar Shukla, Est Dev Patel, and Subrata Kumar Hydrodynamics of Two-Phase Immiscible Flow in T-Junction Microchannel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Akepogu Venkateshwarlu and Ram Prakash Bharti Experimental Study of Onset of Nucleate Boiling from Submerged Ribbon Heaters of Varying Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 John Pinto, Janani Srree Murallidharan, and Kannan Iyer Numerical Study of Bubble Growth on a Hydrophilic Surface . . . . . . . . . 287 Abhishek K. Sharma and Shaligram Tiwari Frequency Analysis of Direct Contact Condensation Using the Wavelet Transform During the Vertical Steam Injection on the Subcooled Water Pool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Saurabh Patel and Parmod Kumar A Numerical Analysis of Flat-Fan Spray Injection into Coflow of Air . . . 313 Shirin Patil, Kiran Kumar, Srikrishna Sahu, and Ravindra G. Devi Capillary Rise in the Interstices of Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Chitransh Atre, Aditya Manoj, and Baburaj A. Puthenveettil A Study of Flow Patterns Near Moving Contact Lines Over Hydrophobic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Charul Gupta, Anvesh Sangadi, Lakshmana Dora Chandrala, and Harish N. Dixit An Experimental Investigation of an Effect of Swirl Flow Field and the Aerodynamic Force on the Droplet Breakup Morphology . . . . . . 351 Pavan Kumar Kirar, Surendra Kumar Soni, Pankaj S. Kolhe, and Kirti Chandra Sahu Droplet Growth and Drop Size Distribution Model for Dropwise Condensation on Hydrophobic Tubular Surfaces . . . . . . . . . . . . . . . . . . . . . 361 Waquar Raza, Ramesh Narayanaswamy, and K. Muralidhar
viii
Contents
Analysis of Deformation Effects on Falling Spray Droplet Motion Under Postulated Sodium Spray Fire Scenario in SFR . . . . . . . . . . . . . . . . 375 S. Muthu Saravanan, P. Mangarjuna Rao, and S. Raghupathy Particle Filtration in Suspension Droplet Breakup . . . . . . . . . . . . . . . . . . . . 387 Kishorkumar Sarva, Tejas G. murthy, and Gaurav Tomar Dynamic Characteristics of Submicron Particle Deposited on the Charged Spherical Collector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Abhishek Srivastava, Bahni Ray, Mayank Kumar, Debabrata Dasgupta, Rochish Thaokar, and Y. S. Mayya Accommodating Volume Expansion Effects During Solid–Liquid Phase Change—A Comparative Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 Keyur Kansara and Shobhana Singh Self-Similar Velocity Profiles in Granular Flow in a Silo with Two Asymmetrically Located Exits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Yashvardhan Singh Bhati and Ashish Bhateja Droplet Impact and Spreading Around the Right Circular Cone: A Numerical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Prakasha Chandra Sahoo, Jnana Ranjan Senapati, and Basanta Kumar Rana Effect of Direct Current Electrowetting on Dielectric on Droplet Impingement Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 K. Niju Mohammed, A. Shebin, E. Mohammed Haseeb, P. S. Tide, Franklin R. John, Ranjith S. Kumar, and S. S. Sreejakumari Evolution of Ferrofluid Droplet Deformation Under Magnetic Field in a Uniaxial Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Debdeep Bhattacharjee, Arnab Atta, and Suman Chakraborty Time-Dependent Droplet Detachment Behaviour from Wettability-Engineered Fibers during Fog Harvesting . . . . . . . . . . . 463 Arijit Saha, Arkadeep Datta, Arani Mukhopadhyay, Amitava Datta, and Ranjan Ganguly Stability Analysis from Fourth-Order Nonlinear Multiphase Deep Water Wavetrains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Tanmoy Pal and Asoke Kumar Dhar Drop Size and Velocity Distributions of Bio-Oil Spray Produced by Airblast Atomizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 Surendra Kumar Soni and Pankaj S. Kolhe Numerical Investigation of Oil–Water Two-Phase Flow Through Sudden Contraction Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Soham Mahindar, Mushtaque Momin, and Mukesh Sharma
Contents
ix
Validation of the Time Model in Gas–Liquid Horizontal Pipe Flow . . . . . 513 Tarannum Mujawar and Jyotirmay Banerjee Experimental Investigation on the Influence of Bed Height and Bed Particle Size on Bed Expansion for a Bubbling Fluidized Bed . . . . . . . . . . 527 D. Musademba and Prabhansu Study on Escapes Probability of Gas Bubble in Surge Tank Using Water Model Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 P. Lijukrishnan, Indranil Banerjee, S. Manikandan, S. Rammohan, V. Vinod, and S. Raghupathy Mathematical Modelling and Optimization of Cylindrical Heat Pipe . . . 545 Dinesh Kumar Jain and A. V. Deshpande Anomalous Motion of a Sphere upon Impacting a Quiescent Liquid: Influence of Surface Wettability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 Prasanna Kumar Billa, Tejaswi Josyula, and Pallab Sinha Mahapatra Analysis of Fission Gas-Fuel Particle Dispersion in a Voided Triangular Channel Under Sustained PCM Conditions . . . . . . . . . . . . . . . 567 B. Thilak, P. Mangarjuna Rao, and S. Raghupathy Experimental Investigation of Droplet Spreading on Porous Media . . . . . 577 Anushka, Prashant Narayan Panday, Prasanta Kumar Das, and Aditya Bandopadhyay Investigation of the Liquid Sheet Breakup Dynamics in Like-On-Unlike Impinging Injectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 Aditi Sharma, Bikash Mahato, P. Ganesh, and K. P. Shanmugadas Numerical Investigation of Mist Flow Characteristics in a Hexagonal Fuel Rod Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 Ayush Kumar Rao, Shivam Singh, and Harish Pothukuchi Droplet Impact on a Superheated Concave Surface Having a Curvature Ratio of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 B. S. Renjith, K. Niju Mohammed, and Ranjith S. Kumar Numerical Study on the Jet Breakup of Molten Nuclear Fuel in the Coolant in Dripping Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 M. Chandra Kumar, A. Jasmin Sudha, V. Subramanian, and B. Venkatraman Electromagnetohydrodynamic (EMHD) Flow Actuation with Patterned Wettability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 Apurav Tambe, Shubham Agarwal, and Purbarun Dhar
x
Contents
Mixing in a Size Segregated Fluidized Bed: Simulations and Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 Harshal G. Gamit, Kamal Kishor Pandey, S. Srinivas, and Manaswita Bose Drop Impact on a Deep Pool: A Revisit to the Large Bubble Entrapment Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 Yarra Chiranjeevi Nikhil, Akash Anand, and Hiranya Deka 2D Numerical Simulation of the Electrospraying Process of a Viscoelastic Liquid in an Ambient, Highly Viscous Liquid . . . . . . . . . 667 Vimal Chauhan, Shyam Sunder Yadav, and Venkatesh K. P. Rao Deformation Dynamics During Complete Rebounding During Impact of a Falling Droplet of Varied Surface Tension on a Sessile Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 Pragyan Kumar Sarma and Anup Paul Real-Time Strengthening of Natural Convection and Dendrite Fragmentation During Binary Mixture Freezing . . . . . . . . . . . . . . . . . . . . . 691 Virkeshwar Kumar, Shyamprasad Karagadde, and Kamal Meena Study of Bubble Growth on a Heated Vertical Surface: Influence of Axial Flow Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 Nikhil Chitnavis, Harish Pothukuchi, and B. S. V. Patnaik Comparative Study of Droplet Impact Characteristics with Various Viscous Liquids: A Study of Both Miscible and Immiscible Droplet Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 Kollati Prudhvi Ravikumar, Abanti Sahoo, and Soumya Sanjeeb Mohapatra Parametric Study on Marangoni Instability in Two-Layer Creeping Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 Ankur Agrawal and P. Deepu Measurement of Force in Granular Flow Past Cylindrical Models for Various Inclination Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 Aadarsh Kumar, Deepika Chimote, Aqib Khan, Yash Jaiswal, Rakesh Kumar, and Sanjay Kumar Experimental Interfacial Reconstruction and Mass Transfer Modelling of a Slug Bubble During Co-current Flow in a Millimetric Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Lokesh Rohilla, Ravi Prakash, Raj Kumar Verma, and Arup Kumar Das Novel and Efficient Superhydrophilic Surface for Improved Critical Heat Flux in Heat Pipe Applications . . . . . . . . . . . . . . . . . . . . . . . . . 759 Pradyumna Kodancha, Siddhartha Tripathi, and Vadiraj Hemadri
Contents
xi
Effects of Wettability on the Flow Boiling Heat Transfer Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769 Akash Priy, Israr Ahmad, Manabendra Pathak, and Mohd. Kaleem Khan Fluid-Structure Interaction Computational Study to Assess the Usage of Asymmetric Canard as Yaw Control Device for a Generic Fighter Aircraft . . . . . . . . . . . . . . . . . 783 V. Sundara Pandian, R. J. Pathanjali, B. Praveen Kumar, Muralidhar Madhusudan, and Dharmendra Narayan Physiological FSI Study for Phonoangiography-Based Rupture Risk Prediction in Abdominal Aortic Aneurysms . . . . . . . . . . . . . . . . . . . . . 799 Sumant R. Morab, Janani S. Murallidharan, and Atul Sharma Impact of Building Configurations on Fluid Flow in an Urban Street Canyon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811 Surendra Singh, Lakhvinder Singh, and S. Jitendra Pal Experimental and Theoretical Analysis of Flow-Induced Vibration of Cantilevered Flexible Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825 Shubham Giri, V. Kartik, Amit Agrawal, and Rajneesh Bhardwaj Flow-Induced Vibration of an Elastically Mounted Cylinder Under the Influence of Downstream Stationary Cylinder . . . . . . . . . . . . . . 839 Abhishek Thakur, Atul Sharma, and Sandip K. Saha
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.
xiii
Multiphase Flow
Comparative CmFD Study on Geometric and Algebraic Coupled Level Set and Volume of Fluid Methods Orkodip Mookherjee, Shantanu Pramanik, and Atul Sharma
1 Introduction Computational multi-Fluid Dynamics (CmFD) is a branch of Computational Fluid Dynamics (CFD) which involves a study on multi-physics fluid dynamics, heat transfer, and phase change processes occurring among multiple fluids. For the CmFD, accurate prediction of the interface plays a vital role in simulating a variety of twophase phenomena like droplet breakup and coalescence, droplet evaporation, droplet combustion, spray atomization, etc. As compared to CFD for a single phase flow, CmFD for two-phase flow involves additional challenges on the discontinuity of material properties across the interface and mass, momentum, and energy transport at the interface. Various CmFD methods have been developed to address these challenges which can be broadly classified as Lagrangian and Eulerian methods. The present work deals with the Eulerian approach, which involves capturing the interface on a fixed mesh. Common Eulerian methods are Volume Of Fluid (VOF) method [4], Level Set (LS) method [14], and Coupled Level Set and Volume Of Fluid (CLSVOF) method [13]. In CLSVOF method, the interface is represented by a level set function φ and the conservation of mass is guaranteed by a volume fraction field F. The key role of any VOF-type scheme is the numerical approximation of the advection fluxes to update the fluid volume fractions. Based on the computation of fluxes, there are two types of VOF method: algebraic and geometric methods. An algebraic VOF method, like that of Hirt and Nichols [4], does not require any explicit geometrical reconstruction of the interface in the solution procedure. Thus, these VOF methods are quite straightforward and easy to implement. A geometric VOF method, on the other hand, consists O. Mookherjee (B) · S. Pramanik Department of Mechanical Engineering, NIT Durgapur, Durgapur 713209, India e-mail: [email protected] A. Sharma Department of Mechanical Engineering, IIT 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_1
3
4
O. Mookherjee et al.
of two steps: advecting the interface, and, identifying its new position (geometrical reconstruction). These features make the computational implementation reasonably complex; the complexity increases further for 3D problems. An overall comparison between algebraic VOF methods [10] and geometric VOF methods reveal that the results obtained from algebraic methods are better than Simple Line Interface Calculation (SLIC) VOF method but inferior to most of the Piecewise Linear Interface Calculation (PLIC) VOF methods [8]; particularly for complex flow fields. However, development of new algebraic interface capturing methods are justified because they are relatively much simpler and robust than a PLIC type VOF scheme. An algebraic interface capturing scheme using the hyperbolic tangent function, Tangent of the Hyperbola for INterface Capturing (THINC) was first developed by Xiao et al. [15]. The smooth step-like nature of the hyperbolic tangent function makes it suitable for interpolating the advection fluxes of the volume fraction field and is effective in eliminating numerical diffusion and oscillations. Numerical experiments [16] reveal that the THINC scheme coupled with the Weighted Line Interface Calculation (WLIC) framework possesses adequate accuracy and has a more robust performance as compared to existing high-resolution scheme-based algebraic methods. Considering the advantages and disadvantages of algebraic and geometric type of CLSVOF methods, a direct comparison between them on a variety of multiphase flow problems is not available in the present literature. Therefore, the objective of this work is to present a numerical methodology of the PLIC-based geometric CLSVOF and WLIC-THINC scheme-based algebraic CLSVOF methods along with the incompressible Navier–Stokes equations on a co-located grid using the balanced-force approach. An in-house code is developed and performance of both the numerical techniques are studied on standard CmFD benchmark problems like dam break and bubble rise.
2 Governing Equations The present work considers two different fluids, separated by an interface, and each fluid is assumed to be incompressible and immiscible. A single fluid formulation is used, where the same set of mass and momentum conservation equations are used for both the fluids. To capture the interface, CLSVOF method requires solution of separate advection equations for the level set field φ and volume fraction field F. The advection equations are given as, ∂⎡ + ∇ · (u⎡) − ⎡∇ · u = 0 ∂t
(1)
Comparative CmFD Study on Geometric and Algebraic Coupled Level …
5
where ⎡ = φ for level set advection equation and ⎡ = χ, a characteristic function, for advection of volume fraction field. The volume fraction FP is defined as the cell averaged value of χ given as, 1 FP = χ P = ΔxΔy
xe yn χ dxdy
(2)
xw ys
The mass and momentum conservation equations are given as ∇·u=0
(3)
∂(ρu) + ∇ · (ρuu) = −∇ p + 2∇ · [μ D] + ρ g + σ k∇ F ∂t
(4)
where u is the velocity vector, ρ is the density, μ is the viscosity, p is the hydrodynamic pressure, g is the acceleration due to gravity, σ is the surface tension coefficient, k is the curvature of the interface, and D is the rate of deformation tensor defined as ∇u + ∇u T /2. Following the work of Hong and Walker [5], a piezometric pressure P is defined as P = p − ρg · x
(5)
Implementing this in Eq. 4, the modified Navier–Stokes equation is given as ∂(ρu) + ∇ · (ρuu) = −∇ P + 2∇ · [μ D] + {(ρ0 − ρ1 )g · x + σ k}∇ F ∂t
(6)
Due to the piezometric formulation, the pressure and gravity forces appear as gradient quantities and can be treated identically in the discrete level. This treatment allows a perfect force balance between them in the scenarios where they are competing against each other. The fluid properties are represented as a function of volume fraction given by ρ = ρ1 F + ρ0 (1 − F),
(7)
μ = μ1 F + μ0 (1 − F)
(8)
where the subscripts 1 and 0 represent fluid 1 and 0, respectively.
6
O. Mookherjee et al.
3 Numerical Methodology Finite volume method is used to discretize the governing equations, presented in Sect. 2. A co-located grid arrangement is followed, where all the solution variables (velocity, pressure, volume fraction, and level set) are defined at the centroid of each control volume (cell).
3.1 CLSVOF Advection Equations The generalized CLSVOF advection equation (Eq. 1) is discretized using an operator splitting algorithm [13], given as ∗
n
⎡P = ⎡P − n+1
⎡P
G nxe − G nxw ∗ ue − uw Δt + ⎡ P Δt, Δx Δx
∗
= ⎡P −
G ∗yn − G ∗ys Δy
∗
Δt + ⎡ P
vn − vs Δt Δy
(9) (10)
The flux for the volume fraction advection equation in the x direction G xe is given as yn
xe −u e Δt
G xe = −
χ Pup (x, y)dxdy ys
(11)
xe
where Pup is defined as Pup =
P if u e ≥ 0 E if u e < 0
(12)
Similarly advection fluxes in the y direction can be obtained. For the level set advection equation, a second-order ENO scheme [14] is used to determine the advection fluxes. Finally, Strang-splitting [13] is employed to minimize the error by dimensional splitting. For the PLIC-CLSVOF method, the characteristic function (χ ) for the volume fractions is the sharp unit Heaviside function given as χ (x, y) =
1 for fluid 1 at the point (x, y) 0 for fluid 1 at the point (x, y)
(13)
Comparative CmFD Study on Geometric and Algebraic Coupled Level …
7
A piecewise linear segment is geometrically constructed which defines the interface between the two fluids. The orientation and position of the interface are determined by the normal vector (calculated from the smooth level set field) and the volume fraction field, respectively. The advection fluxes are obtained by calculating the area under the linear interface segment. A detailed description of the method is presented in the literature [12, 13]. For the THINC-CLSVOF method, a hyperbolic tangent function is used as the characteristic function; given as χx P
1 x − xw − x˜ P 1 + αx tanh β = 2 Δx
(14)
where αx represents the direction of the interface; given as αx =
1 if n x p ≥ 0 −1 if n x p < 0
(15)
where n is the interface normal and β is a smoothness parameter for the characteristic function. A larger value of β results in a sharp function thereby reducing the thickness of the diffused interface. In the present work, β = 2.3 which corresponds to three mesh smoothing. Further, x˜ P Δx is the distance between xw and the interface, indicating position of the interface in a two-fluid cell. The value of x˜ P can be obtained from the cell volume fraction as FPn
1 = Δx
xe χ P (x, x˜ P )dx
(16)
xw
By using such a smoothened characteristic function χ , the volume fraction field F becomes smoother than that of the PLIC-CLSVOF method. This feature is useful for the implementation of interfacial body forces in a continuum framework. Also, interface reconstruction is not required because the fluxes for volume fraction advection equation can be directly obtained by analytically integrating χx within the required limits. The accuracy of the present THINC-based algebraic method is further improved by coupling it with a WLIC framework, where the shape of the interface is weighted by using the weights calculated from the interface normal n given as, χ P = ωx P (n P )χx p + ω y P (n P )χ yp
(17)
where ωi P =
|n i P | | | |n x P | + |n y P |
(18)
8
O. Mookherjee et al.
are the weights, and χx and χ y are the characteristic functions of the vertical and horizontal interface, respectively. n x and n y are the x and y component of the interface normal n. Further details are available in [16]. After the solution of level set advection equation, the level set field is reinitialized to preserve it as the signed distance function. This is achieved by solving an Eikonal equation, given as |∇φ| = 1
(19)
where the above equation is solved by using a fast sweeping method [18].
3.2 Navier–Stokes Equations The Navier–Stokes equation is solved using a semi-explicit projection method, wherein the pressure and interfacial body force terms are considered implicitly with all other terms as explicit. The discretized momentum equation is given as − (ρu)nP (ρu)n+1 P = −∇ · (ρuu)n − ∇ P n+1 Δt + ∇ · (2μ D)n + {(ρ0 − ρ1 )g · x + σ k}∇ F n+1
(20)
In the predictor step of the pressure projection method, the pressure and body force terms are not considered in the momentum equation; and the predicted cell center velocity u∗P is given as ρ
u∗P − unP + ∇ · (ρuu)n = ∇ · (2μ D)n Δt
(21)
Momentum advection of the interfacial cells is achieved by the product of volume fraction-based advected mass and corresponding velocity at the donor cell face [1]. Subtracting Eq. (21) from Eq. (20), the corrected cell center velocity un+1 P is given as ρ
− u∗P un+1 P = −∇ P n+1 + {(ρ0 − ρ1 )g · x + σ k}∇Fn+1 Δt
(22)
A proper balance between the interfacial and pressure forces in the co-located grid arrangement is achieved by using a balanced-force method [2]. The pressure field for the next time step is obtained from the pressure Poisson equation, which is derived from the divergence of Eq. (22) utilizing the continuity equation (Eq. 3); given as
Comparative CmFD Study on Geometric and Algebraic Coupled Level …
∇·
Δt ∇P ρ
n+1
Δt = ∇ · u + n+1 {(ρ0 − ρ1 )g · x + σ k}∇ F n+1 ρ ∗
9
(23)
3.3 Solution Algorithm A generalized solution algorithm for the present algebraic and geometric CLSVOF methods is presented as follows. 1. Initialize the velocity, pressure, volume fraction, and level set fields. 2. Calculate the geometric properties of the interface (normal and curvature). 3. Solve the prediction equation (Eq. 21) for the predicted velocity u∗P at the cell centers. 4. Solve the pressure Poisson equation (Eq. 23) for the pressure field P. 5. Solve the correction equation (Eq. 22) for the final velocity un+1 at the cell P centers. 6. Advect the interface by solving the advection equations for the characteristic function χ and level set function φ. 7. Reinitialize the level set field. 8. Stop the simulations if the termination criteria is satisfied, or else repeat the process from step 2. Computationally, the most expensive step in the above algorithm is solution of the pressure Poisson equation which is solved using a preconditioned GMRES method [11].
4 Results and Discussion For a relative performance study of the present PLIC-CLSVOF and WLIC-THINCCLSVOF methods, static droplet test for the surface tension model, and two sufficiently different benchmark CmFD problems on dam-break simulation and bubble rise are considered. In the dam-break problem involving collapse of a liquid column, gravitational force plays the dominating role. Contrarily, in the bubble rise problem, both the capillary and gravitational forces dictate the shape of the rising bubble.
4.1 Static Droplet To compare the performance of the surface tension model, standard benchmark test of a static drop in equilibrium without any external forces is considered. Parameters of the test problem are same as that of Francois et al. [2]. The computational domain is a
10 Table 1 Magnitude of maximum spurious velocities |u max | obtained after one and 50 time steps. The density ratio is 10 and the time step is 10−3
O. Mookherjee et al.
T
PLIC
WLIC-THINC
Francois et al
Δt
6.4 × 10−3
9.6 × 10−3
4.87 × 10−3
50Δt
1.38 × 10−1
2.73 × 10−1
1.63 × 10−1
square of size 8 having a droplet of radius 2 at the center. Simulations are performed using a mesh size of 40 × 40. The interface curvature is numerically calculated from the LS field. Table 1 depicts the comparison of maximum magnitude of spurious velocity computed after one and 50 time steps by both the CLSVOF methods with the work of Francois et al. [2]. It is observed from the table that the magnitude of spurious velocities obtained from the algebraic CLSVOF method is slightly more than the geometric method, although these results are well validated against the work of Francois et al. [2].
4.2 Dam Break In this problem, an initially static square water column confined in the corner of a rectangular cavity, as depicted in Fig. 1, collapses under the action of gravitational force. The parameters of the problem along with the boundary conditions are chosen according to the experimental setup of Martin and Moyce [7], shown in Fig. 1. Water is considered as the reference fluid 1 and air as the reference fluid 0. Physical properties for this problem (in SI units) are ρ1 = 1000, ρ0 = 1.2, μ1 = 1.139×10−2 , and μ0 = 1.78 × 10−4 . An experimental investigation was done by Martin and Moyce [7] to measure the temporal evolution of the leading edge distance of the surge front which evolves to fill the container as the water column collapses. Simulations are performed using a 128 × 32 uniform mesh. The result of both the CLSVOF methods along with the experimental values are presented in Fig. 2 for various non-dimensional times, √ T = t g/a. Here, a is the initial width of the water column and Z is the leading edge distance from the left wall of the cavity. As compared to the experimental and numerical results in the literature [3, 9, 17], Fig. 2 shows an excellent performance of the present CLSVOF methods. Since the PLIC-CLSVOF method incorporates a sharp-interface formulation, while the interface is inherently diffused in WLICTHINC-CLSVOF method, their performance can be better analyzed in the scenarios where the topology of the interface will undergo rapid deformation. This is evident in the next test problem.
Comparative CmFD Study on Geometric and Algebraic Coupled Level …
11
Fig. 1 Computational setup of the dam-break simulation depicting initial configuration of the interface, size of the domain, and the boundary conditions
Fig. 2 Comparison of temporal evolution of the leading edge distance between present CLSVOF methods and existing experimental and numerical results
4.3 Buoyant Rise of a Bubble A bubble, having a radius of 0.25 and centered at (x, y) = (0.5, 0.5), in an initially quiescent medium, is considered here as shown in Fig. 3. The size of the computational domain and the boundary conditions are also presented in the figure. Physical properties for this problem (in SI units) are ρ1 = 1000, ρ0 = 1, μ1 = 10, μ0 = 0.1, g = 0.98, and σ = 1.96. Further, the density and viscosity ratio are 1000 and 100, respectively, which are large enough for the shape of the bubble to be considered within the skirted and dimpled ellipsoidal-cap regimes indicating that breakup may occur. Simulations are performed using a 75 × 150 uniform mesh and the results are presented in Fig. 4 in terms of the temporal evolution of shape of the bubble and compared with the numerical work of Hysing et al. [6]. Contrasting results are obtained from the present CLSVOF methods, as can be expected, due to their dissimilar treatment of the interface. In the PLIC-CLSVOF method, the breakup of the droplet and resulting formation of the satellite drops occur naturally due to the sharp treatment of the interface. Consequently, the obtained results validate well with the work of Hysing et al. where a similar sharp treatment of the interface was used. For the WLIC-THINC-CLSVOF method, a delayed breakup of the droplet is seen in the figure since the interface is diffused within three to four mesh cells. This results
12
O. Mookherjee et al.
in the formation of filament-like structures which tend to become an integral part of the bubble; thus, hampering its shape as it rises upward. To ascertain the efficacy of the CLSVOF methods quantitatively, Fig. 5 presents validation of the instantaneous rise velocity of the bubble with the work of Hysing et al. [6]. The performance of both the CLSVOF methods is in good agreement with the published results.
Fig. 3 Computational setup of the bubble rise simulation depicting initial configuration of the interface, size of the domain, and the boundary conditions
Fig. 4 Comparison of temporal evolution of the interface using the PLIC-CLSVOF (blue line) and WLIC-THINC-CLSVOF (green line) schemes at: (a) t = 1.2; (b) t = 2.2; (c) t = 2.6; (d) t = 3.0. Red line represents the work of Hysing et al. [6]
Comparative CmFD Study on Geometric and Algebraic Coupled Level …
13
Fig. 5 Comparison of instantaneous rise velocity of the bubble between present CLSVOF methods and Hysing et al. [6]
5 Conclusions Coupled Level Set and Volume Of Fluid (CLSVOF) methods can be broadly categorized into two types: geometric type methods and algebraic type methods. Accurate interface construction is a crucial step of the geometric methods which involves significant numerical complexity. On the other hand, in an algebraic method the interface is numerically diffused within a few mesh cells which makes them relatively straightforward to design and implement. In the present study, a framework of both geometric type Piecewise Linear Interface Calculation (PLIC) CLSVOF method and algebraic type Weighted Line Interface Calculation-Tangent of the Hyperbola for INterface Capturing (WLIC-THINC) CLSVOF method are presented along with the numerical technique for solution of the incompressible Navier–Stokes equation in a co-located grid using the balanced-force method. Relative performance of the two CLSVOF methods is studied on standard two-phase flow benchmark problems: dam break and bubble rise. The key observations are as follows. • For the dam-break problem, similar results are obtained from both the CLSVOF methods, which are in excellent agreement with the existing experimental and numerical results. • Simulations of the bubble rise problem, however, reveal discernible differences between the results; although validation against existing numerical results are quite satisfactory. • This is attributed to the fact that in the PLIC-CLSVOF method, a sharp interface is maintained which can accurately mimic the droplet breakup phenomenon; and thus, can capture the dynamics precisely. • For the WLIC-THINC-CLSVOF method, the interface is numerically diffused across three to four mesh cells causing a delayed breakup of the droplet; and thus, degrades the accuracy of the solution.
14
O. Mookherjee et al.
Nomenclature D F g k P u t β Γ σ ϕ χ Ω ()
Rate of deformation tensor Volume fraction Acceleration due to gravity (ms−2 ) Interface curvature (ms−1 ) Piezometric pressure (Nm− 2 ) Velocity vector (ms−1 ) Time (s) Interface thickness parameter– Advected property Surface tension coefficient (Nm− 1 ) Level set Characteristic function Interface weights Volume-averaged quantity
References 1. Arrufat T, Crialesi-Esposito M, Fuster D, Ling Y, Malan L, Pal S, Scardovelli R, Tryggvason G, Zaleski S (2021) A mass-momentum consistent, volume-of-fluid method for incompressible flow on staggered grids. Comput Fluids 215:104785 2. Francois MM, Cummins SJ, Dendy ED, Kothe DB, Sicilian JM, Williams MW (2006) A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. J Comput Phys 213(1):141–173 3. Gada VH, Sharma A (2011) On a novel dual-grid level-set method for two-phase flow simulation. Numer Heat Transf, Part B: Fundamentals 59(1):26–57 4. Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39(1):201–225 5. Hong W-L, Walker DT (2000) Reynolds-averaged equations for free-surface flows with application to high-Froude-number jet spreading. J. Fluid Mech 417:183–209 6. Hysing S-R, Turek S, Kuzmin D, Parolini N, Burman E, Ganesan S, Tobiska L (2009) Quantitative benchmark computations of two-dimensional bubble dynamics. Int J Numer Meth Fluids 60(11):1259–1288 7. Martin JC, Moyce WJ (1952) Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal plane. Philos. Trans. Roy. Soc. London A 244:312–324 8. Pilliod Jr JE, Puckett EG (2004) Second-order accurate volume-of-fluid algorithms for tracking material interfaces. J Comput Phys 199(2):465–502 9. Raessi M, Pitsch H (2012) Consistent mass and momentum transport for simulating incompressible interfacial flows with large density ratios using the level set method. Comput Fluids 63:70–81 10. Rudman M (1997) Volume-tracking methods for interfacial flow calculations. Int J Numer Meth Fluids 24(7):671–691 11. Saad Y, Kesheng W (1996) DQGMRES: a direct quasi-minimal residual algorithm based on incomplete orthogonalization. Numer Linear Algebra with Appl 3(4):329–343 12. Son G, Hur N (2002) A coupled level set and volume-of-fluid method for the buoyancy-driven motion of fluid particles. Numer Heat Transfer: Part B: Fundament 42(6):523–542
Comparative CmFD Study on Geometric and Algebraic Coupled Level …
15
13. Sussman M, Puckett EG (2000) A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows. J Comput Phys 162(2):301–337 14. Sussman M, Smereka P, Osher S (1994) A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 114(1):146–159 15. Xiao F, Honma Y, Kono T (2005) A simple algebraic interface capturing scheme using hyperbolic tangent function. Int J Numer Meth Fluids 48(9):1023–1040 16. Yokoi K (2007) Efficient implementation of THINC scheme: a simple and practical smoothed VOF algorithm. J Comput Phys 226(2):1985–2002 17. Yu C-H, Wen HL, Gu ZH, An RD (2019) Numerical simulation of dam-break flow impacting a stationary obstacle by a CLSVOF/IB method. Commun Nonlinear Sci Numer Simul 79:104934 18. Zhao H (2005) A fast sweeping method for Eikonal equations. Math Comput 74(250):603–627
Numerical Modelling of a Binary Droplet on Solid Surface Mradul Ojha, Lalit Kumar, and Rajneesh Bhardwaj
1 Introduction The study of sessile evaporating droplet is an important issue in the scientific community because of its practical implications in various industrial applications, which include ink-jet printing, electronic cooling, fuel combustion, and natural phenomena like rain, snow, and dew generation. The evaporation behaviour of sessile droplets depends on the combined interplay of solid–liquid interaction and liquid-ambient conditions. The interaction of solid and liquid takes into account by the effects governing surface wettability [1], temperature [2] and roughness, etc., whereas the liquid-ambient state includes the composition of liquid (pure or multi-component system [3], ambient temperature, and humidity [4], etc. The pure liquid sessile droplet evaporation dynamics in the context of uniform wettability (i.e. hydrophilic) were initially studied by Deegan et al. [5]. They show a non-uniform evaporative flux at the liquid–vapour interface, with the maximum evaporative flux at the contact line because of the pinned nature of the liquid–vapour interface. Therefore, the author proposed a constant contact radius (CCR) evaporation mode. A similar study by Patil et al. [6] on hydrophobic surfaces introduces a constant contact angle (CCA) mode, which shows a continuous slipping of the contact line, keeping the contact angle constant. The studies also show a mixed mode evaporation at the latter stage of a pure liquid evaporating sessile droplets, irrespective of their
M. Ojha Centre for Research in Nanotechnology and Science, IIT Bombay, Mumbai 400076, India L. Kumar Department of Energy Science and Engineering, IIT Bombay, Mumbai 400076, India R. Bhardwaj (B) 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_2
17
18
M. Ojha et al.
surface wettability. However, the mixed mode is distinctly observed for the binary droplet during most of their evaporation time, as observed in following publication. Ozturk and Erbil [7] show the experimental results of 2 µl binary droplets evaporating under different relative humidity and ambient temperature conditions. They also found a nonlinear decrement of volume over time irrespective of humidity, as suggested in previous literature. The other important effect is water vapour adsorption from the environment on pure ethanol droplets. Gurrala et al. [8] experimentally investigate the evaporation dynamics of the ethanol–water binary droplet at different substrate temperatures. They also concluded a nonlinear trend in the evaporation process under ambient conditions. But at an elevated temperature, a linear volume decrement with time was observed, like a pure fluid with higher volatility. Shi et al. [9] conducted the evaporation of the binary liquid droplet of water and ethanol on a poly (tetrafluoroethylene) surface. The study suggested the changes in the contact angle, drop height, and contact diameter which is divided into three stages. The former stage is governed by ethanol and the latter by water. At the same time, the intermediate stage shows the most significant effect of the binary drops. The available literature is mostly on experimental investigation for estimating the nonlinear variation of volume over time. A few concentrate on numerical and analytical study for the evaporation dynamics of the binary liquid sessile droplet like Ozturk and Erbil [7] proposed an model, where the mass loss expression proposed by Hu et al. [1] is used for binary droplet evaporation, with modification in diffusion coefficient, molecular mass, and saturation pressure. Diddens et al. [3] also proposed a finite element model to incorporate the effects of surface tension, mass density, diffusion coefficient, and activity. This makes the modelling more complex. Since most industrial applications are under ambient conditions, the model presented above requires high computational time and power. The above proposed model can be relaxed by using different dimensionless groups that define the dominant phenomena and the variable region of the evaporating binary sessile droplet. The present model was an extension of the model proposed by Kumar and Bhardwaj [10]; for the pure liquid sessile model to the binary liquid sessile droplet. The model simplifies by considering the effects of the activity coefficient, diffusion coefficient, and density on the diffusion-limited evaporation model. The present model also includes the dimensionless number to simplify the species transport equation for the binary sessile droplet.
2 Numerical Modelling The section consists of numerical modelling of the binary liquid droplet evaporation in the ambient environment. This section introduced with the geometric generation of the droplet, which is followed by a governing equation, boundary condition and algorithm.
Numerical Modelling of a Binary Droplet on Solid Surface
19
Fig. 1 Sessile droplet on a hydrophobic surface, b hydrophilic surface
2.1 Geometry Description The droplet formed during the experiments is reproduced for the finite element modelling using the MATLAB PDE tool. The characteristic length of the initial droplet volume deposited on the substrates is always less than the capillary length of the liquids used. Thus, the spherical cap assumption for the droplet is valid for the geometry generation. To verify the spherical cap assumption of the droplet, Bond number [2] (Bo = ρgh(0, t)R/γ ) is calculated for the different cases, and obtained less than 0.1. Thus, it is safe to approximate the interface of the droplet using the spherical cap approximation. The spherical cap droplet contact angle (θ ) and volume (V (h(0, t), R)) are calculated using the expression below: tan
h(0, t) θ = 2 R
V (h(0, t), R) = π
h(0, t) 2 3R + h(0, t)2 6
(1) (2)
The relevant geometric parameters, such as, the height of the liquid–vapour interface at the droplet apex, h(0, t) and the wetted radius (R) are taken from experiments. Their definitions are depicted in Fig. 1. As the droplet on substrate is uniformly surrounded by ambient condition. Therefore, the net diffusion of vapour in azimuthal plane is negligible. Thus, the droplet surrounding is made in an axis-symmetric cylindrical coordinate system, as shown in Fig. 2. Here, the shaded area represents the droplet region in the axis-symmetric plane and the meshed area represents the surrounding region.
2.2 Governing Equation The model employs species transport equation in the region surrounding the binary liquid sessile droplet, where vapour diffuses from the vicinity of the liquid–vapour interface into the ambient.
20
M. Ojha et al.
Fig. 2 Schematic of sessile droplet and the boundary condition
The species transport equation is expressed as follows [11]: ∂ci + u.∇ci = ∇ D(i,air) ∇ci ∂t
(3)
The first term on the left of Eq. 3 shows the transient term with ci is the concentration of the component, the second term on the left shows the convective term with (u) is the velocity field in the surrounding, and on the right is the diffusion term with (Di, air ) is the diffusion of the component (i ) in air. As there is no external heating or mass flow of air in the vicinity of the droplet, mass loss due to natural and forced convection (u = 0) is negligible in the surrounding [2, 5]. It should also be noted that vapour diffusion through the surrounding air is very slow; the time for vapour to adjust to the new shape of the droplet is of the therefore, R2 O tevp ∼ Di,air , which is much shorter than the total evaporation time, which is h(0,t) of the O t f ∼ 0.2ρ R csat [1], where, csat and Di,air are saturation concentration Di,air and diffusion coefficient of the respective fluids. Therefore, the resulting ratio of the tevp 5csat two times scale t f ∼ ρ for water is approx. 1.1523 ∗ 10−4 and for ethanol is approx. 9.0521 ∗ 10−4 is very very less than one for the fluids under consideration. As a result, the surrounding environment is in a quasi-steady state of evaporation at all times [1, 5]. Therefore, the species transport defined in Eq. 3 is simplified in the form of the Laplace equation as follows: ∇ 2 ci = 0
(3)
Numerical Modelling of a Binary Droplet on Solid Surface
21
2.3 Boundary Conditions for the Binary Liquid Sessile Droplet As for the interface, evaporating cooling is not considered in the present study because the experiments are conducted under ambient conditions, so the difference in temperature across the interface is of the order of (1 ◦ C). There is order (1.2 ◦ C) temperature difference between the contact line temperature and the apex temperature of the sessile water droplet interface [10], and 2.4 ◦ C for sessile ethanol droplet [12]. Therefore, the temperature difference at interface for the binary droplet made up from water and ethanol lies between the above temperatures. Therefore, a constant temperature interface is considered. This results in constant saturation concentration in the vicinity of the liquid–vapour interface. As for binary liquid droplet, adding two miscible solvents alters the partial pressures of each component from their pure states. This altered partial pressure is estimated by Raoult’s law, which is stated for an ideal mixture of components, but the mixture of ethanol and water is non-ideal in nature [13]. The non-ideal affection of the species is estimated by the activity of respective species using the ‘Aerosol Inorganic–Organic Mixtures Functional groups Activity Coefficients’ (AIOMFAC) Model [14, 15]. Therefore, Raoult’s law was modified by taking component activity into account. This changes Raoult’s law as follows [16]: pLGi = γi xi psati
(4)
where i = 1, 2, . . . n for the number of component, γi is the activity coefficient of the component at specific mixture composition, and xi is the mole fraction of the component in the liquid phase. psati is the saturation pressure that corresponds to the pure state of the component, and pLGi is the partial pressure of the component at the interface of binary liquid droplet. The following estimations are used for the component’s number of moles in the binary liquid droplet: ni =
νi Vtotal ρi Mi
(5)
where n i , νi and Vtotal are the number of moles, volume fraction of component in liquid phase and total volume of the droplet, respectively. There after, using Eq. 5, the mole fraction for ethanol in the binary liquid droplet is estimated as follows: xE =
nE n E + nW
(6)
The ‘Aerosol Inorganic–Organic Mixtures Functional groups Activity Coefficients’ (AIOMFAC) Model [16, 14] is used to estimate the activity coefficient of ethanol and water at the temperature of 25 °C. The variation of activity coefficient with mole fraction of ethanol in liquid phase can be seen in Fig. 3.
22
M. Ojha et al.
Fig. 3 Activity coefficient of ethanol and water for ethanol–water binary mixture at different ethanol composition (x E ), where markers are from AIOMFAC model and continuous line is curve fitting
The following curve fitting equations have been used to approximation the activity coefficient for ethanol and water: γ E = 3.538e−8.937x E + 1.508e−0.4626x E
(7)
γW = 0.798e−5.825x W + 1.883e−0.6738x W
(8)
where x E and x W are the mole fraction of ethanol and water in the liquid phase. As we solve the governing equation in the form of concentration gradient across the surrounding domain. Therefore, the concentration of the component in the vicinity of the liquid–vapour interface ciLG of the droplet uses the expression below [16,14]: ciLG = γi xi
psati Mi = γi xi csati Ru T
(9)
where psati , Mi , Ru , and T are saturated vapour pressure of the ith compomolecular mass of the component, universal gas constant nent at 3interface, 8.314 m Pa K−1 mol−1 and temperature, respectively. Here, the saturated vapour pressure variation with temperature is calculated using Antoine equation [17]. The saturated vapour pressure is converted to saturated concentration using an ideal gas equation, see Eq. (9). The variation of the saturation concentration of the pure liquid component with temperature is approximated by a forth order polynomial given below: csati = Ca T 4 + Cb T 3 + Cc T 2 + Cd T + Ce
(10)
where the temperature in degree Celsius and constant Ca , Cb , Cc , Cd , and Ce used is given in table 2: The saturation concentration calculated using the Eq. (10) for water and ethanol at the temperature of 25 °C is 0.023 and 0.1461 kg/m3 , respectively, which is consistence with previous literature [18, 19].
Numerical Modelling of a Binary Droplet on Solid Surface Table 1 Thermodynamic properties of water and ethanol
Solvent
Density kg/m3 Molecular mass (g/mol) Diffusion coefficient m2 /s
Table 2 Coefficient values for saturation concentration equation for water and ethanol
23
Water
Ethanol
998
784
18.015
46.04
2.51 ∗ 10−5
1.356 ∗ 10−5
Solute
Water
Ethanol
Ca
4.168 ∗ 10−9
2.008 ∗ 10−8
Cb
−3.204 ∗ 10−8
1.428 ∗ 10−7
Cc
1.755 ∗ 10−5
1.033 ∗ 10−4
Cd
2.284 ∗ 10−4
0.001308
Ce
0.005173
0.03879
The other boundary conditions for the ambient state and the substrate are as follows: 1. A Dirichlet boundary condition for water vapour at the ambient state is defined as follows: c∞W = Rh csatW
(11)
where c∞W is the water vapour concentration in the surrounding air, Rh is the ambient relative humidity, and csatW is the saturation concentration of water. Similarly, the concentration of ethanol in the ambient state is zero. c∞ E = 0
(12)
where c∞ E is the concentration of the ethanol in ambient state. 2. A Neumann boundary condition with no penetration of components at the substrates is applied
∂ci ∂z
=0
(13)
z=± h2
The schematics for the boundary condition and the computational domain are shown in Fig. 2.
2.4 Algorithm The solution of the governing (see Eq. 3) provides the spatial variation of the component concentration in the computational domain, from which the concentration
24
M. Ojha et al.
gradient (∇ci ) across the interface generates an evaporative flux ( ji ) as follows: ji = − Di, air ∇ci .n
(14)
where n represents the surface normal to the interface. This results in estimating, the mass loss (m) ˙ due to diffusion of the component using evaporative flux as follows: m˙ =
ji .dA
(15)
where dA represents the elemental surface area of the interface. After estimating the evaporative mass loss of species, it is converted to respective volume loss for the calculation of the volume left Vileft of each component for the successive time step (∇t) as follows: Vileft = νi .Vtotal −
m.∇t ˙ ρi
(16)
As the volume of the component left in the droplet is known, the new total volume (see Eq. 17) of the droplet and the updated volume fraction (see Eq. 18) of the droplet are estimated to be used in the next iteration as follows: Vtotal = VEleft + VWleft
(17)
VEleft VEleft + VWleft
(18)
ν EK +1 =
where VEleft and VWleft denote the amounts of ethanol and water left in the liquid phase, and (k + 1)th is the next iteration. The next step is to find updated contact radius (R) or contact angle (θ ), for next iteration, which depends on the mode of evaporation observed in the experiments, mean constant contact radius (CCR) or constant contact angle (CCA). For meshing a linear triangular mesh elements are generated through the computational domain (see Fig. 2) from the liquid–vapour interface to the ambient state. The liquid–vapour interface is simultaneously refined, which ensures an accurate estimate of the concentration gradient. A mesh-independent study was carried out to increase the model’s accuracy. The grid independence study is conducted for 50 % (V/V) concentration of ethanol at the contact angle of 45° with contact radius as 1.72 mm at the temperature of 25 °C. Therefore, the L-2 norm is calculated with respect to most refine grid size [10]. As a result, there are 7118 elements discretizing the computational domain as given in Fig. 4.
Numerical Modelling of a Binary Droplet on Solid Surface
25
Fig. 4 Grid convergence study, L-2 norm error in concentration is plotted with respect to different tested grids
3 Code Validation To validate the present model, we compare our computational model with the experimental results of previously published data on pure and binary sessile droplets evaporation. First we compare the variation of volume with time for Ozturk and Erbil [4]. The author studied the effect of humidity on the evaporation dynamics of ethanol and water binary droplet. The study shows the nonlinear variation of the volume over time irrespective of the humidity. The respective published data is the variation of 2 µl initial volume with different composition of ethanol in water over time with relative humidity and temperature as 52 ± 2% and 25 °C, respectively. We simulated the present model at same experimental condition as aforesaid with a CCA mode for evaporation. Figure 5 shows the comparison of the present model and the experiment are well. Shi et al. [9] also conducted a similar study on the wetting and evaporation behaviour of water–ethanol sessile droplet on PTFE surface. The author also shows the nonlinear variation of mass loss over time due to high volatility of ethanol. The Fig. 5 Comparison of Ozturk and Erbil [4] experimental results with present model for volume variation with time. E: Experiment denotes the Ozturk and Erbil experimental results
26
M. Ojha et al.
Fig. 6 Comparison of Shi et al.[9] experimental results with present model
author also suggests that a depinning with the contact line’s contraction for high ethanol concentration leads to a mixed evaporation mode. The study of mixed mode of evaporation (for contact line dynamics) is not included in the present study. Therefore, a little variation between experiments and the proposed model is observed for high ethanol-concentrated binary droplets (see Fig. 6).
4 Conclusion This work investigates the evaporation dynamics of a sessile binary droplet on both hydrophilic and hydrophobic surfaces. The geometric configuration of the sessile droplet in prior-art experiments is generated in the axis-symmetric cylindrical coordinate system. A diffusion-limited mass transfer equation is employed for the computational domain using the Dirichlet boundary condition of saturated concentration at the liquid-vapour interface and the product of relative humidity to saturation concentration at the ambient condition. The linear triangular mesh elements are generated in the computational domain. The governing equation is solved using the Finite element method in the MATLAB PDE tool. A non-uniform mesh element is employed, with finer mesh at the interface and coarse mesh at the far field. The fine mesh at the interface ensured accurate concentration gradient estimation. A mesh-independent study was carried out to increase the model’s accuracy. As a result, 7118 elements are discretizing the computational domain. We compare our computational model with the available experimental results on pure and binary sessile droplets. We observe a nonlinear trend in the volume of the evaporating droplets with time, which is typical for all the values of relative humidity chosen herein. We attribute this observation to the significant difference in the volatility of water and ethanol that constitutes the binary liquid mixture. The faster evaporation rate of ethanol dominates the droplet mass loss process at the initial stage of drying. Thereafter, the water
Numerical Modelling of a Binary Droplet on Solid Surface
27
component remains, which evaporates at a later stage. Hence, the aforesaid nonlinearity stems from a transition from two separate regimes of droplet evaporation: the first is limited by ethanol drying, and the second is limited by the water component.
References 1. Hu H, Larson RG (2002) Evaporation of a sessile droplet on a substrate. J Phys Chem B 106(6):1334–1344 2. Hu H, Larson RG (2005) Analysis of the effects of Marangoni stresses on the micro-flow in an evaporating sessile droplet, Langmuir 21(9):3972–3980. PMID: 15835963 3. Diddens C, Kuerten JGM, Van der Geld CWM, Wijshoff HMA (2017) Modelling the evaporation of sessile multi-component droplets. J Colloid Interface Sci 487(2017), 426–436 4. Ozturk T, Yildirim Erbil H (2018) Evaporation of water-ethanol binary sessile drop on fluoropolymer surfaces: influence of relative humidity. Colloids Surfaces a: Physicochem Eng Aspects 553:327–336 5. Deegan RD, Bakajin O, Dupont TF, Huber G, Nagel SR, Witten TA (1997) Capillary flow as the cause of ring stains from dried liquid drops, Nature 389(6653)827–829 6. Patil ND, Bhardwaj R () Evaporation of a sessile microdroplet on a heated hydrophobic substrate. 7. Ozturk T, Yildirim Erbil H (2020) Simple model for diffusion- limited drop evaporation of binary liquids from physical properties of the components: ethanol–water example. Langmuir 36(5):1357–1371. PMID: 31909624 8. Gurrala P, Katre P, Balusamy S, Banerjee S, Chandra Sahu K (2019) Evaporation of ethanolwater sessile droplet of different compositions at an elevated substrate temperature. Int J Heat Mass Transfer 145:118770 9. Shi L, Shen P, Zhang D, Lin Q, Jiang Q (2009) Wetting and evaporation behaviours of water– ethanol sessile drops on PTFE surfaces, surface and interface analysis: an international. J Dev Develop Appl Tech Anal Surfaces, Interfaces Films 41(12–13):951–955. 10. Kumar M, Bhardwaj R (2018) A combined computational and experimental investigation on evaporation of a sessile water droplet on a heated hydrophilic substrate. Int J Heat Mass Transfer 122:1223–1238 11. Bozorgmehr B, Murray BT (2021) Numerical simulation of evaporation of ethanol–water mixture droplets on isothermal and heated substrates ACS Omega 6(19):12577–12590 12. Ye S, Wu C-M, Zhang L, Li Y-R, Liu Q-S (2018) Evolution of thermal patterns during steady state evaporation of sessile droplets. Exper Thermal Fluid Sci 98:712–718 13. Yu Borodulin V, Letushko VN, Nizovtsev MI, Sterlyagov AN (2019) The experimental study of evaporation of water–alcohol solution droplets. Colloid J 81(3):219–225 14. Zuend A, Marcolli C, Booth AM, Lienhard DM, Soonsin V, Krieger UK, Topping DO, McFiggans G, Peter T, Seinfeld JH (2011) New and extended parameterization of the thermodynamic model aiomfac: calculation of activity coefficients for organic-inorganic mixtures containing carboxyl, hydroxyl, carbonyl, ether, ester, alkenyl, alkyl, and aromaticfunctional groups. Atmospheric Chem Phys 11(17):9155–9206 15. Zuend A, Marcolli C, Luo BP, Peter T (2008) A thermodynamic model of mixed organicinorganic aerosols to predict activity coefficients. Atmospher Chem Phys 8(16):4559–4593 16. Diddens C (2017) Detailed finite element method modelling of evaporating multi-component droplets. J Comput Phys 340:670–687 17. Vargaftik NB (1975) Handbook of physical properties of liquids and gases- pure substances and mixtures, 2nd edn. 1
28
M. Ojha et al.
18. Diddens C, Tan H, Lv P, Versluis M, Kuerten JGM, Zhang X, Lohse D (2017) Evaporating pure, binary and ternary droplets: thermal effects and axial symmetry breaking. J Fluid Mech 823:470–497. 19. Upadhyay G, Bhardwaj R (2021) Colloidal deposits via capillary bridge evaporation and particle sorting thereof. Langmuir 37(41):12071–12088
Characterization of the Spray Flow Field at the Exit of a Pintle Injector Sanjay Kumar Gupta, Aman Bakshi, Rachit Bundiwal, and K. P. Shanmugadas
1 Introduction Liquid rocket engines are being developed currently, focusing on reusability and high thrust applications. Proper thrust control is one of the important requirements for soft-landing and orbital maneuvering of the engines. The engines should be able to restart multiple times for the reusable application. Thrust control is primarily achieved by controlling the oxidizer and fuel flow rates and by using throttleable injectors [1]. Proper mixture formation and species placement within the combustor are important to dictate the combustion efficiency, ISP, heat transfer characteristics, and stability. Any defect in the injector operation can result in different anomalies such as soot formation on the injector plate, clogging, hot streaks, chamber damage, thrust pulsations, and thermoacoustic instabilities. Pintle injectors are the throttleable injectors in which the thrust control is achieved by controlling the flow rate of fuel/oxidizer by adjusting the pintle movement [2]. Pintle injectors were used in the lunar descent module of the Apollo rockets (TRW make) and are also used in the Merlin engines of Falcon 9 rockets by SpaceX. The pintle injector provides excellent atomization and flame stabilization characteristics. Compared to other coaxial injectors, which are arranged in large numbers, a single pintle injector covers the entire combustion zone. Even at low propellant flow rates, a pintle injector can regulate both the injection area and the velocity to maintain acceptable spray conditions. Methane and liquid oxygen (METHALOX) are considered as one of the potential propellant combinations for such applications [3]. Methane has good thermal conductivity, and high specific heat which makes it suitable for regenerative cooling. Furthermore, it is non-toxic and inexpensive, and it can also be produced on other planets by Sabatier reaction. S. K. Gupta · A. Bakshi · R. Bundiwal · K. P. Shanmugadas (B) Department of Mechanical Engineering, IIT Jammu, Jammu and Kashmir 181221, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_3
29
30
S. K. Gupta et al.
Even though the current pintle injectors are providing reasonable performance, the formation of the hot streaks with hydrocarbon fuels, non-uniform heat flux in the radial direction, and higher heat flux to the chamber wall is the disadvantages related to pintle injectors. Hence, design modifications are required to develop pintle injectors with compact sprays suiting for a given combustion chamber volume without compromising the atomization characteristics.
2 Literature Review and Objectives Many previous investigations are reported on the development and characterization of the pintle injector. Austin and Heister examined the behavior of a hole-injectiontype pintle injector for a wide range of overall momentum ratios and the characteristic lengths of the chamber [4]. Using unsymmetrical dimethylhydrazine and N2O4, Yue et al. analyzed the pintle injector combustor flow using numerical simulation, and they observed two recirculation regions in the chamber [5]. A simplified axisymmetric pintle injector known as the planar pintle injector was the basis of an experimental study by Sakaki et al. in which they were able to capture the CH chemiluminescence imaging using [6]. Son et al. investigated a pintle injector in which a liquid sheet injected from the pintle nozzle is ruptured by a gas jet from an annular gap. They evaluated different spray parameters and developed correlations [7]. Based on detailed experiments and numerical simulation, Fang et al. determined how the key structural characteristics of the LOX/methane pintle injectors affect the spray cone angles and combustion performances [8]. Cheng et al. carried out the theoretical analysis, numerical simulations, and testing to accurately predict the spray angle of liquid–liquid pintle injectors [9]. The present work discusses the development of a novel pintle injector using additive manufacturing methods. The spray structure is characterized by using laser flow diagnostic techniques to obtain the primary and secondary atomization characteristics. A cut section view of the pintle injector which is designed and developed during this work is shown in Fig. 1.
3 Materials and Methods 3.1 Design of the Pintle Injector The design of pintle injector is performed following the design procedure given in [10]. In addition to the existing design, the liquid and gas flow is made to swirl in counter-rotating directions. This would impart swirl momentum to both phases and help to achieve better atomization and spray structure. The injector has two parts: the
Characterization of the Spray Flow Field at the Exit of a Pintle Injector
31
Fig. 1 Cut section view of the pintle injector
pintle and the main body. The pintle body had two manifolds one for liquid and other for the gas. Calculations are done for the diameter of all the openings, mass flow rate, exit velocity, and injection pressure inside the injector. The optimum mixture ratio is calculated as 3.4 by doing the CEA analysis for the required thrust conditions of 1 kN. The optimum mixture ratio is selected considering a chamber temperature of 3000 K and pressure of 25 bar. The design of the pintle injector considers gaseous methane flow through the annulus and liquid oxygen flow through the pintle passage. After calculating the mixture ratio, a scaled-down experimental conditions are estimated considering air and water as the test fluids. The exit area for both fuel and oxidizer was calculated and the pintle diameter of 20 mm is considered such that experiments can be conducted within 10 bar upstream injection conditions. The pintle opening distance is maintained as 1 mm. The geometry is modeled using a CAD modeling software Solidworks. On completion of the design, the model was 3D printed in multi-jet printing (MJP) method using M2R-WT material. The final injector hardware is shown in Fig. 2.
32
S. K. Gupta et al.
Fig. 2 3D-printed pintle injector
3.2 Test Conditions In the present work, experiments are conducted at atmospheric ambient conditions. Water and air are used as oxidizer and fuel, respectively. Test setup mainly consists of two different lines, i.e., the liquid line and the gas line. Liquid is pressurized using compressed air and its flow is regulated using a needle valve. Gas flow is controlled using Alicat MFC. The liquid line is connected to the liquid manifold of the injector and the gas line is connected to the gas manifold of the injector. The experimental arrangement is shown in Fig. 3. Mixture ratio (MR) is defined as the ratio of the mass flow rate of oxidizer to the mass flow rate of fuel. For maintaining a constant mixture ratio of 3.4, mass flow rate of liquid is varied from 39.28 to 62.5 g/s, and mass flow rate of air is varied from 11.55 to 18.38 g/s as given in Table 1. Tests are also conducted by varying the mixture ratio as given in Table 2.
3.3 Diagnostic Techniques In the present work, spray visualization is done using direct laser sheet imaging. To capture the average velocity field, spray PIV is done in which a high-speed LDY-300
Characterization of the Spray Flow Field at the Exit of a Pintle Injector
33
Fig. 3 Experimental arrangements
Table 1 Test conditions with same mixture ratio Case
Injection pressure of water (bar)
Mass flow rate of water (mw ) (g/s)
Mass flow rate of air (ma ) (g/s)
Mixture ratio = mw /ma
1
0.5
39.28
11.55
3.4
2
1
48.535
14.28
3.4
3
1.5
49.73
14.63
3.4
4
2
55.41
16.30
3.4
5
2.5
62.13
18.27
3.4
6
3
62.5
18.38
3.4
Table 2 Test operating conditions with different mixture ratios Case Mass flow rate of water (mw ) Mass flow rate of air (ma ) (g/ Mixture ratio MR = mw /m (g/s) s) 1
39.28
17.85
2.2
2
39.28
11.55
3.4
3
39.28
9.82
4
4
39.28
8.73
4.5
PIV laser is used. Using a 1500 mm spherical convex lens and -75 mm cylindrical concave lens, a laser sheet is created with 1 mm thickness and it covers the mid-plane of the injector. High-speed photron camera with a 100 mm lens is used to capture the images. To visualize the flow and study the spray characteristics, laser is set at a frequency of 8 kHz and 16,000 images are captured at a frame rate of 8000 fps. Spray structure, cone angle variation, and penetration length are measured for different cases.
34
S. K. Gupta et al.
To study the average velocity field using spray PIV, the laser is set at 5 kHz and 20,000 images are captured at a frame rate of 5000 fps. The laser and the camera are synchronized using a pulse generator. Experiments are carried out at a delta t of 14 µs.
4 Results and Discussion 4.1 Structure of the Pintle Injector Spray The operating conditions are varied from Case 1 to Case 6 as given in Table 1 by increasing the liquid and gas flow rates while keeping the mixture ratio constant. Direct laser sheet imaging provides detailed variation in spray structure and the primary atomization for all the cases. Instantaneous images showing the variation in the spray structure are shown in Fig. 4. Time-resolved images showed the primary and secondary atomization processes at the injector exit. At the pintle exit, a fine droplet formation occurs, and a finely atomized spray is ensured from the injector. Apart from this, a ligament shedding phenomenon is observed at the injector exit. These ligaments are originated from an accumulated liquid bulk at the injector’s outer lip, which swirls around the tip. These liquid bulk extends further and shed into ligaments and droplets from multiple locations from the injector exit. These ligament shedding results in the formation of bigger liquid droplets.
4.2 Variation of Cone Angle and Penetration Length In the present work, different sheet parameters, namely cone angle and penetration length are quantified at selected operating conditions as given in Table 2. The operating conditions are varied from Case 1 to Case 4 by increasing the air flow rate and keeping the liquid flow rate constant to get different mixture ratio values. Direct laser sheet imaging is done to estimate the variation of the cone angle and penetration length for different mixture ratio cases. Spray cone angle and penetration length are measured from the time-averaged images using image J software. The cone angle is defined as included angle measured between two lines that mark the spray periphery, as shown in Fig. 5. Spray penetration represents the distance till which the bulk of droplets are penetrating. Both of these quantifications are very important in the designing of pintle injector. A wider cone angle increases the possibility of higher wall heat flux and a larger penetration length affects the net combustion efficiency and combustor length. Figure 6 shows the variation in cone angle and penetration with the change in mixture ratio. Figure 7 shows the variation of penetration length and cone angle for different cases (Cases 1–6). The increase in the mixture ratio from 2.2 to 4.5
Characterization of the Spray Flow Field at the Exit of a Pintle Injector
35
Fig. 4 Instantaneous images at different test cases
Case1
Case2
Case3
Case4
Case5
Case6
resulted in an increase in the spray cone angle from 60 to 72°. The higher liquid momentum resulted in a larger spread of the spray. This also resulted in the reduction of penetration length. An increase in the liquid and gas flow rates with fixed MR resulted in a smaller cone angle, whereas no specific trend is observed in the case of penetration length.
4.3 Droplet Velocity Field at the Exit of the Injector The droplet velocity field at the exit of the injector is captured using spray PIV measurements. Figure 8 shows an average velocity field at the exit of the injector for Case 1. Droplet velocities are maximum in the spray cone region where the droplet density is maximum. Peak velocities are varying from 12 to 25 m/s as the flow rates are increased (Cases 1–6).
36
S. K. Gupta et al.
Cone angle
Cone angle ( °)
Fig. 6 Variation of cone angle (°) and penetration length (mm) with the mixture ratio
75
Penetration length 75
70
70
65
65
60
60
55
55
50
50 2
3 4 Mixing ratio
Penetration length (mm)
Fig. 5 Average image showing cone angle and penetration length
5
The axial velocity field at the exit for Case 1 is shown in Fig. 9. Peak axial velocities are in the range of 10–20 m/s. The liquid phase dominates the spray structure and no recirculating flow patterns are observed. Even though the finer droplets are recirculated to the center region, the net velocity is positive in all cases. Variation of axial velocity at 10 mm below the injector outlet for case 1 is shown in Fig. 10. Variations of the axial velocity for different cases are shown in Fig. 11. An asymmetrical spray velocity is observed as the flow rates are increased. The asymmetry in the spray structure is primarily due to injector manufacturing. Even a slight imperfection created in the internal passage due to the support material or
Characterization of the Spray Flow Field at the Exit of a Pintle Injector
75
Penetration length 70
70
65
Penetration length(mm)
Cone angle
Cone angle ( °)
Fig. 7 Variation of cone angle (°) and penetration length (mm) for different cases (considering same mixture ratio = 3.4)
37
65
60
60
55
55
50
50 45
45 1
2
3
4
5
6
Cases
Fig. 8 Time-averaged resultant droplet velocity field at the exit of pintle injector for Case 1
Fig. 9 Time-averaged axial droplet velocity field at the exit of pintle injector for Case 1
38
S. K. Gupta et al. 16
Velocity (m/s)
Fig. 10 Variation of axial velocity at 10 mm below the injector outlet for case 1
14 12 10 8 6 4 2 0
-60
-40
-20
0
20
40
60
Position (mm)
improper cleaning process can have a significant effect on the spray. Hence, proper care needs to be incorporated while printing the injector hardware in plastic. Design modifications are planned to correct these errors and experiments are in progress.
8 5 2 -50
-40
-30
-20
-10
-1
0
10
20
30
40
50
-4 -7
Vy (m/s)
-10 -13 -16 -19 -22
case1 case2 case3
-25 -28 -31 -34 -37
case4 case5 case6
-40
Position (mm) Fig. 11 Comparison of axial velocities for different cases at 10 mm below the injector outlet
Characterization of the Spray Flow Field at the Exit of a Pintle Injector
39
5 Conclusions In this study, a liquid-centered bi-swirl pintle injector is designed and developed using additive manufacturing techniques. Laser diagnostic techniques are implemented to capture the spray characteristics. The spray structure is captured using direct laser sheet imaging and cone angle and penetration length are quantified. High-speed PIV experiments are conducted to capture the velocity field and the resultant velocity field data is presented. The pintle injector develops a uniform spray at the exit, and periodic droplet/ligament shedding is observed at the injector exit. The velocity field shows the dominance of the liquid phase on the overall flow field. Further efforts are undertaken to improve manufacturing accuracy and spray quality. Acknowledgements The experimental work is carried out at National Centre for Combustion Research and Development (NCCRD), IIT Madras. Authors thank NCCRD for providing the diagnostic equipment and Mr. Aravind IB for helping with the experiments.
Nomenclature CEA MJP MR mw ma Vy
Chemical Equilibrium Application Multi-jet printing Mixture ratio Mass flow rate of water (g/s) Mass flow rate of air (g/s) Axial velocity (m/s)
References 1. Bazarov VG, Yang V (1998) Liquid-propellant rocket engine injector dynamics. J Propul Power 14(5):797–806 2. Dressler G, Bauer J (2000) TRW pintle engine heritage and performance characteristics. In: 36th AIAA/ASME/SAE/ASEE joint propulsion conference and exhibit, Jul 2000, p 3871 3. Zong N, Yang V (2007) Near-field flow and flame dynamics of LOX/methane shear-coaxial injector under supercritical conditions. Proceedings of the Combustion Institute, 31(2):2309– 2317 4. Austin BL, Heister SD, Anderson WE (2005) Characterization of pintle engine performance for nontoxic hypergolic bipropellants. J Propul Power 21(4):627–635 5. Yue CG, Chang XL, Yang SJ, Zhang YH (2011) Numerical simulation of a pintle variable thrust rocket Engine. In: International Workshop on Computer Science for Environmental Engineering and Eco Informatics, Jul 29, 2011. Springer, Berlin, Heidelberg, pp 477–481 6. Sakaki K, Kakudo H, Nakaya S, Tsue M, Isochi H, Suzuki K, Makino K, Hiraiwa T (2015) Optical measurements of ethanol/liquid oxygen rocket engine combustor with planar pintle Injector. In: 51st AIAA/SAE/ASEE joint propulsion conference, p 3845
40
S. K. Gupta et al.
7. Son M, Yu K, Koo J, Kwon OC, Kim JS (2015) Effects of momentum ratio and weber number on spray half angles of liquid controlled pintle injector. J Therm Sci 24(1):37–43 8. Fang XX, Shen CB (2017) Study on atomization and combustion characteristics of LOX/ methane pintle injectors 136:369–379. Acta Astronautica 9. Cheng P, Li Q, Xu S, Kang Z (2017) On the prediction of spray angle of liquid-liquid pintle injectors 138:145–151. Acta Astronautica 10. Son M, Radhakrishnan K, Koo J, Kwon OC, Kim HD (2017) Design procedure of a movable pintle injector for liquid rocket engines. J Propul Power 33(4):858–869
Experimental Investigation of Vapour Bubble Condensation in Subcooled Water Using Different Nozzles Samarendu Biswas, Aranyak Chakravarty, and Mithun Das
1 Introduction The direct contact condensation (DCC) of steam water is extensively used in different industrial fields for its high heat transfer coefficient. It is moreover used in nuclear reactor safety systems. During accidental conditions, such as coolant loss, steam pipe break, exceed reactor steam pressure limit, the safety or relief valve is opened and steam is injected into the subcooled system through a sparger. In this way, heat and pressure are released from steam. At the end of the stage of the pressure-releasing process, bubbling and chugging regimes will appear for low mass flux. In these regimes, pressure oscillations occur due to low-frequency high amplitude of steam which may damage the safety system. Therefore, the study of bubble condensation dynamics of the steam-submerged jet will be helpful for nuclear safety systems.
2 Literature Review and Objective In recent decades, the DCC of a bubble has been performed to understand condensation regimes by many researchers [1–5]. Chan and Lee [1] developed a condensation regime domain for a vertically downward steam nozzle using a wide range of steam mass flux of 1–175 kg/(m2 s) and the subcooled water temperature of 40–90 °C. Currently, Yang et al. [3] also established a regime domain for a vertical upward nozzle and studied pressure oscillation parameters for low steam mass flux. They considered mass flux and subcooled water temperature 8.34–16.71 kg/(m2 s) and 40–85 °C, respectively. Later the same group [6] studied the buoyancy effect on pressure oscillation amplitude and frequency of vertical upward steam jets from the S. Biswas · A. Chakravarty · M. Das (B) School of Nuclear Studies and Application, Jadavpur University, Kolkata 700106, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_4
41
42
S. Biswas et al.
injector with an 8 mm inner diameter. Liu et al. [4, 7] experimentally investigated the characteristics of subsonic steam jet condensation pressure oscillations characteristics for different vessel pressure. They used a relatively large vertical downward steam outlet with a 28 mm inner diameter. In the chugging region, the pressure oscillation intensity in the pipe enhanced with increasing vessel pressure but no significant effects were found on the overall pressure oscillation intensity in the pool. Zhang et al. [8] have experimentally performed to study the bubble dynamic parameters and frequency of steam direct contact condensation. The bubble is formed from a vertical downward pipe with a 26 mm inner diameter at low steam mass flux. They calculated the non-dimensional maximum bubble diameter and identified condensation regimes. Alden et al. [9] studied the plume stability of steam during condensation from the small-sized nozzle (diameter = 2.4 mm) into a crossflow of subcooled water for higher mass fluxes. In summary, the above-mentioned many research works on steam water direct contact condensation have been done on the relatively larger and fixed diameter of the steam jet. There are few experimental studies on the small size of steam jets with different sizes. The objectives of this experiment are as follows: A vertical upward steam jet of a vapour bubble was used in this experiment to assess the physics of flow patterns and interface morphology of the bubbling regime. Three separate nozzles, each with a diameter of 1.7, 2.7, and 3.5 mm, and the temperature of pool water is between 60 and 80 °C with 5 °C intervals, are used to study the behaviour of steam bubble formation. The frequency of bubble formation, the maximum bubble area at the stage of bubble development, and the frequency of bubble collapse are investigated for various pooled temperatures and nozzle types.
3 Materials and Methods A schematic of the experimental set-up is shown in Fig. 1. A heat source for steam generation, mineral water, steam pipes, control valves, a rectangular shape subcooled water pool, a temperature measurement device, an electric heater for heating the pool water, a high-speed camera, and other elements is included in the experimental setup. The pool water container measurements are 100 mm by 104 mm by 303 mm. Three different nozzle diameters 1.7 mm, 2.7 mm, and 3.5 mm are employed in this experiment. Initially, an airtight vessel with prefilled water is heated using an induction heater to produce steam. To prevent air from being entrapped in the steam bubble, experimental data is gathered after a period of time. To maintain and regulate the water temperature uniformly, a chiller and an electric heater are used. T-type thermocouples are installed to measure the water and steam temperature. Steam is supplied into the subcooled water tank through an injection nozzle. The steam bypass regulating valve is used to control steam mass flux until the smooth detachable bubble comes out from the injector. The condensation processes of steam bubbles
Experimental Investigation of Vapour Bubble Condensation …
43
Fig. 1 Schematic diagram of the experimental set-up
are captured using a high-speed video camera Phantom VEO E-310L(7302) with the help of a backlighting system to enhance the picture.
3.1 Image Processing Method To obtain steam bubble dynamic parameters, images of the experiment are captured by a high-speed camera with a frame rate of 11,000 frames/s and a resolution of 512 × 512 pixels. Captured images are processed with the help of ImageJ and MATLAB software. MATLAB is used to resize the captured picture, while ImageJ is used to measure the bubble area. Although bubble production in subcooled water creates a three-dimensional image, the area of the bubble in two-dimensional regions is calculated by counting pixels and finding out the bubble’s equivalent diameter.
4 Results and Discussion 4.1 Single Bubble Condensation Figure 2 shows the complete process of a bubble’s growth and separation and collapse which is discharged from a 1.7 mm nozzle and captures a frame gap is 5.45 ms. As shown in Fig. 2, the first red-marked bubble undergoes maximum growth and the next frame bubble forms necking separation and in the next three frames, the bubble
44
S. Biswas et al.
Fig. 2 Development of a single bubble, expansion, separation, and collapse
completely collapsed. From necking separation to the collapse of a bubble, it takes around 22 ms. If a bubble’s neck separation is its starting point (as shown in the blue marked box), near about 76 ms is required to complete the growth of the bubble for the 1.7 mm nozzle. Similar results are found for other nozzles.
4.2 Bubble Formation Frequency Here, bubble formation frequency is calculated on basis of the number of bubbles separated from an injector in a second. Bubble frequency for different nozzles with various pool temperatures is shown in Fig. 3. For a given pool temperature, frequency
Experimental Investigation of Vapour Bubble Condensation …
45
Fig. 3 No. of bubble formed with respect to time for different nozzles
reduces with increasing nozzle diameter. On the other hand, bubble frequency decreases with increasing pool temperature for all nozzles. From the figure, it is observed that the maximum bubble formation frequency is found for low pool temperature and smaller sizes of the nozzle. At 60 °C pool temperature, the bubble formation frequency for the 1.7 mm nozzle is 157 bubbles/sec.
4.3 Maximum Bubble Radius Maximum bubble formation images for different nozzles with different pool temperatures are shown in Fig. 4. It is obvious that a bigger bubble is formed for a larger diameter nozzle and higher pool temperature. From the area of the maximum bubble regime, we calculate the equivalent bubble diameter. We also calculate the equivalent diameter for other regimes which are discussed in the next sections.
4.4 Effect of Nozzle Diameter and Pool Temperature on Bubble Collapsing Duration Figures 5 and 6 show the effect of nozzle diameter and pool temperature on maximum bubble formation to completely collapsing time duration, respectively. For a given pool temperature, bubble collapsing time is maximum for the 3.5 mm nozzle. The time duration for this nozzle is about 14.2 ms for 75 °C pool temperature.
46
S. Biswas et al.
Fig. 4 Maximum bubble formation pictures
Fig. 5 Effect of nozzle diameter on bubble collapsing duration at 75 °C pool temperature
For a given nozzle diameter, bubble diameter, as well as bubble formation duration increase with pool temperature as shown in Fig. 6. The maximum bubble diameter and collapsing duration for the 1.7 mm nozzle are found 4 mm and 6 ms, respectively. Similar trends are found for other nozzles and pool temperatures.
Experimental Investigation of Vapour Bubble Condensation …
47
Fig. 6 Effect of pool temperature on bubble collapsing duration for the 1.7 mm nozzle
5 Conclusions To analyse the characteristics of the vapour bubble, an experimental investigation on the direct condensation of seam and water in the unstable bubbling regime is conducted. Three different nozzle diameters of 1.7 mm, 2.7 mm, and 3.5 mm have been utilized to explore the morphology of the upward steam bubble interface with subcooled water for the temperature range of 60 to 80 ºC. Following analysis of data on bubble area, bubble equivalent diameter, and frequency of bubble production are studied. The maximum bubble frequency is found for the 1.7 mm diameter nozzle at a low pool temperature (60° C). Maximum bubble formation to completely collapsing time duration is found in the larger diameter of the nozzle and comparatively higher subcooled pool temperature. Acknowledgements This work is financially supported by Jadavpur University Research Grant (Ref. No.: S-3/120/22), Jadavpur University, Kolkata. The authors would also like to acknowledge the support from AMRA Lab, Dept. of Power Engineering, Jadavpur University and use of SERBfunded (Grant No. CRG/2019/005887) high-speed Camera.
References 1. Chan CK, Lee CK (1982) A regime map for direct contact condensation. Int J Multiph Flow 8(1):11–20
48
S. Biswas et al.
2. Chong D, Zhao Q, Yuan F, Cong Y, Chen W, Yan J (2015) Experimental and theoretical study on the second dominant frequency in submerged steam jet condensation. Exp Thermal Fluid Sci 1(68):744–758 3. Yang Q, Qiu B, Chen W, Chong D, Liu J, Yan J (2020) Experimental investigation on the condensation regime and pressure oscillation characteristics of vertical upward steam jet condensation with low mass flux. Exp Thermal Fluid Sci 1(111):109983 4. Liu X, Xie X, Meng Z, Zhang N, Sun Z (2021) Characteristics of pool thermal stratification induced by steam injected through a vertical blow down pipe under different vessel pressures. Appl Therm Eng 1(195):117169 5. Qiu B, Liu J, Yan J, Chong D, Wu X (2017) Experimental investigation on the driving force and energy conversion in direct contact condensation for steam jet. Int J Heat Mass Transf 1(115):35–42 6. Yang Q, Qiu B, Chen W, Zhang D, Chong D, Liu J, Yan J (2021) Experimental study on the influence of buoyancy on steam bubble condensation at low steam mass flux. Exp Thermal Fluid Sci 1(129):110467 7. Liu X, Yu M, Li W, Yu P, Meng Z, Sun Z, Zhang N, Ding M (2022) Experimental investigation of the pressure oscillations induced by subsonic steam jets under different vessel pressures. Nucl Eng Des 15(395):111867 8. Zhang D, Qiu Z, Tong L, Cao X (2022) Experimental investigation on the steam condensation dynamic characteristics at low steam mass flux. Ann Nucl Energy 1(170):108980 9. Alden ZR, Maples GD, Dressler KM, Nellis GF, Berson A (2021) Plume stability during direct contact condensation of steam in a crossflow of subcooled water, at high mass flux and with a small nozzle diameter. J Heat Transf 143(9)
VOF Simulations of Evaporation and Condensation Phenomenon Inside a Closed-Loop Thermosyphon Vivek K. Mishra, Saroj K. Panda, Biswanath Sen, M. P. Maiya, and Dipti Samantaray
1 Introduction The nuclear fuels prior to loading in reactors are stored in nuclear fuel storage vault. The fuels contain fissile materials, which generate decay heat during storage in the vault. Conventionally, cooling air is supplied to the vault for efficient heat removal, and the temperature inside the vault is maintained below 65 °C to maintain the integrity of the vault and ensure long-term storage of the fuel during normal operation [1]. Since it is difficult to monitor the temperature and air flow inside the vault, CFD simulations are effective to monitor these aspects of the nuclear fuel storage vault [2, 3]. Further, it is utmost important to maintain the temperature inside the vault within 90 oC during station blackout [1]. The concrete structure of the vault can degrade if the decay heat is not removed, and the longevity of the fuel storage can be affected. Therefore, an efficient passive cooling arrangement is required to effectively cool the fuels and concrete structure inside the vault. Thermosyphon is an effective heat removal device that works on principles of both conduction and phase change. It comprises a sealed metallic tube with working fluid inside it and the heat is transferred through cyclic evaporation and condensation of working fluid. Therefore, an additional external electrical element is not required to operate the thermosyphon. Thermosyphon can be considered as a heat sink or a passive cooling system for the fuel storage vault. Therefore, it becomes important to
V. K. Mishra · D. Samantaray Homi Bhabha National Institute, Anushaktinagar, Mumbai 400094, India S. K. Panda · B. Sen (B) Indira Gandhi Centre for Atomic Research, Tamil Nadu, Kalpakkam 603102, India e-mail: [email protected] M. P. Maiya Department of Mechanical Engineering, Indian Institute of Technology Madras, Tamil Nadu, Chennai 600036, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_5
49
50
V. K. Mishra et al.
study the temperature and mass transfer distribution during steady state operation in an environment which is replica of fuel storage vault.
2 Literature Review and Objective The multiphase liquid–vapor phase change process has been investigated by different researchers using volume-of-fluid (VOF) approach, level set (LS), and combination of both the methods. The initial CFD simulations of boiling using VOF method were conducted by Schepper et al. [4]. The simulations investigated the boiling of hydrocarbon in a steam cracker. The results simulated using VOF approach were in good agreement with their previously obtained experimental data [5]. Further studies [6–10] on the simulation of boiling paved way for research and analysis of heat pipes using CFD. A wickless heat pipe also known as thermosyphon is widely used due to its passive cooling applications, reliability, cost-effectiveness and ease of manufacturing. The effect of heat flux and filling ratio on a thermosyphon which used water as a working fluid was investigated using experiments and simulations by Alizadehdakhel et al. [6]. Further, Lin et al. [7] developed a VOF-based CFD model which was able to correctly predict the thermal conductivity using temperature at evaporator and condenser end along with heat load. The miniature oscillating heat pipe using water as a working fluid. The effect of evaporator length and inner diameter of the thermosyphon was varied to analyze its heat transfer characteristics in terms of thermal conductivity. Jeong et al. [8] proposed a hybrid thermosyphon to extract heat from a spent fuel storage facility to avoid thermal degradation and maintain its structural integrity for long-term storage of spent fuel. With the increased requirement of safety in nuclear power plants and maturity in design, the application of thermosyphon in nuclear plant is important. In a study of nuclear reactor similar to Fukushima No 1 plant, Mochizuki et al. [10] investigated the effect of a loop heat pipe to remove the residual heat from the reactor for its safe shutdown. Xiong et al. [11] experimentally studied a heat pipe for effectively removing heat from a spent fuel storage vault during an accident. In the present study, a thermosyphon is suggested as a passive cooling system for dry storage of a nuclear fuel storage vault. The design is based on the heat generated by the storage vault. To evaluate the performance of the thermosyphon, experiments were performed to obtain thermal conductivity. In addition to this, CFD simulation of a wickless heat pipe that takes into account every aspect of the heat transfer phenomena occurring inside the heat pipe which has also been reported. CFD models can also reduce the quantity of experimentation. Therefore, in this paper, a thorough CFD modeling of two-phase flow and heat transfer processes during the operation of a straight thermosyphon has been undertaken.
VOF Simulations of Evaporation and Condensation Phenomenon Inside …
51
Fig. 1 Schematic representation of experimental setup
3 Experimental 3.1 Experimental Setup The schematics of the experimental apparatus for our present investigation is shown in Fig. 1. The thermosyphon was manufactured using a 2 m long copper tube with 0.022 m outer diameter, and 0.001 m of thickness. The evaporator where the heating is provided is 0.3 m long and the condenser section through which heat is dissipated is 0.2 m in length. The heat is provided to the evaporator using ceramic heaters. The condenser side is exposed to the environment to obtain natural cooling. In the middle, adiabatic region of 1.5 m length is insulated using insulation of glass wool and fiberglass cloth. The temperature is measured using thermocouples attached to the surface of the thermosyphon.
3.2 Experimental Procedure Before initiation of the experiment, the vacuum pump is connected to the thermosyphon. The vacuum created removes the non-condensable gases and reduces the vapor pressure inside the tube. The working fluid (water) is filled in the thermosyphon and the volume of filled water is carefully measured. The pressure gauge connected to the thermosyphon indicates the pressure in the tube. A constant temperature is maintained at the evaporator end using the heater. The reading was taken around
52
V. K. Mishra et al.
1.5 h after achieving a stable temperature at the thermosyphon. The experiment was repeated to confirm the correctness of the experimental data.
4 Numerical Methods 4.1 Governing Equations The two-phase flow and phase change in the thermosyphon is modeled using VOF multiphase models. The mass, momentum, and energy equations for multiphase VOF model are calculated as. Mass conservation equation: ∂(αρ) + ∇.(αρu) = Ms ∂t
(1)
Momentum conservation equation: ∂(ρu) + ∇.(ρuu) = −∇ p + ∇.[μ(∇u + ∇u T )] + ρg + S ∂t
(2)
Energy conservation equation: ∂(ρe) + ∇.[(ρe + p)u] = −∇(k∇T ) + S H ∂t
(3)
where density (ρ) of liquid phase is a function of temperature, which is given as ρ = 859 + 1.252T − 0.00264T 2
(4)
The surface tension varies with the temperature and is represented as σ = 0.098 − 1.845 × 10−5 T − 2.3 × 10−7 T
(5)
The equation for turbulent kinetic energy (k) is defined as ∂ u jk νt ∂k ∂k ∂ ν+ + + = ∂t ∂x j ∂x j σk ∂ x j ∂u j ∂u i ∂u i 2 νt − kδi j + −ε ∂x j ∂ xi 3 ∂x j The specific rate of dissipation (E) equation is defined using
(6)
VOF Simulations of Evaporation and Condensation Phenomenon Inside …
53
νt ∂ε ∂ε ∂ u j ε ε ∂ ν+ + c1 = + ∂t ∂x j ∂x j σε ∂ x j k ∂u j ∂u i ∂u i ε2 2 νt + − c2 − kδi j ∂x j ∂ xi 3 ∂x j k
(7)
2
The eddy/turbulent viscosity (μt ) was calculated using μt = ρCμ kε , where C 1 = 1.44; C 2 = 1.92; C μ = 0.09; σ k = 1 and σε = 1.3
4.2 Geometry and Boundary Conditions The 2D model is built using ANSYS design modeler. In the fluid region, uniform and non-uniform grids are generated using “ANSYS-Meshing” module. The evaporator region of the thermosyphon is specified as constant temperature of 60 °C. The condensers are specified as 27 °C, with a heat transfer coefficient of 7 W/m2 -K. A heat flux of 0 W/m2 -K is applied to the adiabatic region of the thermosyphon. The entire evaporator region is patched with water and entire pressure of 450 mm-Hg is applied to the volume of thermosyphon.
5 Results and Discussion The current numerical simulations were conducted on a 2D model of the thermosyphon to analyze the heat and phase change phenomenon in terms of temperature distributions along the height of thermosyphon and boiling and condensation phenomena near evaporator and condenser, respectively.
5.1 Validation The temperature distributions along the length of the thermosyphon obtained from experiments are used to validate the numerical data. The predicted temperature distributions are compared with the experimentally obtained data (see Fig. 2) and found to be in good agreement with each other. The experimentally validated CFD model was used to carry out all further simulations.
54
V. K. Mishra et al.
Fig. 2 Comparison of experimental measured temperature with simulated results
5.2 Temperature Distributions The temperature contours at different time duration, during startup and steady state operation, are presented in Fig. 3. The temperature distributions inside the thermosyphon are useful to understand the heat transfer process. Initially, the operating pressure and temperature were specified as saturation values. The temperature of the evaporator was increased as the constant temperature was applied to the evaporator walls. The difference in temperature at the evaporator end leads to boiling due to heat transfer on the wall of the evaporator. The vapor flows from the evaporator to the condenser section as observed at 100 and 500 s. A heated region at the condenser is obtained at 1000 s. Later, the heat transfer process in the evaporator becomes nearly uniform.
5.3 Evaporation The time evolution of the vapor phase obtained at the evaporator end is presented in Fig. 4. The rise of the vapor phase in the thermosyphon is responsible for heat transfer and an increase in temperature inside the thermosyphon. It is observed that initially the liquid pool is kept quiescent and the vaporization phase is reached. The vapor starts to form when the saturation temperature is reached. Further, the liquid at the top vaporizers before the liquid at the bottom. It is because the phase change is affected by the local pressure and hydrostatic pressure of the liquid. The difference in pressure at the bottom and nucleation site is difficult to be obtained. The pressure difference makes it difficult to obtain the nucleation sites in the lower region.
VOF Simulations of Evaporation and Condensation Phenomenon Inside …
55
Fig. 3 Variations in temperature with time along the height of the thermosyphon
Fig. 4 Time evolution of water volume fraction distributions in the thermosyphon
5.4 Condensation of Water Film The converse of boiling takes place at the condenser section. The condensate film obtained at the condenser and adiabatic region is presented in Fig. 5. The liquid film falls through the adiabatic region of the thermosiphon (see Fig. 5a). The liquid in the evaporator region is replenished by the continuous flow of thin liquid film as shown in Fig. 5b.
56
V. K. Mishra et al.
Fig. 5 Condensed water film profile near a condenser and b along the adiabatic walls
5.5 Conductivity Calculations The resistance offered to heat transfer in the thermosyphon is calculated using the formula Rth = ΔT /Q, where Q is the applied heat load, given as Q = V × I—losses. The thermal resistance obtained for the thermosyphon using the experiment was 0.04 K/W, whereas the thermal resistance obtained using the numerical simulation was nearly 0.034 K/W. The difference in the predicted result was obtained due to the presence of non-condensable gases in the thermosyphon and improper insulation around the thermosyphon which resulted in heat losses.
6 Conclusions In the present study, heat and mass transport phenomena inside a two-phase closed loop thermosyphon are simulated which used water as a working fluid. The heat transfer inside the thermosyphon was due to the combined effect of conduction and phase change of the working fluid. The results of the CFD simulations show that the proposed CFD model was able to predict the temperature distribution along the length of the thermosyphon successfully. The temperature distribution inside the thermosyphon becomes uniform after 1000 s after a steady state is achieved. The boiling phenomenon in the evaporator region of the thermosyphon shows that the water boils at a higher rate near the liquid–vapor interphase due to the hydrostatic pressure difference between the two regions. The condensed liquid on the condenser wall flows through the evaporator and charges the pool of water in the evaporator. The thermal resistance was calculated to be nearly 0.38 K/W. The disparity between the experimental and numerically obtained values of the thermosyphon was due to the
VOF Simulations of Evaporation and Condensation Phenomenon Inside …
57
presence of non-condensable gases in the tube and insufficient insulation around the thermosyphon. The present study will be helpful to understand the heat and phase change phenomenon taking place inside the thermosyphon which can be used to improve the heat removal from the fuel storage vault. Acknowledgements One of the authors, Vivek K. Mishra, is grateful to the Department of Atomic Energy for the DGFS fellowship. All the authors thank the Engineering Services Group for their help during the commissioning of the experimental setup.
Nomenclature e I k Q Rth S T u V σ ρ β ε
Energy term (W) Current (A) Turbulence kinetic energy (m2 /s2 ) Power supplied (W) Thermal resistance of thermosyphon (K/W) Source term (W/m3 ) Temperature (K) Supply velocity (m/s) Voltage (V) Surface tension of the liquid (N/m) Density of air (kg/m3 ) Coefficient of thermal expansion (1/K) Specific rate of dissipation
References 1. Design of Concrete Structures Important to Safety of Nuclear Facilities, Atomic Energy Regulatory Board, India (2001) 2. Alyokhina S (2018) Thermal analysis of certain accident conditions of dry spent nuclear fuel storage. Nucl Eng Technol 50(5):717–723 3. Mishra VK, Panda SK, Sen B, Maiya MP, Rao BPC (2022) Numerical analysis of forced convection heat transfer in a nuclear fuel storage vault. Int J Therm Sci 173:107429 4. Schepper SCKD, Heynderickx GJ, Marin GB (2009) Modeling the evaporation of a hydrocarbon feedstock in the convection section of a steam cracker. Comput Chem Eng 33(1):122– 132 5. Schepper SCKD, Heynderickx GJ, Marin GB (2008) CFD modeling of all gas–liquid and vapor–liquid flow regimes predicted by the Baker chart. Chem Eng J 138(1–3):349–357 6. Alizadehdakhel M, Rahimi M, Alsairafi AA (2010) CFD Modeling of Flow and Heat Transfer in a Thermosyphon. Int Commun Heat Mass Transf 37:312–318 7. Lin Z, Wang S, Shirakashi R, Winston Zhang L (2013) Simulation of a miniature oscillating heat pipe in bottom heating mode using CFD with unsteady modelling. Int J Heat Mass Transf 57(2):642–656
58
V. K. Mishra et al.
8. Jeong YS, Bang IC (2016) Hybrid heat pipe based passive cooling device for spent nuclear fuel dry storage cask. Appl Therm Eng 96:277–285 9. Mochizuki M, Singh R, Nguyen T, Nguyen T (2014) Heat pipe based passive emergency core cooling system for safe shutdown of nuclear power reactor. Appl Therm Eng 73(1):699–706 10. Xiong Z, Wang M, Gu H, Ye C (2015) Experimental study on heat pipe heat removal capacity for passive cooling of spent fuel pool. Ann Nucl Energy 83:258–263
Hydrodynamics of Droplet Generation Under Squeezing Regime in a T-junction Cylindrical Microfluidic System Pratibha Dogra and Ram Prakash Bharti
1 Introduction The interest in the study of fluid–fluid interaction in the microfluidic systems has escalated over the years because of its prodigious potential to accentuate the processes coupled with medical and industrial relevance [1]. The two-phase microfluidic systems are the most rudimentary and crucial domains in multiphase flows. The research in this area involves various aspects like droplet generation [2], movement [3], coalescence [4], etc. The manipulation and production of micron sized droplets involves employment of different geometric configurations and combinations, intrinsic properties of phases involved and external forces like magnetic and electrical fields. The generation and manipulation of these micron sized droplets is complex in terms of the production controllability, hence profound understanding of the droplet behavior is crucial, especially the breakup behavior. Apart from various geometric configurations available like co-flow, flow focusing devices [5, 6], the symmetric T-junction is an elementary domain for investigation of droplet breakup in microfluidics because of its ability to produce monodisperse droplets and simple geometry [7–9]. Use of microfluidic geometry to elucidate the instability leading to formation of the droplets was first done by [9] using T-shaped geometry with water/ oil as dispersed and continuous phases. They saw various dynamic patterns evolving because of the geometry and interaction of both phases. The authors suggested that the droplet formation is dominated by the balance between the capillary force and the tangential shear force. A detailed experimental study of the T-junction configuration [7] identified the droplet formation under squeezing regime and predicted that the mechanism is directly related to the geometrical confinement in which droplet is forming. De Menech et al. [10] elucidated three distinct regimes for formation P. Dogra · R. P. Bharti (B) Complex Fluid Dynamics and Microfluidics (CFDM) Lab, Department of Chemical Engineering, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_6
59
60
P. Dogra and R. P. Bharti
of droplets in T-junction configuration, i.e., squeezing, dripping and jetting and concluded that droplet breakup in the squeezing regime is the result of pressure rise in the upstream. Christopher et al. [9] performed the experiments for the fluids with a higher viscosity in a T-junction microchannel. They reported that the droplet size and generation frequency ( f ) have a power relation with Capillary number (Ca). The effect of wetting properties on droplet shapes was studied by [11]. They predicted that the wetting behavior affects the pressure balance during the droplet formation. Both experimental and numerical effects of channel wettability on the droplet length was done by Bashir et al. [12]. They elucidated the strong impact of the contact angle on the droplet generation and pinch-off time. Kovalev et al. [13] have experimentally observed the 3D flow structures in droplet generation in rectangular cross section T-junction at low viscosity ratios (10–3 ) of two phases. Boruah et al. [14] studied the effect of surface wettability on droplet dynamics. They did numerical investigation on the displacement behavior of a dispersed liquid column flowing through a microfluidic T-junction. They reported that surface wettability is a dominant factor in determining the interface evolution. Li et al. [10] numerically studied the impact of ribs on the droplet dynamics. They performed studies on the T-junction with varying size of ribs and found that incorporating ribs in T-junction offers good control over droplet formation. Later, they performed experiments on the same geometry and gave the scaling laws for the slug and the droplet length. Kalantarifard et al. [15] have explored the theoretical and experimental limits of monodispersed droplet generation. They explored underlying reasons for variation in droplet size. The dependence of bubble shape and size on the inertial force of the dispersed phase was presented by [16]. They gave scaling law for droplet size in terms of the dispersed phase inertial force. Study on interface evolution and droplet pinch-off was done by [17]. The authors explored the different pinch-off mechanisms under the squeezing regime at wide range of flow rate ratio (0.1 < Qr < 10). In most of the studies, it has been predicted that the inertial force effects are negligible when talking about the droplet generation under the squeezing regime. There are many scaling laws available predicting the droplet size, but these are not universal, and hence the study of droplet generation under squeezing regime under different parameters becomes important in order to develop the correlations that can be universally applied. This work presents the effects of the inertial force at different contact angle and channel dimensions on the droplet generation.
2 Mathematical Modeling 2.1 Problem Statement The problem involves the two-phase flow of Newtonian immiscible fluids under laminar, isothermal conditions through a T-junction cross-flow microfluidic system, as shown in Fig. 1. The diameter of both the main channel, carrying continuous phase
Hydrodynamics of Droplet Generation Under Squeezing Regime …
61
Fig. 1 Representation of the cylindrical T-junction microfluidic system
and the side channel, carrying dispersed phase is taken same (Dd = Dc = 140 μm and 280 μm). The length of main channel is L = (L u + Dd + L d ) = 5280 μm and of side channel is L s = 1000 μm. The continuous and dispersed phases flow through their respective channels at flow rates equal to Qc μl/s and Qd μl/s. At the channel outlet, static pressure condition is employed, and at the channel boundaries, no-slip condition is imposed. The mass transfer and phase change effects along the interface are ignored, also the contact angle is static and the surface tension acting at the fluid interface is assumed to be constant.
2.2 Governing Equations The simplified continuity and momentum equations that governs the underlying physics involving multiphase flow are as follows: ⇀
∇·V =0 ρ(Φ)
⇀
∂V ∂t
⇀
(1)
⇀
+ V ·∇ V = −∇ p + ∇ · τ + Fσ
(2)
⇀
where V and p are the velocity vector and pressure fields, respectively, and τ is the deviatoric stress tensor. ⇀
τ = 2μ(Φ) D
(3)
⇀
where D the rate of deformation tensor is defined as follows. ⇀ ⇀ ⇀ D = 21 (∇ V ) + (∇ V )T
(4)
62
P. Dogra and R. P. Bharti
For elucidating the topological changes at the interface separating the both fluids, conservative level set method is employed which is given by Eq. (5). dφ dt
⇀
+ V .∇φ = γ ∇.φ '
'
φ = [∈ls ∇φ − φ(1 − φ)n]
(5) (6)
where φ is level set variable whose value lies between 0 (pure CP) to 1 (pure DP). ∈ ls , and γ are level set parameters which accounts for re-initialization and stabilization of level set function φ, and interface thickness. The properties of phases like density (ρ) and dynamic viscosity (μ) for the two-phase system are the function of level set variable (φ) defined as: α(φ) = αc + (αd − αc )φ
(7)
where α is the fluid property and the subscripts ‘c’ and ‘d’ refer to the continuous and dispersed phases, respectively. For accounting the surface tension forces between the two immiscible phases, the continuum surface force model [4] is employed as follows: Fσ = σ κδ(φ)n
(8)
where σ is the surface tension coefficient (N/m), δ(φ) is the Dirac delta function which is approximated as δ(φ) = 6(|∇φ|)|φ(1 − φ)|, κ = R −1 = −(∇ · n) is the mean curvature, and n is unit normal.
2.3 Computational Approach The computational model is built in a CFD solver, COMSOL multi-physics using finite element method. For each simulation, the time step (Δt) of 10−4 s is selected to find the iterative solution for two phase flow for following parameters: 1. Mesh attributes: Linear, non-uniform, unstructured, fine tetrahedral mesh, maximum mesh element size δ max = 30.6 μm with element count N e = 137,278 and degrees of freedom (DOF) = 158,122 2. Level set parameters: γ = 1 m/s, ∈ ls = 11.85 μm. 3. Flow governing parameters: ρ r = 0.998; θ = 120°, 135°; Qr = 0.667 and 1; μr = 3.5; Cac < 10−4 ; σ = 10 mN/m.
Hydrodynamics of Droplet Generation Under Squeezing Regime …
63
3 Results and Discussion The results elucidated in this study are taken after the first droplet formation. The results are presented in terms of the phase, pressure, velocity magnitude and recirculation zone contours.
3.1 Phase Profiles The phase flow contours predicting the second droplet pinch-off and breakup time and its movement along the channel at two different contact angles (θ = 120° and 135°) and flow rate ratios under squeezing regime are presented in Fig. 2. The onset of droplet formation is marked by the evolution, followed by steady growth of the dispersed phase into the continuous phase. Because of this phenomenon competing forces, i.e., the shear as well as the pressure buildup takes place which is followed by droplet pinch-off. Since the study is performed under the squeezing regime (Cac < 10–3 ), the shear stress effects exerted on the interface of the emerging dispersed phase are negligible, because of which the dispersed phase grows into the continuous phase until it completely blocks the main channel resulting into the pressure buildup at the upstream resulting into droplet formation. From the phase contours, as the contact angle is increased from θ = 120° to 135°, there is change in the curvature of the droplet at the rear end. For θ = 120°, the curvature is more convex at the rear end at the pinch-off stage. As the contact angle is increased, the curvature is more on the concave side. This is because, at the lower contact angle, the interaction of the fluid and the surface is more, hence more volume of the dispersed phase is interacting with the channel wall resulting into convex rear end. The effect of changing the flow rate of the dispersed phase on the droplet breakup, for both the contact angle is not much pronounced but the time required by the droplet to stabilize for θ = 120° is 6 folds more as the dispersed phase flow rate is decreased from 15 to 10 μl/min. But the same pattern is not followed when θ = 135°. This may be attributed to the reason that, as the contact angle is decreased more dispersed phase is entering continuous phase resulting in more time to stabilize.
3.2 Droplet Length The data for the droplet lengths at the pinch-off time and the stabilizing time is presented in Table 1. From the table, it can be predicted that at low value of the contact angle (θ = 120°), the breakup time is comparatively less than that for θ = 135° because at low contact angle more surface and interface interaction occur leading to less breakup time and larger droplet size. Also, there is no effect of increasing the flow rate of dispersed phase on the length of the droplet when θ = 120° but on
64
P. Dogra and R. P. Bharti
Fig. 2 Phase profiles for the breakup and stable droplet formation stages at various flow rate ratios for θ = 120° Table 1 Droplet length and breakup time θ
Channel dia. (μm)
Flow rate ratio
Breakup time (s)
Breakup length (μm)
Stable time (s)
Stable droplet length (μm)
120°
140
0.667
0.182
302.7
0.338
321.5
120°
140
1
0.1824
302.1
0.2074
321.5
135°
140
0.667
0.183
324
0.229
314.3
135°
140
1
0.1831
304
0.227
324.1
120°
280
0.667
1.389
621.4
1.657
680.2
120°
280
1
1.392
615.9
1.602
687.4
increasing the contact angle, there is significant effect of the flow rate of the dispersed phase on the droplet size.
3.3 Instantaneous Pressure Profiles This section elucidates the instantaneous pressure profiles in the dispersed, continuous and the droplet phases. The probe points for measuring the pressure in dispersed, continuous and droplet phases are taken at 300 μm upstream and downstream of the junction as shown in Fig. 1. The point pressures for all the phases, i.e., pdp , pcp , pdrop are converted into non-dimensional quantities using viscous shear term as: P = p/ μDc uc c
(10)
Hydrodynamics of Droplet Generation Under Squeezing Regime …
65
Fig. 3 Point pressure profiles in dispersed, continuous and droplet phases during droplet formation at θ = 120° and Qr = 1 for D = 140 μm
The droplet formation is marked by the entry of the dispersed phase into the continuous phase (filling stage) until it expands to the entire channel cross section, followed by the elongation of the dispersed phase into downstream (necking) and hence the pinch-off. The point pressure profiles for all the phases are elucidated in Fig. 3. As the dispersed phase tip starts entering the main channel, there is resistance in the flow of the continuous phase resulting into the pressure buildup in the upstream section of the channel. This pressure buildup continuous till the channel is completely blocked by the dispersed phase. Till this point, the dispersed phase elongated thread reaches its critical thickness and pinches off resulting in the sudden drop in pressure in the dispersed phase. From the Fig. 4, it can be seen that, during the entire process of droplet generation, there is no substantial change in the dispersed phase pressure values except for the point when there is sudden drop in the pressure values because of the pinch-off. The aforementioned trend is shown for both the flow rate ratios and the contact angle values. This result is reconcilable with the assumptions used by Garstecki et al. [7]. From Fig. 5, it can be seen that there is negligible effect of changing the dispersed phase flow rate on the point pressure profile, but the effect of increasing the contact angle is profound. As discussed before, as the contact angle is increased, there is change in the rear end curvature of the droplet from convex to more concave. Thus, at the rear end, there exists less pressure at lower contact angle (θ = 120°).
3.4 Velocity Profiles and Recirculation Zones In the squeezing regime, during the droplet formation, the impact of pressure build up is more substantial as compared to the shear stress values. The droplets formed are in shape of plugs that inhabit the entire cross section of the main channel.
66
P. Dogra and R. P. Bharti
Fig. 4 Point pressure profiles in dispersed phase at various contact angles (θ) and flow rate ratios Qr for D = 140 μm
Fig. 5 Point pressure profiles in continuous phase at both contact angles (θ) and flow rate ratios Qr for D = 140 μm
In the first two contours in Figs. 7, 8 and 10, the droplet phase slowly emerges into the main channel, forming a thread which grows gradually as the continuous phase proceeds downstream. This causes implosion for the continuous phase resulting into the formation of recirculation zones. From Fig. 9, it can be predicted that the fluctuation in the velocity magnitude is more pronounced when the continuous phase flow rate is higher keeping the channel dimension and contact angle same. This is because at high Qc , the flow of continuous phase around the dispersed phase will be more pronounced leading to higher intensity recirculation zones as shown in first two contours in Figs. 7 and 8. The onset of the necking stage is marked by the departure of the thread tail from the upstream side as shown in Fig. 6c marked by the first crest, i.e., point (i) in velocity magnitude (Fig. 9). The necking proceeds until the droplet detachment starts marked by the stage when the neck width reaches its critical value. Right after the droplet detachment, a large recirculation zone evolve, caused by the flinching of the dispersed phase thread and the surface tension forcing the droplet to form the stable curvature. The large recirculation corresponds to the sudden velocity
Hydrodynamics of Droplet Generation Under Squeezing Regime …
67
spike as shown with point (ii) in Fig. 9. Also, on comparing Figs. 7 and 10, it can be seen that the more recirculation zones are formed when the channel dimensions are increased two fold keeping the flow rate constant. This can be because of the increased Reynolds number.
Fig. 6 Point pressure profiles in droplet phase at both contact angles (θ) and flow rate ratios Qr for D = 140 μm
Fig. 7 Recirculation zones formation during dispersed phase evolution and droplet pinch-off for D = 140 μm at Qr = 0.667
68
P. Dogra and R. P. Bharti
Fig. 8 Recirculation zones formation during dispersed phase evolution and droplet pinch-off for D = 140 μm at Qr = 1 Fig. 9 Comparison of velocity magnitude over time for Qd = 15 μl/min and 10 μl/min with fixed Qc = 15 μl/min
4 Conclusions In the present study, a cylindrical microfluidic cross-flow T-junction geometry is studied for two flow rate ratios and contact angles. It can be elucidated that, variation in the contact angle does not have very substantial effect on the droplet shape, size but there is significant change in the time taken by the droplets to stabilize and bring in equilibrium curvature. Also, it can be predicted that because of the change in the droplet curvature owing to lower contact angle, the magnitude of the pressure drop in the continuous phase is substantially low.
Hydrodynamics of Droplet Generation Under Squeezing Regime …
69
Fig. 10 Recirculation zones formation during dispersed phase evolution and droplet pinch-off for D = 280 μm at Qr = 1
References 1. Jena SK, Bahga SS, Kondaraju S (2021) Prediction of droplet sizes in a T-junction micros channel: Effect of dispersed phase inertial forces. Phys Fluids 33:032120 2. Lei L, Zhao Y, Chen W, Li H, Wang X, Zhang J (2021) Experimental studies of droplet formation process and length for liquid–liquid two-phase flows in a microchannel. Energies 14:1341 3. Garstecki P, Fuerstman MJ, Stone HA, Whitesides GM (2006) Formation of droplets and bubbles in a microfluidic T-junction scaling and mechanism of break-up. Lab on Chip 6:437– 446 4. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. J Comput Phys 100:335–354 5. Anna SL, Bontoux N, Stone HA (2003) Formation of dispersions using “flow focusing” in microchannels. Appl Phys Lett 82:364–366 6. Whitesides GM (2006) The origins and the future of microfluidics. Nature 442:368–373 7. Thorsen T, Roberts RW, Arnold FH, Quake SR (2001) Dynamic pattern formation in a vesiclegenerating microfluidic device. Phys Rev Lett 86:4163 8. Cramer C, Fischer P, Windhab EJ (2004) Drop formation in a co-flowing ambient fluid. Chem Eng Sci 59:3045–3058 9. Christopher GF, Noharuddin NN, Taylor JA, Anna SL (2008) Experimental observations of the squeezing-to-dripping transition in T-shaped microfluidic junctions. Phys Rev E 78:036317 10. Li X, He L, He Y, Gu H, Liu M (2019) Numerical study of droplet formation in the ordinary and modified T-junctions. Phys Fluids 31:082101 11. Venkateshwarlu A, Bharti RP (2022) Interface evolution and pinch-off mechanism of droplet in two-phase liquid flow through T-junction microfluidic system. Colloids Surf A Physicochem Eng Asp 642:128536 12. Umbanhowar P, Prasad V, Weitz DA (2000) Monodisperse emulsion generation via drop break off in a coflowing stream. Langmuir 16:347–351
70
P. Dogra and R. P. Bharti
13. Kovalev V, Yagodnitsyna AA, Bilsky AV (2018) Flow hydrodynamics of immiscible liquids with low viscosity ratio in a rectangular microchannel with T- junction. Chem Eng J 352:120– 132 14. Bashir S, Rees JM, Zimmerman WB (2014) Investigation of pressure profile evolution during confined microdroplet formation using a two-phase level set method. Int J Multiph Flow 60:40– 49 15. Kalantarifard, Alizadeh-Haghighi E, Saateh A, Elbuken C (2021) Theoretical and experimental limits of monodisperse droplet generation. Chem Eng Sci 229:116093 16. De Menech M, Garstecki P, Jousse F, Stone HA (2008) Transition from squeezing to dripping in a microfluidic T-shaped junction. J Fluid Mech 595:141–161 17. Wu Y, Fu T, Zhu C, Ma Y, Li HZ (2014) Bubble coalescence at a microfluidic T-junction convergence: from colliding to squeezing. Microfluid. Nanofluidics 16:275–286
Coalescence of Disc-Shaped Falling Droplets Inside Quiescent Liquid Media Deepak Kumar Mishra, Raghvendra Gupta, and Anugrah Singh
1 Introduction Droplet coalescence is an important phenomenon observed in nature as well as in industries, where two miscible droplets come into contact and form a single droplet. [1] Many scientific studies have been done in the past to investigate the different variety of coalescence, such as a single droplet to bulk liquid, two drops in the air [2], two droplets in another liquid media present in the surrounding, etc. [3, 4] Total interfacial area reduces during coalescence, and it is governed by interfacial tension between fluids. High-speed flow visualization and analytical and numerical investigation of coalescence of droplets in different environments are rich in literature [5]. Confined droplets, due to their high interfacial area density, have shown tremendous application in emulsion, droplet microreactor, liquid–liquid contactors, underground oil, and gas reservoir multiphase flow, etc. Droplet coalescence also occurs in such confined channels, which affects the heat and mass transfer rate, mixing, and reaction rates. [6] Droplets acting as a reactor are commonly known as droplet microreactors. A series of droplets are introduced into the microreactor, having a continuous fluid stream. In such fluids, the mixing inside the droplets is important, e.g. to determine the kinetic rate of reaction accurately in flow chemistry. Droplet coalescence can occur in such droplet reactors, which affects the design aspects of the reactors. Therefore, studying the flow physics of such confined droplet coalescence is important. D. K. Mishra (B) · R. Gupta · A. Singh Department of Chemical Engineering, IIT Guwahati, Guwahati, Assam 781039, India e-mail: [email protected] R. Gupta Jyoti and Bhupat Mehta School of Health Sciences and Technology, IIT Guwahati, Guwahati, Assam 781039, India Centre for Sustainable Polymers, IIT Guwahati, Guwahati, Assam 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_7
71
72
D. K. Mishra et al.
Numerous investigations into droplet coalescence over liquid surfaces and droplet coalescence in surrounding gaseous media are well-documented in the literature, but little is known about the flow physics inside the droplet during coalescence in surrounding liquid media. In this study, due to the confinement of the channel in one direction, disc-shaped droplet coalescence inside another liquid medium is investigated using the particle image velocimetry technique. Flow physics of film drainage, film rupture, and connection growth is shown qualitatively.
2 Objective This study investigates the physics of disc-shaped droplet coalescence falling under the effect of gravity inside continuous liquid media using particle image velocimetry. It investigates the movement of liquid film between droplets when the large droplet approaches the smaller drop, merging of interface and then the growth of the connection of droplet interface till a new larger droplet of circular disc shape forms and continues to fall. Recirculation inside the droplets during coalescence is also observed qualitatively.
3 Materials and Methods A syringe pump (Holmarc HO SPLF 2) is used to inject droplets of different sizes by controlling their volume at different time intervals. After sufficient time, a large droplet is injected after a small droplet to ensure that coalescence occurs only after the droplets achieve their respective terminal velocities. A high-intensity non-flickering continuous LED light is used for illumination. A schematic diagram of the flow loop of the experiment is shown in Fig. 1. Confinement of the channel is achieved by using two glass slides (76.2 mm × 25.4 mm × 1 mm) attached to two 1 mm thick glass slides using UV super glue (Excel Impex). Silicone sealant is used at one opening to close the channel. The confined channel used in the experiment is shown in Fig. 2. A superhydrophobic coating of 0.2% by vol. of octadecyltrichlorosilane (OTS) in toluene is applied to the front and back glass walls to minimize the wall resistance that prevents the drop from falling but at the same time enable us to visualize the 2D motion of a disc-shaped droplet. Glass slides are first cleaned with acetone and isopropyl alcohol followed by nitrogen wash and then treated with the ozone before the coating of OTS to make homogeneous coating throughout the glass surface. Silicone oil is used as a continuous phase and glycerol-water mixture for the dispersed phase. The density of aqueous glycerol solution (74% glycerol) is 1200 kg/m3 and viscosity is 0.035 Pa s. The density of silicon oil is 971 kg/m3 , and viscosity at 25 °C is 0.0371 Pa s. Silicon oil is a more viscous fluid, due to which the circular shape of the droplet is maintained.
Coalescence of Disc-Shaped Falling Droplets Inside Quiescent Liquid …
73
Fig. 1 Schematic diagram of the experimental set-up
Fig. 2 Confined channel used in the experiment showing the inlet where droplets are injected
At first, a drop of 18 µl volume was injected, followed by a second drop of 12 µl volume. Spherical droplets become disc-shaped inside the confined channel. Starting from the rest position at the top of the channel, droplets slowly fall under the effect of gravity surrounded by silicon oil. Terminal velocity is achieved after falling some distance due to buoyancy and drag forces acting in an upward direction and gravitational force in the downward direction. Due to the higher terminal velocity of the larger diameter droplet, it approaches the smaller drop and collides to form a new bigger drop. Poly Methyl Methacrylate (PMMA) particles (Dynoseeds Ltd.) are used as seeding particle for droplet, which is neutrally buoyant and follows the flow faithfully. Images are captured at 150 fps with a resolution of 1280 × 1024 using a high-speed camera (Photron UX50).
74
D. K. Mishra et al.
4 Results and Discussion Seeded droplets falling through silicon oil inside a confined channel under the effect of gravity achieve terminal velocity well before merging. Two droplets of different sizes dropped from rest position at the top of the channel and central location from side walls. Larger diameter droplet will achieve higher terminal velocity than smaller droplet. The higher terminal velocity of larger droplets causes the collision and merging with the smaller droplet. Spacing between the droplet injection kept such a way that coalescence occur after both the droplets achieve their respective terminal velocities. Figure 3 shows the time evolution of the coalescence phenomenon of two discshaped droplets where both have achieved their terminal velocities. Figure 3i indicates that a larger droplet is approaching towards smaller droplet due to its higher terminal velocity. Figure 3ii shows the film rupture between droplet interfaces and surrounding liquid drain away from the coalescence as the distance between droplets keeps reducing, causing the liquid between the droplets to move out. Figure 3iii and iv shows that oil drain away from the merging point and the growth of connection of the newly formed interface. After the merging of droplets, the newly formed interface connection starts to reshape due to surface tension force acting at the interface. Figure 3v and vi shows the end of oil drainage from the merging point and maximum growth of the interface, resulting in a circular interface boundary of the new droplet. Reshaping occurs until the balance of interfacial tension force at the interface is established. Yellow colour marking lines show the flow direction near the merging point of the droplets.
5 Conclusions The coalescence of two disc-shaped droplets falling inside liquid media under the influence of gravity was studied in a one-way confined channel using particle image velocimetry. The larger droplet, due to its higher terminal velocity, approached the smaller droplet, collided, and merged to form a new droplet. Time evolution images of the coalescence recorded and analysed. Images show that coalescence occurred in three stages. As the larger droplet reached the smaller droplet, the liquid film between the droplets leaked outside till the merging point. Furthermore, interface rupture occurred, and a new interface appeared. The newly formed interface grew, pushing the liquid outside till the shape of the droplet became circular. A new larger droplet was also disc-shaped due to the confinement of the channel. Qualitative images were shown only to depict coalescence, and further processing will be done to generate velocity vectors inside the droplet during coalescence.
Coalescence of Disc-Shaped Falling Droplets Inside Quiescent Liquid … Fig. 3 Seeded droplet at different positions (i) Larger droplet approaching smaller droplet (surrounding liquid drainage) (ii) Droplet coalescence (film rupture) (iii) and (iv) Connection growth (v) Maximum growth (vi) Reshaping to the circular disc-shaped larger droplet
75
76
D. K. Mishra et al.
Acknowledgements We gratefully acknowledge the FIST Grant SR/FST/ETII-071/2016(G) from Department of Science and Technology, Government of India.
References 1. Aarts DGAL, Lekkerkerker HNW (2008) Droplet coalescence: drainage, film rupture and neck growth in ultralow interfacial tension systems. J Fluid Mech 606:275–294 2. Pak CY, Li W, Steve Tse YL (2018) Free energy and dynamics of water droplet coalescence. J Phys Chem C 122:22975–22984 3. Rahman MM, Lee W, Iyer A, Williams SJ (2019) Viscous resistance in drop coalescence. Phys Fluids 31:1–10 4. Basheva ES, Gurkov TD, Ivanov IB, Bantchev GB, Campbell B, Borwankar RP (1999) Size dependence of the stability of emulsion drops pressed against a large interface. Langmuir 15:6764–6769 5. Saifi AH, Tripathi MK (2020) Distinct coalescence behaviors of hot and cold drops in the presence of a surrounding viscous liquid. Phys Fluids 32:1–10 6. Deka H, Biswas G, Sahu KC, Kulkarni Y, Dalal A (2019) Coalescence dynamics of a compound drop on a deep liquid pool. J Fluid Mech 866:1–11
Effects of Power Law Fluid Characteristics on Core-Annular Flow in a Horizontal Pipe Sumit Tripathi
1 Introduction High viscosity non-Newtonian fluids have several industrial applications including food, mining, petroleum, paint, cosmetic and pharmaceutical industries [1, 2]. Most of such fluids have power law characteristics and they are often required to be transported through pipelines. However, the huge viscosity and power law characteristics makes them difficult to simply pump and transport through pipes. A promising solution to transport such fluids in pipelines is using a core-annular flow (CAF) method where such high viscosity fluids are surrounded by a low viscosity fluid, provided both the fluids are immiscible. The water-lubricated CAF method has been extensively studied for transporting the high viscosity petroleum oils [3–11]. Some recent studies have also explored the pipe flow of non-Newtonian fluids in the CAF arrangement [12–17]. Li et al. proposed a theoretical model to predict pressure drop and thickness of the annular fluid for a gas–liquid pair [13]. Picchi et al. also performed modelling studies on CAF for shear thinning fluids in capillary tubes as well as in horizontal and inclined pipes [14]. Sahu has performed liner stability analysis on CAF having a pair of Newtonian (core) and non-Newtonian (annular) fluids that are separated by a mixing layer and studied the role of yield stress of annular fluid on the CAF stability [15]. Usha and Sahu performed a similar study on linear stability analysis of the CAF arrangement with Newtonian (core) and Herschel-Bulkley (annular) fluid pair [16]. Overall, the role of power law characteristics of both the fluids in the CAF arrangement is relatively less explored in the available literature. In this work, theoretical velocity profiles are presented by considering both the fluids as power law fluids in a perfect CAF arrangement. The effects of flow-behaviour and consistency indices of both the fluids on velocity profiles are presented in detail. S. Tripathi (B) Department of Mechanical and Aerospace Engineering, Institute of Infrastructure, Technology, Research and Management, Ahmedabad 380026, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_8
77
78
S. Tripathi
2 Flow Equations An axisymmetric perfect core-annular flow (PCAF) with two incompressible and immiscible fluids is considered. The flow is caused by the pressure difference imposed across the two ends of the pipe. Both the fluids (core and annular) are assumed to be power law fluids with viscosities given by Eqs. (1) and (2), respectively. The flow is assumed to be steady and fully developed with zero radial and azimuthal velocity. The fluid velocity in the z-direction is denoted by u z , while η is the interface height. The core and annular fluids are assumed to be power law fluids following the Ostwald–de Waele relationship. The relationships between shear stress and shear rate for core and annular fluids, respectively, are given by: τr(1) z τr(2) z
n 1 du (1) z = μ1 − dr n 2 du (2) z = μ2 − dr
(1)
(2)
where μi and n i are the consistency and flow-behaviour indices. Here, i = 1, 2 represents Fluid 1 and Fluid 2, respectively.
2.1 Core Fluid The azimuthal velocity of both the fluids are assumed to be zero, thus the force balance in the z-direction on a fluid element of radius r, as shown in Fig. 1a, gives: τr(1) z =
χr 2
(3)
Here, χ = − ddzp , now using Eqs. (1) and (3), we get: du (1) χ 1/n 1 1/n 1 z =− r dr 2μ1 or, u (1) z
χ 1/n 1 n1n+1 n1 =− r 1 + C1 n1 + 1 2μ1
where C 1 is a constant, which is to be determined using boundary conditions.
(4)
Effects of Power Law Fluid Characteristics on Core-Annular Flow …
79
Fig. 1 Fluid elements in a core and b annular fluids
2.2 Annular Fluid Similarly, as shown in Fig. 1b, the pressure force acting on an annular element (thin ring) can be balanced with the shear force on this element, which gives: (2) (r + dr )τr(2) z |r +dr − r τr z |r = rχ dr
Taking limits dr → 0 and using Eq. (2) for the annular fluid, we get: du (2) χr C2 1/n 2 z =− + dr 2μ2 r μ2 where C 2 is a constant, which is to be determined using boundary conditions.
2.3 Boundary Conditions The boundary conditions for this flow system are [12]:
(5)
80
S. Tripathi (2) τr(1) z = τr z atr = η (2) u (1) z = u z atr = η
u (2) z
(6)
= 0at r = R
Here, no-slip boundary condition is used at the pipe wall, while at the interface, both the fluids are assumed to have the same velocity and shear stress values. Velocity profiles of the PCAF system can be obtained by solving Eqs. (4) and (5) using the boundary conditions shown in Eq. (6) as [12]: ⎧ 1/n 2 n2 +1 1/n 1 n1 +1 n 2 +1 n 1 +1 ⎪ χ χ n1 2 n2 − η n2 n1 − r n1 ⎪ R + η for 0 ≤ r ≤ η ⎨ n 2n+1 2μ2 n 1 +1 2μ1 u z (r ) = 1/n 2 n2 +1 n 2 +1 ⎪ χ 2 ⎪ R n2 − r n2 for η ≤ r ≤ R ⎩ n 2n+1 2μ2
(7) where χ is the pressure gradient, R is the pipe radius, and η is the interface height.
3 Results and Discussion In this section, the effects of non-Newtonian fluid characteristics of core and annular fluids on the CAF velocity profiles are discussed using the flow-equations developed in the previous section. A few cases with variation of flow-behaviour and consistency indices are demonstrated and it is observed that, as the viscosity of the core fluid increases, the dependency on its non-Newtonian characteristics decreases. For very high viscosity of the core fluid (typically, > 1000 cP), the velocity profiles depend only weakly on the non-Newtonian characteristics and are almost the same as that for a Newtonian core. Further, it is seen that the low viscosity annular fluid is very sensitive to the flow-behaviour index. The pressure gradient (χ ) and pipe dimensions are kept the same in all the cases discussed below.
3.1 Increasing Viscosity of Core Fluid, Both Fluids Being Newtonian The effect of increasing viscosity of the core fluid is shown in Fig. 2. Here, both the fluids are assumed to be Newtonian (n 1 = n 2 = 1). The annular fluid has the viscosity of water, while the viscosity of the core fluid is increased up to 1000 times as compared to the viscosity of water. The non-dimensional radius and axial velocity are plotted on y and x axes, respectively. It can be observed that at high viscosity the core fluid almost behaves like a solid, with negligible deformation and relatively flat velocity profile.
Effects of Power Law Fluid Characteristics on Core-Annular Flow …
81
Fig. 2 Velocity profiles with increasing viscosity of the core fluid, χ = 254.5 Pa m−1 , R = 1.5 mm, η = 1.256 mm, the viscosity units are Pa sn
3.2 Varying Flow-Behaviour Index of Low Viscosity Core Fluid The effect of varying the flow-behaviour index of the core is shown in Fig. 3. Here, both the fluids have low viscosities, and the consistency indices of both the fluids are kept constant (μ1 = μ2 = 0.001 Pa sn ). The annular fluid is assumed to be Newtonian with flow-behaviour index of n2 = 1, while that of the core, n1 is varied. It can be seen that increasing the power law index corresponds to somewhat similar effects as increasing the consistency index of the core fluid. However, this effect becomes negligible once the consistency index of the core fluid is very high, as discussed in the next section.
3.3 Varying Flow-Behaviour Index of Highly-Viscous Core Fluid Effects of changing the flow-behaviour index of a high consistency index core fluid are shown in Fig. 4. It can be seen from Figs. 3 and 4 that the power law behaviour of the core fluid has an effect on velocity profiles only up to certain range of consistency index. If the core fluid has very high consistency index, the velocity profile depends weakly on its non-Newtonian characteristics.
82
S. Tripathi
Fig. 3 Velocity profiles with varying the flow-behaviour index of low viscosity core fluid, χ = 254.5 Pa m−1 , R = 1.5 mm, η = 1.256 mm, μ1 = μ2 = 0.001 Pa sn
Fig. 4 Velocity profiles with varying the flow-behaviour index of highly-viscous core fluid, χ = 254.5 Pa m−1 , R = 1.5 mm, η = 1.256 mm, μ1 = 1 Pa sn , μ2 = 0.001 Pa.s
3.4 Varying Flow-Behaviour Index of Low-Viscous Annular Fluid CAF is only useful if we have a low viscosity fluid in the annular region, lubricating the high viscosity core fluid. However, the annular fluid may also have non-Newtonian characteristics. The effect of varying the flow-behaviour index of the annular fluid is shown in Fig. 5. It can be seen that since the annular fluid has low consistency index, it is very sensitive to the flow-behaviour index. This suggests that if we use a low
Effects of Power Law Fluid Characteristics on Core-Annular Flow …
83
Fig. 5 Velocity profiles with varying the flow-behaviour index of annular fluid, χ = 254.5 Pa.m−1 , R = 1.5 mm, η = 1.256 mm
viscosity shear thinning fluid at the annulus, the pumping pressure, and therefore the wall shear stress can be further reduced.
4 Conclusions The presented model clearly indicates the dependency of velocity profiles on the power law characteristics of the core and annular fluids. The higher the viscosity of the core fluid, the more it behaves like a solid and the lesser is the dependency on its power law index. Further, the core fluid power law index plays a significant role when there are less viscosity differences between the core and annular fluids. Additionally, the CAF velocity profiles are very sensitive to the flow-behaviour index of the annular fluid, provided it has low consistency index. The presented study can further be extended to develop relevant pressure drop models for a non-Newtonian core-annular fluid pair. Acknowledgements The author would like to thank Prof. Amitabh Bhattacharya (IIT Delhi), Prof. Ramesh Singh (IIT Bombay) and Prof. Rico Tabor (Monash University) for their support in carrying out this work.
84
S. Tripathi
References 1. Zhang S, Zhu Y, Hua Y, Jegat C, Chen J, Taha M (2011) Stability of surfactant-free high internal phase emulsions and its tailoring morphology of porous polymers based on the emulsions. Polymer 52(21):4881–4890 2. Foudazi R, Masalova I, Malkin AY (2011) Flow behaviour of highly concentrated emulsions of supersaturated aqueous solution in oil. Rheol Acta 50(11–12):897–907 3. Ghosh S, Mandal TK, Das G, Das PK (2009) Review of oil water core annular flow. Renew Sustain Energy Rev 13(8):1957–1965 4. Rodriguez OMH, Bannwart AC (2006) Experimental study on interfacial waves in vertical core flow. J Petrol Sci Eng 54(3):140–148 5. Ooms G, Vuik C, Poesio P (2007) Core-annular flow through a horizontal pipe: hydrodynamic counterbalancing of buoyancy force on core. Phys Fluids 19(9):092103 6. Bannwart C (2001) Modeling aspects of oil–water core–annular flows. J Petrol Sci Eng 32(2):127–143 7. Arney MS, Bai R, Guevara E, Joseph DD, Liu K (1993) Friction factor and holdup studies for lubricated pipelining -I. experiments and correlations. Int J Multiphase Flow 19(6):1061–1076 8. Bensakhria YP, Antonini G (2004) Experimental study of the pipeline lubrication for heavy oil transport. Oil and Gas Sci Tech 59(5):523–533 9. Hu HH, Joseph DD (1989) Lubricated pipelining: stability of core-annular flow. part 2. J Fluid Mech 205:359–396 10. Joseph DD, Bai R, Chen KP, Renardy YY (1997) Core-annular flows. Annu Rev Fluid Mech 29(1):65–90 11. Tripathi S, Tabor RF, Singh R, Bhattacharya A (2017) Characterization of interfacial waves and pressure drop in horizontal oil-water core-annular flows. Phys Fluids 29(8):082109 12. Tripathi S, Bhattacharya A, Singh R, Tabor RF (2015) Lubricated transport of highly viscous non-newtonian fluid as core-annular flow: a cfd study. Procedia IUTAM 15:278–285 13. Li H, Wong TN, Skote M, Duan F (2013) A simple model for predicting the pressure drop and film thickness of non-newtonian annular flows in horizontal pipes. Chem Eng Sci 102:121–128 14. Picchi D, Ullmann A, Brauner N (2018) Modeling of core-annular and plug flows of newtonian/ non-newtonian shear-thinning fluids in pipes and capillary tubes. Int J Multiph Flow 103:43–60 15. Sahu KC (2019) Linear instability in a miscible core-annular flow of a newtonian and a bingham fluid. J Nonnewton Fluid Mech 264:159–169 16. Usha R, Sahu KC (2019) Interfacial instability in pressure-driven core-annular pipe flow of a newtonian and a herschel–bulkley fluid. J Nonnewton Fluid Mech 271:104144 17. Tripathi S, Tabor RF, Singh R, Bhattacharya A (2020) Experimental studies on pipeline transportation of high internal phase emulsions using water-lubricated core-annular flow method. Chem Eng Sci 223:115741
Evaporation Dynamics of Bidispersed Colloidal Suspension Droplets on Hydrophilic Substrates Under Different Relative Humidity and Ambient Temperature Mahesh R. Thombare, Suryansh Gupta, and Nagesh D. Patil
1 Introduction Sessile droplet evaporation is one of the most ubiquitous and widely observed phenomena in everyday life. It essentially pervades a vast majority of the natural and man-made processes. The complexity associated with the drying phenomenon of droplets is enormous, owing to the involvement of a host of different inter-related coupled processes. Researchers from diverse backgrounds have been conducting various experimental and numerical studies in a quest to comprehend the different facets of the droplet drying phenomenon. Especially, the study on evaporating sessile droplets containing colloidal particles has attracted a lot of attention among the scientific community. In the past, researchers have tried to explore various interesting aspects encompassing the droplet evaporation such as internal flow patterns, particle–substrate/particle–particle interactions, effect of colloidal particles (type, size, shape and concentration), Contact Line dynamics (CL), substrate wettability, deposit morphology, etc. Most of the previous works attempted to build a comprehensive understanding of the coupled effects, inherently dictating the dynamics which could, aid in exercising the desired control. Owing to the influential role of evaporating droplets in several key processes of scientific and industrial significance, there exist innumerable applications in multiple areas such as ink-jet printing [1], nanotechnology[2], biomedicine [3], coating[4], etc. To gain useful insights on this subject, some of the notable works in the literature pertaining to the evaporation
M. R. Thombare · S. Gupta · N. D. Patil (B) Department of Mechanical Engineering, IIT Bhilai, Raipur 492015, India e-mail: [email protected] M. R. Thombare Department of Mechanical and Aerospace Engineering, IIT Hyderabad, Hyderabad 502285, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_9
85
86
M. R. Thombare et al.
dynamics of colloidal suspension droplets have been reviewed, emphasizing the key findings these studies have established.
2 Literature Review and Objective In the seminal work by Deegan and co-workers [5–7], the cause behind the formation of ring-like deposition obtained from the evaporating coffee droplets was first unveiled. They reported the existence of a non-uniform evaporation flux across the droplet’s liquid–gas interface. During this, the droplet assumed a spherical cap shape while operating in the constant contact angle mode of evaporation. It was also found that the highest rate of evaporation occurred near the CL thereby, generating an outward capillary flow towards it to replenish the depleted volume. Interestingly, the suspended particles were also transported along with this flow thus, distributing the particles to form an outer ring. Uno et al. [8] reported in their work that the particles accumulate in thin layers near the droplet’s periphery causing an inhibition of the CL movement (self-pinning). Moreover, an increased rate of evaporation in the upper layers promoted particle adsorption to the substrate leading to the Coffee-Ring (CR) deposit pattern. However, in a later study by Hu and Larson [9], it was shown that, despite having the highest evaporation rate and a pinned CL, there still exists a possibility for an alteration in the CR-like deposition. As, it was noted that, for the evaporating droplets involving recirculatory Marangoni flows, the particles are advected towards the centre forming a ring with inner deposits. Hence, this necessitates the elimination or curtailing of such flows, to obtain the CR deposit patterns. In the past, much of the attention had been directed towards finding ways to alter the CR deposit pattern into the uniform deposition, due to its desirability in various applications such as ink-jet printing, coating, etc. As demonstrated in the work by Majumder et al. [10], Marangoni flow causes the advection of particles from the CL region towards the droplet’s apex, via the liquid–gas interface. This causes a homogenous density of particles within the bulk thus, suppressing the CR pattern giving rise to a uniform deposition. An alternative route for realising uniform deposit pattern is by, influencing the droplet’s evaporation kinetics or the mutual interaction forces between the substrate–particle, particle–particle, and particle–fluid. As per the findings reported by Bigioni et al. [2], augmenting the rate of evaporation accompanied by an attractive particle interaction with the droplet’s interface, leads to a compact uniform monolayer of nanocrystals. In the study by Li et al. [11], it was revealed that, by controlling the evaporation rate it is possible to intercept the outward-moving particles at the descending liquid–gas interface, provided, the interface shrinkage rate exceeds the average rate of particle diffusion. This, in turn, leads to interlocking of the particles at the surface thus, forming a viscous quasi-solid layer finally yielding a uniform deposit pattern. Another way to influence the deposit patterns is by influencing the nature and extent of the different interaction forces as mentioned earlier. To this end, Bhardwaj
Evaporation Dynamics of Bidispersed Colloidal Suspension Droplets …
87
et al. [12] demonstrated the effect of modulating particle–particle and particle– substrate interaction by varying the pH of the solution, leading to the deposit pattern modification. Their findings suggested that, at lower pH values there exists an attractive particle–substrate interaction for the particles lying in the vicinity of the substrate and hence, forms a ring with uniform inner deposition. Whereas, at high pH value, the particle–substrate interaction turned strongly repulsive, leading to a ring-like deposition, without any trace of particles elsewhere in the ring’s interior. Other aspects such as particle shape, size, type, concentration and multi-sized colloidal dispersion also exert a significant influence on the morphology of the dried patterns. Based on this premise, some of the relevant works are discussed to underscore the influential role of the aforementioned factors. Yunker et al. [13] reported that with the inclusion of the ellipsoidal particles into the colloidal droplet suspension containing spherical particles, a disruption in the formation of CR-type deposit patterns was manifested. It was also found that, under a favourable particle aspect ratio, the particles aggregated themselves into a loosely packed arrangement at the droplet’s free surface which, inhibited the incoming particles migrating towards the CL. Pertaining to the effect of variation in the particle size on the final deposit morphology, Choi et al. [14] examined the dried deposit patterns obtained from the evaporating droplets containing polystyrene beads (1 and 6 µm) and hollow glass spheres (9–13 µm). In the presence of large-sized polystyrene particles, depinning of the CL rendered the particles to move along with it to form an inner deposition. While the small-sized particles induced pinning of the CL resulting in the formation of the CR pattern. For the glass hollow spheres containing a mix of both large and small particles, CL depinned and led to an inner deposit pattern. Researchers have also investigated the effect of multi-sized particles suspended in evaporating liquid droplets, on the final deposit morphology. Jung et al. [15] performed experimental investigation on the evaporating droplets containing bidispersed colloidal particles (500 and 5 µm). It was observed that the nanoparticles were deposited near the CL promoting the pinning effect whereas, microparticles were transported towards the droplet’s interior due to the action of surface tension force of the liquid. Eventually, it led to the separation of particles wherein, the larger particles segregated near the centre surrounded by a ring of the smaller particles. Similar findings were also reported that involved the size-based separation of particles near the CL region [16]. There are other important factors such as the ambient conditions that play a significant role in influencing the droplet evaporation dynamics. However, in the literature, only few studies have investigated this aspect to understand its influence on the final deposition. Chhasatia et al. [17] performed an experimental investigation to study the effect of RH on the final dried patterns. Their findings suggested that the evaporation rate of the colloidal suspension droplet was found to decrease with increasing humidity. In addition, droplet spreading seemed to enhance at a lower contact angle giving rise to larger deposition. Sefiane et al. [3] examined the effect of ambient temperature on the deposit pattern obtained from the evaporating droplets containing TiO2 and Al2 O3 nanoparticles. The deposit pattern in case of the Al2 O3 nanoparticles remained indifferent to the variation in the ambient temperature. However, for the TiO2 nanoparticles, the droplet evaporation under elevated ambient temperature
88
M. R. Thombare et al.
yielded an outer ring along with concentric rings in the interior. Askounis et al. [18], examined the effect of atmospheric pressure on the evaporating droplets containing SiO2 particles. They reported an increase in the evaporation rate due to the enhancement of the water-vapour’s effective diffusion coefficient. At 750 mbar pressure, the droplet showcased stick–slip behaviour and thus, formed a stick–slip pattern. Whereas at 500 mbar ring-like pattern emerged which, further transformed into a wider and thicker ring as, the pressure was reduced to 250 mbar. Based on the literature review, it is found that studies pertaining to the influence of ambient conditions on evaporating colloidal droplet suspensions are relatively scarce. And, to the best of our knowledge, there does not exist any study that, examined the effect of ambient air temperature and RH on the dynamics of, evaporating droplets containing bidispersed colloidal particles over hydrophilic glass substrates. Moreover, this study will be of particular interest to the scientific community for its potential application requiring, the manipulation of the deposit patterns by controlling the ambient conditions. The salient objective(s) for the current study are as follows: • To study the effects of the ambient air temperature and RH on the evaporating aqueous droplets containing bidispersed colloidal particles. • To examine the variations in the morphology of the final deposit patterns obtained under different conditions and, to ascertain the underlying reasons causing it. • To find out any discernible distinction obtained from the size-based sorting of particles near the contact line, resulting from the variation of ambient conditions.
3 Materials and Methods 3.1 Preparation of Colloidal Solution and Droplet Generation: An aqueous colloidal suspension of fluorescent polystyrene microbeads (density ~ 1005 kg /m 3 ) having a concentration of 2.6% (w/v), for two different particle sizes 0.5 and 1 µm were procured from Polysciences Inc. Using this solution, desired solutions having a particle concentration of 0.1% (w/v) were obtained by diluting it with deionized water, for both the particle sizes. Then, equal volumes from each of these as prepared solutions were mixed to obtain the required bidispersed particle solution. To ensure uniform dispersion of particles in the solution, without any sedimentation or agglomeration, sufficient sonication was performed (about 1 h) at every stage of preparation. Droplets of 1.5 ± 0.3 µL volume were generated utilizing a variable volume micropipette (Range: 0.2–2.0 µL, Resolution: 0.01 µL) and deposited gently on the hydrophilic glass substrate (75 × 15 × 1.2 mm). Before performing each experiment, substrates were sequentially cleaned with IPA and DI water and subsequently, dried for 20 min by placing in a hot oven maintained at 100°C. The contact angle measurement was performed using the Advance Drop Shape Analyzer
Evaporation Dynamics of Bidispersed Colloidal Suspension Droplets …
89
(Krüss, DSA25). The equilibrium contact angle of 1.5 µL volume sessile droplets on the glass substrate was found to be 30 ± 3°.
3.2 Experimental Set-Up The experiments were performed on the Advance Drop Shape Analyzer (Krüss, DSA25). As depicted in the Fig. 1, it consists of a climate-controlled chamber (TC30 Temperature Controlled Chamber, range: 5°C to 90°C, HC4210 Humidity Control Unit, range: 15–85%), a high-speed camera (CM4210 optics module with 6.5 × manual zoom, Field of View: 3.2 × 3.2 to 18.5 mm, maximum fps: 2000, Image and Pixel resolution: 1920 × 1200 and 5.86 µm/pixel), monochromatic LED light source (Wavelength, 470 nm) and a temperature sensor. Using the high-speed camera, side visualization of the temporal evolution of evaporating droplets was acquired at 50 fps. The experiments were performed under different ambient temperatures (Ta = 25 and 50 °C) and RH (30, 50, and 70%). To ensure the repeatability of results, each experiment was conducted at least thrice. Post-processing of the recorded videos was performed using Virtual Dub (for image extraction). Further, to obtain the droplet-related parameters (diameter, height, volume and contact angle), an in-house MATLAB [19] routine was implemented that analysed and processed the images based on the pixel information. The images of the final deposit morphologies were obtained at lower and higher magnifications by using the fluorescence microscopy (Olympus IX73, Objective lens: 5X to 40X) and scanning electron microscopy (GEMINI SEM 500 KMAT, Carl Zeiss Inc.), respectively. For the quantitative profile characterization of the final deposit patterns, 3D optical profilometer (S neox, SENSOFAR Inc.) was utilized. The measurement of the ring profiles was performed at four different azimuthal locations and then later averaged, to obtain the final ring profile.
4 Results and Discussion Experimental results pertaining to the evaporation of microlitre droplets containing bidispersed colloidal particles have been discussed here. For a given ambient air temperature (Ta = 25 and 50 °C), the droplet evaporation was carried out at three different RH viz. 30, 50, and 70%. The variation of the droplet-related parameters, such as height, volume, contact angle and wetted diameter, normalized by their respective initial values, has been plotted against the normalised time, as shown in the Fig. 2. Based on the variation of the wetted diameter and the contact angle (shown graphically in the Fig. 2a–e), it is found that the droplet evaporation follows a constant contact radius mode of evaporation in all the cases. Moreover, it is found that there does not exist any substantial difference in this trend under different ambient conditions and ambient temperatures.
90
M. R. Thombare et al.
Fig. 1 Photograph of the experimental facility used for studying the dynamics of droplet evaporation
Fig. 2 Plots representing the temporal variation of the normalised droplet related parameters at different ambient air temperatures (Ta = 25 and 50 °C) and RH (30, 50 and 70%)
However, discernible differences are evident in the deposit morphologies as depicted in the Fig. 3. The fluorescence micrographs in the first row shows the variations in the dried patterns, with increasing RH at a given ambient air temperature. (Ta = 25 °C). The dried patterns shown in the first row of the Fig. 3a–c exhibit a clear transition with increasing RH from, a ring with non-uniform inner deposition to a ring with uniform inner deposition. For Ta = 50 °C, similar deposit patterns are manifested, except for, an increase in the intensity of the particles lying towards
Evaporation Dynamics of Bidispersed Colloidal Suspension Droplets …
91
Fig. 3 Fluorescence microscopy images (Top row, at 5 × magnification) of the final deposition patterns obtained at Ta = 25 °C for different RH a 30, b 50, c 70%. SEM characterisation of the CL region is highlighted in the red box at a magnification of 3000X (middle row) and 10000x (bottom row)
the ring’s interior (Fig. 4a–c top row). For quantitative analysis, the variation of the measured ring profiles with respect to the radial position is shown in the Fig. 5. At Ta = 25 °C (Fig. 5a), it can be observed that the ring width and height decreases as the RH is increased. Although, similar trend is conspicuously visible at Ta = 50 °C (Fig. 5b) as well for different RH but, the corresponding values of the ring width and height are relatively diminished. The above experimental findings clearly establish, the influential role of the ambient conditions in causing distinctive variations in the final deposit patterns. Interestingly, one of the studies [20] in the literature have also reported similar deposit patterns obtained from, the evaporating colloidal droplet suspension on heated hydrophilic substrates. Their findings revealed the prevalence of Marangoni recirculation at elevated substrate temperatures leading to, the coffee ring (thick ring with uniform inner deposition) to coffee eye (thin ring with non-uniform inner deposition) transition. This fact alludes to the existence of Marangoni recirculation for the present case as well, owing to the similarity in the deposit patterns. As per our analysis, the potential mechanism initializing the development of Marangoni flow can be attributed to, the enhancement in the evaporation flux as RH is decreased. As a result, the evaporative cooling rate will be augmented near the droplet’s apex, giving rise to, a depression in the liquid–vapour interfacial temperature as compared
92
M. R. Thombare et al.
to the CL. Hence, a surface tension gradient will be established leading to a circulatory thermal-Marangoni flow, directed along the liquid–vapour interface from the CL towards the apex. Consequently, it aids in the inward advection of the particles from the CL towards droplet’s centre which, competes against the particle transportation via evaporatively driven outward capillary flow. Hence, this elucidates the underlying physics dictating the formation of the ring with non-uniform inner deposition at lower RH. On the contrary, at higher values of RH, the evaporation flux is suppressed substantially, which is also evident from the longer evaporation durations observed during the experiments. Therefore, a significant difference in the interfacial temperature is not produced due to, the inhibition of the evaporation driven cooling at the droplet’s apex. As per the earlier discussions on the effect of the ambient air temperature it is found that, the particle number density in the droplet’s interior increased at Ta = 50 °C, as compared to Ta = 25 °C. This can be ascribed to the increased drying rate which leads to, an enhancement in the interfacial evaporative cooling at the apex, as compared to the CL. Thus, a steeper gradient in the surface tension will be developed between them generating, a relatively stronger thermalMarangoni recirculation. As a result, more particles will be advected away from the CL thus, yielding a rise in the number of particles deposited in the droplet’s interior (first row of Fig. 4a,b).
Fig. 4 Fluorescence microscopy images (Top row, at 5 × magnification) of the final deposition patterns obtained at Ta = 50 °C for different RH a 30, b 50, c 70%. SEM characterisation of the CL region is highlighted in the red box at a magnification of 3000X (middle row) and 10000x (bottom row)
Evaporation Dynamics of Bidispersed Colloidal Suspension Droplets …
93
Fig. 5 Optical profilometry of the ring profile showing the height and width variation with respect to the radial location for different RH (30%. 50% and 70%) at given ambient temperature a Ta = 25 °C, b Ta = 50 °C
Another important aspect regarding the deposit morphology is the size-based sorting of the particles near the CL. To this end, SEM characterization of the final deposition is performed to investigate the respective particle arrangement (as shown in the images in second (3000x) and third row (10000x) of Figs. 3 and 4). The region highlighted within the red box is viewed at a higher magnification to differentiate between the different sized particles. It is observed in all cases that, segregation of the 0.5 and 1 µm particles is evident near the CL wherein, 0.5 µm particles lie in the vicinity of the CL followed by the 1 µm particles. Further away, a mixed zone consisting of both the particles is formed spanning till the entire width of the ring. For the case of Ta = 25 °C, it is seen that there did not exist any substantial variation in the particle sorting. However, some visible differences can be sighted for the deposit patterns obtained from the droplets dried at Ta = 50 °C. At a RH of 70%, the number of 0.5 µm particles near the CL is relatively higher compared to the deposition obtained at 30% and 50% (third row of Fig. 4a–c). Therefore, it can be inferred from these findings that, there exists a possibility to alter the sorting of particles by controlling the ambient conditions. However, this work offers only preliminary insights on this aspect since, single bidispersed particle combination is considered. The future work will be focused upon studying the effect of the ambient conditions in influencing the particle sorting by, considering different particle combinations of varying sizes.
94
M. R. Thombare et al.
5 Conclusions Experimental investigations on the evaporation of droplets containing bidispersed (0.5 and 1 µm) colloidal particles on hydrophilic glass substrates are performed, under different ambient conditions. Various ambient air temperature (Ta = 25 and 50 °C) and RH (30, 50, and 70%) is considered, to examine their effect on the evaporation dynamics and the final deposit morphology. Based on the experimental findings following important inferences can be drawn: • Due to the pinned contact line, the droplets follow constant contact radius mode of evaporation. It is observed that, the normalised variations in the droplet related parameters remained, mostly unaffected for all the cases. • For a given ambient temperature, the evaporation flux increased as the RH is lowered leading to, a shorter evaporation duration and vice versa. Moreover, with an increase in the ambient air temperature, the evaporation rate is seen to get further augmented. • Significant changes in the dried deposit patterns are observed under various ambient conditions. As per our findings with increasing RH, the deposit morphology transitioned from, a thin ring with a non-uniform inner deposit to a thick ring with a uniform inner deposit. A similar trend is witnessed for both the ambient temperatures, except for, a higher particle number density in the inner region at Ta = 25 °C compared to Ta = 50 °C, as the RH is reduced. • The ring profile characterisation of the dried deposit patterns reveals that there is an increase in the width and height of the ring with the increasing RH, for a given ambient temperature. Further, at Ta = 50 °C, there exists a decrement in the corresponding values of the ring width and height for different RH, as compared to Ta = 25 °C. • The underlying cause for the variations in the deposit morphologies is attributed to the development of recirculatory thermal-Marangoni flows at lower RH, besides evaporatively driven capillary flow. Since, the evaporation rate is enhanced as RH is reduced inducing, a higher evaporative cooling at the droplet’s apex relative to the CL. However, at higher RH, these circulations are insignificant since the evaporation rate is suppressed which inhibits the creation of any strong interfacial temperature differential. • In the presence of this recirculation, an inward advection of the particles occurs thus, forming a ring with non-uniform inner deposit. In addition, it is found that this advection intensity is further aggravated at higher ambient temperature. • Size-based sorting of particles is prevalent near the CL in all the cases. At Ta = 25 °C, for different RH, the CL region apparently lacked any substantial difference in the particle sorting. However, at Ta = 50 °C, for RH of 70%, the number density of 0.5 µm particles is comparatively higher, as compared to the case with RH of 30 and 50%. Acknowledgements N.D.P. gratefully acknowledges the financial support from Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India,
Evaporation Dynamics of Bidispersed Colloidal Suspension Droplets …
95
New Delhi, by grant number SRG/2020/001947, for developing the experimental setup used in the present work. M.R.T. was also supported by a JRF fellowship under this grant.
References 1. de Gans B-J, Duineveld PC, Schubert US (2004) Inkjet printing of polymers: state of the art and future developments. Adv Mater 16(3):203–213 2. Bigioni TP, Lin X-M, Nguyen TT, Corwin EI, Witten TA, Jaeger HM (2006) Kinetically driven self assembly of highly ordered nanoparticle monolayers. Nat Mater 5(4):265–270 3. Sefiane K (2010) On the formation of regular patterns from drying droplets and their potential use for bio-medical applications. J Bionic Eng 7(4):S82–S93 4. Kim H, Boulogne F, Um E, Jacobi I, Button E, Stone HA (2016) Controlled uniform coating from the interplay of Marangoni flows and surface-adsorbed macromolecules. Phys Rev Lett 116(12):124501 5. Deegan RD, Bakajin O, Dupont TF, Huber G, Nagel SR, Witten TA (1997) Capillary flow as the cause of ring stains from dried liquid drops. Nature 389(6653):827–829 6. Deegan RD, Bakajin O, Dupont TF, Huber G, Nagel SR, Witten TA (2000) Contact line deposits in an evaporating drop. Phys Rev E 62(1):756–765 7. Deegan RD (2000) Pattern formation in drying drops. Phys Rev E 61(1):475–485 8. Uno K, Hayashi K, Hayashi T, Ito K, Kitano H (1998) Particle adsorption in evaporating droplets of polymer latex dispersions on hydrophilic and hydrophobic surfaces. Colloid Polym Sci 276(9):810–815 9. Hu H, Larson RG (2006) Marangoni effect reverses coffee-ring depositions. J Phys Chem B 110(14):7090–7094 10. Majumder M, Rendall CS, Eukel JA, Wang JYL, Behabtu N, Pint CL, Liu T-Y, Orbaek AW, Mirri F, Nam J, Barron AR, Hauge RH, Schmidt HK, Pasquali M (2012) Overcoming the “coffee-stain” effect by compositional marangoni-flow-assisted drop-drying. J Phys Chem B 116(22):6536–6542 11. Li Y, Yang Q, Li M, Song Y (2016) Rate-dependent interface capture beyond the coffee-ring effect. Sci Rep 6(1):24628 12. Bhardwaj R, Fang X, Somasundaran P, Attinger D (2010) Self-assembly of colloidal particles from evaporating droplets: role of DLVO interactions and proposition of a phase diagram. Langmuir 26(11):7833–7842 13. Yunker PJ, Still T, Lohr MA, Yodh AG (2011) Suppression of the coffee-ring effect by shapedependent capillary interactions. Nature 476(7360):308–311 14. Choi Y, Han J, Kim C (2011) Pattern formation in drying of particle-laden sessile drops of polymer solutions on solid substrates. Korean J Chem Eng 28(11):2130–2136 15. Jung J-Y, Kim YW, Yoo JY (2009) Behavior of particles in an evaporating didisperse colloid droplet on a hydrophilic surface. Anal Chem 81(19):8256–8259 16. Wong T-S, Chen T-H, Shen X, Ho C-M (2011) Nanochromatography driven by the coffee ring effect. Anal Chem 83(6):1871–1873 17. Chhasatia VH, Joshi AS, Sun Y (2010) Effect of relative humidity on contact angle and particle deposition morphology of an evaporating colloidal drop. Appl Phys Lett 97(23):231909 18. Askounis A, Sefiane K, Koutsos V, Shanahan MER (2014) The effect of evaporation kinetics on nanoparticle structuring within contact line deposits of volatile drops. Colloids Surf A Physicochem Eng Asp 441:855–866 19. Dash A, Bange PG, Patil ND, Bhardwaj R (2020) An image processing method to measure droplet impact and evaporation on a solid surface. S¯adhan¯a 45(1):287
Comparative CFD Analysis of Heat Transfer and Melting Characteristics of the PCM in Enclosures with Different Fin Configurations Arun Uniyal and Yogesh K. Prajapati
1 Introduction In the current scenario, demand of uninterrupted energy supply is continuously increasing. It is due to the fact that day to day activities and life style of the mankind are mainly governed by energy consuming appliances. It is well known that thermal energy in one of the most usable forms of energies required in various applications. Over the period of time several techniques have been developed to store the thermal energy that holds the potential to fulfil the continuous energy supply. Available thermal energy received through the solar radiation is available in abundant amount, if it could be stored judicially during the sunshine hours this stored energy can provide the heat according to demand. Phase Change Materials (PCM) have been recognized as the materials that can store and release the energy by changing its phase. Such materials have gained more attention in recent times due to their large latent heat capacity and economical aspects. PCMs have wide range of applications, that is, electronic industry, textile industry, battery thermal management system, solar energy storage [1–3].
2 Literature Review and Objective It is worth mentioning that usually PCM have low-thermal conductivity. Hence, in the literature, consistent efforts have been made towards increasing the thermal conductivity of the PCM [9, 11]. A. Uniyal (B) Department of Mechanical Engineering, NIT Uttarakhand, Srinagar, Garhwal 246174, India e-mail: [email protected] Y. K. Prajapati Department of Mechanical Engineering, BIT Sindri, Dhanbad, Jharkhand 828123, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_10
97
98
A. Uniyal and Y. K. Prajapati
Besides, insertion of fins is one of the most effective techniques for enhancing the thermal conductivity inside the PCM enclosure. Several researchers have made significant contribution in this field. Ji et al. [4] have proposed the inclined fins in the rectangular domain of the PCM. They observed significant improvement in the melting rate if the fin was placed inclined at an angle of −15° consequently, 62.7% time was saved in complete melting of the PCM. Chen et al. [5] have used double triangular fins in a rectangular section using nanocomposite. Results revealed that volume fraction of 1% of nanoparticle exhibits the best performance and saved 184 s time compared with pure PCM. Samakoush et al. [6] examined the combination of triangular and rectangular fins. Results show that in a case with six proposed fins witness higher melting rate and reduces the melting period by 57.56% as compared to no fin configurations. Additionally, literature survey confirms that availability of fins in the PCM domain certainly helps in enhancing the heat transfer characteristic of the PCM. Henceforth, most of the researchers have proposed different types of fin configurations to further improve the performance. In the present work, comparative study has been performed to compare the performance of different fin configuration while keeping the equal fins heat transfer area in all the cases. Please note that four different fin configurations, that is, single rectangular, double rectangular, triple rectangular and triangular fins are used in the square domain to understand the melting characteristics, enhancement ratio and impact of natural convective currents on melting characteristics of PCM.
3 Geometrical Description Different fin configurations of the rectangular and triangular fins inside the square domain (size = 20 mm) of the PCM have been presented in a Fig. 1. Five cases have been considered where fins surface area in each case is constant (30 mm2 ). In order to maintain the equal fin surface area, fin length and width are varied in the domain. Except 4th case, length of the fin is taken as 15 mm and whereas, fin thickness is kept 1 and 2 mm. In the case 5, base of the triangular fins are 2 mm. Paraffin wax is taken as phase change material whose property has been given in Table 1. PCM is filled inside the square domain of dimension 20 mm. A constant temperature source (350 K) is applied on left side edge of the domain and rest sides of the domain are adiabatic in nature.
3.1 Numerical Methodology and Governing Equation Two-dimensional (2D) numerical simulation has been conducted using ANSYS Fluent. Phase change process of the PCM has been modelled using enthalpy porosity
Comparative CFD Analysis of Heat Transfer and Melting …
99
Case 1
Case 2
Case 3
Case 4
Case 5
Fig. 1 Schematic diagram of different cases in square PCM domain (all dimensions are in mm)
method [8]. Following assumptions were considered during in the numerical simulation: (a) Liquid PCM is assumed as incompressible and Newtonian fluid. (b) Thermophysical properties of the PCM remains constant. (c) The volume of the PCM remain constant before and after the melting process (d) No heat loss occurs from the square domain. It is understood that in the melting process of the PCM, natural convection will play significant role hence, it has been considered by introducing the Boussinesq
100
A. Uniyal and Y. K. Prajapati
Table 1 Thermophysical properties of paraffin wax [10]
Parameter
Value
Thermal conductivity [W/m–K]
0.21
Heat of fusion [kJ/kg]
189
Density
[kg/m3 ]
795/920
Specific heat [kJ/kg–K]
2/2.15
Dynamic viscosity of liquid paraffin [kg/m–s]
0.023
Thermal expansion coefficient [1/K]
0.0003085
equation. Following governing equations have been used in the present numerical model. ρ ρ= β T − Tliquids + 1
(1)
The continuity may be written as: ∂ − → ρ + ∇.(ρV ) = 0 ∂t
(2)
Equation below represents the momentum equation: − ∂ − →− → − → → → (ρV ) + ∇.(ρ V V ) = μ∇ 2 V − ∇ρ + ρ − g + S ∂t
(3)
− → In the above equation, S denotes the source term. This term has been added to consider the natural convection in the melting process. (1 − λ)2 − → − → × V S = −Amushy λ3 + e
(4)
where Amushy is the mushy zone constant which helps in measuring the damping behaviour. The value of Amushy varies from 104 to 107 . Following energy equation is solved: − ∂ → (ρH) + ∇. ρ V H = ∇.(k∇T) ∂t
(5)
The total enthalpy contains both sensible as well as latent heat: h total = h sen + h lat Sensible and latent heat have been calculated from the following equations:
(6)
Comparative CFD Analysis of Heat Transfer and Melting …
101
T h sen = h r e f + c p
dT
(7)
Tr e f
h lat =
n i=1
λi h s f
(8)
where λ is the liquid fraction, its value varies from 0 to 1 for each computational cell and it is considered as mushy zone when λ varies between 0 and 1. λ = 0 indicates that cell is in solid state while λ = 1 indicates that cell consists of liquid state.
3.2 Initial and Boundary Conditions In the present study numerical model, a constant temperature source of 350 K has been applied on the left side of square domain. PCM is kept at initial temperature of 303 K. Gravity force has been applied along Y-axis. As mentioned in the above section, ANSYS Fluent is used for the simulation, SIMPLE scheme and first-order implicit method are used for the pressure velocity coupling and transient formulation, respectively. Pressure correction equation has been solved by PRESTO scheme. Momentum and energy equations are discretized by second-order upwind scheme. Convergence criteria of are set at 10−4 , 10−4 , 10−6 are set respectively for continuity, momentum and energy equations.
3.3 Grid Independency Test and Model Validation In order to perform the grid independency test four distinct element sizes 0.2 mm, 0.15 mm, 0.1 mm, 0.08 mm have been taken. Figure 2a describes the variation of liquid fraction with respect to time for different size of elements. It can be seen that with varying element size, liquid fraction is almost similar and maximum variation in results is found to be 1.48%., Hence, grid size of 0.1 mm is taken for all the simulations. The current numerical model has been validated with the experimental work of Kamkari et al. [7]. Identical geometry of Kamkari et al. has been generated and Lauric acid is considered as the PCM. Besides, similar initial and boundary conditions have been used. Results have been compared in Fig. 2b, it may be seen that present model closely predicts the liquid fraction with Kamkari et al. with maximum deviation of 7–10%.
102
A. Uniyal and Y. K. Prajapati
110
100
90 80 70
0.08 mm 0.10 mm
60
0.15 mm
50
Liquid fraction (%)
Liquid fraction (%)
100
80 60 40 Present work Kamkari et al. (2014)
20
0.20 mm
0
40 0
200
400
600
800
1000
0
50
100
Time (Sec)
Time (Min)
(a)
(b)
150
200
Fig. 2 a Grid independency test. b Validation of the present work with Kamkari et al. [7]
4 Results and Discussion The comparative results of all the five configurations have been discussed to comprehend the heat transfer rate, melting characteristics of the PCM and enhancement ratio. Besides, different contour plots are drawn to visualize the stream lines patterns and phase change process of the PCM in the domain.
4.1 Melting Characteristics of the PCM In order to understand the charging (melting) characteristics of the phase change material, it is necessary to predict the energy storage rate. In the literature, several attempts have been made to expedite the melting rate which results in decrease in energy storing time. Figure 3 illustrates the liquid fraction of the PCM with respect to time for all the five cases at the fixed wall temperature of 350 K. Figure 3 shows that owing to different melting rates, slopes of the curves are dissimilar for different cases. First configuration without any fin deliberates the least slope as compared to other cases. It indicates that melting rate of the PCM is slowest in this domain. Unavailability of the fins lowers the heat transfer from the heated wall to the remaining portion of the domain. Therefore, melting process is delayed consequently, it takes 2073 sec to melt the complete PCM. However, slopes of the other curves representing finned configuration are considerably different. It may be seen that curves are quite steeper at the beginning and with passage of time it flattens which indicates that melting process is rapid at the beginning, however, slow melting occurs at the end. It is understood that heat is being propagated from the left to the right of the square enclosure. PCM is first melted at
Comparative CFD Analysis of Heat Transfer and Melting …
103
Fig. 3 Variation of liquid fraction with respect to time for each case
the left side of the domain and gradually solid–liquid interface moves away from heat wall. Inclusion of fins expedites the heat dissipation from heated wall to remaining portion of the domain which leads to faster melting Since the fin length is restricted up to 15 mm, the portion of the enclosure located far away from the fins receives less heat hence; slow melting of the PCM was experienced. It has been reported that initially (up to 100 sec), cases 3, 4 and 5 have nearly same melting rate which is ≈ 30% higher than case 1. Whereas, melting rate of the PCM in 2nd domain is only ≈ 15% higher as compared to case 1. Since 2nd domain holds only single fin which is mainly concentrated to the middle of the PCM domain. Further, it may be noted that case 3 and 5 show nearly similar trends and both of them have been melted completely (100% melting) within 820 seconds which is 60.44% higher than the case 1. Among the finned configurations, least performance is given by case 4. This is mainly due the presence of shorter fins which restricts the heat transfer up to the middle portion of the domain, beyond it, slower heat transfer occurs through PCM. In order to have deep insights of the melting phenomenon of the PCM, liquid fraction contours are shown in Fig. 4 at different time intervals. Please note that blue and red colours represent solid and liquid phases of the PCM, respectively. Uniform thickness of the melted PCM has been observed in case 1 within 100 secs. Over the period of time, amount of melted PCM increases in the entire domain. Additionally, it can be seen that the thickness of melted PCM considerably reduces from the top to the bottom of the PCM domain. It is due to the evolution of natural convective currents in the enclosure as a result of buoyancy effect. Further Fig. 4 clearly depicts that PCM melting process is considerably higher in cases 3 and 5. Longer fins, better approachability and locations of the fins in the enclosure assisted to expedite the melting process in both the cases.
104
A. Uniyal and Y. K. Prajapati
Fig. 4 Liquid fraction contours of different cases at different time intervals
4.2 Enhancement Ratio Enhancement Ratio (ER) is an indicator which shows the impact of fins inside the PCM enclosure. It provides the necessary information about liquid fraction of the PCM existing in the finned structure to the liquid fraction available in without fin configuration. Figure 5 illustrates the ER for all the cases with respect to time. It’s noteworthy that case 1 is considered as reference PCM domain for the calculations of ER and assigned zero value of ER Except case 1, all the cases have noticeable value of enhancement ratio which is being varied with time. Figure 5 shows that value of ER for all the cases is increasing up to 500 s because by this time heat transfer mainly occurs through fins. Later on heat is transferred through PCM only. The highest value of ER is 63.6, 62.9, 43.8, 41.4 for the cases of 3, 5, 2 and 4, respectively. After certain period of time, all the curves are merged together because effect of fins has been diminished due to restriction of length.
4.3 Evolution of Stream Lines To understand the characteristics of the streamlines formed during the melting process of the PCM, Fig. 6 has been drawn. It illustrates the streamlines in all the five cases evolved at time 400 s. Case 1 has only one vortex formation in the melted zone
Comparative CFD Analysis of Heat Transfer and Melting …
105
Fig. 5 Enhancement ratio with respect to time for all the cases
which continuously expands with time while in the finned enclosures, 3–4 eddies are formed. It is clearly visible that fins are mainly responsible for increasing the number of vortex formation. Figure 6 also shows that pattern of eddies is different based on number of fins and their geometries. Besides, vortices are more prominent close to the fins and there is no such pattern observed in the solid region of the PCM. It is worth mentioning that due to the formations of vortices, heat transfer via convective mode is significantly affected. It is clearly depicted that case 3 and 5 have a greater number of short and long loops of the streamlines due to which their melting rate is higher while, constrained length of the loops has been observed with case 4 due to which its melting rate is quite low as compared with other cases.
5 Conclusions In the present study, two-dimensional transient numerical simulation has been performed for the square domain of PCM integrated with distinct fin configurations keeping equal heat transfer surface area of the fins. Impact of the various fin configurations is examined to understand the melting phenomenon of the PCM. Furthermore, evolution of the stream lines and enhancement ratio have also been obtained for each case. Following conclusion can be obtained from the present study. • Fin arrangement, dimension and its location in the domain significantly affects the melting rate of the PCM. • Cases 3 and 5 facilitate highest melting performance which is 60.4% higher than the case 1. • Case 3 observes the maximum enhancement ratio whereas, least value is obtained with case 4.
106
A. Uniyal and Y. K. Prajapati
Fig. 6 Streamline contours of the different cases at time interval of 400 s
• Owing to gravity, natural convective current generates in the enclosures, which is mainly responsible for the convective heat transfer.
Nomenclature A cp ER g h k T
Area[m2 ] Heat capacity[J/kg-K] Enhancement ratio[-] Gravity [m/s2 ] Enthalpy[kJ/kg] Thermal conductivity[W/m–K] Temperature [K]
Comparative CFD Analysis of Heat Transfer and Melting …
t ρ β λ
107
Time[sec] Density of air[kg/m3 ] Thermal expansion coefficient[–] Liquid fraction[–]
References 1. Mondal S (2008) Phase change materials for smart textiles—an overview. Appl Therm Eng 28(11–12):1536–1550 2. Rajabifar B (2015) Enhancement of the performance of double layered microchannel heatsink using PCM slurry and nanofluid coolants. Int J Heat Mass Transf 88:627–635 3. Khanna S, Reddy KS, Mallick TK (2018) Optimization of finned solar photovoltaic phase change material ( finned pv pcm ) system. Int J Therm Sci 130:313–322 4. Ji C, Qin Z, Low Z, Dubey S, Choo FH, Duan F (2018) Non-uniform heat transfer suppression to enhance PCM melting by angled fins. Appl Therm Eng 129:269–279 5. Chen SB et al (2021) Combined effect of using porous media and nano-particle on melting performance of PCM filled enclosure with triangular double fins. Case Stud Therm Eng 25:100939 6. Masoumpour-Samakoush M, Miansari M, Ajarostaghi SSM, Arıcı M (2021) Impact of innovative fin combination of triangular and rectangular fins on melting process of phase change material in a cavity. J Energy Storage 103545 7. Kamkari K, Shokouhmand H (2014) Experimental investigation of phase change material melting in rectangular enclosures with horizontal partial fins. Int J Heat Mass Transf 78:839–851 8. V. V. R. & P. C., “A Fixed grid numerical modelling methodology for convection diffusion mushy region phase change problems,” Int. Jounal Heat Mass Transf., vol. 30, no. 8, pp. 1709– 1719, 1978. 9. Mat S, Al-Abidi AA, Sopian K, Sulaiman MY, Mohammad AT (2013) Enhance heat transfer for PCM melting in triplex tube with internal-external fins. Energy Convers Manag 74:223–236 10. Olfian H, Soheil S, Ajarostaghi M, Farhadi M (2020) Melting and solidification processes of phase change material in evacuated tube solar collector with u-shaped spirally corrugated tube. Appl Therm Eng 116149 11. Safari V, Abolghasemi H, Kamkari B (2021) Experimental and numerical investigations of thermal performance enhancement in a latent heat storage heat exchanger using bifurcated and straight fins. Renew Energy 174:102–121
Influence of Contact Line Velocity Implementation in Dynamic Contact Angle Models for Droplet Bouncing and Non-bouncing Dynamics on a Solid Substrate Priyaranjan Sahoo, Javed Shaikh, Nagesh D. Patil, and Purnendu Das
1 Introduction The experimental study and numerical modelling of droplet impact on solid substrate [1–3], has heavily attracted researchers’ interest for the last two decades because of its extensive implementations in industrial applications such as electrohydrodynamic inkjet printing [4], self-cleaning [5], spray cooling [6], etc. In the last few years, researchers have been developing hydrophilic, hydrophobic, and super hydrophobic surfaces having a static contact angle in the range of 0°–90°, 90°–150°, and 150°–180°, respectively. The hydrophilicity (hydrophobicity) is achieved by increasing (lowering) the surface energy with the introduction of a regularly patterned or irregular microstructure in the surface, which leads to the Wenzel (Cassie Baxter) wetting state. Upon impact on the hydrophilic surface, the droplet generally shows non-bouncing spreading behaviour, but on the hydrophobic surface, both bouncing and non-bouncing natures are noticed depending on droplet diameter, impact velocity, and viscosity of the liquid [7]. There are numerous potential applications for both the bouncing and non-bouncing natures of droplet impact over a surface. In inkjet printing, the sticking of ink droplets because of their post-impact non-bouncing nature helps in uniform ink deposition [8]. On the other hand, to avoid the ice formation and accumulation on aircraft surfaces due to the sticking of supercooled liquid, the bouncing effect of droplet impact is desired [9].
P. Sahoo (B) · N. D. Patil · P. Das Department of Mechanical Engineering, IIT Bhilai, Raipur 492015, India e-mail: [email protected] J. Shaikh Department of Mechanical Engineering, IIT 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_11
109
110
P. Sahoo et al.
2 Literature Review and Objective Hao Li et al. [10] studied the dynamic behaviour of an impacting droplet over a LASER patterned superhydrophobic surface to establish the concept of impact velocity independence of contact time for a bouncing droplet. Xuan Zhang et al. [11] investigated the reduction of contact time in the case of impact over a moving superhydrophobic surface experimentally. Wang et al. [12] and Jung and Bhushan [13] experimentally studied the bouncing and non-bouncing nature of impact depending on static contact angle, diameter of drop, and impact velocity. Compared to experimental studies, there is very little numerical study of droplet impact. Nagesh D. Patil et al. [14] studied droplet impact numerically, using the level set method, and showed that the accuracy of the solution increased when a dynamic contact angle based boundary condition was implemented. In their study, they used the Yokoi model [3] of dynamic contact angle. The wetting contact line velocity (UCL ) was calculated by implementing the FOU scheme to obtain velocity (from the NS solver) of the neighbour grids. The objective of this work is to compare the accuracy of the result when the SOU and QUICK schemes are used instead of FOU to calculate the wetting contact line velocity by validating with the published literature.
3 Materials and Methods To simulate the impact, a 2D axisymmetric cylindrical computational domain of 3 × 3 units of non-dimensional length is used. With a grid size of 0.0151 × 0.0151 and a stability criterion approved time step of 1 × 10–4 . The initial diameter of the drop is taken as a length scale and the impact velocity is taken as a velocity scale to non-dimensionalize the problem. As shown in Fig. 1, an axisymmetric boundary condition is used in the axis of symmetry, no slip boundary condition is used at the interface of drop and surface (except at the wetting contact line), and an outflow boundary condition is used in the other two computational domain boundaries. The level set function, a normal distance sign function, is used to distinguish heavier fluids (liquid) from lighter ones (air), having positive values inside the liquid, zero at the interface, and a negative value outside the droplet. The boundary conditions of the level set function (F) are also shown in Fig. 1 below.
3.1 Numerical Methodology The computational domain is divided into uniform grids of size 0.0151 × 0.0151. The U velocity, V velocity, and pressure control volumes are arranged as per the staggered grid arrangement and level set grid points are assigned as per the DGLSM arrangement. A detailed discussion is in [13]. In the solution step, first momentum
Influence of Contact Line Velocity Implementation in Dynamic Contact …
111
Fig. 1 2D axisymmetric cylindrical computational domain with all boundary conditions for a droplet impacting with a velocity of V0
(NS) Eq. (2) and the continuity Eq. (1) are solved to obtain the velocity field using the SIMPLE method. The surface tension at the interface control volumes is modelled as a volumetric term instead of a surface term and added to the momentum equation. The velocity of level set grid points is calculated by interpolating the velocity obtained from the NS solver. Then the level set advection Eq. (3) is solved. After each time step, a reinitialization equation [14] is solved to maintain the normal sign distance property of the level set function before moving to the next time step. − → ∇·U =0 ∂ ρm U ∂τ
T 1 + ∇ · ρm U U = − ∇ P + ∇ · μm ∇ U + ∇ U Re ρm ˆ 1 k nδ ˆ ε (∅) − j+ 2 Fr we ∂φ − → + U · ∇∅ = 0 ∂τ
(1)
(2) (3)
112
P. Sahoo et al.
*all the equations are in non dimensionalized form.
3.2 Dynamic Contact Angle Modelling and Contact Line Velocity Approximation Patil et al. [14] showed that applying dynamic contact angle (θd ) based boundary condition (refer Fig. 1) in level set advection equation increases the accuracy of the solution. In the present work, the Yokoi model (4) [3], based on Tanner’s law, is used to calculate θd as ⎧
1/ ⎪ ⎪ 3 μU C L ⎪ ⎨ min θ st + , θ adv if U C L ≥ 0 γ θd =
1/3 ⎪ ⎪ μU C L ⎪ ⎩ max θ st + γ k , θ rec ifU C L < 0
(4)
rec
where θstatic , θadv and θrec are equilibrium, static advancing, and receding contact angles, respectively. μ and γ are viscosity and surface tension of the liquid, k adv and k rec are material-related constants, and UCL is the contact line velocity. Because UCL is a critical input parameter to the dynamic contact angle model as well as the slip boundary condition (at wetted radius), the method used to approximate its value has a significant impact on solution accuracy. Sikalo et al. [15] approximated the contact line velocity by differentiating the wetted radius. Griebel et al. [16] used the velocity parallel to the substrate of the very first grid along the Z direction near to the contact point as the velocity of the contact line, which resembles the FOU scheme. In our work, two different cases of contact line velocity estimation, SOU and QUICK approximation, are implemented, details of which have been shown below in the Fig. 2. While formulating the contact line velocity by SOU and QUICK scheme, an assumption is made that the liquid air interface passes exactly through the middle of the level set grid just above the interfacial grid (iif,1 ) and its right adjacent grid (iif+1,1 ).
4 Results and Discussion For the simulation, we considered two different cases of droplet impact, that is, of bouncing and non-bouncing fate, already present in the published literature [13]. The three schemes (FOU, SOU, and QUICK) of contact line velocity approximation are implemented in the numerical model for both cases. The numerically obtained results of the variation of droplet height and wetted diameter with time are compared to the published experimental data of a droplet of 2 mm diameter impacting with a
Influence of Contact Line Velocity Implementation in Dynamic Contact …
113
Fig. 2 Approximation of contact line velocity: FOU, SOU and QUICK scheme
velocity of 0.44 m/s over two surfaces of n-hexatriacontane coated silicon with θeq = 91°, θadv = 141°, θreced = 54° (non-bouncing) and nanostructured super hydrophobic with θeq = 158°, θadv = 165°, θreced = 142° (bouncing).
4.1 Comparison of Experimental and Numerically Obtained Droplet Wetted Diameters with Non-Bouncing Mood In Fig. 3, the variation of droplet diameter (non-dimensionalized) with time is plotted for a droplet of 2 mm in diameter impacting with a velocity of 0.44 m/s on a surface with θeq = 91°, θadv = 141° and θreced = 54°. After the impact, the droplet spreads rapidly due to the inertia force. till 6 ms. This spreading leads to an increase in the surface area of the droplet and gives rise to a converse surface force responsible for retracting the droplet shape. Hence the wetted diameter decreases after 6 ms. Again, due to inertia induced by gravitational force on the liquid above the surface, the droplet spreads and the surface
114
P. Sahoo et al.
Fig. 3 Comparison of the numerically calculated time-varying wetted diameter with the experimental result [13] of a non-bouncing droplet impacting with a velocity of 0.44 m/s on a surface with θeq = 91°, θadv = 141° and θreced = 54°
area increases, and the above phenomena repeats, resulting in the oscillating nature of the droplet wetted diameter. The amplitude of oscillation of droplet wetted diameter decreases progressively as the oscillation dampens due to viscous dissipation with time. The numerically obtained results by SOU and FOU approximated contact line velocity, implemented as a boundary condition to DGLSM, show a good match with the above discussed trend as well as the experimental results of Jung and Bhushan [13]. But for QUICK approximation, the result initially shows a better match with experimental data till 5 ms, after which it gives erroneous solutions.
4.2 Comparison of Experimental and Numerical Maximum Droplet Heights with Non-Bouncing Mood In Fig. 4, the variation of droplet height (non-dimensionalized) with time is plotted for the case as stated in Sect. 4.1. After impact, the droplet spreads due to inertia force till 6 ms and the diameter increases, which leads to a rapid decrease in height. After 6 ms, the droplet contracts due to dominance of the surface force and the droplet height increases till 13 ms, then again it decreases and attains an oscillation the same as that of the diameter of the droplet, the phenomena of which is already
Influence of Contact Line Velocity Implementation in Dynamic Contact …
115
Fig. 4 Comparison of numerically calculated time-varying droplet height with experimental results [13] of a non-bouncing droplet impacting with a velocity of 0.44 m/s on a surface with θeq = 91°, θadv = 141° and θreced = 54°
discussed in Sect. 4.1. The numerically obtained results by SOU and FOU approximated contact line velocity, implemented as a boundary condition to DGLSM, match the experimental results of Jung and Bhushan [13] in this case as well. But the QUICK approximation is erroneous as it shows a max height of zero. The erroneous computation of the result using the QUICK scheme as shown in Figs. 3 and 4 is due to the location of the iif+1,1 th LS grid point. For the non-bouncing nature of impact where the contact angle is smaller, the iif+1,1 th LS grid point predominantly lies outside the heavier fluid. It is discussed more in detail towards the end of this section.
4.3 Comparison of Experimental and Numerically Calculated Droplet Wetted Diameters with Bouncing Mood In Fig. 5, the wetted diameter of the droplet (non-dimensionalized) is plotted with time for a droplet with an initial diameter of 2 mm impacting with a velocity of 0.44 m/s on a surface with θeq = 158°, θadv = 165°, θreced = 142°. As similar to the non-bouncing nature of impact, in this case also, the droplet spreads rapidly just after the impact because of the inertia and contracts thereafter because of the surface tension force, so the wetted diameter increases rapidly then decreases. Because of
116
P. Sahoo et al.
Fig. 5 Numerical vs. experimental result [13] of temporal variation in the wetted diameter of a bouncing droplet with an impact velocity of 0.44 m/s and an initial diameter of 2 mm on a surface with θeq = 158°, θadv = 165°, θreced = 142°
the higher equilibrium contact angle of the surface, the surface force and inertia force during contraction overcome the wetting force and the droplet bounces. In the bouncing instant, the wetted diameter of the droplet attains zero value. Unlike the non-bouncing droplet, this case shows a good match with experimental results of Jung and Bhusan [13] for SOU, FOU, as well as QUICK scheme based contact line velocity approximation, which is used as a boundary condition in DGLSM.
4.4 A Comparison of Experimental and Numerically Calculated Droplet Maximum Height with Bouncing Mood The Fig. 6 shows the variation of the maximum height of a droplet for the same case as mentioned in Sect. 4.3. The height decreases till < 5 ms then increases converse to the diameter of the droplet. Unlike the non-bouncing nature of impact, for the bouncing nature of droplet impact, the QUICK scheme produces results with acceptable accuracy as presented in Figs. 5 and 6, similar to SOU and FOU. The results show that both the FOU and SOU schemes for contact line velocity approximation give nearly equal matches of the numerical data to the experimental
Influence of Contact Line Velocity Implementation in Dynamic Contact …
117
Fig. 6 Numerical vs. experimental result [13] of temporal variation in the maximum height of a bouncing droplet with an impact velocity of 0.44 m/s and an initial diameter of 2 mm on a surface with θeq = 158°, θadv = 165°, θreced = 142°
[13] for both bouncing and non-bouncing droplets. But the QUICK scheme approximation gives a good match for the bouncing case, that is, for surfaces having a higher equilibrium contact angle, and erroneous results for surfaces having a lower equilibrium contact angle. As presented in Fig. 2, the contact line velocity is approximated using the velocity of the level set grid points iif,1 for FOU, iif-1,1 and iif,1 for SOU, iif-1,1 , iif,1 and iif+1,1 for the QUICK scheme. The velocities at LS grid points are obtained by solving the Navier–Stokes equation, and the solution is dependent on physical properties like density and viscosity of the medium. In the case of lower equilibrium contact angle, the right adjacent level set grid (iif+1, 1 ) to the level set grid just above the interfacial grid (iif,1 ) lies outside the droplet which leads to a severe jump in physical properties like density and viscosity. For the non-bouncing nature of impact, the droplet initially holds a higher contact angle, which entraps iif+1, 1 inside the droplet. Later, as the droplet spreads, the contact angle decreases and the grid point lies outside. As a result, the QUICK scheme initially shows a good match with experimental data, but later gives erroneous result plots, as shown in Figs. 3 and 4. But for the bouncing case, the equilibrium contact angle of the surface is higher, which entraps the right adjacent level set grid (iif+1, 1 ) inside the droplet and restricts the severe jump of physical properties. For that reason, the QUICK scheme approximates contact line velocity as good as the FOU and the SOU approximation. Hence, the dynamic contact angle, which is used as a boundary condition for the
118
P. Sahoo et al.
Fig. 7 Orientation of LS grid points iif+1,1 , iif,1 and iif-1,1 for lower and higher equilibrium contact angle
level set advection equation, is obtained correctly. Figures 5 and 6 show that the QUICK scheme gives a good match of numerical data with experimental for the entire duration. The locations of the LS grid points for the two different cases would be more clear from the Fig. 7.
5 Conclusions Implementation of FOU, SOU, and QUICK scheme approximations of contact line velocity, used as boundary conditions to level set function, is investigated for droplet impact study over two different wettability surfaces. It is found that the SOU and FOU approximations give nearly a good match of numerically obtained wetted diameter and max height of droplet with experimental data for both cases, but the QUICK scheme gives a good match for surfaces having a higher equilibrium contact angle (>150°) but is not suitable for surfaces with a lower equilibrium contact angle. Acknowledgements N.D.P. gratefully acknowledges the financial support under start-up-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.
References 1. Gada VH, Sharma A (2011) On a novel dual-grid level-set method for two-phase flow simulation. Numer Heat Transf Part B: Fundament 59(1):26–57 2. Gada VH (2012) A novel level-set based CMFD methodology in 2D/3D Cartesian and cylindrical coordinates and its application for analysis of stratified flow and film boiling. IIT Bombay 3. Yokoi K, Vadillo D, Hinch J, Hutchings I (2009) Numerical studies of the influence of the dynamic contact angle on a droplet impacting on a dry surface. Phys Fluids 21(7):072102 4. Galliker P, Schneider J, Eghlidi H, Kress S, Sandoghdar V, Poulikakos D (2012) Direct printing of nanostructures by electrostatic autofocussing of ink nanodroplets. Nat Commun 3(1):1–9 5. Lu Y, Sathasivam S, Song J, Crick CR, Carmalt CJ, Parkin IP (2015) Robust self-cleaning surfaces that function when exposed to either air or oil. Science 347(6226):1132–1135 6. Jia W, Qiu HH (2003) Experimental investigation of droplet dynamics and heat transfer in spray cooling. Exp Therm Fluid Sci 27(7):829–838
Influence of Contact Line Velocity Implementation in Dynamic Contact …
119
7. Patil ND, Shaikh J, Sharma A, Bhardwaj R (2022) Droplet impact dynamics over a range of capillary numbers and surface wettability: assessment of moving contact line models and energy budget analysis. Phys Fluids 34(5):052119 8. Hoffman H, Sijs R, de Goede T, Bonn D (2021) Controlling droplet deposition with surfactants. Phys Rev Fluids 6(3):033601 9. Lin Y, Chen H, Wang G, Liu A (2018) Recent progress in preparation and anti-icing applications of superhydrophobic coatings. Coatings 8(6):208 10. Li H, Zhang K (2019) Dynamic behavior of water droplets impacting on the superhydrophobic surface: both experimental study and molecular dynamics simulation study. Appl Surf Sci 498:143793 11. Zhang X, Zhu Z, Zhang C, Yang C (2020) Reduced contact time of a droplet impacting on a moving superhydrophobic surface. Appl Phys Lett 117(15):151602 12. Wang Z, Lopez C, Hirsa A, Koratkar N (2007) Impact dynamics and rebound of water droplets on superhydrophobic carbon nanotube arrays. Appl Phys Lett 91(2):023105 13. Jung YC, Bhushan B (2009) Dynamic effects induced transition of droplets on biomimetic superhydrophobic surfaces. Langmuir 25(16):9208–9218 14. Patil ND, Gada VH, Sharma A, Bhardwaj R (2016) On dual-grid level-set method for contact line modeling during impact of a droplet on hydrophobic and superhydrophobic surfaces. Int J Multiph Flow 81:54–66 15. Šikalo Š, Wilhelm HD, Roisman IV, Jakirli´c S, Tropea C (2005) Dynamic contact angle of spreading droplets: experiments and simulations. Phys Fluids 17(6):062103 16. Griebel M, Klitz M (2014) Simulation of droplet impact with dynamic contact angle boundary conditions. In: singular phenomena and scaling in mathematical models, Springer, Cham, pp 297–325
Droplet Impact and Wetting on a Micropillared Surface Yagya Narayan and Rajneesh Bhardwaj
1 Introduction Superhydrophobic surfaces engineered with micro/nano features are ubiquitous due to their important properties like self-cleaning [1, 2], anti-icing [3], and dropwise condensation [4]. In general, the surface morphology is bioinspired by nature; for instance, taro [5] and lotus leaves [6] that possess self-cleaning properties. These properties can influence the droplet wetting and impact behaviour with substrate, and may result in slippery and low hysteresis surfaces [7–10]. A microtextured surface is fabricated by developing grooves/pillars by physical or chemical processes [5, 11]; wherein the characteristics are defined on the basis of surface height correlations. A droplet spreads and settles on a surface in a metastable configuration based on the degree of wettability for a given solid–liquid system [12]. A droplet on physically textured surfaces principally has two states, that is, Wenzel and Cassie state. In Wenzel state, the wetting front follows all the topographical variations of surface. Minimum energy required to form unit area solid–liquid interface by replacing its solid–vapour interface component, yields the following equation for the contact angle [13, 14]; cosθrw = r cosθe
(1)
where r is the roughness factor, the ratio of the actual liquid–solid contact area to the total projected area. In the case of Cassie state, the droplet only wets the top area of pillars and thus forming a void space between them. The energy minimization process provides an expression for contact angle as follows [13, 14]; cosθrc = φs cosθe + φs − 1
(2)
Y. Narayan · R. Bhardwaj (B) 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_12
121
122
Y. Narayan and R. Bhardwaj
where φs is the ratio of liquid–solid contact area (or droplet pillar contact area) to the total projected area. Numerous studies [15, 16] have demonstrated the effects of elasticity [17], roughness [18] and physicochemical properties of surfaces [19, 20]. In the context of droplet impact, impact on micro-textured surface provides different regimes such as complete rebound, partial rebound, and no-rebound [20], and effects like droplet splashing [18], spreading, receding and bouncing [21] on interaction with substrate. A droplet acquires kinetic energy over the height of the fall, which gets converted into surface energy and loses in the form of viscous dissipation/heat generation. The interplay of energies decides the bouncing and non-bouncing behaviour (see Patil et al. [20] and review by Yarin [21]). Extensive research has been done on droplet-bouncing and non-bouncing surfaces with hydrophobic and superhydrophobic properties. The deformation in the droplet’s shape on impact [22, 23] and comparison to its impact on hydrophilic surfaces have also been reported [24]. Patil el al. [20] studied the droplet impact on a micro-textured surface with square pillars arranged in a square arrangement and three regimes, namely partial bouncing, complete bouncing, and non-bouncing were reported. A brief literature review shows that there are several open research questions. For instance, do wetting characteristics change if micropillars are arranged in two different profiles with equal liquid–solid contact area fraction φs ? Does Weber number at which droplet starts to change its wetting behaviour from Cassie state depends on the arrangement of pillars? To this end, the objective of this paper is to measure contact angle and droplet wetting diameter on a microtextured surface with two different pillar arrangements and to analyse droplet impact on each surface. The ratio of liquid–solid contact area to the total projected area has been kept constant, as shown in Fig. 1.
Fig. 1 Optical microscopic image of pillars arranged in (a) square profile, (b) hexagon profile. The profile of one cell and corresponding pitch lengths have been illustrated
Droplet Impact and Wetting on a Micropillared Surface
123
2 Methodology In this work, micro-textured surfaces were manufactured by standard photolithography techniques. Five sets of surfaces were manufactured with the same liquid–solid contact area fraction φs in each set. The characteristic distance defining a single cell is called pitch. Figure 1 shows one set of surfaces; a surface with pillars arranged in square profile (a) on the left and with hexagonal profile (b) on the right side. We consider square and hexagonal profile of pitch distance l and a, respectively. The relation between these distances can be found by equating φs for both surfaces (considering one cell), as follows; πr 2p l2
=
2πr 2p √ 3 3a 2 2
,
(3)
where r p is the radius of the pillar. The only variable in the expression used by Patankar (Eq. 2) is φs (equilibrium contact angle is taken as a constant). According to this equation, the contact angle value for surfaces with equal φs are the same. However, this expression does not account for pillar arrangement and concludes that the droplet’s shape is the same for different pillar arrangements. The relation between l and a for equal φs can be written as follows; 2l a= √ . 3 3
(4)
The total number of cells involved can be calculated by dividing √ the total wetted 2 area π R 2 by the area of a single cell. The areas of a cell are l 2 and 3 23a in the case of square and hexagonal profiles, respectively.
2.1 Fabrication Process for Making Micropillared Surfaces The standard photolithography technique was used for the fabrication of micropillared surfaces. The polished side of a 2-inch Si wafer was spin-coated with SU-8 2025 epoxy photoresist polymer. Firstly, Si wafer was RCA cleaned and placed in a furnace for wet oxidation to remove suspended particles and prevent soluble components at a later stage. Secondly, the wafer was spin-coated at 500 rpm for 10 s, followed by 2300 rpm for 40 s. Thirdly, the wafer was gently baked for 3 min at 65o C followed by 8 min at 95o C. A double-sided aligner (EVG620, EV Group Inc.) was used to align the iron oxide coated glass mask with square-shaped patterns using a Laser Writer (LW405, Microtech Inc.). Additionally, the wafer was subjected to UV light for two to three minutes at an intensity of 160 mJ/cm2 . Fourthly, the post-baking procedure includes heating at 65o C for 1 min and 95o C for the next 6 min, followed by atmosphere cooling at ambient temperature. After that, samples were cleaned with
124
Y. Narayan and R. Bhardwaj
isopropanol and processed for 5–6 min in SU-8 photo developer. Finally, the hard baking step includes 10-min baking at 120o C. These surfaces were coated with 10 nm Pt coating for comparison with previously published data [20] and increasing durability of surfaces. Optical images of surfaces were taken using an optical microscope (Olympus MX-40, objective lens 4X); representative surfaces with pitch distance of 50 and 43.9 µm for square and hexagonal profiles, respectively (Fig. 1), for all the surfaces diameter of pillars is 20 µm. The error in diameter and pitch distance are ± 1 µm and ± 2 µm, respectively, which is used for error bars in the result section. The pitch distances taken for the square profile are 40, 50, 60, 70, and 80 µm. For hexagonal arrangements, the corresponding values are 35, 43.9, 52.6, 61.4, and 70.2 µm. Each surface’s pillar diameter is constant, that is 20 µm. Figure 1 shows optical microscopic images of pillars arranged in (a) square profile and (b) hexagon profile with the same solid area fraction. A red polygon indicates the geometry of one cell in each part of the figure, which differs in two cases.
2.2 Generation and High-Speed Visualization of a Droplet Figure 2 shows the setup for (a) visualization of droplet behaviour on the surface and (b) high-speed imaging of droplet impact on the surface of interest. A deionized water droplet of diameter 1.7 ± 0.05 mm, generated using a 31 gauge needle syringe (BD Inc.), was used for experiments. Isopropanol liquid was used for cleaning surfaces. After drying out, the impact was analysed with a high-speed camera (NXA3S3 IDT Inc.), assembled with a microscopic objective lens (Qioptiq Inc.) having a focal length of 9.5 cm. This phenomenon was recorded at 1000 frames per second (fps) to ensure sufficient frames for all information about the phenomenon. Required velocities are generated via droplets falling under gravity. The difference between the surface and the needle tip height varied from 2 to 15 mm. This height adjustment is ensured by a height-adjustable precision table (Holmarc Inc.). The velocity thus obtained varies from 0.14 to 0.54 m/s, corresponding to the Weber number from 0.46 to 6.885. Images were rendered and analysed using ImageJ software to measure contact angle and wetted diameter. The angle between the surface and the tangent to the droplet at the three-phase contact point is analysed and reported as the contact angle (as shown in Fig. 3). The contact angle of the flat surface comes out to be 92o . The contact angle and wetted diameter uncertainty are ± 2° and ± 0.03 mm, respectively. To ensure repeatability, experiments were repeated at least three times.
Droplet Impact and Wetting on a Micropillared Surface
125
Fig. 2 Experimental set-up depicting components used for: (a) droplet behaviour visualization (b) impact visualization
Fig. 3 Contact angle and wetted diameter comparison between pillars arranged in (a) square profile with pitch 60 μm and (b) hexagonal profile with pitch 52.4 μm. Zoomed view of contact line, which shows air space between pillars of the microtextured surface, is also shown on the left-top of each part. In case of square profile, air space is clearly visible, however, due to positioning of pillars in hexagon, only light coming from beyond is visible
126
Y. Narayan and R. Bhardwaj
3 Results and Discussion 3.1 Wetting Characteristics Figure 3 illustrates droplet characteristics over square profile (a) with pitch 60 µm, and hexagonal profile (b) of pitch 52.4 µm. The two test cases has an equal value of liquid–solid contact area fraction φs . On the left-top side of each part, a zoomedin view of the three-phase contact point is added, which depicts air space between pillars, confirming the presence of Cassie state. The contact angle on the two surfaces are almost same. Figure 4 illustrates values of contact angle and wetted diameter as a function of liquid–solid contact area fraction for pillars arranged in square and hexagonal profiles. On decreasing φs from 0.2 to 0.05, the contact angle increases for square profiles from 93o to 141o . Similarly, droplets on hexagonal profiles follow the same trend, but at φs = 0.05, the droplet on square profiles exhibits the Wenzel state. In contrast, the value of contact angle at this φs , for hexagonal profiles is larger, implying Cassie state. The theoretical trend plotted with experimental data confirms this hypothesis. The corresponding values of wetted diameter are also shown. The sharp jump at φs = 0.05 indicates the Wenzel condition, which means Cassie state does not exist at and beyond this φs for the square profile.
3.2 Effect of Weber Number for Non-bouncing Condition Further, we performed droplet impact experiments on the two profiles. Figure 5 shows a comparison of the Weber number (We) for the two profiles at which the impacting droplet does not bounce. Beyond this Weber number, either the droplet shows partial bouncing leaving a satellite droplet on the surface or it directly enters into the Wenzel state. Below this Weber number, the droplet always bounces on the surface. In all cases, hexagonal structure shows a larger We than square structure. The value of We is significantly larger at the largest pitch (or lowest φs ). In this case, the droplets go to the Wenzel state in square structure, however, it does not in case of hexagonal profile.
4 Conclusions In conclusion, the present study provides fundamental insights into the wetting characteristics of micro-textured surfaces when the arrangement of pillars is modified. A regular hexagon possesses general characteristics because a circle can be closely approximated into a hexagon, incorporating all the available substrate area. We analysed droplet contact angle and wetting diameter concerning the arrangement of
Droplet Impact and Wetting on a Micropillared Surface
127
Fig. 4 Comparison of theoretical and experimental values of contact angle in square and hexagonal arrangement (y-axis left) is shown. A comparison of wetted diameter between square and hexagonal arrangement (y-axis right) is shown. Values of theoretical contact angle is adopted from Patankar [13]. Points depicted by circles indicate experimental Wenzel states Fig. 5 Comparison of points when droplet does not bounce completely and transits into Wenzel state afterward. Points depicted by circles are Wenzel states
128
Y. Narayan and R. Bhardwaj
pillars, that is, when pillars are arranged in square and hexagon profiles. These pillars were manufactured by standard photolithography. The hexagonal profile shows better results than the square profile by continuing into Cassie state at lower φs , when the square arrangement fails to do so. The contact angle and wetted diameter in both cases are almost the same; however, the droplet exhibits the Wenzel state at high pitch distances (or low φs ) in the case of square arrangement.
Nomenclature r φs θe θrw θrc rp R l a
Roughness factor Liquid–solid contact area fraction Equilibrium contact angle at flat surface [rad] Contact angle in case of Wenzel state [rad] Contact angle in case of Cassie state [rad] Radius of micropillar [µm] Wetting radius of a droplet [µm] Pitch distance in case of square profile [µm] Pitch distance in case of hexagonal profile [µm]
References 1. Blossey R (2003) Self-cleaning surfaces—virtual realities. Nat Mater 2(5):301–306 2. Nishimoto S, Bhushan B (2013) Bioinspired self-cleaning surfaces with superhydrophobicity, superoleophobicity, and superhydrophilicity. RSC Adv 3(3):671–690 3. Liu K, Jiang L (2012) Bio-inspired self-cleaning surfaces. Annu Rev Mater Res 42:231–263 4. Xuemei C, Jun W, Ruiyuan M, Meng H, Nikhil K, Shuhuai Y, Zuankai W (2011) Nanograssed micropyramidal architectures for continuous dropwise condensation. Adv Funct Mater 21(24):4617–4623 5. Kumar M, Bhardwaj R (2020) Wetting characteristics of colocasia esculenta (taro) leaf and a bioinspired surface thereof. Sci Rep 10(1):1–15 6. Zhang M, Feng S, Wang L, Zheng Y (2016) Lotus effect in wetting and self-cleaning. Biotribology 5:31–43 7. Martin V, Yuxi Z, Noor AJ, Leyla S, Tohid FD (2019) Liquid-infused surfaces: a review of theory, design, and applications. Acs Nano 13(8):8517–8536 8. Bruno A, Jacco HS (2020) Statics and dynamics of soft wetting. Ann Rev Fluid Mech 52:285– 308 9. Laxman KM, Nagesh DP, Rajneesh B, Adrian N (2017) Droplet bouncing and breakup during impact on a microgrooved surface. Langmuir 33(38):9620–9631 10. Manish K, Rajneesh B, Kirti Chandra S (2020) Wetting dynamics of a water droplet on micropillar surfaces with radially varying pitches. Langmuir 36(19):5312–5323 11. Lena A, Kari M, Mika S (2020) Precise fabrication of microtextured stainless steel surfaces using metal injection moulding. Prec Eng 62:89–94 12. Abraham M, Claudio DV, Stefano S, Alidad A, Jaroslaw WD (2017) Contact angles and wettability: towards common and accurate terminology. Surf Innov 5(1):3–8
Droplet Impact and Wetting on a Micropillared Surface
129
13. Neelesh AP (2003) On the modeling of hydrophobic contact angles on rough surfaces. Langmuir 19(4):1249–1253 14. David Q (2008) Wetting and roughness. Ann Rev Mater Res 38(1):71–99 15. Christophe J, Sigurdur TT (2016) Drop impact on a solid surface. Ann Rev Fluid Mech 48(1):365–391 16. Khojasteh D, Kazerooni M, Salarian S, Kamali R (2016) Droplet impact on superhydrophobic surfaces: a review of recent developments. J Ind Eng Chem 42:1–14 17. Hardt S, McHale G (2022) Flow and drop transport along liquid-infused surfaces. Annu Rev Fluid Mech 54:83–104 18. Hao J (2017) Effect of surface roughness on droplet splashing. Phys Fluids 29(12):122105 19. Chenlu Q, Fan Z, Ting W, Qiang L, Dinghua H, Xuemei C, Zuankai W (2022) Pancake jumping of sessile droplets. Adv Sci 9(7):2103834 20. Nagesh DP, Rajneesh B, Atul S (2016) Droplet impact dynamics on micropillared hydrophobic surfaces. Experiment Therm Fluid Sci 74:195–206 21. Alexander LY et al (2006) Drop impact dynamics: splashing, spreading, receding, bouncing. Ann Rev Fluid Mech 38(1):159–192 22. Christophe C, C´edric B, Denis R, David Q (2004) Maximal deformation of an impacting drop. J Fluid Mech 517:199–208 23. Li X, Mao L, Ma X (2013) Dynamic behavior of water droplet impact on microtextured surfaces: the effect of geometrical parameters on anisotropic wetting and the maximum spreading diameter. Langmuir 29(4):1129–1138 24. Antonini C, Amirfazli A, Marengo M (2012) Drop impact and wettability: from hydrophilic to superhydrophobic surfaces. Phys Fluids 24(10):102104
Multiphase Modelling of Thin Film Flow Over the Vertical Plate Under Gravity in Pseudo-Laminar Region N. Shiva, Nilojendu Banerjee, and Satyanarayanan Seshadri
Abbreviations Nomenclature κ n X Y v m ˙ Cf A τ wall V ρ F P t α μ g σ Lc
Curvature at interface (m− 1 ) Normal vector X coordinate (mm) Y coordinate (mm) Kinematic viscosity of film (m2 /s) Mass flow rate of film (kg/s) Skin friction coefficient Amplitude of wavy film (mm) Wall shear stress (MPa) Y-Velocity of film (m/s) Density (kg/m3 ) Force (N) Pressure (Pa) Time (s) Phase fraction Dynamic viscosity (Pas) Gravity (m/s2 ) Surface tension coefficient (N/m) Characteristic length (m)
N. Shiva (B) Department of Mechanical Engineering, NIT Puducherry, Karaikal 609609, India e-mail: [email protected] N. Banerjee · S. Seshadri Department of Applied Mechanics, IIT Madras, Chennai 600036, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_13
131
132
δ V k ω Ω R Re Fr Ca Bo
N. Shiva et al.
Film thickness (mm) Volume of film (m3 ) Bulk modulus (Pa) Wave frequency (s− 1 ) Wave number (m− 1 ) Radius of curvature (m) Film Reynolds number Film Froude number Film Capillary number Film Bond number
Subscripts 1, 2 Y m X l, g max o avg
Phase 1, Phase 2 Vary along Y axis Mixture Vary along X axis Liquid, Gas Maximum/At Interface Free stream of film Average
1 Introduction The efficiency of condenser, boiler, and evaporator used within various thermal power plants, heat pumps, and refrigeration systems is significantly influenced by liquid film flow. In all such applications, the liquid passes three regimes: laminar, pseudolaminar, and turbulent regions. In a laminar region, viscous force dominates over gravity force, and the film thickness grows along its plate length. In a pseudo-laminar region, viscous and gravity forces alternately dominate, and waves are produced at the interface. A pseudo-laminar region can be seen when the film Reynolds number is above 30. Hence, the film flows in a pseudo-laminar regime found in most applications. In the wavy film region, the average film thickness is transient in nature; as a result, average heat transfer coefficient also changes with time. Hence, predicting the vapour mass flow rate at the boiler and evaporator outlet is difficult. Numerous researchers performed several experiments along with mathematical models to describe the dynamical behaviour of waves using linear stability analysis and long wave approximation methods [1, 2]. However, developing an analytical model that assists numerical simulation is still necessary. This helps to better understand force distribution as a function of film thickness and change in wave amplitude
Multiphase Modelling of Thin Film Flow Over the Vertical Plate Under …
133
with time and space, variation of film velocity profile, and wall shear stress along the plate. The formation of the thin film on the surface acts as a heat transfer barrier, limiting the efficiency of the boiler and evaporator in turning liquid into vapour in heat pump systems and steam power plants. This results in the entry of incompressible fluid inside the compressor leading to wear and tear of parts as lubrication oil is diluted. Due to this, blockage takes place in the various parts of the suction lines of the compressor. In the same way, liquid corrodes the turbine blades, which reduces the lifespan of the turbine and compressor. This behaviour becomes more complicated, especially in the pseudo-laminar region. This is because the wavy nature of the film creates non-uniform pressure distribution over various parts of the engineering systems, leading to the failure of the components in a frequent manner. Therefore, it is necessary to conduct an in-detail investigation, particularly in the pseudo-laminar region, to understand the dynamic nature of the wavy film along with heat transfer.
2 Literature Review and Objective Film flow is a topic of research due to the effect of complex, unstable wave physics at a low Reynolds number on the heat transfer coefficient. Many researchers are still developing the profile of the liquid film flowing over the surface. Ackerberg [3] developed a mathematical model for wall shear stress, film thickness, and velocity profile for boundary layer flow over the vertical plate without considering the surface tension effect, wavy nature at the interface, viscosity and gravity effect on the velocity profile and wall shear stress. Portalski [4] conducted an experiment on the film flow over vertical plates and concluded that surfactants added to the plate reduce the wave and ripples on the liquid surface and validated his results with the Nusselt mathematical equation for a liquid film thickness and velocity profile over a vertical plate for the laminar region. Portalski [4] also validated his results with Kapitsa’s mathematical model for Reynolds number at the wave inception and average film thickness.S. Portalski [4], in his literature, mentioned about Nusselt’s maximum velocity to average film velocity ratio as 1.5, and Kapitsa’s theory which showed that wave creation starts at low Reynolds numbers Re ≈ 6. The waves produced during the film flow were mathematically modeled by Kheshgi and Scriven [1], demonstrating that whether the wave wavelength is less than or greater than the crucial wavelength determines whether it grows or decays. [1] Orr-Sommerfeld analysis, asymptotic expansion method, and Navier–Stokes systems direct solutions were carried out, showing the generation of spatial and temporal disturbances during the film flow under gravity. The main objective of the present work is to analyse the behaviour of thin film flow while considering all the forces like viscosity, gravity, surface tension, wall shear, and pressure. Two approaches need to be undertaken to define film behaviour. The first approach is the analytical approach, in which mathematical modelling must be done to observe the distribution of forces with wavy film thickness and amplitude of the wave, variation of film velocity profile, and wall shear stress along the plate. The
134
N. Shiva et al.
second approach is the simulation approach, in which the multiphase VOF method present in the CFD package ANSYS Fluent can be used to observe how the film Reynolds number, film thickness, wall shear stress, and velocity profile of the film varies along the plate length. Focusing on these two goals allows for a comprehensive understanding of the velocity profile of the film, the temporal and spatial variations in film thickness, and the impact of different forces on the flow of the film over a vertical plate within the pseudo-laminar region.
3 Materials and Methods The method of simulation approach and analytical modelling that may offer a comprehensive behaviour of film flow over the vertical plate under gravity is described in this part.
3.1 Simulation Modelling 3.1.1
Geometry and Mesh
With ANSYS design modular, geometry for 2-D analysis was produced. In the simulation, the plate length is assumed to be 60 mm, and the domain’s total thickness is 61 mm. One domain area is designated for film, and the other for air. 1 mm is taken as the initial film thickness. After creating a 2-D surface using the sketch tool, the face split option was utilized to split the domain into the air and film zones, as shown in Fig. 1(a). ANSYS ICEM CFD software was used for meshing. Blockstructured quadrilateral grid type was used in the simulation. The mesh grid size controlled by the inflation layer is depicted in Fig. 1(b). Table 1 contains information about mesh. Wall Y + was kept less than 1, and simulation was carried out using a different number of nodes 8000, 32,400, 40,500, 48,600, 72,900 for film-free stream Reynolds number 4000, and with 48,600 total nodes, grid independence was achieved. Similarly, simulation was carried out using a different number of nodes 8000, 24,300, 32,400 for film free stream Reynolds number 2000, and with 24,300 total nodes, grid independence was achieved. For simulation, the scaled residual was used with different convergence criteria 10–3 and 10–6 ; it was observed that governing equations for both cases converge at the same value. Hence, convergence criteria 10–3 are sufficient for converging the solution. Therefore, results of 48,600 and 24,300 nodes with convergence criteria 10–3 for film free stream Reynolds number 2000 and 4000 were taken into account for analysis.
Multiphase Modelling of Thin Film Flow Over the Vertical Plate Under …
(a)
135
(b)
Fig. 1 CFD model for film-free stream Re = 2000 and Re = 4000. (a) Domain and (b) Mesh
Table 1 Mesh details for film free stream Reynolds number 4000 and 2000 Reynolds number 2000
4000
Grid size
0.002 mm (Near wall distance)
0.002 mm (Near wall distance)
Aspect ratio
1–8 (film region)
1–5 (film region)
Growth ratio
1.2
1.2
Mesh control
Inflation layer (First layer thickness) Inflation layer (First layer thickness)
Total nodes
24,300
48,600
Total elements
24,684
49,284
3.1.2
Governing Equations
Both fluids are assumed to be incompressible in the multiphase VOF model. The following is the continuity equation: (− →) ∇. V = 0
(1)
During the film flow, the gravitational force acts in the downward direction, and it makes the liquid thinner during the flow. The viscous force acts upward and tries to make the film thicker. Other force, such as surface tension, also acts at the interface. Hence momentum equation that can be written for the film as follows: ( ) ∂ ρm V→ ∂t
( )] ) [ ( + ∇ · ρm V→ V→ = −∇ P + ∇ · μm ∇ V→ + ∇ V→ T + ρm g→ + F→
(2)
136
N. Shiva et al. 12 ρm κ1 Δα1 ) n = ∇α1 F→ = (σ0.5(ρ 1 +ρ2 ) n nˆ = |n| κ = ∇ · nˆ
} (3)
To track the interface of the two phases, the volume fraction equation derived from the continuity equation neglecting mass transfer is as follows: ( ) ∂αi + ∇ · αi V→ = 0 ∂t ∑2
(4)
αi = 1
(5)
i=1
Mixture density and viscosity for the two phases are calculated in each cell by the equation given: ρm = ρ2 α2 + (1 − α2 )ρ1 μm = μ2 α2 + (1 − α2 )μ1
} (6)
At the interface, shear stress between the two phases remains equal to maintain the continuous behaviour of the interface. Hence the interface equation is given by: μl
| | ∂ Vg || ∂ Vl || = μ g ∂ X | X =δl ∂ X | X =δl
(7)
Non-dimensional parameters used to describe the flow are given by: Re =
˙ (4m) (4Vo δ) = ν μl
(8)
Vo Fr = √ gδ
(9)
Ca =
(μl Vo ) σ
) ( ΔρgL 2c Bo = σ
3.1.3
Model and Schemes
Details of the set-up used in the Fluent are provided in Table 2.
(10)
(11)
Multiphase Modelling of Thin Film Flow Over the Vertical Plate Under …
137
Table 2 Fluent set-up for simulation Solver set
Solver
Pressure based
Space
2-d
Time
Unsteady state
Multiphase model
VOF
Viscous model
k-omega (SST)
Primary phase
Water (Liquid)
Secondary phase
Air
Pressure–velocity coupling
Scheme
PISO
Spatial discretization
Gradient
Least square cell based
Pressure
PRESTO
Momentum
Second order upwind
Volume fraction
Geo-Reconstruct
Turbulent kinetic energy
First order upwind
Specific dissipation rate
First order upwind
Model Phase
3.1.4
Boundary Condition
Details of boundary conditions adopted in this simulation are provided in Fig. 2 and in Table 3.
3.2 Analytical Modelling 3.2.1
Distribution of Force as a Function of Film Thickness
The motion of the given liquid on the vertical surface is governed by the mutual dominance of shear, surface tension, and gravity forces [5]. So, it is essential to study their effects on the dynamic behaviour of the film. This can be done with the help of the order of magnitude analysis. Ca =
Fviscous Fsurface tension
(
δ 2 ρl g = σ
)
⎫ → ρl DDtV = −∇ P + ρl g→ + μl ∇ 2 V→ ⎪ ⎪ ⎪ 2 ⎪ μl ∇ 2 V→ ≈ ρl g→ ⇒ μl ∂∂ XV2 ≈ ρl g ⎬ 2 ρ gδ μl δV2 ≈ ρl g ⇒ V ≈ ( lμl ) ⎪ ⎪ ⎪ δ2 ρ g ⎪ ⎭ Ca = (μσl V ) = ( σ l )
(12)
(13)
138
N. Shiva et al.
Fig. 2 Boundary conditions for film Re = 2000 and Re = 4000
Table 3 Boundary conditions for simulation
Free stream velocity (Liquid)
Case 1
Case 2
0.5 m/s
1 m/s
Velocity inlet (Air)
0.005 m/s
Wall
No slip wall (V = 0)
Symmetry
Zero normal velocity and gradient at a symmetry plane
Pressure Outlet
0 Pa (Gauge Pressure)
) ( ΔρgL 2c Bo = = Fsurface tension σ ( 2 ) δ ρl Fviscous Ca ) = =( Bo Fgravity Δρ L 2c Fgravity
(14)
(15)
Equation 15 shows that film thickness variation depends on the viscous to gravity ratio.
Multiphase Modelling of Thin Film Flow Over the Vertical Plate Under …
3.2.2
139
Distribution of Forces Within the Film as a Function of Change of Amplitude with Time and Space
We can write the bulk modulus elasticity of the given liquid in the form as: k = −V
dP
(16)
dV
V =YX
(17)
dY dX + X Y V ] [ dY dX + −d P = k X Y dV
=
(18) (19)
Figure 3 shows the wavy interface of the film. Assume, X = A sin(ωt − ΩY ) , V =
dA dX = X A
ω ω dY = ⇒ dY = dt dt Ω Ω
(21)
σ Fsurface tension = 2R 2π R 2 [ ] ωdt 2 dA + Fsurface tension = 2kπ R A ΩY [ ] dA ωdt Finertia − Fviscous = 2kπ R 2 + A ΩY −d P =
3.2.3
(20)
(22) (23) (24)
Velocity Profile of the Film
We can approximate the velocity profile within the film as cubic in nature. This can be expressed as shown below: ( ) ( )2 ( )3 X X X VX + a2 = a0 + a1 + a3 Vmax δ δ δ At, X = 0, VX = 0 ⇒ a0 = 0,
At, X = δ,
∂ VX =0 ∂X
(25)
140
N. Shiva et al.
Fig. 3 Wavy nature at the interface of the film
[
] ∂ 2 VX At X = 0, ρl g + μl = 0 At, X = δ, VX = Vmax ∂ X2 ] ] [ [ ρl gδ 2 (a2 + 1) (3 − a2 ) , a3 = a2 = − , a3 = − 2μl Vmax (2) (2) ⎫ ⎧ ( ) ( )3 ( ) ρl gδ 2 ⎪ ⎪ X X ⎪ ⎪ ⎪ ⎪ 3 − − ⎪ ⎪ ⎨ δ δ (2μl Vmax ) ⎬ VX 1 [ ( ) = ( )2 ( )3 ] ⎪ Vmax 2⎪ X X X ⎪ ⎪ ⎪ ⎪ +2 − − ⎪ ⎪ ⎭ ⎩ δ δ δ ∂ VX ρl gδ 2 = 0 , Vmax = ∂X 6μl [( ) ( ) ( ) ] X X 2 1 X 3 ρl gδ 2 − VX = + 2μl δ δ 3 δ {δ Vavg =
VX d X
0
δ
, Vavg =
ρl gδ 2 8μl
Vmax = 1.333 Vavg
3.2.4
Wall Shear Stress Along the Plate
In this section, the wall shear stress equation for the film is obtained.
(26)
(27)
(28)
(29)
(30) (31)
Multiphase Modelling of Thin Film Flow Over the Vertical Plate Under …
141
4ρl Vavg δ μ ( l ) ∂ VX ρl gδ = μl = ∂ X X =0 2
ReY = τwall
Cf = τwall =
τwall 1 ρ V2 2 l avg ρl5 g 3 δ 7 8Re2Y μl4
=
(32)
16ρl2 gδ 3 Re2Y μl2 (33)
4 Results and Discussion Simulation for free stream film Reynolds number 2000 and 4000 were carried out and results were obtained for analysis.
4.1 Grid Independency Test Figure 4a and b show the wall shear stress variation along the plate length. The grid independence test was done for film-free stream Reynolds numbers 2000 and 4000 with different numbers of nodes 24,300, 32,400, 48,600, and 72,900. Wall shear stress is used for the grid independency test because if wall shear stress is computed correctly, the boundary layer of the flow will also be calculated accurately.
(a)
(b)
Fig. 4 Grid independency test for the different number of nodes for film-free stream Reynolds number (a) 2000 and (b) 4000
142
N. Shiva et al.
Fig. 5 Validation of simulation data with Kapitsa [4] Eq. 34 for average film thickness variation with film-free stream Reynolds number
4.2 Validation with Kapitsa Equation The Kapitsa [4] proposed average film thickness Eq. 34 as a function of film free stream Reynolds number for the pseudo-laminar region of the film flowing over a vertical plate under gravity. Simulation data shows a 1.13% average error with Kapitsa Eq. 34, as shown in Fig. 5. [
δavg
2.4ReVo ν 2 = g
]1/3 (34)
4.3 Variation of Film Thickness Along the Plate Length In Fig. 6a and b, there is a bulge initially, and thickness decreases at a later stage. When the film encounters the wall, there are two counter-acting forces: viscous and gravity. Viscosity makes the liquid thicker, and gravity makes the liquid thinner. Initially, wall shear stress is too high; as a result, the viscous effect is more dominant over the gravitational force. As the viscous influence is more prevalent over gravity, film starts to accommodate at some point such that thickness is increased, and continuously mass is increased at that point; as a result, the gravity effect increases, and after some time, it overcomes the viscous effect. At some point, there can be an equilibrium
Multiphase Modelling of Thin Film Flow Over the Vertical Plate Under …
(a)
143
(b)
Fig. 6 Variation of the thickness of the film along plate length for film-free stream Reynolds number (a) 2000 and (b) 4000
state between viscous and gravity force, but as potential energy is maximum, it is a type of unstable equilibrium. Disturbance in an unstable equilibrium is created by capillary pressure due to surface tension force. When the fluid is dominated more by gravity force, film thickness decreases; as a result, mass of continuous film decreases, gravity force reduces after some distance, and again viscous dominates the flow. In this manner, viscous and gravity forces dominate each other throughout the flow, creating waves on the free surface. Equation 15 shows that as the thickness increases, the viscous force increases, and the gravity force decreases. As thickness decreases, gravity force increases, and viscous force decreases.
4.4 Variation of Film Reynolds Number Along the Plate Length In Fig. 7a and b, we can see that in the viscous dominant region mass flow rate of the film increases; hence film Reynolds number increases, whereas in the gravity dominant region, the mass flow rate of the film decreases because of the favourable nature of the gravity force on the film flow along the plate axial direction. As a result, the film Reynolds number decreases. Equation 8, clearly demonstrates this fact.
4.5 Velocity Profile of Film Simulation data in Fig. 8a shows a 1.8% average error with analytical Eq. 31 and a 12% average error with Nusselt [4] velocity ratio of 1.5. Simulation data in Fig. 8b shows a 3% of average error with analytical Eq. 29. In a practical scenario, shear
144
N. Shiva et al.
(a)
(b)
Fig. 7 Variation of film Reynolds number along plate length for film-free stream Reynolds number (a) 2000 and (b) 4000
(a)
(b)
Fig. 8 Comparison of simulation data with (a) Nusselt velocity ratio [4], Analytical Eq. 31, and (b) Analytical Eq. 29 for velocity profile of film-free stream Reynolds number 2000
exerted by air on the film interface in the pseudo-laminar region will be negligible; as a result, simulation data and analytical model slightly vary at the film interface, as illustrated in Fig. 8b.
4.6 Variation of Wall Shear Stress Along the Plate Length Equation 33 shows that wall shear stress decreases along the plate as the film Reynolds number increases. After some distance, inertial and viscous force attains stable equilibrium near the wall and the denominator term is more dominant over the numerator term in Eq. 33; as a result, wall shear stress becomes nearly constant, as shown in Fig. 9.
Multiphase Modelling of Thin Film Flow Over the Vertical Plate Under …
145
Fig. 9 Variation of wall shear stress along plate length for film-free stream Reynolds number 2000
4.7 Variation of Film Thickness with Time This section describes the variation of film thickness with respect to time at two different locations for a particular film free stream Reynolds number. As the initial film is more viscous force dominated, which dampens oscillation, the amplitude doesn’t significantly increase at Y = 10 mm, as illustrated in Fig. 10a and Eq. 24. As demonstrated in Fig. 10b and Eq. 24, the film thickness amplitude increases significantly at Y = 50 mm due to the greater dominance of inertial force.
(a)
(b)
Fig. 10 Variation of film thickness with time at (a) Y = 10 mm and (b) Y = 50 mm for film-free stream Reynolds number 4000
146
N. Shiva et al.
5 Conclusions • The present study shows the unstable behaviour of film over a vertical plate in a pseudo-laminar region. In this work, an analytical expression for the distribution of forces as a function of film thickness and change in amplitude with time and space was developed. The study has additionally developed an analytical model to describe the velocity profile of the film and the corresponding wall shear stress. • A simulation was carried out with the multiphase VOF method of ANSYS Fluent and compared with the developed analytical expression. • Simulation data were validated with the Kapitsa film thickness equation, which shows a 1.13% of average error. • Analytical expression of velocity profile shows 3% of average error with simulation data and ratio of maximum velocity to the average velocity of the film was found nearly 1.333, which can be observed in simulation data with 1.8% of average error but it shows 12% of average error with Nusselt’s value which is 1.5. • An analytical expression for the distribution of forces and simulation data shows that inertial force grows the waves, viscous force damps the waves, and continuous dominancy of viscous and gravity force results in the creation of waves. Acknowledgements The first author would like to thank the second authors for their unwavering support and direction during the Project at IIT Madras.
References 1. Kheshgi HS, Scriven LE (1987) Disturbed film flow on a vertical plate. Phys Fluids 30(4):990– 997 2. Liu WM, Chen COK (2022) Stability analysis of thin power-law fluid film flowing down a moving plane in a vertical direction. Fluids 7(5):167 3. Ackerberg RC (1968) Boundary-layer flow on a vertical plate. Phys Fluids 11(6):1278–1291 4. Portalski S (1964) Velocities in film flow of liquids on vertical plates. Chem Eng Sci 19(8):575– 582 5. Ajaev VS (2012) Interfacial fluid mechanics. Springer, New York
Evolution of Marangoni Thermo-Hydrodynamics Within Evaporating Sessile Droplets Arnov Paul and Purbarun Dhar
Nomenclature V P T D K t ρ μ Lh R βT M c J θ0 γ Ta Va γT a,w (subscript)
Velocity (m/s) Pressure (Pa) Temperature (T) Diffusity of water in air (m2 /s) Thermal conductivity (W/mk) Time (s) Density (kg/m3 ) Viscosity (N.s/m2 ) Latent heat of vaporization (J/kg) Universal gas constant (J/mol.k) Thermal expansion coefficient (K− 1 ) Molecular weight (kg/mol) Concentration (kg/m3 ) Evaporative flux (kg/m2 s) Initial contact angle of droplet (°) Surface tension (N/m) Overall temperature inside droplet (K) Overall velocity inside droplet (m/s) Temperature co-efficient of surface tension (Nm/K) Air and water phase
A. Paul (B) · P. Dhar Department of Mechanical Engineering, Hydrodynamics and Thermal Multiphysics Lab (HTML), Indian Institute of Technology 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_14
147
148
A. Paul and P. Dhar
1 Introduction Evaporation of sessile droplets continuous to be a topic of ongoing research because of its relevance to many natural processes and modern practical applications in the field of heat and mass transfer. Among natural processes, it plays a crucial role in fog, dew and, rain formation mechanism [1, 2]. It is also relevant to other practical applications such as inkjet printing [3], spraying of pesticides [4], anti-icing/ anti-fogging devices in automobile and air crafts [5]. Also, many modern micro scale processes [3, 6] require precise control over deposition pattern of dissolve solute particles which in turn depends on internal advection of liquid molecules during evaporation. Moreover, when the internal flow field is strong enough to induce Stefan flow in the gaseous domain due to shear at the liquid–vapour interface, it may modulate the evaporation process itself as discussed in literature. Hence, an in-depth knowledge about the evaporation mechanism and associated fluid flows of drying droplets under different conditions is of immense importance for efficient designing and development of practical droplet-based devices.
2 Literature Review and Objective In general, based on shape and geometry droplet literature is broadly classified into sessile and pendant droplets. A pendant droplet simply suspends in a gaseous medium, and evaporation takes place uniformly at a rate proportional to droplet diameter [7]. Whereas a sessile droplet, resting over a solid substrate remains in equilibrium with the gaseous phase surrounding it. The evaporative behaviour of such droplets is more complex, influenced by wetting properties of liquid, three-phase interaction [8], ambient conditions [9, 10], geometry [11–13] and thermo-physical properties [14] of the underlying solid substrate. For instance, a sessile droplet evaporates mostly via constant contact radius mode on high surface energy substrates where for the super-hydrophobic (SH) surfaces constant contact angle mode or mixed mode of evaporation is noticed [8]. These different modes of evaporation may significantly affect the internal advection of fluid particles. In case of pinned contact line over hydrophilic surfaces an outward radial flow was reported by Deegan et al. [15]. This flow is mainly generated due to replenishment the liquid that is being removed from the edge of droplet due to higher evaporative flux near the three-phase contact line. This outward flow of liquid plays a crucial role in transporting dissolved solute particles towards the periphery (forming coffee stain rings). In addition to wetting state, the internal advection dynamics of a drying drop is dictated by a number of factors such as direction of heat flow from substrate [16], geometrical parameters [17], and thermal effects [18]. For instance, the altered wetting state on curved surfaces can appreciably modulate the internal flow field depending upon the nature of curvature. For a concave surface the circulation strength is mitigated whereas the same is strongly intensified on convex geometry [17].
Evolution of Marangoni Thermo-Hydrodynamics Within Evaporating …
149
On heated surfaces [18], the temperature distribution inside droplet is substantially modulated due to the influence of evaporative cooling. This tampered temperature profile may generate buoyancy driven advection or thermal Marangoni flow which ultimately modify the magnitude and pattern of the internal circulation velocity. Also, the strength of internal circulation may be appreciably controlled when droplets are seeded with suitable buoyant particles and external non-tactile body forces (such as electric and magnetic fields) are present in the system [19]. This internal hydrodynamics when sufficiently intensified may itself affect the evaporation mechanism. In such cases, the strong circulation velocity generates shear at the interface which ultimately results in displacing the vapour filed layer with the fresh air in the gaseous domain thus augmenting the evaporation rate [19]. The brief literature summarized above substantiates the influential role of wetting state and substrate temperature on the internal advection of drying droplets. The above discussion also put forward the strong dependence of the evaporation mechanism on the internal hydrodynamics. However, only a few works have addressed the internal flow pattern during the evaporation process in transient stage despite the same may occupy a large fraction of the droplet lifetime. For instance, Maatar et al. [20] showed that the transient state accounts for 80% of droplet lifetime in evaporation of 3-methylpentane droplets on heated surfaces while in isothermal conditions it occupies about 33% of the evaporation time [21]. Recently, Chen et al. [22] probed the internal advection pattern during initial transient regime of rapidly evaporating droplets. However, none of the previous studies analysed the role of contact angle on transient evolution of internal circulation strength of an evaporating sessile droplet. In present study, we numerically model the internal flow field and corresponding temperature profile during the transient stage of an evaporating water droplets for different wettability conditions. The droplet volume is considered to be identical while the contact angle is varied over a wide range to mimic the geometry of a sessile droplet over substrates having different surface energies. The governing differential equations for the transient heat and mass transfer phenomenon are numerically solved in fully coupled manner to investigate the evolution of internal flow field and corresponding temperature distribution.
3 Mathematical Formulation 3.1 Governing Equations We consider an axisymmetric water droplet with pinned contact line evaporating in quiescent ambient as shown in Fig. 1. The transient problem is mathematically formulated in axisymmetric r-z coordinate system based on Arbitrary Lagrangian– Eulerian (ALE) framework [23] with the following assumptions: • The fluid flow is laminar and incompressible.
150
A. Paul and P. Dhar
Fig. 1 Schematic and meshing of the computation domain for droplet-ambient system in r-z coordinate
• Droplet contact line is pinned. • The thermo-physical properties of fluids are constant unless explicitly specified such as temperature variation of density and surface tension. Under these conditions the continuity and momentum eqn. in liquid and gas phase is given as: ∇.V = 0 ρ
∂V + (Vc .∇)V ∂t
= ∇ − p I + μ ∇V + (∇V )T + Fg
(1) (2)
where ρ, μ, p, t, g are density, viscosity, pressure, time and gravitational constant, V c is the convection velocity defined as difference between mesh velocity and material velocity in ALE framework, Fg = (ρr + Δρ)g is the gravity force that depends on change in density brought about by change in temperature in liquid phase: Δρ = βT ρw,r (T − Tr ), where β T is thermal expansion coefficient, T is the temperature, and subscript w and r denotes water phase and reference state. In present study, the ambient condition is considered as the reference state. For gaseous phase, density is a function of both temperature and vapour concentration given as:
Evolution of Marangoni Thermo-Hydrodynamics Within Evaporating …
Δρ =
pa,r Ma pa Ma + cMw − + cr Mw RT RTr
151
(3)
where pa = po -cRT is partial pressure of air, po is total atmospheric pressure, c is the molar vapour concentration, R is the universal gas constant and M a and M w is molar mass of air and water respectively. To model the natural convection phenomenon, Boussinesq approximation is used assuming change in density does not tamper the incompressible flow assumption [24]. The mass diffusion at liquid–vapour interface is govern by convection–diffusion eqn. given as: ∂c + (Vc .∇)c = ∇[D∇c] + S ∂t
(4)
where D is the vapour diffusion coefficient and S is source term for the concentration. In the present study, S = 0. The heat transport eqn. in liquid and gas phase is given as:
∂T + (Vc .∇)T ρ ∂t
= ∇[k∇T ]
(5)
where C p is specific heat and k is the thermal conductivity.
3.2 Initial and Boundary Conditions We apply constant temperature boundary condition at solid–liquid interface and at far ambient denoted as T s and T a respectively. In present study we consider T s = T a = 15 °C. The liquid–vapour interface is in saturated vapour condition (C s ) while the far field ambient has a relative humidity Φ. We apply no slip boundary condition for velocity and no penetration condition for vapour across the solid–liquid interface for liquid domain and gaseous domain respectively. The initial numerical values of velocity, temperature and relative humidity in present study is consider as 0 m/s, 15 °C, and 50% respectively. Next, we apply stress balance in normal and tangential direction across the liquid– vapour interface: / n · σa − n→ · σw ) · n→ = γ ro · n→ (→
(6)
n · σa − n→ · σw ) · t→ = −γT ∇g T (→
(7)
where ∇g = (I − n→ · n→ T )∇ is the surface gradient operator, γ , σ, t→, n→, r o , γ T are the surface tension, stress tensor, unit vector along tangential and normal direction, overall curvature radius and temperature coefficient of surface tension respectively.
152
A. Paul and P. Dhar
Next, we apply the mass balance across liquid–vapour interface since the same cannot store any mass given as: 1 1 Vg − Vl = J − ρw ρa J = n→ · −D∇c + Vg · c Ml
(8) (9)
where V g , V l is the velocity in gas and liquid phase and J is evaporative mass flux. This evaporated liquid mass takes away latent heat of vaporization. As a result, the liquid–vapour interface acts as heat sink for the internal liquid given as: n k g (∇T )g − kl (∇T )l = −J L h
(10)
where L h is the latent heat of vaporization.
3.3 Numerical Scheme The governing equations along with the boundary conditions are numerically solved in a fully coupled manner using commercial software COMSOL Multiphysics 4.4 that provides iterative solution based on finite element method. The discretization in the finite element formulation was done using Lagrange linear shape elements for both velocity and pressure fields. The meshing operation was performed using triangular mesh with the finest mesh size (about 200 times finer than contact radius) being located near the liquid–vapour interface and at the contact line (see Fig. 1). These considerations ensure accuracy in solving heat and mass transport equations and surface tension gradient. The computation domain is considered to have a radius of 100 times the initial radius of droplet as shown in Fig. 1. Moreover, an automatic remeshing operation was adopted to reduce the mesh distortion during evaporation process. In present study, the minimum mesh quality was set to 0.2. The tolerance value was set at 0.001 for all variables. For the grid independent test, we vary the grid numbers from 19,067 to 33,167 and the change in evaporation rate was less than 0.08%. The validation of the above model is done by comparing present results with the experimental observations of Paul et al. [11] as shown in Fig. 2. We simulate the droplet evaporation problem following two approaches: (i) No internal advection: we neglect the advection of fluid in the mass transport equation and hence the process is considered as diffusion driven evaporation; (ii) With internal advection (IA): we take into account the transport of fluid particles due to buoyancy effects and Marangoni advection. It is noticed that, the advection model predicts the instantaneous droplet volume more accurately while the no advection model under predicts the evaporation rate.
Evolution of Marangoni Thermo-Hydrodynamics Within Evaporating …
153
Fig. 2 Comparison of present numerical model with the experimental results on transient droplet volume
4 Results and Discussion 4.1 Temporal Variation of Internal Velocity Field and Temperature Distribution Figure 3 illustrates the spatial distribution of the internal circulation velocity and corresponding thermal contours inside a sessile droplet at different time instants. It is observed that, on hydrophilic surfaces (initial contact angle, θ 0 = 30°), multiple vortices (multi-vortex pattern) are formed during initial transient regime of evaporation before the internal flow pattern becomes time invariant. In this stage, the total number of vortices firstly increases to maximum value at t = 4 s, then reduces to a stable value (single vortex) at latter stages. The final stable vortex consists of a large recirculating flow entraining liquid particles from the peripheral region to the apex along liquid–vapour interface. These findings behind the dynamic evolution of the circulation flow may be attributed the internal thermal imbalance generated due to evaporation of droplets. The diffusion model predicts the evaporative flux to be maximum near the contact line for a sessile droplet evaporating over hydrophilic surfaces [15]. As a result, the liquid–vapour interface at peripheral region is cooled to a greater extent at the very beginning of evaporation (t = 0.3 s). This thermal disparity due to evaporative cooling may cause thermal Marangoni flow and buoyancy driven advection. However, the convection patterns due to these two mechanisms are opposite to each other. Initially, the relatively cold fluid at liquid–vapour interface descends towards the periphery while the hotter fluid from the droplet bulk moves upward due to buoyancy. This condition triggers interfacial advection of fluid from to the apex to periphery (as observed at t = 0.3 s). However, as evaporation progress, the temperature gradient at the liquid–vapour interface sets off Marangoni advection that entrains liquid from the peripheral region to the apex. These two counteracting flow mechanisms are responsible for generation of vortex flows during the transient regime of evaporation. With progressing time, the non-uniform evaporative flux along
154
A. Paul and P. Dhar
Fig. 3 Transient evolution of the internal velocity field and corresponding temperature distribution of a sessile droplet evaporating over hydrophilic surfaces (θ 0 = 30°)
the internal convective flow cause multiple local maxima in interfacial temperature. These temperature extrema at different locations give rise to local thermal Marangoni cells which ultimately result in formation of multiple vortices as observed at t = 4 s in Fig. 3. This local distribution of vortices is advantageous for bulk motion of liquid and hence felicitate the mixing process at initial transient regime. Finally, at latter stages (t = 10 s) the temperature distribution becomes monotonically decreasing along liquid–vapour interface from the contact line to central region. As a result, the Marangoni convection becomes unidirectional along decreasing temperature thus forming single recirculation flow (one vortex pattern) inside the droplet. These observations at this stable condition are in line with the previous studies on internal circulation behaviour of droplets at steady state [16]. Also, it is noticed that the internal temperature at the stable state (t = 10 s) is unevenly distributed at different vertical and horizontal locations with the colder fluid mass being at the top central region. Hence, buoyancy-driven advection may have significant role in the overall internal circulation flow in addition to Marangoni convection. Figure 4 shows the transient variation of internal velocity field and corresponding temperature distribution of an evaporating sessile droplet over SH surfaces (initial contact angle, θ 0 = 150°). It can be noticed that at very beginning (i.e. when the convection current is negligible) the temperature at the apex region is reduced to a
Evolution of Marangoni Thermo-Hydrodynamics Within Evaporating …
155
greater extent due to large evaporative flux resulting from diffusion driven evaporation [15]. This cooling effect remains localized at the central core and thus induces liquid motion in the central vertical column due to buoyancy effects (as observed at t = 0.5 s). However, the effect of thermal Marangoni flow during this initial regime is relatively small due to little variation in temperature along liquid–vapour interface (as seen in t = 0.3 and 0.5 s). As a result, the buoyancy driven advection becomes dominant and the fluid particles descend downward forming a single vortex flow with maximum velocity being localized at the central core. As the evaporation process progress, the buoyancy driven convection elevate the thermal diffusion process inside the liquid droplet. As a result, the temperature distribution becomes more uniform at intermediate stages (as observed at t = 1.1 s). This phenomenon ultimately alleviates the driving gradient for buoyancy mechanism. Also, the interfacial temperature gradient at this stage is too weak to stimulate Marangoni advection. As a result, the strength of internal circulation velocity at this intermediate stage (t = 1.1 s) is greatly reduced compared to early stages. Also, it can be mentioned that the continuous evaporative cooling along with the internal advection gradually leads to nearly uniform temperature distribution inside the droplet with the progressing evaporation time. In such cases, although the buoyancy effects become negligible, steep thermal gradient is induced at liquid–vapour
Fig. 4 Temporal evolution of the velocity field and corresponding temperature distribution inside a sessile droplet evaporating over SH surfaces (θ 0 = 150°)
156
A. Paul and P. Dhar
interface near the peripheral region (as noticed at t = 15 s). This condition ultimately set off thermal Marangoni advection thus forming single recirculation flow (one vortex pattern) inside the droplet.
4.2 Influence of Contact Angle on the Internal Circulation Velocity and Temperature In this subsection, we portray the transient thermal response and internal circulation velocity of the evaporating droplets as a whole for different contact angles. Figure 5 illustrates the role of initial contact angle on the temporal evolution of overall temperature (T a ) and internal circulation velocity (V a ) of evaporating droplets. It can be noticed that there are several peaks and valleys in the velocity plot on hydrophilic surfaces. This is mainly due to the formation and evolution of localized circulation cells at different locations. On SH surfaces, the circulation velocity initially reaches to a local peak value followed by a step decrease at the intermediate stage (t = 1.1 s) and then gradually increases to a stable value with the progressing evaporation time. This behaviour may be attributed to the initial dominance of buoyancy force followed by alleviation of same as discussed earlier. Finally, at successive stages, thermal gradient near the contact line becomes significant thereby manifests higher velocity due to Marangoni flow. Also, it is noted that the velocity scale is increased in a consistent manner with the increase in contact angle. On SH surfaces the overall circulation velocity becomes almost an order of magnitude higher as compared to the hydrophilic surfaces. The overall temperature (T a ) plot shows consistently decreasing trend at initial stages and finally reaches stable state with progressing time (see Fig. 5) for different contact angles of the evaporating droplets. This stable state is reached at much early stages for the hydrophilic surfaces compared to the SH surface. This is mainly due to large heat conduction path inside droplet on SH surfaces which delays stable temperature distribution inside the droplet. Also, the overall stable temperature for the SH surface is much lower than the hydrophilic case. This is mainly due to altered geometry of the droplet over these surfaces. For the hydrophilic surface, the large solid– liquid interfacial area and small droplet height (i.e., conduction path) cause higher overall temperature inside the droplet. Now for an identical volume the droplet placed on SH surfaces the maximum height of liquid column is increased with a simultaneous decrease in solid–liquid contact area. These conditions ultimately reduce the overall stable temperature of the bulk fluid inside droplet.
Evolution of Marangoni Thermo-Hydrodynamics Within Evaporating …
157
Fig. 5 Transient evolution of the overall internal circulation velocity and temperature variation inside sessile droplets for different contact angles
5 Conclusions In present study we numerically probe the temporal evolution of internal hydrodynamics and temperature distribution profile of drying sessile droplets. A transient evaporation model based on ALE framework is employed for this purpose. We vary contact angle and other geometrical parameters keeping the volume of droplet constant to understand the influence of wettability of underlying substrate. The key insights are given as: • On hydrophilic surfaces multiple vortices are formed at the transient stage due to non-uniform evaporative flux across liquid–vapour interface. This cause uneven evaporative cooling which set off thermal Marangoni and buoyancy driven advection inside droplet. Eventually, as temperature distribution across liquid–vapor interface becomes monotonic a single large recirculation vortex is formed from the peripheral region to the apex. • For SH surfaces, the buoyancy effect becomes dominant at initial stages that cause single large recirculating flow encompassing the whole liquid domain inside the droplet. Finally at latter stages, as the temperature field reaches a nearly uniform state the Marangoni effect predominates over buoyancy forces.
158
A. Paul and P. Dhar
• The overall temperature inside the droplet is significantly lower for SH surface due to large conduction path. This condition cause strong thermal gradient across the droplet interface near periphery that set off Marangoni advection. As result, the velocity scale is obtained as an order of magnitude higher for the SH surfaces as compared to hydrophilic surfaces.
References 1. Tardif R, Rasmussen RM (2010) Evaporation of nonequilibrium raindrops as a fog formation mechanism. J Atmos Sci 67(2):345–364 2. Ucar IO, Erbil HY (2012) Use of diffusion controlled drop evaporation equations for dropwise condensation during dew formation and effect of neighboring droplets. Colloids Surf, A 411:60–68 3. Park J, Moon J (2006) Control of colloidal particle deposit patterns within picoliter droplets ejected by ink-jet printing. Langmuir 22(8):3506–3513 4. Yu Y, Zhu H, Frantz JM, Reding ME, Chan KC, Ozkan HE (2009) Evaporation and coverage area of pesticide droplets on hairy and waxy leaves. Biosyst Eng 104(3):324–334 5. Lee S, Kim DI, Kim YY, Park SE, Choi G, Kim Y, Kim HJ (2017) Droplet evaporation characteristics on transparent heaters with different wettabilities. RSC Adv 7(72):45274–45279 6. Karlsson S, Rasmuson A, Björn IN, Schantz S (2011) Characterization and mathematical modelling of single fluidised particle coating. Powder Technol 207(1–3):245–256 7. Picknett RG, Bexon R (1977) The evaporation of sessile or pendant drops in still air. J Colloid Interface Sci 61(2):336–350 8. Bormashenko E, Musin A, Zinigrad M (2011) Evaporation of droplets on strongly and weakly pinning surfaces and dynamics of the triple line. Colloids Surf, A 385(1–3):235–240 9. Sefiane K, Wilson SK, David S, Dunn GJ, Duffy BR (2009) On the effect of the atmosphere on the evaporation of sessile droplets of water. Phys Fluids 21(6):062101 10. Paul A, Dhar P (2022) Interactive evaporation of neighboring pendant and sessile droplet pair. J Heat Transfer 144(12):121603 11. Paul A, Khurana G, Dhar P (2021) Substrate concavity influenced evaporation mechanisms of sessile droplets. Phys Fluids 33(8):082003 12. Paul A, Dhar P (2021) Evaporation kinetics of sessile droplets morphed by substrate curvature. Phys Fluids 33(12):122010 13. Paul A, Dash RK, Dhar P (2023) Phenomenology and kinetics of sessile droplet evaporation on convex contours. Int J Therm Sci 187:108194 14. Dunn GJ, Wilson SK, Duffy BR, David S, Sefiane K (2009) The strong influence of substrate conductivity on droplet evaporation. J Fluid Mech 623:329–351 15. Deegan RD, Bakajin O, Dupont TF, Huber G, Nagel SR, Witten TA (1997) Capillary flow as the cause of ring stains from dried liquid drops. Nature 389(6653):827–829 16. Al-Sharafi A, Yilbas BS, Sahin AZ, Ali H, Al-Qahtani H (2016) Heat transfer characteristics and internal fluidity of a sessile droplet on hydrophilic and hydrophobic surfaces. Appl Therm Eng 108 17. Dhar P, Khurana G, Anilakkad Raman H, Jaiswal V (2019) Superhydrophobic surface curvature dependence of internal advection dynamics within sessile droplets. Langmuir 35(6):2326–2333 18. Kumar S, Medale M, Brutin D (2022) Numerical model for sessile drop evaporation on heated substrate under microgravity. Int J Heat Mass Transf 195:123150 19. Dhar P, Jaiswal V, Chate H, Maganti LS (2020) Control and modulation of droplet vaporization rates via competing ferro-and electro-hydrodynamics. arXiv preprint arXiv:2006.00750 20. Maatar A, Chikh S, Ait Saada M, Tadrist L (2015) Transient effects on sessile droplet evaporation of volatile liquids. Int J Heat Mass Transf 86:212–220
Evolution of Marangoni Thermo-Hydrodynamics Within Evaporating …
159
21. Chen YH, Hu WN, Wang J, Hong FJ, Cheng P (2017) Transient effects and mass convection in sessile droplet evaporation: the role of liquid and substrate thermophysical properties. Int J Heat Mass Transf 108:2072–2087 22. Chen Y, Hong F, Cheng P (2020) Transient flow patterns in an evaporating sessile drop: a numerical study on the effect of volatility and contact angle. Int Commun Heat Mass Transfer 112:104493 23. Yang K, Hong F, Cheng P (2014) A s coupled numerical simulation of sessile droplet evaporation using arbitrary Lagrangian-Eulerian formulation. Int J Heat Mass Transf 70:409–420 24. Sobac B, Brutin D (2012) Thermal effects of the substrate on water droplet evaporation. Phys Rev E Stat Nonlinear Soft Matter Phys 86(2 Pt 1):021602
Investigation of Liquid Vaporization Characteristics at Low–Pressure Conditions Sarvjeet Singh, Jaydip Basak, Prodyut Chakraborty, and Hardik Kothadia
Nomenclature A Ps P Pv Pv,0 Tw Tw,0 Te ΔT mw,v Vw Vw,0 t cp hf,g h ρ NEF
Area of flash chamber (m2 ) Saturation pressure (kPa) Flash chamber pressure (kPa) Vacuum tank pressure (kPa) Initial Vacuum tank pressure (kPa) Water temperature (°C) Initial Water temperature (°C) Equilibrium temperature (°C) Degree of superheat (°C) Water mass vaporized (g) Water volume (L) Initial Water volume (L) Time (s) Specific heat (kJ/kgK) Latent heat of the vapor (kJ/kg) Volumetric heat transfer coefficient (kW/m3 K) Water density (kg/m3 ) Non equilibrium fraction
S. Singh · J. Basak · P. Chakraborty · H. Kothadia (B) Department of Mechanical Engineering, IIT Jodhpur, Jodhpur, Rajasthan 342030, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_15
161
162
S. Singh et al.
1 Introduction Flash evaporation is identified as a rapid cooling technique when a liquid is subjected to an abrupt drop in its pressure below its saturation point. The liquid under the mentioned situation quickly vaporizes, resulting in a significant temperature drop, which provides the cooling effect. Moreover, the flash evaporation cooling method is commonly adopted practice for a wide variety of science and engineering applications. A primary method of cooling, which has various practical uses in everyday life, includes seawater desalination, recovering waste heat, drying food, and treating contaminated waste like nuclear waste and sludge [1–3]. Moreover, vacuum flash spray cooling is an important technology in aerospace thermal control that relies on liquid flash evaporation to achieve rapid equipment cooling [4]. Most researchers are interested in the flashing of the water pool and spray. Miyatake et al. [5, 6] conducted their study on the liquid pool at low pressure conditions. They call this process static flash evaporation. They used a non-equilibrium fraction (NEF) which describes variation in the water film temperature and can be used to determine the completion of the flash process in an experiment. They also introduced the coefficient defining the flash evaporation rate. Gopalakrishnan et al. [7] defined dimensionless factors which affect rates of flash evaporation in liquid pools and experimental data. These non-dimensional parameters are the Prandtl number, hydrostatic head, and Jakob number. Peterson et al. [8] investigated flash evaporation in liquid pools that were quickly depressurized. Their experimental analysis is done by using water and Freon-11 as flashing liquid. No bubble formation occurs at low superheat, but violent bubble formation occurs at higher superheat. Zhang et al. [9] examined the effect of steam carrying in static flashing phenomena considering the NaCl solution and fresh- water as a liquid medium. The study concluded that there is an increase in the steam carrying ratio with the increasing value of vacuum and the initial pool temperature. Saury et al. [10] investigated the effects of depressurization rate and water film height on flashed mass and NEF ascent in an experimental analysis of flash evaporation. The influence of non-isothermal flashing droplet characteristics on vacuum flash spray cooling was numerically simulated and experimentally investigated by Cheng et al. [11]. Isao et al. [12] developed a theoretical model that can predict the change in droplet temperature when there is a sudden drop in pressure. Sun et al. [13] proposed a phase change model and applied it to simulate evaporation and condensation problems with volume-of-fluid (VOF) method in the FLUENT code. The process of flashing leads to a higher rate of vaporization flow compared to simple evaporation [14]. In order to use it wisely, a deeper grasp of it is required. With this aim and objective in mind, experimental apparatus was put together along with analytical work in our laboratory, in order to develop a model and analyze the changes in a number of different factors such as NEF, evaporated mass, and heat transfer coefficient. The results of this study will contribute to providing enriched knowledge about the mechanism of flash evaporation and help in the development of control technologies for rapid cooling.
Investigation of Liquid Vaporization Characteristics at Low–Pressure …
163
2 Experimental Setup and Procedure The complete experimental setup with all the components is shown in Fig. 1. In brief, the low-pressure vaporization setup consists of a custom fabricated cuboidal flashing chamber having a length and breadth of 0.135 m, a height of 0.23 m and is sealed at each side with 20 mm thick insulated foam. Also, a vacuum tank is used to maintain the low pressure during the experiment. Both tanks are connected to each other with the help of a ball valve and pipes. A heating source is utilized to maintain the temperature up to desired condition. The temperature and the pressure of the system is measured with the help of K-type thermocouples and pressure transducers. The inlet and the drain valves are provided in the flash chamber to fill the water and drain after the experiment. The flash chamber is filled again after each experiment. An electronic weighing machine has been opted to measure the mass of evaporated liquid. A data acquisition system is used to record all the variations of transient variables during the process. Before performing each experiment, the flash chamber is filled to the desired level of liquid height or volume and then heated up to a certain temperature. Meanwhile, vacuum tank is evacuated using a vacuum pump to a certain vacuum pressure. All the sensing instruments are properly checked and calibrated to avoid all types of errors in the results. After achieving all the initial conditions, a rapid connection has been made between the tanks by opening the ball valve. A rapid pressure drop is taking place inside flash chamber which was initially at atmospheric condition. The whole liquid came into a superheat metastable state and tried to regain the stable state. Conversion of liquid to vapors takes place following the primary bulk boiling phenomenon. Thus, the flashing process takes place. There is heat and mass transfer taking place during the process until it regains the equilibrium state. The readings are taken after the system obtains a stable state. All the experiments are repeated multiple times to check the authenticity of the low vaporization process.
3 Performance Characteristics The cumulative mass of vaporized water mw,v at a given time instant is calculated using an energy balance on water inside the flash tank as Eq. 1. Junjie [15] proposed a formulation to evaluate the heat transfer coefficient during the flashing phenomenon. ' ' Here h is defined as the overall intensity of flash heat transfer in flash duration time. m w,v (t) =
m w (t)C p (Tw (t − δt) − Tw (t)) + m w,v (t − δt) h fg
(1)
T (t) − Te T (t) − Te = Tw,o − Te ΔT
(2)
N E F(t) =
h = ρC p
1 − N E F(t) t
(3)
164
S. Singh et al.
Fig. 1 Photograph of the experimental setup
Vacuum
Vacuum
Transmitter
4 Results and Discussion In this chapter, we performed an experimental study to understand the phase change characteristics of the liquid when exposed to a low-pressure condition which also called as a flashing condition. For this purpose, a fully insulated cuboidal tank is fabricated. Distilled water is used as the liquid medium during the experiments. Figure 2 (a and b) illustrates the variation of pressures (Flash chamber, saturation, and vacuum tank) which are initially at P = 101.33 kPa, Pv,0 = 6.33 kPa and 20.63 kPa, T w,0 = 85 °C and V w,0 = 1L. When the ball valve between the two tanks is abruptly opened, the pressure in the flash chamber is drastically reduced. The vacuum tank’s pressure is simultaneously disturbed and reduced until an equilibrium level is reached in both tanks. The water in the flash tank reaches a superheated condition because of the fast pressure change inside the flash chamber, which causes the flashing phenomenon and causes the water pool’s temperature Tw to change over time. For each experiment, the time variation of the water temperature Tw and of the different pressures (P and Pv ) within the flash tank were measured. Antoine equation [16] is used to calculate the saturation pressure from the water temperature. When a liquid is exposed to a low-pressure condition at some initial temperature of liquid than the liquid gains a new state higher than its saturation state. Figure 3 shows the evolution of saturation temperature (T s ) and water pool instantaneous temperature T (t) with time. It is clear from the figure that there is a decrease in the magnitude of the water temperature taking place at the initial stage of the flashing. After some instances, water temperature T (t) coincides with the saturation temperature (Ts)
Investigation of Liquid Vaporization Characteristics at Low–Pressure …
165
Fig. 2 Evolution of different pressures
because both the tanks pressure come in equilibrium. When the initial temperature and the saturation temperature, which correspond to the equilibrium pressure Pe, are different temperatures, superheat (ΔT ) results.
166
S. Singh et al.
Fig. 3 Evolution of water film temperature and key parameters
4.1 Temperature Characteristics In the present section, the influence of the different parameters on pool temperature is shown. Figure 4a and b shows the evolution of the water pool temperature for four different values of the initial temperature at 1L of water volume and two different values of vacuum tank pressure (6.33 kPa and 20.63 kPa). It has been clearly observed that the initial temperature has far less of an impact on temperature evolution than vacuum tank pressure. It is attributed to the fact that at low value of vacuum tank pressure and high initial water temperature impose a large degree of superheat. Due to larger superheat, a larger liquid- vapor phase change takes place and flashing phenomenon happens very violently. At the free surface of the liquid a complete disruption occurs, and the vapor bubble formation takes place throughout the whole volume of water. At the initial period, the water temperature dropped quickly, and bubbles generated violently in the liquid. Later, the temperature drop slowed down to nearly zero. The temperature trend agrees with that of others’ research such Miyatake et al. [5, 6], Saury et al. [10].
4.2 Mass Evaporated (mw,v ) In this section, we have presented the statistics of the mass of water vaporized due to the low-pressure exposure of the water pool. There is a phase change taking place which results in the production of the vapors. This is one of the characteristics of the low pressure phenomenon to enhance the vaporization rate. Figure 5a and b
Investigation of Liquid Vaporization Characteristics at Low–Pressure …
167
Fig. 4 Water temperature evolution for several initial vacuum tank pressures
presents the vapors production at different value of initial vacuum tank pressure Pv,0. Equation 1 is used to calculate the magnitude of water vapor generation. From Fig. 5a and b, it is concluded that the water vapor production takes place in a logarithmic manner irrespective of initial conditions. It is also observed that the amount of water vaporized significantly depends on the magnitude of the initial vacuum tank pressure and the initial temperature of the liquid pool. To avoid the double dependence of mass of water vaporized on parameters. The behavior of mass of water vaporized can be explained with a single parameter i.e., degree of superheat.
168
S. Singh et al.
Fig. 5 Evolution of the mass of water vaporized for several initial vacuum tank pressure
Investigation of Liquid Vaporization Characteristics at Low–Pressure …
169
4.3 Nonequilibrium Fraction (NEF) In this section, we have presented the dimensionless parameter i.e., NEF to reduce the dependency on the dimensional parameters. Miyatake et al. [5, 6] presented this dimensionless number. It is used to determine the degree of flash evaporation completion. Figure 6a and b shows the variation of NEF using Eq. 2 along time for varying initial vacuum tank pressure. The exponential decay of the NEF is obtained in all the cases. The NEF decreases over time and comes to a steady state after achieving the equilibrium pressure. A small value of NEF indicates the higher amount of vapor produced in the flashing. So, this parameter is used to know the completeness of the process. It is evident from the figures that at a higher value of superheat and lower initial value of vacuum tank pressure results in low value of NEF Which contribute a large drastically in the flashing process.
4.4 Heat Transfer Coefficient (h) During the flash evaporation of the water pool full bulk of the liquid take part in the flash not only the steam-liquid part so the volumetric heat transfer coefficient “h” has been considered. Heat transfer coefficient during transient flash evaporation is calculated by using Eq. 3. According to equation, heat transfer coefficient is presented in Fig. 7 for different set of the experiments. As the flashing begins, value of the “h” reaches peak and it gradually drops down with the time increase. This is attributed to the uneven descent of the non-equilibrium value. Results of Fig. 7a and b also suggests that a higher value initial vacuum tank pressure can weaken the intensity of flashing heat transfer. But as time increases, the influence of initial water film superheat on the average intensity of flashing heat transfer during flash becomes weaker. Moreover, it can also be predicted from Fig. 7a and b at a high value of ΔT a quick increase in heat transfer coefficient is obtained and it drops with the decrease in the superheat.
5 Conclusions In this chapter, we have presented an experimental study the influence of the initial vacuum tank pressure and initial temperature of the liquid at constant volume of the liquid is analyzed on different thermo-properties. The utmost significant inferences associated with this investigation are as follows: • The dynamic process of rapid evaporation at low pressure is analyzed. The trend of the obtained results is similar to the results of the previous work in this area. Magnitude-wise the results are not similar due to the unlike setup design, process parameters and the environmental conditions.
170
Fig. 6 NEF versus time under different initial vacuum tank pressure
S. Singh et al.
Investigation of Liquid Vaporization Characteristics at Low–Pressure …
Fig. 7 Heat transfer coefficient evolution calculated by Eq. (3)
171
172
S. Singh et al.
• When the flash starts, the pressure of the flash tank starts to drop in an exponential manner and adjacently the pressure inside the vacuum tank increases but after some both came in equilibrium. Pressure. • Superheat is the main driving force behind the flashing phenomenon. The value of NEF lets us determine the completion of the flashing process and phase change. • A logarithmic trend of mass evaporation is observed during flashing, and it is greatly affected by the pressure conditions. • Heat transfer coefficient during flashing is found to be high at the beginning of the flashing and gradually decreases with time. • The current study is helpful in the various industrial applications like nuclear plants, desalination, and waste heat recovery.
References 1. Aoki I (1995) Water flash evaporation under low pressure conditions. Prev Heat Mass Transf 6(21):518 2. Bartak J (1990) A study of the rapid depressurization of hot water and the dynamics of vapour bubble generation in superheated water. Int J Multiph Flow 16(5):789–798 3. Muthunayagam A, Ramamurthi K, Paden J (2005) Low temperature flash vaporization for desalination. Desalination 180(1–3):25–32 4. Wang J-X, Li Y-Z, Zhang H-S, Wang S-N, Mao Y-F, Zhang Y-N, Liang Y-H (2015) Investigation of a spray cooling system with two nozzles for space application. Appl Therm Eng 89:115–124 5. Miyatake O, Murakami K, Kawata Y, Fujii T (1973) Fundamental experiments with flash evaporation. Heat Transfer Jpn Res 2(4):89–100 6. Miyatake O, Fujii T, Hashimoto T (1977) An experimental study of multi-stage flash evaporation phenomena. Heat Transfer Jpn Res 6(2):25–35 7. Gopalakrishna S, Purushothaman V, Lior N (1987) An experimental study of flash evaporation from liquid pools. Desalination 65:139–151 8. Peterson R, Grewal S, El-Wakil M (1984) Investigations of liquid flashing and evaporation due to sudden depressurization. Int J Heat Mass Transf 27(2):301–310 9. Zhang D, Chong D, Yan J, Zhang Y (2012) Study on steam-carrying effect in static flash evaporation. Int J Heat Mass Transf 55(17–18):4487–4497 10. Saury D, Harmand S, Siroux M (2002) Experimental study of flash evaporation of a water film. Int J Heat Mass Transf 45(16):3447–3457 11. Cheng W-L, Peng Y-H, Chen H, Hu L, Hu H-P (2016) Experimental investigation on the heat transfer characteristics of vacuum spray flash evaporation cooling. Int J Heat Mass Transf 102:233–240 12. Satoh I, Fushinobu K, Hashimoto Y (2002) Freezing of a water droplet due to evaporation—heat transfer dominating the evaporation–freezing phenomena and the effect of boiling on freezing characteristics. Int J Refrig 25(2):226–234 13. Sun D, Xu J, Chen Q (2014) Modeling of the evaporation and condensation phase-change problems with fluent. Numer Heat Transf, Part B: Fundamentals 66(4):326–342 14. Singh S, Chakraborty PR, Kothadia HB (2023) Experimental study on energy transformation of static liquid pool during flash evaporation. Appl Therm Eng 220:119712 15. Junjie Y, Dan Z, Daotong C, Guifang W, Luning L (2010) Experimental study on static/ circulatory flash evaporation. Int J Heat Mass Transf 53(23–24):5528–5535
Effect of Surface Structures on Droplet Impact Over Flat and Cylindrical Surfaces Saptaparna Patra, Avik Saha, and Arup Kumar Das
1 Introduction The lotus leaf has unique superhydrophobic and self-cleaning characteristics [1]. This attribute of the lotus leaf has subsequently been used to fabricate artificial surfaces which can display similar superhydrophobic and self-cleaning properties. Thus, the wetting of superhydrophobic surfaces has attracted a lot of research work in recent years. Researchers have fabricated many superhydrophobic surfaces with contact angles greater than 150° [2]. The chemical and physical properties of the solid substrate have an important role in deciding the droplets’ behavior after the impact [3]. Introducing textures on the substrate can also influence the movement of the droplet [4]. With the advent of new manufacturing technologies and microfabrication, it has also become possible to fabricate complex superhydrophobic substrate structures [5]. Droplet dynamics are largely governed by an interplay between the inertia effects during the initial impact, followed by which the viscous and surface tension forces become more dominant. This interplay continues till equilibrium is achieved between these forces [6]. Generally, the superhydrophobic property of structured surfaces can be attributed to the air trapped in between the structures, which reduces the inertia effect of the droplet during impact. However, when the droplet impacts with a high enough energy, it can remove the trapped air, neutralizing the superhydrophobicity to some extent [7]. Due to the wide-ranging occurrences and applications of droplet impact in nature, it becomes essential to be able to model the dynamics optimally. One of the most common droplet impact studies applications is spray cooling, which allows high heat transfer rates, and is predominant in the electronics and semiconductor industries [8]. S. Patra Department of Mechanical Engineering, NIT Durgapur, Durgapur 713209, India A. Saha (B) · A. K. Das Department of Mechanical and Industrial Engineering, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_17
173
174
S. Patra et al.
The anti-icing properties of the superhydrophobic surfaces can also be utilized in colder climates, especially for power transmission equipment and aerospace industries [9–11]. Some other commonly encountered applications of droplet dynamics include wetting during fuel injection [12, 13], inkjet printing [14], anti-corrosion [15], pesticide spraying [16], and many more.
2 Literature Review and Objective Two main wetting phenomena are observed during droplet impact on structured surfaces. The droplet penetrates the entire structured geometry by expelling the air, which is the Wenzel State [17]. When the droplet rests on the structured surface, not wholly penetrating the gaps, it is known as the Cassie–Baxter state [18]. Bhardwaj et al. experimentally studied the droplet impact hydrodynamics on flat and micropillar hydrophobic surfaces and observed non-bouncing, partial bouncing, and complete bouncing as possible outcomes and the transition from the Cassie–Baxter state to the Wenzel State. [19]. Many researchers have experimentally studied droplet impact on superhydrophobic flat and curved surfaces. It was experimentally observed that the impact velocity of the droplet on the superhydrophobic surface has an important role in the subsequent droplet morphologies [20]. Antonini et al. found that the droplet spreading time is independent of the impact velocity for moderate Weber numbers, with the surface contact angle being the more decisive factor [21]. Experimental investigation of droplet impact on solid surfaces has revealed six possible outcomes: deposition, prompt splash, corona splash, receding breakup, partial rebound, and complete rebound [22]. Ding et al. studied the droplet impact on a superhydrophobic surface with a single circular pillar and reported droplet rebound and splashing for various values of Weber numbers [23]. Li et al. numerically simulated droplet impact on a flat solid superhydrophobic surface and observed a doughnut-shaped breakup regime [24]. A relationship between molecular dynamics and experimental investigation of droplet impact on a pillared superhydrophobic surface. They observed that the impact velocity does not affect the contact time from 0.31 to 1.71 m/s. [25]. Abolghasemezaki et al. experimented on droplet impact on structured cylindrical surfaces to reduce contact time, which was independent of the impact velocity [26]. Khojasteh et al. observed the impact on flat and spherical superhydrophobic surfaces. They found that the predicted spreading factor followed the experimental results for flat surfaces but not spherical ones [27]. A more comprehensive review of recent advances in droplet hydrodynamics on a superhydrophobic surface can be found in [28]. The main objective of this paper is to study the hydrodynamics of a droplet after its impact on flat and cylindrical structured surfaces. Gauthier et al. have investigated the impact of a water droplet on superhydrophobic textured surfaces employing experimental techniques. They used a nickel wire on a polished aluminum surface to generate a superhydrophobic surface with stripes [29]. Many studies have also
Effect of Surface Structures on Droplet Impact Over Flat and Cylindrical …
175
observed the effects of a superhydrophobic pillared surface [30–32] on the droplet dynamics, but correlation and comparison between them are rare in literature. Similarly, many researchers have studied the droplet impact on cylindrical structured surfaces, but no comparison exists between them. In this paper, we have numerically simulated the impact of a droplet on a hydrophobic flat and cylindrical surface and then the same surfaces with pillars and stripes, thus rendering the surfaces superhydrophobic.
3 Materials and Methods 3.1 Mathematical Modeling The Volume of Fluid (VOF) approach has been used to trace the droplet’s movement after its impact on the hydrophobic surfaces. VOF is a well-established modeling method that effectively tracks the interface of the phases. To model the drop impact, the conservation of momentum (Eq. 1) and conservation of mass (Eq. 2) equations for incompressible fluid have been solved using the framework of OpenFOAM. The volumetric surface tension force has been accounted for in the source term of the momentum equation ∂ρ + ∇.(ρu) = 0 ∂t
(1)
∂(ρu) + ∇.(ρuu) = ∇. µ ∇u T + ∇u − ∇ p + ρg + Fs ∂t
(2)
The volumetric surface tension is calculated using the Continuum Surface Tension model as depicted in Eq. 3, where σ is the coefficient of surface tension and κ is the mean curvature of the free surface (Eq. 4). Fs = σ κ∇α
(3)
∇α κ = −∇. |∇α|
(4)
In the VOF model, the volume fraction (α) is used to capture the interface. The α = 1 denotes a water-filled cell, and α = 0 denotes an air-filled cell. This α is solved from the. VOF advection equation as depicted in Eq. 5. All the thermophysical properties, such as density and viscosity, are calculated using the phase fraction equation as depicted in Eq. 6 and 7. The various properties of the phases are shown in Table 1.
176 Table 1 Properties of the phases
S. Patra et al.
Phase
Viscosity
Density
Surface Tension
Water
0.00089 Pa s
998 kg/m3
0.07 N/m
Air
1.81e-5 Pa s
1.225 kg/m3
∂α + ∇.(αu) = 0 ∂t
(5)
ρ = αρliquid + (1 − α)ρgas
(6)
μ = αμliquid + (1 − α)μgas
(7)
3.2 Geometric Modeling Several geometries were modeled for our study. A spherical droplet of water is dropped on a solid substrate (flat and curved) with an initial velocity (Fig. 1a, b). The initial velocity was set so that the impact velocity was 1 m/s. On the flat surfaces, two different structures, pillars, and stripes were added, and the results were subsequently compared with a hydrophobic plain flat surface. The initial spherical droplet diameter for the flat surfaces was assumed to be 1.6 mm. A spherical droplet of diameter D0 = 2.4 mm was dropped on the curved surfaces of diameter D = 4 mm. The ratio of the diameters D* is fixed as 1.67. Three different structured curved surfaces were studied: pillared, circumferential stripes, and axial stripes, as shown in Fig. 1, and the corresponding results were compared with that of a hydrophobic curved surface without any structures. The equilibrium contact angle was set at 120°.
3.3 Validation of Numerical Model Experimental data obtained by Wang et al. [33] has been used to verify the solution methodology and the accuracy of our model. Here we compare the droplet morphologies as obtained experimentally with our numerical model. A similar validation has also been performed by Khojasteh et al. [27]. The model has been set up using a droplet radius of 1 mm, which impacts the superhydrophobic surface (with a static contact angle of 163 degrees) with a velocity of 0.56 m/s. The Weber number is 4.36. As observed in the experiment, the droplet flattens into a pancake shape upon impact and then retracts and rebounds, thus confirming our numerical simulations. The morphology comparisons are shown in Fig. 2, and they appear to be in good agreement.
Effect of Surface Structures on Droplet Impact Over Flat and Cylindrical …
177
(a)
(b) Fig. 1 Geometry, domain discretization, and boundary conditions for a flat structured surface and b cylindrical structured surface
(a)
(b) Fig. 2 a Experimental investigation by Wang et al. [33]. b Numerical simulation results obtained by our model
178
S. Patra et al.
4 Results and Discussion 4.1 Impact of Droplet on Structured Flat Surfaces Upon impact of the droplet on the flat structured substrate, we observed a complete rebound in both the cases (Striped and Pillared), whereas, on the plain hydrophobic surface, we obtained a partial rebound (which is per what has been reported in the literature [34]) (Fig. 3c). When the droplet strikes the surface, the kinetic energy is converted into interfacial energy and viscous dissipation, and the droplet starts to spread. However, the viscous dissipation can be neglected as we deal with the droplet on a millimeter scale. The structures then obstruct this spreading. The droplet on the pillared surface initially deforms and forms a multi-layered pancake-like structure (2 ms) by completely impinging the gaps between the pillars (Wenzel State) (Fig. 3a). In the striped surface, such a pancake-like structure is not seen; instead, the droplet cannot completely impinge the gaps after 2 ms (Transition between the Wenzel and Cassie-Baxter States). The droplet spreads to a maximum diameter at the 3 ms mark, which begins to retract on both surfaces. Interestingly, an air pocket is observed during the retraction stages on both surfaces, which is more prominent on the striped surface (Fig. 3b). The main forces behind the retraction of the droplet are the surface tension and capillary forces. On the flat hydrophobic surface, the droplet spreads symmetrically. The spreading is intuitively higher on the flat surface because there are no obstructions. The droplet spreading in the transverse (henceforth referred to as the XDirection) and the direction into the plane (henceforth referred to as the Z-direction) of Fig. 3, for the pillared surface, is also symmetrical. The spreading in the X-direction in the pillared surface is more than the striped surface because of the unrestricted passage of the droplet over the pillars. Because the stripes prevent spreading in the X-Direction, the spreading is more in the Z-Direction on the striped surface.
(a)
(b)
(c) Fig. 3 Front view of a Droplet impact on the pillared surface. b Droplet impact on the striped surface. c Droplet impact on the plain hydrophobic surface
Effect of Surface Structures on Droplet Impact Over Flat and Cylindrical …
179
The spreading in the X and Z-Directions with time for the three surfaces have been depicted in Fig. 4b, c, respectively. The spreading during the initial impact is quantified using the normalized maximum spreading diameter (Dmax /D0 ). The normalized diameter at any instant is calculated as the ratio between instantaneous droplet diameter and the original droplet diameter (D0 ). The post-impact bouncing dynamics are shown in Fig. 4d. After reaching the maximum spreading diameter, the droplets start to retract as the surface energy begins to convert into kinetic energy., thus giving the droplet an upward velocity, and the droplet rebounds from the surface. After subsequent bounces, as more and more energy is dissipated to the surface, the droplet’s rebound height reduces (Fig. 4a). This rebound height is highest for the plain surface, followed by the striped and pillared surfaces, and this can be attributed to the fact that the highest amount of energy is dissipated as surface energy during the impact on the pillared surface. The contact time with the surface and the structures is highest for the plain hydrophobic surface, followed by the pillared and striped surface.
4.2 Impact of Droplet on Structured Cylindrical Surfaces When the droplet impacts the structured cylindrical surface, we observe two main phenomena: complete rebound and partial rebound with droplet splitting. The four geometries under investigation are a hydrophobic cylinder, cylinder with axial stripes, cylinder with circumferential stripes, and cylinder with pillars. The droplet morphologies at various time stamps are shown in figure. It is shown that on the cylindrical pillared surface, a complete rebound occurs. On the axially and circumferentially striped cylinders, as well as on the hydrophobic cylinder, rebound with droplet splitting is observed. During the impact, the major forces involved are gravity, inertial, and surface tension forces. The effects of viscous dissipation are neglected again due to the size of the droplet. Since the diameter of the cylinder is larger than the initial diameter of the droplet, the droplet cannot completely encompass the cylinder in the azimuthal direction. Hence, upon spreading the droplet overhangs the cylinder, maintaining downward inertia. Subsequently, when the droplet starts retracting, the middle region of the droplet gains kinetic energy upward, whereas the overhanging parts have downward inertia, thus stretching the droplet. When this stretching overcomes the surface tension, the droplet splits. In the case of the cylinder with the axial stripes, the droplet stretches in the axial direction more than the azimuthal direction. This causes an earlier splitting (Figs. 5b, 6b). In the cylinder with circumferential stripes, the obstruction prevents stretching in the axial direction, and hence more spreading is observed in the azimuthal direction. This brings about splitting a little later (during the rebound) (Figs. 5c, 6c). The split droplets later recombine. In the cylinder with pillars, the availability of space allows the droplet to spread more evenly, leading to lesser stretching; hence, no splitting is observed. In the plain hydrophobic cylinder, the spreading occurs symmetrically. It is observed that the angle of wrap around the cylinder increases with time. The
180
S. Patra et al.
Fig. 4 a Height of the droplet. b Spreading in X-Direction. c Spreading in Z-Direction. d Normalised maximum spreading diameter
(a)
(b)
(c)
(d)
Effect of Surface Structures on Droplet Impact Over Flat and Cylindrical …
181
reason behind it is that the droplet has split, and the split droplets move downwards in opposite directions. A similar observation can be made for the axial strips. Whereas for the circumferential strips, since the droplet splitting occurs during rebound, the wrap angle keeps decreasing as the droplet bounces back. In the pillared surface, no splitting is observed, and the droplet usually bounces back, and hence the angle of wrap decreases (Fig. 7a). The height of the droplet is depicted in Fig. 7b. It is observed for the hydrophobic cylinder and the axially striped cylinder that after impact, the height of the droplet reduces. This is because the split droplets fall down under the effect of gravity. On the other hand, the droplets split during the rebound period and later recombine for the circumferentially striped cylinder, which is why the height increases. For the pillared cylinder, no splitting is observed, the droplet rebounds and the height keeps increasing. The contact times with the surface are also observed to be the highest for the pillared cylinder and the hydrophobic cylinder, followed by the axially striped cylinder, and lowest for the circumferentially striped cylinder. Interestingly, the droplet cannot penetrate the structures in the axially striped cylinder and remains at a transition between the Cassie-Baxter and Wenzel states.
(a)
(b)
(c)
(d) Fig. 5 Top view of impact on a Hydrophobic cylinder. b Cylinder with axial stripes, c Cylinder with circumferential stripes. d Cylinder with pillars
182
S. Patra et al.
(a)
(b)
(c)
(d) Fig. 6 Front view of the impact on a Hydrophobic cylinder. b Cylinder with axial stripes. c Cylinder with circumferential stripes. d Cylinder with pillars
5 Conclusion In this present work, an effort is made to study the droplet impact dynamics on flat and cylindrical structured surfaces. Numerical simulations have been performed using OpenFOAM. The presence of structures on the hydrophobic surfaces effectively renders the surfaces superhydrophobic. We have studied a striped and pillared flat surface and compared the resulting dynamics with a plain hydrophobic surface. It is observed that the structures obstruct spreading: they promote spreading in the direction of the stripes and restrict it in the transverse direction. The pillars, on the other hand, prevent spreading symmetrically. Thus in the pillars, the spreading is symmetrical. The contact time of the droplet with the surface is observed to be higher for the pillared surface than the striped surface, and thus we can also conclude that a higher amount of energy is dissipated to the pillared surface than the striped surface. The contact time with the hydrophobic flat surface is the highest. During the initial impact, the droplet completely impinges the gaps between the structures (Wenzel State), which later transitions into the Cassie-Baxter state during subsequent impacts after bouncing. The splitting of droplets on plain and structured cylindrical surfaces is also investigated. The droplet is split in all three geometries except for the pillared cylinder. The contact time is also the highest for the pillared cylinder. Interestingly, splitting is observed on the circumferentially striped cylinder, and the split droplets recombine later on rebound.
Effect of Surface Structures on Droplet Impact Over Flat and Cylindrical … Fig. 7 a Angle of wrap around the cylinder. b Height of the droplet with time
(a)
(b)
Nomenclature ρ μ α D D0 Dmax D* u
Density of phase (kg/m3 ) Viscosity of phase (Pa s) Phase fraction (–) Diameter of cylinder (mm) Initial diameter of droplet (mm) Maximum spreading diameter (mm) Diameter ratio D/D0 Velocity (m/s)
183
184
g σ Fs κ
S. Patra et al.
Accelaration due to gravity (m/s2 ) Coefficient of Surface tension (N/m) Force due to surface tension (N) Mean curvature of free surface (–)
References 1. Barthlott W, Neinhuis C (1997) Purity of the sacred lotus, or escape from contamination in biological surfaces. Planta 202:1–8 2. Das S, Kumar S, Samal S, Mohanty S, Nayak S (2018) A review on superhydrophobic polymer nanocoatings: recent development and applications. Ind Eng Chem Res 57 3. Kannan R, Sivakumar D (2008) Impact of liquid drops on a rough surface comprising microgrooves. Exp Fluids 44:927–938 4. Malouin BA Jr, Koratkar NA, Hirsa AH, Wang Z (2010) Directed rebounding of droplets by microscale surface roughness gradients. Appl Phys Lett 96(23) 5. Wang D, Sun Q, Hokkanen MJ, Zhang C, Lin FY, Liu Q, Zhu SP, Zhou T, Chang Q, He B, Zhou Q (2020) Design of robust superhydrophobic surfaces. Nature 582:55–59 6. Bussmann M, Mostaghimi J (1999) On a three-dimensional volume tracking model of droplet impact. Phys Fluids 11:1406–1417 7. Shen Y, Tao J, Tao H, Chen S, Pan L, Wang T (2015) Relationship between wetting hysteresis and contact time of a bouncing droplet on hydrophobic surfaces. ACS Appl Mater Interfaces 8. Yan Z, Zhao R, Duan F, Wong T, Toh K, Choo K, Chan P, Chua Y (2011) Spray cooling 9. Sarkar D, Farzaneh M (2009) Superhydrophobic coatings with reduced ice adhesion. J Adhes Sci Technol 23:1215–1237 10. Wang N, Xiong D, Deng Y, Shi Y, Wang K (2015) Mechanically robust superhydrophobic steel surface with anti-icing. UV-durability, and corrosion resistance properties. ACS Appl Mater Interfaces 11. He H, Guo Z (2021) Superhydrophobic materials used for anti-icing Theory, application, and development. iScience 24 12. Liu YC, Farouk T, Savas AJ, Dryer FL, Avedisian CT (2013) On the spherically symmetrical combustion of methyl decanoate droplets and comparisons with detailed numerical modeling. Combust Flame 160:641–655 13. Moreira ALN, Moita AS, Panão M (2010) Advances and challenges in explaining fuel spray impingement: how much of single droplet impact research is useful. Progr Energy Combust Sci 14. Yusof A, Keegan H, Spillane CD, Sheils O, Martin C, O’Leary J, Zengerle R, Koltay P (2011) Inkjet-like printing of single-cells. Lab Chip 11:2447–2454 15. Vazirinasab E, Jafari R, Momen G (2017) Application of superhydrophobic coatings as a corrosion barrier: a review. Surface Coatings Technol 16. Massinon M, Lebeau F (2012) Experimental method for the assessment of agricultural spray retention based on high-speed imaging of drop impact on a synthetic superhydrophobic surface. Biosys Eng 112:56–64 17. Wenzel RN (1936) Resistance of solid surfaces to wetting by water. Ind Eng Chem 28(8):988– 994 18. Cassie ABD, Baxter S (1944) Wettability of porous surfaces. Trans Faraday Soc 40:546–551 19. Patil N, Bhardwaj R, Sharma A (2015) Droplet impact dynamics on micropillared hydrophobic surfaces. Exp Thermal Fluid Sci 74 20. Tsai P, Pacheco S, Pirat C, Lefferts L, Lohse D (2009) Drop impact upon micro- and nanostructured superhydrophobic surfaces. Langmuir: ACS J Surfaces Colloids 25
Effect of Surface Structures on Droplet Impact Over Flat and Cylindrical …
185
21. Antonini C, Amirfazli A, Marengo M (2012) Drop impact and wettability: from hydrophilic to superhydrophobic surfaces. Phys Fluids 24 22. Rioboo R, Marengo M, Tropea C (2001) Outcomes from a drop impact on solid surfaces. Atom Sprays 11:155–166 23. Ding S, Hu Z, Dai L, Zhang X, Wu X (2021) Droplet impact dynamics on single-pillar superhydrophobic surfaces. Phys Fluids 33(10):102108 24. Li W, Wang J, Zhu C, Tian L, Zhao N (2021) Numerical investigation of droplet impact on a solid superhydrophobic surface. Phys Fluids 33(6):063310 25. Li H, Zhang K (2019) Dynamic behavior of water droplets impacting on the superhydrophobic surface: both experimental study and molecular dynamics simulation study. Appl Surf Sci 498:143793 26. Abolghasemibizaki M, McMasters RL, Mohammadi R (2018) Towards the shortest possible contact time: droplet impact on cylindrical superhydrophobic surfaces structured with macroscale features. J Colloid Interface Sci 521:17–23 27. Khojasteh D, Bordbar A, Kamali R, Marengo M (2017) Curvature effect on droplet impacting onto hydrophobic and superhydrophobic spheres. Int J Comput Fluid Dyn 31:1–14 28. Khojasteh D, Kazerooni M, Salarian S, Kamali R (2016) Droplet impact on superhydrophobic surfaces: a review of recent developments: J Ind Eng Chem 42 29. Gauthier A, Symon S, Clanet C, Quéré D (2015) Water impacting on superhydrophobic macrotextures. Nat Commun 6:1–6 30. Zhang R, Farokhirad S, Lee T, Koplik J (2014) Multiscale liquid drop impact on wettable and textured surfaces. Phys Fluids 26(8):082003 31. Tsai P, Hendrix M, Dijkstra R, Shui L, Lohse D (2011) Microscopic structure influencing macroscopic splash at high Weber number. Soft Matter 7 32. Lee J, Hwang S-H, Yoon S-S, Khang D-Y (2019) Evaporation characteristics of water droplets in Cassie, Wenzel, and mixed states on superhydrophobic pillared Si surface. Colloids Surf A 562:304–309 33. Wang Z, Lopez C, Hirsa A, Koratkar N (2007) Impact dynamics and rebound of water droplets on superhydrophobic carbon nanotube arrays. Appl Phys Lett 91(2):023105 34. Parihar V, Chakraborty S, Das S, Chakraborty S, Dasgupta S (2020) Role of anisotropic pinning and liquid properties during partial rebound of droplets on unidirectionally structured hydrophobic surfaces. Chem Eng Sci 230
Numerical Investigation on Bubble Dynamics Using DOE Approach for Cavitation Machining Process Amresh Kumar, Tufan Chandra Bera, and B. K. Rout
1 Introduction Cavitation is a hydrodynamic phenomenon that forms vapour pockets (vapour or/ both gases mixed in liquid) in flowing liquid in the low-pressure zone. Since the nineteenth century, many researchers and engineers have been involved in eliminating the harmful effects of cavitation that cause the failure of fluid machinery and drop in its efficiency. On the contrary, cavitation benefits in many ways, such as cleaning off the oxides, permitting the good bond in acoustic soldering, tunnelling through rocks, and agitating the rate of chemical reactions. Several studies highlight the benefits of cavitation in treating a range of pollutants (including organic compounds) [1, 2], synthesis of bio-diesel, medical ultrasonography, and groundwater cleaning. Controlled hydrodynamic cavitation is used very effectively in improving the fatigue strength of the machined material through cavitation peening [3, 4]. Therefore, to harness the erosive aspect of cavity bubbles in the machining of materials and material processing industries, a new non-traditional machining method has been proposed.
2 Literature Review and Objective Cavitation machining (CM) is a non-traditional machining (NTM) technique. In CM, the cavity bubble will generate at the entrance of the orifice and expand, consequently flowing downstream and collapsing near the specimen surface. The collapse of the cavity bubble produced a high impact of implosion pressure and creates an erosion pit A. Kumar (B) · T. C. Bera · B. K. Rout Department of Mechanical Engineering, Birla Institute of Technology & Science, Pilani Campus, Pilani, Rajasthan 333031, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_18
187
188
A. Kumar et al.
on the specimen surface [5, 6]. The dynamic process of the cavity bubble generates a high-velocity microjet, leading to an erosion pit on the specimen [7, 8]. Recently cavitation-based machining is gaining more attention due to its ability to enhance the internal surface integrity of additively manufactured products [9–11] and the materials having a high strength-to-weight ratio. Several studies and numerical simulations have been published on the dynamics of a cavity bubble in a flow domain using finite volume, finite difference, and boundary element methods [12, 13]. Many numerical studies reveal the phenomena of cavity bubble’s inception followed by its growth and collapse in water using Navier–Stokes equations with appropriate cavitation model and Rayleigh–Plesset equation [14–16]. Sonde et al. [17] studied the dynamic parameters of a cavity bubble in a flow domain using the Keller–Miksis model of bubble dynamics considering the effect of the initial radius of the bubble, initial pressure inside the bubble, and ambient pressure of the flow domain on implosion pressure and concluded that the dynamics of a cavity bubble plays a vital role in cavitation peening. After a thorough literature survey, this can be concluded that the parameter affecting most in CM is the magnitude of implosion pressure generated due to the collapse of a cavity bubble depending on the dynamics of the cavity bubble. The factors associated with the dynamics of a cavity bubble are downstream pressure, carrier fluid density, initial radius, and initial pressure inside the bubble. Process parameters optimisation for CM is a less explored area. Therefore, it is necessary to analyse the influence of process parameters of bubble dynamics on the implosion pressure, and this is the first attempt to identify the best feasible input conditions for the maximum implosion pressure. Therefore, a full factorial design is used in the present research to determine the significant input factors for the desired output response. The paper is presented in the following ways. Section 3 discusses the detailed methodology, while Sect. 4 shows the results and analysis. In the end, Sect. 5 concludes the paper.
3 Numerical Model and Methodology In this research, the detailed numerical analysis of a cavity bubble has been discussed. Then, a full factorial design is considered to study the effect of controllable and uncontrollable factors to simulate the response. The numerical model for the present research is discussed in Sect. 3.1. The factors responsible for the implosion pressure were identified based on the present model. To understand the impact of these factors on the response, experiments are conducted using various treatment combinations of the design. For generations of combinations, the levels of control factors are chosen at three different levels, and the uncontrollable factors are determined separately though these factors are difficult to control during experiments.
Numerical Investigation on Bubble Dynamics Using DOE Approach …
189
3.1 Numerical Model The Keller–Miksis [18] model for bubble dynamics is considered, and the simulations are conducted after identifying suitable numerical time integration techniques. The Keller–Miksis model traces the implosion pressure precisely on the collapse of a cavity bubble because it considers the compressibility of the carrier fluid. However, the present investigation focussed on the implosion pressure, as this response is critical for predicting material erosion in CM. The K-M model as in Eq. (1), R is the radius of the cavity bubble, C is the velocity of sound in liquid, ρ is the density of the liquid, and P pressure on the surface of cavity bubble wall. The output response of cavity bubble dynamics is such as the evolution of bubble radius with respect to time (dR/dt), bubble wall velocity (m/s), and implosion pressure (MPa) can be determined from this model shown in (1). 2 1 ˙ R˙ R ˙ R˙ 1 ˙ ¨ 1− + 1.5 R 1+ P+ RR 1 − R = P C 3C ρ C ρC Ro 3γ 4μ ˙ 2σ − P = Pv + Pg − R − P∞ R R R
(1)
(2)
Similarly, in (2), Pv is vapour pressure inside the cavity bubble at its working temperature, Pg is the partial gas pressure in the cavity bubble, R0 is the initial bubble radius, σ and μ are surface tension and viscosity of the liquid, respectively, P∞ is ambient pressure far from the cavity bubble, γ is the heat capacity ratio, and its value will be equal to 1.4. Numerical simulation is carried out for the cavity bubble to trace the implosion pressure of a cavity bubble on its collapse. The cavity bubble is assumed to be initially filled with non-condensable gases. Gas inside the bubble is an ideal gas and follows a polytropic relation in its expansion and compression. The cavity bubble is taken as an adiabatic and isentropic system. The polytropic coefficient (γ ) is chosen as 1.4. Substituting the new variable dR/dt = V to transform (1) into singleorder ordinary differential equation (ODE) and the equation is solved by using the fourth-order Runge–Kutta method (RK4) with an initial condition R(0) = R O , and dR/dt = 0. The time step value is decreased up to the relative error of 10–6 to solve the single-order differential equation.
3.2 Parameters Selection Referring to Eqs. (1) and (2), four factors, such as initial radius, gaseous pressure inside the bubble, ambient pressure, and carrier fluid density, are important parameters influencing implosion pressure. Two out of four factors are control factors, which means an operator can control it in real time. The complete process is shown in the
190
A. Kumar et al.
Fig. 1 P—Diagram of bubble dynamics
P-diagram, Fig. 1. It is challenging to control the initial pressure inside the bubble as it depends on the amount of mixed air in the carrier fluid. Brennen [19] proposed the relationship between the initial pressure inside the bubble and air content in the carrier fluid, shown in (3). Pg = 69 × α
(3)
where Pg is the initial pressure inside the bubble in Pascal (Pa) and α is the air content in fluid in parts per million (ppm). According to [19], the α value of pressurised fluid can be as low as 3 ppm and it requires a lot of time and resources to decrease the air concentration below this value. Theoretically, at atmospheric pressure, the saturation is about α = 15 ppm. Therefore, three values of α are considered in the present work, i.e. 3.5, 7 and 14 ppm, with increasing initial diameter of a cavity bubble of 100 μ (0.0001 m), 500 μ (0.0005 m), and 1 mm (0.001 m), respectively. Subsequently, the effects of uncontrollable factors on implosion pressure are examined. Selected input factors and responses are inputted in the Minitab 18 software. The ANOVA and analysis of mean (ANOM) techniques are applied to identify the statistically significant factor and its relationship with the response factor.
Numerical Investigation on Bubble Dynamics Using DOE Approach …
191
Table 1 Controllable factors with their levels Factors
Symbol
Level 2
Level 3
Downstream pressure (MPa)
A
0.1
0.2
0.3
B
998.2
1005.3
1019.6
Fluid density
(Kg/m3 )
Level 1
Table 2 Sets of uncontrollable factors Uncontrollable factors
Initial bubble radius (m)
Pressure inside bubble (Pa)
Set 1
0.0001
242
Set 2
0.0005
483
Set 3
0.001
966
3.3 Factorial Design The full factorial design available in the design of experiments (DOE) is selected, in this case, it is 32 designs to create the factor combinations for which numerical experiments are performed. The factors with their corresponding levels are tabulated in Table 1, and the set of uncontrollable factors which are kept constant during full factorial design experiments are tabulated in Table 2. These uncontrollable factors are fed as input along with controllable factors in Eq. (1). Subsequently, simulated responses are fed to the software for ANOVA and ANOM. These techniques are used to identify the influence of the main effects with the help of Minitab 18 software. In this case, the impact of all the considered parameters on implosion pressure is obtained.
4 Results and Discussion This section discusses the results of the computational method using the design of experiments.
4.1 Significant Parameters The resulting output of ANOVA with three different sets of uncontrollable factors is presented in Tables 3, 4, and 5, respectively. Table 3 shows the detailed ANOVA table model, error percentage, and contribution of factors. Similarly, Tables 4 and 5 presented the source, the contribution, and the P-value for identifying significant control factors on output response. The result reveals that the downstream pressure significantly affects the output response, i.e. implosion pressure. In contrast, the
192
A. Kumar et al.
Table 3 ANOVA table with set 1 of uncontrollable factors Source
DF
Seq. SS
Contribution (%)
Adj. SS
Adj. MS
P-value
Model
4
8,89,392
100
8,893,922
2,22,348
0.00
Linear
4
8,89,392
100
8,89,3922
2,22,348
0.00
A
2
8,89,237
99.98
8,89,2376
4,44,618
0.00
B
2
1546
0.02
1546
773
0.04
Error
4
406
0.00
406
102
Total
8
8,89,432
100
Table 4 ANOVA table with set 2 of uncontrollable factors Source
DF
Seq. SS
Contribution (%)
P-value
A
2
3,67,369
99.98
0.000
B
2
404
0.01
0.071
Table 5 ANOVA table with set 3 of uncontrollable factors Source
DF
Seq. SS
Contribution
P-value
A
2
123,177
99.99%
0.000
B
2
80
0.01%
0.112
density of carrier fluid is a less significant factor in the implosion pressure. Moreover, the density of fluid helps in the nucleation and generation of cavity bubbles, which directly affects the material removal process. The main effect plot is presented in Figs. 2, 3 and 4. The maximum implosion pressure is produced with the third level of downstream pressure (0.3 MPa) with set 1 of uncontrollable factors, i.e. initial bubble radius (0.0001 m) and initial pressure inside the bubble (242 Pa), respectively, generate the highest implosion pressure followed by the second and third set of uncontrollable factors, i.e. 3350.73 MPa, 1981 MPa, and 1064.83 MPa respectively. However, an increase of α value 3.5 to 7 in the carrier fluid leads to a reduction in output response, i.e. implosion pressure of about 41%. Later confirmation test has been carried out with the obtained optimal conditions with the lower and higher level of insignificant factors, i.e. the density of carrier fluid.
4.2 Confirmation Study The implosion pressure on the collapse of a cavity bubble with both the carrier fluid density levels is 3328.11 and 3379.23 MPa, respectively. Therefore, the maximum implosion pressure on the collapse of a cavity bubble is as follows: (1) the third level
Numerical Investigation on Bubble Dynamics Using DOE Approach …
Fig. 2 Main effect plot with the first set of uncontrollable factors
Fig. 3 Main effect plot with the second set of uncontrollable factors
193
194
A. Kumar et al.
Fig. 4 Main effect plot with the third set of uncontrollable factors
of downstream pressure is 0.3 MPa; (2) the density of carrier fluid is 1019.6 kg/m3 ; (3) the initial radius of bubble 100μm (0.0001 m); and (4) the pressure inside the bubble is 242 Pa. The magnitude of implosion pressure slightly exceeds to 0.85%. Moreover, the simulated results from the numerical computation have been verified with the results available in the article [17], and the comparative results are presented in Fig. 5. In that study, the authors studied the effect of initial pressure inside the bubble at 242, 483, and 966 Pa with downstream pressure of 0.2 MPa. Hence, it can be concluded that the results of the present study are in good agreement with the results available in the literature. Therefore, the implosion pressures from the current research are suitable for machine materials like mild steel, chromium–molybdenum alloy steel, nickel-based (Inconel 625, Inconel 718, etc.), cobalt-based, and iron-based super-alloys (SS 660B, CG 27, etc.) that have yield strengths in the range of 275 to 1200 MPa. However, this efficient method can also be used in industries to synthesise the micro- and nanoparticles from the parent materials. The current work also validates that the implosion pressure behaves linearly with the downstream pressure, as shown in Fig. 6. While fluid density will have less impact on implosion pressure, as shown in Fig. 7, it helps in the bubble nucleation process and the implosion of a greater number of bubbles increases the material removal rate during CM. This analysis is carried out to validate the above results and to analyse the effect of the input control factor on the output response separately.
Numerical Investigation on Bubble Dynamics Using DOE Approach …
195
Fig. 5 Result validation (Present result validation with Sonde et al. [17])
Fig. 6 Effect of downstream pressure on implosion pressure
5 Conclusions This current research presents input factors to maximise the implosion pressure for cavitation machining. ANOVA and ANOM main effect plot for implosion pressure results also reveal that the downstream pressure of a flow domain is the most significant factor in cavitation machining. However, initial pressure inside the bubble is
196
A. Kumar et al.
Fig. 7 Effect of fluid density on implosion pressure
also one of the critical factors, but the human operator in real time cannot control this. While the density of carrier fluid is a less significant control factor, this will help initiate cavity bubbles. In addition, the present research provides the feasibility of machining materials with high strength-to-weight ratios, like nickel-based, titanium-based, and iron-based alloys.
References 1. Braeutigam P, Wu ZL, Stark A, Ondruschka B (2009) Degradation of BTEX in aqueous solution by hydrodynamic cavitation. Chem Eng Technol 32:745–753 2. Kalumuck KM, Chahine GL (2000) The use of cavitating jets to oxidize organic compounds in water. ASME J Fluids Eng Trans 122:465–470 3. Soyama H, Saito K, Saka M (2002) Improvement of fatigue strength of aluminum alloy by cavitation shotless peening. ASME J Eng Mater Technol 124:135–139 4. Klumpp A, Lienert F, Dietrich S, Soyama H, Schulze V (2017) Surface strengthening of AISI4140 by cavitation peening. In; 13th International conference on shot peening (ICSP-13) 18–21 Sept 2017, Montreal, Canada, pp 440–445 5. Marcon A, Melkote SN, Castle J, Sanders DG, Yoda M (2016) Effect of jet velocity in co-flow water cavitation jet peening. Wear 360–361:38–50 6. Duraiselvam M, Galun R, Siegmann S, Wesling V, Mordike BL (2006) Liquid impact erosion characteristics of martensitic stainless steel laser clad with Ni-based intermetallic composites and matrix composites. Wear 261:1140–1149 7. Franc JP, Michel JM (2005) Fundamentals of cavitation. Kluwer Academic Publisher, USA 8. Cheng F, Ji W (2017) Cavitation erosion of a single bubble in water as a kind of dynamic damage. Proc Inst Mech Eng Part J: J Eng Tribol 231:1383–1389 9. Polishetty A, Shunmugavel M, Goldberg M, Littlefair G, Singh RK (2017) Cutting force and surface finish analysis of machining additive manufactured titanium alloy Ti–6Al–4V. Procedia Manuf 7:284–289
Numerical Investigation on Bubble Dynamics Using DOE Approach …
197
10. Nagalingam AP, Thiruchelvam VC, Yeo SH (2019) A novel hydrodynamic cavitation abrasive technique for internal surface finishing. J Manuf Process 46:44–58 11. Nagalingam AP, Yuvaraj HK, Santhanam V, Yeo SH (2020) Multiphase hydrodynamic flow finishing for surface integrity enhancement of additive manufactured internal channels. J Mater Process Technol 283:1–21 12. Dadvand A, Khoo BC, Shervani-Tabar MT, Khalilpourazary S (2012) Boundary element analysis of the droplet dynamics induced by spark-generated bubble. Eng Anal Boundary Elem 36:1595–1603 13. Fu Z, Popov V (2015) The ACA-BEM approach with a binary-key mosaic partitioning for modelling multiple bubble dynamics. Eng Anal Boundary Elem 50:169–179 14. Qin Z, Bremhorst K, Alehossien H, Meyer T (2007) Simulation of cavitation bubbles in a convergent—divergent nozzle water jet. J Fluid Mech 573:1–25 15. Peng G, Shimizu S, Fujikawa S (2011) Numerical simulation of cavitating water jet by a compressible mixture flow method. J Fluid Sci Technol 6:499–509 16. Koukouvinis P, Bergeles G, Gavaises M (2015) A cavitation aggressiveness index within the Reynolds averaged Navier Stokes methodology for cavitating flows. J Hydrodyn 27:579–586 17. Sonde E, Chaise T, Boisson N, Nelias D (2018) Modeling of cavitation peening: jet, bubble growth and collapse, micro-jet and residual stresses. J Mater Process Technol 262:479–491 18. Keller JB, Miksis M (1980) Bubble oscillations of large amplitude. The J Acoust Soc Am 68:628–633 19. Brennen C (1969) The dynamic balances of dissolved air and heat in natural cavity flows. J Fluid Mech 37:115–127
Effect of Surface Tension on the Thermal Performance of Pulsating Heat Pipe with and Without Surfactant Vaishnavi K. Wasankar and Pramod R. Pachghare
1 Introduction Electric circuit used in electronic devices is continual undergoing miniaturization due to requirement of compact space leads to high heat flux. Akachi introduced first pulsating heat pipe (PHP) in 1990 as a substitute to conventional heat pipe for effective cooling purpose [1]. PHP is a wickless, two-phase device having an ability to transfer heat with a simple design and economical cost of production has several applications besides electronic cooling, such as thermal control of spacecraft, cryogenics, and heat exchangers. PHP mostly comprises of interconnecting tubes that are only partially filled with the working fluid. There are essentially major three components of a conventional PHP such as evaporator, adiabatic, and condenser section, respectively. Heat input is absorbed by the working fluid in an evaporator portion, and this heat is subsequently rejected to condenser for working fluid cooling. The working fluid continuously condenses in the condenser section and continuously evaporates in the evaporator, generating a pulsing, non-equilibrium vapor pressure field that promotes fluid motion between adjusting tube sections and causes an uneven distribution of liquid plug and vapor bubble. Sensible heat of the liquid plug and latent heat of the vapor bubble both contribute to heat transmission [2]. Basically, three parameters, physical properties of working fluid, the geometric parameters, and the operating parameter, are crucial for the proper PHP. Geometric parameters of PHP are as follows: the inner diameter, the length of the evaporation and condensation section, the number of turns, the cross-section shape of the PHP etc.; physical PHP characteristics are the thermodynamic and hydrodynamic characteristic of the working fluid, such as surface tension, viscosity, latent heat, etc.; and operational parameters are filling ratio, heat input, V. K. Wasankar (B) · P. R. Pachghare Department of Mechanical Engineering, Government College of Engineering, Amravati 444604, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_19
199
200
V. K. Wasankar and P. R. Pachghare
inclination angle, etc. Researchers have utilized different variety of working fluids such as nanofluids [3], ferrofluids, binary fluids [4], and mixtures [5] to enhance the thermal performance and heat transfer characteristics of the PHPs [6].
2 Literature Review and Objective According to the available literature, the tube’s interior diameter has the biggest impact on the PHP system. The normal operation of PHP is based on the formation of the vapor slugs and liquid plugs. A dimensionless quantity known as the Bond number, which determines the critical diameter criterion. It indicates the relative strengths of gravity and surface tension, which determine how the two come into being. In comparing the heat transmission capabilities of PHPs with inner diameters of 2 mm and 1 mm, Charoensawan et al. [7] observed that the thermal resistance of the PHP with tube diameter of 2 mm was 10% lower than that of the other. Filling ratio has a significant impact on PHP’s thermal performance, according to research by Sameer Khandekar et al., that for a given heat, FR of 25–65% causes self-sustained pulsating action demand for throughput. Khandekar et al., that for a given heat, FR of 25–65% causes self-sustained pulsating action. Eö = BO =
2 g Dcri (ρliq − ρvap ) σ
(1)
The overall degree of freedom and the bubble pumping action was not sufficient above this range to produce satisfactory performance [8]. Additionally, PHP that is only partially filled performs better thermal performance [9]. PHP’s ability to transfer heat is constrained by the working fluid’s inadequate transport capabilities. Knowing a working fluid’s limitations helps to choose a working fluid according to application. Han et al. [10] addressed the possibility that a fluid with a higher latent heat at a higher heat input would be more capable of transporting heat. The self-rewetting behavior of distilled water and the self-pulsating action of PHP are both improved by the addition of a small amount of alcohol [11]. As the temperature difference between the evaporator and condenser reduces, the use of self-rewetting fluid causes a reduction in thermal resistance [12]. While hydrophilic surfaces have stronger thermal resistance at higher temperatures, hydrophobic surfaces have lower thermal resistance at lower temperatures. With more liquid plugs inside it, PHP with hydrophobic surfaces has better wettability, which increases thermal resistance [13]; properties of alcohol are listed in Table 1. Experiments are conducted by Pachghare and Mahalle [5], Patel and Mehta [14], and others to investigate the impact of pure fluids such as pure water, acetone, methanol, and ethanol and their binary mixture on the thermal performance of PHP. It was found that water-based mixtures had lower thermal resistance than their pure working fluid counterparts. There is no discernible difference between PHP working with pure and binary mixture working fluids in terms of overall heat resistance. The formation, development, coalescence,
Effect of Surface Tension on the Thermal Performance of Pulsating … Table 1 Properties of pure fluids at 25 °C and 1 atm [14]
201
Property
DI water Ethanol Acetone
Boiling point (°C)
100
78
56
Liquid density (kg/m3 )
997.1
785.5
784.6
Liquid-specific heat (kJ/kg °C)
4.183
2.513
2.56
Latent heat (kJ/kg)
2257
846
518
dP/dT (Pa/°C)
1946
4299
6262
0.8905
1.0820
0.3166
71.97
22.28
22.29
Dynamic viscosity (Pa s) ×
10–3
Surface tension (N/m) × 10–3
and detachment processes of vapor bubbles are significantly influenced by surface tension of the working fluid. It also has an impact on PHP’s hydrodynamics and heat transfer limit [15]. A chemical substance known as a surfactant [16] is used to lower surface tension. Fluid made on the addition of a surfactant is more stable, and alter the properties of the vapor–liquid interface, which alters the boiling mechanism. Additionally, it reduces bubble size and boosts bubble density to improve boiling heat transfer [17]. Patel et al. [14] and Wang et al. [18] conducted experiments utilizing various water-based mixtures with varying concentrations of surfactant and compared their results with their pure fluid yields better performance. On the other hand, thermal resistance decreases at low filling ratios and low heat input as expected. The effect of surfactant improves the working fluid’s thermal performance [19–22]; nevertheless, the optimal surfactant concentration is still unknown. Additional studies came to the conclusion that a water-based mixture in a 1:1 ratio provides better results than other proportions [23]. This information was beneficial while choosing a working fluid for an experiment [24]. In this review of the literature, an effort is made to examine the impact of surface tension fluctuation on pure and binary fluid of a PHP due to addition of the surfactant. Check the performance experimentally and compare the outcomes with its pure fluid surfactant. Even a little improvement in PHP’s thermal performance and heat dissipation rate will advance the system’s suitability for use in a variety of applications.
3 Experimental Setup The experimental setup used to investigate thermal performance is shown schematically in Fig. 1. A copper capillary tube with a 2 mm diameter is meandering into two turns and filled to a constant ratio of 50% with the working fluids. To track the impact of the input heat flux, the power is changed from 0 to 120 W in steps of 10 W. The adiabatic section is covered with insulating asbestos thread, and the tubes in the evaporator, condenser, and adiabatic section are 50 mm, 100 mm, and 80 mm long, respectively.
202
V. K. Wasankar and P. R. Pachghare
Fig. 1 Schematic diagram of the setup
The oil is heated with the aid of a coil heater in the evaporator section since oil bath heating is used there for heating purposes. By employing synthetic insulation, the thermocouples that are fixed inside the evaporator portion are kept out of touch with the oil. A dimmer stat component installed inside the control panel provides the input power to the heaters. Four type K thermocouples mounted in each section are used to measure the temperatures in the evaporator, condenser, and adiabatic sections, while two thermocouples are mounted to measure the temperatures in the cooling water inlet and exit. Water bath cooling is provided in the condenser portion with suitably controlled inflow and outflow rates. A box (170 * 100 * 70 mm3 ) made of acrylic sheet encloses both the evaporator and condenser parts. The condenser section’s cooling water flow rate is kept constant at 1.302 kg/min. Working fluids are made using SDS (CH3 (CH2 )11OSO3 Na), with a water–acetone solution serving as the base fluid. A binary mixture is given a calculated amount of surfactant, such as 500 or 1000 ppm, to arrange the solutions at room temperature. The opposing thermophysical properties remain similar at a very low concentration of the additive, despite a significant change in surface tension of the base working fluid with the addition of the surfactant.
3.1 Experimental Procedure • The first step within the experimentation is to evacuate and filling of the capillary tube at ambient temperature, i.e., 25 ± 3 °C.
Effect of Surface Tension on the Thermal Performance of Pulsating …
203
• The copper tube is first evacuated using a vacuum pump that creates a vacuum of roughly 70–75 mm of Hg. The vacuum pump is first linked to the filling valve, and then the evacuation is completed. • Utilizing a burette and syringe, the necessary volume of working fluid is filled through the filling valve after the proper level of vacuum has been reached, and then the filling valves are closed. • A small cooler pump is used to deliver tap water to the condenser section. • Once the transparent acrylic sheet-clad condenser shell was filled with water, the inlet and outlet valves were adjusted to obtain the necessary flow rate of cooling water, which was then measured using a beaker and a stopwatch. • The dimmer stat assembly that is provided is used to supply the controlled power. • The temperature of the oil rises as a result of heating it, and heat input is given to the heater unit until a steady state is established, at which point the temperature at various PHP locations is recorded. • The readings are recorded once the system reaches steady state. The aforementioned process is repeated for various power inputs and varied working fluid concentrations.
4 Results and Discussion In all, two distinct tests are carried out on PHP using pure and binary fluids to examine the impact of different surfactant concentrations on its thermal performance. Effect on PHP displays results with the help of several graphs. In general, thermal resistance— the difference in temperature between the evaporator and condenser sections—can be used to assess the thermal performance of PHP. R=
(Te − Tc ) Q in
(2)
where Qin represents the net heat input power to the PHP at the evaporator when thermal losses of 10% are taken into account (Fig. 2).
4.1 Thermodynamic Behavior of PHP Equation shows that the thermal resistance is directly influenced by the evaporator temperature (2). The discussion, therefore, begins with the impact of surface tension fluctuation on evaporator temperature. Figure 3 shows depicts the overall impact of a 1000 ppm solution and 500 ppm surfactant. Due to a low fluid inventory in the PHP and a high proportion of the working fluid being converted to vapor state, a low filling ratio causes the PHP to display a high evaporator temperature. The dynamics of the bubbles inside the loop are impacted by changes in surface tension. The
204
V. K. Wasankar and P. R. Pachghare
Fig. 2 Photograph of the setup
evaporator section is predicted to produce more vapor because the reduced surface tension encourages the fast nucleation of vapor bubbles. This increases the pumping motion within the PHP. As a result, with a particular heat input, the surfactant solution exhibits earlier. In general, the thermal resistance is higher if there is a larger temperature difference between the evaporator and condenser part. The average evaporator temperature and average condenser temperature for the water–acetone surfactant solution increase 100 Average Evaporator Temperature (°C)
Fig. 3 Average evaporator temperature versus heat input for water-based binary fluid at various surfactant concentration
500 ppm 90
1000 ppm
80 70 60 50 40 30
0
20
40 60 80 Heat Input (W)
100
120
Effect of Surface Tension on the Thermal Performance of Pulsating …
70 Average Condensor Temperature (°C)
Fig. 4 Average condenser temperature Vs heat input for water-based binary fluid at various surfactant concentration
205
500 ppm 1000 ppm
60
50
40
30
20
0
20
40 60 80 Heat Input (W)
100
120
linearly as heat input increases, as can be observed in Figs. 3 and 4. Greater density gradients in the PHP tubes are caused by the evaporator’s temperature increasing. The thermal resistance of PHP to surfactant-added water–acetone mixture decreases with increasing heat input. This is due to the simultaneous reduction of wall friction and concomitant reduction of thermal resistance compared to heat transfer. Due to the high saturation temperature and high specific heat, the evaporator tube wall temperature is high only for binary fluids. Figure 5 shows that the thermal resistance for a water–acetone surfactant solution declines with increasing heat input for all heat inputs; however, it is significantly higher for heat inputs between 500 and 1000 ppm surfactant solution. The trend in two-phase hydrodynamics is in 1000 ppm surfactant. Solution is not qualitatively different from the investigation of behavior in 500 ppm solution. However, there were some significant quantitative changes. As the surfactant concentration rises from 500 to 1000 ppm, quantity of bubbles in the PHP grows. Figure 5 shows that the rate of decreasing thermal resistance was found to be much higher for 500 ppm than for 1000 ppm. The graph’s nature is linearly declining; however, the water–acetone mixture’s nature is different for 1000 ppm surfactant solution. Figure 6 compares Manoj Kumar’s work and current work on thermal resistance. It is clear that the thermal resistance trends of two working fluids are essentially the same at 500 ppm and 1000 ppm solution. In a study by Manoj Kumar et al., thermal resistance is large at low input and then gradually decreases, but already reaches the lowest level of current research even at low infeed’s. From current research, 1000 ppm appears to have the lowest thermal resistance, with a value of 0.833 °C/W. In addition, the drying limit has been shown to increase with increasing surfactant concentration (it decreases with higher surface tension) in order to reduce the surface
206
1.5
500 ppm 1000 ppm
Thermal Resistance (°C/W)
Fig. 5 Average-thermal resistance versus heat input for water-based binary fluid at various surfactant concentration
V. K. Wasankar and P. R. Pachghare
1.3
1.1
0.9
0.7
0
20
40 60 80 Heat Input (W)
100
120
tension. As a result, pure distilled water (in Manoj Kumar et-al.’s study) reaches the dry out condition earlier because of its high surface tension, which needs a more heat to maintain the same regime of high surfactant solution. Beyond a specific surfactant concentration limit, the surface tension of the solution rises rather than falls. Water has the highest thermal resistance among all of the fluids. Hence, avoid the usage of surfactants at high concentrations. Pure working fluids at 1000 ppm solution, while water–acetone has the lowest thermal resistance with the same concentration. A significant reduction was observed at high heat inputs in pure water. 3.5
Thermal Resistance (°C/W)
Fig. 6 Comparison of present work result with Manoj Kumar et al.’s work result at various surfactant concentration
Present work (500 ppm)
3
Present work (1000 ppm)
2.5
Manoj Kumar et-al (500 ppm) [15] Manoj Kumar et-al (1000 ppm) [15]
2 1.5 1 0.5
0
20
40
60
80
Heat Input (W)
100
120
Effect of Surface Tension on the Thermal Performance of Pulsating …
207
5 Conclusions 1. The flow configuration of the two-phase mixture significantly changes on the addition of surfactant. After the filling procedure, it causes tremendous foaming which provide more stable fluid and modify its vapor–liquid interfacial properties which changes boiling mechanism. 2. The PHP typically starts in the evaporator with a low heat input when there is low surface tension. Two-phase flow likewise undergoes an early oscillatory to circulation transition. 3. Reduction of surface tension starts PHP at a low heat input within the evaporator. Surfactant reduces the evaporator temperature for an equivalent heat input. 4. As more heat is added, thermal resistance for pure as well as binary fluids with and without surfactant diminishes. Lower thermal resistance was obtained in each instance of their water-based binary combination with surfactant as compared to pure fluid. In comparison with pure fluid with surfactant solution, a water-based mixture, namely a water–acetone mixture with surfactant yields the best outcome about 17%. Acknowledgements The research has been financially supported by AICTE, New Delhi, under RPS (Grant No. 8-129/FDC/RPS (POLICY-l) 2019-20 Dated: 14/08/2020). The authors express their sincere thanks to all for support provided.
Nomenclature A Cp Di Eo¨ FR PHP ρl ρv g K
ΔT T sec Qin Qout Rth Te Tc Bo
Area (m2 ) Specific heat (KJ/Kg K) Inner Diameter (mm) Evotos Number (–) Filling Ratio (–) Pulsating Heat Pipe (–) Density of Liquid (kg/m3 ) Density of Vapor (kg/m3 ) Acceleration due to gravity (m/s2 ) Thermal Conductivity (W/m°C) Evaporator–Condenser (°C) Section Temperature Difference (W) Heat Input at Evaporator Section (W) Heat Input at Evaporator Section Thermal Resistance (°C/W) Average Evaporator Temperature (°C) Average Condenser Temperature (°C) Bond number (–)
208
σ
V. K. Wasankar and P. R. Pachghare
Surface Tension (N/m)
References 1. Akachi H (1990) Actronics Kabushiki Kaisha, structure of a heat pipe. U.S. Patent 441-41 2. Ma H. Oscillating heat pipe, mechanical and aerospace engineering. University of Missouri Columbia, MO, USA 3. Qu J, Wu H-Y, Cheng P (2010) Thermal performance of an oscillating heat pipe with Al2 O3 – water nanofluids. Int Commun Heat Mass Transfer 37(2010):111–115 4. Pachghare PR, Mahalle AM (2013) Effect of pure and binary fluids on closed loop pulsating heat pipe on thermal performance. Procedia Engineering 51:624–629 5. Pachghare PR, Mahalle AM (2014) Thermo-hydrodynamics of closed loop pulsating heat pipe: an experimental study. J Mech Sci Technol 28(8):3387–3394 6. Raghurami D (2017) Performance analysis of pulsating heat pipe using various fluids. Int J Eng Manuf Sci 7:281–291 7. Charoensawan P, Khandekar S, Groll M, Terdtoon P (2003) A closed loop pulsating heat pipes Part A: parametric experimental investigations. Appl Therm Eng 23:2009–2020 8. Khadekar S, Groll M (2004) An insight into thermo-hydraulic coupling in pulsating heat pipes. Int J Thermal Sci 43(1):1320. https://doi.org/10.1016/S12900729(03)00100-5 9. Zhang Y, Faghri A (2008) Advances and unsolved issues in pulsating heat pipe. Heat Transfer Eng 29:1320. https://doi.org/10.1080/01457630701677114 10. Han H, Cui X, Zhu Y, Sun S (2014) A comparative study of the behaviour of working fluids and their properties on the performance of pulsating heat pipe (PHP). Int J Thermal Sci 82:138–147 11. Singh B, Kumar P (2021) Heat transfer enhancement in pulsating heat pipe by alcohol-water based self-rewetting fluid. Thermal Sci Eng Progr 22:100809 12. Fumoto K, Kawaji M, Kawanami T (2008) Experimental study on a pulsating heat pipe with self-rewetting fluid 13. Wang J, Xie J, Liu X (2020) Investigation of wettability on performance of pulsating heat pipe. Int J Heat Mass Transfer 14. Patel VM, Gaurav, Mehta HB (2017) Influence of working fluids on start-up mechanism and thermal performance of a closed loop pulsating heat pipe. Appl Thermal Eng 110:1568–1577. https://doi.org/10.1016/j.applthermaleng.2016.09.017 15. Morgan AI, Bromley LA, Wilke CR (1949) Effect of surface tension on heat transfer in boiling. Ind Eng Chem 41(12):2767–2769. https://doi.org/10.1021/ie50480a025 16. Yang YM, Maa JR (1983) Pool boiling of dilute surfactant solutions. J Heat Transfer 105(1):190–192. https://doi.org/10.1115/1.3245541 17. Hetsroni G, Zakin JL, Lin Z, Mosyak A, Pancallo EA, Rozenblit R (2001) The effect of Surfactant on bubble growth, wall thermal patterns and heat transfer in pool boiling. Int J Heat Mass Transf 44(2):485–497 18. Wang XH, Zheng HC, Si MQ, Han XH, Chen GM (2015) Experimental investigation of effect of surfactant on performance of heat transfer of PHP. Int J Heat Mass Transfer 83:586–590 19. Xing M (2016) Experimental study on thermal performance of a pulsating heat pipe with surfactant aqueous solution. Int J Heat Mass Transfer 94:119–137 20. Wang J, Xie J, Liu X (2019) Investigation on the performance of closed loop pulsating heat pipe with surfactant. Appl Thermal Eng 10 21. Kumar M, Kant R, Das AK, Das PK (2019) Effect of surface tension variation of the working fluid on the performance of a closed loop pulsating heat pipe. Heat Transfer Eng 40(7):509–523. https://doi.org/10.1080/01457632.2018.1436390 22. B Markal, R Varol, Investigation of the effect of miscible and non-miscible fluids on the thermal performance of pulsating heat pipe. Heat Mass Transfer
Effect of Surface Tension on the Thermal Performance of Pulsating …
209
23. Zhu Y, Cui X, Han H, Sun S (2014) The study on the differences of the start-up and heat transfer performances of PHP with water-acetone mixture. Int J Heat Mass Transfer 77:834–842 24. Wasankar V, Pachghare P (2020) Effect of surface tension on the thermal performance of pulsating heat pipe: a review. Int J Res Eng IT Soc Sci 10(06):29–34. ISSN 2250-0588 (Impact factor: 6.565)
Study of Physical Characteristics of a Bi-porous Composite Capillary Wick for a Flat Miniature Loop Heat Pipe Toni Kumari and Manmohan Pandey
1 Introduction LHP is a two-phase passive heat transfer device that removes a high heat and transfers it over a long distance with a small temperature change from the heat source to the heat sink. In the 1970s, Russian scientist Maydanik proposed the first LHP for aircraft [2]. LHPs are extensively employed in a wide range of cooling applications, such as the cooling of modern electronic equipment, aircraft, and thermal control systems for spacecraft [3, 4]. Since LHP is driven by a capillary force, making the device passive requires no external power source and has no mechanical parts. As a result, LHP is an extremely reliable, auto-control device that requires no maintenance and it is gravity-independent. A schematic of LHP is shown in Fig. 1. It is composed of an evaporator, a condenser, a compensation chamber, and transport lines. A schematic of LHP is shown in Fig. 1. It is composed of an evaporator, a condenser, a compensation chamber, and transport lines. The evaporator consists of a capillary wick, which is a crucial element of LHP. Dissipated heat from the electronic device is taken by the evaporator surface which transfers heat to the working fluid. Evaporation of working fluid on the wick surface develops capillary pressure, which helps to circulate the fluid inside the loop, during circulation [5]. Heat gets rejected in the condenser. After that liquid returns to the reservoir, also called CC. However, in actual operation, only a portion of the dissipated heat is utilized to evaporate the working fluid. While the remaining dissipated heat might transfer back to the compensation chamber due to thermal conduction between the wick and wall, called heat leak. The heat leak could reduce the evaporation rate and increase the evaporator wall temperature. Therefore, it leads to start-up problems [6, 7] and temperature overshoot and undershoot.
T. Kumari (B) · 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_20
211
212
T. Kumari and M. Pandey
Fig. 1 Schematic diagram of LHP [1]
2 Literature Review and Objective Wu et al. [8] successfully applied two methods to deal with heat leaks (a) increasing the thickness of the wick (b) using a low thermal conductivity of wick material. Choi et al. [9] proposed a miniature loop heat pipe (mLHP) for electronic application and found that to dissipate high heat flux by maintaining evaporator temperature below 70º C, it needs low thermal conductivity of wick material. Therefore, to study the heat leak, many researchers worked on less conductive wick materials, such as titanium, nickel, and polytetrafluoroethylene [10–12]. Although low thermal conductivity wick material abates the heat leak problem [13], it also reduces the heat transport capacity. Singh et al. [14] compared copper with nickel wicks and found copper wick has higher heat leak. But they also found copper wicks have superior thermal performance than nickel wicks. Similar results were observed by Siedel et al. [15] and Zhou et al. [16]. A composite wick made of low and high thermal conductivity materials was proposed by some authors [17, 18] in order to take benefit of both high and low thermal conductivity wick materials. The wick’s pore size also has a conflicting impact on the performance of LHP, just like the wick’s thermal conductivity showed in the above literature review [19]. The small size of pores generates large capillary force. However, this also increases the flow resistance and chance of dry-out. Large pores facilitate fluid flow but weaken the sample strength and pumping capability. To solve the aforementioned issue, a biporous wick and a bi-porous composite wick were proposed [20, 21]. Chen et al. [22] studied the performance of bi-porous wick. They found that the device could start at a heat load as low as 2.5 W without any temperature oscillation and temperature overshoot. Wu et al. [23] designed a double-layer bi-porous wick. The outer layer of the wick was bi-porous and had large pores size, which helps in vapour escape. The inner layer of the wick was mono-porous and had small pores size, increasing the strength and pumping capability of the wick. The wick enhanced the maximum heat load by 67% compared to the mono-porous wick. Based on the current literature review, it can be said that the characterization of wicks has been studied, but the detailed characterization of the bi-porous composite
Study of Physical Characteristics of a Bi-porous Composite Capillary …
213
wick is limited in number compared to simple wicks. There are limited studies that considered the wettability and capillary rise of the bi-porous composite wick. Since wicks are the heart of loop heat pipes, it is important to implement and better understand the characteristics of composite bi-porous wicks. But, maintaining a precise balance of porosity, permeability, wettability, and pore radius are very challenging. This study presents the characterization studies such as capillary rise, wettability, porosity, pore radius, permeability, and XRD of the bi-porous copper–aluminium composite wick for a flat miniature loop heat pipe. Wettability, pore radius, and capillary rise are directly related to capillary pumping force, which play a major role to make the device passive. Therefore, current studies focussed on these factors in detail, which helped in analysing the performance of manufactured wick.
3 Methodology The first stage of manufacturing a wick is selecting the powder. The selection of powder is determined by a number of factors, including thermal conductivity, particle size, and compatibility with working fluid. Aluminium powder (Sigma- Aldrich) and copper powder (Sigma-Aldrich) with average particle sizes of less than 10 µm and less than 425 µm, respectively, were selected in the current study. As shown in Fig. 2, the Cu and Al powders were thoroughly combined with the PVA solution, a binding agent, at a volume ratio of 20%. In order to create bi-porous wicks, PVA also worked as a pore former during sintering. The powder was compacted using a mild steel reusable die and punch, which is more affordable than the conventional practice. To develop a composite layer, each powder was placed in the mould one at a time. The aforementioned setup was held in the UTM for 13 min under the compaction force of 160 KN. Then, the compressed samples are sintered in a muffle furnace at different temperatures for different time duration. At a higher temperature, the PVA in the sample evaporates and forms pores. In the furnace, the sintered wick sample was left to slowly cool for about 12 h. Then, the compressed samples are sintered in a muffle furnace at different temperatures for different time duration. At a higher temperature, the PVA in the sample evaporates and forms pores. In the furnace, the sintered wick sample was left to slowly cool for about 12 h.
4 Characterization of Wick Parameters 4.1 Capillary Rise Capillary rise is an important aspect to analyse the capillary performance of a porous wick in LHP. Figure 3 shows the visual images of the meniscus rising in the bi-porous composite wick, captured by an infrared (IR) camera (FLIR A655SC), at different
214
T. Kumari and M. Pandey
Fig. 2 The fabrication process of the bi-porous composite wick
times. Initially, the meniscus arose at a rapid rate, but after some time, as a result of the gravity impact of the lifted liquid, meniscus velocity began to decline. A physical test for the capillary rise in the wick was done as shown in Fig. 4. First, some liquid in a beaker was placed on the weighing balance machine and a reading was taken. Then, the liquid (acetone) surface was touched perpendicularly by the wick bottom surface. The liquid started rising along the wick due to the capillary suctions, and variation of reading in the weighing machine was recorded with time. By the conservation of mass, the machine reading difference at zero time and after reaching a steady state, reflected the capillary pumping mass. Since liquid acetone has been used, therefore, the evaporated acetone during testing has to be considered, as it easily evaporates at ambient conditions. The height of the capillary rise (H) and the capillary pumping mass (m) are correlated as given in Eq. (1). The calculated value of capillary height was found to be approximately 9.5 mm.
Study of Physical Characteristics of a Bi-porous Composite Capillary …
215
Fig. 3 Thermal (IR) images of capillary rise of the composite wick with acetone fluid at different times in second. a 0, b 15, c 120, d 300
Fig. 4 Schematic diagram of the capillary rise setup of bi-porous composite wick
H=
m ρ.ε.Aw
(1)
216
T. Kumari and M. Pandey
4.2 Wettability Contact angle (CA) is the angle between the solid–liquid and liquid–vapour interface. It indicates the wettability and the capillary force of a surface. Generally, two methods are used to calculate the static contact angle, such as the sessile drop method and the pendant drop method. In the present work, it was measured by a three-point tangent method using the sessile drop method with a volume of 4 µL. Figure 5 shows the images of drops formed on both sides of the porous composite wick surfaces (Al and Cu) with acetone and methanol fluids. Expression for the capillary force by the Young–Laplace equation is given below: ΔPc =
2σ cos θ r
(2)
From Eq. 2, it can be seen that capillary force depends upon surface tension (σ ), contact angle (θ ), and effective pore radius (r). The capillary force could be increased by increasing the surface tension of the fluid, decreasing the contact angle, and decreasing the effective pore radius. Also, a lower contact angle has higher wettability of fluid for the given surface. Therefore, acetone is a better choice of working fluid than methanol for the present working fluid.
Fig. 5 Contact angle images on wicks surfaces with different working fluids a aluminium versus acetone, 19.2°, b aluminium versus methanol, 42°, c copper versus acetone, 24.4°, d copper versus methanol, 53.3°
Study of Physical Characteristics of a Bi-porous Composite Capillary …
217
4.3 Porosity The porosity of the sintered wicks was measured by the Archimedes method [24]. In Archimedes’s method, three independent weights are required: (1) dry weight (md ), (2) weight of the saturated wick in the air (ma ), and (3) weight of the saturated wick in the liquid (mw ). First, the dry weight of the wick was measured using an electronic weighing machine. Then, the sample was soaked in liquid (deionized water) at 75 °C for about 8 h, till it got saturated. The porosity of the sample is calculated by the following relation (3): ε=
Vvoid ma − md = Vtotal ma − mw
(3)
where V void is the volume of the void and V total is the total volume of the sample. The porosity was found to be 42%. It was found that the porosity of the wick depends upon sintering parameters such as sintering temperature and sintering time. We found that by increasing either sintering temperature or time while keeping other parameters fixed, the porosity of the wicks decreases.
4.4 Permeability The permeability of the wick was determined by the Carman–Kozeny formula [24]: k=
d 2 ε3 180(1 − ε)2
(4)
The permeability of the wick was calculated to be about 0.031 µm2 .
4.5 Pore Size FESEM was used to study the surface morphology of the sintered wick such as pore size and pore structures. Figures 6 and 7 show the SEM morphology of the Al wick sample and Cu wick sample, respectively. Statistical analysis was done using ImageJ software, it was found that the pore size of the Al side wick varies from 2.36 to 10 µm, and the average pore size was calculated to be 6.82 µm. The pore size range of Cu side wick varies from 0.070 to 1.160 µm, and the average was found to be 0.38 µm.
218
T. Kumari and M. Pandey
Fig. 6 a Morphologyof the aluminium wick, b corresponding pore size distribution
Fig. 7 a Morphologyof the copper wick, b corresponding pore size distribution
4.6 XRD Analysis Figure 8 shows the X-ray diffraction (XRD) analysis of Al powder, Cu powder, composite sintered wick from the Al side, and composite sintered wick from the Cu side. Figure 8 clearly shows there is no other peak apart from Cu, Al, and some small. peaks of their oxides. Therefore, there is no residual amount of pore-forming agent (PVA) in the sintered wick. Some tiny peaks of metal oxides were seen in the sintered wick, which is good for LHP. It decreases the thermal conductivity of the wick, hence decreasing the heat leak and improving the performance of LHP.
Study of Physical Characteristics of a Bi-porous Composite Capillary …
219
Fig. 8 XRD analysis of Al powder, Cu powder, and composite wick
5 Conclusions Characterization of a flat, bi-porous copper–aluminium composite wick, 50 mm diameter, 4 mm thick, was done. Capillary rise, wettability, pore size, porosity, and permeability of the wick were determined. • It was found that acetone has a lower contact angle, making it a better choice of working fluid compared to methanol. • The capillary rise was visualized with the help of an IR camera and a 9.5-mm capillary rise was obtained using mass conservation. • The wick’s porosity and permeability were found to be 42% and 0.031 µm2 , respectively. • The pore sizes of Cu side wick and Al side wick vary from 0.07 µm to 1.16 µm and 2.365 µm to 10 µm, respectively. • The average pore size of the Al side wick and the Cu side wick was 6.82 µm and 0.38 µm, respectively. • The present bi-porous composite wick will effectively reduce heat leaks and is anticipated to enhance the LHP’s start-up performance and rate of evaporation. Acknowledgements The SEM imaging and XRD analysis were conducted at the Central Instruments Facility of IIT Guwahati.
220
T. Kumari and M. Pandey
Nomenclature LHP PVA SEM UTM CC d K CA Aw
Loop heat pipe Polyvinyl alcohol Scanning electron microscope Universal testing machine Compensation chamber Average powder diameter Permeability Contact angle The cross-section area of wick
References 1. Kumari T, Nashine C, Pandey M (2022) Fabrication of a cost-effective bi-porous composite wick for loop heat pipes. In: North-east research conclave. Springer, Singapore, pp 117–124 2. Ahmed S, Nashine C, Pandey M (2022) Thermal management at microscale level: detailed study on the development of a micro loop heat pipe. Micro Nano Eng 100150 3. Ahmed S, Pandey M, Kawaji M (2022) Loop heat pipe design: an evaluation of recent research on the selection of evaporator, wick, and working fluid. J Thermal Sci Eng Appl 14(7):070801 4. Siedel B, Sartre V, Lefèvre F (2015) Literature review: steady-state modelling of loop heat pipes. Appl Therm Eng 75:709–723 5. Coso D, Srinivasan V, Lu MC, Chang JY, Majumdar A (2012) Enhanced heat transfer in biporous wicks in the thin liquid film evaporation and boiling regimes. J Heat Transf 134(10) 6. Joung W, Yu T, Lee J (2008) Experimental study on the loop heat pipe with a planar bifacial wick structure. Int J Heat Mass Transf 51(7–8):1573–1581 7. Wu SC, Gu TW, Wang D, Chen YM (2015) Study of PTFE wick structure applied to loop heat pipe. Appl Therm Eng 81:51–57 8. Wu SC, Peng JC, Lai SR, Yeh CC, Chen YM (2009) Investigation of the effect of heat leak in loop heat pipes with flat evaporator. In: 2009 4th international microsystems, packaging, assembly and circuits technology conference. IEEE, pp 348–351 9. Choi J, Sano W, Zhang W, Yuan Y, Lee Y, Borca-Tasciuc DA (2013) Experimental investigation on sintered porous wicks for miniature loop heat pipe applications. Exp Thermal Fluid Sci 51:271–278 10. Vasiliev L, Lossouarn D, Romestant C, Alexandre A, Bertin Y, Piatsiushyk Y, Romanenkov V (2009) Loop heat pipe for cooling of high-power electronic components. Int J Heat Mass Transf 52(1–2):301–308 11. Huang BJ, Huang HH, Liang TL (2009) System dynamics model and startup behavior of loop heat pipe. Appl Therm Eng 29(14–15):2999–3005 12. Wu SC, Lee TJ, Lin WJ, Chen YM (2017) Study of self-rewetting fluid applied to loop heat pipe with PTFE wick. Appl Therm Eng 119:622–628 13. Wang S, Zhang W, Zhang X, Chen J (2011) Study on start-up characteristics of loop heat pipe under low-power. Int J Heat Mass Transf 54(4):1002–1007 14. Singh R, Akbarzadeh A, Mochizuki M (2009) Effect of wick characteristics on the thermal performance of the miniature loop heat pipe. J Heat Transf 131(8) 15. Siedel B, Sartre V, Lefèvre F (2013) Numerical investigation of the thermohydraulic behaviour of a complete loop heat pipe. Appl Therm Eng 61(2):541–553
Study of Physical Characteristics of a Bi-porous Composite Capillary …
221
16. Zhou W, Ling W, Duan L, Hui KS, Hui KN (2016) Development and tests of loop heat pipe with multi-layer metal foams as wick structure. Appl Therm Eng 94:324–330 17. Xu J, Zhang L, Xu H, Zhong J, Xuan J (2014) Experimental investigation and visual observation of loop heat pipes with two-layer composite wicks. Int J Heat Mass Transf 72:378–387 18. Xin G, Zhang P, Chen Y, Cheng L, Huang T, Yin H (2018) Development of composite wicks having different thermal conductivities for loop heat pipes. Appl Therm Eng 136:229–236 19. Yeh CC, Chen CN, Chen YM (2009) Heat transfer analysis of a loop heat pipe with biporous wicks. Int J Heat Mass Transf 52(19–20):4426–4434 20. Li H, Liu Z, Chen B, Liu W, Li C, Yang J (2012) Development of biporous wicks for flat-plate loop heat pipe. Exp Thermal Fluid Sci 37:91–97 21. Mathews AJ, Ranjan S, Inbaoli A, Kumar CS, Jayaraj S (2021) Optimization of the sintering parameters of a biporous copper-nickel composite wick for loop heat pipes. Materials Today: Proceedings 46:9297–9302 22. Chen BB, Liu W, Liu ZC, Li H, Yang JG (2012) Experimental investigation of loop heat pipe with flat evaporator using biporous wick. Appl Therm Eng 42:34–40 23. Wu SC, Huang CJ, Chen SH, Chen YM (2013) Manufacturing and testing of the double-layer wick structure in a loop heat pipe. Int J Heat Mass Transf 56(1–2):709–714 24. Dullien FAL (1991) Porous media: fluid transport and pore structure. Academic Press, New York
Influence of Air Injection on Cavitation in a Convergent–Divergent Nozzle Pankaj Kumar, Santosh Kumar Singh, Jaisreekar Reddy, and Mihir Shirke
1 Introduction Cavitation occurs when there is growth and burst of cavities in any liquid. As the pressure in a liquid flow falls below the saturation vapour pressure, the liquid is unable to withstand the tensile stress and form bubbles [1]. Understanding the physics underpinning the two-phase cavitation flow phenomena is critical in order to lessen the detrimental effects of cavitation or maximize its positive effects in nonchemical reactors, waste management and hydrodynamic cavitation. Since CD nozzle has the highest contraction ratio, it results in a generation of broader cavitation range. In that respect, investigating cavitation dynamics in basic geometries like CD nozzles is the best way to get there [2, 3]. Vena Contracta is the point where the static pressure is at its lowest and the velocity is at its highest. As a result, massive dynamic heads can form at the throat (Vena Contracta), lowering the static pressure to dangerously low levels [4]. When at the Vena Contracta pressure reaches a critical level, dissolved gas diffuses into the accessible nuclei, promoting the formation of nuclei (submicron bubbles). Reducing the exit pressure lowers the static pressure at the Vena Contracta to the liquid’s vapour pressure, forming a vapour cavity. Ukon [5] and Arndt et al. [6], respectively, carried out air injection analysis with a stationary foil to explore the cavitation dynamics interaction with air. Cavitation erosion testing showed that air injection was extremely successful in minimizing erosion. Pham et al. [7] used high-speed video and transducers which detect pressure fluctuations to analyse irregularities in cavitation and the control of these instabilities by means of air injection. A regulated number of bubbles were injected into a transparent venture nozzle to explore its effects on cavitation in jet fuels used for aviation P. Kumar · S. K. Singh (B) · J. Reddy · M. Shirke Department of Mechanical Engineering, SRM Institute of Science and Technology, Kattankulathur, Chennai 603203, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_21
223
224
P. Kumar et al.
[8]. They discovered that by injecting gas, the position of cavitation inception may be altered. By employing time-resolved X-ray densitometry measurements, a set of trails were conducted to explore how a stable sheet cavity can periodically transform to shredding cloud cavitation [9]. Hence, it is necessary to explore the effect of various techniques of injection of air, as it can prove to be an exceptional factor to enhance the cavity formation. In this paper, the cavitation phenomena are analysed with the help of cavity structures. Cavitation phenomena is explored with and without air injection in CD nozzle. Pressure distribution at different stages of cavitation are obtained and analysed with vapour fraction and turbulent kinetic energy. The symmetry of structure with characteristic length and frequency is noted.
2 Numerical Scheme The working medium consists of water and vapour as a homogenous mixture model. The continuity equation is as follows: δ (ρm ) + ∇.(ρm .→ um) = 0 δt
(1)
where ρm is the mixture density, and u→m is the mass-averaged velocity. The corresponding momentum equation for the mixture flow is written as follows: δ T (ρm u→m ) + ∇.(ρm u→m u→m ) = −∇p + ∇.[μm (∇ u→m + ∇ u→m )] δt + ρm g→ + F→
(2)
where μm is viscosity of mixture, g→ is the acceleration due to gravity, and F→ is the external force on body. The velocity components in the above equation are replaced by the mean and instantaneous values in Reynolds averaged (RANS) methods, u→ = u→ + u'. After that, an ensemble average is calculated. The impacts of turbulence are represented by additional words as a result of the averaging procedure. δxδ j (−ρu i u j ) is known as the Reynolds stress equation. The incorporated Reynolds stress terms necessitate the use of extra mathematical models in order to close the momentum equation. RSM is one method, which entails solving distinct transport equations for each of the added Reynolds stresses. The SST k-ω model has been used as the basis for this research. The turbulent viscosity, t, in the k- SST model is defined as follows: μt =
1 ρk ω max 1,SF αa1 ω
(3)
Influence of Air Injection on Cavitation in a Convergent–Divergent Nozzle
225
where α is a Reynolds number-dependent input parameter. Following that, the transport equations for k and ω are expressed as follows: δ δ δk δ ⎡k + G k − Yk + Sk (ρk) + (ρkμi ) = δt δxi δx j δx j
(4)
δ δ δω δ (ρω) + (ρωμi ) = = ⎡ω + G ω − Yω + Sω δt δxi δx j δx j
(5)
and
Y stands for turbulence and dissipation, ⎡ for effective diffusivity, G stands for generation. Sk and Sω are source terms that have been defined. The Rayleigh–Plesset equation is used to model cavitation, which is a mass transfer mechanism. The Singhal cavitation model has been adopted in this study due to its capacity for the consideration of air, water, and water vapour. Multiphase can be accounted for using this model. Water is used as the primary phase, and water vapour and air are used as secondary phases in the simulations. The inlet boundary condition was velocity and pressure outlet at outlet with operating pressure was kept at 80 kPa. Transient state simulation has been used here, with time step size in the order of 10−5 s. The SIMPLE algorithm was used to tackle the pressure velocity coupling problem. The momentum, k, and ω were discretized using the second-order method, while vapour transport was discretized using the first-order technique. To discretize pressure, the PRESTO! (Pressure Staggering Option) system was adopted. Cavitation number is used as a criterion for comparing cavitating device performance. The cavitation number is calculated as follows: Cavitation Number (σ ) =
P2 −Pv ρv 2 2
P2 = Upstream fluid pressure, Pv = Vapour pressure, ρ = Fluid density, v = Fluid velocity at throat. The local vapour fraction is determined using a vapour mass fraction ( f ) transport equation in the Singhal cavitation model. The rate of vapour phase condensation (Rc ), the rate of evaporation (Re ). These terms are defined as follows in the model: ∂ (ρmix f ) + ∇.(ρmix u→mix f ) = ∇.(⎡∇ f ) + Re − Rc ∂t (α) The vapour volume fraction: α= f
ρmix ρVapoor
(6)
226
P. Kumar et al.
Modified Rayleigh–Plesset equation: ∂ (ρmix f ) + ∇.(ρmix u→mix f ) = (n4π )1/3 ∂t ρvapor. ρliq 2 PB − P∞ 1/2 + (3α)2/3 ρmix 3 ρliq
(7)
(Re ) mass source term Re = C 1
1.0,
√ k
σ
1 2 pvapor − p 2
ρvapor ρliq 1 − fv − fg 3 ρliq
(8)
2.1 (8) (Rc ) mass sink term. Rc = C 2
√ max 1.0, k σ
ρvapor ρliq
2 pvapor − p 1/2 fv 3 ρliq
(9)
Cavity dynamics model: 3γ R0 d2 R 3 dR 2 1 2σ . R 2 + = [ PB + dt 2 dt ρliq R R 4μmix dR 2σ − . − P∞ − R R dt
(10)
The chosen geometry is depicted in Fig. 1 which is similar to the simulation carried by Doltade et al. [10]. Further, air is injected here used to extend this study. The CD nozzle has a 2 mm throat diameter and a 20 mm pipe diameter. The CD nozzle has a half converging angle of 22.61° and a half diverging angle of 6.4°. The parallel and flat sides of the CD regions. To produce a properly developed stream and eliminate any entry or escape effects, the CFD model was extended by 60 mm before the intake and 200 mm after the outlet. The mesh has an impact on the simulation’s accuracy, convergence so structured mesh with fine near throat and coarse near inlet and outlet was adopted. The following factors were used to optimize the mesh, keeping the element quality closest to 1. The best fit after iterating the sizing was obtained by setting it to a maximum of 2.5e−4 mm and minimum of 1.5e−4 mm. The grid independence study was carried out to optimum mesh size between 35,000, 55,000, 72,000. It is found that medium mesh size 55,000 is sufficient to capture the flow structures. There is no significant variation between 55,000 and 72,000. The element
Influence of Air Injection on Cavitation in a Convergent–Divergent Nozzle
227
Fig. 1 Computational domain a full view, b axisymmetric view
size has been set to 2.7e−4 mm which represent number of nodes are 55,000. The meshing was chosen two-dimensional axisymmetric domain. Using Ansys Fluent 18.2, simulations were done in transient state. Three different velocities were considered to get inception, sheet cavitation and cloud cavitation, i.e. 0.125, 0.15, 0.35 m/s. For Aerated flow, two different air velocities were chosen 0.6 and 0.8 m/s. The operating pressure was taken to be 80,000 Pa. Numerical simulations are compared with the simulation of Doltade et al. [10] at similar conditions. The velocity magnitude is observed 40 m/s, and vapour fraction value is 0.8 as shown in Fig. 2 which is well matching with the reference paper, and hence, numerical scheme is found suitable to capture the cavity dynamics.
3 Results and Discussion Three separate regimes can be defined depending on the cavitation number: inception cavitation (σ /σ I = 1), sheet cavitation (σ /σ I = 0.77, 0.75), and cloud cavitation (σ / σ I = 0.65) appear as the cavitation number lowers where (σ i ) is inception case and its value is 1.693. There is an increase in cavity length with the lowering of cavitation number. This is also further extended with air injection. There is an attempt for fast Fourier transformation of the variation of cavity length with time and found peak frequency for aerated case is lower that the peak frequency of non-aerated case. This represents that the life cycle time is more for the aerated case as shown in Fig. 3 which is consistent with cavity elongation. Further, it is also noticed here that with air injection the cavitation peak intensity value also increases. The different cavity length observed are listed in Table 1. Vapour fraction and turbulent kinetic energy (TKE) contour for different cases are plotted and shown in Fig. 4. At the cavitation number close to 1, it is difficult to see the cavity and mostly research observed through acoustic nature. With increase in the velocity, cavitation number reduces and cavity length can be visualized. These are
228
P. Kumar et al.
Fig. 2 Validation of numerical simulation, a velocity magnitude, b vapour fraction where (a, b) Reference paper [10] and (a', b' ) current simulation
shown in Fig. 4 (σ /σ I = 0.77, 0.75, 0.65). There is a consistent relation found between vapour fraction and turbulent kinetic energy. There is a significant reduction of TKE for the zone of formation of vapour fraction which in turn reveal that cavitation produces at the vortex core where velocity is minimum.
4 Conclusions The impact of important design and operational aspects of cavity dynamics was studied numerically. Three different cavitation regimes were investigated in this paper: (1) inception cavitation, (2) ‘sheet cavitation’, and (3) ‘cloud cavitation’. Air bubbles have been injected to further aerate those regimes. The cavitation number was seen reducing while the flow discharge velocity has been kept constant for all flows. When decreasing the cavitation number, there is a prominent increase in the cavitation intensity, i.e. cavitation volume, cavity length. The turbulent kinetic energy shows increase with decrease in cavitation number. Increase in air mass flow rate the cavity increases, decreasing cavitation number, increasing length and bubble intensity. As the air injection rate rises, so does the size of the associated cavity and the shedding bubble cluster. In FFT analysis, the peak frequency took a drop after introduction of air which concludes that there is an increase in cavitation intensity.
Influence of Air Injection on Cavitation in a Convergent–Divergent Nozzle
229
Fig. 3 Frequency spectrums for cavitation number 0.65 a without air, b with air flow velocity 0.6m/ Table 1 Accuracy of aerodynamic-independent variables
Cavitation number
Cavity length (mm)
(σ /σ I = 1)
0.7
(σ /σ I = 0.77)
3
(σ /σ I = 0.75)
7
(σ /σ I = 0.65)
12
(σ /σ I = 0.65)
16
230
P. Kumar et al.
Fig. 4 Typical snapshot of vapour fraction and turbulent kinetic energy at different cavitation number
The method and results given here can be used to design and improve venturi as hydrodynamic cavitation devices.
Nomenclature P2 P2 ρm v ν Rc Re g→ (α) μm F→ σ u→m CD
Upstream fluid pressure Vapour pressure Mixture density Fluid velocity at throat Kinematic viscosity Vapour phase condensation Rate of evaporation Acceleration due to gravity Vapour volume fraction Viscosity of mixture External body force Cavitation number Mass-averaged velocity Convergent–divergent
References 1. Brennen CE (2015) Cavitation in medicine. Interface Focus 5(5):20150022 2. Sarvothaman VP, Simpson AT, Ranade VV (2018) Modelling of vortex based hydrodynamic cavitation reactors. Chem Eng J 377:119639
Influence of Air Injection on Cavitation in a Convergent–Divergent Nozzle
231
3. Carpenter J, Badve M, Rajoriya S, George S, Saharan VK, Pandit AB (2017) Hydrodynamic cavitation: an emerging technology for the intensification of various chemical and physical processes in a chemical process industry. Rev Chem Eng 33:433–468 4. Tomov P, Khelladi S, Ravelet F, Sarraf C, Bakir F, Vertenoeuil P (2016) Experimental study of aerated cavitation in a horizontal venturi nozzle. Exp Thermal Fluid Sci 70:85–95 5. Ukon Y (1986) Cavitation characteristics of a finite swept wing and cavitation noise reduction due to air injection. In: Proceedings of the international symposium on propeller and cavitation, vol 23, pp 383–390 6. Arndt REA, Ellis CR, Paul S (1995) Preliminary investigation of the use of air injection to mitigate cavitation erosion. J Fluids Eng 117(3):498 7. Pham TM, Larrarte F, Fruman DH (1999) Investigation of unsteady sheet cavitation and cloud cavitation mechanisms. J Fluids Eng 121(2):289–296 8. Dunn PF, Thomas FO, Davis MP, Dorofeeva IE (2010) Experimental characterization of aviation-fuel cavitation. Phys Fluids 22(11):117102 9. Ganesh H, Makiharju SA, Ceccio SL (2016) Bubbly shock propagation as a mechanism for sheet-to-cloud transition of partial cavities. J Fluid Mech 802:37–78 10. Doltade SB, Dastane GG, Jadhav NL, Pandit AB, Pinjari DV, Somkuwar N, Paswan R (2019) Hydrodynamic cavitation as an imperative technology for the treatment of petroleum refinery effluent. J Water Process Eng 29:100768. Lamb H. Hydrodynamics, 6th edn. Cambridge University Press 1932 (Reprinted by Dover, 1945)
Investigation of the Droplet Impingement on a Hydrophobic Surface with a Fixed Particle K. Niju Mohammed, P. S. Tide, Franklin R. John, A. Praveen, and Ranjith S. Kumar
1 Introduction Droplet impingement studies play a pivotal role in ex- plaining a wide spectrum of activities including natural phenomena, chemical and industrial processes, and biological applications. These include studies on phenomena like rainfall [16], soil erosion [4] and droplet impact on lotus leaves [5]. Practical applications like combustion [13], anti- icing [11], spray painting [6], inkjet printing [9] and trickle bed reactors [7] play a vital role in industries. Biological applications include biochemical analysis [17], anti-microbial [8] processes and so on. Single droplet impingement studies are significant as it deals with the droplet spreading characteristics and contact time with the surface [14]. The major nondimensional parameters which governs the problem include impact Weber number and spread factor. Another influential factor in the droplet impact studies is the residence time, which is the time period for which the droplet is in contact with the surface. Distinct phenomena occur during the droplet impingement on solid surfaces such as pinning, rebounding and fragmentation [3]. Various factors such as droplet viscosity, density, surface tension, velocity and surface texture affect the result of droplet impact. Studies on homogeneous surfaces with uniform wettability are well-documented. Droplet impact on hydrophilic, hydrophobic and super hydrophobic flat and spherical surfaces are studied in detail [1, 2, 12]. Even though compound wettability surfaces have worthwhile applications, literature review implies that the fundamental K. Niju Mohammed (B) · P. S. Tide · F. R. John · A. Praveen Department of Mechanical Engineering, School of Engineering, Cochin University of Science and Technology, Cochin, Kerala, India e-mail: [email protected] K. Niju Mohammed · A. Praveen · R. S. Kumar Department of Mechanical Engineering, College of Engineering Trivandrum, Trivandrum, Kerala, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_22
233
234
K. Niju Mohammed et al.
knowledge about the behaviour of droplet impingement on compound wettability surfaces is scarce. The objective of the present work is to understand the fundamental processes involved in the droplet impingement on compound wettability surface. The compound wettability surface has wide spectrum of applications such as droplet-based cell culturing, proteomics, antifogging, selective cooling or heating, self-cleaning, etc. [15]. In the current work, a comparative study on hydrodynamics of droplet impact on hydrophobic surface fixed with and without hydrophilic spherical particle at different Weber numbers has been investigated. The study brings out the influence of surface wetness on droplet impact hydrodynamics. The variable wettability surfaces have significant effect on spreading characteristics and the resulting droplet impact dynamics. Unlike other studies in the literature, our study focuses on the combined influence of hydrophilic sphere and hydrophobic surface which acts a compound wettability surface on droplet impingement. Also the influence of varying Weber numbers on droplet collisions has been investigated to have a thorough understanding of the underlying phenomenon. Note that the particle size is 1 mm which is less than the droplet diameter of 2.7 mm. The ratio of diameter of particle to diameter of droplet above one has been done previously by other researchers, but the ratio below one has not seen attempted [10].
2 Methodology Test specimens consist of (1) simple hydrophobic slide and (2) hydrophobic slide attached with hydrophilic sphere. At first, a hydrophobic surface is made using an acrylic slide coated with Neverwet™, a commercial hydrophobic coating. Acrylic slide is first cleaned with acetone and then with water and dried for 2–3 min. The Neverwet base coating is applied on top of the cleaned acrylic slide and kept for 30 min. After that, two coats of top coating is applied above the base coated surface and allowed to dry for half an hour. Thus, the hydrophobic surface is ready for conducting experiments. For the realisation of hydrophobic surface with hydrophilic particle, the spherical-shaped mustard seed is attached on the top of hydrophobic slide. The experimental setup mainly consists of test specimens, droplet dispensing system, high-speed camera, backlight for illumination and a computer for data storage and analysis. The schematic diagram of test setup is shown in Fig. 1. Experiments are carried out isothermally at a temperature of 28 °C and atmospheric pressure. Deionised water is used as the working fluid for the experiments with a density of 998 kg/m3 . Deionised water droplets were generated using a burette fitted with a flat tipped hypodermic needle. The droplet dispensed from the needle under the influence of gravity is of the size of 2.7 mm. A high-speed camera (Hi-Spec 2, Fastec imaging, USA) equipped with a Tamron 90 mm f/2.8 macro-lens is used to capture the droplet impact morphology evolution sequence. The images are recorded at 4500 frame per second. An LED back light
Investigation of the Droplet Impingement on a Hydrophobic Surface …
235
Fig. 1 Schematic diagram of experimental setup
illumination system is employed, and the images are captured to understand the physics of droplet impingement. To calibrate the length measurement, a photograph of a measuring scale is converted into frame pixels employing the open software Image J V 1.53 a from NIH, USA. The calibration error in measurement of pixel is found out to be 14.18 pixels with a length scale of 1.0 mm. The diameter of the droplet is found to be 2.7 mm by using this technique. The experiments are carried out for different heights of impact quantified by non-dimensional Weber number given by We = ρv2 D0 , where ρ is the density of the fluid, v is the droplet impact velocity, D0 is the initial diameter of the droplet, and σ is the surface tension of the fluid. The velocity of impact is the velocity at which the drop strikes the surface, and the residence time is the time duration between impact and rebound of the droplet. The velocity of impact and residence time of the droplet are estimated from the analysis of spatial and temporal data. The static contact angle for the droplet on hydrophobic surface is estimated to be 146°.
3 Results and Discussion The monodispersed droplets produced from the needle are impinged on the hydrophobic surface with and without particle separately. The impact of droplet is done at a temperature of 28 °C for the Weber numbers ranging from 5.2 to 109.The droplet deformation parameters are quantified from the images of high-speed camera. The initial droplet diameter is measured to be 2.7 mm. The impact velocity and residence time were calculated from the images depicting both spatial and temporal data. The impact velocity depends on the height of the droplet impact.
236
K. Niju Mohammed et al.
(a) 0ms
2ms
6ms
8ms
10ms
12ms
0ms
4ms
8ms
12ms
14ms
17.5ms
0ms
2ms
6ms
8ms
10ms
12ms
(b)
(c)
Fig. 2 Comparison of morphological evolution of droplet impingement on flat hydrophobic surface at Weber number a 5.20, b 42.25 and c 109 (Scale bar indicates 2.2 mm)
The monodispersed droplet after hitting the hydrophobic surface spreads and attains maximum diameter (Dmax ). After reaching the maximum diameter, the droplet retracts and undergo three different outcomes based on various Weber numbers. The morphological evolution of droplet with time is shown in Fig. 2. The distinct regimes visible in this scenario for different Weber numbers are (1) rebounding, (2) jetting and (3) fragmentation. For low Weber numbers of 5.2 and 12.6, rebounding of droplets is observed. For medium Weber numbers of 42.3 and 79.3, jetting of pilot drops is observed from the liquid mass. For higher Weber numbers of 94.1 and 109, fragmentation of droplet happens. During the rebounding state, the retraction of droplet occurs after maximum spreading and bounces of the surface without pinning. Jetting of secondary droplet takes place along with the rebounding phenomenon. Fragmentation or splashing of droplet at higher Weber numbers occurs due to high momentum of droplet and the resultant increases in kinetic energy. Consequently, the inertia forces dominate over surface tension forces and droplet fragments. The temporal evolution image sequence of droplet impact on spherical particle for various Weber numbers are shown in Fig. 3. For low Weber number of 5.2 and medium Weber number of 42.3, as the droplet impinges on this compound wettability surface, it spreads to maximum diameter and then retracts. During retraction, liquid droplet forms a liquid pillar like structure and it oscillates for some time until it becomes a stagnant droplet. The droplet has a tendency to rebound but remained pinned on the surface due to the high wettability or hydrophilic nature of the spherical particle. For higher Weber number of 109 splashing is observed. The droplet disintegrates into secondary droplets while trying to spread. This is due to increase in kinetic energy at higher Weber number similar to droplet impingement on flat plate. Spreading factor is an important parameter in single droplet impact dynamics. Spreading factor, β = D/D0 , is given by the ratio of instantaneous droplet diameter (D) during the course of impact to the diameter of the droplet before impact (D0 ). As in Fig. 4, the impact of droplet on the flat surface leads to increase in diameter with the evolution of time. When the Weber number is larger (We = 42.25), the spreading factor increases due to the increased potential energy. This develops larger inertia
Investigation of the Droplet Impingement on a Hydrophobic Surface …
237
(a) 0ms
4ms
8ms
12ms
16ms
20ms
0ms
4ms
8ms
12ms
16ms
20ms
0ms
4ms
8ms
12ms
16ms
20ms
(b)
(c)
Fig. 3 Comparison of morphological evolution of droplet impingement on sphere embedded hydrophobic surface at Weber number a 5.20, b 42.25 and c 109
forces which leads to greater spreading than at lower Weber number (We = 5.20). For sphere embedded surfaces also for higher Weber number (We = 42.25), the spread factor shows a rise in the early stage due to higher potential energy compared to the energy at lower Weber number (We = 5.20).
4 Conclusions An experimental investigation on droplet impingement was performed, and images of the droplet impact on flat hydrophobic surface with and without particle were captured. The nature of droplet impact regime shifts from bouncing in the pure hydrophobic flat surface to pinning in the compound wettable hydrophilic spherical particle embedded hydrophobic surface. For lower Weber numbers, bouncing and jetting of droplet takes place on hydrophobic flat surfaces, whereas pinning occurs on compound wettability surface. At higher Weber numbers, splashing occurs on both surfaces. Based on parameters such as spreading factor and Weber number, distinct regimes of droplet impact such as pinning and splashing are observed on compound wettability surface.
Fig. 4 Variation of spreading factor with time for a flat surface for We = 5.20 and 42.25 and b sphere embedded hydrophobic surface for We = 5.20 and 42.25
238 K. Niju Mohammed et al.
Investigation of the Droplet Impingement on a Hydrophobic Surface …
239
Nomenclature We β D D0 t
Weber number Spreading factor Instantaneous drop diameter Initial drop diameter Time in milliseconds
References 1. Aria AI, Gharib M (2014) Physicochemical characteristics and droplet impact dynamics of superhydrophobic carbon nanotube arrays. Langmuir 30(23):6780–6790 2. Banitabaei SA, Amirfazli A (2017) Droplet impact onto a solid sphere: Effect of wettability and impact velocity. Phys Fluids 29(6):062111 3. Chen L, Xiao Z, Chan PCH, Lee Y-K, Zhigang L (2011) A comparative study of droplet impact dynamics on a dual-scaled superhydrophobic surface and lotus leaf. Appl Surf Sci 257(21):8857–8863 4. Dunkerley D (2021) Rainfall drop arrival rate at the ground: a potentially informative parameter in the experimental study of infiltration, soil erosion, and related land surface processes. CATENA 206:105552 5. Ensikat HJ, Ditsche-Kuru P, Neinhuis C, Barthlott W (2011) Superhydrophobicity in perfection: the outstanding properties of the lotus leaf. Beilstein J Nanotechnol 2(1):152–161 6. Garbero M, Vanni M, Fritsching U (2006) Gas/surface heat transfer in spray deposition processes. Int J Heat Fluid Flow 27(1):105–122 7. Gunjal PR, Ranade VV, Chaudhari RV (2005) Dynamics of drop impact on solid surface: experiments and VOF simulations. AIChE J 51(1):59–78 8. Habib S, Zavahir S, Abusrafa AE, Abdulkareem A, Sobolcˇiak P, Lehocky M, Vesela D, Humpol´ıcˇek P, Popelka A (2021) Slippery liquid-infused porous polymeric surfaces based on natural oil with antimicrobial effect, Polymers 13(2):206 9. Huang Y, Jiang L, Li B, Premaratne P, Jiang S, Qin H (2020) Study effects of particle size in metal nanoink for electrohydrodynamic inkjet printing through analysis of droplet impact behaviors. J Manuf Process 56:1270–1276 10. Liu X, Zhang X, Min J (2019) Maximum spreading of droplets impacting spherical surfaces. Phys Fluids 31(9):092102 11. Lv J, Song Y, Jiang L, Wang J (2014) Bio-inspired strategies for anti-icing. ACS Nano 8(4):3152–3169 12. Meng K, Jiang Y, Jiang Z, Lian J, Jiang Q (2014) Impact dynamics of water droplets on cu films with three-level hierarchical structures. J Mater Sci 49(9):3379–3390 13. Moreira ALN, Moita AS, Panao MR (2010) Advances and challenges in explaining fuel spray impingement: How much of single droplet impact research is useful? Prog Energy Combust Sci 36(5):554–580 14. Simhadri Rajesh R, Naveen PT, Krishnakumar K, Kumar Ranjith S (2019) Dynamics of single droplet impact on cylindrically-curved superheated surfaces. Exp Thermal Fluid Sci 101:251– 262 15. Satpathi NS, Malik L, Ramasamy AS, Sen AK (2021) Drop impact on a superhydrophilic spot surrounded by a superhydrophobic surface. Langmuir 37(48):14195–14204 16. Verma AS, Castro SGP, Jiang Z, Teuwen JJE (2020) Numerical investigation of rain droplet impact on offshore wind turbine blades under different rainfall conditions: a parametric study. Compos Struct 241:112096
240
K. Niju Mohammed et al.
17. Wu T, Luo Z, Ding W, Cheng Z, He L (2017) Monodisperse droplets by impinging flowfocusing. Microfluid Nanofluid 21(8):1–6
Effect of Impact Velocity on Spreading and Evaporation of a Volatile Droplet on a Non-porous Substrate Amit Yadav, Allu Sai Nandan, and Srikrishna Sahu
Nomenclature d max h te R˙ regime
Maximum spread diameter [mm] Free fall height of the droplet [cm] Evaporation time [sec] Average rate of spreading in a regime
[mm/sec]
1 Introduction Understanding the simultaneous spreading and evaporation of a droplet after its impact on a surface is important in several fields of interest ranging from industrial to biologic applications like transdermal drug delivery, cooling applications, and inkjet printing [1–3]. In the past, numerous studies have been reported on the evaporation of a pinned sessile droplet and found that the evaporation rate strongly depends on the base radius, concentration difference of saturated vapor at the air–liquid interface and ambient air, and contact angle [4–6]. A mathematical model for the evaporation rate of a pinned sessile droplet appears in Eq. (1) [1] as follows: dV (t) = −α f (θ )R(t) dt
(1)
V is droplet volume, and R is the base radius of the sessile droplet, both of which vary with time ‘t.’ f (θ ) is a function of the contact angle θ , and α is a constant that depends on the ambient and liquid temperature, properties of the liquid, and its vapor. A. Yadav (B) · A. S. Nandan · S. Sahu Department of Mechanical Engineering, IIT Madras, Madras 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_23
241
242
A. Yadav et al.
Numerical simulations of the same process reveal that the mass flux of evaporation is highest at the three-phase contact line [4]. In the previous works on the simultaneous spreading and evaporation of a droplet on a non-porous substrate, a theoretical model was developed to predict the droplet base radius, contact angle, and the mass evaporated with time. It was assumed that the evaporation rate in the spreading case is the same as Eq. (1). The time evolution of the predicted droplet size agreed well with the experimental data [7]. In the case of complete wetting liquids, it was experimentally deduced that the mass evaporated varies linearly with time, the same as observed for a pinned droplet [8]. In the case of simultaneous spreading and evaporation on a non-porous substrate, two basic processes are in action, viz. spreading due to surface forces (interfacial forces at the liquid–gas and three-phase contact line) and contraction due to evaporation. In a complete wetting case (Young’s equilibrium contact angle is zero), initially, the droplet spreads due to the surface forces. Although evaporation is present during this stage, it is not substantial. As the droplet spreads to a limiting radius, contraction due to liquid evaporation is dominant compared to surface forces, and the droplet vaporizes completely after some time [7]. In most previous studies, the evaporation of a sessile droplet (both pinning and spreading cases) was considered where the droplet was gently placed on the substrate. However, in many practical applications like spray cooling, inkjet printing, spray painting, etc., the droplets impact the surface with a finite velocity. In the previous studies, little attention is given to analyze the impact velocity effect on spreading and evaporation dynamics. In this work, the objective is to experimentally study the variation in evaporation time and spreading of a droplet due to its velocity of impact on a non-porous substrate. The droplet impact is visualized using an optical setup. The images are processed to evaluate and analyze the time variation of droplet size.
2 Materials and Methods 2.1 Experimental Setup Figure 1 shows a schematic of the experimental setup. A syringe pump (NE-1000/ Syringe One) was used to generate a single droplet in each experiment. The inner diameter of the syringe is 1 mm for all experiments, so the droplet size is almost similar in all cases. Ethanol (99.9% pure) was used as the working liquid. The droplet’s weight produced by the syringe pump was measured using an analytical balance (Contech CAI-304). The initial weight of the ethanol droplet was found to be 3.8 ± 0.1 mg with a diameter of 2.1 ± 0.03 mm. A flat and smoothened glass plate 50 mm × 50 mm × 4 mm was used as the substrate. The droplet was allowed to fall onto the substrate from different heights, h (= 6.5, 20, and 32 cm). Droplet free fall height was restricted to 32 cm to avoid splashing. For h (= 0 and 6.5 cm),
Effect of Impact Velocity on Spreading and Evaporation of a Volatile …
243
Fig. 1 Schematic of the experimental setup for droplet impact and evaporation study
manual intervention was done to remove stand holding needle as soon as the droplet was released. Experiments were also conducted for the sessile droplet case, where the droplet was gently placed on the substrate. For each height, the experiment was repeated four times. A DSLR camera (Nikon D7000) was used to record the droplet images from the top at 24 fps. A DC LED light source (power: 12W) with a diffuser plate was used to illuminate the sessile droplet from the top.
2.2 Image Processing From the recorded video, the images are extracted and processed using MATLAB to obtain the time evolution of droplet size, as explained below and depicted in Fig. 2.
Fig. 2 Image processing of a droplet image
244
A. Yadav et al.
i. Subtract the background image from the droplet image to get the background subtracted image. ii. Convert background subtracted image to grayscale. iii. Binarize the grayscale image. iv. regionprops function in MATLAB is used to get an equivalent circular diameter in pixels. v. Calibration has been done to obtain the spatial resolution of the image. Examples of droplet image, background subtracted image, and binary image are illustrated in Fig. 2. The initial time t = 0 s corresponds to the time when the droplet impacts the substrate. The time t = t e , refers to the total time of droplet evaporation obtained by considering the instant when the total intensity of the binarized image is zero. To validate the method used to find the evaporation time using image processing, the evaporation time was measured by time tracking the weight of the droplet using the analytical balance. It was observed that the time taken for 97% of droplet mass evaporation according to the balance reading matched with the evaporation time obtained using the image processing method.
3 Results and Discussion 3.1 Regimes of Droplet Impact Outcome The process following the droplet impact can be described in terms of four regimes, as shown in Fig. 3, which are (i) inertial regime, (ii) spreading, (iii) slow contraction, and (iv) rapid contraction. This is explained below. Inertial Regime: Just after the droplet impacts the surface, a large part of its kinetic energy is utilized to accelerate the fluid in the radial direction. Due to this, a sudden increase in its size can be observed from the sudden rise in droplet size, as shown in Fig. 3. The inertial regime time scale is of the order of 10 ms. Spreading: After viscous forces dampen inertial forces, droplet spreads due to the action of interfacial forces at the three-phase contact line. The spreading stage continues until the maximum size, as depicted in Fig. 3. In this regime, evaporation of the droplet is not substantial [7]. Slow Contraction: From Fig. 3, we can observe that its size does not change considerably after the droplet spreads to the maximum diameter. This event can be seen as a curve with a slight curvature in Fig. 3. It might be because of spreading rate due to interfacial forces at the three-phase contact line which is similar to the contraction rate due to evaporation. Rapid Contraction: The evaporation rate becomes significant at the contact line, and the droplet vaporizes. It appears as if the droplet is contracting [7].
245
d(mm)
Effect of Impact Velocity on Spreading and Evaporation of a Volatile …
t(s)
Fig. 3 Variation of diameter with time for droplet impact height, h = 32 cm
3.2 Influence of Droplet Height/Velocity on Impact Outcome In Fig. 4, images of the droplet are shown at different time instants for varying free fall heights. The overall evolution of a droplet after impact and the different regimes of the outcome are similar to Fig. 3. Nevertheless, the difference occurs in the rate of change of droplet size and evaporation time, as discussed below. Table 1 shows the average rate of change in droplet size ( R˙ regime ) for different regimes at different free fall heights. Positive and negative signs indicate expansion and contraction, respectively. R˙ regime is calculated as shown below:
Fig. 4 Images of a droplet at different time instants with different free fall heights
246
A. Yadav et al.
dfinal − dinitial R˙ regime = t dfinal − Diameter at the end of a regime dintial − Diameter at the start of a regime t − Time interval of a regime. From Table 1, one can notice that, as h increases R˙ I increases. R˙ II is the highest for no impact case (h = 0 cm), while all other impact cases have similar average spreading rates, for R˙ III and R˙ IV are not showing any particular trend with change in h; however, both are least for no impact case. Figure 5 presents a comparison of the time variation of droplet diameter for different impact heights and for a case when a droplet is gently placed on the substrate. It can be seen from these plots that after the inertial regime, all the curves follow a similar trend till maximum spreading is achieved. However, after that, the plot corresponding to h = 0 cm diverges from other plot trends taking more time to evaporate than in other cases. Table 1 Variation of the average rate of change in size in different regimes Spreading R˙ II (mm/s)
Slow contraction R˙ III (mm/s)
Rapid contraction R˙ IV (mm/s)
0
124.6 ± 3.2
0.79 ± 0.01
− 0.09 ± 0.01
− 3.5 ± 0.2
6.5
165 .7± 2.5
0.50 ± 0.05
− 0.19 ±0.03
− 6.1 ± 0.7
20
184.6 ± 1.4
0.46 ± 0.01
− 0.14 ±0.01
− 4.3 ±0.2
32
190.3 ± 3.7
0.46 ±0.05
− 0.22 ±0.07
− 4.2 ±0.4
d(mm)
h(cm)
Inertial R˙ I (mm/s)
t(s)
Fig. 5 Variation of diameter with time for different free fall heights
Effect of Impact Velocity on Spreading and Evaporation of a Volatile … Table 2 Variation of te and dmax with free fall height for droplet impact
247
S. No.
h(cm)
te (sec)
dmax (mm)
1
0
35.1 ± 2.0
26.2 ± 0.5
2
6.5
29.9 ± 0.5
24.8 ± 1.0
3
20
31.0 ± 0.1
24.3 ± 0.2
4
32
31.1 ± 0.3
24.5 ± 1.1
Table 2 shows the variation in evaporation time (te ) and maximum spread diameter (dmax ) with impact height (h). One can see from the table that varying impact heights do not lead to any considerable change in the evaporation time or maximum spread diameter, compared to the sessile droplet case (h = 0 cm), with about a 10% reduction in te . It is observed for the droplet impact cases.
4 Conclusions The consequence of finite surface impact velocity on the spreading and evaporation of an ethanol droplet is experimentally investigated. Three different droplet impact velocities corresponding to different heights of free fall were considered. In addition, a sessile droplet (h = 0 cm) is also studied. Four different regimes, namely inertial regime, spreading, slow contraction, and rapid contraction, are observed for all the cases. We observed there is no considerable variation of te and dmax with change in free fall height for all impact cases, but for the evaporation time, te and maximum diameter are consistently smaller compared to the sessile droplet. Thus, finite impact velocity causes faster evaporation of a droplet compared to the case where the droplet is gently placed on the substrate. Further investigations are currently being carried out to understand the physics behind the observations and perform the experiments on a broader parameter space by varying substrate porosity, temperature, and droplet composition.
References 1. Kathe K, Kathpalia H (2017) Film forming systems for topical and transdermal drug delivery. Asian J Pharm Sci 12(6):487–497. https://doi.org/10.1016/j.ajps.2017.07.004 2. Kim J (2007) Spray cooling heat transfer: the state of the art. Int J Heat Fluid Flow 28(4):753–767. https://doi.org/10.1016/j.ijheatfluidflow.2006.09.003 3. Lohse D (2021) Annual review of fluid mechanics fundamental fluid dynamics challenges in inkjet printing. https://doi.org/10.1146/annurev-fluid-022321 4. Hu H, Larson RG (2002) Evaporation of a sessile droplet on a substrate. J Phys Chem B 106(6):1334–1344. https://doi.org/10.1021/jp0118322 5. Birdi KS, Vu DT, Winter A (1989) A study of the evaporation rates of small water drops placed on a solid surface [Online]. Available: https://pubs.acs.org/sharingguidelines
248
A. Yadav et al.
6. Shi L, Shen P, Zhang D, Lin Q, Jiang Q (2009) Wetting and evaporation behaviors of waterethanol sessile drops on PTFE surfaces. Surf Interface Anal 41(12–13):951–955. https://doi.org/ 10.1002/sia.3123 7. Lee KS, Cheah CY, Copleston RJ, Starov VM, Sefiane K (2008) Spreading and evaporation of sessile droplets: universal behaviour in the case of complete wetting. Colloids Surf A Physicochem Eng Asp 323(1–3):63–72. https://doi.org/10.1016/j.colsurfa.2007.09.033 8. Starov V, Sefiane K (2009) On evaporation rate and interfacial temperature of volatile sessile drops. Colloids Surf A Physicochem Eng Asp 333(1–3):170–174. https://doi.org/10.1016/j.col surfa.2008.09.047
Study of Liquid–Vapor Oscillating Nature in a U-Shaped Tube for a Pulsating Heat Pipe Anoop Kumar Shukla, Est Dev Patel, and Subrata Kumar
Nomenclature A Ap B cp cv h H M mv P pv R t T Xp
Dimensionless amplitude of pressure oscillation Cross-sectional area of the tube [m2 ] Dimensionless amplitude of displacement Specific heat at constant pressure [J/kg-K] Specific heat at constant volume [J/kg-K] Heat transfer coefficient [W/m2 K] Dimensionless heat transfer coefficient Dimensionless mass of vapor plugs Mass of vapor plugs [kg] Dimensionless vapor pressure Vapor pressure [Pa] Gas constant [J/kg-K] Time [s] Temperature [K] Dimensionless displacement of liquid slug
Greek Symbols νe θ Θ ψ
Effective viscosity [m2 /s] Dimensionless temperature Dimensionless temperature Dimensionless parameter defined by Eq. (33)
A. K. Shukla (B) · E. D. Patel · S. Kumar Department of Mechanical Engineering, IIT Patna, Patna 801106, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_24
249
250
γ ω τ τp ρ ωo
A. K. Shukla et al.
Ration of specific heats Dimensionless forced angular frequency Dimensionless time Shear stress [N/m2 ] Density [Kg/m3 ] Dimensionless natural angular frequency
Subscript 1 2 e h c l v p
Left vapor plug Right vapor plug Evaporator Heating Condenser Liquid Vapor Plug
1 Introduction While on one hand, the usage of micro-electronics is increasing, on the other hand, an increase power–decrease size scenario is in demand. Therefore, it has become necessary to move toward such electronic thermal management devices that are compact as well as highly effective. Because of its simple structure, quick thermal response, and highly effective heat transfer capabilities, pulsating heat pipe (PHP) has found a prominent place among the many existing modern electronic cooling technology methods. Compared to conventional heat pipes, there is no need of wicktype structure inside PHP. Hence, there is an absence of countercurrent flow which arises due to vapor and condensate flow in the system [1]. A PHP is a passive device with two phases that can transfer large amounts of heat. One end of PHP receives heat from the evaporator section and transfers it to the other end, called the condenser section, by the pulsating action of the liquid slug. Due to the random volumetric distribution of liquid in each section of PHP, the pressure drop in each section will be different when heat is applied to the PHP, which eventually works as the driving force for pulsations inside the tube. Thus, a PHP is a thermally driven non-equilibrium heat transfer device. The diameter of PHP is one of the most critical parameters for its successful operation. PHP’s internal diameter has to fall within a certain range in order for distinct liquid and vapor plugs to form. If it is too big, the liquid will merge to form one single liquid column.
Study of Liquid–Vapor Oscillating Nature in a U-Shaped Tube …
251
Miyazaki and Akachi [2] conducted an experimental investigation to determine the interdependence between pressure oscillation and oscillatory flow. Miyazaki and Arikawa [3] analyzed the wave velocity of oscillatory flow and found that this velocity is reasonably consistent with what Miyazaki and Akachi predicted in their study. Tong et al. [4] conducted their flow visualization experiment to investigate the slug–plug formation and their flow mechanism in closed-loop PHP. They had taken methanol as working fluid and reported that a certain minimum critical heat input is required to start the flow of fluid in PHP, and the circulation of working fluid achieved at 60% FR. During the start-up period, the oscillations that occur in PHP have a large amplitude. Shafii et al. [5] proposed a mathematical model of looped and unlooped PHPs having multiple liquid slugs and vapor plugs. They claimed that the total number of vapor plugs eventually drops to the total number of evaporator sections present in the setup, whatever the initial number of vapor plugs in the system. The heat transfer in PHP occurs mainly due to sensible heat exchange, and latent heat transfer (condensation and evaporation) is responsible for oscillation in the system. Zhang et al. [6] investigated the liquid–vapor oscillating flow in a capillary U-shaped tube numerically. First, the momentum equation for a single liquid slug and energy and mass equations for two vapor plugs are non-denationalized and then solved numerically using an implicit scheme. Only pressure, wall shear, and gravity forces were considered in this study. The initial movement of the liquid slug had no discernible impact on the amplitude of oscillations. The influence of gravity has very less impact on oscillatory flow due to smaller cross-section of tube. Zhang et al. [7] conducted their experiment using three working fluids: deionized water, ethanol, and FC-72 in a 1.18-mm-diameter copper tube with three turns to record the thermal oscillations of the thin wall surface. Due to lower latent heat of vaporization of FC-72, the amplitude of thermal oscillation is much smaller and movement of oscillation is faster. Cheng et al. [8] presented a mathematical model for a closed-loop PHP having four loops with multiple numbers of liquid slugs and vapor plugs. The equation of motion is analogous to the spring-mass-damper system. The natural frequencies of the system vary with the liquid slugs and vapor plugs distribution in the system. The average speed of liquid slugs was also calculated. Kim et al. [9] studied the effect of evaporator and condenser temperature fluctuations during oscillating flow in a U-shaped section for the PHP. They found that with the increase in amplitude and frequency of the of wall temperature, the frequency of liquid slugs oscillations decreases. A detailed study considering surface tension, pressure losses at bends, and capillary effect in a U-shaped PHP was presented by Dilawar et al. [10]. The heat transfer rate reduced due to pressure losses at the bends. The reduction in the amplitude of oscillations and an increase in the length of the adiabatic channel also reduce thermal performance of PHP. Shaha et al. [11] studied the performance of a quartz glass single-loop heat pipe having 4 mm internal diameter, with water as a working fluid, and found that at FR 30–60%, PHP failed to function in a horizontal position. There always exists a minimum critical power below which no fluid movement occurs in PHP. An evaporative liquid film model was introduced by Nemati et al. [12], to study the heat transfer performances of a U-shaped tube PHP having a 1 mm diameter with ethanol and water as working fluid.
252
A. K. Shukla et al.
Under the same working conditions, the ethanol gives a larger value of amplitude of oscillations as compared to water. The filling ratio and evacuation pressure also play an important role in the start-up and transient thermal performance of the closed-loop pulsating heat pipe [13]. A new approach to solve the non-dimensionalized governing equation using MATLAB is proposed in this present study. Vapor mass equations are being written considering the length of liquid column in the evaporator section instead of taking the length of the vapor zone in this study.
2 Theoretical Model The U-shaped tube having two vapor plugs serving as the spring and a liquid slug between them act as mass is shown in Fig. 1. The tube has diameter, d, and length, 2L. Both sides of the tube represent the evaporator zone, each of length L h . The middle section denotes the condenser section of length 2L c . Initial temperature at the evaporator section is Te , and the condenser section is at Tc . The liquid slug is initially at the bottom of the U-shaped tube and has a length of L p . In the absence of any disturbances, the entirety of the liquid section resides in the condenser section, while the vapor occupies the evaporator sections. However, when a disturbance occurs and a liquid slug is displaced by a distance x_p (considered positive for the right tube), this process enables a portion of the liquid in the right column to interact with the evaporator, while simultaneously allowing a portion of the vapor in the left column to come into contact with the condenser. Now, due to this, boiling takes place only in the right side, and condensation takes place in the left side. This increases the vapor density in the right side and the vapor pressure while decreasing vapor and pressure in the left side of the liquid mass. The pressure difference starts the oscillations of the liquid mass in both the directions. The friction and inertia forces also make sure that the motion does not subside. Following assumptions were helped to model the oscillatory flow and heat transfer: • The working fluid always follows the saturation condition, and vapor behaves as an ideal gas. • The heat transfer only causes the evaporation or condensation. • The heat transfer coefficient of evaporation and condensation is constant and equal. • The surface tension is considered negligible at the interface of liquid and vapor. • The temperature gradient between evaporator and condenser section is neglected. • The pressure loss at the bend is ignored in the analysis.
Study of Liquid–Vapor Oscillating Nature in a U-Shaped Tube …
253
Fig. 1 Miniature U-shaped tube
2.1 Governing Equations The motion of liquid slug can be analyzed using the mass, momentum, energy, and ideal gas equation. Momentum equation can be written as for the liquid mass from Fig. 1: Ap ρl
d2 xp = ( pv1 − pv2 )Ap − 2ρl g Ac xp − π d L p τp dt 2
(1)
Arranging the Eq. (1), give d2 xp 2g 32ve dxp Δp + + 2 xp = 2 dt Lp ρl L p d dt
(2)
Now, using the energy equation for vapor (effectively thermodynamics first law) d(m v1 cv Tv1 ) dm v1 π dxp = cp Tv1 − pv d 2 dt dt 4 dt
(3)
d(m v2 cv Tv2 ) dm v2 π dxp = cp Tv2 + pv d 2 dt dt 4 dt
(4)
Rearranging Eqs. (3) and (4) gives,
254
A. K. Shukla et al.
m v1 cv
dTv1 dm v1 π dxp = RTv1 − pv d2 dt dt 4 dt
(5)
m v2 cv
dTv2 dm v1 π dxp = RTv1 + pv d2 dt dt 4 dt
(6)
Now, using the ideal gas law for vapor zone π pv1 L h + xp d 2 = m v1 Rg Tv1 4 π pv2 L h + xp d 2 = m v2 Rg Tv2 4
(7) (8)
Differentiating Eq. (7) w.r.t time will give d pv1 π 2 π dxp dm v1 dTv d L h + xp + pv1 d 2 = m v1 R + RTv 4 dt 4 dt dt dt
(9)
Substituting Eq. (5) into Eq. (9) RTv1
d pv1 π cv dm v1 π dxp = d2 + pv1 d 2 L h + xp dt 4 cp dt 4 dt
(10)
Substituting the value of RTv1 from Eq. (7) into Eq. (10) dxp 1 1 d pv1 1 1 dm v1 = + m v1 dt γ pv1 dt L h + xp dt
(11)
Integrating the above Eq. (11), the mass of vapor at the left side is as follows: 1 γ L h + xp m v1 = C1 pv1
(12)
Putting Eq. (12) into the Eq. (7), we get the temperature of left side vapor zone: Tv1 =
π d 2 (γ γ−1) p 4C1 R v1
(13)
Similarly, for the right side to reveal 1 γ L h − xp m v2 = C2 pv2
Tv2 =
π d 2 (γ γ−1) p 4C2 R v2
(14)
(15)
Study of Liquid–Vapor Oscillating Nature in a U-Shaped Tube …
255
Here, C1 &C2 are integration constants. For symmetric section, the integration constants will be same; hence. C1 = C2 = C. Now, the motion of liquid and vapor lead to the change in the mass of the vapor, and it can be formulated as: dm v1 = dt
dm v2 = dt
h c π d xp (Tv1 −Tc ) h fg h π d x (T −T ) − e ph fg e v1
−
h e π d xp (Te −Tv2 ) h fg h c π d xp (Tv2 −Tc ) h fg
xp > 0 xp < 0 xp > 0 xp < 0
(16)
(17)
2.2 Non-dimensionalization of Governing Equations First, governing equation are non-dimensionalized. The reference state of system is used to non-dimensionalize the same. The temperature of the vapor section is taken as T0 , while the pressure is taken as p0 and the displacement of the liquid section is taken as xpo at reference state. Constants C1 &C2 can be found using Eqs. (13) and (15) C1 = C2 =
π d 2 (γ γ−1) p 4RT0 0
(18)
At the reference state, mass of vapor sections are hence as follows: m v10 =
π d2 p0 L h + x p 4RT0
(19)
m v20 =
π d2 p0 L h − x p 4RT0
(20)
Hence, the average mass of the vapor sections is as follows: m0 =
m v10 + m v20 π d2 p0 L h = 2 4RT0
(21)
Now, substituting the non-dimensionalize Eqs. (18) and (21) into Eqs. (12)–(15) gives m v1 = m0
pv1 p0
γ1
L h + xp Lh
(22)
256
A. K. Shukla et al.
Tv1 = T0 m v2 = m0
Tv2 = T0
pv1 p0
pv2 p0
(γ γ−1) (23)
γ1
pv2 p0
L h − xp Lh
(24)
(γ γ−1) (25)
Adding a few other definitions to reduce complexity Tv1 Tv2 pv1 pv2 , θ2 = , P1 = , P2 = , T0 T0 p0 p0 xp m v1 m v2 , M2 = , Xp = M1 = m0 m0 Lh θ1 =
(26)
This reduces the above Eqs. (22)–(25) to 1 M1 = P1γ 1 + X p
(γ −1) γ
θ1 = P1
1 M2 = P2γ 1 − X p
(γ −1) γ
θ2 = P2
(27) (28) (29) (30)
Putting all the non-dimensional variables into Eq. (2) and making nondimensionalized time τ=
νe t d2
(31)
and Eq. (2) becomes d Xp d2 Xp + ωo2 X p = ψ(P1 − P2 ) + 32 dτ 2 dτ
(32)
Here, ωo2 and ψ are dimensionless quantities and can be expressed as ωo2 =
2gd 4 p0 d 4 , ψ = L p ve2 ρl L p L h ve2
(33)
Study of Liquid–Vapor Oscillating Nature in a U-Shaped Tube …
257
Substituting parameters from Eqs. (26)-(31) into Eqs. (16) and (17) (
−Hc X p (θ1 − θc ) xp > 0 −He X p (θe − θ1 ) xp < 0 ( d M2 He X p (θe − θ2 ) xp > 0 = Hc X p (θ2 − θc ) xp < 0 dτ
d M1 = dτ
(34)
(35)
Here, 4h C RT02 d 4h e RT02 d , He = , p0 h fg ve p0 h fg ve Te Tc θe = , θc = To To
Hc =
(36)
The reference temperature To is taken as the average of Te and Tc , then θc = 1 − Θ, θe = 1 + Θ Here, Θ=
Te − Tc Te + Tc
Initial Conditions The reference state itself is chosen as initial state of the system and these are as follows: X p = X 0, τ = 0
(39)
θ1 = θ2 = 1, τ = 0
(40)
P1 = P2 = 1, τ = 0
(41)
M1 = 1 + X 0 , M2 = 1 − X 0 , τ = 0
(42)
2.3 Numerical Solution of Non-dimensionalize Equations The differential equation Eq. (32) represents forced vibration equation. The pressure difference between both the vapor plugs is as follows:
258
A. K. Shukla et al.
ΔP = A cos ωτ
(43)
Then, the solution of Eq. (32) becomes X p = Bo e
−16τ
/ cos
ωo2
− 16τ + φo
ψ A cos(ωτ − δ) + / 2 ωo2 − ω2 + 1024ω2
(44)
Here, Bo and φo are constant that depends upon the initial conditions of vibratory motion. The phase difference between ΔP and xp is δ which can be given as tan δ =
32ω − ω2
ωo2
(45)
The first term of Eq. (44) represents the complimentary function of differential equation, which takes into account the effect of the initial conditions of vibratory motion, and the second term represents the particular integral of differential equation. As the time step increases the energy of system which is associated with initial condition dissipates through viscous damping, then the whole response of system is governed by pressure difference stated in Eq. (43) and after sufficiently large time steps, Eq. (44) becomes X p = B cos(ωτ − δ)
(46)
Here, B = /
ψA 2 ωo2 − ω2 + 1024ω2
(47)
To solve the dimensionless governing equation, the variables pressure, temperature, mass, and displacement have been calculated at the different time steps. For the calculations, Runge–Kutta fourth-order method is employed. The variable dt which is the dimensionless time step is set to 0.0001, and it can be further decreased, but this will result in increase in computation power. The numerical approach used in this solution is described as follows: 1. Initially by putting all the initial conditions in Eq. (32) will give the position at next time step. 2. Now, to calculate the rate of change in mass of vapor with respect to Δτ for the initial time step using Eqs. (34) and (35). 3. Now, by adding the initial mass with the change in mass as obtained above, we get the mass for the next time step.
Study of Liquid–Vapor Oscillating Nature in a U-Shaped Tube …
259
4. Now, the mass of the next time step will be used to calculate the temperature and pressure for the next time step using Eqs. (27)–(30). 5. Now using this pressure, we can find the position for the next time step again using Eq. (32), and using this temperature, we can find the mass change for the next time step using Eqs. (34) and (35). Doing this process over and over, we keep on getting the next values of the mass, temperature, pressure, and position of the liquid column.
3 Results and Discussion The results are validated with Shafii et al. [5] model. Figure 2 shows the validation results of displacement x p of the liquid slug. It can be seen that the results are well agreed with Shafii et al. [9] model. These results are obtained by taking d = 3.34 mm, L p = 0.2 m, L c = L h = 0.1 m, Tc = 293.15 K, Te = 396.55 K, and h e = h c = 200 W/m2 . Non-dimensional parameters ωo2 = 1.2 × 104 , ψ = 1.2 × 105 , Θ = 0.15 and He = Hc = 3000 have been used. Mass addition in liquid slug is taking place due to condensation of vapor, and mass addition in vapor is taking place due to vaporization of liquid is taken in this model. Figure 3 compares the initial displacement of liquid slug, and dimensionless mass, pressure, and temperature of vapor plugs with the results of Zhang et al. [6]. Figure 3a shows the dimensionless displacement of the liquid slug. Compared with existing model, it can be seen that the amplitude of the oscillations has increased, while frequency of oscillations decreased in this model. Hence, heat transfer will be comparatively more, in this model. On the other hand, it can be seen from Fig. 3f–i that the maximum temperature and pressure of vapor plugs in the system are decreased, and the minimum temperature and pressure are reduced.
Fig. 2 Validation with Shafii et al. [5] model
260
A. K. Shukla et al.
Fig. 3 Comparison of parameter variations between the current model and an existing model
Study of Liquid–Vapor Oscillating Nature in a U-Shaped Tube …
Fig. 3 (continued)
261
262
A. K. Shukla et al.
Fig. 3 (continued)
The maximum temperature obtained from these plots is more than the evaporator temperature Te , and it is because of compression of vapor in the miniature tube. This compression of vapor results, due to the inertia of liquid slug during motion, which is basically spring-like compression which happens during oscillating spring-mass system. It can also be observed that there is a phase difference of around π between the temperatures and pressures in the two vapor plugs. A smooth plot having no irregularity and no bulge has been obtained. The most significant observation can be derived from the dimensionless mass plots. In Fig. 3b– e, a comparison is presented between the dimensionless mass variations over dimensionless time, as obtained in our current model and in the model proposed by Zhang et al. [7]. This analysis reveals that in our study, the variations in the masses of both vapor plugs exhibit a similar behavior as that seen in the pressure and temperature plots over time. Essentially, this indicates that when the vapor mass on one side is increasing, it is simultaneously decreasing on the other side. This phenomenon arises from the concurrent processes of liquid evaporation and vapor condensation occurring in both columns of the U-shaped tube. Figure 4 illustrates a typical relative variation of various non-dimensional parameters over dimensionless time, showcasing how pressure and temperature change with the displacement of the liquid slug. Consequently, this plot offers a clear and insightful physical interpretation of the processes taking place within the U-shaped tube. Figure 5a–c shows the gravitational effect on the displacement of liquid slug and temperature and pressure of vapor plugs inside the U-shaped tube As observed from the figure, the influence of gravity on various performance parameters of the PHP is generally not substantial. Nevertheless, a slight deviation can be seen in the figures. This phenomenon can be explained by considering the actual mass of liquid present within the capillary tube. Due to the extremely small diameter of the capillary tube, the mass of the liquid slug inside the tube is quite negligible, thereby resulting in a minimal impact of gravity. In practical scenarios, it is the surface tension force that predominantly prevails over gravitational forces in such devices.
Study of Liquid–Vapor Oscillating Nature in a U-Shaped Tube …
Fig. 4 Variation of different dimensionless parameters with dimensionless time
Fig. 5 Effect of gravity on the different operating parameters
263
264
A. K. Shukla et al.
4 Conclusions The present study investigates the pulsating behavior of a liquid slug within a Ushaped miniature tube, while also considering the influence of gravity on various parameters. In this study, a new approach for mass transfer equation for both the vapor plugs is introduced. Unlike previous models that only considered evaporation and condensation phenomena for vapor plugs, this model takes into consideration the evaporation of the liquid slug in the evaporator and the condensation of vapor plug in the condenser. By incorporating these additional factors, it addresses the rate of change in mass of the vapor plugs at both ends of the miniature tube, a consequence of the evaporation of liquid and condensation of vapor. This variation in mass of the vapor plugs results in a pressure difference across the tube, ultimately leading to the pulsating motion of the liquid slug. In the existing literature on PHP, it is evident that there are primarily three factors responsible for the changes in pressure within the vapor plugs. These factors include temperature variations, mass changes during condensation and evaporation, and the compression and expansion of vapor plugs during their motion, which can be likened to a spring-like effect. This model comprehensively accounts for all three pressure variations in the vapor plugs, resulting in a smooth and uniform plot of pressure and temperature without any irregularities.
References 1. Akachi H (1996) PHPs. In: Proceedings of the 5th international heat pipe symposium 2. Miyazaki Y (1996) Heat transfer characteristics of looped capillary heat pipe. In: Proceedings of the 5th international heat pipe symposium, pp 378–383 3. Miyazaki Y (1999) Oscillatory flow in oscillating heat pipe. In: Proceedings of the 11th international heat pipe conference, pp 367–372 4. Tong BY, Wong TN, Ooi KT (2001) Closed-loop PHP. Appl Therm Eng 21(18):1845–1862 5. Shafii MB, Faghri A, Zhang Y (2001) Thermal modeling of unlooped and looped PHPs. J Heat Transfer 123(6):1159–1172 6. Zhang, Y, Faghri A, Shafii MB (2002) Analysis of liquid–vapor pulsating flow in a U-shaped miniature tube. Int J Heat Mass Transf 45(12):2501–2508 7. Zhang XM, Xu JL, Zhou ZQ (2004) Experimental study of a PHP us ing FC-72, ethanol, and water as working fluids. Exp Heat Transf 17(1):47–67 8. Cheng P, Ma H (2011) A mathematical model of an oscillating heat pipe. Heat Transfer Eng 32(11–12):1037–1046 9. Kim S, Zhang Y, Choi J (2013) Effects of fluctuations of heating and cooling section temperatures on performance of a PHP. Appl Therm Eng 58(1–2):42–51 10. Dilawar M, Pattamatta A (2013) A parametric study of oscillatory two-phase flows in a single turn PHP using a non-isothermal vapor model. Appl Therm Eng 51(1–2):1328–1338 11. Saha N, Das PK, Sharma PK (2014) Influence of process variables on the hydrodynamics and performance of a single loop PHP. Int J Heat Mass Transf 74:238–250 12. Nemati R, Shafii MB (2018) Advanced heat transfer analysis of a U-shaped PHP considering evaporative liquid film trailing from its liquid slug. Appl Thermal Eng 138:475–489
Study of Liquid–Vapor Oscillating Nature in a U-Shaped Tube …
265
13. Patel ED, Kumar S (2021) The impact of variation in filling ratios, evacuation pressure, and heat input on thermal performance of PHP. IEEE Trans Compon, Packag Manuf Technol 12(2):259–269
Hydrodynamics of Two-Phase Immiscible Flow in T-Junction Microchannel Akepogu Venkateshwarlu and Ram Prakash Bharti
1 Introduction In the contemporary research world, microfluidics has gained so much importance because of its versatile applications in diverse fields of science and engineering like pharmaceuticals, DNA analysis, protein encapsulation, synthesis of new materials, biomedical reagents, drug delivery and RT-PCR tests [1–8]. All these applications involve fundamentally the hydrodynamics of the droplet generation. There are various geometries available to generate the droplets, namely co-flow, flow focusing, and cross-flow or T-junction. The T-junction microfluidic devices are very popular due to its distinctive advantages like ease of control over the droplet size and production, simplicity toward geometrical construction. The interplay of the forces between the phases is described by defining the dimensionless numbers: capillary number (Cac ), ratio of viscous force to the interfacial tension force; Reynolds number (Rec ), ratio of the inertial force to the viscous force. At microscale, the interfacial and viscous forces dominate the flow physics [9–12]. Thorsen et al. [13] firstly conducted the experiments in T-junction microfluidic device to produce the droplets and observed different flow patterns due to the geometrical effects. Nisisako et al. [14] reported that the droplet size has an inverse relation with the continuous phase flow rate for an oil–water combination. Garstecki et al. [15] have done experiments for a several combinations of geometries and fluid properties in T-junction microchannel and developed an empirical correlation to predict the droplet length for the squeezing regime (Cac < 10−2 ) as L = (α + β Q r )wc , which is independent of the physical properties. Depending upon the value of capillary number, the flow regimes are classified [16–18] as squeezing, dripping and jetting. It is also reported that droplets are A. Venkateshwarlu · R. P. Bharti (B) Complex Fluid Dynamics and Microfluidics (CFDM) Lab, Department of Chemical Engineering, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_25
267
268
A. Venkateshwarlu and R. P. Bharti
monodispersed and stable when its length is larger than the width of the horizontal channel [15, 19–21]. Inertial effects are dominant at higher capillary numbers (Cac > 10−2 ), and the flow becomes either jetting or two-layered flow [15, 17, 22]. Several other researchers [19, 23–33] have performed experiments and numerical studies to understand the hydrodynamics of the droplet generation. Although many attempts have been made to understand the hydrodynamics of droplet formation in a T-junction microchannel, a single correlation that combines the influence of interfacial tension and shear force to elucidate the flow regimes classification is still challenging. In our previous study [34], an empirical correlation is developed to describe the classification of flow regimes and droplet zones based on the transition capillary number for a broad range of the flow governing parameters like flow rate ratios and capillary numbers. In the current study, the droplet formation and its dynamics are described by the pressure evolutions for various flow regimes at a constant contact angle.
2 Problem Description and Methodology The T-junction microchannel is constructed by primary or horizontal channel (continuous phase, CP) and vertical channels (dispersed phase, DP), wherein two-phase laminar flow is taking place, as shown in Fig. 1. The vertical channel (width, wdp ) intersects the primary channel (width, wcp ) at 90° angle to form a T-junction at a distance of L up . The length of the vertical channel is L dp , and the downstream length of the channel is L ds . Hence, the total length of the channel, L cp = L up + wdp + L ds , , is sufficient to avoid the end effects. Both the fluids are assumed to be immiscible, non-reactive, equal density, incompressible, at isothermal conditions. The channel walls are assumed to be solid by employing noslip boundary condition. The continuous (CP) and dispersed phases (DP) enter at their respective inlets with the flow rates, Q cp , and Q dp ; and flow rate ratio Q r = Q dp /Q cp . The outlet is open to atmosphere, i.e., p = 0. The surface tension is assumed to be
Fig. 1 Schematics of T-junction microfluidic device
Hydrodynamics of Two-Phase Immiscible Flow in T-Junction …
269
◦
static (θ = 135 ), and the Marangoni effects are ignored. Hence, the interfacial tension and contact angle are constant throughout the study. The governing equations used to solve the physics given above are as follows: ∇ ·V=0 ρ(φ)
∂V + V · ∇V = −∇ p + ∇ · τ + Fst ∂t
(1) (2)
∂φ (3) + V · ∇φ = γ ∇ · ( ls ∇φ − φ(1 − φ)n) ∂t ∇φ Here, τ = μ(φ) ∇V + (∇V)T , n = |∇φ| , V represents the velocity field, and p, pressure, respectively. The conservative level set method is a robust scheme to capture the topology of the interface in the two-phase flow, and it is relatively easy to implement [17, 18, 35–37]. The level set function (φ) ranges from 0 to 1; 0–0.5 for the continuous phase, 0.5–1 for the dispersed phase and at the fluid–fluid interface, φ = 0.5. γ is the reinitialization parameter, and εls is the parameter to control the interface thickness. The force acting between the two immiscible phases is given by the continuum surface force (CSF) model [38] as follows: Fst = σ κδ(φ)n Here, σ denotes the surface tension coefficient (N/m), δ(φ) is the Dirac delta function that is approximated by a smooth function, δ(φ) = 6|∇φ||φ(1 − φ)|, n is unit normal, and κ = R −1 = −(∇ · n) is the mean curvature. The physical properties of the fluid at any point is given below: S(φ) = Scp + Sdp − Scp φ Here, S = μ, and ρ, i.e., viscosity and density. Numerical parameters: wcp = wdp = 100 μm; L up = L dp = 900 μm; L cp = 4000 μm; ρcp = ρdp = 1000 kg/m 3 ; the viscosity of the dispersed phase, μdp = 0.001 Pa.s; flow rate of the dispersed phase, Q dp = 0.14 μL/s [15, 39]; 1/10 ≤ Q r ≤ 10; capillary number, 10−4 ≤ Cac ≤ 1; the Reynolds number, Rec = 1/10; the = 5 μm; time step, t = 100 μs. relative tolerance, 5×10−3 ; γ = 1 m/s; εls = h max 2 The governing equations are solved using the COMSOL multiphysics, which is based on the finite element method. The entire computational domain is discretized into non-uniform, unstructured, linear triangular mesh. The mesh independence study and the validation of the solution approach are already established previously [32, 34].
270
A. Venkateshwarlu and R. P. Bharti
3 Results and Discussion The transition capillary number (Car,trans ) is introduced previously [34] to classify the droplet and non-droplet zones, i.e., Car,trans = (Cad,trans /Cac,trans ) = β Q 2r . The droplet zone consists of three flow regimes: squeezing, transition-1, and dripping. Similarly, the non-droplet zone consists of transition-2, parallel and jetting flow regimes. The results are presented in the subsequent sections, which are in terms of the phase contours and time evolutions of pressure profiles of the flow regimes to describe the flow dynamics.
3.1 Phase Profiles of the Flow Regimes The phase profiles (refer to Fig. 2) reveal that the flow governing parameters (Cac and Q r ) substantially affect the flow patterns. The squeezing regime is observed at the lower values of the capillary numbers (Cac < 10−2 ), as shown in Fig. 2(I) for Cac = 10−4 , Q r = 1. It is observed that the droplet breakup happens exactly at the right corner of the Tjunction and droplet size is larger than the width of the primary channel. Further, the droplets being formed are monodispersed in nature. In transition-1 flow regime, the droplet breakup location slowly starts shifting toward the downstream of the primary channel, as shown in Fig. 2(II) for Cac = 10−2 , Q r = 1. In the dripping flow regime, the droplet size reduces to the width of the primary channel and breakup location is
Fig. 2 Phase contours in droplet zone (I–III) and non-droplet zone (IV-VI)
Hydrodynamics of Two-Phase Immiscible Flow in T-Junction …
271
away from the T-junction, as shown in Fig. 2(III) for Cac = 2×10−2 , Q r = 1/2. The droplet formation is certain in the squeezing, transition-1 and dripping flow regimes. On moving further, the transition-2 flow regime is seen between the dripping and parallel flow regimes, as shown in Fig. 2(IV) for Cac = 5 × 10−2 , Q r = 1. A few droplets are observed to form at first, and as time passes on, the dispersed phase appears to flow parallel to the channel. The dispersed phase entirely occupies the downstream of the channel and flows parallel to the continuous phase in a parallel flow regime, as shown in Fig. 2(V) for Cac = 10−3 , Q r = 5. The jetting flow regime occurs at high capillary numbers, wherein the inertial force dominates, and the flow becomes a thread-like structure, as shown in Fig. 2(VI) for Cac = 1, Q r = 1/10.
3.2 Pressure Evolutions of the Flow Regimes The time evolution of the pressure profiles in CP and DP elucidates the droplet breakup mechanism in the microchannel, as shown in Figs. 3 and 4. The locations of pcp and pdp are point A (L up − wcp /2, wcp /2) and point B (L up + wcp /2, −wdp /2), respectively, (refer to Fig. 1). It can be observed that the droplet formation in the squeezing regime is attributed solely due to rise in pcp of the upstream of the primary channel (refer to Fig. 3a). pcp is constant during the filling stage and slowly rising to the maximum value which allows the development of the droplet phase. DP shows a dominating behavior over the CP, i.e., pcp > pdp in the squeezing regime. Nevertheless, it is noted that pdp ≈ pcp at the pinch-off point. In the transition-1 regime, pcp is increasing in the squeezing regime up to a threshold value as the breakup location slowly shifts toward the downstream, as shown in Fig. 3b. There is drop in pcp and pdp at the breakup point. The droplet formation cycle repeats for an interval of time. In the dripping regime, pcp drops in the initial stage, then suddenly increases up to a maximum value and then drops at the droplet breakup point, as shown in Fig. 3c. This behavior is arising due to the balance of the shear stress acting on the droplet from the surrounding fluid, interfacial tension force and force due to pressure in the upstream. In the transition-2 flow regime, both pcp and pdp are increasing with the same rate up to a maximum value and then drop at the breakup point, Fig. 4a. It can be noticed that pdp is slightly greater than pcp . Hence, there is a droplet formation for certain time, and on increasing the time, pcp and pdp are decreasing gradually. Hence, there is no formation of droplet, and both the phases are flowing parallel to each other. In the parallel flow regime, there is a pressure buildup ( pcp ) due to the obstruction of the CP flow by the DP, as shown in Fig. 4b. However, pdp has a greater dominance over pcp for the entire duration of the flow. Hence, there is possibility to droplet formation and the DP continue to fill the entire downstream of the channel. In the jetting regime, it can be found that pcp is dominant over pdp at every time instant, which is contrasting behavior to the other flow regimes discussed before, i.e., pcp > pdp , as shown in Fig. 4c. This is solely because of the inertial effects.
272
A. Venkateshwarlu and R. P. Bharti
Fig. 3 Pressure profiles for the droplet zone (I-III) in the continuous ( pcp ) and dispersed phases ( pdp )
4 Conclusions The formation of droplets and their hydrodynamics for a two-phase immiscible system in a T-junction microchannel are thoroughly studied computationally using the COMSOL multiphysics software. A transition capillary number that classifies the flow regimes and droplet zones are elucidated using the phase profiles as a function of time. It is found that there is a clear distinction between the zones in the phase contours: droplet and non-droplet. The mechanism that describes the dynamic behavior of the flow regimes is further discussed using the pressure evolutions in the DP and CP. In the squeezing regime, pressure in CP and DP is cyclic; at regular intervals of time, the pressure profiles’ cycle repeats. However, the pressure evolution in other flow regimes is different from the squeezing regime. It is concluded that the pressure developed in the DP shows domination over the CP in all the flow regimes except for the jetting-type flow regime.
Hydrodynamics of Two-Phase Immiscible Flow in T-Junction …
273
Fig. 4 Pressure profiles for the non-droplet zone (IV-VI) in the continuous ( pcp ) and dispersed phases ( pdp )
References 1. Barnes HA (1994) Rheology of emulsions—a review. Colloids Surf, A 91:89–95. https://doi. org/10.1016/0927-7757(93)02719-U 2. Whitesides GM, Stroock AD (2001) Flexible methods for microfluidics. Phys Today 54(6):42– 48. https://doi.org/10.1063/1.1387591 3. Maan AA, Schroën K, Boom R (2011) Spontaneous droplet formation techniques for monodisperse emulsions preparation—perspectives for food applications. J Food Eng 107(3–4):334– 346. https://doi.org/10.1016/j.jfoodeng.2011.07.008 4. Kaminski TS, Garstecki P (2017) Controlled droplet microfluidic systems for multistep chemical and biological assays. Chem Soc Rev 46(20):6210–6226. https://doi.org/10.1039/c5cs00 717h 5. Kulju S, Riegger L, Koltay P, Mattila K, Hyväluoma J (2018) Fluid flow simulations meet high-speed video: computer vision comparison of droplet dynamics. J Colloid Interface Sci 522:48–56. https://doi.org/10.1016/j.jcis.2018.03.053 6. Mansard V, Mecca JM, Dermody DL, Malotky D, Tucker CJ, Squires TM (2016) Collective Rayleigh-Plateau instability: a mimic of droplet breakup in high internal phase emulsion. Langmuir 32(11):2549–2555. https://doi.org/10.1021/acs.langmuir.5b04727
274
A. Venkateshwarlu and R. P. Bharti
7. Gerecsei T, Ungai-Salanki R, Saftics A, Derényi I, Horvath R, Szabo B (2022) Characterization of the dissolution of water microdroplets in oil. Colloids Interfaces 6(1):14. https://doi.org/10. 3390/colloids6010014 8. Rabiee M, Ghasemnia NN, Rabiee N, Bagherzadeh M (2021) Microfluidic devices and drug delivery systems. Biomed Appl Microfluidic Devices, Chapter 7, 153–186. https://doi.org/10. 1016/B978-0-12-818791-3.00013-9 9. Ho C, Tai Y (1998) Micro-electro-mechanical-systems (MEMS) and fluid flows. Annu Rev Fluid Mech 30:579–612. https://doi.org/10.1146/annurev.fluid.30.1.579 10. Bharti RP, Harvie DJE, Davidson MR (2008) Steady flow of ionic liquid through a cylindrical microfluidic contraction–expansion pipe: electroviscous effects and pressure drop. Chem Eng Sci 63(14):3593–3604. https://doi.org/10.1016/j.ces.2008.04.029 11. Nge PN, Rogers CI, Woolley AT (2013) Advances in microfluidic materials, functions, integration, and applications. Chem Rev 113(4):2550–2583. https://doi.org/10.1021/cr300337x 12. Shang L, Cheng Y, Zhao Y (2017) Emerging droplet microfluidics. Chem Rev 117(12):7964– 8040. https://doi.org/10.1021/acs.chemrev.6b00848 13. Thorsen T, Roberts RW, Arnold FH, Quake SR (2001) Dynamic pattern formation in a vesiclegenerating microfluidic device. Phys Rev Lett 86(18):4163–4166. https://doi.org/10.1103/Phy sRevLett.86.4163 14. Nisisako T, Torii T, Higuchi T (2002) Droplet formation in a microchannel network. Lab Chip 2:24–26. https://doi.org/10.1039/B108740C 15. Garstecki P, Fuerstman MJ, Stone HA, Whitesides GM (2006) Formation of droplets and bubbles in a microfluidic T-junction—scaling and mechanism of break-up. Lab Chip 6(3):437– 446. https://doi.org/10.1039/B510841A 16. Gupta A, Murshed SMS, Kumar R (2009) Droplet formation and stability of flows in a microfluidic T-junction. Appl Phys Lett 94:164107. https://doi.org/10.1063/1.3116089 17. Bashir S, Rees JM, Zimmerman WB (2011) Simulations of microfluidic droplet formation using the two-phase level set method. Chem Eng Sci 66(20):4733–4741. https://doi.org/10. 1016/j.ces.2011.06.034 18. Wong V-L, Loizou K, Lau P-L, Graham RS, Hewakandamby BN (2017) Numerical studies of shear-thinning droplet formation in a microfluidic T-junction using two-phase level-SET method. Chem Eng Sci 174:157–173. https://doi.org/10.1016/j.ces.2017.08.027 19. Anna SL (2016) Droplets and bubbles in microfluidic devices. Annu Rev Fluid Mech 48:285– 309. https://doi.org/10.1146/annurev-fluid-122414-034425 20. Bashir S, Rees JM, Zimmerman WB (2014) Investigation of pressure profile evolution during confined microdroplet formation using a two-phase level set method. Int J Multiph Flow 60:40– 49. https://doi.org/10.1016/j.ijmultiphaseflow.2013.11.012 21. De Menech M, Garstecki P, Jousse F, Stone HA (2008) Transition from squeezing to dripping in a microfluidic T-shaped junction. J Fluid Mech 595:141–161. https://doi.org/10.1017/S00 2211200700910X 22. Christopher GF, Noharuddin NN, Taylor JA, Anna SL (2008) Experimental observations of the squeezing-to-dripping transition in T-shaped microfluidic junctions. Phys Rev E 78(3):036317. https://doi.org/10.1103/PhysRevE.78.036317 23. Pit AM, Duits MHG, Mugele F (2015) Droplet manipulations in two phase flow microfluidics. Micromachines 6(11):1768–1793. https://doi.org/10.3390/mi6111455 24. Martino C, deMello AJ (2016) Droplet-based microfluidics for artificial cell generation: a brief review. Interface Focus 6(4):20160011. https://doi.org/10.1098/rsfs.2016.0011 25. Zhu P, Wang L (2017) Passive and active droplet generation with microfluidics: a review. Lab Chip 17(1):34–75. https://doi.org/10.1039/C6LC01018K 26. Doufène K, Tourné-Péteilh C, Etienne P, Aubert-Pouëssel A (2019) Microfluidic systems for droplet generation in aqueous continuous phases: a focus review. Langmuir 35(39):12597– 12612. https://doi.org/10.1021/acs.langmuir.9b02179 27. Han W, Chen X (2021) A review on microdroplet generation in microfluidics. J Braz Soc Mech Sci Eng 43:247. https://doi.org/10.1007/s40430-021-02971-0
Hydrodynamics of Two-Phase Immiscible Flow in T-Junction …
275
28. Wu J, Yadavali S, Lee D, Issadore DA (2021) Scaling up the throughput of microfluidic dropletbased materials synthesis: a review of recent progress and outlook. Appl Phys Rev 8(3):031304. https://doi.org/10.1063/5.0049897 29. Venkateshwarlu A, Bharti RP (2022) Effect of contact angle on droplet generation in a Tjunction microfluidic system. Recent Trends Fluid Dyn Res (Book series: Lecture Notes in Mechanical Engineering), chapter 11, 137–147. https://doi.org/10.1007/978-981-16-6928-6_ 11 30. Dhondi P, Venkateshwarlu A, Bharti RP (2022) Effect of shear rate on non-Newtonian droplet generation in T-junction microfluidic system. Recent Trends Fluid Dyn Res (Book Series: Lecture Notes in Mechanical Engineering), chapter 11, 161–172. https://doi.org/10.1007/978981-16-6928-6_13 31. Samadder S, Venkateshwarlu A, Bharti RP (2022) Effects of inertial force and interfacial tension on droplet generation in a T-junction microfluidic system. Recent Trends Fluid Dyn Res (Book series: Lecture Notes in Mechanical Engineering), chapter 14, 173–182. https://doi. org/10.1007/978-981-16-6928-6_14 32. Venkateshwarlu A, Bharti RP (2022) Interface evolution and pinch-off mechanism of droplet in two-phase liquid flow through T-junction microfluidic system. Colloids Surf, A 642:128536. https://doi.org/10.1016/j.colsurfa.2022.128536 33. Venkateshwarlu A, Bharti RP (2023) Effects of surface wettability and flow rates on the interface evolution and droplet pinch-off mechanism in the crossflow microfluidic systems. Chem Eng Sci 267:118279. https://doi.org/10.1016/j.ces.2022.118279 34. Venkateshwarlu A, Bharti RP (2021) Effects of capillary number and flow rate on the hydrodynamics of droplet generation in two-phase cross-flow microfluidic systems. J Taiwan Inst Chem Eng 129:64–79. https://doi.org/10.1016/j.jtice.2021.07.045 35. Amiri HAA, Hamouda AA (2013) Evaluation of level set and phase field methods in modeling two phase flow with viscosity contrast through dualpermeability porous medium. Int J Multiph Flow 52:22–34. https://doi.org/10.1016/j.ijmultiphaseflow.2012.12.006 36. Olsson E, Kreiss G (2005) A conservative level set method for two phase flow. J Comput Phys 210(1):225–246. https://doi.org/10.1016/j.jcp.2005.04.007 37. Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton Jacobi formulations. J Comput Phys 79(1):12–49. https://doi.org/10.1016/ 0021-9991(88)90002-2 38. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modelling surface tension. J Comput Phys 100(2):335–354. https://doi.org/10.1016/0021-9991(92)90240-Y 39. Soh GY, Yeoh GH, Timchenko V (2016) Numerical investigation on the velocity fields during droplet formation in a microfluidic T-junction. Chem Eng Sci 139:99–108. https://doi.org/10. 1016/j.ces.2015.09.025
Experimental Study of Onset of Nucleate Boiling from Submerged Ribbon Heaters of Varying Width John Pinto, Janani Srree Murallidharan, and Kannan Iyer
Nomenclature Cp k Tw Tv T sat p∞ y δ ρ σ
Specific heat [J/kg K] Thermal conductivity [W/mK] Heater temperature [K] Vapour temperature [K] Saturation temperature [K] Ambient pressure [N/m2 ] Distance from the heater [m] Thermal boundary layer thickness Density [kg/m3 ] Surface tension [N/m]
[m]
1 Introduction Boiling is the most preferred mode of heat transfer for high heat flux applications. This is why boiling heat transfer is extensively studied. Prior to the ONB, heat is transferred from the wall to the fluid by transient conduction and then convection. Thus, with increase in wall heat flux, the wall temperature increases. The onset of boiling increases the heat transfer to the fluid considerably and hence slows the increase in the wall temperature. This makes onset of nucleate boiling a crucial aspect J. Pinto (B) · J. S. Murallidharan Department of Mechanical Engineering, IIT Bombay, Mumbai 400076, India e-mail: [email protected] K. Iyer Department of Mechanical Engineering, IIT Jammu, Jammu & Kashmir 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_26
277
278
J. Pinto et al.
to be studied. Knowing the ONB point is important as it dictates whether a system should be designed based on purely convective heat transfer calculation or purely boiling heat transfer calculations. ONB heat flux is also important in the design of cryogenic equipment or fuel lines. In such applications, the formation of vapour must be avoided, and hence, ONB heat flux gives the maximum permissible heat leak [1]. In applications such as immersion cooling of microelectronics, the maximum chip temperature must be kept below the safe operating temperature. This requires that ONB should start well below this value. The knowledge of ONB for different surface geometry, materials and operating conditions will help in improving the design of such systems.
2 Literature Review and Objective There are many studies on the onset of boiling in the literature. These studies focus on either the prediction of ONB from the available models [2–4] or the effect of ageing and wettability on boiling [5–7] or on the effect of micro-, nanostructures or etching/laser texturing on boiling [8–12]. Howell and Siegel [2] measured the wall superheat at ONB for bubbles nucleating from artificial cavities on an SS 410 strip of size 101.6 mm × 12.7 mm × 1.5875 mm. They used Hsu’s [13] criteria for onset and found that Hsu’s criteria suggest that the larger cavities sizes present in study (r > 150 µm) will not nucleate. To overcome this discrepancy, they modified the Hsu’s criteria taking into account evaporation and condensation effects for larger cavities. The modified criterion was able to then predict the onset from these large cavities. This criterion did not give an upper limit to the cavity that can nucleate, and the largest cavity that can nucleate will thus be limited by its vapour retention capability. Wiebe and Judd [3, 14] measured the thickness of the thermal boundary layer at onset for water at different subcooling on a 50.8-mm-diameter copper block. They used this thickness to calculate the wall superheat at ONB using Hsu’s criteria and found that it matches the trend of the experimental data well. Basu et al. [4] measured the wall superheat and heat flux at ONB for flow boiling condition on copper block (304.8 mm × 31.75 mm × 38.1 mm) and Zircaloy-4 rod bundle (3 × 3 arrangement, 11.1 mm OD, 14.29 mm pitch and 0.15 mm wall thickness, heated length 910 mm). They found that wall superheat was underpredicted by the available models in the literature. They attributed this effect due to the wettability difference. They suggested that with increased wettability, the size range of the available cavities decreases as the fluid floods the cavities due to the lower contact angle. They proposed a new model which takes into account the wettability effect on the available cavities. This new model compensates for the effect of wettability. Theofanous et al. [5] studied boiling on titanium thin film (140 nm) on borosilicate glass (20 mm × 40 mm × 0.13 mm). They found that ageing of the heaters decreased the wall superheat at ONB and increased the nucleation site density. The oxides formed due to ageing increased the nucleation site density and helped reduce the wall superheat at ONB. Bourdon et al. [6, 7] studied the influence of wettability alone on boiling by using highly smooth
Experimental Study of Onset of Nucleate Boiling from Submerged …
279
surfaces (bronze, 80 mm diameter thick with 24.8 nm thin layer of gold and glass). The wettability was modified by grafting technique. They found that increased wettability alone can trigger boiling and that wall superheat at ONB is lower for hydrophobic surface. Thus, boiling can be triggered by applying hydrophobic dots on the surface. The effect of laser texturing, etching, micro-, nanostructures is to reduce the wall superheat at ONB [8–12]. Early onset is reported for structured surfaces as compared to plain surfaces. Prior to onset of boiling, the heat is transferred to the fluid by natural convection and post-onset by nucleate boiling. Studies are available in the literature that look at the effect of heater size on natural convection and on boiling. Various researchers [15– 18] have looked at the effect of heater material and thickness heat transfer. (√ on boiling ) kρCp have higher heat It was found that the materials having higher effusivity transfer coefficient. For effect of thickness, two behaviours were found. Materials of higher conductivity that are limited by their thermal capacity have lower heat transfer coefficient for lower thickness, whereas for materials having lower thermal conductivity, resistance to lateral conduction is the dominating factor, and hence, HTC increases with decreasing thickness. Various studies are available that look at the effect of width of the heater on natural convection and boiling heat transfer [19– 23]. Boiling heat transfer is reported to be unaffected by the change in the width of the heater. However, natural convection is enhanced by the reduction in width. An enhanced convection is likely to delay the ONB. The studies on ONB available in the literature do not focus on the heater side factors like the thickness, size and material of the heater. However, the literature review highlights that the heater side factors influence the natural convection and boiling phenomenon. From the available data on ONB, it is difficult to interpret the effect of such parameters on ONB as each study involves a heater with different roughness and wettability. In order to understand the effect of these parameters on ONB, a systematic study is needed which will try to isolate the effect of other parameters so that the effect of heater side factors on ONB is clearly seen. The present study is an effort towards answering these unanswered questions.
3 Methodology Experiments on boiling of water at atmospheric pressure on ribbon heaters for subcooling ranging from 5 to 20 °C were undertaken. Cylindrical glass container (φ 190 mm × 100 mm) was used as the boiling vessel. Deionized water was used as the working fluid. A 1-kW heater was used to maintain the bulk fluid at the desired temperature. A K-type thermocouple was used to measure the bulk temperature. Visualization of bubbles was done using a Canon DSLR EOS 550D camera at 50 fps and a resolution of 1280 × 720 pixels using a 100-mm f/2.8 Canon macro lens. LED light was used as a light source. GW Instek PSW 30-108 Series Programmable DC Power Supply was used to provide power to the test heater in constant current mode. The test heaters were made from stainless steel 316 ribbon, wire EDM cut to
280
J. Pinto et al.
Fig. 1 Schematic diagram of the test set-up
get different widths. The ribbons had uniform thickness and length of 0.2 mm and 88 mm, respectively, whilst the width varied from 2 to 10 mm in steps of 2 mm. The strips were polished with silicon carbide abrasive paper (800 to 2000 grit) to get a uniform surface finish. Voltage supplied to the test heater was measured by potential taps placed on the heater 60 mm apart. The current supplied was read from the current monitoring pins of the analogue output of the power supply. Keysight DAQ 970A was used for data measurement and logging. The schematic of the set-up is shown in Fig. 1. Before the experiment, the bulk fluid was heated and made to boil vigorously for 15 min. Then, the power supplied to the bulk heater was adjusted to get the desired subcooling. Once the desired subcooling was reached, the heating of the test heater was started. To maintain steady state conditions, the current to the heater was increased in steps and held constant for 45 s by loading a test script in the power supply. 45 s was found to be the optimum time that did not influence ONB. The current steps were calculated such that the power increase step was always 10% and that the heat flux steps were always same for heaters of different widths. During heating, the heater was visualized for ONB. A stopwatch was used to note the time of ONB from the start of heating. From this, the heat flux and temperature at the point of ONB could be calculated. At least three sets of data for the same bulk temperature were taken to see the repeatability and spread of the data. Before switching to the next bulk temperature, the bulk fluid was changed and the boiling vessel was cleaned. The test heaters were handled carefully and cleaned with IPA before the experiment to avoid any surface contamination. In experiments, ONB is difficult to measure especially at high subcooling. This is because at high subcooling, stationary bubbles form on the heater at lower heat fluxes and remain attached to the heater. Such bubbles depart only when the heat flux is increased. In the present experimental study, such bubble formation is not
Experimental Study of Onset of Nucleate Boiling from Submerged …
281
characterized as ONB. ONB is considered to occur when a bubble ebullition cycle is seen as suggested by Howell and Siegel [2]. The heat flux supplied to the heater was calculated from the electrical power and the surface area of the heater. The temperature of the heater was derived from the measured resistance of the ribbon. The heater resistance (V/I) was noted for different bulk temperatures during calibration in a water bath. A linear relationship between resistance and temperature was derived which was used to estimate the heater temperature during experiments. The maximum uncertainties in temperature and heat flux are ± 1 °C and ± 2%, respectively.
4 Results and Discussion In this section, the measured heat flux and wall temperature data at ONB are presented for different bulk temperatures. The effect of bulk temperature and width of the heater on these parameters are discussed below.
4.1 Heat Flux Variation with Subcooling and Heater Width The heat flux at ONB and the wall temperature at ONB for different bulk temperatures and width of the heater were measured. The results of the heat flux at ONB are shown in Fig. 2. The error bands shown reflect the range of uncertainty and are based on the maximum uncertainty in measurement and the standard deviation of the multiple tests (σ ). The maximum between these is taken as the error bar or range of uncertainty. For each heater width, the experiment was repeated thrice. The heat flux at ONB increases with the increase in subcooling. This trend is same for any width. This result is in line with the findings in the literature. With decrease in bulk temperature, more heat is needed to bring the fluid to the nucleation state. The effect of width is more evident when the width is smaller. The heat flux at ONB for the 2-mmwidth heater is around 100 per cent more than the 10-mm-width heater for all bulk temperatures. This can be attributed to the fact that decrease in heater size enhances the convective heat transfer. An enhanced convective heat transfer delays the onset of boiling and leads to a higher heat flux at ONB [24]. The heat fluxes at ONB for 4 mm, 6-mm- and 8-mm-width heaters are very close to the 10-mm-width heater. This highlights that the effect of increased convective heat transfer for decreased width does not take effect until the width is considerably smaller.
282
J. Pinto et al.
Fig. 2 Variation of heat flux at ONB with subcooling for different widths
4.2 Wall Superheat Variation with Subcooling and Heater Width The results of the wall superheat at ONB are shown in Fig. 3. The error bars describe the range of uncertainty based on repeatability and measurement uncertainty. Generally, the wall superheat increases with the increase in subcooling and then tends to remain constant. This behaviour is similar to the ones for water observed in the literature [25, 26]. The wall superheat shows a significant increase with a decrease in the heater width. Reduction in the width of the heater enhances the convective heat transfer and hence reduces the thermal boundary layer thickness. The reduced thermal boundary layer thickness increases the activation temperature of the cavities on the heater and hence leads to a higher wall superheat at ONB as shown in Fig. 4. This effect when coupled with the effect of subcooling increases the wall temperature significantly at lower bulk temperatures. Figure 5 shows the variation of wall superheat and heat flux at ONB for different widths. There is a sharp increase in the wall superheat and heat flux at ONB for the 2-mm-width heater in comparison with the other heaters. This shows that the flow field generated around a 2-mm-width heater is significantly different than larger width heaters. Study is undergoing to understand this observed transitional behaviour.
Experimental Study of Onset of Nucleate Boiling from Submerged …
283
Fig. 3 Variation of wall superheat at ONB with subcooling for different widths Fig. 4 Effect of thermal boundary layer thickness on wall superheat at ONB
5 Conclusions The results of this study can be summarised as follows: • Increasing subcooling increases the heat flux at ONB. The ONB heat flux for 20 K subcooling is 130% more than for 5 K subcooling. • Increasing subcooling increases the wall superheat at ONB. The wall superheat initially increases with subcooling then remains constant. The trend for subcooling higher than 20 K needs to be examined to verify the presence of this asymptotic behaviour.
284
J. Pinto et al.
Fig. 5 Variation of wall superheat and heat flux at ONB with heater width for 5 K subcooling
• Decreasing width increases heat flux at ONB. The heat flux at ONB for 2-mmwidth heater is around 100% more than for 10-mm-width heater for all subcooling. • Decreasing width increases wall superheat at ONB. The wall superheat at ONB for 2-mm-width heater is around 50% more than for 10-mm-width heater for all subcooling. • There is a sudden increase in ONB heat flux and wall superheat for 2-mmwidth heater in comparison with other heaters. This phenomenon is under further investigation.
References 1. Frost W, Dzakowic GS (1969) Manual of boiling heat-transfer design correlations. Accessed: Apr 05 2022 [Online]. Available: https://apps.dtic.mil/sti/citations/AD0698323 2. Howell J, Siegel R (2021) Incipience, growth, and detachment of boiling bubbles in saturated water from artificial nucleation sites of known geometry and size. Accessed: Oct 05, 2021 [Online]. Available: https://ntrs.nasa.gov/api/citations/19660008286/downloads/196600 08286.pdf?attachment=true 3. Wiebe JR, Judd RL (1971) Superheat layer thickness measurements in saturated and subcooled nucleate boiling. J Heat Transf 93(4):455–461. https://doi.org/10.1115/1.3449845 4. Basu N, Warrier GR, Dhir VK (2002) Onset of nucleate boiling and active nucleation site density during subcooled flow boiling. J Heat Transf 124(4):717–728. https://doi.org/10.1115/ 1.1471522 5. Theofanous TG, Tu JP, Dinh AT, Dinh TN (2002) The boiling crisis phenomenon part I: nucleation and nucleate boiling heat transfer. Exp Therm Fluid Sci 26(6–7):775–792. https:// doi.org/10.1016/S0894-1777(02)00192-9
Experimental Study of Onset of Nucleate Boiling from Submerged …
285
6. Bourdon B, Di Marco P, Rioboo R, Marengo M, De Coninck J (2013) Enhancing the onset of pool boiling by wettability modification on nanometrically smooth surfaces. Int Commun Heat Mass Transf 45:11–15. https://doi.org/10.1016/J.ICHEATMASSTRANSFER.2013.04.009 7. Bourdon B, Rioboo R, Marengo M, Gosselin E, De Coninck J (2012) Influence of the wettability on the boiling onset. Langmuir 28(2):1618–1624. https://doi.org/10.1021/LA203636A/ ASSET/IMAGES/LA203636A.SOCIAL.JPEG_V03 8. Duan L, Liu B, Qi B, Zhang Y, Wei J (2020) Pool boiling heat transfer on silicon chips with non-uniform micro-pillars. Int J Heat Mass Transf 151:119456. https://doi.org/10.1016/J.IJH EATMASSTRANSFER.2020.119456 9. Liu B, Yang X, Jie Z, Wei J, Li Q (2022) Enhanced pool boiling on micro-nano composited surfaces with nanostructures on micro-pin-fins. Int J Heat Mass Transf 190:122812. https:// doi.org/10.1016/J.IJHEATMASSTRANSFER.2022.122812 10. Lee CY, Zhang BJ, Kim KJ (2012) Morphological change of plain and nano-porous surfaces during boiling and its effect on nucleate pool boiling heat transfer. Exp Therm Fluid Sci 40:150–158. https://doi.org/10.1016/J.EXPTHERMFLUSCI.2012.02.011 11. Sitar A, Može M, Crivellari M, Schille J, Golobiˇc I (2020) Nucleate pool boiling heat transfer on etched and laser structured silicon surfaces. Int J Heat Mass Transf 147:118956. https://doi. org/10.1016/J.IJHEATMASSTRANSFER.2019.118956 12. Serdyukov V, Starinskiy S, Malakhov I, Safonov A, Surtaev A (2021) Laser texturing of silicon surface to enhance nucleate pool boiling heat transfer. Appl Therm Eng 194:117102. https:// doi.org/10.1016/J.APPLTHERMALENG.2021.117102 13. Hsu YY (1962) On the size range of active nucleation cavities on a heating surface. J Heat Transf 84(3):207–213. https://doi.org/10.1115/1.3684339 14. Wiebe JR (1970) Temperature profiles in subcooled nucleate boiling. Accessed: Apr 08, 2022. [Online]. Available: https://macsphere.mcmaster.ca/handle/11375/20780. 15. Sharp R (1964) The nature of liquid film evaporation during nucleate boiling 16. Magrini U, Nannei E (1975) On the influence of the thickness and thermal properties of heating walls on the heat transfer coefficients in nucleate pool boiling. J Heat Transfer 97(2):173–178. https://doi.org/10.1115/1.3450337 17. Grigoriev VA, Pavlov YM, Ametistov YV, Klimenko AV, Klimenko VV (1977) Concerning the influence of thermal properties of heating surface material on heat transfer intensity of nucleate pool boiling of liquids including cryogenic ones. Cryogenics (Guildf) 17(2):94–96. https://doi.org/10.1016/0011-2275(77)90105-9 18. Chuck TL, Myers JE (1978) The effect of heater plate thickness on boiling heat-transfer coefficients. Int J Heat Mass Transf 21(2):187–191. https://doi.org/10.1016/0017-9310(78)902 22-3 19. Park KA, Bergles AE (1987) Natural convection heat transfer characteristics of simulated microelectronic chips. J Heat Transfer 109(1):90–96. https://doi.org/10.1115/1.3248074 20. Kwark SM, Amaya M, Kumar R, Moreno G, You SM (2010) Effects of pressure, orientation, and heater size on pool boiling of water with nanocoated heaters. Int J Heat Mass Transf 53(23–24):5199–5208. https://doi.org/10.1016/J.IJHEATMASSTRANSFER.2010.07.040 21. Rainey KN, You SM (2001) Effects of heater size and orientation on pool boiling heat transfer from microporous coated surfaces. Int J Heat Mass Transf 44(14):2589–2599. https://doi.org/ 10.1016/S0017-9310(00)00318-5 22. Lee WR, Lee JY (2014) Effect of flow instability on pool boiling and CHF of thin flat plate heater PCB 36(12):1028–1036. https://doi.org/10.1080/01457632.2015.979118 23. Kim JH, You SM, Pak JY (2006) Effects of heater size and working fluids on nucleate boiling heat transfer. Int J Heat Mass Transf 49(1–2):122–131. https://doi.org/10.1016/J.IJHEATMAS STRANSFER.2005.08.001
286
J. Pinto et al.
24. Carey V (2018) Liquid-vapor phase-change phenomena: an introduction to the thermophysics of vaporization and condensation processes in heat transfer equipment, Second. CRC Press 25. Goel P, Nayak AK, Kulkarni PP, Joshi JB (2017) Experimental study on bubble departure characteristics in subcooled nucleate pool boiling. Int J Multiph Flow 89:163–176. https://doi. org/10.1016/J.IJMULTIPHASEFLOW.2016.10.012 26. Narayan S, Singh T, Srivastava A (2020) Experiments on pool boiling regimes and bubble departure characteristics of single vapor bubble under subcooled bulk conditions. Exp Therm Fluid Sci 111(April 2019):109943. https://doi.org/10.1016/j.expthermflusci.2019.109943
Numerical Study of Bubble Growth on a Hydrophilic Surface Abhishek K. Sharma and Shaligram Tiwari
Nomenclature α m˙ U σ ϑ kl δs ρ μ E kt p Se T g
Volume fraction Volumetric mass transfer rate kg m− 3 s− 1 Velocity vector ms− 1 Surface tension coefficient Nm− 1 Kinematic viscosity m2 s−1 Thermal conductivity of liquid W m− 1 K− 1 Interface Dirac delta function Density kg m− 3 Viscosity kg m− 1 s− 1 Energy per unit mass J kg− 1 Thermal conductivity W m− 1 K− 1 Pressure N m− 2 Source term J m−3 s−1 Temperature K Acceleration due to gravity ms− 2
1 Introduction Nucleate pool boiling is an efficient mode of heat transfer in which a large amount of heat transfer can be achieved within a low degree of superheat. It covers the broader area of applications such as distillation, cooling of nuclear reactors thermal desalination, chemical industries and refrigeration systems. This mode of heat transfer also A. K. Sharma (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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_27
287
288
A. K. Sharma and S. Tiwari
proved to be effective in data centre cooling and microelectronic devices where a high heat dissipation rate is required within a small change of temperature. There are so many equipment on the systems working in microgravity condition like space station and space shuttle, where it is very critical to attain efficient heat transfer. So, it is of extreme importance to further advance the study associated with this phenomena considering both terrestrial and microgravity conditions.
2 Literature Review and Objective Several studies have been performed in order to find empirical correlations and models for different bubble dynamics parameters such as bubble waiting time, Dd and nucleation site density that govern the heat transfer process during nucleate pool boiling. Son et al. [1] performed numerical simulations to study the dynamics of bubble on a horizontal surface using level set method of interface capturing. They predicted the influence of wall superheat and Ca on growth and departure of bubble and observed an increment in bubble size with increase in superheat level and Ca . Using the same method of interface capturing, Abarajith and Dhir [2] investigated the consequence of change in Ca on bubble dynamics by considering two different fluids water and PF-5060. They reported that the Dd increases with an increment in Ca for both the fluids. In addition, they also stated that the Dd for water is more than that of PF-5060 at all values of contact angles. Wu and Dhir [3] used moving mesh method incorporated with level set function to study the effect of subcooling at various gravity levels and reported that the influence of subcooling is more significant at microgravity level as compared to the terrestrial gravity conditions. By further extending this work, Wu and Dhir [4] investigated the effect of noncondensables on dynamic behaviour of a single bubble and stated that the effect of noncondensables is considerable under reduced gravity conditions. Phan et al. [5] performed numerical study to predict the effect of wettability modulation on mechanism of nucleation and heat transfer behaviour by using nanocoating technology to vary surface wettability for water. They reported that the decrease in the Ca leads to an increment in Dd and reduction in departure frequency. Focusing on heat transfer behaviour, they observed that the heat transfer coefficient (HTC) deteriorates with reduction in Ca for surfaces having weak wettability. On the other hand, for surfaces with high wettability, this effect gets reversed. Gong and Chang [6] used lattice Boltzmann method to numerically investigate the effect of superheat, surface wettability and heater size on pool behaviour under saturated condition. They found an enhancement in Dd with the increase in superheat and decrease in surface wettability. In addition, they also noticed that the nucleation site density increases with the rise in heater size, superheat level and Ca . In spite of a number of research works performed in this field, there is still a scatter in the reported results. This is mainly because of different models and correlations used to govern the complex phenomena associated with the process of nucleate pool boiling. Thus, in order to have a better understanding of the impact of different
Numerical Study of Bubble Growth on a Hydrophilic Surface
289
parameters, the present work focuses on the numerical computation of a single vapour bubble on a hydrophilic surface in which the effect of variation in Ca , wall superheat and gravity level on the growing bubble have been investigated.
3 Methodology 3.1 Governing Equations In the present work, VOF method given by Hirt and Nichols [7] is employed to perform the numerical simulation. Here, both the vapour and liquid phases are considered to be incompressible in the bulk region. A volume function α is defined, which represents the distribution of phase in each cell. Here, αl and αv represent the volume fraction of liquid and vapour, respectively, in each cell. In the two-phase cell, conservation of mass is achieved by solving the following equations. ∂αl 1 + ∇.(U αl ) = m˙ l ∂t ρl
(1)
∂αv 1 + ∇.(U αv ) = m˙ v ∂t ρl
(2)
αl + αv = 1
(3)
Based on the formulation of single fluid flow and using the surface tension model given by Brackbill et al. [8], the modified momentum equation is as follows. ∂ (ρ(F)U ) + ∇.(ρ(F)UU ) ∂t )] ( [ ˆ s = −∇ p + ρ(F)g + ∇. μ(F) ∇U + ∇U T + σ k nδ
(4)
Here, nˆ is unit normal vector to the interface, and k is mean interface curvature. The effect of change in phase at the vapour–liquid interface and heat transfer associated with micro layer evaporation is taken by introducing as a source terms in the following energy equation. ∂ (ρ E) + ∇.(U (ρ E + p)) = ∇.μ(kt ∇T ) + Se ∂t
(5)
In this model, a function F is used to represent the volume fraction of fluid in each control volume. This function F is defined as follows: ( ) 1, x ∈ liquid F(x) = (6) 0, x ∈ vapor
290
A. K. Sharma and S. Tiwari
At the interface, this function takes the value between zero and one. Thermophysical properties, density and viscosity are calculated as follows: ρ(F) = ρl F + ρv (1 − F)
(7)
μ(F) = μl F + μv (1 − F)
(8)
In comparison with the bubble size, the region of microlayer between an expanding bubble and the heat transfer surface is extremely thin whose dimension lies in the range of micrometres. As a result, employing the VOF approach to simulate the microlayer region is challenging, so special handling is required to include the effect of microlayer evaporation in order to meet the required physical condition. In the present work, microlayer model given by Cooper and Lloyd [9] is used to calculate the initial thickness of microlayer. According to this model, the initial microlayer thickness at a particular radial location is given as follows: √ δ0 = 0.8 ϑt R
(9)
Here, tR is the time taken by the contact line to travel to that radial location. Heat transfer through the microlayer is computed on the basis of one-dimensional heat conduction by using the following equation: qml = kl
Tw − Tsat δ(r )
(10)
Here, qml is the local heat transfer through the microlayer, and δ(r ) is the local thickness of microlayer. Saturated-interface-volume phase change model given by Pan et al. [10] is used to consider the phase change process at the interface of liquid and vapour.
3.2 Computation Domain and Boundary Conditions In the present work, a two-dimensional domain is considered. In order to reduce the cost of computation, axisymmetric model is used in which the height and radius of the domain are 4.5 mm and 2 mm, respectively. The description of phases is done on the basis of volume fraction of the vapour. In Fig. 1, liquid is present where vapour fraction is zero as depicted in blue colour, and the presence of vapour is shown in red colour with volume fraction of 1. Condition of no-slip boundary is imposed at the outer radial wall and heating surface. The outer boundary is maintained at atmospheric pressure by imposing pressure outlet boundary condition.
Numerical Study of Bubble Growth on a Hydrophilic Surface
291
Fig. 1 Computation domain
3.3 Numerical Technique and Grid Independent Study In the present work, ICEM CFD 2021 R1 is used to generate the grid in the computational domain. Commercial software ANSYS Fluent 2021 R1 is employed to solve the flow and energy equations based on finite volume method. VOF method is adopted to capture the vapour–liquid interface. The effect of evaporation of liquid underneath the microlayer in the microscopic region is taken by using user-defined function (UDF). Geo-reconstruct scheme is used for the spatial discretization of volume fraction. PISO algorithm is used for performing pressure–velocity coupling. Discretization of convective terms of energy and momentum equation are done by second-order upwind scheme, and pressure term is discretized by PRESTO! scheme. Discretization of gradient is done by least square cell-based scheme. Grid independent study is performed by taking three different grid sizes, and the result is compared in the form of Dd for all the cases. It can be observed from Table 1 that the variation in Dd between the second and third case is insignificant as compared to the corresponding variation between the first and second case. On the basis of above analysis, grid size of 0.01 mm with a grid number of 90,000 is adopted for performing the computation in the current study. The time step of 10–6 s has been taken for the present simulation.
292
A. K. Sharma and S. Tiwari
Table 1 Grid independent study S. No.
Minimum edge length (mm)
Bubble departure diameter (mm)
56,446
0.012
1.564
2
90,000
0.01
1.575
3
112,500
0.008
1.578
1
No. of grids
3.4 Validation In order to validate the present work, simulation is performed by applying the same boundary conditions which was taken by Allred et al. [11]. A constant heat flux of 3W/cm2 is provided to the heating surface. Initialization of vapour bubble is done by placing a seed bubble of radius 0.27 mm. The solid heating surface is initialized at a temperature of 378.15 K. Hydrophilic surface is considered for which both receding contact angle (θ rec ) and advancing (θ adv ) are small. The value of receding and advancing contact angle is taken as 30° for the present case. The result of the simulation in the form of bubble morphology at different normalized time t* (bubble growth time normalized by departure time) is compared with similar result of Allred et al. [11, 12] in Fig. 2, and it is found that the present simulation accurately replicates the growth of bubble as observed in previous experimental and numerical work of Allred et al. [11, 12]. t*=0
0.2
0.6
1 Smooth aluminum θrec =5° and θadv=29° Experimental [11]
Allred et al. [12] θrec =30° and θadv=30°
Present work θrec =30° and θadv=30°
Fig. 2 Progression of bubble growth with normalized time
Numerical Study of Bubble Growth on a Hydrophilic Surface
293
4 Results and Discussion The influence of two different wall superheat temperatures 12 K and 9 K under normal gravity condition is evaluated for the vapour bubble at contact angles of 30° and 45° with the heating surface. Microgravity condition (0.1 g) with contact angle of 30° is also considered. Results are analysed in the form of Dd , wall heat flux, growth period and progression of bubble growth with time.
4.1 Bubble Morphology Figures 3, 4 and 5 show the morphology of bubble evolution with time for different cases considered in present study. It can be stated from Figs. 3 and 4 that the trend of morphological evolution of bubble at a particular Ca is similar for both the wall superheat temperatures, whereas at a fixed temperature, a slight variation in the trend of bubble morphological evolution is noticeable with the change in Ca . It is also observed that the Dd and growth time increase with an increment in the value of the Ca . This is attributed to the fact that the increase in Ca leads to an increment in the perpendicular component of surface tension, which acts as an opposing force for the bubble detachment from the surface. This enforces the bubble to grow more and more so that the buoyancy force increases to a value which can oppose the surface tension force so as to promote the departure of bubble. It can be stated from Fig. 5 that the lateral spreading of bubble before departure is comparatively more at lower wall superheat temperature under reduced gravity condition.
(a) ΔT=9 K
(b) ΔT=12 K Fig. 3 Progression of bubble growth at Ca of 45° under normal gravity
294
A. K. Sharma and S. Tiwari
(a) ΔT=9 K
(b) ΔT=12 K Fig. 4 Progression of bubble growth at Ca of 30° under normal gravity
(a) ΔT=9 K
(b) ΔT=12 K Fig. 5 Progression of bubble growth at Ca of 30° under microgravity
4.2 Bubble Equivalent Radius The influence of superheating on bubble growth parameter in terms of bubble radius is presented in Figs. 6, 7 and 8 for different values of gravity level and Ca . Figure 6 elucidates that the effect of an increment in wall temperature is to enlarge the bubble size and decrease the time of bubble growth before departure. The growth rate of the bubble depends upon the difference in temperature between the vapour and liquid phase which is directly influenced by the superheating provided to the
Numerical Study of Bubble Growth on a Hydrophilic Surface
295
Fig. 6 Bubble growth curve for Ca of 45° under normal gravity
Fig. 7 Bubble growth curve for Ca of 30° under normal gravity
surface. Liquid temperature surrounding the vapour gets elevated with increase in wall temperature, which leads to an enhancement in rate of evaporation at the interface of vapour and liquid. The influence of increase in wall temperature is also to make the evaporation rate in the microlayer region to be more intense. The combined effect of both macroscopic and microscopic heat transfer processes has crucial influence in increasing the radius of the bubble as the wall temperature is increased. It is evident from the Figs. 7 and 8 that the impact of increase in wall temperature to increase the radius of the bubble radius is more under microgravity conditions in
296
A. K. Sharma and S. Tiwari
Fig. 8 Bubble growth curve for Ca of 30° under microgravity
the later stage before the departure. This demonstrates that the gravity level has a significant influence on dynamic behaviour of bubble growth.
4.3 Bubble Base Radius (Br ) For wall superheat temperature of 12 K, Figs. 9 and 10 show the modulation of Br with time at contact angles of 45° and 30°, respectively, under normal gravity. Figure 11 presents the corresponding parameter at a contact angle of 30° under microgravity condition (0.1 g). During the preliminary period of bubble growth, the increment in Br is linear for both contact angles, but at a later stage, the decrease in base radius at low Ca is faster as compared to the high Ca (Figs. 9 and 10). The peak value of the base radius is more at a lower Ca . An increment in the value of Ca makes the bubble to stay on the heating surface for a large time before departure. On comparing the variation of Br for normal and microgravity conditions at a wall superheat of 12 K (Figs. 10 and 11), it is observed that the Br is comparatively more under normal gravity level. To understand the impact of superheating on bubble growth and associated heat transfer for different values of gravity level Ca , bubble growth parameters (growth period, Dd ) and wall heat flux at the time of departure are shown in Tables 2, 3 and 4. It can be observed from Tables 2 and 3 that the effect of an increase in wall superheat temperature on wall heat flux is more at a lower value of Ca as compared to the corresponding increment at higher value of Ca . On comparing Tables 3 and 4,
Numerical Study of Bubble Growth on a Hydrophilic Surface
297
Fig. 9 Variation of Br at a Ca of 45° under normal gravity
Fig. 10 Variation of Br at a Ca of 30° under normal gravity
it can be stated that the increment in wall heat flux with the increase in superheating is slightly more on normal gravity as compared to the microgravity condition.
5 Conclusions In the present work, numerical simulation is performed to study the growth of the bubble and associated heat transfer in nucleate pool boiling. The effect of providing different wall superheats to the heating surface is investigated at various gravity levels to see its influence on bubble morphology. The impact of varying the Ca is also
298
A. K. Sharma and S. Tiwari
Fig. 11 Variation of Br at a Ca of 30° under microgravity
Table 2 Predicted wall heat flux and bubble growth at Ca of 45° under normal gravity Wall superheat ∆T(K) Growth period (m s) Departure diameter Dd (mm)
Wall heat flux (W/m2 )
9
33
2.73
34,447
12
30
2.82
52,180
Table 3 Predicted wall heat flux and bubble growth at Ca of 30° under normal gravity Wall superheat ∆T(K) Growth period (m s) Departure diameter Dd (mm)
Wall heat flux (W/m2 )
9
13
2.57
44,305
12
11
2.40
77,852
Table 4 Predicted wall heat flux and bubble growth at Ca of 30° under microgravity Wall superheat ∆T(K) Growth period (m s) Departure diameter Dd (mm)
Wall heat flux (W/m2 )
9
30
3.38
34,423
12
15
3.20
65,785
studied. It can be stated from the result that, for a particular value of wall superheat, reduction in the gravity level leads to an increment in Dd and departure time. The variation in morphological evolution of bubble with the increase in temperature of the heating surface is more significant under microgravity as compared to the normal gravity condition. The increment in wall heat flux with the increase in superheat value
Numerical Study of Bubble Growth on a Hydrophilic Surface
299
is found to be more at a lower Ca as compared to the corresponding enhancement at a higher value of Ca .
References 1. Son G, Dhir VK, Ramanujapu N (1999) Dynamics and heat transfer associated with a single bubble during nucleate boiling on a horizontal surface. J Heat Transfer 121(3):623–631 2. Abarajith HS, Dhir VK (2002) A numerical study of the effect of contact angle on the dynamics of a single bubble during pool boiling. In ASME International Mechanical Engineering Congress and Exposition 3638:467–475 3. Wu J, Dhir VK (2010) Numerical simulations of the dynamics and heat transfer associated with a single bubble in subcooled pool boiling. J Heat Transf 132(11) 4. Wu J, Dhir VK (2011) Numerical simulation of dynamics and heat transfer associated with a single bubble in subcooled boiling and in the presence of noncondensables. J Heat Transf 133(4) 5. Phan HT, Caney N, Marty P, Colasson S, Gavillet J (2009) Surface wettability control by nanocoating: the effects on pool boiling heat transfer and nucleation mechanism. Int J Heat Mass Transf 52(23–24):5459–5471 6. Gong S, Cheng P (2015) Lattice Boltzmann simulations for surface wettability effects in saturated pool boiling heat transfer. Int J Heat Mass Transf 85:635–646 7. Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39(1):201–225 8. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. J Comput Phys 100(2):335–354 9. Cooper MG, Lloyd AJP (1969) The microlayer in nucleate pool boiling. Int J Heat Mass Transf 12(8):895–913 10. Pan Z, Weibel JA, Garimella SV (2016) A saturated-interface-volume phase change model for simulating flow boiling. Int J Heat Mass Transf 93:945–956 11. Allred TP, Weibel JA, Garimella SV (2021) The role of dynamic wetting behavior during bubble growth and departure from a solid surface. Int J Heat Mass Transf 172:121167 12. Allred TP, Weibel JA, Garimella SV (2018) Enabling highly effective boiling from superhydrophobic surfaces. Phys Rev Lett 120(17):174501
Frequency Analysis of Direct Contact Condensation Using the Wavelet Transform During the Vertical Steam Injection on the Subcooled Water Pool Saurabh Patel and Parmod Kumar
Nomenclature DCC CWT DWT DW ms
Direct contact condensation Continuous wavelets transform Desecrate wavelet transform Daubechies wavelet transform Milliseconds
1 Introduction Direct contact condensation (DCC) is a phenomenon in which vapor comes direct contact with another fluid and condensate. Steam injection into the pool filled with subcooled water generally occurs in many industries like nuclear reactors [1], boilers, and direct contact heat exchangers [2]. During the loss of coolant accidents, it can be possible a large amount of steam has condensed into the suppression pool due to the smaller size of the boiling water reactor (BWR). The structural loads act on the boiling water reactor due to the sudden condensation of a large amount of steam into the suppression pool. The root cause for the loads acting on the suppression pool is pressure fluctuations during the steam injection. These pressure oscillations are associated with the different frequencies. Frequency analysis is vital for the safety purposes of the design of BWR and the prevention of accidents like Fukushima in Japan [3].
S. Patel · P. Kumar (B) School of Mechanical and Material Engineering, IIT Mandi, Kamand 175075, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_28
301
302
S. Patel and P. Kumar
2 Literature Review and Objective Although many experimental investigations have been performed in order to get the physical essence of the direct contact condensation phenomenon; however, most studies focus on the shape of the bubble at the pipe outlet, regime map, and average heat transfer coefficient. The regime maps are plotted by different researchers [4–7] on the basis of mass flow rate and pool subcooled temperature in the direct contact condensation. Still, no one can reach a consensus due to the highly transient nature of the phenomenon and different experimental conditions. The different regimes are involved during the direct contact condensation mainly named as chugging, oscillatory bubble, ellipsoidal jet, and conical jet. The chugging regimes are further classified internal chugging and external chugging. Chan and Lee [4] reported that the different frequencies are involved in different regimes in the vertical downward injection of steam into the pool. This investigation has been performed in the mass flux range of 0–175 kg/m2 -sec. The frequencies associated with the ellipsoidal bubble, oscillatory bubble, and oscillatory cone jet detachment are 11, 7, and 40 Hz, respectively. In the chugging regime, bubble detachment frequencies are between 1 and 3 Hz. In another experimental investigation, Aya et al. [8, 9] revealed that the lower frequency is in the range of 2–8 Hz, middle frequency is 15 Hz, and higher frequency is in the range of 100-150 Hz for their setup. Youn et al. [10] reported in an experimental investigation that frequency increases with the mass flow rate and is not as much affected by pool subcooled temperature in the chugging regime. The mass flow rate and the subcooled temperature range are 10–80 kg/m2 -sec and 30–80 °C, respectively, in the horizontal injection of steam in the pool. Hong et al. [11] investigated that dominant frequency increases with mass flow rate up to 300 kg/m2 -sec in the condensation oscillation regime. However, the dominant frequency decreases when the mass flow rate increases above 300 kg/m2 -sec. Researchers have seen the first dominant frequency in direct contact condensation but occasionally the second dominant frequency for pressure oscillation. Qiu et al. [12] reported that the variation in the steam plume and the evolution and burst of the larger bubble are the main causes of first and second dominant frequencies, respectively. Tang et al. [13] revealed that when a bubble condensation occurs in the pool, there are three modes: shape oscillation, transition, and capillary wave regimes. The dominant frequency for sound is in the range of 150–300 Hz for every condensation regime. Additionally, several investigations are concentrated on determining the penetration length of the steam jet into the water pool. Qiu et al. [14] reported that the penetration length of the steam jet increases for the sonic and supersonic steam inject with the steam mass flux and pool water temperature. Xu and Guo [15] revealed that the penetration length of the steam jet increases with pool water temperature and steam mass flux. However, it decreases with enhancing of the Reynolds number. In addition, studies [16, 17] also delineate the flow pattern and heat transfer characteristics in the DCC phenomenon. Patel et al. [18] performed the numerical simulation using the pattern reorganization algorithm to find the bubble volume and frequency in the chugging regime.
Frequency Analysis of Direct Contact Condensation Using the Wavelet …
303
The previous studies of frequency analysis of pressure spikes have been done using fast Fourier transformation (FFT). The limitation of the FFT of time domain frequency analysis is that the signal frequency is not well localized due to the two events not being well separated from each other. The uncertainty of two events in the frequency and time is high in FFT. In the present study, the time domain analysis of the frequency of pressure spikes and acoustic sound of bubbles due to the collapse has been done using the continuous wavelet transform (CWT) in the MATLAB [19]. The frequency analysis with the time is crucial in the nuclear industry for safer operation and structural design. The steam is injected in the vertical downward orientation in the experiment, and the range of the steam mass flux and pipe submergence depth is 10–30 kg/hr and 1–9 cm, respectively.
3 Experimental Setup and Methodology The schematic diagram of the experimental setup used in the present work is shown in Fig. 1. The experimental facility comprises a boiler, high-speed camera, steam lines, two solenoid valves, a steam mass flow meter, and a water pool. The non-IBR boiler (make: Hi-Therm boiler Ltd., mode: EW-36) is used to heat the water. The boiler consists of a boiler tank, four heating rods, temperature, and pressure regulator. The water is heated in the boiler tank until it reaches boiling. The cylindrical shape boiling tank is fully insulated. The steam generated inside the boiler tank is supplied through the steam lines in the water pool. The pressure and temperature regulators control the pressure and temperature inside the boiler. In the present investigation, the cutoff pressure and temperature of the boiler are set up at 5 bar and 150 °C, respectively. The ball valve in the steam line is used to regulate the steam mass flow rate. The steam is supplied in the pool, whose size is 76 cm in length, 61 cm in width, and 76 cm in height. The steam flow rate measurement has been taken from the steam mass flow meter (make: Honeywell; model: SMV 800 multivariate transmitter) inserted in the pipelines. The presence of non-condensable gasses in the pipeline may affect the phenomenon. The non-condensable gasses inside the steam lines are removed by the solenoid valve 1 through the steam injected into the pipe before the experiments start. A half-inch-steel pipe was used in the experiments, which has exceptional properties during high load conditions. The pressure transducer (make: Kistler; type: 6061B; range: 0–250 bar) has been installed at 15 cm from the outlet of the pipe. The temperature of the steam and pool water in the pipe is monitored by K-type sheathed duplex thermocouples. Furthermore, an omnidirectional hydrophone (make: Kavone Tech. Pvt. Ltd.; range: 100 Hz to 4 kHz; sensitivity: − 170db ± 3db) has been used to capture the amplitude of the acoustic sound due to bubble collapse and pressure spikes inside the water pool. For achieving the steady-state condition of steam mass flow rate, steam is first passed through a drain line using solenoid valve 1 for about 90 s and then it injected in the water pool. Solenoid valve two opens, and solenoid valve 1 closes over 90 s.
304
S. Patel and P. Kumar
Fig. 1 a Schematic diagram of the experimental setup and b experimental setup in the ThermoFluid Laboratory at IIT Mandi
Meanwhile, simultaneously, the high-speed camera (make: Photron; model SA1.1) is triggered to capture the phenomenon. A DC light is used for proper illumination of the visualization window during the high-speed photography. The frame rate per second of the camera in which video is recorded is 125 fps, and the total capture time is 43.26 s. The output of all the sensors is recorded using a data acquisition system (make: Gems sensors; model: DNA AI 277) at 1000 Hz. For image processing and wavelet analysis, the MATLAB® toolbox is used. For repeatability, each experiment is performed five times. The study has been performed at the mass flow rates of 10–30 kg/hr with the uncertainty of ± 1 kg/hr and four submergence depths of pipe varying from 1 to 9 cm at a step of 4 cm.
Frequency Analysis of Direct Contact Condensation Using the Wavelet …
305
3.1 Method of Analysis 3.1.1
Continuous Wavelet Transform
Wavelet analysis is a newly established techniques, which is used to examine the data in different time and frequency resolutions. At low frequencies, wavelet transform gives good resolution of frequency. The probe signal comprises slowly varying components interspersed with abrupt changes. The continuous wavelet transform (CWT) provides the joint time–frequency visualization of the probe signal. In the present study, the Morse wavelet is used to analyze data. The CWT function which is based on the spread of wavelet energy determines the largest and smallest scale for the investigation. In the wavelet scalogram, the white dashed line shows the cone of influence, and with this region, the cone estimates are reliable. The CWT of a function f(t) at a scale m (where m belongs to a set of positive real numbers) and translational value n (where n is any real number) is given in integral form as follows: Fw (m, n) =
1
∞
1
m2
−∞
(t − n) dt f (t) m
In the above equation, ψ(t) is known as the mother wavelet, a continuous function of the frequency and time domain. Many researchers have previously used wavelet analysis [20–22] to understand the dynamics of the multiphase flows. In liquid–gas flows, wavelet analysis is seldom used as the multiscale components are involved in the dynamics of such flows.
3.1.2
Daubechies Wavelet
Daubechies 4 wavelet method has been used to analyze acoustic sound signals captured by the hydrophone in the present investigation. For a given width, the Daubechies wavelets have the most vanishing moments. After the decomposition of the wavelet, the range of frequency decreases while the range of scale increases. In the present investigation, the signal has been decomposed into five levels.
4 Results and Discussion Several observations are made from the visual analysis of Fig. 2. It shows that the shape of the bubble frequently changes in a chaotic manner. The DCC cycle (bubble formation, growth of bubble, elongation, and rupture) repeats in a periodic manner during the whole phenomenon. Due to the elongation and rupture of the bubble, pressure oscillations occur in the pipe. These oscillations are associated with the range
306
S. Patel and P. Kumar
0s
0.09 s
1.3 s
4s
4.05 s
4.3 s
1.4 s
5s
3.7 s
5.16 s
8s
11.7 s
12 s
13.1 s
13.5 s
15 s
16 s
17 s
17.64 s
17.9 s
Fig. 2 Spatio-temporal bubble evolution at mass flow rate 30 kg/hr and pipe submergence depth of 9 cm
of frequency. Therefore, the frequency analysis became necessary to understand the DCC phenomenon properly. In addition, because of the bubble rupture, the pressure decreases suddenly inside the pipe, reaching near about vacuum pressure. Due to a vacuum in the pipe, water suction happens to the pipe. During the whole DCC cycle, the pressure also keeps changing so the analysis of pressure signal and its associated frequency became crucial.
4.1 Continuous Wavelet Transformation (CWT) Analysis of Pressure Signal Different pressure oscillation patterns appeared when the steam was injected into the pipe depending on the energy balance between the injected steam and pool water. The dynamic nature of the DCC phenomenon made the pressure characteristics immensely complicated. Therefore, finding the dominant frequency of the pressure signal is not easy. CWT analysis of pressure signal at different mass flow rates and 1 cm depth is shown in Fig. 3. From the visual analysis of the magnitude scalogram of the pressure signal, peak frequency increases with the mass flow rate at a constant
Frequency Analysis of Direct Contact Condensation Using the Wavelet …
307
depth. The reason associated with an increase in the frequency with the mass flow rate is that the momentum transfer rate becomes higher at a higher mass flow rate and the DCC cycle repeats very fast. In addition, it is also observed that the peak frequency shifts toward the left side with the time. The reason for shifting of the left side of the peak frequency is the bubble formation, and collapse occurs early with the time. The range of the peak frequency due to the pressure oscillation is 0–0.1 Hz, and sampling frequency is 1000 Hz for present experimental investigation.
4.2 Analysis of Acoustic Sound Signal The discrete wavelet transform (DWT) of the acoustic sound signal recorded by the hydrophone is depicted in Fig. 4. In the analysis, the signal has been decomposed into one approximation (a5 ) and five details (d1 , d2 , d3 , d4, and d5 ). Here, signal d5 is at the low-frequency band, and d1 is at the high-frequency band. Since the signal has been decomposed at different time resolutions and frequency bands, it is expected to give interfacial insight at different mass flow rates. The approximation a5 is at a higher resolution level and frequency band around 0–200 Hz, corresponding to the bubble detachment and collapse, as shown in Fig. 4. The approximation a5 shows less fluctuation in the signal at a mass flow rate of 10 kg/hr, and fluctuations increase as the mass flow rate increases at the same pipe submergence depth. At higher mass flow rate, the pressure oscillations are more due to the periodic rupture of the bubble. The detail signal d5 shows the same order of magnitude at every mass flow rate in the low-frequency range. It shows the sources of the peak frequency of the acoustic sound are the same as the low-frequency range. The details signal d1 is a high-frequency component, which shows the order of magnitude is higher at a mass flow rate of 10 kg/hr. The similar order of magnitude is found at mass flow rates of 20 and 30 kg/hr. It shows the large magnitude of pressure waves generated due to the periodic bubble collapse at the mass flow rate of 10 kg/hr. The larger magnitude of the pressure wave is due to the violent chugging which is observed at the low mass flow rate, and it gets reduced with an increase in the mass flow rate. DWT of acoustic sound at a steam mass flow rate of 20 kg/hr and 5-cm pipe submergence depth is depicted in Fig. 5. At the high resolution and low frequency, the order of magnitude is the same as the higher submergence depth shown in Fig. 4(b). The detail factor d1 also shows the same behavior, which can be observed at the higher submergence depth at a constant mass flow rate.
308
S. Patel and P. Kumar
(a)
(b)
(c) Fig. 3 Wavelet scalogram of pressure signal for 1 cm submergence depth of pipe and different mass flow rates: a 10 kg/hr, b 20 kg/hr, and c 30 kg/hr
Frequency Analysis of Direct Contact Condensation Using the Wavelet …
309
(a)
(b) Fig. 4 Discrete wavelet transform of acoustic sound at pipe submergence depth 9 cm and different mass rates a 10 kg/hr, b 20 kg/hr, and c 30 kg/hr
310
S. Patel and P. Kumar
(c) Fig. 4 (continued)
Fig. 5 Discrete wavelet transform of acoustic sound at steam mass rates 20 kg/hr and 5-cm pipe submergence depth
Frequency Analysis of Direct Contact Condensation Using the Wavelet …
311
5 Conclusions The experimental study of the vertical steam injection into the pool has been performed to analyze the DCC phenomenon. The significant findings associated with bubble evolution, frequency due to pressure oscillation, and acoustic sound with different steam mass flow rates are summarized as follows: 1. The bubble’s shape continuously changes with time due to the unstable nature of the DCC phenomenon. The stages of bubble evolution are formation, growth, elongation, and rupture. 2. The CWT scalogram shows that the peak frequency of the pressure increased with mass flow rate and shifted toward the left with time. The magnitude of peak frequency decreases with an increase in the mass flow rate. The peak frequencies lie in the range of 0–0.1 Hz. 3. The DWT of acoustic sound obtained from the hydrophone showed a fluctuation in signal increase as the mass flow rate enhanced. It is also found that higher order of magnitude is obtained at a lower mass flow rate due to the violent chugging at a lower mass flow rate at constant pipe submergence depth. 4. The DWT of acoustic sound shows the pipe submergence depth with little impact on the order of magnitude and fluctuation in the signal at a constant mass flow rate. Acknowledgements The authors acknowledge the financial support the SERB, Government of India, provided for research project SRG/2019/000362.
References 1. Guleria, S.D. and Kumar, P.,2021. Experimental analysis of direct contact condensation during vertical injection of steam on subcooled water pool. In Proceedings of the 26thNational and 4th International ISHMT-ASTFE Heat and Mass Transfer Conference December 17–20, 2021, IIT Madras, Chennai-600036, Tamil Nadu, India. Begel House Inc.. 2. Qiu B, Tang S, Yan J, Liu T, Chong D, Wu X (2014) Experimental investigation on pressure oscillations caused by direct contact condensation of sonic steam jet. Exp Thermal Fluid Sci 52:270–277 3. Accident TFD (2015) Non-serial Publications. IAEA, Vienna 4. Arinobu, M., 1980. Studies on the dynamic phenomena caused by steam condensation in water (No. NUREG/CP--0014 (VOL. 1)). 5. Chan CK, Lee CKB (1982) A regime map for direct contact condensation. Int J Multiph Flow 8(1):11–20 6. Wu XZ, Yan JJ, Pan DD, Liu GY, Li WJ (2009) Condensation regime diagram for supersonic/ sonic steam jet in subcooled water. Nucl Eng Des 239(12):3142–3150 7. Gregu G, Takahashi M, Pellegrini M, Mereu R (2017) Experimental study on steam chugging phenomenon in a vertical sparger. Int J Multiph Flow 88:87–98 8. Aya I, Nariai H, Kobayashi M (1980) Pressure and fluid oscillations in vent system due to steam condensation,(I) Experimental results and analysis model for chugging. J Nucl Sci Technol 17(7):499–515
312
S. Patel and P. Kumar
9. Aya I, Kobayashi M, Nariai H (1983) Pressure and fluid oscillations in vent system due to steam condensation,(II) high-frequency component of pressure oscillations in vent tubes under at chugging and condensation oscillation. J Nucl Sci Technol 20(3):213–227 10. Youn DH, Ko KB, Lee YY, Kim MH, Bae YY, Park JK (2003) The direct contact condensation of steam in a pool at low mass flux. J Nucl Sci Technol 40(10):881–885 11. Hong SJ, Park GC, Cho S, Song CH (2012) Condensation dynamics of submerged steam jet in subcooled water. Int J Multiph Flow 39:66–77 12. Qiu B, Yan J, Liu J, Chong D, Zhao Q, Wu X (2014) Experimental investigation on the second dominant frequency of pressure oscillation for sonic steam jet in subcooled water. Exp Thermal Fluid Sci 58:131–138 13. Tang J, Yan C, Sun L, Li Y, Wang K (2015) Effect of liquid subcooling on acoustic characteristics during the condensation process of vapor bubbles in a subcooled pool. Nucl Eng Des 293:492– 502 14. Qiu B, Yan J, Liu J, Chong D (2015) Experimental investigation on pressure oscillation frequency for submerged sonic/supersonic steam jet. Ann Nucl Energy 75:388–394 15. Xu Q, Guo L (2016) Direct contact condensation of steam jet in crossflow of water in a vertical pipe. Experimental investigation on condensation regime diagram and jet penetration length. Int J Heat Mass Transf 94:528–538 16. Gulawani SS, Joshi JB, Shah MS, RamaPrasad CS, Shukla DS (2006) CFD analysis of flow pattern and heat transfer in direct contact steam condensation. Chem Eng Sci 61(16):5204–5220 17. Gulawani SS, Dahikar SK, Mathpati CS, Joshi JB, Shah MS, RamaPrasad CS, Shukla DS (2009) Analysis of flow pattern and heat transfer in direct contact condensation. Chem Eng Sci 64(8):1719–1738 18. Patel G, Tanskanen V, Hujala E, Hyvärinen J (2017) Direct contact condensation modeling in pressure suppression pool system. Nucl Eng Des 321:328–342 19. MathWorks, MATLAB Wavelet Toolbox, R2020a. 20. Ellis N, Briens LA, Grace JR, Bi HT, Lim CJ (2003) Characterization of dynamic behaviour in gas–solid turbulent fluidized bed using chaos and wavelet analyses. Chem Eng J 96(1–3):105– 116 21. Ellis N, Bi HT, Lim CJ, Grace JR (2004) Influence of probe scale and analysis method on measured hydrodynamic properties of gas-fluidized beds. Chem Eng Sci 59(8–9):1841–1851 22. Ren J, Mao Q, Li J, Lin W (2001) Wavelet analysis of dynamic behavior in fluidized beds. Chem Eng Sci 56(3):981–988
A Numerical Analysis of Flat-Fan Spray Injection into Coflow of Air Shirin Patil, Kiran Kumar, Srikrishna Sahu, and Ravindra G. Devi
1 Introduction The need for large amount of intake air in gas turbines for power generation introduces small size contaminants in the air into the engine. This results in their deposition on the compressor blade surfaces despite the use of air filters. The particle adherence leads to the degradation of the blade surface and foulant buildup [1]. Fouling in compressor blades due to dirt deposition is a significant issue as it impedes the compressor’s performance and degrades the overall engine efficiency. The compressor fouling contributes to 70–85% of all performance losses during the gas turbine operation [2]. The online water washing approach is an effective alternative for compressor blade cleaning and optimizes the time span between offline washing, thereby minimizing engine shutdowns. In online water washing systems, a series of flat-fan nozzles are typically used at the engine bell mouth to inject water sprays into the inflowing air. Whilst the previous studies on online water washing systems primarily focused on the liquid injection system, the evolution of spray characteristics has not gained much attention. Since the spray momentum just before impact on the compressor blades eventually determines the foulant removal efficiency, the spray characteristics of the injected liquid are important in achieving effective cleaning. Water droplet size is one of the critical parameters in determining the efficiency of online washing. However, optimizing the injector operating conditions is not straightforward, mainly due to the trade-off between blade cleaning effectiveness and material erosion. Large size droplet impingement enhances the possibility of the blade surface stresses and failure [3], whilst very small droplets follow the air stream and do not clean the blade surface S. Patil (B) · K. Kumar · S. Sahu Department of Mechanical Engineering, IIT Madras, Chennai 600042, India e-mail: [email protected] R. G. Devi Baker Hughes, Bengaluru, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_30
313
314
S. Patil et al.
[4]. Injection of washing fluid across the airstream avoids the formation of spray sheet centrifuging towards the compressor casing [5]. Flat-fan nozzles distribute the small size droplets into turbulent zones of relatively low air speed in front of the engine and are not affected by the compressor centrifugal forces. Therefore, a study of spray characteristics in the presence of a coflow is essential in optimizing the online water washing system. A parametric investigation carried out at varying coflow air velocities and initial liquid temperatures can provide insight into the influence of spray-flow interaction and liquid preheating on droplet characteristics prior to the blade impingement [6]. There are few works which carried out Computational Fluid Dynamics (CFD) studies to characterize spray from flat-fan atomizers [7, 8]. However, the influence of coflow air and preheated liquid jet on the evolution of spray characteristics has not received much attention in the literature. In the present study, our focus is to numerically simulate the flat-fan atomizer, which injects liquid within the wind tunnel in the presence of coflow air. For this purpose, the discrete phase modelling is adopted. The continuous air flow is treated in the Eulerian framework, whilst injected droplets are tracked in the Lagrangian framework. The droplet breakup, collision and coalescence are also modelled. The coflow air velocity within the wind tunnel is varied over a wide range from 0 to 70 m/s. The liquid temperature is varied from 300 to 333 K. The grid independence study is conducted. The validation of the simulation results is carried out with the in-house experimental data. The spray structure is characterized using cone angle and axial variation of different spray properties such as characteristic droplet velocity and temperature are reported for a range of operating conditions.
2 Numerical Methodology 2.1 Computational Domain and Mesh Figure 1 shows the computational domain for the present study. The length and width of the domain are 500 and 150 mm, as indicated. Gravity is considered in the negative X direction. The flat-fan atomizer is located at the centre of the X–Z plane corresponding to Y = 0 mm. The atomizer injects liquid droplets parallel to the X–Y plane. Hexahedron structured mesh is considered throughout the domain with aspect ratio and orthogonality as 1. Meshes with the different number of cells (2.1–3.8 million cells) have been considered to carry out a grid independence study, which will be discussed later in Sect. 2.4. All simulations are carried out using Ansys Fluent [9].
A Numerical Analysis of Flat-Fan Spray Injection into Coflow of Air
315
Fig. 1 Computational domain for the present study. Boundary conditions for the domain are also shown
2.2 Governing Equations The gas phase conservation equations are solved in the Eulerian reference frame in terms of mean flow field quantities, based on Reynolds averaging. Equation 1 shows Reynolds averaging continuity equation. The vapour mass from droplet evaporation is represented in the gas phase continuity equation as the volumetric source term (S m ) given by Eq. 2. ∂(ρu i ) ∂ρ + = Sm ∂t ∂ xi Sm =
1 m˙d V n
(1) (2)
where m˙d represents the evaporation rate of a typical droplet, n is the number of droplets within the cell and V is the volume of the computational cell. The source term S m also appears in the species equation for the liquid vapour. The Reynolds averaged momentum equation is given by Eq. 3. The term −ρu i u j represents the Reynolds stresses arising due to air turbulence. The momentum exchange from the liquid droplet to the air appears as a source term (S f ) given by Eq. 4. ∂u i ∂ ∂(ρu i ) ∂ ρu i u j ∂P ∂ μ + + −ρu i u j + S f =− + ∂t ∂x j ∂x j ∂x j ∂x j ∂x j Sf =
1 (m˙d u i − FD,u i − FP i ) V n
(3) (4)
316
S. Patil et al.
where the first term on the right-hand side represents the momentum exchange due to droplet evaporation and, FD,u i and FP i represent the drag force and pressure gradient force experienced by the droplets in the ith direction, respectively. Equation (5) shows the energy equation where E, k eff and τi j,e f f are the total energy, effective thermal conductivity and deviatoric stress tensor, respectively. S h is the source term denoting the heat exchange to the droplet from the gas phase given by Eq. 6. ∂(ρ E) ∂u i (ρ E + P) ∂ T∞ ∂ ke f f + = + u i (τi j,e f f ) + Sh ∂t ∂ xi ∂x j ∂x j Sh =
1 (m˙d C pd (T∞ − Td ) + m˙d Hlat V n
(5) (6)
where C pd is the droplet heat capacity, H lat is the latent heat of the liquid, T∞ and T d are the gas phase and droplet temperatures, respectively. The species conservation is given by Eq. 7. Here, Y k is the local mass fraction of the k th species and J k is the diffusion mass flux of the k th species given by Eq. 8. ∂ − → (ρYk ) + ∇.(ρu i Yk ) = −∇. J k + Sm,k ∂t μt − → ∇Yk Jk = − ρ Dk,m + Sct
(7) (8)
where Dk,m is the diffusion coefficient for the k th species and Sct and μt are the turbulent Schmidt number and turbulent viscosity, respectively. Since only two species are present, i.e., water vapour and air, the species equation needs to be solved only for the water vapour, as the non-reacting flow situation is applicable to the present study. The primary breakup of the injected liquid into droplets is accounted by selecting the corresponding model in Ansys. Forces acting on a droplet are presented in Eq. 9. → ud ρd π d 3 d − − → − → − → − → − → = F D + F P + F V M + F SL + F T P 6 dt
(9)
− → − → where F D is the drag force, F P is the force acting on the droplet due to pres− → − → sure gradient force, F V M is the virtual mass force, F SL is the Saffman lift force, − → F T P is the Thermophoretic force. The Taylor Analogy Breakup (TAB) model is utilized to account for secondary droplet breakup. The droplet collision is also suitably modelled. The two-way coupling between the dispersed and the continuous phases has been considered. The details on the modelling approach can be found in [10, 11].
A Numerical Analysis of Flat-Fan Spray Injection into Coflow of Air
317
Table 1 Operating conditions and other simulation parameters required to model primary atomization of the liquid sheet Operating conditions Liquid temperature, T liq (K)
300 and 333
Liquid flow rate (kg/s)
0.0166
Coflow air velocity, V co (m/s)
0, 30, 50 and 70
Simulation parameters Injection type
Flat-fan atomizer
Particle type
Droplet (Water)
Spray half cone angle (α)
37.5o
Orifice width (mm)
1.2
2.3 Operating and Boundary Conditions The inlet and outlet of the simulation domain correspond to the X–Z plane at Y = 0 and Y = 500 mm, respectively (see Fig. 1). Remaining faces of the domain (i.e., in X– Y and Y–Z planes) are considered as solid walls on which no-slip boundary condition is prescribed. Table 1 lists the operating conditions for the present study such as mass flow rate through the atomizer, inlet air coflow velocity (V co ) and atomizing liquid temperature (T liq ). Additionally, other simulation parameters required to model the primary atomization of the liquid sheet are also listed in Table 1.
2.4 Grid Independence Study Table 2 shows that the arithmetic mean diameter (AMD) and mean droplet velocity (V d ) are not sensitive even if the number of cells in a mesh is increased significantly from 2.1 to 3.8 million. Hence, the mesh with 2.1 million cells is considered for the rest of the simulations. Though Table 2 corresponds to operating condition, V co = 30 m/s and T liq = 300 K, it is noted that similar results are observed for other cases. Table 2 Grid independence study at Y = 450 mm for the case V co = 30 m/s and T liq = 300 K Number of cells in million
AMD (µm)
Mean droplet velocity, V d (m/s)
2.1
88
25.3
2.8
89
25.1
3.8
88
25.2
318
S. Patil et al.
2.5 Validation The validation of the present simulation results has been done against the in-house experiments by Kumar et al. [6] who measured spray characteristics within a wind tunnel. The computational domain shown in Fig. 1 mimics the test section of the wind tunnel in the experiments. Kumar et al. [6] reported measurement of AMD and V d at axial distance, Y = 250 mm for different lateral distances (X) from X = 37 mm to X = 112 mm. Accordingly, the numerically calculated average droplet size and velocity are compared to the experimental measurements at Y = 250 mm. Table 3 shows the comparison between simulation and experimental results for T liq = 300 K and V co is 0, 30, 50 and 70 m/s. It can be observed that decent validation is obtained for all cases. Though not shown here, good validation was also obtained for the cases with T liq = 333 K.
3 Results and Discussion In this section, we discuss the different simulation results such as spray structure, axial evolution of spray parameters and influence of preheated liquid on the atomization process for flat-fan atomizers. Table 3 Validation for the present simulation results for different values of V co at T liq = 300 K
Parameter
Simulation
Experimental
AMD (µm)
131
144
V d (m/s)
8.6
6.9
AMD (µm)
131
144
V d (m/s)
23.6
24.3
AMD (µm)
126
136
V d (m/s)
31
31.3
AMD (µm)
113
138
V d (m/s)
44.7
37.3
V co = 0 m/s
V co = 30 m/s
V co = 50 m/s
V co = 70 m/s
The results are validated with Kumar et al. [6]
A Numerical Analysis of Flat-Fan Spray Injection into Coflow of Air
319
3.1 Spray Structure Figure 2 shows the spray contour plots for droplet size considering different coflow velocities (V co = 0–70 m/s) for T liq = 300 K. It can be observed that droplet size varies in the range (≈5–500 µm), which is in agreement with the experimental results from Kumar et al. [6]. At higher coflow velocity (i.e., V co = 70 m/s, Fig. 2d), considerable droplet breakup is observed as compared to other cases, where V co = 0–50 m/s (Fig. 2a–c). It can be noted that the maximum droplet size decreases when V co is increased from 0 to 70 m/s. However, the maximum droplet size decreases by 10 and 20% when V co is increased from 30–50 to 50–70 m/s, respectively. This highlights that the atomization of the droplets is influenced due to the presence of coflow air, especially at higher coflow velocities. Figure 2 also clearly demonstrates that the spray structure becomes narrower as V co increases from 0 to 70 m/s. This is quantified by measuring the half cone angle (α) of the spray and its variation with V co is shown in Fig. 3. It is observed that for V co = 0 m/s, α is 37.5°, which is also the input value of α in simulation (see Table 1). As expected, α continues to decrease with an increase in V co due to the aerodynamic forces exerted by the coflow air on the spray droplets. However, α does not decrease linearly with V co , instead α decrease by 15 and 35%. When V co is increased from 30–50 to 50–70 m/s, respectively. This observation is similar to the arguments noted earlier for droplet size in Fig. 2. It is interesting to note the positive correlation between maximum droplet size and cone angle for V co = 0–70 m/s.
3.2 Axial Evolution of Spray Parameters In this section, we discuss the axial variation of AMD and mean droplet velocity V d , which are vital spray parameters. It is noted here that in order to evaluate average spray parameters at each axial location, only those droplets are considered that fall within the window of specified size. For instance at an axial location, Y = 50 mm, the droplets are considered within the window from Y = 40–60 mm (i.e., ΔY = 20 mm) and X = 10–140 mm (i.e., ΔX = 130 mm, refer Fig. 1 for co-ordinate axis). For other axial locations as well, the same window size (i.e., ΔX and ΔY ) is maintained. Figure 4 shows the variation of AMD with Y considering different coflow air velocities. It is observed that overall AMD decreases as the spray evolves axially downstream. However, close to the atomizer exit, AMD is observed to increase initially. This is because the spray interacts with the wall at Y ≈ 100–170 mm for V co = 30–70 m/s. This means that in the region Y = 50–100, the spray is still expanding in the cross-stream (X) direction and, hence, droplets are closely packed. This provides a favourable condition for droplet–droplet coalescence which tends to increase AMD in the region Y = 50–100 mm, specifically for V co = 50–70 m/s. For V co = 0–30 m/ s, the spray has already expanded enough in the region Y > 50 mm due to which coalescence probability is lower and accordingly AMD doesn’t increase.
320
S. Patil et al.
(a) Vco = 0 m/s
(b) Vco = 30 m/s
(c) Vco = 50 m/s
(d) Vco = 70 m/s
Fig. 2 Spray contour plots for droplet size for different air coflow velocities when T liq = 300 K for a V co = 0 m/s, b V co = 30 m/s, c V co = 50 m/s, d V co = 70 m/s
Further, it is observed that at each axial location, AMD is comparable for different coflow velocities (i.e., V co = 30–70 m/s). This is in contrast to the observation reported in the context of Fig. 2, where maximum droplet size was observed to decrease with V co . Also, considerable daughter droplets all over the simulation domain were observed in Fig. 2d for V co = 70 m/s, which could have led to the decrease in AMD as compared to V co = 30–50 m/s. However, it is worth noticing that coalescence is also occurring simultaneously as the spray evolves axially downstream and though maximum droplet size decreases with V co , AMD remains similar over the range of V co = 30–70 m/s. It is noted that for V co = 0 m/s, AMD decreases
A Numerical Analysis of Flat-Fan Spray Injection into Coflow of Air
321
Fig. 3 Variation of half cone angle (α) with V co
Fig. 4 Variation of AMD with axial distance (Y ) for different values of V co
till Y = 200 mm and afterwards, it continues to increase axially downstream. This clearly highlights that the presence of surrounding coflow air significantly influences the spray evolution. Figure 5 shows the variation of mean droplet velocity (V d ) with axial distance downstream. For V co = 0 m/s, the droplets decelerate axially downstream till Y = 250 mm due to the absence of any external assistance from surrounding airflow. Beyond Y = 250 mm, V d remains almost constant. Now considering the trend of droplet velocity for V co = 30–70 m/s, it can be observed that V d gradually increases with axial distance downstream and at each axial location, V d is observed to be proportionally higher with V co . Moreover, the slope of the curve also increases as V co increases from 30 to 70 m/s which suggests that droplet acceleration increases for higher coflow air velocity. Additionally, it is noticed that the V d considerably lags behind the corresponding inlet coflow air velocity for all cases. This means that, there are some larger droplets in the flow and because of their inertia, they are still not able to catch up with the surrounding air flow velocity. This is clearly highlighted in Fig. 6, which shows scatter plot between droplet size and velocity for
322
S. Patil et al.
Fig. 5 Variation of mean droplet velocity, V d with axial distance (Y ) for different values of V co
Fig. 6 Scatter plot between droplet size and velocity for V co = 70 m/s
V co = 70 m/s, where droplet velocity decreased with the increase in droplet size. Similar plots were observed for other cases but not shown here. For obtaining Fig. 6, the droplets in region Y = 400–500 mm are considered such that droplets got sufficient time to interact with the airflow. It is observed that droplet velocities vary in the range from 35 to 65 m/s and smaller droplets (≈50–100 µm) can reach velocities 60–65 m/s whilst larger droplets (≈350–450 µm) have velocities 35–45 m/s. This suggests that the considered V co is not enough to make all the droplets faithfully follow the airflow.
3.3 Effect of Liquid Preheating Figures 7 and 8 show the variation of AMD and V d with axial distance respectively for T liq = 300 and 333 K, when V co = 30 m/s. It is observed that AMD and V d showed a negligible difference either for atomizing liquid at atmospheric temperature or
A Numerical Analysis of Flat-Fan Spray Injection into Coflow of Air
323
with the preheated condition. This again highlights that eventually, spray parameters such as AMD and V d are strongly governed by the surrounding coflow air velocities rather than any temperature of atomizing liquid. Further, it can be mentioned that droplet evaporation is not influenced significantly with preheated liquid at 333 K as compared to the case of T liq = 300 K as the AMD and V d remains almost similar for both cases (Figs. 7 and 8). Figure 9 shows the variation of droplet temperature (T drop ) with axial distance when T liq = 333 K and V co = 30–70 m/s. It can be observed that droplet temperature decreases with Y for all cases. This is expected that as the surrounding coflow air is maintained at 300 K and as the preheated droplets are convected downstream of the domain, the droplet loses heat to the surrounding airflow. However, it is interesting to note that though droplet temperature is comparable for V co = 30 and 50 m/s, droplet temperature falls significantly for V co = 70 m/s. This is similar to the observation in Fig. 2 (for maximum droplet size, breakup morphology) and Fig. 3 (for cone angle) as T drop shows a non-linear trend with an increase in V co . Fig. 7 Variation of AMD with axial distance (Y ) for different values of T liq with V co = 30 m/s
Fig. 8 Variation of V d with axial distance (Y ) for different values of T liq with V co = 30 m/s
324
S. Patil et al.
Fig. 9 Variation of T drop with axial distance (Y ) for V co = 30, 50 and 70 m/s
4 Conclusions The present paper reports a numerical study of spray injection from a flat-fan atomizer into surrounding coflow air, which is relevant to online water washing systems in gas turbines. The Eulerian and Lagrangian reference frames are adopted to simulate coflow air and transport of spray droplets, respectively. The discrete phase model is implemented where droplet collision, coalescence and droplet breakup are also accounted. The effect of surrounding coflow air velocity (V co = 0–70 m/s) and preheated liquid (T liq = 333 K) on the spray structure and axial evolution of different spray parameters (AMD, V d ) are studied in detail. Due to the aerodynamic forces exerted by the coflow air, the spray tends to become narrower and the spray cone angle decreases with the increase in coflow air velocity. Though significant daughter droplets were observed for V co = 70 m/s, AMD was found to be almost similar over the range of V co = 30–70 m/s for different axial locations. This suggests that along with droplet breakup, droplets are also undergoing coalescence within the simulation domain for V co = 70 m/s due to which AMD remains similar to cases V co = 30– 50 m/s. On the other hand, mean droplet velocity was strongly governed by coflow air velocity. But due to the presence of larger droplets and owing to their inertia, the mean droplet velocity considerably lagged behind the inlet coflow air velocity. There is the negligible influence of liquid preheating either on AMD or V d . Further, the droplet temperature decreased with an increase in V co , especially when V co is varied from 50 to 70 m/s.
References 1. Diakunchak IS (1992) Performance deterioration in industrial gas turbines. ASME J Eng Gas Turbines Power 114:161–168 2. Scheper GW, Mayoral AJ, Hipp EJ (1978) Maintaining gas turbine compressors for high efficiency. Power Eng 82(8):54–57
A Numerical Analysis of Flat-Fan Spray Injection into Coflow of Air
325
3. Hornak SS, Lowdermilk RS, Miller RA (1980) Removable wash spray apparatus for gas turbine engine. U.S. Patent 4,196,020 4. Mustafa Z (2006) Analysis of droplets in compressor gas turbines, PhD Thesis: Cranfield University, UK 5. McDermott P (1991) Method and apparatus for cleaning a gas turbine engine. U.S. Patent No. 5,011,540 6. Kumar K, Chaudhari V, Sahu S, Devi RG (2021) Investigation on flat-fan spray characterization in high-speed air Coflow for gas turbine online water washing application. In: Proceedings of the ASME 2021 gas turbine India conference 7. Kashani A, Parizi H, Mertins KH (2018) Multi-step spray modelling of a flat fan atomizer. Comput Electron Agric 144:58–70 8. Fesal SNM, Fawzi M, Omar Z (2017) A numerical analysis of flat fan aerial crop spray. In: IOP conference series: materials science and engineering, vol 243, no 1, p 012044, IOP Publishing 9. ANSYS FLUENT Theory Guide, 2013. Release 18.2, 15317 (November) 10. Muddapur A, Jose JV, Sahu S, Sundararajan T (2020) Spray dynamics and evaporation in a multihole GDI injector for pulsatile and split injection at elevated ambient pressure and temperature conditions. Fuel 279:118057 11. Muddapur A, Srikrishna S, Sundararajan T (2020) Spray dynamics simulations for pulsatile injection at different ambient pressure and temperature conditions. Proceed Inst Mechan Eng Part A J Power Energy 234(4):500–519
Capillary Rise in the Interstices of Tubes Chitransh Atre, Aditya Manoj, and Baburaj A. Puthenveettil
Nomenclature g σ μ ρ R Θ h t tc L(G) A(G) Hc D C P Q
[ ] Gravity ms−2 [ ] Surface tension Nm−1 Dynamic viscosity[ [Pa − s] ] Density of liquid Kgm−3 Radius of the capillary [m] Static contact angle [°] Height of the meniscus [m] Time [sec] Time scale [sec] Length of one curve y [m] [ ] Cross-sectional area of the interstice m2 Length scale along z-direction [m] Diameter of the capillary [m] Constant [[−] ] Pressure Nm−2 [ ] Volume flux along z-direction m3 s−1
C. Atre (B) · B. A. Puthenveettil Department of Applied Mechanics, IIT Madras, Chennai 600036, India e-mail: [email protected] A. Manoj Department of Mechanical Engineering, IIITDM Kancheepuram, Chennai 600127, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_31
327
328
C. Atre et al.
1 Introduction Capillary action plays an important role in many applications such as flow through porous media, textile industries, ink printing, microfluidics etc. as well as in natural process like liquid flow in soils, trees and leaves. In such flows, the main driving force for a liquid to move is provided by the surface tension. Capillary rise in a circular capillary has the bulk fluid and the meniscus rise at the same rate but in a closed geometry which include corners, the bulk fluid and the corner fluid rise at different rates. Corner rise experiments in a linear profile created by two vertical plates were done by Higuera et al. [1]. They found that at initial stages of rise, the gravity effects are negligible and at the later stages, the viscous effects will dominate. Using lubrication approximation, they showed that the corner meniscus height to be proportional to the cube root of time. Dong et al. [2] did experiments to study the capillary rise in a capillary with a square cross-section and validated with their theory. They found that the imbibition rates are in proportion to the (σ/μ)1/2 and the velocity of imbibition is proportional to D 1/2 , where D is the tube size. Previously, Ponomarenko et al. [3] performed experiments in a quadratic corner and developed a law which mentions that the height of the corner meniscus varies as t 1/3 and to be independent of the radius of the corner, even though they did not verify this independence. Recently, Zhou and Doi [4] developed a theory for small corner rise between curved surface profiles. Their outcomes support the results given by [3]. The most commonly occurring geometry having circular regions and corners is the interstices in between an array of tubes or rods. The dependence of corner rise in such a geometry on time and, especially on the radius of the tubes or rods is not known. In the present study, we perform experiments to measure the rise of liquid in the corners in interstices.
2 Experimental Setup We use three different sizes of circular capillaries having outer diameters of 700, 1380 and 10, 000μm. Figure 1 shows a schematic diagram of the experimental setup. For each of the above diameters, we used three borosilicate circular glass tubes each of 10 cm in length held together by heat shrink sleeves to create a three cornered interstice shown in the Fig. 2a. We used water mixed with blue ink to visualize the rise. The properties for water are given in the Table 1. The capillary bundle was fixed vertically and a container having DI water mixed with ink was slowly brought up into contact with the bottom part of the capillary bundle with the help of traverse. As the bottom surface of the capillary touches the water surface, water rises in the interstice. The bulk fluid as well as corner fluid rises with high rates initially with the rate of rise decreasing. When the bulk meniscus reaches equilibrium height, the corner meniscus still continues to rise due to the fact that a wetting liquid tries to cover as much area
Capillary Rise in the Interstices of Tubes
329
Fig. 1 a Schematic representation of experimental setup. b Experimental image of water rise inside interstice of capillaries of OD = 1380µm
of solid surface to attain minimum interfacial energy. We record the videos of this rise using NIKON D5300 camera with 50fps with TAMRON Macro lens of focal range 180–400 mm. A LED light source was used for the back lighting. The capillary height in the corners of each capillary bundle was measured as a function of time from those videos. The Fig. 1b shows an image frame of rise inside an interstice created by three capillaries of 1380μm OD. The bulk meniscus and the corner meniscus tip can be seen in zoomed image.
3 Methodology The inset in the Fig. 3 shows the variations of the liquid level in the corners for each of the capillary bundle with time. There are two types of regimes that we observe. An initial regime where h ∼ t 4/5 and a later regime where h ∼ t 1/3 . We now focus on the latter regime and find our scaling relation that collapses the data in this latter regime. In Fig. 2a, the shaded region represents the interstice which is created by joining the three circular capillaries. This interstice contains 3 sharp corners. Considering one
330
C. Atre et al.
Fig. 2 a Schematic view of the interstice formed by three circular capillaries. b Horizontal crosssectional view of meniscus in the interstice c 3-D view of meniscus inside the interstice
Table 1 The values of the parameters 0.071
μ(Pa − s)
) ( ρ K g/m 3
θ (deg)
8.4 × 10−4
997
45
corner, let the two surfaces that form the corner be defined by the function y = ±E(x)/2 where, since the tube cross-section is circular,
(1)
Capillary Rise in the Interstices of Tubes
331
Fig. 3 Variation of the dimensionless corner height with the dimensionless time for three different capillaries
) ( √ E(x) = 2 R − R 2 − x 2
(2)
Let the bottom of the meniscus be at z = 0 and the tip of the meniscus at z = Z m . Figure 2b represents a cross-sectional view of interstice. Figure 2c shows 3-D view with the Cartesian coordinates. Our main objective is to find the corner meniscus profile equation with respect to time. Using Onsager’s principle [4], we develop a Rayleighian function relation defined by ( ) ( ) ( ) R G˙ = F˙ G˙ + Φ G˙
(3)
˙ where G(z, t) = ∂G(z, t)/∂t, with G(z, t) being the distance ( )from the edge of corner to meniscus surface at time instant t (see Fig. 2b) and F˙ G˙ represents the rate of ˙ change of free energy when the interface ( is) moving at the rate G(z, t). . Hereinafter, (dot) indicates derivative w.r.t. time. Φ G˙ represents half of the rate of dissipation ( ) ˙ t) is to be finally obtained by minimizing the function R G˙ . Area of energy. G(z, the interstice region, A(G), in the limits 0 < x < G and |y| < E(x)/2 ʃG A(G) =
E(x)dx.
(4)
0
Using Eq. (2) in (4), √ A(G) = 2RG − G R 2 − G 2 + R 2 sin−1 G/R. Let A' (G) = ∂ A/∂G which from (6), is,
(5)
332
C. Atre et al.
) ( √ A' (G) = 2 R − R 2 − G 2 .
(6)
Hereinafter, ' indicates derivative w.r.t. G. Now using mass conservation equation, ∂ A(G)/∂t = A' G˙ = −∂ Q/∂z
(7)
where Q(z, t) is the volume flux of fluid flowing in the z-direction. To evaluate F˙ in (3), we find the free energy of the system as the sum of gravitational energy and interfacial energy. ʃzm {ρg A(G) · z − 2L(G)σ cos θ }dz
F(G) =
(8)
0
where σ is the surface tension, θ is the contact angle and L(G) is the length of the curve for 0 < x < G, as shown in the Fig. 2b ʃG L(G) =
/ 1+
( ) 1 dE(x) 2 dx 4 dx
(9)
0
After substituting (2) in (9), L(G) = Rsin−1 (G/R)
(10)
)−1/2 ( G2 L (G) = 1 − 2 R
(11)
and therefore, '
From (8), ˙ F(G) =
ʃzm
( ) ˙ ρg A' z − 2L ' σ cosθ Gdz.
(12)
0
Integrating (12) by parts and using Eq. (7), we get, F˙ =
) ) ( ʃZ m ( ∂ L ' /A' σ cosθ Qdz. ρg − 2 ∂z
(13)
0
We now obtain the expression for the dissipation function φ in (3) using lubrication approximation. The velocity in z-direction vz is determined by Stoke’s equation.
Capillary Rise in the Interstices of Tubes
333
) ( 2 ∂ ∂2 ∂P − ρg μ + 2 vz = − ∂x2 ∂y ∂z
(14)
with the boundary conditions are vz = 0 at boundary y = ±E(x)/2. .. Solving (14) analytically, we get a parabolic profile for the flow as, [ ) ] ( 2y 2 3 vz (x, y, z, t) = v z (x, z, t) 1 − , 2 E(x)
(15)
where v z (x, z, t) is the y-averaged velocity. The flux is calculated by integrating the velocity over the area as, ʃG(z) Q(z) =
E(x)v z (x, z, t)dx.
(16)
0
In channel flow, v z (x, z, t) = C(z, t)E 2 (x)
(17)
where C(z, t) is proportionality constant, then ʃG(z) Q(z) =
E 3 (x)C(z, t)dx
(18)
0
In other words, Q(z) = B(G)C(z, t)
(19)
where, ʃG B(G) =
E 3 (x)dx.
(20)
0
Evaluating B(G) using (2) and (20) B(G) = −15R 4 arcsin(G/R) + 32R 3 G √ √ − 17R 2 G R 2 − G 2 + 2G 3 R 2 − G 2 − 8RG 3 . Expanding (21) as a Taylor series for small G/R, we get,
(21)
334
C. Atre et al.
B(G) ≃
1 G7 7 R3
(22)
The dissipation function Φ is 1 Φ= 2
ʃZ m ʃG(z) 0
12μ v z (x, z, t)dzdx, E(x)
(23)
0
which using (17) and (19), simplify to
Φ=
1 2
ʃZ m
12μ 2 Q (z)dz B
(24)
0
Substituting (13) and (23) in (3), the Rayleighian becomes ) ) ( ʃZ m ( ʃZ m ∂ L ' /A' 1 12μ 2 σ cos θ Qdz + Q (z)dz R= ρg − 2 ∂z 2 B 0
(25)
0
Using Onsager’s principle, δ R/δ Q = 0δ R/δ Q = 0 ( )) ( ∂ L ' / A' B Q= −ρg + 2σ cosθ 12μ ∂z
(26)
Using (7) and (26), ( ( ) )) ( ∂ L ' /A' B 1 ∂ ˙ G= ' ρg − 2σ cosθ . A ∂z 12μ ∂z
(27)
Inserting the values of A' , B and L ' in (27) from (6), (22) and (11) and taking the binomial approximation, we get the final time evolution equation for the dimensionless meniscus thickness as, [( ) ( ) ( ) )] ( −ρg R 2 1 G 7 ∂(G/R) ∂ 2σ cos θ ∂ G 2 = 1− (28) ∂t 84μ (G/R)2 ∂ z R Rρg ∂z R We choose R as the characteristic length scale in x direction and Hc =
2σ cosθ Rρg
(29)
Capillary Rise in the Interstices of Tubes
335
as the characteristic length scale in the z-direction. The characteristic time is chosen as tc =
84μHc ρg R 2
(30)
Rewriting (28) in dimensionless form ( ( )2 ) 2 ˜ ˜ ˜ ∂ G ∂ ∂ G˜ ∂ G G − 8G˜ − 2G˜ 2 = −7G˜ 4 ∂ z˜ ∂ z˜ ∂ 2 z˜ ∂ t˜
(31)
where, G G˜ = R z z˜ = Hc t t˜ = tc For self-similar solution of (31), we assume ( ) G˜ z˜ , t˜ = F(χ )t˜α
(32)
χ = z˜ t˜β
(33)
where,
Here, α and β are some parameters. Substituting (32) and (33) in (31), we obtain, (
) ( ) ' F ' χβ + Fα t˜(α−1) = −7F 4 F ' t˜(5α+β) − 8F F 2 + F F '' t˜(3α+2β)
(34)
The above equation becomes time independent when, α − 1 = 5α + β = 3α + 2β
(35)
i.e. when α=
−1 −1 &β = 6 3
(36)
At the tip of the corner meniscus, z˜ = Z˜ m and χ = χo . . Using (36) in (33), we then obtain Z˜ m = χo t˜(1/3)
(37)
336
C. Atre et al.
as the scaling for the tip of the corner meniscus. The scaling law implied by (37) is ( )1/3 t Zm = χo Hc tc
(38)
where Hc and tc are given by (29) and (30), respectively. Substituting (29) and (30) in (38) implies that ( Zm = C
t (σ cosθ )2 ρμg
)1/3 (39)
( ) where C = χo 2/(168)1/3
4 Results and Discussion The Fig. 3 shows plot between the dimensionless corner height rise Z˜ m verses dimensionless time t˜. At the later time periods, where gravity and viscous effects become dominant balanced by surface tension force, the measured Z˜ m for R = 690μm and R = 5000μm collapses on to (39). At the initial time stages, the experimental plot shows a steeper slope, the rate of capillary rise at the corner is then faster in this initial regime compared to the later regime. The gravitational effects become negligible here. The experimental results for the smallest two capillaries overlap each other in this initial time period. However, for the larger capillary, there is a shift in this initial regime. Scaling results for this initial regime are yet unknown.
5 Conclusions In the present work, we developed a scaling law for the capillary rise of the liquid in the corner of an interstice created by an array of circular capillary tubes/rods. Using the minimization of sum of free energy and viscous dissipation using Onsager’s principle, )1/3 ( cosθ)2 , independent of the capillary height was shown to scale as Z m = C t (σρμg the radius of the capillary tube/rod. The measurements of capillary heights in the interstitial corners of capillary bundles of 3 sizes showed an initial regime with a higher rate of rise, and a later regime with a lower rate of rise. The proposed scaling law matched the behavior of the corner rise in the later regime. Acknowledgements We gratefully acknowledge the financial support of DST, Government of India through their grant IMP/2018/001167. We acknowledge the support of Usha P Verma Sc ’G’, Advanced Systems Laboratory (ASL), DRDO, and Astra Microwave Products Ltd.
Capillary Rise in the Interstices of Tubes
337
References 1. Higuera FJ, Medina A, Linan A (2008) Capillary rise of a liquid between two vertical plates making small angle. Phys Fluids 20:102–109 2. Dong M, Chatzis I (1995) The imbibition and flow of a wetting liquid along the corners of a square capillary tube. J Colloid Interface Sci 172:278–288 3. Ponomarenko A, Quere D, Clanet C (2011) A universal law for capillary rise in corners. J Fluid Mech 666:146–154 4. Zhou J, Doi M (2020) Universality of capillary rising in corners. J Fluid Mech 900:A29-43
A Study of Flow Patterns Near Moving Contact Lines Over Hydrophobic Surfaces Charul Gupta, Anvesh Sangadi, Lakshmana Dora Chandrala, and Harish N. Dixit
Nomenclature r θ θw θmo φ U ρ μ σ lc
Radial coordinate (m) Angular coordinate (radians) Microscopic contact angle (radians) Macroscopic contact angle (radians) Contact angle (radians) Plate speed (m/s) Density of fluid (kg/m −3 ) Viscosity Surface tension / σ Capillary length ρg
Re
Reynolds number
Ca λ
Capillary number Viscosity ratio μμ BA
ρUlc μ μU σ
1 Introduction A contact line forms at the junction between a solid surface and an interface of two immiscible fluids. Moving contact lines and associated flow patterns in fluid phases are fundamental to several flow configurations, including droplets evaporating and spreading on solid surfaces, inkjet printing, sliding drops on lotus leaves, etc. The C. Gupta · A. Sangadi · L. D. Chandrala · H. N. Dixit (B) Department of Mechanical and Aerospace Engineering, IIT Hyderabad, Hyderabad, Telangana, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_32
339
340
C. Gupta et al.
Fig. 1 The possible flow patterns near the moving contact line: (i) A rolling motion in phase B and a split-streamline motion in phase A, (ii) A split-streamline motion in phase B and a rolling motion in phase A, (iii) The rolling motion in both phases A and B
motion of the contact line over a solid surface generates different flow patterns in both fluid phases, as shown in Fig. 1. For example, by considering a downward motion of the solid plate into a liquid pool of two immiscible fluids, one can observe three kinematically consistent flow patterns near the moving contact line: (i) A rolling motion in phase B and a split-streamline motion in phase A, where the interface moves toward the contact line, (ii) A split-streamline motion in phase B and a rolling motion in phase A, where the interface moves away from the contact line, and (iii) the rolling motion in both phases A and B with no interface movement. Moving contact lines have been studied analytically by several researchers. Among them, Huh & Scriven [1] were the first to mathematically analyze the moving contact line dynamics in the viscous limit considering a flat interface approximation. According to their theory, dynamic contact angle (θd ) and viscosity ratio (λ) are key parameters that characterize the flow patterns. A contact angle is defined as an angle between the interface and the solid surface and measured always( in phase ) B. μA Viscosity ratio λ is a ratio of the viscosity of phase A to that of phase B λ = μ B . A critical viscosity ratio curve (Fig. 2) separates a rolling from a split-stream line flows on θd and λ plane, with the rolling motion occurring primarily in a more viscous phase. The theory argues that the local solution with a no-slip boundary condition becomes singular as one approaches the contact line. Only a few experimental studies deal with moving contact lines despite their fundamental importance [3–5]. Dussan et al. [5] performed experiments with a high viscous drop sliding on an inclined plate. With the help of dye visualizations, they observed a rolling motion at the advancing front of the drop. Using Particle Image Velocimetry (PIV), Chen et al. [3] quantitatively showed a rolling motion in the liquid phase when a solid surface was immersed in a high viscous liquid (PDMS with a viscosity of 60,000 cSt). However, the above experimental studies were at λ 1000
(3)
As mentioned in the previous section, the rigid sphere model is used to evaluate the falling velocity of sodium droplets in the spray fire analysis codes [4–7]. Fig. 2 Force balance on the rigid sphere falling in the gas medium
380
S. Muthu Saravanan et al.
3.2 Model for Liquid Droplet Deformation One of the significant parameters governing the motion of droplets is the drag force arising from the interaction between the droplet surface and the fluid flowing around the droplet. In Eq. (1), the drag coefficient is expressed as a function of Reynolds number, i.e., C d = f (Red ), that accounts for the fluid flow effects and it is applicable for spherical geometry. However, the liquid droplets of bigger size could experience considerable deformation due to aerodynamic forces acting on the droplet surface (depicted in Fig. 3). In the extreme case, the droplet shape will approach that of a disk, for which the drag coefficient is significantly higher than that of a sphere. The droplet deformation can be expressed by the Weber number which is defined as the ratio of the aerodynamic force to the stabilizing surface tension force. The value of drag coefficient also depends on the particle shape, and hence for the liquid droplets, C d = f (Red , We). In order to include the droplet distortion effect in the estimation falling droplet velocity, the following droplet deformation model has been used, which takes account of the aerodynamic force causing the droplet deformation, the restoring surface tension force and the damping viscous force. d2 y C2 ρ u 2 C3 σ C4 μd dy = − y− dt 2 C1 ρd r 2 ρd r 3 ρd r 2 dt
(4)
In the above equation, ‘y’ is the droplet distortion parameter and it is used to obtain variation of drag coefficient due to droplet deformation based on the following correlation. Cd,de f = Cd (1 + 2.632 y)
(5)
The above equation considers linearly varying drag coefficient between the spherical and disk shapes. In the limit of no distortion (i.e., y = 0), Eq. (5) gives the drag Fig. 3 Distortion of spherical liquid droplet as it falls through the gas medium
Analysis of Deformation Effects on Falling Spray Droplet Motion Under …
381
coefficient of a sphere, while at maximum distortion (i.e., y = 1), the drag coefficient corresponding to a disk will be obtained.
4 Results and Discussion The settling velocity of liquid droplets falling in gas medium has been evaluated using the above models. The governing equations of droplet motion (Eq. 2) and droplet deformation (Eq. 4) are solved in a coupled way along with the other constitutive relations (Eqs. 3, 5) using 4th order Runge–Kutta method [15]. The initial conditions for the analysis are u = 0 and y = 0. The values of parameters used in the evaluation are given in Table 1.
4.1 Evaluation of Settling Velocity of Falling Water Droplets Table 2 gives the comparison of settling velocity of water droplets estimated using the model developed and the experimental results available in the literature [13]. The model predictions are in good agreement with the experimental values for various sizes of droplets considered. Table 1 Properties of material used for the analysis
Properties Density
Table 2 Settling velocity of water droplets in air
(kg/m3 )
Water
Sodium
Air
1000
834.65
1.16
Viscosity (10–4 Pa-s)
8.9
2.36
1.81
Surface tension (N/m)
0.0728
0.1587
–-
Droplet size (mm)
Settling velocity (m/s) Experiment
Model predictions
5.76
9.17
9.0
4.24
8.93
8.9
3.12
8.19
8.05
2.67
7.65
7.61
2.29
7.08
7.0
1.97
6.42
6.59
1.45
5.24
5.25
1.06
4.24
4.37
0.58
2.36
2.36
0.27
1.03
1.03
382
S. Muthu Saravanan et al.
Further, the settling velocity of water droplets has been estimated based on the spherical droplet assumption (i.e., solving Eq. 2 alone) and with the droplet deformation effects (i.e., Eqs. 2, 4) and Fig. 4 shows the comparison of these results along with the experimental data. It shows that the settling velocity of larger droplets (Dd > ~1.5 mm) is considerably less than the values predicted based on the spherical droplet model. This is because the larger size droplets deviate from the spherical shape significantly as they experience considerable deformation. However, smaller size droplets (i.e., Dd < 1.5 mm) experience negligible deformation and, hence, their settling velocity could be estimated appropriately based on the spherical droplet assumption. The deformation of 5.76 mm water droplet from the initial spherical shape is depicted in Fig. 5. Fig. 4 Settling velocity of water droplets
Fig. 5 Deformation of 5.76 mm water droplet
Analysis of Deformation Effects on Falling Spray Droplet Motion Under …
383
Fig. 6 Variation of falling velocity of water droplet
The variation of free falling droplet velocity (Dd = 5.76 mm) along the 20 m height is shown in Fig. 6. The figure shows comparison of experimental data [14] with the predictions of rigid sphere model and along with the droplet deformation effects. The predictions of the developed model show very good agreement with the experimental data. The result clearly shows that the droplet deformation has effect on the droplet motion for most of the falling length. Such a large falling distance and bigger droplet sizes can be envisaged in accidental sodium spray fire scenarios.
4.2 Evaluation of Settling Velocity of Sodium Droplets The validated model has been used to evaluate the settling velocity of falling sodium droplets of different sizes. The settling velocity of liquid sodium droplets has been estimated with the deformation effects and also based on the spherical droplet assumption (Fig. 7) and these results are given in Table 3. As stated earlier, the droplet deformation effect is more pronounced on the settling velocity of larger sodium droplets and it is indicated by the droplet distortion parameter. The deformation of 9 mm sodium droplet is shown in Fig. 8. For smaller droplets, the spherical droplet assumption is valid as the value of ‘y’ is close to zero. The above results brought out the effect of droplet deformation on the velocity of falling sodium droplets and developed model is incorporated into the sodium fire analysis code for the realistic simulation of postulated sodium spray fire events in the safety evaluation of SFR.
384
S. Muthu Saravanan et al.
Fig. 7 Settling velocity of sodium droplets
Table 3 Settling velocity of sodium droplets Droplet diameter (mm)
Spherical droplets
With deformation effect
Cd
y
C d,def
u (m/s)
u (m/s)
0.50
1.47
1.8
0.0005
1.48
1.8
1.00
0.79
3.4
0.0036
0.80
3.4
2.00
0.48
6.2
0.0221
0.52
6.0
3.00
0.44
8.0
0.0512
0.50
7.5
4.00
0.44
9.2
0.0845
0.54
8.3
5.00
0.44
10.3
0.1221
0.59
8.9
6.00
0.44
11.3
0.1626
0.63
9.4
7.00
0.44
12.2
0.2049
0.68
9.8
8.00
0.44
13.0
0.249
0.74
10.1
9.00
0.44
13.8
0.2943
0.79
10.3
5 Conclusions A numerical model has been developed to evaluate the velocity of falling sodium droplets pertaining to postulated spray fire scenarios in SFR. The classical model for spherical particle motion has been coupled with a model for liquid droplet deformation that determines the drag coefficient dynamically accounting for the variations in droplet shape. The developed numerical model has been validated using the settling velocity data of free falling water droplet experiments in the literature. The model predictions were found to be in very good agreement with the experimental results. The validated model has been used to evaluate the settling velocity of sodium droplets of different sizes and these results revealed the effect of droplet deformation on the falling velocity of sodium droplets. The developed model is useful in the analysis of postulated sodium spray fire scenarios in SFR.
Analysis of Deformation Effects on Falling Spray Droplet Motion Under …
385
Fig. 8 Deformation of 9 mm sodium droplet
References 1. Malet JC et al (1981) Potential results of spray and pool fires. Nucl Eng Des 68:195–206 2. Yi Z et al (2021) Droplet generation during spray impact onto a downward-facing solid surface. Experiment Therm Fluid Sci 126:110402 3. Heisler M, Morewitz HA (1979) An investigation of containment pressurization by sodium spray fires. Nucl Eng Des 68:219–224 4. Tsai SS (1980) The NACOM code for analysis of postulated sodium spray fires in LMFBRs, NUREG/CR-1405. Brookhaven National Laboratory, USA 5. Scholtyssek W, Murata KK Sodium spray and jet fire model development within the CONTAINLMR code. In: International topical meeting on the safety of advanced reactors, April 18–20 6. Yamaguchi A, Takata T, Okano Y (2001) Numerical methodology to evaluate fast reactor sodium combustion. Nucl Technol 136(3):315–330 7. Muthu Saravanan S, Mangarjuna Rao P et al 2016) NAFCON-SF: Asodium spray fire code for evaluating thermalconsequences in SFR containment. Annal Nuclear Energy 90:389–409 8. Spalding DB (1955) Some fundamentals of combustion. Butterworths ScientificPublications, London, Gas Turbine Series 9. Morewitz HA (1979) Sodium spray fires. Nucl Eng Des 55:275–281 10. Hughes RH, Gilleland ER (1952) Chem Eng Prog 48:497 11. Liu AB, Mather D, Reitz RD (1993) Modelling the effects of drop drag and breakup on fuel sprays, SAE technical paper 930072 12. McCabe L, Warren J, Smith C, Peter H (2001) Unit operations of chemical engineering, McGraw Hill, New York, USA 13. Gunn R, Kinzer GD (1949) The terminal velocity of fall for water droplets in stagnant air. J Meteorol 6:243–248 14. Otis J (1941) Laws, Measurements of the fall velocity of water drops and rain drops. In: Transactions of American geophysical Union, pp 709–721 15. Steven C, Raymond C Numerical methods for engineers, McGraw Hill education
Particle Filtration in Suspension Droplet Breakup Kishorkumar Sarva, Tejas G. murthy, and Gaurav Tomar
Nomenclature Q Dn Dp φ i
Flow rate ml/m Nozzle diameter mm Particle diameter μm Particle volume fraction Successive droplet breakup number
1 Introduction Particle-laden fluid flows are common in several industrial applications such as fuel cells, 3D printing, energy–density combustion fuels. These flows can show complex functional and rheological properties. Uniform distribution of particles is desired during the processing of suspensions to achieve homogeneous properties in the final product. Effect of presence of particles on the interfacial behavior of a suspension has gained much interest lately due to its applications in the upcoming 3D printing technologies. Furbank and Morris [1] studied the dripping and jetting dynamics of suspensions. Subsequently, the canonical problem such as pendent drop formation, particle-laden drop breakup have also been studied [2–5]. K. Sarva (B) Interdisciplinary Centre for Energy Research, Indian Institute of Science, Bangalore, India e-mail: [email protected] T. G. murthy Department of Civil Engineering, Indian Institute of Science, Bangalore, India G. Tomar Department of Mechanical Engineering, Indian Institute of Science, Bangalore, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_36
387
388
K. Sarva et al.
Different stages of dripping dynamics can be classified into (i) formation of pendant droplet with homogeneous suspension, (ii) initiation of the neck formation leading to an inhomogeneous distribution of particles in the drop and (iii) catastrophic breakup of the neck and formation of the drop. In each stage, the dynamics is governed by the local particle packing fraction (effective shear viscosity). Neck formation stage has fewer particles and thus the effective viscosity is reduced, thereby leading to formation of a particle less neck which eventually leads to the formation of the droplet [3, 6]. Due to the evolution of the initially homogeneous suspension into an inhomogeneous suspension during the drop formation, the dynamics of the dripping of particle-laden fluids is significantly different from that of the pure Newtonian fluid. The understanding of the particle volume fraction has not been completely understood. Furbank and Morris [1] have shown that at large volume fraction, the satellite drop will disappear for a particular particle diameter and volume fraction. It has also been shown that the presence of a single particle in the filament formed during necking changes the breakup length and time [2, 7]. Unlike in the Newtonian fluid dripping, the stresses and dynamic behavior of suspensions are very complicated. The forces exerted by the particles, particle size, concentration significantly affect the drop formation. The evolution of the neck with a particle is found to show power law variation in time (h α (t-t0 )2/3 ) [8]. However, Zhao et al. [6], Thievenaz and Sauret [5] showed that there are essentially three stages through which neck evolves. In the initial thicker neck stage, the neck region has more or less uniform particle distribution. As the neck begins to thin, the particles segregate forming clumps at the two ends of the neck. In the final stage, neck extends further forming a particle less liquid filament. Zhao et al. [6] further showed that the universal scale proposed in [8] is not valid during the entire evolution of the neck. Based on the different stages, power law variation is found to be a function of the particle distribution. In the transition and liquid stage, the neck diameter varies linearly. A new scaling at the transition stage was proposed by [5] recently to predict the transition stage of the evolution and estimated the effect of particle volume fraction and polydispersity on the scaling analysis. However, successive droplet evolution of a suspension fluid at different particle and flow parameters has not been studied. Although [1] identified the effect of volume fraction on the satellite droplet, dripping to jetting transition, there is no study on the complex dripping dynamics of a suspension that is observed in Newtonian cases [9, 10]. In this preliminary study, we show the effect of particle volume fraction and particle diameter on the successive droplet breakup using experimental methodology introduced in the next section.
Particle Filtration in Suspension Droplet Breakup
389
2 Methodology The present study utilizes a basic experimental setup shown in the Fig. 1. It is developed for visualizing successive droplet breakup and to capture details of droplet evolution between two droplet breakups. The main components of the setup are drop generating module and shadowgraphy module. The former mainly contains a syringe pump mounted with a syringe of 29.4 mm barrel diameter, a nozzle, a tank to isolate the drop from surrounding effects and to collect fluid dispensed during the experiments. At the top of isolating tank, nozzle of diameter Dn = 5.3 mm is fixed vertically to the barrel. The shadowgraphy module consists of Nova S9 high-speed camera mounted with an appropriate lens arranged on a tripod separated from the former module. A diffuser sheet is placed between the tank and illumination from LED lamp to perform shadowgraphy. The particle-laden fluid utilized in the present study is prepared in such a way that, polystyrene particles are neutrally buoyant by suspending in aqueous glycerol solution (ρ = 1050 kgm−3 ). The viscosity of the continuous fluid is μ = 1.9 mPas and its surface tension is σ = 70 mNm−1 . The details of the parameters in the present study are provided in the Table 1. In non-dimensional terms, the dynamics are defined based on four parameters, namely the Bond, Bo = ρg R 2 /σ , Weber, W e = ρ RU 2 /σ , 1/4 Kapitza, K a = 3ϑ(gρ 3 /σ 3 ) as defined in Rubio-Rubio et al. [11] and nozzle-toparticle diameter ratio (Dn /Dp ). The sequence of suspension preparation such as preparing aqueous glycerol solution, measuring particle volume fraction, mixing the solution with particle using magnetic stirrer and duration of mixing are standardized. Once the suspension is prepared, the solution is kept in isolation to note the particle settling time. This allowed us to capture the time required to perform an experiment before the particles settle in the solution. The droplet generating module can dispense large number of drops at each flow rate. The nozzle diameter that is chosen in the present work allows dispensing a slurry prepared with particle diameter from 80 μm to as large as 900 μm. The sequence of
Fig. 1 Schematic of the experimental test rig used to study the dripping dynamics
390 Table 1 Experimental conditions of neutrally buoyant suspension prepared with aqueous solutions of glycerol
K. Sarva et al.
Ka
We
Bo
Dn /Dp
0.013
0.021
1.05
35.33
0.013
0.021
1.05
5.8
At these system conditions, successive droplet evolution is observed at different volume fractions (φ)
images for droplet evolution is captured by the camera with 6400 frames per second, a single pixel size being 3.75 μm. As the present aim of the experiment focuses on the macroscopic and microscopic details of suspension droplet evolution and breakup events, a larger vertical resolution of 512 pixels is chosen with width of 256 pixels. A typical experimental run includes recording the successive pinch-off points at a constant flow rate by connecting the syringe pump to the needle. To understand effect of flow rate on the suspension droplet pinch off, the experiments are conducted at each flow rate and images are recorded using shadowgraphy technique. The recorded images are post-processed, using in-house MATLAB code written to track the drop tip length (L b ) and drop volume (V ) at each time as it evolves. Periodicity between successive droplet breakup is measured using time period between the two drops (δT ) by looking at maximum number of breakup points at a given flow rate. The details of the effect of volume fraction and particle diameter are explained in the next section.
3 Results and Discussion Dripping of a moderately viscous Newtonian liquid at Q = 30 ml/m flow rate from a Dn = 5.3 mm nozzle is shown in the Fig. 2. Here, the droplet evolution is axisymmetric as the droplet evolves until both primary (Lp ) and secondary (Ls ) droplet breakup. The time required for the primary breakup condition is shown at the top of each profile. At the primary droplet breakup with time indent 0 s, a long filament connecting the primary drop and the liquid connecting nozzle is shown, which results in the formation of a secondary droplet at 0.0065 s. As the primary droplet breakup occurs, the filament starts retracting toward the nozzle. During the retraction stage, multiple neck formation is seen starting from the tip of the filament. Finally, it breaks at the minimum neck formed in the liquid connected to the nozzle. Figure 3 shows the successive droplet breakup profiles of suspensions of particle diameter Dp = 80 μm and increasing the volume fractions φ = 5, φ = 30 and φ = 60%. With increase in the particle volume fraction, the breakup length of primary and secondary droplet is seen increasing as shown in the Fig. 4. As the secondary droplet evolves, the number of neck formations is observed to reduce with increase in the particle volume fraction. It is also observed that the part of the filament with least thickness increases in length, which may cause further breakup of the filament. These effects may be attributed to the increase in the effective
Particle Filtration in Suspension Droplet Breakup
391
Fig. 2 Primary and secondary droplet evolution and breakup of a Newtonian droplet are shown at Q = 30 ml/m
viscosity of the suspension. To understand the effect of particle diameter Dp , φ = 20% suspension is prepared with 900 μm particles. The suspension droplet evolution with Dp = 80 μm and Dp = 900 μm is shown in the Figs. 5 and 6, respectively. The time required for breakup event to happen for a φ = 20% suspension in the Fig. 5 is increased compared to the pure continuous phase shown in Fig. 2. However, the breakup structure is axisymmetric as it evolves for smaller particle Dp. In the case of larger particle Dp as shown in the Fig. 6, particle–particle and particle-filament interaction is seen to be stronger due to which the retraction of the filament is asymmetric. In this figure, it is seen that different stages exist as the droplet evolves, such as homogeneous particle distribution, neck formation and retraction stages. As the drop evolution reaches the neck formation stage, the filament length scale is comparable to the particle Dp . At this stage, particle filtration, grazing, angle collision events are observed. In the retraction stage, as the filament moves toward the nozzle, particles will have head-on collisions, which cause the filament to form asymmetric profile. It is our future interest to study the particles effect on the filament retraction dynamics.
4 Conclusions In this work, we have presented the effect of particle volume fraction and particle size on the successive droplet breakup. It is shown that, at smaller particle diameters, increase in effective viscosity causes an increase in the breakup length scale. Particle– particle interaction and particle-filament interaction are stronger at larger particle
392
K. Sarva et al.
Fig. 3 Effect of volume fraction on the breakup structure with increase in the volume fraction for the parameters Q = 30 ml/m, Dp = 80 μm. With increase in the particle volume fraction, filament necking has been modified Fig. 4 Breakup length of successive droplets with increasing volume fraction is shown. In general, the droplet breakup length for both primary and secondary droplets is increasing with the volume fraction
Particle Filtration in Suspension Droplet Breakup
393
Fig. 5 Primary and secondary droplet evolution of 20% volume fraction suspension with 80 μm particle diameter at Q = 30 ml/m
Fig. 6 Evolution of suspension droplet with φ = 20% for the particle diameter Dp = 900 μm and flow rate of Q = 30 ml/m
394
K. Sarva et al.
diameters. Local events such as grazing, angle collision and head-on collision events result in the filament destabilization and subsequent rapid breakup. The filament retraction post-breakup will be altered due to these local events, resulting in changes in the breakup length and breakup time of successive droplets.
References 1. Furbank RJ, Morris JF (2004) An experimental study of particle effects on drop formation. Phys Fluids 16(5):1777–1790 2. Bertrand T, Bonnoit C, Clément E, Lindner A (2012) Dynamics of drop formation in granular suspensions: the role of volume fraction. Granul Matt 14(2):169–174 3. Bonnoit C, Bertrand T, Clément E, Lindner A (2012) Accelerated drop detachment in granular suspensions. Phys Fluids 24(4):043304 4. Furbank RJ, Morris JF (2007). Pendant drop thread dynamics of particle-laden liquids. Int J Multiphase Flow 33(4):448–468 5. Thiévenaz V, Sauret A (2022) The onset of heterogeneity in the pinch-off of suspension drops. Proceed Nat Acad Sci 119(13) 6. Zhao H, Liu HF, Xu JL, Li WF, Lin KF (2015) Inhomogeneity in breakup of suspensions. Phys Fluids 27(6):063303 7. van Deen MS, Bertrand T, Vu N, Quéré D, Clément E, Lindner A (2013) Particles accelerate the detachment of viscous liquids. Rheologica Acta 52(5):403–412 8. Miskin MZ, Jaeger HM (2012) Droplet formation and scaling in dense suspensions. Proceed Nat Acad Sci 109(12):4389–4394 9. Ambravaneswaran B, Phillips SD, Basaran OA (2000) Theoretical Analysis of a dripping faucet. Physic Rev Lett 85(25):5332–5335 10. Coullet P, Mahadevan L, Riera CS (2005) Hydrodynamical models for the chaotic dripping faucet. J Fluid Mech 526:1–17 11. Rubio-Rubio M, Taconet P, Sevilla A (2018) Dripping dynamics and transitions at high Bond numbers. Int J Multiph Flow 104:206–213
Dynamic Characteristics of Submicron Particle Deposited on the Charged Spherical Collector Abhishek Srivastava, Bahni Ray, Mayank Kumar, Debabrata Dasgupta, Rochish Thaokar, and Y. S. Mayya
Nomenclature ρa μa a
ρp p
σa Qp E V FD τP FG F DEP FE
Density of air [kg/m3 ] Dynamic viscosity of air [kg/m− s ] Relative permittivity of air Density of particle [mm] Relative permittivity of particle Electrical conductivity of particle [S/m] Charge on a particle [c] Electric field in domain [V/m] Electric potential in domain [V] Viscous drag force [N] Partial response time [s] Gravitational force [N] Dielectrophoretic force [N] Electrostatic force [N]
A. Srivastava · B. Ray (B) · M. Kumar · D. Dasgupta Department of Mechanical Engineering, IIT Delhi, New Delhi 110016, India e-mail: [email protected] R. Thaokar · Y. S. Mayya DepartmentofChemical Engineering, IIT Bombay, Powai, Mumbai, Maharastra 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_37
395
396
A. Srivastava et al.
1 Introduction Particulate matter (PM), a primary source of air pollution, is harmful to human health [1, 2]. The “submicron zone” comprises particles ranging in size from 0.1 to 1 µ. This poses a risk to the lungs of humans [3]. The collection process can be facilitated by inertial impaction, interception, gravity, electrostatic attraction, diffusion, and temperature gradient motion [4]. Inertia causes a particle carried by an air stream to follow the stream yet hit the obstruction [5]. Therefore, inertial forces are normally most significant for aerosol collection in scrubbers and high-speed filter. This is most dominating process for larger particles. In interception, the particle’s trajectory may pass close enough to the collector for its surface to touch it. Similarly, slowly moving air particles can settle on a collecting surface due to gravity. Brownian diffusion becomes prominent below 0.1 µ [6]. However, submicron particles are not affected by inertia or diffusion. Due to their poor inertial characteristics and low mobility, submicron particles are difficult to remove from gas streams. In this size range, charged collector scrubbing is necessary due to the effectiveness of electrical interactions. Impact scrubbing that is electrically augmented increases the relative velocity between the collector and the particles [7]. Consequently, their likelihood of capture increases. The schematic of the different capture mechanism is shown in Fig. 1. Inertia, electrostatic forces, and fluid friction are the primary factors that govern the outcomes of fine particle collisions with collectors (and are captured). In order to have a conceptual understanding of this topic, it is required to use equation of motion to determine the characteristic displacement, acceleration, and drift time of the particulate matter towards the collector. This study investigates how electrostatic forces, gravitational forces, dielectrophoretic forces, and viscous forces all influence Fig. 1 Particle capture mechanism [8]
Dynamic Characteristics of Submicron Particle Deposited …
397
the trajectories of individual particles. Consequently, the purpose of this work is to investigate the fundamental effects of electrostatic charge on fluid dynamics, particle movement, and collection by a spherical charged collector.
2 Literature Review and Objective Particles of submicron size in the Green’s-Field gap or the accumulation mode make up a large portion of particulate matter. According to observations made by Jaworek et al. submicron-sized particles whose mobility is controlled by air drag or molecular forces are extremely challenging to capture. Therefore, the inertial force has no effect on the dynamics of the particles [9]. This results in a significant decrease in the deposition efficiency of conventional particle removal systems such as wet scrubbers and cyclones [10]. In a similar study, Lear et al. found that the inertial impact and diffusion of submicron particles had minimal impact. Therefore, the effectiveness of electrical interactions, electro scrubbing is a viable option for particles of this size range [7]. Additionally, Ziwen Zuo et al. discovered that due to electrostatic forces, particles actively deposit on charged droplets, therefore extending the collection region of a single droplet through inertial collision [11]. The remarkable efficiency of electrostatic wet deposition has been demonstrated both theoretically [6, 12–19] and practically [20–25] in previous research. In a significant number of earlier studies [26–29], researchers explored the particle trajectories as well as the collection efficiency of charged collectors. Lee et al. examined charged particle trajectories in a Lagrange reference frame using a discrete phase model (DPM) that incorporated interception as an important deposition mechanism. They then used the DLVO theory to calculate the interaction energies between particle and collector surfaces [26]. In a similar manner, Zuo et al. performed a numerical analysis of the paths of charged particle trajectories around a charged collector while taking into consideration three different impact modes. In addition to this, the trajectories that were calculated numerically agreed with his experimental finding [27]. In a similar manner, Jaworek et al. examined aerosol particle trajectories with the aid of his numerical model for a limited range of the Coulomb number and the Reynolds number [28]. Their primary objective was to determine the collision trajectories. As a direct consequence, Jaworek again conducted an experimental investigation of the particle trajectories in order to validate the previously mentioned numerical model [29]. Furthermore, many researchers have incorporated various forces into their numerical models to predict collection efficiency and particle trajectories, but the evolution of the dynamic characteristics of individual particles striking around the circumference of collector still have not been explored. These findings come from research that ranges from the most recent to the oldest studies [6, 11, 28, 30]. The knowledge of drift time, characteristic response to respective forces under consideration will provide insights into how the location or position of a particle placed in a field that is not uniform affects the capture of an individual particle by a charged pendant collector. Therefore in order to address these research gaps, the present
398
A. Srivastava et al.
study leverages a previously developed mathematical model [11]. Under the impact of a non-uniform electric field, the model will enable us in getting insight into the dynamic characteristics of a particle initially placed at various positions within the computational domain.
3 Methodology This section explains the configuration of the model followed by the set of governing equations.
3.1 Model Geometry Figure 2 shows the model schematic, which depicts a pendant charge collector of diameter (D) creating a non-uniform electric around it. The collector is positioned in still air (i.e., u = 0), which has density (ρa ), dynamic viscosity (μa ), and relative permittivity (εa ). Silica glass particles with dimension (d p ), density (ρ p ), relative permittivity (ε p ), and electrical conductivity (σ p ) are randomly distributed across the base of the model across a length H. The shortest distance between the ground electrode and the collector’s base is H–D. Because particles have opposite polarity charges, they drift towards the collector along the electric line of force and thus get captured.
Fig. 2 Model geometry representation with particles at the bottom at t = 0. (All dimensions in mm)
Dynamic Characteristics of Submicron Particle Deposited …
399
3.2 Basic Governing Equations The numerical approach follows the movement of particles in the gas phase without considering particle interactions. The flow of air is laminar, Newtonian, and incompressible. The fundamental governing equations of air flow are mass and momentum conservation equations. ρa ∇u a = 0,
(1)
ρa (u a .∇)u a = ∇. − pa I + u a (∇u a + ∇u aT + ρa g.
(2)
Similarly, the solution for electric field is determined by applying Gauss’ law of electrostatics to the entire domain The necessary equation are as follows: ∇.(ε 0 ε r V ) = 0,
(3)
E = −∇ V .
(4)
Furthermore, each particle moves independently from the others, Newton’s equation of motion may be used to trace the trajectories of each of these particles. dxp = up , dt mp
(5)
du p = FD + FG + FD E P + FE , dt
(6)
Table 1 Values of parameter involved in analysis Parameters
Value
Density of air (ρ a )
1.2 kg/m3
Dynamic viscosity of air (μa )
1e-5 Pa-s
Relative permittivity of air (εa )
1
Density of particles (ρ p )
1000 kg/m3
Diameter of particles (d p )
10 µm
Relative permittivity of particles (ε p )
7.6
Electrical conductivity of particles (σ p )
1E-13 S/m
Electrical potential of collector (V_0)
1000 V
Charge on particle (Qp )
4.75e-16 C
Diameter of spherical charged collector (D)
1.26 mm
Distance from hinge point of collector to the particle plate (H)
2 mm
Span of collector plate (L)
4 mm
400
A. Srivastava et al.
where FD =
ρ p d 2p 1 m p u − u p , /textwher e τ p = τP 18μ ρp − ρ , ρp εa − ε p 3 = 2πr p ε0 (ε a ) ∗ ∇|E|2 , εa + 2 ∗ ε P FG = m p g
FD E P
FE = Q P E.
(7) (8) (9) (10)
4 Results and Discussion 4.1 Validation of Results To validate the numerical model, the existing experimental data of Zuo et al. [11] is compared. The numerical data for particle velocity agrees quite well with the existing experimental data. The Pearson correlation coefficient between numerical data and experimental data is 0.9261, indicating a strong correlation between experimental and numerical data (Fig. 3).
Fig. 3 Particle velocity approaching charged collector compared experimentally and numerically
Dynamic Characteristics of Submicron Particle Deposited …
401
Fig. 4 Electric potential distribution due to charged collector
4.2 Electric Potential and Electric Field Norm Distribution Within the Domain The potential distribution and electric field distribution around the pendant collector are given by the solutions of Eqs. (3) and (4). Solving the Laplace equation for electric potential (V) inside the given domain yields the potential distribution. As indicated in Fig. 2, the spherical collector’s boundary was held at a constant potential (V_0), and the flat plate electrode was grounded. The distribution of the electric field (E) is determined by taking the gradient of the scalar electric potential (V) shown in Figs. 4, 5.
4.3 Trajectories of Particle Drifting Towards Charged Collector The trajectory of particles attracted by a spherical charged collector is depicted in the Figure 6 above. Each particle is assigned a unique index, which aids in explaining the particle’s kinematic and dynamic characteristics. The particle index on each symmetrically half side is kept similar because they will all behave similarly in terms of magnitude.
402
A. Srivastava et al.
Fig. 5 Electric field norm distribution due to charged collector Fig. 6 Charged collector particle trajectories
4.4 Analysis of Forces Acting on Particle Drifting Towards the Charged Collector 4.4.1
Analysis of Norm of Electrostatic Forces
Each particle encounters a varied amount of electrostatic force due to non-uniform electric field distribution. Based on the initial position of particle on collection plate or ground electrode, particle 5 will experience greatest electrostatic force throughout its course, while particle 1 would experience minimal electrostatic force. Figure 5 shows that the electric field norm is significant along the lower portion of collector. As a result, particles (particles 3–5) in the region of strong electric field experience the strongest electrostatic force relative to particles (particle 1–2) lying farther away on the ground plate with respect to collector.
Dynamic Characteristics of Submicron Particle Deposited …
403
Fig. 7 Evolution of Electrostatic Force acting on each particle along its trajectory
Furthermore, the change in the strength of the electric field norm is directly related to whether the electrostatic force on each particle increases or decreases along its trajectory. Therefore, Fig. 5 makes it clear how the electric field norm changes along the electric line of forces, and Fig. 7 shows how the electric field variation affects electrostatic forces along the particle trajectory.
4.4.2
Analysis of Magnitude of Viscous Drag Along the Particle Trajectory
Figures 7 and 8 show that the scale of drag force acting on a particle is nearly identical to electrostatic force. As a result, viscous drag plays a significant role in determining a particle’s fate. Since the air is still, the typical fluctuation of drag force will be exactly proportional to the magnitude of particle velocity. The drag force curves for particles 3, 4, and 5 have inflection points at 0.017, 0.014, and 0.013 s, respectively. This is because the value of net positive difference between electrostatic and viscous drag attains minima which compels the slope of acceleration to change from negative to positive. It implies that the velocity of particles 3, 4, and 5 will continue to increase with time until they are trapped, hence increasing the magnitude of drag force which is obviously consistent with Fig. 8. However, after 0.021 s, the net difference in magnitude of electrostatic and drag forces becomes negative for particles 1 and 2. As a result of the decrease in particle velocity magnitude after 0.021 s, drag force displays a declining pattern. After a while, the difference in electrostatic and drag force magnitude becomes positive, causing the drag force to increase again until the particle is captured.
404
A. Srivastava et al.
Fig. 8 Evolution of drag force acting on each particle along its trajectory
4.4.3
Analysis of Dielectrophoretic Force
Figure 9 shows that the scale of DEP force is an order of magnitude lower than that of drag and electrostatic force. As a result, the DEP force plays allow influence in particle capture However, particles closer to the collector experience greater DEP force than particles farther away on the collecting plate. In a stronger electric field zone, for particles 3, 4, 5, DEP will aid in overcoming gravitational force after lift-off, whereas for particles 1 and 2, it will be too weak to even overcome gravity force. Fig. 9 Evolution of dielectrophoretic force acting on each particle along its trajectory
Dynamic Characteristics of Submicron Particle Deposited …
4.4.4
405
Analysis of Gravitational Force
Gravitational forces are two orders of magnitude less significant than electrostatic and drag forces and thus have little role in particle fate.
5 Conclusions The present study illustrates how a particle behaves dynamically with respect to each of the forces that are being taken into consideration. According to the results of the analysis, the electrostatic and viscous drag are the fundamental forces that determine the trajectory of the particle. Both dielectrophoretic forces and gravitational forces have a negligible impact. Particles which are located most closely to the collector are subjected to the greatest electrostatic and drag forces, whereas particles located farther are subjected to the least amount of force. Because the particles’ starting velocity was zero, there will be zero viscous drag at the moment of take-off. Consequently, the electrostatic attractive force that a particle experience will be at its highest value, and it will drift towards the collector with its highest value of initial acceleration while it is being lifted off the ground. Based on this, we are able to conclude that electrostatic force is the single key reason for the lifting of particles. In light of the fact that the particles have a size of submicron, the path that they take will be along the electric line of forces. However, particles of a bigger size will continue along their inertial trajectory and depending on where on the collecting plate they were first located, they may or may not be collected.
References 1. Kumar P, Pirjola L, Ketzel M, Harrison RM (2013) Nanoparticle emissions from 11 non-vehicle exhaust sources-a review. Atmos Environ 67:252 2. van Donkelaar A, Martin RV, Spurr RJD, Burnett RT (2015) High-resolution satellite derived PM2.5 from optimal estimation and geographically weighted regression over North America. Environ Sci Technol 49:10482 3. Di Natale F, Carotenuto C, D’Addio L, Jaworek A, Krupa A, Szudyga M, Lancia A (2015) Capture of fine and ultrafine particles in a wet electrostatic scrubber. J Environ Chem Eng 3(1):349–356 4. Strauss W (1966) Industrial gas cleaning. Pergamon Press, London 5. Ranz WE, Wong JB (1952) Impaction of dust and smoke particles on surface and body collectors. Ind Eng Chem 44(6):1371–1381 6. George HF, Poehlein GW (1974) Capture of aerosol particles by spherical collectors: electrostatic, inertial, interception and viscous effects. Environ Sci Technol 8(1):46–49 7. Lear CW, Krieve WF, Cohen E (1975) Charged droplet scrubbing for fine particle control. J Air Pollut Control Assoc 25(2):184–189 8. Fitch, E.C (n.d.) Filtration mechanics-how filters capture particles. Machinery Lubrication. https://www.machinerylubrication.com/Read/31252/filtration-particles-capture
406
A. Srivastava et al.
9. Jaworek A, Krupa A, Sobczyk AT, Marchewicz A, Szudyga M, Antes T, Carotenuto C (2013) Submicron particles removal by charged sprays. Fundament J Electrostat 71(3):345–350 10. Jaworek A, Balachandran W, Krupa A, Kulon J, Lackowski M (2006) Wet electro scrubbers for state of the art gas cleaning. Environ Sci Technol 40(20):6197–6207 11. Zuo Z, Wang J, Huo Y, Liu H, Xu R (2016) Particle motion induced by electrostatic force of a charged droplet. Environ Eng Sci 33(9):650–658 12. Kraemer HF, Johnstone HF (1955) Collection of aerosol particles in presence of electrostatic fields. Ind Eng Chem 47(12):2426–2434 13. Nielsen KA, Hill JC (1976) Capture of particles on spheres by inertial and electrical forces. Ind Eng Chem Fundam 15(3):157–163 14. White HJ (1977) Electrostatic precipitation of fly ash. J Air Pollut Control Assoc 27(1):15–22 15. Wang* HC, Stukel JJ, Leong KH (1986) Charged particle collection by an oppositely charged accelerating droplet. Aerosol Sci Technol 5(4):409–421 16. Jaworck A, Krupa A, Adamiak K (1996) Particle trajectories and collection efficiency of submicron particles on a charged spherical collector. In: IAS’96 conference record of the 1996 IEEE industry applications conference thirty-first IAS annual meeting, vol 4, pp 2036–2047 17. Yang HT, Viswanathan S, Balachandran W, Ray MB (2003) Modeling and measurement of electrostatic spray behavior in a rectangular throat of pease-anthony venturi scrubber. Environ Sci Technol 37(11):2547–2555 18. Zhao H, Zheng CG (2008) Modeling of gravitationalwet scrubbers with electrostatic enhancement. Chem Eng Tech 31:1824 19. Carotenuto C, Di Natale F, Lancia A (2010) Wet electrostatic scrubbers for the abatement of submicronic particulate. Chem Eng J 165(1):35–45 20. Pilat MJ, Jaasund SA, Sparks LE (1974) Collection of aerosol particles by electrostatic droplet spray scrubbers. Environ Sci Technol 8(4):360–362 21. Metzler P, Weiß P, Büttner H, Ebert F (1997) Electrostatic enhancement of dust separation in a nozzle scrubber. J Electrostat 42(1–2):123–141 22. Jaworek A, Krupa A, Adamiak K (1998) Submicron charged dust particle interception by charged drops. IEEE Trans Ind Appl 34(5):985–991 23. Balachandran W, Jaworek A, Krupa A, Kulon J, Lackowski M (2003) Efficiency of smoke removal by charged water droplets. J Electrostat 58(3–4):209–220 24. Jaworek A, Balachandran W, Lackowski M, Kulon J, Krupa A (2006) Multi-nozzle electrospray system for gas cleaning processes. J Electrostat 64(3–4):194–202 25. Krupa A, Jaworek A, Sobczyk AT, Marchewicz A, Szudyga M, Antes T (2013) Charged spray generation for gas cleaning applications. J Electrostat 71(3):260–264 26. Li W, Li H, Shen S, Cui F, Shen B, Huang Y (2018) Study of particle rebound and deposition on fibre surface. Environ Technol 27. Zuo Z, Wang J, Huo Y, Xu R (2017) Numerical study of particle motion near a charged collector. Particuology 32:103–111 28. Jaworek A, Adamiak K, Krupa A (1997) Deposition of aerosol particles on a charged spherical collector. J Electrostat 40:443–448 29. Jaworek A, Adamiak K, Krupa A, Castle GSP (2001) Trajectories of charged aerosol particles near a spherical collector. J Electrostat 51:603–609 30. Nielsen KA, Hill JC (1976) Collection of inertialess particles on spheres with electrical forces. Ind Eng Chem Fundam 15(3):149–157
Accommodating Volume Expansion Effects During Solid–Liquid Phase Change—A Comparative Study Keyur Kansara and Shobhana Singh
Nomenclature C h sl He k m T u, v x, y ρ δ l s t
Specific heat of PCM (J/kg K) Latent heat of fusion (J/kg) Height of enclosure (m) Thermal conductivity of PCM (W/m K) Mass of PCM (Kg) Temperature (K) Velocity component in x, y direction (m/s) Space coordinates (m) Density of PCM (kg/m3 ) Position of solid–liquid interface (m) Liquid Solid Time (s)
1 Introduction and Literature Review The electronic components, devices, or detectors working inside the satellite payload demands rigorous thermal and structural requirements for efficient working in the stringent space environment. The recurrent transient working nature of several subsystems of communication, optical, microwave, and remote sensing satellites necessitates the development of a periodic transitory thermal control system. The K. Kansara · S. Singh (B) Department of Mechanical Engineering, IIT Jodhpur, Jodhpur 342037, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_38
407
408
K. Kansara and S. Singh
inherent transient working nature makes the phase change materials an ideal candidate for the thermal management of satellite subsystems. The isothermal energy storage capability of PCM fulfills the requirement of maintaining the electronic components between the operating ranges during various mission stages. The implementation process of PCM-based thermal control system is complicated by the low thermal conductivity of PCM and designing a leak-proof space compatible containment system. It is essential to incorporate both thermal and structural considerations while designing the containment system for PCM. The structural considerations demand the system to be leak-proof and able to sustain static and dynamic stresses. Additionally, the container must be thermally protected from a stringent space environment without impairing the system’s performance. The PCMs have the propensity to expand/shrink during melting and solidification processes, respectively. Hence, the expansion or shrinkage of PCMs during the melting and solidification process may adversely affect the thermal performance of the PCM-based thermal management system. Therefore, it is necessary to accommodate the volume expansion/shrinkage of PCM during the design process of the containment system. The conversion of the material from solid to liquid phase highly influences the thermophysical properties such as thermal conductivity, specific heat, viscosity, and density. The convection–diffusion phenomena govern the solid–liquid phase change process. The difference in the density of the solid and liquid phases is responsible for inducing natural convection currents and volumetric expansion during the phase change process. However, it is challenging to completely accommodate the density difference of both solid and liquid phases in the analytical and numerical models. Moreover, the mass conservation at the solid–liquid interface should take into account the volume change of the computational domain. The first-ever analytical solution of the solid–liquid phase change process also considered a constant density in solving the ice freezing problem known as Stefan Problem [1]. In fact, the conventional analytical and numerical models available to simulate the phase change process often neglect the volume expansion effects considering the density of PCM constant in both solid and liquid phases. Reviewing the experimental works conducted to study the solid–liquid phase change process, the volume change of the PCM during melting and solidification should be accounted given the significant influence of volume expansion on the thermal performance of PCM-based TCS. Beckermann and Viskanta [2] conducted an experimental study on the melting of gallium inside a rectangular enclosure accommodating the volume change effects by constructing a small hole in the top wall of the enclosure allowing the liquid gallium to enter the test cell. Maintaining the quantity of gallium inside the enclosure entirely filled throughout the experiment minimized the effect of volume expansion on the development of natural convection flow patterns. Subsequently, Braga and Viskanta [3] experimentally analyzed the impact of density extremum on the solidification process of water inside a rectangular enclosure. In the study, the rectangular cavity was designed to have a small (~ 3 mm) gap near the top wall to include the expansion of water without affecting the solidification of water. Moreover, Zhang et al. [4] and Zongqin et al. [5] used the same strategy of providing
Accommodating Volume Expansion Effects During Solid–Liquid Phase …
409
an empty space inside the container for allowing the PCM to expand/shrink without affecting the melting/solidification cycle. The volume expansion of PCM not only affects the thermal performance of the system but also significantly influences the mechanical performance. The velocity variation inside the container and mass conservation at the solid–liquid interface is typically the most difficult part of numerical modeling the volume expansion. Conti [6] studied the variation in the domain pressure during the planar solidification of a finite slab by developing a thermo-mechanical system. The top wall of the container was modeled as an elastic wall allowed to move as a consequence of the change in the PCM volume during the melting/solidification process. Assis et al. [7] conducted a numerical and experimental investigation on PCM melting inside a spherical shell, assuming a constant PCM density for each phase and a linear variation in the density within the mushy zone region. A large portion of the cavity near the top wall was filled with air to accommodate the expansion of PCM during the melting process. The solid–liquid phase change process was numerically modeled using the Volume of Fluid approach for demonstrating the moving air-PCM interface. Shmueli et al. [8] also adopted a similar approach to model the volume expansion of the PCM inside a vertical cylindrical tube. However, the author considered a negligible impact of the pressure variation within the system, which may lead to an inaccurate prediction of the melting time. Furthermore, Ho et al. [9] numerically demonstrated the volume expansion effect on PCM melting by enabling the unrestricted movement of the top wall while the remaining walls of the container were imposed with no-slip boundary conditions. Similarly, Bilir et al. [10] and Wang et al. [11] considered the assumption of constant PCM density in both liquid and solid phases to be valid in their analysis of PCM melting inside the enclosure. The extensive literature review on different mass accommodation methods authenticates the importance of considering the effect of volume expansion during the solid– liquid phase change process. However, the literature mentioned above assumes the density of PCM constant in both solid and liquid phases, which may not be an appropriate assumption in space applications. The density variation throughout the melting cycle in solid, liquid, and mushy zone regions is responsible for inducing natural convection currents. Hence, the present numerical model fully accounts for the PCM density change with domain temperature during completely solid, liquid, and mushy zone regions. Moreover, selecting a suitable mass accommodation method is essential for the numerical model to consider the difference in the density of PCM. Consequently, the present work conducts a comparative analysis of two mass accommodation methods, namely an open boundary and a free/movable surface, to incorporate the volume expansion during the melting cycle of PCM. The problem under investigation, physical model, mathematical model, governing equations, initial and boundary conditions, numerical solution procedure, model validation, and definitions of dimensionless performance parameters are described in Sect. 2, while Sect. 3 presents a comprehensive discussion of the results of the study. Lastly, the essential findings of the present work are listed in Sect. 4.
410
K. Kansara and S. Singh
2 Problem Description and Methodology The problem under consideration is to analyze and compare the effect of volume expansion during the melting of PCM inside an enclosure using two different mass accommodation methods, namely an open boundary and a free/movable surface. A detailed description of the computational domain and numerical methodology adopted to capture the convection–diffusion phenomena is discussed in the subsequent sections.
2.1 Physical Model A two-dimensional computational domain consisting of a square enclosure filled with paraffin-based PCM is considered in the present work. The schematic diagram of both the mass accommodation methods, i.e., an open surface and a free/movable surface, is presented in Fig. 1a and b, respectively. The open boundary method consists of a port at the top wall of the container through which the mass can enter or leave the domain to accommodate the volume expansion of PCM. The port size is estimated to be 5% of the cavity height according to the variation in the density of PCM. However, the second mass accommodation method considers the top wall of the enclosure as a free/movable surface instead of an open boundary, as shown in Fig. 1b. The isothermally heated left wall of the cavity is of a dimensional length H e (20 mm), maintaining a constant temperature of T h (345 K) throughout the melting process, while the right wall of the enclosure, acting as a heat sink is kept at a constant temperature of T c (300 K). The bottom wall of length L e (20 mm) of the enclosure is kept adiabatic throughout the melting process. The height (H e ) and length (L e ) of the enclosure are evaluated, ensuring minimal boundary effects on the melting of the PCM near the heated wall. The domain is filled with paraffin-based PCM, whose melting temperature range is T m = 319.15 − 321.15 K, which lies within the operating range of the selected space application. The initial temperature of the entire domain is set to 300 K, less than the melting temperature of PCM, keeping the PCM in a solid state at the initiation of the melting process. The melting cycle of the PCM inside the domain runs for 3600 s for each case selected for the study.
2.2 Numerical Methodology The convection–diffusion-driven melting process of the PCM inside the computational domain is simulated utilizing the methodology based on the enthalpy-porosity approach reported by Brent and Voller [13]. The detailed procedure to model the
Accommodating Volume Expansion Effects During Solid–Liquid Phase …
411
Fig. 1 Schematic diagram of the computational domain along with the boundary conditions for two mass accommodation methods a an open boundary and b a free/movable surface
solid–liquid phase change, including governing equations, scaling factors, nondimensional variables, and numerical solution procedure, can be found in the author’s previous reported study [14]. In the present section, the incorporation of density difference and mass flux term in the governing equations are briefly explained. The density difference in both solid and liquid phases is responsible for the PCM’s expansion or shrinkage during the melting and solidification processes. Since the density of PCM in the solid and liquid phases is different, the numerical model should satisfy the mass conservation at the solid–liquid interface (Table 1). The mathematical model employed to solve the governing equations takes into account the following assumptions, Table 1 Temperature-dependent thermophysical properties of PCM [12] Property
Values
Phase change temperature (K)
319.15 (Ts ) − 321.15 (Tl )
Latent heat (J/kg)
173,400
Specific heat (J/kg K)
2890
Thermal conductivity (W/m K)
0.21 if T ≤ Ts 0.21 −
0.09(T −Ts ) (Tl −Ts )
if Ts < T < Tl
0.12 if T ≥ Tl Density (kg/m3 )
750 if T ≤ Ts
Dynamic viscosity (kg/m s)
if T ≥ Tl 0.001 exp −4.25 + 1790 T 750 0.001(T −319.15)+1
412
K. Kansara and S. Singh
1. In the liquid state, PCM is assumed to be an incompressible Newtonian fluid. 2. The numerical model anticipates the negligible effect of convection and radiation heat losses to the environment. Energy Equation: The following equations represent the energy conservation at the solid–liquid interface during the melting process of PCM for accommodating the volume expansion effects, ∂ ∂ ∂ Ts ∂ Ts ∂ Ts ks + ks = ρs C ps , ∂x ∂x ∂y ∂y ∂t ∂ Tl ∂ Tl ∂ Tl ∂ ∂ ∂ Tl ∂ Tl +v + kl + kl = ρl C pl u . ∂x ∂x ∂y ∂y ∂x ∂y ∂t
(1) (2)
The continuity and energy conservation equations described by Hahn and Ozisik [1] need to be modified as follows to take into account the impact of volume expansion during the melting of PCM, dδ = ρl vl , dt ∂ Ts ∂ Tl ∂ Ts ∂ Tl dδ + − kl + = ρs h sl . ks ∂x ∂y ∂x ∂y dt (ρs − ρl )
(3) (4)
In the first approach to accommodate differences in the PCM density in the numerical model, the enclosure is designed with a port open to the atmosphere (i.e., p0 = 0) on the top wall that allows the PCM to flow in or out of the system. However, the open boundary mass accommodation method is complex to design for space applications with a closed system since the total domain mass changes with time. Hence, the second approach contemplates the top wall of the enclosure as a free surface (∂u x /∂ y = 0), which moves corresponding to the volume change of PCM. Hence, the total mass of the domain is described as. m (t) = ρs δ(t) + ρl [H (t) − δ(t)],
(5)
ρl δ(t). H (t) − Hi = 1 − ρs
(6)
2.3 Solution Procedure The governing equations that describe the evolution of natural convection inside the computational domain during the melting process are implicitly solved using a
Accommodating Volume Expansion Effects During Solid–Liquid Phase …
413
double-precision finite volume-based ANSYS Fluent 2020 R2. The SIMPLE algorithm is used for the discretization of pressure–velocity coupling. The PRESTO scheme is employed to discretize the pressure correction term. The advection terms in the momentum and energy equations are approximated using second-order upwind scheme. A first-order implicit formulation is utilized to integrate the unsteady terms of the governing equations. The discretized momentum, pressure, and energy equations are iterated with the under relaxation factors of 0.7, 0.3, and 1, respectively, setting up the absolute residual criterion to 1 × 10–3 , 1 × 10–4 , and 1 × 10–8 for mass, momentum, and energy conservation, respectively. Moreover, the grid and time step independence test reveal that the element and time step size of 0.1 mm and 0.05 s are reasonable for running further simulations, ensuring high accuracy and less computational time for the numerical solver.
2.4 Numerical Model Validation The numerical model employed in the present study is validated by comparing it with the experimental data obtained by Gau and Viskanta [15], as described comprehensively in the author’s previous study. Figure 2 compares the location of the phase interface with time between the present numerical model and experimental results published by Gau and Viskanta. It is depicted in Fig. 2 that there are minor discrepancies between the current model and published data, which may be attributed to the radiation and convection losses. Conclusively, the present numerical model predicts the melt front with less than 1.5% discrepancy values. 2 min
Fig. 2 Comparison of the phase interface at various times between the present prediction and experiment by Gau and Viskanta [15]
6 min
10 min
19 min
1.00
y/W
0.80 0.60 0.40 0.20 0.00 0.00
Gau & Viskanta Present Study 0.20
0.40
0.60 x/L
0.80
1.00
414
K. Kansara and S. Singh
3 Results and Discussion For the purpose of accommodating the effect of volume expansion during the melting of PCM inside an enclosure, numerical simulations are carried out utilizing two different mass accommodation methods. The current section of the work describes the effect of volume expansion on the heat transfer and flow characteristics of PCM. In order to understand the impact of volume expansion on the melting rates of PCM, the variation in liquid fraction value with time using both the mass accommodation methods (i.e., an open boundary and a free/movable surface) is presented in Fig. 3. For the comparison purpose, the variation in liquid fraction with time for the case which neglects the volume expansion effect is also illustrated in Fig. 3. It can be observed from Fig. 3 that the melting rate increases swiftly with time when the volume expansion effects are considered negligible, while the rate of increase of liquid fraction is slower for the open boundary and free/movable surface cases comparatively. It is noticeable from Fig. 3 that the difference in the rate of liquid fraction is minimal at the beginning of the melting process for each case because the melting process is governed by conduction heat transfer, with convection exerting little influence on the overall heat transport from source to PCM. As time advances, the role of the natural convection heat transfer in PCM melting becomes influential as convection currents start to develop, inducing strong buoyancy forces. The expansion of PCM limits the development of convection currents and significantly slows down the melting process. Looking into Fig. 3 further to distinguish the change in the liquid fraction among the case without volume expansion and an open boundary provides a significant discrepancy of 8.75% at the end of the melting process (t ~ 3600 s). However, the same difference in the liquid fraction value reduces to 8.3% at the end of the melting process (t ~ 3600 s) between the case without volume expansion and a free/movable surface. On a comparative note, 95.7% of the PCM is melted at the end of the melting process (t ~ 3600 s) when the volume expansion effects are considered negligible. At the same time, the exact value is reduced to 87.48% and 83.78% for the case of an open boundary and a free/ movable surface. 1 0.8 Liquid Fraction
Fig. 3 Variation in average liquid fraction with time for different mass accommodation cases
0.6
Without VE
0.4
An Open Boundary
0.2
A Free/Movable Surface
0 0
600
1200
1800 Time (s)
2400
3000
3600
Accommodating Volume Expansion Effects During Solid–Liquid Phase … 0.6 0.5 H(t) - H(i) (mm)
Fig. 4 Variation in the height of the top wall of the enclosure with time for each mass accommodation case
415
0.4 0.3
An Open Boundary
0.2
A Free/Movable Surface
0.1 0 0
600
1200
1800 Time (s)
2400
3000
3600
To better understand the effect of volume expansion on the melting process of PCM, the variation in the height of the enclosure with time for both cases (an open boundary and a free/movable surface) is plotted in Fig. 4. It is depicted in Fig. 4 that the height of the enclosure gradually starts to increase as the melting process begins. As the melting progresses, the cavity height further increases and becomes continual after approximately 1003 s for both cases (an open boundary and a free/movable surface). It is viable to observe that both mass accommodation methods predict an almost perpetual increment in the cavity height with time. The maximum variation in the enclosure height is observed to be 0.58 mm for an open boundary case and 0.57 mm for the free/movable case. The reason for the slower melting rate of PCM can be attributed to the change in enclosure height because of the volume expansion of PCM during the melting cycle. Moreover, the free/movable surface method is imposed with a shear boundary condition method at the top wall of the enclosure. As a consequence of the change in the volume of PCM, the top wall of the enclosure attains the velocity and starts to displace from the beginning of the melting process. The variation in the velocity of the top wall with time is shown in Fig. 5. It can be seen in Fig. 5 that the top wall velocity increases continuously and reaches the maximum value of 0.78 mm/ s at 566 s, decreasing again up to 0.7 mm/s and becomes constant afterward till the end of the melting process. This increment in the velocity of the top wall allows the PCM to expand and reduces the undesirable rise in the pressure inside the enclosure. The analysis of the melt front movement and flow distribution inside the enclosure gives an insight into the formation of convection currents that drive the heat transfer process from source to PCM for each mass accommodation case. Figure 6 shows a comparative analysis of PCM liquid fraction and melt front movement at the instances of 300 s, 600 s, 1200 s, 1800s, 2400 s, 3000 s, and 3600 s. The initial temperature of the domain is 300 K which is less than the melting temperature (319.15 to 321.15), keeping the PCM filled inside the enclosure in solid form before the melting process begins. At time t > 0, the solid PCM adjacent to the left wall (heat source) starts melting, and the solid–liquid interface gradually moves toward the heat sink as the melting progresses for each case. The strength of the natural convection gradually increases as the solid–liquid interface quickly shifts toward the heat sink. It is worth
416
K. Kansara and S. Singh
Top Wall Velocity (mm/s)
0.8 0.6 0.4 0.2 0 0
600
1200
1800 2400 Time (s)
3000
3600
Fig. 5 Velocity distribution of the top wall of the enclosure with time for the free/movable mass accommodation case
noting that the gravity force acts in the same direction as the heat flow pulling the liquefied PCM downwards, which strengthens the domination of buoyancy force on the progression of the melting process over time. The presence of a port to accommodate the volume expansion makes the PCM melting slower compared to a free/ movable surface method, as shown in Fig. 6. Once the solid PCM in the top portion of the cavity completely melts, the PCM at the bottom of the enclosure begins to melt at a faster rate. Looking into the contours further, around ~ 70% of the PCM melts within 1200 s for each case, while the remaining ~ 30% takes another 2400 s to melt, still leaving a small portion of the solid PCM near the heat sink. Also, it is evident from Fig. 6 that the convection currents get consolidated in the liquid region, and the solid–liquid interface moves swiftly toward the heat sink. However, the presence of a constant temperature wall near the heat sink prevents the PCM from completely melting. In addition, the liquid–solid interface can be observed getting diffused with time, especially after 1200 s near the bottom portion of the enclosure. The reason for the spread liquid–solid interface can be attributed to the conduction-dominated melting process at the bottom of the enclosure.
4 Conclusions In this paper, the effect of the volume expansion on the solid–liquid phase change process using two different mass accommodation methods (an open boundary and a free/movable surface) is numerically investigated. The computational domain consists of an enclosure filled with paraffin-based PCM. The essential findings of the present work are reported in this section. • Convection current growth is restricted by PCM expansion, which thus considerably slows the melting process.
Accommodating Volume Expansion Effects During Solid–Liquid Phase …
A free/movable surface
3600s
2400s
1200s
300s
An open boundary
417
Fig. 6 Comparison of melt front movement inside the enclosure with time for each volume expansion method
• The maximum discrepancy in the liquid fraction value between using an open boundary and a free/movable surface compared to the case, which considers the volume expansion effect negligible, is approximately 8.75% and 8.3%, respectively. • Utilizing an open boundary and a free/movable surface melts 87.48% and 83.78% of PCM filled inside the enclosure. In comparison, 95.7% of the PCM melted during the without volume expansion case. • The maximum variation in the enclosure height is observed to be 0.58 mm for an open boundary case and 0.57 mm for the free/movable case. • Around 70% of the PCM melts within 1200 s, while the remaining 30% takes an additional 2400 s to melt, still leaving a small amount of the solid PCM close to the heat sink.
418
K. Kansara and S. Singh
References 1. David WH, Ozisik MN (2002) Heat conduction, 3rd ed. Wiley 2. Beckermann C, Viskanta R (1989) Effect of solid subcooling on natural convection melting of a pure metal. J Heat Transfer 111(2):416–424. https://doi.org/10.1115/1.3250693 3. Braga SL, Viskanta R (1992) Effect of density extremum on the solidification of water on a vertical wall of a rectangular cavity. Exp Therm Fluid Sci 5(6):703–713. https://doi.org/10. 1016/0894-1777(92)90114-K 4. Zhang Z, Bejan A (1990) Solidification in the presence of high Rayleigh number convection in an enclosure cooled from the side. Int J Heat Mass Transf 33(4):661–671. https://doi.org/ 10.1016/0017-9310(90)90165-Q 5. Zongqin Z, Bejan A (1989) Melting in an enclosure heated at constant rate. Int J Heat Mass Transf 32(6):1063–1076. https://doi.org/10.1016/0017-9310(89)90007-0 6. Conti M (1995) Planar solidification of a finite slab: effects of the pressure dependence of the freezing point. Int J Heat Mass Transf 38(1):65–70 7. Assis E, Katsman L, Ziskind G, Letan R (2007) Numerical and experimental study of melting in a spherical shell. Int J Heat Mass Transf 50(9–10):1790–1804. https://doi.org/10.1016/j.ijh eatmasstransfer.2006.10.007 8. Shmueli H, Ziskind G, Letan R (2010) Melting in a vertical cylindrical tube: numerical investigation and comparison with experiments. Int J Heat Mass Transf 53(19–20):4082–4091. https:// doi.org/10.1016/j.ijheatmasstransfer.2010.05.028 9. Ho CJ, Liu KC, Yan WM (2015) Melting processes of phase change materials in an enclosure with a free-moving ceiling: an experimental and numerical study. Int J Heat Mass Transf 86:780–786. https://doi.org/10.1016/j.ijheatmasstransfer.2015.03.063 10. Bilir L, Ilken Z (2005) Total solidification time of a liquid phase change material enclosed in cylindrical/spherical containers. Appl Therm Eng 25(10):1488–1502. https://doi.org/10.1016/ j.applthermaleng.2004.10.005 11. Wang S, Faghri A, Bergman TL (2010) A comprehensive numerical model for melting with natural convection. Int J Heat Mass Transf 53(9–10):1986–2000. https://doi.org/10.1016/j.ijh eatmasstransfer.2009.12.057 12. Hong Y, Ye WB, Du J, Huang SM (2019) Solid-liquid phase-change thermal storage and release behaviors in a rectangular cavity under the impacts of mushy region and low gravity. Int J Heat Mass Transf 130:1120–1132. https://doi.org/10.1016/j.ijheatmasstransfer.2018.11.024 13. Brent AD, Voller VR, Reid KJ (1988) Enthalpy-porosity technique for modeling convectiondiffusion phase change: application to the melting of a pure metal. Numer Heat Transf 13(3):297–318. https://doi.org/10.1080/10407788808913615 14. Kansara K, Singh VK, Patel R, Bhavsar RR, Vora AP (2021) Numerical investigations of phase change material (PCM) based thermal control module (TCM) under the influence of low gravity environment. Int J Heat Mass Transf 167:120811. https://doi.org/10.1016/j.ijheatmasstransfer. 2020.120811 15. Gau C, Viskanta R (1986) Melting and solidification of a pure metal on a vertical wall. J Heat Transf 108:174–181
Self-Similar Velocity Profiles in Granular Flow in a Silo with Two Asymmetrically Located Exits Yashvardhan Singh Bhati and Ashish Bhateja
1 Introduction An effort to improve the understanding of the flow of grains through an opening in a silo is essential from the viewpoint of industrial applications and due to its resemblance with other similar situations, e.g. pedestrian motion from a narrow exit. The flow rate is a crucial parameter in the design of such systems. A wellknown correlation for predicting the mass flow rate of granules exiting through an outlet is due to Beverloo and co-workers [1]. A modification is recently proposed considering cases where jamming can occur [2]. The Beverloo correlation arguably has a simple form, which can be derived from the dimensional analysis [3]. The correlation involves two fitting parameters, the physical origin of which is unclear [3]. Looking for an expression involving parameters with a clear physical meaning, a recent study by Janda et al. [4] in a quasi-two-dimensional silo demonstrated the selfsimilarity of velocity and solid-fraction profiles at the outlet, which can be employed to estimate the flow rate of granular media. Several works subsequently reported the appearance of the self-similarity of flow profiles in varying scenarios [5–11], which includes grains discharging through a planar silo [5, 8, 10], a cylindrical silo [7], and a hopper [9]. A recent study by Bhateja and Jain [10] extended the self-similarity of flow profiles in centric-orifice silos to those with eccentrically located exits. Interestingly, the self-similar nature of velocity and solid-fraction profiles appears in the flow of elliptical grains in a two-dimensional silo [11]. Furthermore, it is worth noting that the self-similarity originally reported for a monodisperse granular assembly holds for a polydisperse system [7, 8, 10] as well. To the best of our knowledge, all such studies employed a single outlet. The existence of self-similar flow profiles remains to be seen for silos having multiple Y. S. Bhati · A. Bhateja (B) School of Mechanical Sciences, Indian Institute of Technology Goa, Ponda, Goa 403401, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_39
419
420
Y. S. Bhati and A. Bhateja
outlets at the base. This work, therefore, aims to explore the self-similarity of velocity profiles at the exit in a two-dimensional silo with two asymmetrically located outlets, utilizing computations based on the discrete element method [12, 13].
2 Methodology A schematic showing various dimensions of the planar silo is shown in Fig. 1. Particles are considered to be dry and non-cohesive discs of mean diameter d. A size polydispersity of ± 10% is taken into account. We utilize the linear spring-dashpot force scheme [14, 15] to model both particle–particle and wall-particle interactions. The computations are carried out using LAMMPS [16]. The quantities of interest presented here are made dimensionless with respect to the particle diameter d, density ρ, and gravitational acceleration g. Using periodic boundary conditions along y direction, the particles crossing the depth of h = 6 below the outlets re-enter the silo at the top (not shown for brevity), thereby maintaining the fill height H in the steady state nearly equal to its value before the onset of flow. The silo width is W = 100, and the fill height is H ≃ 165. Number of grains corresponding to this fill height is N = 16,500. The spring stiffness for normal and tangential interactions, respectively, are k n = 106 and k t = 2/7 k n . The normal and tangential damping coefficients are γ n = 226.124 and γ t = 0, and the friction coefficient is μ = 0.3. The parameters are same for particle–particle and wall-particle interactions. The time step for numerical integration is ∆t = 10−4 , and simulations are run for 7 million time steps. The data shown here are averaged over 50,000 snapshots in the steady state, with each snapshot recorded in an interval of 100 time steps. The macroscopic flow fields at the exit are generated by following a coarsegraining technique [17, 18] with the Heaviside step function. The mean velocity vector, v, having components vx and vy along the horizontal (x) and vertical (y) directions, respectively, at a point p having a position vector xp is given by
Fig. 1 A schematic illustrating granular flow in a two-dimensional silo with two orifices at the base. The coordinate axes and direction of gravity are also appropriately shown
Self-Similar Velocity Profiles in Granular Flow in a Silo with Two …
421
vx =
np 1 ∑ ci x , n p i=1
(1)
vy =
np 1 ∑ ci y , n p i=1
(2)
where cix and ciy are, respectively, the components of the instantaneous velocity vector ci of particle i along x and y directions, and np is the number of particles located at or within w/2 distance from the point p, fulfilling the following conditions | | |xi − x p | ≤ w/2,
(3)
| | | yi − y p | ≤ w/2.
(4)
and
Here, (x i , yi ) and (x p , yp ) are the coordinates of particle i and point p, respectively. Note that w is the coarse-graining width. In this study, we take w = d.
3 Results and Discussion Here, as illustrated in Fig. 1, we examine granular flow through two exits wherein one orifice is kept stationary at the centre, and another is displaced to the right of the fixed one. The former and latter are labeled as central and right orifice, with sizes Dc and Dr , respectively. Each outlet size ranges between 9 and 15, with an interval of one mean particle diameter. The minimum size of both outlets ensures a continuous discharge without jamming at the outlet [19, 20]. The inter-orifice distance L measures the distance between the nearest corners of the apertures and varies from 1 to 15 in unit increments. Let us now present the variation of mass flow rate m ˙ with L from both outlets in ˙ for two pairs of Dc and Dr , with each Fig. 2. Without loss of generality, we display m set having the extreme values, i.e. Dc = Dr = 9 and Dc = Dr = 15. As displayed in Fig. 2a, the mass flow rate from both outlets is nearly the same for the smallest outlet size. A slight difference is, however, obtained between both outlets with the maximum size, as displayed in Fig. 2b. Furthermore, one commonality observed in both cases is the reduction in m ˙ with L, in line with earlier observations [21]. Recall, as stated in the Introduction, Janda et al. [4] presented the estimation of mass flow rate using the self-similar profiles of vertical velocity and solid fraction at the exit. Here, we examine the self-similar feature of velocity profiles at both outlets, enabling us to extend the observations in a single-orifice system to its multi-outlet counterpart. To this end, we increase the size of one outlet from 9 to 15, keeping that of the neighbouring one constant, for a given inter-orifice distance L. To keep the
422
Y. S. Bhati and A. Bhateja
Fig. 2 Variation of the mass flow rate m ˙ with inter-orifice distance L for a Dc = Dr = 9, and b Dc = Dr = 15
analysis simple and without loss of generality, we show data for two scenarios for L = 1. In the first case, the size of the central orifice is fixed at Dc = 15, whereas the width of the right outlet is kept constant at Dr = 15 in the second scenario. Note that qualitatively similar results are obtained for other such combinations. Figures 3a and 3b present the profiles of vertical velocity vy at the central and right outlets, respectively. Note that the magnitude of vy is plotted. At the outset, as expected, we observe that the velocity increases at a given location within the outlet as its size grows. Furthermore, the profiles are not symmetric about the orifice centre, which is anticipated because of the asymmetric arrangement of the outlets. Next, let us plot the profiles of scaled vertical velocity for all outlet sizes. Figures 3c and 3d present the variation of scaled vertical velocity vy = vy /(gDc )1/2 and vˆ y = vy / (gDr )1/2 with x. The former (vy ) and latter (ˆvy ) denote the scaled velocity at the central and right outlets, respectively. Here, x = 2(x-e)/D, and e denotes the eccentricity of an outlet when measured with respect to the centre of the silo base, which is zero and (Dc + Dr + 2L)/2 for the central and right outlets, respectively. Note that x = − 1 and 1 denote the left and right corners of an outlet, and x = 0 corresponds to its centre. As shown in Figs. 3c and 3d, the data for both cases, vy and vˆ y , collapse separately onto a single curve, demonstrating the self-similarity of velocity profiles. This observation appears non-trivial as for low inter-orifice distances the flow in the vicinity of an orifice is affected by the presence of its neighbour [20, 22].
4 Conclusions In summary, we explored the flow of dry and cohesionless granular media under the influence of gravity in a planar silo with two asymmetrically located outlets. We have two observations, which are stated as follows. First, the difference in the mass flow rate from each outlet is minor in the range of inter-orifice distances considered. Second, the exit velocity scales with the square root of the outlet size for both exits
Self-Similar Velocity Profiles in Granular Flow in a Silo with Two …
423
Fig. 3 Profiles of the vertical velocity vy at a central and b right outlets. Variation of the scaled velocity c vy = vy /(gDc )1/2 and d vˆ y = vy /(gDr )1/2 with x. The data are presented for the inter-orifice distance L = 1. Legends for (a) and (b) are provided in (c) and (d), respectively
together, thereby displaying the self-similar feature of velocity profiles. This observation aligns with previously reported works, especially with a recent one focussing on flow through eccentrically located exits [10]. Acknowledgements Financial support of SERB, Government of India (Grant No. SRG/2019/ 000891 and CRG/2019/003989) is gratefully acknowledged.
References 1. Beverloo WA, Leniger HA, van de Velde J (1961) The flow of granular solids through orifices. Chem Eng Sci 15(3):260–269 2. Mankoc C, Janda A, Arevalo R, Pastor JM, Zuriguel I, Garcimart´ın A, Maza D (2007) The flow rate of granular materials through an orifice. Granular Matter 9(6):407–414 3. Nedderman RM (1992) Statics and kinematics of granular materials. Cambridge University Press 4. Janda A, Zuriguel I, Maza D (2012) Flow rate of particles through apertures obtained from self-similar density and velocity profiles. Phys Rev Lett 108(24):248001
424
Y. S. Bhati and A. Bhateja
5. Zhou Y, Ruyer P, Aussillous P (2015) Discharge flow of a bidisperse granular media from a silo: discrete particle simulations. Phys Rev E 92(6):062204 6. Madrid M, Asencio K, Maza D (2017) Silo discharge of binary granular mixtures. Phys Rev E 96(2):022904 7. Rubio-Largo SM, Maza D, Hidalgo RC (2017) Large-scale numerical simulations of polydisperse particle flow in a silo. Comput Part Mech 4(4):419–427 8. Bhateja A (2020) Velocity scaling in the region of orifice influence in silo draining under gravity. Phys Rev E 102(4):042904 9. Darias JR, Gella D, Fernndez ME, Zuriguel I, Maza D (2020) The hopper angle role on the velocity and solid-fraction profiles at the outlet of silos. Powder Technol 366:488–496 10. Bhateja A, Jain S (2022) Self-similar velocity and solid fraction profiles in silos with eccentrically located outlets. Phys Fluids 34(4):043306 11. Qingqing G, Yuchao C, Chuang Z (2022) Self-similarity of density and velocity profiles in a 2D hopper flow of elliptical particles: discrete element simulation. Powder Technol 402:117338 12. Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Geotechnique 29(1):47–65 13. Mishra BK (2003) A review of computer simulation of tumbling mills by the discrete element method: Part I—contact mechanics. Int J Miner Process 71:73–93 14. Zhang D, Whiten WJ (1996) The calculation of contact forces between particles using spring and damping models. Powder Techol. 88(1):59–64 15. Shäfer J, Dippel S, Wolf DE (1996) Force schemes in simulations of granular materials. J Phys I 6(1):5–20 16. Thompson AP, Aktulga HM, Berger R, Bolintineanu DS, Brown WM, Crozier PS, in ’t Veld PJ, Kohlmeyer A, Moore SG, Nguyen TD, Shan R, Stevens MJ, Tranchida J, Trott C, Plimpton SJ (2022) LAMMPS—a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. Comp Phys Comm 271:108171 17. Goldhirsch I (2010) Stress, stress asymmetry and couple stress: from discrete particles to continuous fields. Granular Matter 12(3):239–252 18. Weinhart T, Hartkamp R, Thornton AR, Luding S (2013) Coarse-grained local and objective continuum description of three-dimensional granular flows down an inclined surface. Phys Fluids 25(7):070605 19. Janda A, Zuriguel I, Garcimartín A, Pugnaloni LA, Maza D (2008) Jamming and critical outlet size in the discharge of a two-dimensional silo. Europhys Lett 84(4):44002 20. Kondic L (2014) Simulations of 2D hopper flow. Granular Matter 16(2):235–242 21. Maiti R, Das G, Das PK (2017) Granular drainage from a quasi-2D rectangular silo through two orifices symmetrically and asymmetrically placed at the bottom. Phys Fluids 29(10):103303 22. Zhang X, Zhang S, Yang G, Lin P, Tian Y, Wan JF, Yang L (2016) Investigation of flow rate in a quasi 2D hopper with two symmetric outlets. Phys Lett A 380(13):1301–1305
Droplet Impact and Spreading Around the Right Circular Cone: A Numerical Approach Prakasha Chandra Sahoo, Jnana Ranjan Senapati, and Basanta Kumar Rana
Abbreviations Nomenclature D0 Dc L h l DCL t F st UO Dc /DO We D/DO Re
Droplet initial diameter (mm) Base diameter of the cone (mm) Height of the cone (mm) Height of the rectangular domain (mm) Width of the rectangular domain (mm) Spreading diameter at the contact line (mm) Time (ms) Volumetric surface tension force (N/m3 ) Initial droplet velocity(m/s) Cone base-to-droplet diameter ratio Weber number Maximum deformation factor Reynolds number
P. C. Sahoo (B) · J. R. Senapati Department of Mechanical Engineering, National Institute of Technology, Rourkela 769008, India e-mail: [email protected] B. K. Rana School of Mechanical Engineering, KIIT Deemed to Be University, Bhubaneswar 751024, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_40
425
426
P. C. Sahoo et al.
Greek Symbols α ρ μ k n σ g θ
Volume fraction of each phase (No unit) Density (Kg/m3 ) Dynamic viscosity (Kg/m.s) Curvature of the interface Unit vector at the interface (No unit) Surface tension (mN/m) Acceleration due to gravity (m/s2 ) Contact angle (0 )
Subscripts liq gas
Liquid Gaseous
1 Introduction The flow behavior and features of a water droplet impinging on different plane surfaces are very common throughout many engineering processes as well as industrial applications like as fuel particles injecting inside an IC engine, gas turbines cooling, casting procedure, ink-jet photocopying, chemical, and nuclear reactor core for avoiding surface overheating. Over the past years, droplet impingement on the various geometrical surfaces, like flat, curved, and irregular substrate has been investigated by experimental procedure, numerical methods, and theoretical modelling. Many researchers have recommended studying the evolution of numerous steps such as free fall, impingement, hanging, and disjoining of droplets from solid targets, as well as the continuous spread rate on the substrate. Various factors, such as solid– liquid angle of contact, impingement velocity of droplet, fluid physical properties, and target-to-droplet diameter ratio will be used to calculate the optimum deformation factor.
2 Literature Review and Objective Rein [1] studied both experimental and theoretical of a liquid droplet impacting on liquid or solid substrates, elucidating various processes such as rebounding and advancing, splattering of a drop of water on solid objects, and even though rebounding, coalescence, and splattering of a water droplet happens on the surface
Droplet Impact and Spreading Around the Right Circular Cone …
427
of the liquid. Lorenceau et al. [2] planned to conduct experiments to evaluate the dynamic behaviour of droplet strikes as well as the liquid capture across thin fibres. They discovered that at higher critical speed, the water droplet is detached from the fibre, but at lower threshold velocity, it is totally trapped by the fibre. Sher et al. [3] examined experimentally a liquid droplet clamped on a relatively thin wire, and they demonstrated the growing area of the droplets across the wire with different parameters such as velocity of impaction, volume of water droplet, and liquid surface or interfacial tension. The optimum spreading and wettability of a water droplet striking on a smooth flat substrate was studied on experimental and numerical methods by Fukai et al. [4]. It had been seen that the spreading factor increases and the maximum splat radius decreases as the impacting speed of a droplet increases and the advancing contact angle increases. Zhang et al. [5] was experimentally considered the greatest spreading factor and impingement of a liquid droplet onto the solid target with the help of CLSVOF method. They discovered that the travelling speed is larger on a wetting surface than on anti-wetting (hydrophobic) surface, and that recoiling occurs when droplet travelling is greatest, close to the surface. A liquid droplet impingement on the tiny spherical substrate was studied on both experimental and theoretical methods by Bakshi et al. [6]. They observed that the deformation factor differed with droplet impingement speed and smooth solid (target-to-droplet) diameter ratio. Droplet impingement on an irregular curved surface with the help of lattice Boltzmann method (LBM) was numerically investigated by Shen et al. [7]. It had been observed that at higher Weber number, the droplet spreading and also splashing behaviour increases on to the surface. A Newtonian liquid droplet impingement on a short fibre by adopting volume of fluid technique was numerically studied by Khalil et al. [8]. They established that when a droplet’s impingement velocity increases, the threshold radius on the substrate lowers. Pasandideh et al. [9] investigated numerically and finally compared them with experimental results of a liquid water droplet impingements and their spreading phenomena on the flat surface. It had been seen that at varying interfacial angle and liquid adhesion of water phase, the encouragement and promotion of a droplet vary across a broad range on a steel surface and that a mathematical co-relationship for estimating the greatest deformation factor was established. Droplet impingements on the spherical target were investigated by experimental, numerical and compared with theoretical results by Liu et al. [10]. They had shown with various parameters such as surface wettability, Weber number (We) and sphere-to-droplet diameter ratio were effect on the maximum deformation factor. The dynamics of droplets impingement on cylindrical super hydrophobic surfaces target was numerically examined by Zhang et al. [11] and compared with their theoretical outcomes. It had been observed that at higher Weber number and geometrical factor like aspect ratio, the maximum deformation factor increases and finally given a mathematical co-relation for determining maximal deformation factor as measured by geometrical aspect ratio, surface wettability, and Weber number. According to the above literature study, experiments and theoretical investigations have been conducted on droplets impinging on solid objects and derived distinct dynamics behaviours such as spreading, rebound, and coalescence. There has been observed that relatively few simulation studies on the droplet impinging process
428
P. C. Sahoo et al.
have been conducted. A droplet impact and subsequent penetration into the right circular conical target are to be solved numerically with the help of VOF method. The separation of droplets from the base of a solid target is not to be discussed earlier; it is to be clearly explained in the current work. The unsteady volume fraction contour of a droplet impingement, deformation and separation of a liquid droplet from the substrate of each stage during the impinging process, and maximal deformation factor on the spreading area were calculated in order to understand the physical significance of fluid flow behavior.
3 Materials and Methods 3.1 The Problem as a Physical Description The droplet impingement on the conical target and analyzing the dynamic behaviour on and around the surface are described in the present work. For knowing details about the maximum deformation factor, we have taken different non-dimensional parameters like Weber number (We), contact angle (θ ), and cone base-to-droplet diameter ratio (Dc /D0 ). Figure 1 shows a rectangular computational domain (l * h) where a liquid droplet is placed on the right circular conical target. The material of the conical surface is to be taken as copper and dimensions of the geometrical target are where (Dc = diameter of the conical base) and (L = length of the target). The cone base-to-droplet diameter ratios vary from 0.5 to 1 with varying droplet diameters from (D0 = 1.5–3 mm) at constant value of base diameter in whole problem. Figure 2 presents the mesh arrangement on right half of the geometry and also the boundary conditions are imposed on all the sides of the computational domain.
3.2 Governing Equations The present problem of a liquid droplet impingement on the right circular conical target is to be solved with the help of volume of fluid technique. To calculate the temporal value of volume fraction and greatest deformation factor of a droplet impinging on the surface, it is more important to solve the conservation equations for both mass as well as momentum in each phase. And also one more transport equation, i.e. the volume fraction equation is integrated while solving the problem numerically. The governing differential equations consisting volume fraction of each phase can be written as follows. ⎧ mesh has no ith phase, ⎨ 0, αi = 0 < αi < 1, mesh holds ith phase and other phases, ⎩ 1, mesh holds by ith phase
(1)
Droplet Impact and Spreading Around the Right Circular Cone …
429
Fig. 1 Representative diagram of computational domain
Fig. 2 Cells with a specific structure condition of the boundary
And it satisfied ∑
αi = 1.
(2)
There are several ways that it can be written that indicate how much liquid (αliq ) there is in each cell of the impinging process for a two-phase flow.
430
P. C. Sahoo et al.
αliq
⎧ mesh obtains gas ⎨ 0, = 0 < αliq < 1, mesh obtains gas and liquid ⎩ 1, mesh obtains liquid αliq + αgas = 1
(3)
(4)
The conservation of mass of the liquid phase can be expressed as ) ( ) ∂( αliq ρliq + ∇. αliq ρliq u = 0. ∂t
(5)
The conservation of momentum of the fluid flow can be expressed as: ] ( ∂ (ρu) + ∇.(ρuu) = −∇ P + ∇.μ[∇u + ∇u)T + ρg + F st . ∂t
(6)
Here, Fst = interfacial surface tension force per unit volume acting on gas–liquid interface, calculated by the continuum surface force (CSF) model12 that is, ρk∇αliq ) , F st = σ ( ρliq + ρgas /2
(7)
where k = curvature of radius at the interface. So, it can be calculated from divergence of the unit normal vector (n) at the interface that is: k = ∇.n,
(8)
where n = gradient of the volume fraction at the interface, can be written as, ∇αliq |. n= | |∇αliq |
(9)
In Eqs. (5) and (6), the fluid properties, such as density (ρ) and viscosity (μ) can be rewritten as: ) ( ρ = αliq ρliq + 1 − αliq ρgas ,
(10)
) ( μ = αliq μliq + 1 − αliq μgas .
(11)
Droplet Impact and Spreading Around the Right Circular Cone …
431
3.3 Boundary Conditions The boundary conditions for determining aforementioned governing differential equations must be taken into account in the current work. We applied no-slip and also no-penetration boundary conditions to the conical target’s base and lateral surfaces, while pressure outlet conditions have been imposed in rectangular computational domain. The meshes are enhanced indigenously, and very fine grids are to be taken near the interface, as shown in Fig. 2. The top side, bottom side, and left side wall of the computational domain are defined as follows: p = p∞ .
(12)
On the surfaces (base and lateral surface) of cone: u = 0, v = 0.
(13)
3.4 Numerical Method The volume of fluid (VOF) method and finite volume-based solver are to be used for solving the present computational work. It is a stationary cell method in which the interface between two non-miscible fluids is formed as an inconsistency in the form of a volume fraction. In the impinging process, the VOF approach is used to determine the mass conservation and also momentum transport of each phase. Here, there are two fluid phases like gas and liquid phases: Newtonian fluids are considered to be incompressible, have constant viscosity, and interfacial tension and the fluid is taken as laminar-flow. Brackbill et al. [12] established a CSF model, where we can compute the most important parameters like surface forces and surface adhesion easily by using this model. The model equations are explained clearly in sub-Sect. 3.2. In the current work, the computational simulations are solved with the help of computational fluid dynamics (CFD) software ANSYS FLUENT 18.1. The PRESTO, second-order upwind, and geo-reconstruct schemes are utilized to discretize the pressure, momentum, and volume fraction in the governing equation. The Semi Implicit Pressure Linked Equation (SIMPLE) method is used for solving the pressure–velocity coupling equation. Herein, the time discretization is the firstorder implicit and all the residuals and Courant number values are taken as 10–4 and 0.25, respectively. The time step changes within the value of 10–6 s and 10−4 s.
432
P. C. Sahoo et al.
Fig. 3 Maximum deformation factor varies with grid cell number
3.5 Grid Independence Test The mesh arrangement with corresponding boundary conditions in the grid layout diagram is displayed in Fig. 2. The cells are structured, and higher density cell lines are created along the path of droplet passage near the target’s spreading area. The grid independence test for computational work of the present problem, where droplet impingements on the apex of the conical target at angle of contact = 30°, aspect ratio, i.e. cone base-to-droplet diameter ratio = 0.5, and impingement velocity of droplet (V = 1 m/s) is shown in Fig. 3. The maximum deformation factors increase steadily as the number of mesh sizes increases, but after a specific extent, the greatest deformation factor gives a constant value after raising the size of cells. We identified that the maximum deformation factor does not change after 52,274 cells. As a result, the rectangular computational domain with 52,274 cells is designated as a gridindependent domain.
3.6 Validation of Current Computational Scheme The current numerical model is to be validated with the available experimental work done by Yun and Kim [13]. This work is performed well. The progress of both continuing to spread and deformation factors, when droplet collides with a flat solid surface is studied by both experimental and computational as displayed in Fig. 4a, b. In the computational work, the highest spreading factor at the interface region gives 1.69 but in experimental research, it appears 1.62, so the percentage change between them is 4.32%. The deformation factor gets 1.91 during the numerical procedure, while in the experimental investigations it is 1.81. Therefore, the percentage variation is 5.52%. The above two plots display that the present computational work is admitted well with earlier existing experimental results.
Droplet Impact and Spreading Around the Right Circular Cone …
433
Fig. 4 a and b Quantitative variation of present numerical work and experimental work of Yun and Kim [13]
(a) Spreading factor vs. time
(b) Deformation factor vs. time
4 Results and Discussion 4.1 Profile of Droplets Over Time: A Temporal Analysis The simulation results of droplet impingement on the conical solid object are discussed in the current work. The contour of a water droplet continuing to spread and hence its subsequent stages of temporal changes along the axial direction of the conical target are as shown in Figs. 5, 6, and 7. Here, we used a stationary cone base diameter (Dc = 1.5 mm) with varying droplet sizes (D0 = 1.5 mm, 2 mm, and 3 mm) which changes the cone base-to-droplet diameter ratio and also carried a constant value of contact angle (θ = 300 ) and impingement velocity (V = 1 m/s) of a droplet for the current analysis. Free fall, spread, encapsulation, and detachment of a droplet from the target are observed during impinging process. The approach stage is defined as the time when the droplet just touches the apex of the conical surface and has shown a round shape structure (t1 = 0 ms). Then after, it enters the right circular cone, the droplet spreads maximum on the apex of the target in each cases of the aspect ratio, the reason behind that is the inertia force is more when compared
434
P. C. Sahoo et al.
with surface tension force. On the next encapsulation stage, the droplet is completely enveloped on the cone. So, here in, the droplet lengthens more in the axial compared with the radial direction even though surface tension forces greatly exceed inertia forces. Again, the time (t4 = 4.4 ms) with cone base-to-droplet diameter ratio = 0.5 and impact velocity = 1 m /s during the oscillation stage before completing the detachment period, the droplet is in pendant shape. Finally, during the detachment stage, the droplet is kept separate from the bottom of the conical target at (t5 = 5.9 ms). By comparing Figs. 5, 6, and 7, it shows that the interaction of time between the droplet approach and detachment period is less and also maximum amount of liquid film adheres at the cone base with higher cone base-to-droplet diameter ratio.
t1 = 0.0 ms
t2 = 1.7 ms
t3 = 2.7 ms
t4 = 3.7 ms
t5 = 4.7 ms
Fig. 5 Subsequent stages of droplet impingement on cone base-to-droplet diameter ratio = 1, V = 1 m/s
t1 = 0.0 ms
t2 = 2 ms
t3 = 3.0 ms
t4 = 4 ms
t5 = 5 ms
Fig. 6 Subsequent stages of droplet impingement on cone base-to-droplet diameter ratio = 0.75, V = 1 m/s
Droplet Impact and Spreading Around the Right Circular Cone …
t1 = 0.0 ms
t2 = 2.4 ms
t3= 3.4 ms
t4 = 4.4 ms
435
t5 = 5.9 ms
Fig. 7 Subsequent stages of droplet impingement on cone base-to-droplet diameter ratio = 0.5, V = 1 m/s
4.2 Weber Number and Contact Angle are Affected by the Deformation Factor The greatest deformation factor varies with the Weber number (We) and contact angle values as depicted in Fig. 8. With increasing in impingement velocity or Weber number of a droplet for a particular value of geometrical aspect ratio such as cone base-to-droplet diameter ratio and angle between the interfacial tangential forces, it shows the deformation factor increases. And also the deformation factor gives higher value on the solid surface at lower contact angles and higher initial impingement velocity. Figure 9 displays the variation of greatest deformation factor is to be analyzed with different values of contact angles at the interfacial line for a constant Weber number. At contact angle (θ = 30°), it indicates that the deformation factor is more due to low interfacial surface tension force than the higher contact angle (θ = 150°). As a result, as the contact angle raises, so does the deformation factor. The target Fig. 8 Deformation factor distribution with Weber number
436
P. C. Sahoo et al.
Fig. 9 Deformation factor distribution with contact angle
surface is more hydrating at lower contact angles called hydrophilic and in contrast, less wetting on the solid surface at higher contact angle is known as hydrophobic.
5 Conclusions The impingement of a liquid droplet on the right circular cone is investigated computationally during this research using the VOF method. The various parameters, like on cone base-to-droplet diameter ratio, contact angle between the tangents of two surfaces, and impingement velocity of a liquid droplet are affected on the greatest deformation factor. The following conclusions could be drawn as a result: 1. The computational work provides several stages, namely (approach, impingement, and deformation on apex of the target, encapsulation and detachment from the base of target) during the whole impinging process. Their detailed hydrodynamic behaviours and temporal evolution of volume fraction contours are described in detailed in each process. 2. A constant relationship between cone base-to-droplet diameter ratio and greatest deformation factor has been developed. 3. Droplets approach deform more quickly on a fixed Weber number surface when the cone base-to-droplet diameter ratio is greater. With decrease in the cone baseto-droplet diameter ratio, it deforms maximum on the solid surface and delays on the side as well as detaches from the bottom of the cone base.
References 1. Rein M (1993) Phenomena of liquid drop impact on solid and liquid surfaces. Fluid Dyn Res 12(2):61–93
Droplet Impact and Spreading Around the Right Circular Cone …
437
2. Lorenceau É, Clanet C, Quéré D (2004) Capturing drops with a thin fiber. J Colloid Interface Sci 279(1):192–197 3. Sher E, Haim LF, Sher I (2013) Off-centered impact of water droplets on a thin horizontal wire. Int J Multiph Flow 54:55–60 4. Fukai J, Shiiba Y, Yamamoto T, Miyatake O, Poulikakos D, Megaridis CM, Zhao Z (1995) Wetting effects on the spreading of a liquid droplet colliding with a flat surface: experiment and modeling. Phys Fluids 7(2):236–247 5. Zhang YJ, Li P (2016) Spreading simulation of droplet impact on solid surface with CLSVOF method and its experimental verification. Braz J Phys 46(2):220–224 6. Bakshi S, Roisman IV, Tropea C (2007) Investigations on the impact of a drop onto a small spherical target. Phys Fluids 19(3) 7. Shen S, Bi F, Guo Y (2012) Simulation of droplets impact on curved surfaces with Lattice Boltzmann method. Int J Heat Mass Transf 55(23–24):6938–6943 8. Khalili M, Yahyazadeh H, Gorji-Bandpy M, Ganji DD (2016) Application of volume of fluid method for simulation of a droplet impacting a fiber. Propuls Power Res 5(2):123–133 9. Rozhkov A, Prunet-Foch B, Vignes-Adler M (2002) Impact of water drops on small targets. Phys Fluids 14(10):3485–3501 10. Banitabaei SA, Amirfazli A (2017) Droplet impact onto a solid sphere: effect of wettability and impact velocity. Phys Fluids 29(6) 11. Arogeti M, Sher E, Bar-Kohany T (2019) Drop impact on small targets with different targetto-drop diameters ratio. Chem Eng Sci 193:89–101 12. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. J Comput Phys 100(2):335–354 13. Yun S, Kim I (2019) Spreading dynamics and the residence time of ellipsoidal drops on a solid surface. Langmuir
Effect of Direct Current Electrowetting on Dielectric on Droplet Impingement Dynamics K. Niju Mohammed, A. Shebin, E. Mohammed Haseeb, P. S. Tide, Franklin R. John, Ranjith S. Kumar, and S. S. Sreejakumari
Nomenclature θ θ0 εr ε0 d γ LV V We v
Final contact angle (°) Initial contact angle (°) Relative permittivity Permittivity of vacuum (F/m) Dielectric thickness (μm) Surface tension (N/m) Voltage (V) Weber number Characteristic velocity (m/s)
1 Introduction For a few decades, a considerable growth has been witnessed in the field of microfluidics relating to engineering and medical applications [1]. Surface force is more dominant than pressure and body forces at microscale level due to the greater surface area K. Niju Mohammed · P. S. Tide · F. R. John Department of Mechanical Engineering, SchoolofEngineering, Cochin University of Science and Technology, Cochin, Kerala, India K. Niju Mohammed · A. Shebin (B) · E. Mohammed Haseeb · R. S. Kumar Micro/Nanofluidics Research Laboratory, CollegeofEngineering, Trivandrum, Kerala 695016, India e-mail: [email protected] S. S. Sreejakumari Materials Science and Technology Division, CSIR NIIST, Thiruvananthapuram, Kerala 695019, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_41
439
440
K. Niju Mohammed et al.
Fig. 1 Schematic of electrowetting on dielectric
to volume ratio [2]. As a result, the manipulation of liquid droplets on the microscale is a tedious task. On the microscale, droplets can be conveniently controlled by creating gradients of parameters like chemical affinity, thermo-capillarity, sound wave, etc. But due to low cost, fine controllability, high rate of switching, and precision, electrostatic actuation draws attention. Electrowetting and dielectric are the two main mechanisms involved in electrostatic actuation [3, 4]. Electrowetting is a physical phenomenon of changing droplet shape by the application of voltage between droplet and surface it is placed. This phenomenon is obvious when we employ electrode surfaces covered with dielectric substances. Figure 1 depicts a typical arrangement of EWOD. The multi-electrode system is also used for manipulating droplet motion [5]. The contact angle decreases when a voltage is applied. In recent years, the concept of digital microfluidics (DMF) [6] has attracted many researchers in various fields. In DMF, discrete droplets are used instead of continuous flow. It is being employed in a wide spectrum of applications in many scientific domains, from engineering to the biological sciences, including variable focus lenses [7], display technologies [8], fiber optics [9], and lab-on-a-chip applications [10]. EWOD can be used as a platform for performing basic operations of DMF such as generating, splitting, merging, and transporting droplets [11]. Among many other applications, EWOD can be used in hot spot cooling for more efficient and targetspecific thermal management of electronics [12]. In this study, experimental analysis of droplet impingement on the EWOD surface has been carried out. Here, a droplet is allowed to free fall onto a hydrophobic surface and analyze the interaction between the surfaces. The water contact angle (WCA) of the substrate (145°) is measured by using a goniometer. The hydrophobic test specimen is prepared using a spray coater and a droplet impingement experiment is done. We conducted a parametric study by varying the droplet diameter, droplet impingement height, and applied voltage.
Effect of Direct Current Electrowetting on Dielectric on Droplet …
441
Fig. 2 Illustrative representation of experimental setup
2 Methodology Figure 2 provides an illustrative representation of the experimental setup. The deionized (DI) water droplets were generated from the water reservoir attached with a flat-headed needle. By regulating the valve on the reservoir, the droplet is formed on the tip of the needle. When the gravitational force exceeds the adhesive and cohesive forces, the droplet fell from the needle. The EWOD chip (1'' × 3'' ) is composed of a copper-clad substrate with a gap of 1/64 inch (0.396 mm) at the middle. On this, a PDMS-Silica nanocomposite was spray-coated using a 1 mm diameter nozzle [13, 14]. It acts as both hydrophobic as well as dielectric coating. The spray-coated substrate was kept at room temperature for curing for at least 24 h. To record the droplet impingement, a high-speed camera was employed at a frame rate of 4500 frames per second. The camera was regulated by the Hi-Spec Camera software, to save the images of the droplet impingement process. A high-voltage DC source is used to give the required electric signal. The temperature for the experimental condition is set to 26 °C. The terminals of the high-voltage DC source were associated to electrodes by a wire. Initially, the droplet impingement without applying direct current to the EWOD chip was recorded as the reference case. In order to study the droplet impact phenomena, various DC voltages were subsequently applied to the chip. To verify the accuracy and repeatability of the experimental data, each experiment was carried out thrice for each DC voltage.
3 Results and Discussion A reference droplet impingement experiment was conducted at 0 V to understand the droplet behavior at different Weber numbers. The Weber number indicates the ratio of the inertial force to the surface tension force. The Weber number We = ρν 2 D0 /σ,
442
K. Niju Mohammed et al.
Fig. 3 Regimes of droplet impingement on EWOD at 0V
where ρ is the fluid density (kg/m3 ) and v is the characteristic velocity (m/s), D0 is the diameter of droplet (m), and σ is the surface tension of the fluid (N/m). Figure 3 displays bouncing phenomena from We = 0 to We = 15.48, jetting till We = 77.4, and the remaining is splashing. The droplet diameter was 2.76 mm and the contact angle of the PDMS-Silica nanocomposite coat is 145°. From this reference experimental data, we selected We = 20.02 for the study. The impingement velocity was 0.73 m/s. The jetting phenomenon occurred during the droplet impingement on the virgin hydrophobic substrate.
3.1 Dynamics of Droplet Impingement on EWOD Chip Without Voltage The spreading and retraction of the impingement of the droplet were noted with the aid of the high-speed camera as shown in Fig. 4. Just before collision, the droplet maintained an almost spherical shape. When impinged on the substrate, initially the droplet spreads out due to the inertial force. During this period, the droplet area gradually increases to reach maximum value at 3.8 ms, mimicking a “round pudding” shape. When the surface tension force exceeds the inertial force, the retraction stage begins. Here spreading diameter of the droplet is reduced gradually and attained its peak height at 14.7 ms. During retraction, necking is observed. As shown in Fig. 4, jetting occurred at 16 ms. Necking results from a conflict between a capillary retraction that takes place when the bulbous end increases, owing to surface tension and a pressure-driven flow from the jet that feeds the size of the top droplet. As the jet is stretched thinner due to inertial forces in the axial direction, the neck thins and eventually ruptures which ejects the satellite droplet upward due to the abrupt disconnection from the main body of the jet [15]. As shown in Fig. 4, the small droplet stayed in the air and the large droplets clung to the surface, yielding an elongation.
Effect of Direct Current Electrowetting on Dielectric on Droplet …
443
Fig. 4 Droplet impingement process at We = 20.02 on EWOD chip without voltage
During jetting, the small droplet stays in the air and the large droplet gets adhered to the substrate leading to elongation at 17.6 ms. It again breaks into large and small droplets. Here small droplets adhere to the surface. The large droplet hung on the air until its energy is being consumed, and then rests on the hydrophobic substrate. This is in line with the findings reported in [16–19].
3.2 Dynamics of Droplet Impingement on EWOD Chips with Voltage When DC voltages are applied on the EWOD chip, the droplet impingement same as that of the reference case (0 V, We = 20.02) was recorded and is shown in Fig. 5. For a DC voltage of 250 V applied on EWOD, from 0 to 3.8 ms, the initial droplet
444
K. Niju Mohammed et al.
spreading dynamics upon impact as shown in Fig. 5 was similar compared with the scenario without voltages shown in Fig. 4. When the maximum area is reached by the droplet, the spreading diameter of the droplet upon voltage application was larger than that in the no voltage case. The spreading droplet shape was not as smooth and flat compared with the reference case due to the electrostatic force produced by the EWOD. At 3.9 ms to 4.1 ms, the droplet made varying contact angles with the left and right sides. This contact angle variation is induced by a Laplacian pressure difference generated in the droplet which resulted in the radial shift of the droplet to occur. At 15 ms time frame, the droplet entirely shifted to the right electrode resulting in breaking off the electric circuit in the EWOD chip. Eventually, the droplet was not regulated by EWOD anymore and the droplet gets stabilized at 31 ms. In the retraction phase, the droplet dynamics and shape were distinct from the no voltage case depicted in Fig. 5. The water droplet does not separate from EWOD test specimen during the retraction process until it completely shifts to one electrode. During the entire retraction phase, the droplet is adhered to the EWOD chip without jetting and it then gets stabilized. Fig. 5 Droplet impingement process at We = 20.02 on EWOD chip with a DC voltage of 250 V
Effect of Direct Current Electrowetting on Dielectric on Droplet …
445
Fig. 6 Variation in droplet height with voltage over time
3.3 Effect of Voltage on Droplet Height and Diameter on Droplet Impact Dynamics. The inhibition of jetting of the droplet took place by the application of DC voltage to the EWOD as seen in Fig. 5. To understand the effect of voltage on droplet dynamics, the droplet impingement trials were performed at different voltages. The height and diameter of the impacting droplet are the two important parameters for detailing the energy conversion during the impingement process. The kinetic energy was transformed into surface energy and the droplet spreads radially. Figure 6 shows the variation of the height of the droplet at various voltages. The height is defined in this case as the vertical length of the droplet from the EWOD chip to the top of the droplet. At 0 ms, the droplet touched the EWOD chip and its height was nearly the diameter of the droplet (2.76 mm). As shown in Fig. 4, at no voltage condition, the impingement of the droplet undertakes spreading, and the maximum spreading diameter (5.98 mm) was obtained between 3.5 and 4.3 ms. At the same time, it reaches a minimum height (0.928 mm) from the surface. These conditions hold good for a duration of 2.3 ms. In this phase, the droplet velocity is reduced to zero and it attains maximum spreading by converting the kinetic energy of droplet into its surface energy. The minimum height was 0.928 mm and the maximum droplet diameter without voltage was 5.98 mm as given in the Figs. 6 and 7, respectively. Here τ is the nondimensionalized resident time and expressed as τ = D0 /t Vim [20], where Vim is the droplet impact velocity. The maximum spreading phase of the droplet was followed by retraction and jetting, as a result of which the droplet diameter continued to reduce and reaches a minimum and the height becomes maximum. On application of DC voltage to the EWOD, the spreading diameter and height of the droplet shows distinct behavior from the droplet impact without voltage, i.e., the maximum droplet diameter is increased, and
446
K. Niju Mohammed et al.
Fig. 7 Variation in diameter of the droplet with voltage over time
the maximum height gets decreased, e.g., Fig. 7 shows that applying a DC voltage of 300 V results in a maximum droplet diameter of 7.78 mm, which is 30% larger compared with that of the impinging droplet diameter without the application of voltage (5.98 mm), and in the case of height also shows significant changes. A new difference arises in the retraction phase. The diameter of the droplet impinging the DC-EWOD showed some peaks and troughs when the droplet contacted the surface of EWOD, a change in the contact angle occurred following the Young–Lippmann equation. cos θ = cos θ0 +
ε0 εr V2 2γ L V d
(1)
where θ 0 is the initial contact angle, θ is the final contact angle, εr is the relative permittivity, ε0 is the permittivity of vacuum, d is the dielectric thickness, and γ LV is the surface tension between liquid and air. When different values of DC voltages were applied, the contact angle of the droplet decreased according to Eq. (1). Table 1 gives the variation of contact angle with voltages. Table 1 Variation of contact angle with voltages Voltage (V)
Theoretical contact angle (°)
Nature of surface
0
145
Nearly superhydrophobic
150
117
Hydrophobic
200
100
Hydrophobic
250
79
Hydrophilic
300
50
Hydrophilic
Effect of Direct Current Electrowetting on Dielectric on Droplet …
447
When the DC voltage was zero, the contact angle of the impinging droplet was not affected by EWOD. When the DC voltage was applied to the EWOD chip, induced a fluctuation in the diameter of the impinging droplet. As a result, its retraction phase could not proceed normally, causing the droplet to adhere to the hydrophilic surface created due to electrostatic forces and as explained by the Young-Lipmann equation. As the voltage is decreased from 300 to 150 V, the droplet diameter continued to reduce because of the decrease in the intensity of voltage as shown in Fig. 9. This reduction in voltage also changes the contact angle of the impinging droplet. On further reducing the voltages to a low value like 50 V, the diameter of the impinging droplet was similar to the case without voltage. As the DC signal was continuously applied on the EWOD chip, in 3.9 ms, the droplet made a different contact angle on the right and left sides and this varying contact angle resulted in a Laplacian pressure difference across the droplet which resulted in motion in radial direction. At 15 ms, the droplet completely covered the right electrode, and the electric circuit in the EWOD configuration was cut off. As a result, the droplet was not controlled by EWOD and gets stabilized after a partial rebound nearly at 31 ms. Figure 8 shows the variation in the maximum height of the droplet in accordance with different voltages. hmax is the maximum vertical distance from the EWOD chip to the top of the droplet. At low voltage, jetting is observed. As the applied voltage increases hmax decreases and jetting gets inhibited. Figure 9 shows the maximum diameter of droplet spread the droplet. d max increases due to an increase in hydrophilic nature imparted by a rise in voltage. Fig. 8 Variation of h max with voltage
448
K. Niju Mohammed et al.
Fig. 9 Variation of d max with voltage
4 Conclusions In this study, direct current electrowetting on dielectric (DC-EWOD) regulates the motion of a droplet impinging on a hydrophobic substrate. At weber number of 20.02, jetting is inhibited by the application of direct current to the EWOD and the droplet got shifted to one side. On applying the DC voltage on the EWOD chip, the diameter of droplet spreading got increased and the droplet height got decreased during the spreading and retraction processes, respectively. The results showed that the magnitude of the voltage had a dominating effect on jetting inhibition. The results help to provide a reference for future research on applications, including electronic cooling, inkjet printing, and lab-on-chip applications. Acknowledgements The authors would like to acknowledge CSIR-NIIST Pappanamcode for helping with the fabrication of the EWOD chip.
References 1. Lim B, Vavassori P, Sooryakumar R, Kim C (2016) Nano/micro-scale magnetophoretic devices for biomedical applications. J Phys D Appl Phys 50:033002 2. Datta S, Das AK, Das PK (2017) Influence of surface contact angle on uphill motion of droplets due to electrostatic actuation. In: Fluid mechanics and fluid power–contemporary research. Springer, pp 1305–1313 3. Datta S, Kumar P, Das AK (2019) Manipulation of droplets by electrostatic actuation and the related hydrodynamics. J Indian Inst Sci 99(1):121–141 4. Lee J, Moon H, Fowler J, Schoellhammer T, Kim C-J (2002) Electrowetting and electrowettingon-dielectric for microscale liquid handling. Sensors Actuators, A 95(2):259–268. Papers from the Proceedings of the 14th IEEE International conference on microelectromechanical systems
Effect of Direct Current Electrowetting on Dielectric on Droplet …
449
5. Accardo A, Mecarini F, Leoncini M, Brandi F, Di Cola E, Burghammer M, Riekel C, Di Fabrizio E (2013) Fast, active droplet interaction: coalescence and reactive mixing controlled by electrowetting on a superhydrophobic surface. Lab Chip 13:332–335 6. Choi K, Ng AH, Fobel R, Wheeler AR (2012) Digital microfluidics. Annu Rev Anal Chem 5:413–440 7. Doering C, Strassner J, Fouckhardt H (2021) Microdroplet actuation via light line optoelectrowetting (ll-oew). Int J Anal Chem 2021 8. Li J et al (2020) Current commercialization status of electrowetting-on-dielectric (ewod) digital microfluidics. Lab Chip 20(10):1705–1712 9. Vo PQ, Husser MC, Ahmadi F, Sinha H, Shih SC (2017) Image-based feedback and analysis system for digital microfluidics. Lab Chip 17(20):3437–3446 10. Fair RB (2007) Digital microfluidics: is a true lab-on-a-chip possible? Microfluid Nanofluid 3(3):245–281 11. Geng H, Feng J, Stabryla LM, Cho SK (2017) Dielectrowetting manipulation for digital microfluidics: creating, transporting, splitting, and merging of droplets. Lab Chip 17(6):1060– 1068 12. Bindiganavale G, You SM, Moon H (2014) Study of hotspot cooling using electrowetting on dielectric digital microfluidic system. In: 2014 IEEE 27th International conference on micro electro mechanical systems (MEMS). IEEE, pp 1039–1042 13. Bharathidasan T, Narayanan TN, Sathyanaryanan S, Sreejakumari S (2015) Above 170° water contact angle and oleophobicity of fluorinated graphene oxide based transparent polymeric films. Carbon 84:207–213 14. Basu BJ, Bharathidasan T, Anandan C (2013) Superhydrophobic oleophobic PDMS-silica nanocomposite coating. Surf Innov 1(1):40–51 15. Asai B (2022) Outcomes following droplet impact on a hydrophilic substrate: spreading, jetting, and partial rebound. PhD Thesis, Washington State University 16. Tan J, Wang H, Sun M, Tian P, Wang Y, Wang K, Jiang D (2021) Regulating droplet impact on a solid hydrophobic surface through alternating current electrowetting-on-dielectric. Phys Fluids 33(4):042101 17. Nandakumar Chandran K, Naveen P, Abhilash R, Kumar Ranjith S (2021) Theoretical modelling of droplet extension on hydrophobic surfaces. Int J Comput Fluid Dyn 35(7):534–548 18. Rajesh RS, Naveen P, Krishnakumar K, Ranjith SK (2019) Dynamics of single droplet impact on cylindrically-curved super-heated surfaces. Exp Thermal Fluid Sci 101:251–262 19. Ulahannan L, Krishnakumar K, Nair AR, Ranjith SK (2021) An experimental study on the effect of nanoparticle shape on the dynamics of Leidenfrost droplet impingement. Exp Comput Multiphase Flow 3(1):47–58 20. Chen R-H, Chiu S-L, Lin T-H (2007) Resident time of a compound drop impinging on a hot surface. Appl Therm Eng 27(11–12):2079–2085
Evolution of Ferrofluid Droplet Deformation Under Magnetic Field in a Uniaxial Flow Debdeep Bhattacharjee, Arnab Atta, and Suman Chakraborty
Abbreviations Nomenclature Bom Cn Ca λ M μr Re
Magnetic Bond number Cahn number Capillary number Viscosity ratio Mason number Relative magnetic permeability Reynolds number
Subscripts i e
Droplet phase Suspending medium
D. Bhattacharjee (B) · A. Atta Department of Chemical Engineering, IIT Kharagpur, Kharagpur, West Bengal 721302, India e-mail: [email protected] S. Chakraborty 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_42
451
452
D. Bhattacharjee et al.
1 Introduction Immense industrial and biomedical applications (e.g., polymer processing, magnetically controlled optics, ferrofluid-based sensors, biomedical imaging, drug delivery, and diagnosis of malignant tumors) make the droplet magnetohydrodynamics an emerging area of research for the last few decades [1–5]. In the case of the flow field, the droplet deformation depends on the nondimensional capillary number (Ca), which signifies the ratio between viscous force (that tries the droplet to deform) and the interfacial force (that maintains the droplet to its initial spherical shape). Following the pioneering research (on the droplet dynamics of Newtonian fluid under the governance of background flow in a small deformation regime) of Taylor [6], Han and Chin [7] experimented to show the deformation and breakup characteristics of the droplet in extensional flow. Here they considered both the droplet phase and continuous phase to be viscoelastic and showed the effects of different fluid properties (such as viscosity, elasticity, and interfacial tension) on the deformation dynamics. Moreover, droplet breakups in linear flows have been adequately examined numerically as well as experimentally by Stone [8]. Furthermore, Delaby et al. [9] experimentally exhibited the importance of the viscosity ratio on the deformation dynamics of a droplet in immiscible molten polymer blends for higher Ca limits. Minale [10] analytically shows the droplet deformation considering (i) both the phases are Newtonian, (ii) one of the phases is non-Newtonian, and (iii) considering the confined system where wall effects have been considered. Similarly, the sole impact of a uniform magnetic field on the ferro-droplet has been thoroughly addressed in the literature [11–13]. On a brief note, the appearance of a magnetic field makes the ferro-droplet unstable and expands along with the applied magnetic field orientation [14]. In this context, Afkhami et al. [12, 13] analytically showed the dependency of the aspect ratio (ratio between the length of major and minor axes) of the deformed droplet on the magneto-physical properties of the ferrofluid. A complete numerical study on hysteresis phenomena for the ferrodroplet deformation in a uniform magnetic field has been studied by Lavrova et al. [15]. The dynamics of a neutrally buoyant droplet in the presence of the magnetic field or extensional flow are independently reported in the literature. The droplet dynamics in the combined effect of magnetic field and uniaxial extensional flow can provide many physical imaginations on the fluid mixing controllability. Motivated by this, here we propose to see the impact of the uniform magnetic field in the background uniaxial extensional flow on the ferro-droplet dynamics. Considering Stokes flow, the magnetohydrodynamic problem is solved by following numerical simulations in the small deformation limit.
Evolution of Ferrofluid Droplet Deformation Under Magnetic Field …
453
2 Problem Formulation Here, we considered a Newtonian ferrofluid droplet of radius r i , density ρ, viscosity ηi , and magnetic susceptibility χ i , placed in another Newtonian, non-magnetic fluid of viscosity ηe and density ρ (neutrally buoyant) as shown in Fig. 1. The droplet is directed by an externally employed magnetic field H 0 = H0 ez and uniaxial extensional flow U 0 = T0 · X. Here, H0 is the magnitude of the applied magnetic field, T0 is the rate of the strain tensor, and X is the position vector. The far-field strain rate tensor T0 can be defined as: ⎡ ⎤ −1 0 0 G0 ⎣ T0 = 0 −1 0 ⎦ 2 0 0 2
(1)
where G0 is the rate of strain. Further, we considered the surface tension σ is uniform along with the droplet interface.
2.1 Governing Equations The velocity field for an incompressible flow satisfies the continuity and Navier– Stokes equation as [16, 17]: ⎫ ⎪ ∇ ·u = 0 ⎬ ) ( ( ) ∂u + u ∇ u = −∇ p + η ∇ · ∇u + (∇u)T + FST + FM ⎪ ρ ⎭ ∂t Fig. 1 Schematic representation of the problem description
(2)
454
D. Bhattacharjee et al.
Here FST and FM represent the force due to surface tension and magnetic force, respectively. The surface tension force, FST can be described as [18]: ) [ [ ( FST = ∇ · σ I + −nn T δ
(3)
where I, δ, and n are the second-order identity tensor, Dirac delta function, and unit normal to the interface. Similarly, the force due to magnetic field, FM can be defined as [18]: ( ) μm FM = ∇ · τ M = ∇ · μm HHT − (H ·H)I 2
(4)
In the above equation, H, τ M , and μm are the strength of the applied magnetic field, magnetic stress tensor, and the magnetic permeability of the fluid, respectively. To find the magnetic stress tensor, we need to solve the magneto-static Maxwell equations as given below [18]. ∇ ·B = 0 ∇ ×H=0
⎫ ⎪ ⎪ ⎪ ⎬
⎪ M = χm H ⎪ ⎪ ⎭ B = μ0 (H + M)
(5)
where μ0 (μ0 = 4π × 10−7 N/A2 ) is the permeability of the vacuum. B, M, and χ m are magnetic induction, magnetization, and magnetic susceptibility, respectively.
2.2 Nondimensionalization To nondimensionalize, we have considered the following scales: length—r i , velocity—G0 r i , magnetic field strength—H 0 , hydrodynamic stress—ηe G0 , and magnetic stress—μ0 H 0 2 . Here we assumed that the viscous force is ruling over the inertia force, resulting in a low Reynolds number flow. We have tabulated all the dimensionless parameters and nondimensional numbers in Table 1.
3 Numerical Method In this study, we used the phase-field technique to resolve the two-phase interface features. Phase-field model for a system of two incompressible and immiscible fluids can be described as [19]: ( ) ∂ψ(x, t) + u ·∇ψ(x, t) = ∇ Mψ(x,t) ∇G ∂t
(6)
Evolution of Ferrofluid Droplet Deformation Under Magnetic Field …
455
Table 1 Different dimensionless parameters and nondimensional numbers associated with the present analysis Symbol
Details
λ
Viscosity ratio defined as λ = ηi /ηe
μr
Relative magnetic permeability defined as μr,i = μi /μ0 (droplet phase) (we have considered the suspending medium as non-magnetic in this study, i.e., μr,e = μe /μ0 = 1)
Re
Reynolds number defined as Re = ρG 0 ri2 /ηe (ratio between inertial and viscous stress)
Ca
Capillary number defined as Ca = ηe G 0 ri /σ (magnitude of viscous stress relative to capillary stress)
Bom
Magnetic Bond number defined as Bom = ri μ0 |H0 |2 /σ (magnitude of magnetic stress relative to capillary stress)
M
Mason number defined as M = μ0 |H0 |2 /ηe G 0 (magnitude of magnetic stress relative to viscous stress)
where M ψ (x,t) and G denote the interface mobility factor and chemical potential, respectively. Two fluids are distinguished by the definite values of the phase-field parameter ψ(x,t). The values of the phase-field parameter for suspending fluid phase and droplet phase are taken as −1 and +1, respectively; whereas it varies from −1 to +1 at the two-phase interface. The magnetic potential follows Poisson’s equation in the following form [18]: ( ) ∇ · μm ∇φ mag = 0
(7)
where μm is the magnetic permeability of the fluid which can be presented in terms of the phase-field parameter as [ μm =
] [ ] (1 − ψ(x, t)) (1 + ψ(x, t)) μi + μe 2 2
(8)
For simulations, a square channel of size L × L is considered (here L = 10r i to neglect the effect of channel confinement), where the droplet is situated at the center of the left side (axis of symmetry) as shown in Fig. 2.
3.1 Grid Independence and Cahn Number Independence Test To check the precision of the simulation results, the grid independence and Cahn number (Cn) independence assessments are necessary. In our model, the grid size and the Cahn number are similar and close to the two-phase interface. So, an accurate grid independence study automatically confirms a correct Cahn number independence analysis [20]. For the Cahn number independence study, we have compared the
456
D. Bhattacharjee et al.
Fig. 2 Schematic representation of the computational domain
Fig. 3 Alteration of the droplet deformation for different Cahn numbers (Cn) at Ca = 0.05, λ = 1 in the sole presence of uniaxial extensional flow
temporal droplet deformation in presence of uniaxial flow as illustrated in Fig. 3. This figure confirms that the droplet deformation is almost the same for three different values of the Cahn number. Eventually, we considered Cn = 0.04 for the rest of our analysis.
3.2 Model Validation For validation, we have compared the deformation parameter D [where D = (l − b)/ (l + b); l and b are the lengths of the major and minor axis of the deformed droplet,
Evolution of Ferrofluid Droplet Deformation Under Magnetic Field …
457
Fig. 4 Alteration of the droplet deformation at the steady-state (Dα ) with a Capillary number Ca at λ = 1 in uniaxial extensional flow and b Magnetic Bond number Bom for (μr,i , μr,e ) = (1.89, 1) in sole existence of uniform magnetic field
respectively] with the previously reported results for two cases: (a) droplet deformation in existence of uniaxial extensional flow only and (b) droplet deformation in the sole presence of the uniform magnetic field. Figure 4a shows the steady-state droplet deformation (Dα ) for different Ca values at λ = 1, which confirms that our numerical model is in harmony with the simulation of Stone and Leal [21]. Figure 4b depicts the alteration of droplet deformation at the steady-state (Dα ) with the magnetic Bond number (Bom ) for μr,i = 1.89. After comparison, we found that our numerical model is in the same fashion as the experimental results of Afkhami et al. [13] for Bom < 1.5.
4 Results and Discussion In external flow conditions, the viscosity ratio plays a pivotal role in the deformation dynamics of a droplet. Figure 5 confirmed the impact of the viscosity ratio on the droplet deformation in the Stokes flow regime (here, Re = 0.01). It can be seen that at a fixed Ca value (here, Ca = 0.05), as the viscosity ratio (λ) increases from 0.01 to 5, the deformation parameter (D) also increases. Contrariwise, the time taken to reach steady-state deformation is smaller at the lower viscosity ratio. Another important observation is that for λ ≤ 1, the temporal evolution of the droplet deformation is almost independent of the viscosity ratio. More precisely, the droplet deformation plot at λ = 0.01 almost coincides with that at λ = 0.1. Considering these aspects, we considered λ = 1 to see the droplet deformation characteristics for the rest of the analysis. Figure 6a illustrates the transient droplet deformation for (μr,i , μr,e ) = (2, 1) and λ = 1 for different Mason numbers and α = 90°. In absence of a magnetic field (i.e., M = 0), the drop deforms into a prolate shape due to the hydrodynamic stress
458
D. Bhattacharjee et al.
Fig. 5 Effect of viscosity ratio (λ) on the droplet deformation (D) for Ca = 0.05, M = 0, and Re = 0.01
distribution. For all M > 0 at α = 90°, both the magnetic stress and the hydrodynamic stress are acting in the same direction which assists the droplet to deform into a prolate shape. Moreover, the deformation increases with increasing magnetic field strength. Figure 6b depicts the alteration of transient deformation of the ferro-droplet for (μr,i , μr,e ) = (2, 1) and λ = 1 with different Mason numbers at α = 0°. Here the magnetic stress tries to resist the prolate deformation aided by hydrodynamic stress. For this reason, the deformation for all M > 0 is less when compared with M = 0. Fascinatingly, here we observed two distinct droplet shape deformations. For M = 0, 1, and 5, it forms a prolate shape, whereas, for higher values of M (= 10, 20), it deforms into an oblate shape. The prolate shape formation is because of the higher viscous force compared with the magnetic force. While, for M = 10 and 20, the magnetic stress becomes relatively high so that it overcomes the viscous force and deformed the droplet into an oblate shape.
Fig. 6 Alteration of droplet deformation (D) with time for (μr,i , μr,e ) = (2, 1) and λ = 1 for different Mason numbers. a α = 90°and b α = 0°. Other parameters are Ca = 0.05 and Re = 0.01
Evolution of Ferrofluid Droplet Deformation Under Magnetic Field …
459
Fig. 7 Alteration of droplet deformation (D) with time for different relative magnetic permeability of the droplet phase (μr,i ) for a (M, α) = (1, 90°) and b (M, α) = (1, 0°). Other parameters are μr,e = 1, Ca = 0.05, λ = 1, and Re = 0.01
The relative magnetic permeability of a material represents the ability of that material to be magnetized in presence of an external magnetic field. The magnetic force exerted on a ferrofluid droplet directly depends on its relative magnetic permeability. Hence, the relative magnetic permeability has a key role in droplet morphology alteration. Figure 7a represents that for a system with α = 90°, the droplet deformation (D) increases monotonically with the relative magnetic permeability at a fixed magnetic field strength. This observation follows a similar trend as reported in the literature [22]. This is because the magnetic stress attributed to the droplet pole increases with an increase in relative magnetic permeability. Consequently, the magnetic force dominates the surface tension force and thus the droplet becomes more elongated along the direction of the magnetic field and forms a prolate shape with a high aspect ratio. Interestingly, for the system (M, α) = (1, 0°) as shown in Fig. 7b, the droplet morphology evolved from prolate to oblate shape with the increase of relative magnetic permeability from 1 to 10. This is because with the increase of μr,i , the magnetic stress becomes high and it governs the flow dynamics. As a result, the droplet follows the magnetic field direction and formed an oblate shape.
5 Conclusions In summary, the transient dynamics of a ferro-droplet guided by the uniform magnetic field in background uniaxial extensional flow has been studied numerically. Based on the outcomes, the major findings are as follows: • The droplet deforms into a prolate shape for α = 90°. On the contrary, for α = 0°, the droplet may deform into either a prolate or oblate shape depending on the Mason number.
460
D. Bhattacharjee et al.
• For the system with α = 90°, the droplet elongates more with the higher relative magnetic permeability and forms the prolate shape. Whereas, an enhancement of relative magnetic permeability can change the droplet morphology from the prolate to an oblate shape at α = 0°. Acknowledgements D.B. thanks Dr. Somnath Santra for his insight on numerical analyses. S.C. acknowledges the Sir J. C. Bose National Fellowship awarded by the DST, Government of India.
References 1. Garstecki P, Stone HA, Whitesides GM (2005) Mecism for flow-rate controlled breakup in confined geometries: a route to monodisperse emulsions. Phys Rev Lett 94(16):164501 2. Dutz S, Clement JH, Eberbeck D, Gelbrich T, Hergt R, Muller R, Wotschadlo J, Zeisberger M (2009) Ferrofluids of magnetic multicore nanoparticles for biomedical applications. J Magn Magn Mater 321(10):1501–1504 3. Torres-D´ıaz I, Rinaldi C (2014) Recent progress in ferrofluids research: novel applications of magnetically controllable and tunable fluids. Soft Matter 10(43):8584–8602 4. McClements DJ (2015) Emulsion stability, food emulsions. CRC Press, pp 314–407 5. Bhattacharjee D, Atta A, Chakraborty S (2022) Magnetic field altered ferrofluid droplet deformation in the uniaxial extensional flow. B Am Phys Soc 6. Taylor GI (1932) The viscosity of a fluid containing small drops of another fluid. Proc R Soc Lond Ser A 138(834):41–48 7. Chin BH, Han DC (1979) Studies on droplet deformation and breakup. I. Droplet deformation in extensional flow. J Rheol 23(5):557–590 8. Stone HA (1994) Dynamics of drop deformation and breakup in viscous fluids. Annu Rev Fluid Mech 26(1):65–102 9. Delaby I, Ernst B, Germain Y, Muller R (1994) Droplet deformation in polymer blends during uniaxial elongational flow: Influence of viscosity ratio for large capillary numbers. J Rheol 38(6):1705–1720 10. Minale M (2010) Models for the deformation of a single ellipsoidal drop: a review. Rheol Acta 49(8):789–806 11. Rosensweig RE (1985) Ferrohydrodynamics. Cambridge University Press 12. Afkhami S, Renardy Y, Renardy M, Riffle JS, St Pierre T (2008) Field-induced motion of ferrofluid droplets through immiscible viscous media. J Fluid Mech 610:363–380 13. Afkhami S, Tyler AJ, Renardy Y, Renardy M, St Pierre TG, Woodward RC, Riffle JS (2010) Deformation of a hydrophobic ferrofluid droplet suspended in a viscous medium under uniform magnetic fields. J Fluid Mech 663:358–384 14. Liu J, Yap YF, Nguyen NT (2011) Numerical study of the formation process of ferrofluid droplets. Phys Fluids 23(7):072008 15. Lavrova O, Polevikov V, Tobiska L (2005) Equilibrium shapes of a ferrofluid drop. Proceedings in applied mathematics and mechanics, vol 5. Wiley Online Library, pp 837–838 16. Bhattacharjee D, Atta A (2022) Topology optimization of a packed bed microreactor involving pressure driven non-Newtonian fluids. React Chem Eng 7(3):609–618 17. Bhattacharjee D, Chakraborty S, Atta A (2022) Passive droplet sorting engendered by emulsion flow in constricted and parallel microchannels. Chem Eng Process Process Intensif 181:109126 18. Md Hassan R, Zhang J, Wang C (2018) Deformation of a ferrofluid droplet in simple shear flows under uniform magnetic fields. Phys Fluids 30(9):092002 19. Santra S, Mandal S, Chakraborty S (2018) Electrohydrodynamics of confined two-dimensional liquid droplets in uniform electric field. Phys Fluids 30(6):062003
Evolution of Ferrofluid Droplet Deformation Under Magnetic Field …
461
20. Mandal S, Ghosh U, Bandopadhyay A, Chakraborty S (2015) Electroosmosis of superimposed fluids in the presence of modulated charged surfaces in narrow confinements. J Fluid Mech 776:390–429 21. Stone HA, Leal LG (1989) Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J Fluid Mech 198:399–427 22. Ghaffari A, Hashemabadi SH, Bazmi M (2015) Cfd simulation of equilibrium shape and coalescence of ferrofluid droplets subjected to uniform magnetic field. Colloids Surf, A 481:186–198
Time-Dependent Droplet Detachment Behaviour from Wettability-Engineered Fibers during Fog Harvesting Arijit Saha, Arkadeep Datta, Arani Mukhopadhyay, Amitava Datta, and Ranjan Ganguly
1 Introduction Fresh water crisis is one of the most pertinent climate issues of this era, plaguing not only India but also the entirety of the globe as seen in recent events. The situation is terrible in countries like India which accounts for only 4% of the world’s freshwater resources despite having 16% of the world’s population [1]. Nonconventional water sources need to be harnessed to bridge the gap between supply and demand. Fog harvesting is one such water conservation technique that is looked up with much hope. Fogging represents a largely untapped source of water, especially in hilly areas and many industrial zones, the latter primarily referring to the harvesting of cooling tower fog. Cooling tower (CT) happens to be one of the prominent sources of industrial fog. For a 500 MW power plant, the amount of cooling water required is, 54,000 to 60,000 m3 h−1 , from which nearly 3% is lost as vapor and fog from the CT exit [2]. To compensate for this loss, about 900 m3 h−1 of make-up water is needed. Tapping even a mere 1% of this colossal volume means a saving of 9 tons of water in an hour. In a typical fog harvesting configuration, fog droplets, typically in the size range of 4–40 µm, pass through meshes and get captured by the mesh elements. Such meshes or filters are used in large and small-scale atmospheric and industrial water harvesting systems [3, 4]. Geometrical structure of the fog collecting mesh and their surface wettability play major roles in the fog collection efficacy. Intuitively, it may appear that a denser net (i.e., having a large fraction of solid fiber per unit projected area of mesh) would imply that more fog particles would collide with the mesh fiber and gather more water. However, this is not typically the case. Smaller void fraction can also have a shielding effect [5], which would lower the mass flow of the fog A. Saha · A. Datta (B) · A. Mukhopadhyay · A. Datta · R. Ganguly Department of Power Engineering, Advanced Materials Research and Applications (AMRA) Laboratory, Jadavpur University, Kolkata 700106, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_43
463
464
A. Saha et al.
stream through the mesh itself, and rather divert them around the mesh [6–8]. The fraction of the total oncoming fog stream to which the solid fibers pose as obstruction is denoted as aerodynamic efficiency ηaero . The percentage of the fog droplets, whose path ahead of the mesh is geometrically intercepted by the mesh fibers, which actually impinges on the fibers due to inertial impaction, interception and Brownian diffusion is accounted for by the deposition efficiency, or ηdep . It is interesting to note that not all the fog water that deposited on the mesh fiber can be collected. Part of it may be lost due to carryover by the oncoming fog stream, and a part will be lost due to premature dripping [9]. The percentage of water caught by the mesh that drains down the fibers and collects in the water collection manifold at the bottom of the mesh is known as the draining efficiency, or ηdr . The overall collection efficiency, therefore, can be determined by ηcollection = ηcapt × ηdr × ηac
(1)
Commonly deployed FWC meshes consist of fibers woven in orthogonal manner. While the vertical fibers act as a means of sliding the collected fog water, the horizontal fibers pose a challenge to its movement and therefore drainage. An important phenomenon in this regard is the dripping or detachment of deposited liquid from the mesh fiber. Our previous studies have focused on the droplet morphology and detachment criteria from horizontal mesh through numerical simulation [10] and experiments [11], though several important features have further unfolded as our study has progressed. The present work sheds some light on droplet the relatively sparsely characterized phenomenon of droplet detachment from horizontal fiber of varied wettability [12] and diameter, placed in a fog laden flow. Findings of the study are important for designing efficient fog collectors.
2 Materials and Methods We developed three types of fibers, superhydrophilic (SHPL), hydrophobic (HPB) and control. For rendering the surface SHPL, a wet chemical etching process was adopted to change the roughness of commercially available aluminum cylindrical fibers (GRADE-6063). The mesh surfaces were first roughened using sandpapers (P220 Silicon carbide, DV371) to rid it of the oxide protective layer, and then cleansed in acetone and demineralized water respectively using sonication (USB 3.5L H DTC, PCI Analytics) for 15 min each. These surfaces were then treated in a solution of 3 M HCl aqueous (aq.) solution (MERCK Pvt. Ltd.) for 15 to 20 min. This etches the Al surface, creating a micro- and nano-rough texture, rendering the surface SHPL. 2Al(s) + 6HCl(aq) = 2AlCl3 (aq) + 3H2 (g)
(2)
Time-Dependent Droplet Detachment Behaviour …
465
The SHPL surfaces were next passivated in boiling water for 45 to 60 min, which forms nano-böhmite (Al(O)OH) structures on the etched surfaces, improving the surface’s resistance to oxidation in the atmosphere, thus making them durable. To develop hydrophobicity on the mesh surface, the cleansed fibers were dipcoated (NXT dip-KPM, Apex Instruments) (at a coating speed of 3 mm/s) with polydimethylsiloxane (PDMS) (Sigma Aldrich) and cured in an 80 °C furnace for 4 h. Stainless steel (SS304 grade Swent feeder needles from Swastik Enterprise™— India), cleaned in acetone and distilled water, were used as the control fibers. Sessile-droplet water contact angle on these differently wetted fibers was measured using the methodology defined in [11] using a standard goniometer (Holmarc Opto-Mechatronics Ltd). A digital screw gauge (Yuzuki™ digital micrometer, least count: 0.001 mm) was used to measure each needle’s diameter. Before the experiment, each fiber was dry cleaned using a blow drier to evaporate any residual fluid and to blow away any dust that might have been deposited on it. To dispense the fog on the fibers, a medical-grade ultrasonic nebulizer (Yuwell402AI, fog droplet diameter 4–10 µm) was used. The nebulizer’s fogdischarge nozzle was precisely positioned and pointed toward the mesh-fiber axis, as shown in Fig. 1. The distance between the nozzle tip and the needle was fixed and maintained at 2.5 cm. A single-lens digital camera (Nikon D7200 VR with a reversed Nikkon 18– 140 mm lens for macro-imaging), was used to record the droplet detachment event. A back-light LED (Osaka Bi-color dimmable led light OS-LED-308) was used at the rear of the droplet to obtain high-quality photos. ImageJ software is used for different measurements pertaining to the images of pendant droplets, e.g., to determine the contact width for the hanging droplet both just before and after detachment events. Fig. 1 Schematic of the experimental apparatus under the fog condition. Inset shows a typical water droplet pendant upon fog accumulation on the fiber (Scale bar denotes 4 mm). Legend: a fiber, b water droplet, c fog stream, d collection pot
Inset
a b
c
Weighing machine
d
466
A. Saha et al.
3 Results and Discussion The event of droplet detachment from various wettability-engineered fibers is investigated under fogging conditions. The fog stream is allowed to strike the horizontally placed fibers when the deposited liquid forms a liquid film, which finally grows into a pendant droplet at the underbelly of the fiber. Eventually, the droplet grows to a size that its weight exceeds the adhesion force and it falls down. The process of droplet growth and detachment at the location of fog impingement is noticed. During the experiment, the weight of each detached droplet is taken. This revealed an interesting trend. The droplet detachment weight is seen to increase with each detachment event (implying that the size of the detached droplet increases) and then gradually becomes invariant with time (and therefore each detachment event). The variation of detachment weight is shown as a function of droplet number in Fig. 2a, which clearly points to two distinct regimes; viz the transient and the steady regimes. In the transient regime, the fiber is still ‘dry’, where the region of fog impaction is still localized. Isolated droplets could be seen on the fiber (except for SHPL fiber) in this regime. This regime gradually shifts to a steady state with gradual coalescence of the micro-droplets into a film, leading to wetting of the fiber; the region of wetting expands and eventually, the droplets coalesce into a bigger volume of pendant droplet. These two regimes are depicted in Fig. 2a on the control fibers of different diameters (regimes are marked as DRY and WET). This regime shift might be explained by how the fog gradually wets the fiber. In the dry state, the contact line of the droplet is defined as the footprint with which it hangs from the fiber; whereas for the wet state, the contact line has digressed into taking the smaller drops within its vicinity and therefore increasing its footprint and hence adhesion to the fiber. From Fig. 2b we see that the superhydrophilic fibers (SHPL, with CA < 5° measured on a flat plate of similar material and wettability treatment) have the highest detachment weight. The HPB fibers exhibited the least detached droplet weight. In comparison to its “dry” states, the detached droplet weight on wet state marks is just slightly higher (2.93% for 4.36 ± 0.2 mm diameter). SHPL fiber has the highest surface energy compared to the fibers of the other two wettability leading to higher droplet retention capacity. From Fig. 2b, we see that the dependence of the droplet detachment weight with fiber diameter tends to taper off beyond fiber diameter greater than the capillary length of the liquid (2.7 mm for water, denoted in the picture as a vertical dashed line). This may be attributed to the fact that the maximum capillary force acting on the droplet scales as the capillary length scale of the droplet [13]. Apart from fog impaction on the hanging droplet, droplet growth is also driven by Laplace pressure, since smaller drops having higher Laplace pressure “pushes” the liquid into the larger hanging drop from its vicinity, since the latter has a lower Laplace pressure due to larger radius of curvature). Figure 3 denotes such events with pink arrows on control fiber.
Time-Dependent Droplet Detachment Behaviour …
Droplet detatchment weight in un-modified needles
(a)
WET
DRY
0.075
Detatchment weight (g)
Fig. 2 a Detachment weight mapped with every droplet detachment event (droplet number). The droplet detachment weight initially increases (region marked in red, denoted as DRY) and thereafter becomes invariant (region marked in blue, denoted as WET) with droplet detachment event. Fibers of higher diameter hold more water. Different diameters of control surface (SS304) are used for this study. b Comparison of droplet detachment weight from fibers of varied wettability and diameters
467
0.060
0.045
Diameter (mm) 1.27 2.7
0.030 0
10
20
30
40
50
2 4
60
Droplet Number
(b) 0.12
DRY
Detachment weight (g)
0.10
WET HPB CONTROL SHPL
0.08 0.06 0.04 0.02 0
1
2
3
4
5
6
Fiber diameter (mm)
Laplace pressure is the additional pressure a fluid experiences when it is enclosed by a curved surface and is given by the expression ∆p = 2γ L V κm + ∆ρ g→ · →z
(3)
where, ∆ρ denotes the density difference between the liquid and the gas and z is the local elevation of the surface (from the datum) and the mean curvature κ m can be expressed as [14]. κm =
∇ FHess(F)∇ F T − |∇ F|2 Trace(Hess(F)) 2|∇ F|3
(4)
Here, Hess(F) represents the hessian of the surface F(x,y,z) denoting the spatial distribution of the water–air surface [14].
468
A. Saha et al.
SHPL
CONTROL
HPB
DRY
WET
Fig. 3 Droplet detachment from three wettability fibers in both the dry and the wet states captured just before detachment. Contact length (Lc ) is demarcated as [ ] and [ ] in DRY
and WET states respectively. The arrows in pink denote Laplace pressure driven flow from smaller drops from in and around the much larger hanging drop. Scale bar denotes 2 mm
Equation 4 shows that on the same fiber, the droplet with a smaller κ m exerts more excess pressure than that with a larger κ m . Therefore, as the smaller droplets were in close vicinity, touching at their bases by a precursor film [15], they would drain into the larger droplet and coalescence. The space freed by these smaller droplets is regenerated with continuous deposition of fog droplets on the fiber. The contact line in the wet state initially increases as a result of this on-going coalescence of the micro-droplets, but thereafter it attains a steady state, characterized by the invariance toward droplet detachment event (droplet number). This translates to the trend as depicted in Fig. 2a for fibers of different diameter. The variation of the detachment weight with fiber radius for dry and wet conditions on fibers of three different wettability is plotted in Fig. 2b. The surface energy difference is what causes this difference in the detachment volume for different wettability fibers. In comparison to hydrophobic surfaces, hydrophilic surfaces have higher surface energy. A surface with more surface energy can accommodate larger droplets on it. The surface pinning force is another crucial factor that affects the detachment weight. The pinning force effect also rises with fiber diameter. The fiber can hold more water droplets when there is more pinning force acting along its three-phase contact line. As the HPB (PDMS-coated) surface has less surface energy and a weaker pining force (characterized by a lower contact angle hysteresis), it is expected to hold smaller droplets before detachment. Figure 2b also shows the difference between dry state and wet state detachment weight. For the SHPL fiber, the detachment weight curves for dry and regimes almost overlap with each other because the droplet detachment from the horizontal fiber takes place from a water film on the surface for both the dry and wet states. However, when compared to the other two wettability fibers, HPB fiber exhibits the greatest variation, because it resists the formation of a water
Time-Dependent Droplet Detachment Behaviour …
469
film on it in both dry and wet states. However, in the wet state, small droplets are found in the vicinity of the large droplet, which collides with the large droplet at the time of detachment due to Laplace pressure. This phenomenon creates the difference between dry and wet state detachment weight. Another important parameter for determining the variation of detached droplet weight is droplet-fiber adhesive force, which is a function of the liquid surface tension and the contact perimeter that the droplet makes with the fiber. The adhesive force balances the hanging droplet’s weight. In contrast to the other two wettability fibers, the SHPL fiber- droplet contact line is maximum since its pinning force effect is maximum as seen in Fig. 3. This is observed in both the dry and the wet states. The droplet-fiber contact length, L c for HPB fiber is particularly low because of its low surface energy, which prevents the droplet from expanding its perimeters on the fiber. The contact footprint (or the contact length, L c ) of the hanging droplet seems to be a major contributing factor to the trend we see in Fig. 2b. Experimentally, it is quite difficult to explicitly gauge the contact line of the hanging water droplet on a horizontally placed fiber. The contact width marked by horizontal arrows in Fig. 3 ( Red and blue for DRY and WET state respectively) could be taken as an approximation of the contact line, considering the length scale. The maximal width of the growing droplet is measured in both the events. This provides a rough estimate of the detachment weights, given the practical constraints on the pendant droplet imposed by the solid surface geometry and contact line conditions (i.e., a moving or a pinned contact line) [16, 17]. With successive deposition of fog, there is a gradual increase in contact width with fiber diameter for both the regimes, taken just prior to detachment phenomena. This is attributed to the larger footprints of the pendant drop that the fiber can accommodate. The contact width lies in the two extremes for the SHPL and HPB fibers as shown in Fig. 4, with the general trend remaining the same as discussed for control fibers. The contact width differs between the wet and dry states in HPB (18–13.46%) and the SHPL (3–0.5%) cases and is quite smaller than that for the control fibers. For the SHPL fibers, this could be attributed to it having a higher surface energy. It being super hydrophilic, droplet spreads on its surface forming a film of water on the fiber surface, raising the upper limit of contact width. This also attributed to it holding a much higher volume of water during detachment. The formation of this liquid film also explains the overlapping of the contact width data in Fig. 4 (in both the wet and dry regimes) for SHPL cases. There is also no distinct ‘dry’ and ‘wet’ state for the SHPL cases, due to this liquid film formation. This also explains the overlapping of detachment weight data in ‘dry’ and ‘wet’ regimes (Fig. 2b) and high detachment weight for SHPL cases. The observations of the HPB fibers were similar to those made for SHPL fibers, with only one major difference; the contact width for HPB fibers was at the lowest end of the spectrum. This is attributed to its low surface energy, which restricts water to adhere to the surface and inhibits any film formation. This is also supported by the detachment weight data for both dry and wet regimes (Fig. 2b).
470 12
DRY
10
Contact width (mm)
Fig. 4 Variation of contact width of the hanging droplet from differently wettability-engineered fibers of varied diameters just before detachment. Variation of droplet detachment weight with fiber diameter tends to taper off beyond fiber diameter greater than the capillary length of the liquid (2.7 mm for water, shown by the vertical dotted line)
A. Saha et al. WET HPB CONTROL SHPL
8
6
4
2
0
1
2
3
4
5
6
Fiber diameter (mm)
4 Conclusions Liquid droplet detachment from horizontal fibers of varied diameter and varied wettability is studied experimentally in the context of fog harvesting. Liquid deposition is carried out through fog impaction. The role of fiber wettability—superhydrophilic (SHPL), control and hydrophobic (HPB)—on droplet detachment dynamics is explored. It is observed that the SHPL fibers can hold the largest droplets before detachment, followed by the control and HPB fibers, along with the general trend suggesting a higher droplet detachment weight with increase in fiber radius. The dependence of the droplet detachment weight with fiber diameter tends to taper-off beyond fiber diameter greater than the capillary length of the liquid (2.7 mm for water). It is observed that the droplet contact width on the SHPL fibers is more in comparison with the control and HPB fibers, which also supports droplet detached weight trends and is attributed to the balancing act of the adhesive surface tension force to the weight of the pendant droplet. Acknowledgements The authors gratefully acknowledge the funding from DST SERB (Grant No: CRG/2019/005887).
References 1. http://mospi.nic.in/sites/default/files/Statistical_year_book_india_chapters/ch2.pdf. Accessed 05 Jul 2022 2. Ghosh R, Ray TK, Ganguly R (2015) Cooling tower fog harvesting in power plants–a pilot study. Energy 89:1018–1028
Time-Dependent Droplet Detachment Behaviour …
471
3. Das C, Ghosh R, Datta A, Ganguly R. Wettability engineering: a skin deep approach of solving the energy-water nexus. In: Advances in multiphase flows. Begell House Inc. ISBN: 978-156700-504-2 4. Damak M, Varanasi KK (2018) Electrostatically driven fog collection using space charge injection. Sci Adv 4(6):eaao5323 5. de Dios Rivera J (2011) Aerodynamic collection efficiency of fog water collectors. Atmos Res 102(3):335–342 6. Park K-C et al (2013) Optimal design of permeable fiber network structures for fog harvesting. Langmuir 29(43):13269–13277 7. Shi W et al (2018) Fog harvesting with harps. ACS Appl Mater Interfaces 10(14):11979–11986 8. Regalado CM, Ritter A (2016) The design of an optimal fog water collector: a theoretical analysis. Atmos Res 178:45–54 9. Ghosh R, Ganguly R (2020) Fog harvesting from cooling towers using metal mesh: Effects of aerodynamic, deposition, and drainage efficiencies. Proc Inst Mech Eng Part A: J Power Energy 234(7):994–1014 10. Mukhopadhyay A, Dutta PS, Datta A, Ganguly R (2020) Liquid droplet morphology on the fiber of a fog harvester mesh and the droplet detachment conditions under gravity. In: Proceedings of the 8th International and 47th National conference on fluid mechanics and fluid power (FMFP), December 09–11. IIT Guwahati, Guwahati, 781039, Assam, India 11. Datta A, Mukhopadhyay A, Dutta PS, Saha A, Datta A, Ganguly R (2021) Droplet detachment from a horizontal fiber of a fog harvesting mesh, FMFP2021–190. In: Proceedings of the 48th National conference on fluid mechanics and fluid power (FMFP), December 27–29, 2021. BITS Pilani, Pilani Campus, RJ, India 12. De Gennes P-G, Brochard-Wyart F, Quéré D (2004) Capillarity and wetting phenomena: drops, bubbles, pearls, waves, vol 315. Springer, New York 13. Lorenceau É, Clanet C, Quéré D (2004) Capturing drops with a thin fiber. J Colloid Interface Sci 279(1):192–197 14. Goldman R (2005) Curvature formulas for implicit curves and surfaces. Comput Aided Geometric Design 22(7):632–658 15. Bonn D, Eggers J, Indekeu J, Meunier J, Rolley E (2009) Wetting and spreading. Rev Mod Phys 81:739–805 16. Kumar A, Gunjan MR, Raj R (2020) On the validity of force balance models for predicting gravity-induced detachment of pendant drops and bubbles. Phys Fluids 32(10):101703 17. Kumar A, Gunjan MR, Jakhar K, Thakur A, Raj R (2020) Unified framework for mapping shape and stability of pendant drops including the effect of contact angle hysteresis. Colloids Surf A: Physicochem Eng Aspects 124619
Stability Analysis from Fourth-Order Nonlinear Multiphase Deep Water Wavetrains Tanmoy Pal and Asoke Kumar Dhar
1 Introduction There has been substantial interest in the instability analysis of SGW on infinite depth of water. Much of the interest has been fixed on the instability of Stokes waves to sideband perturbations. It has been found that results achieved from cubic NLSE for infinite depth of water do not comply with the exact computations of LonguetHiggins [1, 2]. Considering the perturbation analysis up to fourth order, Dysthe [3] first obtained a remarkable progress on the results. The effect arising from fourthorder equation provides significant progress compare to third-order effects in many ways, and few of these points have been extended by Janssen [4]. Therefore, we may conclude that fourth-order NLSE is an excellent beginning point for performing instability analysis of SGW on deep water. The problem of SGW on deep water, including their stability analysis, has been studied by several authors [4–7] in different contexts. The experiments due to Benjamin and Feir [8] have pointed out that a system of nonlinear waves on infinite depth of water surface may produce incoherence in wave amplitude and wavenumber. The primary mechanisms for this modulational instability have been the matter of many investigations. In particular, the cubic NLSE has been found to be applicable and proposes that solitary envelope wavetrains may occur when dispersive and nonlinear effects are in balance. Onorato et al. [9] have derived the two-coupled NLSE starting from Zakharov’s integral equation. They have considered a weakly nonlinear model that analyses the nonlinear interaction of two wavetrains on infinite depth of water with two different propagation directions and showed that due to the presence of second wave the growth rate of instability (GRI) enhances and the instability region increases. T. Pal (B) · A. K. Dhar Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur 711103, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_44
473
474
T. Pal and A. K. Dhar
Shukla et al. [10] have studied the sideband instability of two obliquely propagating wavetrains on deep water, which are described by two-coupled NLSE, and found that random perturbations can increase to produce nonlinear water wave structures through the nonlinear interaction between a pair of wavetrains. Kundu et al. [11] have also derived the cubic three-coupled NLEE for two wavetrains on finite depth of water and found that GRI for nonlinearly interacting wavetrains on finite depth of water is larger than that for the deep water case. The influence of wind flow for obliquely interacting wavetrains in deep water has been analysed by Senapati et al. [12]. They have pointed out that GRI enhances with the enhancement of wind velocity. On the basis of third-order NLSE for SGW on infinite depth of water, Roskes [13] has made a stability analysis for uniform wavetrains. In that paper, he has presented the results due to nonlinear interaction of two modes in infinite depth of water when modulations occur towards a line along which group velocity projections (GVP) overlap. He has also pointed out that a case of interest occurs when the wave number vectors k1 , k2 for the two modes have equal length and proposed that in the multiphase case, modulations will grow at a faster rate towards this line when 0 ≤ θ < 70.5◦ , where θ indicates the angle between k1 and k2 . In the present paper, we have derived two-coupled fourth-order NLEE for two SGW on deep water which extends the work of Roskes [13]. The present fourth-order analysis gives considerable modification from the third-order analysis in the case of equal amplitudes of the two wavetrains.
2 Governing Equations The free surface of the water in the unperturbed state is considered as z = 0 plane. We take that the two waves with wave numbers k1 and k2 propagate in the (x, y) plane. We consider the x-axis towards a line along which GVP of the two Stokes wavetrains overlap and take the modulations along this axis. For convenience, we use the dimensionless variables by the following transformations: /
k03 ˜ k0 α → α, φ → φ, ˜ (k0 x, k0 y, k0 z) → (x, ˜ y˜ , z˜ ), σ t → t˜, g
(1)
in which k0 is some characteristic wave number and α is the free surface elevation. In subsequent analysis, all these non-dimensional quantities will be expressed with their tilde deleted.
Stability Analysis from Fourth-Order Nonlinear Multiphase Deep Water …
475
For inviscid and incompressible fluid, the governing equations can be expressed as ∇ 2 φ = 0 in − ∞ < z < α,
(2)
φz − αt = φx ax + φ y a y at z = α,
(3)
φt + α = − 21 (∇φ)2 at z = α,
(4)
φ → 0 when z = −∞.
(5)
We consider the solutions of the above-mentioned equations in the form A = A00 +
∞ ∑ ∞ ∑ [ p=0
] Amn exp{i ( pψ1 + qψ2 )} + c.c. ,
(6)
q=0 p=q/=0
in which ψi = ki x x + ki y y − σi t(i = 1, 2) and A means φ and α. Herein, φ00 , α00 , φ pq , α pq and their complex conjugates are slowly varying functions of x1 = εx, y1 = εy, t1 = εt. Further, φ00 , φ pq and their complex conjugates are functions of z, in which ε describes the slow modulations. The linear dispersion relation for the two waves determining the frequency σi (i = 1, 2) are as follows: σi2 = (ki2x + ki2y )1/2 .
(7)
A case of interest is obtained if we take the simple assumption that the wave number of the two waves are same so that k = |k1 |=|k2 | = k0 , the characteristic wave number. Hence, we have k = 1 and (7) then reduces to f (σ ) ≡ σ 2 − 1 = 0.
(8)
3 Evolution Equations for Deep Water On substituting the Expansion (6) for the velocity potential φ (2) and equating the coefficients of expi ( pψ1 + qψ2 ) for ( p, q) [(1, 0), (0, 1), (2, 0), (0, 2), (1, 1), (1, −1)], we have the following equations:
in =
476
T. Pal and A. K. Dhar
d 2 φ pq = ∆2pq φ pq , dz 2
(9)
where [( ∆ pq =
pk1x + qk2x
∂ − iε ∂ x1
)2
( +
pk1y + qk2y
∂ − iε ∂ y1
)2 ]1/2 .
Solving Eq. (9) by using the bottom boundary condition (5), we obtain ( ) φ pq = C pq exp ∆ pq z .
(10)
For convenience we perform the Fourier transform of (2) in the case of ( p, q) = (0, 0). Now the solution takes the following form: φ 00 = C 00 exp(∆00 z).
(11)
Herein, C pq ’s are the functions of x1 , y1 , t1 , and φ 00 is the Fourier transform of φ00 defined by ˚ φ 00 =
)] [( φ00 exp i k x x1 + k y y1 − σ t1 d x1 dy1 dt1 ,
(12)
−∞
where k x and k y are Fourier transform parameters and C 00 depends on k x , k y and σ . We then substitute the Expansion (6) in the Taylor expanded form of the governing Eqs. (3) and (4) about z = 0 and equate the coefficients of expi ( pψ1 + qψ2 ). Now for ( p, q) = [(1, 0), (0, 1), (2, 0), (0, 2), (1, 1), (1, −1), (0, 0)], we get a system of seven equations, in which we put the solutions for φ pq given by Eqs. (10) and (11). To solve the first, second and third sets of equations for ( p, q) = [(1, 0), (0, 1)], [(2, 0), (0, 2), (1, 1), (1, −1)] and [(0, 0)] we introduce the following perturbation expansions for above values of ( p, q) B pq =
∑
ε j B pq j .
(13)
j
Here, B pq denotes C pq , α pq and the index j = 1 for ( p, q) = [(1, 0), (0, 1)] and j = 2 for ( p, q) = [(2, 0), (0, 2), (1, 1), (1, −1), (0, 0)]. We insert (13) in the aforesaid sets of equations to solve C pq for ( p, q) = [(2, 0), (0, 2), (1, 1), (1, −1), (0, 0)]. After eliminating C pq for ( p, q) = [(1, 0), (0, 1)] from first set resulting from (4), we obtain the following equation:
Stability Analysis from Fourth-Order Nonlinear Multiphase Deep Water …
[(
∂ σ + iε ∂t1
)2
477
] ) ∂ m pq − ∆ pq n pq for ( p, q) = [(1, 0), (0, 1)], = −i σ + i ε ∂t1 (14) (
where m pq and n pq are obtained from nonlinear terms. In order to obtain the NLEE, we insert the solutions of various quantities that appear on the right side of Eq. (14) and apply the transformations ζ = x1 − cg t1 , τ = εt1 . Here, cg denotes the group velocity of the basic wavetrains along the x-axis and can be written as )[ ] ( cos θ2 dσ θ . cg = cos = 2 dk k=1 2σ Considering α1 = α101 + εα102 and α2 = α011 + εα012 , we finally obtain fourthorder NLEE for α1 and α2 . i
∂ 2 α1 ∂ 3 α1 ∂α1 + μ11 2 + i μ12 3 ∂τ ∂ζ ∂ζ | 2 | 2) ( ∂α1 = α1 11 |α1 | + 12 |α2 | + i γ11 α1 α1∗ ∂ζ ∗ ∂α ∂α ∂α2 1 + iβ12 α1 α2∗ + iγ12 α12 1 + iβ11 α2 α2∗ ∂ζ ∂ζ ∂ζ ∗ ( ) ) ∂α ∂ ∂ ( α1 α1∗ + δα1 H α2 α2∗ , + iβ13 α1 α2 2 + δα1 H ∂ζ ∂ζ ∂ζ
(15)
and i
∂ 2 α2 ∂ 3 α2 ∂α2 + μ11 2 + i μ12 3 ∂τ ∂ζ ∂ζ | 2 | 2) ( ∂α2 = α2 12 |α1 | + 11 |α2 | + i γ11 α2 α2∗ ∂ζ ∗ ∂α ∂α ∂α1 2 + iβ12 α2 α1∗ + i γ12 α22 2 + iβ11 α1 α1∗ ∂ζ ∂ζ ∂ζ ) ) ∂α1∗ ∂ ( ∂ ( + δα2 H α2 α2∗ + δα2 H α1 α1∗ , + iβ13 α2 α1 ∂ζ ∂ζ ∂ζ
(16)
where θ denotes the angle between k1 and k2 and H refers to one-dimensional Hilbert’s transform given by
478
T. Pal and A. K. Dhar
1 H [┌(ζ )] = P π
ʃ∞ −∞
┌(ζ ) dζ ' (ζ ' − ζ )
(17)
The coefficients μ11 , μ12 , 11 , 12 , γ11 , γ12 , β11 , β12 , β13 and δ are available in Appendix. If we replace α1 by α1 /2 and α2 by α2 /2 in the fourth-order NLEE (15) and (16), the coefficients of third-order terms match to the corresponding coefficients of Roskes [13]. Further, when the second wave is absent and θ = 0, two NLEE (15) and (16) reduce to a single equation equivalent to Eq. (2) of Janssen [4].
4 Modulational Instability Analysis and Results The uniform solutions of two-coupled fourth-order NLEE (15) and (16) are given by ( ) ( ) α j = γ j /2 exp i∆σ j τ for j = 1, 2,
(18)
where γ1 , γ2 represent wave steepness and ∆σ1 , ∆σ2 are the nonlinear frequency shifts given by (
∆σ1 ∆σ2
) =−
1 4
)(
( 11
12
12
11
) γ12 . γ22
(19)
Herein, ∆σ1 , ∆σ2 , represent the frequency shifts, become identical as those of Longuet-Higgins and Phillips [14] if we put β = 180◦ − θ/2 and σ1 = σ2 = σ in their equations. The modulational instability can be developed by introducing small harmonic perturbations in amplitudes and phases so that the uniform solutions (18) can be written as follows: ) ( ) ( )( α j = γ j /2 1 + α 'j expi ∆σ j τ + θ 'j for j = 1, 2,
(20)
where α 'j , θ 'j ( j = 1, 2) stand for infinitesimal perturbations of amplitudes and phases, respectively. Inserting the perturbed solutions (20) in Eqs. (15) and (16), we get a system of four equations by linearizing with respect to α 'j , θ 'j ( j = 1, 2). Performing the Fourier transform of that equations, we get four equations of ' transform quantities α 'j , θ j , which are the Fourier transforms of α 'j , θ 'j ( j = 1, 2), respectively, defined by (
' ' α '1 , θ 1 , α '2 , θ 2
)
1 =√ 2π
ʃ∞ −∞
( ' ' ' ') α1 , θ1 , α2 , θ2 exp(−iλζ )dζ,
(21)
Stability Analysis from Fourth-Order Nonlinear Multiphase Deep Water …
479
where λ is a perturbed wave number. Finally, we have the following nonlinear dispersion relation by considering that the two amplitudes are same (γ1 = γ2 = γ ): {
( ) } P{ + γ(2 /4 (γ) 11 + γ12 + β11 + +β12 + β13 )λ } × P + {γ 2 /4 (γ11(− γ12) + β11 + +β12 − β13})λ = μ11 λ2 μ11 λ2 + γ 2 /4 ( 11 + 12 − 2δλ) ,
(22)
P = Ω − cg λ + μ12 λ3 ,
(23)
˜ + cg λ. Ω=Ω
(24)
{ } P = − γ 2 (γ11 + β11 + β12 )(λ/4) [ { }]1/2 ( ) ± μ11 λ2 μ11 λ2 + γ 2 /4 ( 11 + 12 − 2δλ) .
(25)
where
and
Solving for P, we get
In view of (23) and (25), we have { } Ω = cg λ − μ12 λ3 − γ 2 (γ11 + β11 + β12 )(λ/4) [ ( ) }]1/2 { ± μ11 λ2 μ11 λ2 + γ 2 /4 ( 11 + 12 − 2δλ) ,
(26)
where Ω is the perturbed frequency. Instability occurs when ( ) { μ11 λ2 μ11 λ2 + γ 2 /4 (
11
+
12
} − 2δλ) < 0.
(27)
This condition is fulfilled when θ lies either in the range 0◦ ≤ θ < 70.5◦ or in the range 136.1◦ < θ ≤ 180◦ . From (27), the GRI Ωi , the imaginary part of Ω, becomes { [ ( ) Ωi = μ11 λ2 −μ11 λ2 − γ 2 /4 (
11
+
12
− 2δλ)
}]1/2
.
(28)
At marginal stability Ω is real, and from (26) it is given by / Ω=γ where
11
+ T
12
( 1− √ (
δγ 11 +
12 )T
) ,
(29)
480
T. Pal and A. K. Dhar
T = −2μ11 . The GRI Ωi against the perturbed wave number λ has been drawn in Figs. 1, 2 and 3 for some values of θ lying in the ranges 0◦ ≤ θ < 70.5◦ and 136.1◦ < θ ≤ 180◦ and wave steepness γ . From these figures we find that Ωi enhances with γ . We also observe that Ωi diminishes with the enhancement of θ , when θ lies in the range 0◦ ≤ θ < 70.5◦ , whereas it increases with the increase of θ , when it lies in the interval 136.1◦ < θ ≤ 180◦ . Further, fourth-order result gives a diminish in the growth rate for 0◦ ≤ θ < 70.5◦ , whereas this result gives an increase in the range 136.1◦ < θ ≤ 180◦ . In Fig. 4, we have plotted Ωi against λ for several values of wave steepness γ in the case of single wavetrain. From this figure we have found that Ωi increases with the enhancement of γ . It is to be noted that growth rate for two waves in the case of θ = 0 is much higher than those for single wave as shown from Figs. 1a, 2a, 3a and 4. Finally, in Fig. 5 we have plotted the perturbed frequency Ω at marginal stability against γ for some values of θ lying in the ranges 0◦ ≤ θ < 70.5◦ and 136.1◦ < θ ≤ 180◦ .
Fig. 1 Plot for growth rate Ωi versus λ for γ = 0.1; a θ = 0◦ , 20◦ , 40◦ , 60◦ and b θ = 140◦ , 150◦ , 160◦ . Dashed line indicates the third-order effect and solid line the fourth-order effect
Stability Analysis from Fourth-Order Nonlinear Multiphase Deep Water …
481
Fig. 2 Plot for growth rate Ωi versus λ for γ = 0.2; a θ = 0◦ , 20◦ , 40◦ , 60◦ and b θ = 140◦ , 150◦ , 160◦ . Dashed line indicates the third-order effect and solid line the fourth-order effect
5 Discussion and Conclusions In this paper, the fourth-order NLEE for multiphase deep water wavetrains in infinite depth of water are developed using multiple scale method. The ground for beginning from fourth-order NLEE is that it is an excellent starting point for analysing nonlinear effects in SGW on deep water. We have observed that fourth-order term in evolution equation gives significant change as compare to third-order term. This change is due to the influence of mean flow response corresponding to the terms which are connected to the Hilbert transforms.
482
T. Pal and A. K. Dhar
Fig. 3 Plot for growth rate Ωi versus λ for γ = 0.3; a θ = 0◦ , 20◦ , 40◦ , 60◦ and b θ = 140◦ , 150◦ , 160◦ . Dashed line indicates the third-order effect and solid line the fourth-order effect
Fig. 4 Plot for growth rate Ωi versus λ for γ = 0.1, 0.2, 0.3. Dashed line indicates the third-order effect and solid line the fourth-order effect
Stability Analysis from Fourth-Order Nonlinear Multiphase Deep Water …
483
Fig. 5 Plot of Ω against the wave steepness γ ; a θ = 0◦ , 20◦ , 40◦ , 60◦ and b θ = 140◦ , 150◦ , 160◦ . Dashed line indicates the third-order effect and solid line the fourth-order effect
Acknowledgements The grant of a Senior Research Fellowship provided by CSIR (India) to Tanmoy Pal is gratefully acknowledged.
Appendix The coefficients appearing in Eqs. (15) and (16) μ11 =
1 8
( ) 2 − 3cos2 θ2 , μ12 = 11
cos θ2 (6sin2 θ2 −cos2 θ2 ) , 16
= 2,
= 4(2 − cos θ2 − 2cos2 θ2 + 5cos3 θ2 −2cos4 θ2 − cos5 θ2 )(cos θ2 − 2)−1 , 12
484
T. Pal and A. K. Dhar
γ11 = −6cos θ2 , γ12 = −cos θ2 , ( )( ) θ θ 1 R + Scos β11 = − [2{ cosθ + 2cos 4 2 2 ( θ ) ( ) 2 cos + 2 θ θ θ ) cos 1 + sin2 − 2cos + ( θ2 2 2 2 cos 2 − 2 ) ( θ θ θ θ θ + 1 − cos sinθ sin − cos (7cos − 5 − cosθ sinθ + 2cos sinθ )} 2 2 2 2 2 ) θ ( θ 2θ cos + sin 2 1 − 2cos θ θ θ 2 ( 2θ )2 − cos {2cosθ sin2 − + 2cos3 − 1}], 2 2 2 cos 2 − 2
)( ) ( 1 θ θ β12 = − [2{ cosθ + 2cos R + Scos 4 2 2 ( ) ( ) ( ) 2 cos θ2 + 2 θ θ θ θ θ θ ) cos 1 + sin2 − 2cos + sinθ sin 3 − cos − 4cos2 + ( 2 2 2 2 2 2 cos θ2 − 2 ( ) θ θ θ } + cos 7 + cos − cos3 2 2 2 ( ) ⎧ ⎫ 2 θ 1 − 2cos θ + sin2 θ ⎨ ⎬ sin2θ 2cos θ 2 2 2 ( ) ], + cos −1 + ⎭ 2⎩ 2 cos θ − 2 2
β13
( ) 2cos θ2 cosθ cos θ2 + 2 θ 1 ( θ ) = − [2{sinθ cosθ sin + 4 2 cos 2 − 2 ( ) ( ) θ θ θ θ × 1 + sin2 − 2cos + sinθ sin + cos2 2 2 2 2 ( ) θ θ θ θ θ + 2cos 1 − 2cos } + cos {sinθ sin − 4sin3 2 2 2 2 2 ( ) θ θ 2θ 2cosθ cos 2 1 + sin 2 − 2cos 2 ( θ ) + cosθ − 2 + }], cos 2 − 2 δ = 2cos2 θ2 ,
where ( θ sin2 θ θ θ − sin2 − 1 R = sinθ sin + 2cos + 2 2 2 2 ( )) θ θ 2θ cos 2 1 + sin 2 − 2cos 2 ( θ ) + , cos 2 − 2
Stability Analysis from Fourth-Order Nonlinear Multiphase Deep Water …
485
(( ) )( θ θ 2θ cos − 2 S=( )2 1 + sin − cos 2 2 2 cos θ2 − 2 ( )) θ θ +2 1 + sin2 − 2cos . 2 2 2cos θ2
References 1. Longuet-Higgins M (1978) The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics. Proc Roy Soc Lond A Math Phys Sci 360:471–488 2. Longuet-Higgins M (1978) The instabilities of gravity waves of finite amplitude in deep water II. Subharmonics. Proc Roy Soc Lond A Math Phys Sci 360:489–505 3. Dysthe KB (1979) Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc R Soc Lond A 369:105–114 4. Janssen P (1983) On a fourth-order envelope equation for deep-water waves. J Fluid Mech 126:1–11 5. Stiassnie M (1984) Note on the modified nonlinear Schrödinger equation for deep water waves. Wave Motion 6:431–433 6. Hogan SJ (1985) The fourth-order evolution equation for deep-water gravity-capillary waves. Proc Roy Soc Lond A Math Phys Sci 402:359–372 7. Dhar AK, Das KP (1990) A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water. Phys Fluids A 2:778–783 8. Benjamin TB, Feir JE (1967) The disintegration of wave trains on deep water part 1. Theory. J Fluid Mech 27:417–430 9. Onorato M, Osborne AR, Serio M (2006) Modulational instability in crossing sea states: a possible mechanism for the formation of freak waves. Phys Rev Lett 96:014503 10. Shukla PK, Kourakis I, Eliasson B, Marklund M, Stenflo L (2006) Instability and evolution of nonlinearly interacting water waves. Phys Rev Lett 97:094501 11. Kundu S, Debsarma S, Das KP (2013) Modulational instability in crossing sea states over finite depth water. Phys Fluids 25:066605 12. Senapati S, Kundu S, Debsarma S, Das KP (2016) Nonlinear evolution equations in crossing seas in the presence of uniform wind flow. Eur J Mech B Fluids 60:110–118 13. Roskes GJ (1976) Nonlinear multiphase deep-water wavetrains. Phys Fluids 19:253–1254 14. Longuet-Higgins MS, Phillips OM (1962) Phase velocity effects in tertiary wave interactions. J Fluid Mech 12:333–336
Drop Size and Velocity Distributions of Bio-Oil Spray Produced by Airblast Atomizer Surendra Kumar Soni and Pankaj S. Kolhe
1 Introduction Due to the current environmental regulations on the discharge of harmful emissions from aircraft engines and power-producing plants into the atmosphere, government agencies put a lot of pressure on airline companies and aircraft engine manufacturers to look into the renewable liquid bio-fuels or optimize the current combustion system to reduce harmful emissions. Vegetable oils (VO) are an attractive alternative fuel for power generation due to their cost-effectiveness and ease of availability. The VOs have long hydrocarbon chains with small amounts of triglycerides that make them highly viscous. Therefore, VOs are challenging to atomize. The twin fluid configuration offers an efficient atomization process even for the adverse physical property fuel (viscous fuel) and at low flow rate conditions [1]. The aerodynamic force from the relative velocity destabilized the liquid jet and fragmented the liquid jet into the ligament and bigger droplets. If the aerodynamic potential is sufficient, then the resistive force causes the further breakup of the bigger droplets/ligament into the smaller droplets through the secondary atomization process. The airblast/air-assist atomizer falls under the category of the twin fluid atomizer, which can replace the pressure swirl atomizer. Airblast atomizer employs high kinetic energy of the atomizing air to fragment slowly moving liquid jet/sheets [2–5]. The viscosity has a significant effect on average droplet size, Sauter mean diameter (SMD) [6] and critical Weber number requirements [7]. According to their findings, the atomization process became more difficult as the liquid fluid’s viscosity increased. Airblast (AB) injector was used by Chong and Hochgreb [4] to examine the spray S. K. Soni (B) Department of Mechanical Engineering, IIT Kanpur, Kalyanpur-208016, India e-mail: [email protected] P. S. Kolhe Department of Mechanical and Aerospace Engineering, IIT Hyderabad, Kandi-502285, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_45
487
488
S. K. Soni and P. S. Kolhe
characteristics of rapeseed and Jet-A fuel. They claimed that different physical properties of the liquid fuel led to various droplet size distributions. Straight vegetable oil (VO) sprays generated by an airblast atomizer were studied by Panchasara and Agrawal [8] at various oil temperatures and air-to-liquid mass ratios (ALRs). For qualitative analysis, spray images were captured using laser sheet illumination. Using a Phase Doppler Particle Analyzer (PDPA) technology, quantitative pointwise measurements of droplet sizes and velocities in the complete cone spray were made. Both an increase in oil temperature and an increase in the air-to-liquid mass ratio (ALR) enhance atomization, particularly in the spray’s outer area. A coaxial AB atomizer was used by Urban et al. [2] to examine the spray characteristics of several fuels with varying surface tensions and viscosity. They claimed that although fuel preheating had an impact on droplet size distribution, the aerodynamics produced by atomizing air pressure was a more efficient method for obtaining the required droplet size. The drop size range of the spray ranges from a few microns to a sub-millimeter, with a poly-disperse drop size distribution. The entire set of spray data is difficult to manage. Mean drop size, SMD, and average and RMS velocities were the key spray characteristics used in earlier work to characterize the spray. For one average characteristic, several distributions can be constructed. The Sauter mean diameter (SMD), which makes up about 40% of the cumulative volume fraction, is unable to describe 60% of the larger droplets than the SMD [9]. Those bigger droplets may be harmful from the efficient combustion perspective. In the present study, we have characterized the spray characteristics of the vegetable oil and compared it with the baseline water spray by employing the twin fluid swirl airblast atomizer. Simultaneous drop size-velocity distributions are contrasted when the alternative fuel VO is substituted with water under the same operating conditions. The raster scan PDPA diagnostic is employed in the present investigations, which share the common advantage of being inherently non-intrusive.
2 Experimental Setup Figure 1a, b illustrates the schematics of the spray setup, which consists mainly of an air and liquid reservoir, peristaltic pump, mass flow controller, and commercial atomizer (Delavan-30609-3). The controlled amount of the dried and filtered atomizing air is supplied to the atomizer through the digital mass flow controller (MFC). The MFC-(ALICAT: MCR-100SLPM) is employed in the current study that can deliver the air with the accuracy of ±(0.8% of the reading + 0.2% of the full-scale reading). A positive displacement type of pump: peristaltic pump supplies the liquid fluid through the gearing action. By including the pulse dampener, the gearing action of the pump reduces the fluctuation in the liquid line. The two pressure gauges are installed in the air and liquid streamline near the atomizer to measure the pressure drop in both lines individually. The fixed amount of the liquid, 12 mL/min, is supplied to the atomizer. By altering the air mass flow rate for the constant liquid flow rate,
Drop Size and Velocity Distributions of Bio-Oil Spray Produced …
489
Fig. 1 a Experimental arrangement of the spray setup with the PDPA technique and b crosssectional view of the atomizer
the air-to-liquid mass ratio (ALR) varies in the range from 1 to 4. The choking of the air–liquid mixture limits the increase in the air mass flow rates beyond ALR of 4.
2.1 Measurement Techniques Figure 1a shows the integrated schematic of the PDPA system with the spray setup. The commercially available PDPA system (2C TSI FSA 3500) obtains spatially resolved information on the droplet size distribution and two velocity components simultaneously. The authors suggest to refer our earlier work for other PDPA details and PDPA settings and experimental arrangements [9]. An electronically controlled 3D traverse system (Holmarc) is used to sweep the spray setup assembly in both the transverse and longitudinal directions, while the PDPA is fixed to the optical table. The turbulent nature of the spray requires an adequate sampling size (data rate), carefully selected to obtain acceptable average measurements over time. The software is set up to collect data for measurements up to a time limit of 60 s or for 10,000 valid droplet counts. The axial direction (Z) of 10 to 50 mm with a (∆Z)
490
S. K. Soni and P. S. Kolhe
Table 1 Physical properties of the working fluids Working fluid
Dynamic viscosity μl (mPa·s)
Surface tension, σ (mN/m)
Density, ρ l (kg/m3 )
DI Water
0.78
72
998
Vegetable oil (VO)
4.971
30.6
925
10 mm interval is used for the spatially resolved raster scan measurements. For the water and VO spray, the radial interval (∆r) has moved in steps of 1 mm and 2 mm, respectively. The radial step is selected in the PDPA measurements, so the spray characteristics do not vary significantly. The center of the injector exit diameter is at locations r = 0 mm and Z = 0 mm. In both liquid fluid spray conditions, the Brewster angle setting is enabled for the highest data rate. For the VO and water sprays, the collection optics is positioned at an angle of 44° and 30°, respectively, from the transmission axis. In the current study, the testing fluids for creating the spray are deionized water and edible VO. The viscosity of both testing fluids is measured using the in-house facility of IIT Hyderabad. A rotational rheometer (Model: RheolabQC) is used to measure the viscosity of the two liquid fluids. The fluids’ surface tension and density are extracted from [10]. Details on the physical characteristics of the water and VO fluid are provided in Table 1. The value of 1.2 kg/m3 is considered for air density.
3 Results and Discussion The SMD distribution of VO spray is depicted in Fig. 2, along with a comparison to baseline water spray. The contour map demonstrates that the SMD value is less in the spray’s center and larger droplets are seen at its edges. The tangential momentum arises due to the swirling nature of the spray, which helps to move the bigger droplets in the outer region of the spray. The combined effect of the centrifugal and coalescence increases the droplet diameter, as the axial velocity is also small in the spray outer region (refer to Fig. 2). The size difference between the droplets in the center and those at the spray’s perimeter gradually flattens as axial distance increases (downward). The large droplets near the atomizer exit eventually break into the smaller droplets due to the ongoing secondary atomization process. The droplet size reduces first in the axial downstream direction due to the secondary atomization, reaching a minimum at Z = 30 mm, and then gradually grows in the downstream region. The secondary breakup, which initially results in a decrease in size, is most likely the reason for the drop diameter distribution. Later, the droplet size increases as a result of the combined effects of the droplet coalescence, deceleration, and evaporation phenomena [11, 12]. In comparison with VO spray, the water spray generates smaller droplets with a larger surface area, causing the water droplets to evaporate
Drop Size and Velocity Distributions of Bio-Oil Spray Produced …
491
Fig. 2 Droplet size (SMD) distribution for water and VO sprays
in the surrounding. Because of this, after secondary atomization, the SMD profile grows once again, although this is not the case for the VO spray. The viscosity of the liquid fluid slows the rate at which the surface instabilities grow. [13]. Radial expansion (low visual spray angle), which occurs for VO spray, prevents the liquid film from spreading radially. The probability of multiple droplets at a point increases when the spray cone angle reduces. As a result, when the water spray is changed to VO spray, the droplet diameter rises. The atomization process is influenced by the change in physical characteristics. The finer drop size distribution is produced by the increase in atomizing mass flow rates at constant liquid flow. The SMD distributions depict that the atomizing mass flow rate (ALRs) has a greater impact on the spray size distribution than the viscosity of the liquid. The atomization process for both the spray is considerably enhanced when the ALR is increased. High viscosity results in poor atomization quality, although this issue can be overcome by increasing the air mass flow rate. The larger droplet diameter that was present in the outer region of the VO spray dramatically decreased as the ALR increased. Further downstream of the injector exit, the droplet diameter distribution approaches a similar value for both the spray except the periphery of the spray. The operating condition is chosen to produce droplets in the size range of < 80 µm. To prevent the lean blowout limit during the combustion of liquid fuel, the aforementioned size range is likewise preferred [14]. The smaller droplets (< 80 µm) are desirable as these size droplets will evaporate fast and form a two-phase mixture efficiently. Additionally, these size range droplets help start and sustain the stable combustion operation. The mean axial velocity (U a ) distribution is presented in Fig. 3. At first, liquid droplets that were traveling slowly were transferred momentum from the fastatomizing air. The droplets accelerate and reach the velocity of the core air. Due to the momentum transfer between the droplet and the surrounding air, the core air’s
492
S. K. Soni and P. S. Kolhe
velocity decreases. As a result, the velocity profile flattens as the spray region moves downstream along the spray axis. At the spray centerline, the droplet velocity reaches its maximum value and then monotonically falls to its minimum value at the spray periphery. According to the theory of linear instability and other experimental findings [11], high shear and the interaction of the liquid and gas are advantageous for the creation of small droplets. Due to the smaller radial expansion, most of the VO spray droplets are entrapped in the core region of the spray, resulting in a relatively higher droplet velocity and droplet size. We have observed the maximum droplet velocity (12.05 and 20.22 m/s) and (14.78 and 21.24 m/s) at the spray centerline Z = 10 mm for water and VO spray at ALR of (1 and 1.5), respectively. Near the injector, the VO spray’s droplets travel faster than the water spray’s. The droplet velocity must increase to satisfy the continuity equation due to the reduced radial width. However, the difference in the velocity minimizes in the downstream location. Regardless of the liquid fuel (water/VO), droplet velocity has a positive correlation with rising ALR and a negative correlation with droplet size. For both the spray, the U a has the same profile but a distinct magnitude. Figure 4 shows the correlation between droplet diameter, axial velocity, and swirl velocity at the fixed axial position, Z = 10 mm, and radial position (r) = 2, 4, 6, and 8 mm. The intense swirl velocity occurs at the near nozzle region; hence, we have selected this axial position (Z = 10 mm). These radial locations correspond to where we have observed a significant asymmetry in the swirl velocity. Comparing the scatter plot for the liquid fluid, we can see that the water spray droplets have a substantially high swirl velocity at r = 2 mm. The asymmetry in the swirl velocity dictates the overall effective velocity. The effective velocity is negative in all the measurement locations as the measurement location falls across the centerline (r = 0 mm). The asymmetry gradually diminishes as one move in the radial direction
Fig. 3 Axial velocity distribution for water and VO sprays
Drop Size and Velocity Distributions of Bio-Oil Spray Produced …
493
Fig. 4 Droplet size and velocities (axial and swirl components) correlation at axial position (Z) of 10 mm and radial position (r) of 2, 4, 6, and 8 mm at ALR = 1.5 for water and VO sprays
because a free vortex is created at the spray’s periphery (r = 8 mm) location. The effective swirl velocity varies inversely with the radial locations. The two sprays significantly differ in the effective swirl velocity in the center region of the spray. The water spray has strong swirl momentum, which causes a large radial diffusion than the VO spray. We can see the suppressed swirl velocity for the viscous liquid fluid. The viscous liquid jet breakup process thus involves a small radial diffusion of the spray structure.
3.1 Simultaneous Droplet Size and Velocity Correlation The joint drop size and velocity probability distributions (JPDFs) are created using the simultaneous drop size and drop velocity information. Further, details about the droplet size and velocity JPDFs can be referred to from our previous work [9]. The MATLAB program is written to read the droplet diameter and velocity data as input. In the formulation of JPDFs, the droplet velocity and diameter range are considered from −300 m/s to + 300 m/s and 0 to 200 µm. The centerline variation of drop size and velocity JPDFs for VO spray with baseline water spray for ALR = 1 and 1.5, respectively, is shown in Fig. 5a, b. In the droplet size-velocity correlation, the horizontal and vertical dotted lines are drawn with reference to the average velocity and SMD, respectively. At the low ALR of 1, the peak probability of the dispersed droplets deviates from the mean value. The droplet size and velocity JPDFs show that the smaller droplets have a higher drop velocity than the larger droplets close to the injector zone. The smaller droplets strongly follow the gaseous flow field. The bigger droplets travel with lower velocity arbitrarily due to high inertia. At Z = 10 mm, a few larger droplets have a size larger than 50 µm are present after the breakup process. At the axial location, Z = 20 mm, the larger droplets are rarely present due to the ongoing secondary atomization process. The overall mean droplet
494
S. K. Soni and P. S. Kolhe
Fig. 5 The centerline variations of the droplet size-velocity correlation (JPDFs) at spatial location (i) Z = 10, (ii) 30 and (iii) 50 mm and a ALR = 1 and b 1.5 for water and VO sprays
size thus decreases at this location. There was no more droplet size reduction after Z = 30 mm; hence, the secondary breakdown process stopped. The bigger droplets are present at Z = 50 mm but not at the location (Z = 30 mm). The increase in droplet size may occur due to the combined effect of evaporation and coalescence. The scattered droplet for the VO spray is larger than the SMD for the water spray and falls around the mean velocity. In comparison with the reference value (dotted line value) at Z = 20 mm, the droplets that form after the disintegration are smaller and travel at a lower velocity. The joint probability corresponding to the higher droplet size and velocity has increased for the VO spray once the droplet size and velocity JPDFs are compared at the various axial locations. Because of the higher viscosity, the joint probability of corresponding to the larger droplet size has increased. Poor atomization results from the replacement of the liquid fluid with the viscous liquid fluid. Overall, though, the JPDFs distributions don’t change if the ALR is raised. It causes the JPDFs’ centroid to shift in the direction of smaller droplet sizes and higher velocities. When the ALR has increased, the majority of scattered droplets around the mean value have reached the mean droplet velocity. The probability of the larger droplets decreases with increasing ALR, while the probability of the smaller droplets increases. Increasing ALR improves the secondary atomization process overall. The hollow cone liquid distribution caused due to the swirling action results in a low
Drop Size and Velocity Distributions of Bio-Oil Spray Produced …
495
number count in the center of the VO spray (not shown). The majority of the liquid mass is located in the spray structure’s outer region, with only a minor proportion of it present at the center as smaller droplets. The majority of the droplets are sampled downstream for a fixed time limit. Hence, PDPA sampled fewer droplets in the nearfield central region of the spray within the set value of the time limit of 60 s at ALR = 1.5.
3.2 Secondary Atomization When an airflow of constant relative velocity (U a − U d ) abruptly exposes a liquid droplet of size d, the droplets break up if the shear Weber number (Wes ) is greater than the critical Weber number (Wescr ). The shear Weber number is written as Wes =
ρα (Uα − Ud )2 d σ
(1)
where ρ a , U a , U d , and σ refer to the air density, average air, droplet velocity, and surface tension between liquid and air interface. Based on the value of the shear Weber number, several breakup regimes, such as vibration, bag breakup, multi-mode breakup, catastrophic, etc., have been reported in the previous literature [7, 15]. The definition of the Weber number does not take viscosity into account. Viscosity’s effect is typically described using the Ohnesorge number (Oh). When the viscosity of the liquid droplets approaches zero limits, the critical Weber number (Wescr ) is the potential to break droplets of the diameter d. The critical potential at zero-limit viscosity must be increased with the additional potential necessary to overcome viscous dissipation and break the droplets. The modified critical Weber number (Wem scr ) accounting for the viscous effects is defined in Eq. 2 [15, 16]. c2 Wem scr = Wescr (1 + c1 × Oh ),
(2)
where c1 and c2 are the empirical constants. The critical shear Weber number (Wescr ) is ≈ 12 at Oh tx ). • The point of inflect is observed where the time t y and tx meets at a point where the roll waves transforms into slug. • The sequence of rise in wave crest, formation of jump, crest hitting the top wall of a pipe and slug formation is followed for all the cases of slugs above line TL2 (Fig. 4). • The cycle of convex and flat plateau interface of the wave crest continues till the initiation of jump. The jump initiation starts when the interface is flat and at the offset location from the middle towards windward side. Because drag force gets more area to act upon on the wave interface when it is flat as compared to the convex interface shape. • The jump is observed just before the formation of the slug. As the jump initiates, its velocity increases in the vertical direction and reduces in the horizontal direction. Jump time reduces with the increase in U SL . The time required for the jump is shorter than the time required for the formation of the wave crest. Acknowledgements The financial assistance from Science and Engineering Research Board (SERB), Department of Science and Technology (DST), India (sanction letter no. SB/S3/MIMER/ 0111/2013 dated 23-05-2014), is gratefully acknowledged.
References 1. Vaze MJ, Banerjee J (2011) Experimental visualization of two-phase flow patterns and transition from stratified to slug flow. Proc Inst Mech Eng C J Mech Eng Sci 225:382–389 2. Lighthill MJ, Whitham GB (1955) On kinematic waves I. Flood movement in long rivers. In: Proceedings of the royal society of London. Series A mathematical and physical sciences, vol 229, p 281–316 3. Kadri U, Mudde RF, Oliemans RVA (2007) On the development of waves into roll waves and slugs in gas/liquid horizontal pipe flow. In: ICMF 98 4. Woods BD, Hanratty TJ (1999) Influence of Froude number on physical processes determining frequency of slugging in horizontal gas–liquid flows. Int J Multiph Flow 25:1195–1223 5. Nydal OJ, Pintus S, Andreussi P (1992) Statistical characterization of slug flow in horizontal pipes. Int J Multiph Flow 18:439–453 6. Churchill SW, Churchill SW (1977) Friction-factor equation spans all fluid-flow regimes 7. Andritsos N, Williams L, Hanratty TJ (1989) Effect of liquid viscosity on the stratified-slug transition in horizontal pipe flow. Int J Multiph Flow 15:877–892
Validation of the Time Model in Gas–Liquid Horizontal Pipe Flow
525
8. Kadri U, Mudde RF, Oliemans RVA, Bonizzi M, Andreussi P (2009) Prediction of the transition from stratified to slug flow or roll-waves in gas–liquid horizontal pipes. Int J Multiph Flow 35:1001–1010 9. Thaker J, Banerjee J (2016) On intermittent flow characteristics of gas–liquid two-phase flow. Nucl Eng Des 310:363–377 10. Hurlburt ET, Hanratty TJ (2002) Prediction of the transition from stratified to slug and plug flow for long pipes. Int J Multiph Flow 28:707–729
Experimental Investigation on the Influence of Bed Height and Bed Particle Size on Bed Expansion for a Bubbling Fluidized Bed D. Musademba and Prabhansu
1 Introduction The quality of fluidization has been quantified with reference to the fluctuation of bed and their expansion ratios. Several investigations on the bed expansion ratio have been carried out [1–4] chiefly to describe bubble size and properties, as well as to check the average residence time of the gas in a typical bubbling fluidized bed. Mazumdar and Gunguly defined the bed expansion ratio, R, as the ratio of the difference between the expanded bed (He) and the static bed height (Hs) to the static bed height, i.e. R = (He − Hs)/Hs [5]. The bed expansion ratio (R) is also known as the ratio of the mean of the largest (H2) and smallest (H1) heights of an expanded to the initial static bed height (Hs) for a particular flow rate of gas, i.e. R = (H1 + H2)/2Hs [6]. The bed expansion ratio is one of the most reported properties of fluidized beds [7, 8] and hence should be very important. The knowledge about it is needed due to numerous reasons [9–12] highlighted that the bed expansion ratio is important in the designing of fluidized beds [9]. Hepbasli also noted that a good design for the cyclone separators is entirely dependent on the bed height expansion. It was further added that the automatic adjustments of the heat transfer to different loads on a boiler can be controlled as a function of the bed expansion ratio [13]. It has been noted that the expansion ratio for fluidized beds influences the correct positioning of heat exchanger tube bundles immersed in the fluidized bed, as well as the correct calculation of heat transferred to the furnace walls [12]. Bed expansion ratios give an D. Musademba School of Engineering Sciences and Technology, Chinhoyi University of Technology, 7724 Chinhoyi, Zimbabwe Prabhansu (B) Department of Mechanical Engineering, Sardar Vallabhbhai National Institute of Technology, Surat 395007, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_48
527
528
D. Musademba and Prabhansu
indication of the fluidization and its quality and hence directly affect the heat transfer coefficient [14].
2 Literature Review and Objective The successful and economical design, scale-up, and operation of fluidized bed reactors depend upon the true prediction of its bed hydrodynamics [15]. The hydrodynamic description is necessary for an in-depth understanding of the processes taking place in a fluidized bed. Philippsen et al. found that the knowledge of fluidized bed hydrodynamics is not only important in establishing correct operational parameters but is also useful for making decisions about reactor performance [16]. Zhang et al. investigated Geldart B fine particles (≤20 µm), i.e. Coal-15, GB- 6 and SiO2 -5, and added them to that of A (FCC-76) type to check the behaviour of fluidization [17]. It was found that the distribution of particle size affects the hydrodynamics of the system. Zhou and Zhu and Nguyen and Chian found that a higher conversion rate can be obtained through the addition of additives. It was found that when nano-particles (Group C+) were added to Geldart C powder, the gas-solid holdup increased for the dense phase and decreased for the bubble phase, resulting in more contact and conversion efficiency [18, 19]. The accurate prediction of bed expansion is therefore of prime importance in the design of fluidized beds. The objective of this study was to investigate the effect of bed height and particle size on bed expansion in a bubbling fluidized bed. Investigation on bed expansion properties is very important to expand the application of Geldart B particles in bubbling fluidized beds.
3 Materials and Methods Four grades of white fused alumina (White Aluminium Oxide) loose grains of the Geldart Type-B powders were used as the granular material for bed. The specifications for the material-alumina are given in Table 1. Table 1 Technical specifications for the alumina used in the hydrodynamic studies Grit size
Colour
Average particle size (µm)
Minimum particle size (µm)
Maximum particle size (µm)
Pour density approx. (kg/ m3 )
True density (kg/m3 )
MW54
White
320
210
460
1720
3700
MW60
White
250
177
390
1670
3700
MW80
White
177
125
274
1620
3700
MW100
White
125
74
194
1560
3700
Experimental Investigation on the Influence of Bed Height and Bed …
529
Fig. 1 Schematic of the fluidization and fluid bed heat transfer unit
The investigations were performed under ambient conditions in the Hilton Fluidization and Fluid Bed Heat Transfer Unit H694. The experimental set-up is shown schematically in Fig. 1. The unit consists of a strong glass fluidizing cylinder closed at the bottom portion by an air distributor system, whereas filter assembly at the top end. A pressure probe is fitted to measure the drop in differential pressure across the bed. The fluidizing air the effect of superficial gas velocity on bed expansion for white fused alumina bed particles was investigated first by determining the minimum fluidization velocity for the bed particles. The determinations for the minimum fluidization velocity for all experiments in this current work were carried out according to the standard procedure reported by Davidson and Harrison [20]. The standard procedure was adopted by several investigators [13, 21–24]. The minimum fluidization velocities for the bed materials employed at a bed height of 60 mm are presented in Table 3. The fluidizer was first charged with alumina particles of MW100 grit size to a height of 60 mm using the standard method reported by Canada, McLaughlin, & Staub [25] to determine the bed expansion. The static bed height was determined by applying an airflow rate above the minimum fluidization velocity for a short time so as to achieve a thoroughly mixed bed. The settled bed height was then measured from the transparent scale inscribed along the height of the fluidising column, and this was taken as the static bed height. The airflow rate controller was then adjusted to give an output of 10 l/min, and it was allowed to run for 3 minutes to achieve a steady state. Several measurements were taken for the minimum (hmin) and maximum (hmax) bed heights to determine fluctuation of bed surface at this airflow rate. The airflow
530
D. Musademba and Prabhansu
Table 2 Dimensions of the fluidization and fluid bed heat transfer unit Bed chamber parameter
Dimension
Nominal heightof fluidizing column
220 mm
Nominal diameter of the fluidizing column
110 mm
Cross sectional area
8.66 × 10–9 m2
Volume of the plenum chamber
3.33 × 105 mm3
Type of distributor plate
Perforated plate consisting of six discs of filter paper sandwiched between two stainless gauze discs, all enclosed by a U-shaped rubber ring seal
Table 3 Minimum fluidization velocities attained for bed particles at a height of 60 mm
Mean bed particle size (µm)
Minimumfluidization velocity (m/s)
(MW100) 125
0.06
(MW80) 177
0.09
(MW60) 250
0.11
(MW54) 320
0.128
rate was then raised incrementally in small steps up to 120 l/min at the same static bed height and measurements for hmin and hmax were taken. The bed height was then adjusted to 90 mm and 120 mm, respectively, by adding more MW100 grit size alumina bed material, and the measurement procedure was repeated for each of the new bed heights. The same procedure was repeated for MW80, MW60, and MW54 grit size bed material. The bed expansion ratio was determined from the average for the highest and lowest bed heights to the static bed height for a particular airflow by applying the expansion ratio, R = (hmin + hmax)/2hs [1].
4 Results and Discussion The variation in bed expansion ratio, R, with the gas velocity ratio, Uo/Umf, was determined for three different bed heights using fused alumina bed materials of mean particle diameters of 125–320 µm (Fig. 2a, b). Generally, the bed expansion is negligible at low gas velocity ratios (Fig. 2a, b) thereafter increasing linearly with superficial air velocity to values that seem to be somewhat dependent on the mean particle size diameter of the bed material as well as the bed height (Fig. 2c, d). The bed material remains in a static condition at gas velocity in the range 1–2 for the small particle size (125 and 177 µm), thereafter expanding to ratios between 1.3 and 1.5 depending on bed height (Fig. 2a, b). A low expansion of bed (ratio 1.2–1.4) is observed with large size (250–320 µm) material. The bed, however, remains static up to values of the gas velocity ratios between 5 and 8, thereafter showing marginal expansion depending on the bed height selected.
Experimental Investigation on the Influence of Bed Height and Bed …
531
1.4 60 mm 90 mm 120 mm
60 mm 90 mm 120 mm
1.4
Bed Expansion Ratio (R)
Bed Expansion Ratio (R)
1.6
1.2
1.0
(a)
1.2
1.1
1.0 0
5
10
15
Gas Velocity Ratio (Uo/Umf)
(b)
1.40
1.30 1.25 1.20 1.15 1.10 1.05 1.00
2
4
6
8
Gas Velocity Ratio (Uo/Umf)
1.25
60 mm 90 mm 120 mm
1.20 1.15 1.10 1.05 1.00
0.95 0
(c)
0
1.30
60 mm 90 mm 120 mm
1.35
Bed Expansion Ratio (R)
Bed Expansion Ratio (R)
1.3
2
4
6
8
10
Gas Velocity Ratio (Uo/Umf)
12
0
14
(d)
2
4
6
8
10
12
14
Gas Velocity Ratio (Uo/Umf)
Fig. 2 Comparison of the bed expansion ratio (R) as a function of Gas velocity ratio (Uo/Umf) for bed heights of 60, 90, and 120 mm with particles of the mean diameter of a 125 µm, b 177 µm, c 250 µm and d 320 µm
As can be noted the bed expansion is also inversely proportional to the bed height, with higher bed heights experiencing less bed expansion when compared to smaller bed heights. The initial static bed at low gas velocity ratios may be associated with the packed bed condition of relatively small particle size and low-pressure drop. The effect is more pronounced with large bed particles; here the slip velocity required to produce the all-important drag force is less than the gravitating particle velocity leading only to upward gas flow without any consequentially bed expansion effect. As the gas velocity ratio is increased, the packed bed is transformed into a fluidized state resulting in a linear bed expansion. Similar trends in bed expansion with gas velocity ratio have been reported [26, 27]. This is attributed to the number of gas bubbles formed above the distributor plate. In contrast to low fluidizing gas velocities, there is a considerable high number of bubbles generated at high gas velocities. These bubbles coalesce as they rise up the fluidizing chamber increasing the bed expansion ratio. It has also been observed that the bed expansion ratio is inversely proportional to the increase in bed height [8, 11,
532
D. Musademba and Prabhansu
28]. The decrease is attributed to an increase in bed mass which requires high gas velocities to fluidize it; more so, it offers high bed resistance owing to its increased pressure drop. This has a negative impact on the power required to fluidize the bed under high static bed heights. The relationship of bed expansion with bed height was also reported in previous studies [13, 29]. The authors also observed that bed expansion is inversely proportional to the bed height; they related the decrease in bed expansion ratio to the creation of high resistance to the flow of fluidizing gas owing to the increased static bed. A high bed offers a greater weight than a low bed; this implies that a higher drag force will be required to lift the bed. High operating velocities required for high beds result in the formation of large gas bubbles which creates large voids causing an appreciable reduction in the vertical lift of the particles. The effect is a sudden bed collapse due to the voids, resulting in the reduction of overall height of the expanded bed and consequently bed expansion. The variation of the bed expansion ratio with bed particle size at a bed height of 120 mm is presented in Fig. 3. At a fixed bed height of 120 mm, the bed expansion ratio remains constant up to a value of 3 of the Uo/Umf ratio, thereafter changing with a change in particle size. It was found that with the increase in particle size, the expansion of bed decreases. Beds of smaller particle sizes show a steep bed expansion rise with gas velocity ratio compared to those of larger particle sizes. At the same bed height and operating gas velocities, the bed expansion ratio is thus inversely proportional to particle size. It was also shown that bed expansion decreases with increasing particle size [30]. On the contrary, Glicksman, Yule and Dyrness concluded that no influence on bed expansion could be related to particle size when same excess gas velocity (U-Umf) is applied [2]. This discrepancy can be explained as arising from the different variable parameters used in the analysis. While Glicksman, Yule and Dyrness used the excess gas velocity (U-Umf) for comparing the bed expansion ratio with varying particle 1.6
Bed Expansion Ratio (R)
Fig. 3 Comparison of the bed expansion ratio (R) as a function of gas velocity ratio (Uo/Umf) at bed height of 120 mm for particles of the mean diameter of 125, 177, 250, and 320 µm
125 µm 177 µm 250 µm 320 µm
1.5 1.4 1.3 1.2 1.1 1.0 0
2
4
6
8
10
Gas Velocity Ratio (Uo/Umf)
12
14
Experimental Investigation on the Influence of Bed Height and Bed …
533
sizes; in this work gas velocity ratio (Uo/Umf) was used on the contrary [2]. Using the gas velocity ratio puts the comparison at the same footing since it considers the magnitude by which the gas velocity is applied as opposed to the unequal supply of excess gas velocity. The application of the excess gas velocity criterion does not provide an equal threshold increment for the comparisons given the different minimum fluidization velocities of the material. Since minimum fluidization velocity increases with particle size; it can be noted that the fluidization quality of two beds having different particle sizes cannot be compared on the basis of the same excess gas velocity [27]. Hence for the same bed height and operating gas velocities, the bed expansion ratio decreases with bed particle sizes. This is partly due to a reduction of fines in the bed constituents which are generally lifted up to relatively great heights. There is decreased bubble density when large bed particles are used. The large interstitial spaces formed do not promote bubble formation in the bed. While an increase in the particle size is designed to reduce the bed expansion ratio, the opposite is true when smaller particles are included [31]. In the latter, the interstitial gas flow increases by three orders of magnitude, resulting in an increased bed expansion ratio. Geldart also noted that fine particles have an effect of decreasing the viscosity, thus leading to the formation of numerous gas bubbles which on coalescing results in increased bed expansion [32].
5 Conclusions In this study, the effect on bed expansion ratios by varying the bed particle sizes and bed height is presented. The result indicates that the expansion ratio of bed increased with increasing superficial gas velocity. Within the range of the fluidizing velocities employed in the experiment, the bed expansion ratio was found to be inversely proportional to both bed height and particle size. Smaller bed particles gave rise to a higher bed expansion when compared to larger-sized bed particles. The use of bed particles of relatively large size resulted in reduced bed expansion owing to decreased bubble formation, a process necessary for lifting the bed. Higher gas velocities would be required to lift the bed of larger particles, implying the need for high power in order to fluidize the bed. Fluidized beds of large bed heights and particles generally require high pumping power since it is difficult to fluidize such eds; this entails high operational costs. The design and operation of the heat transfer tubes immersed in fluidized beds and the consequent heat transferred from the bed material were found to be influenced by bed expansion. This parameter has been found to increase with an increase in gas fluidization, bed height of a bubbling fluidized bed and size of bed particles used. Acknowledgements The authors would like to acknowledge the financial support provided by the Department of Science and Technology, New Delhi, India, under Visiting Fellowship grant for the CV Raman International Fellowship for African Researchers 2014. (REF: INT/NAI/CVRF/2014). The authors are also indebted to the Director of the CSIR-CMERI, Durgapur, India, the Director of
534
D. Musademba and Prabhansu
Sardar Vallabhbhai National Institute of Technology Surat, and Chinhoyi University of Technology Management for their support and encouragement.
References 1. Mohanty YK, Roy GK, Biswal KC (2009) Effect of column diameter on dynamics of gas-solid fluidised beds: a Statistical approach. Indian J Chem Technol 16:17–24 2. Glicksman L, Yule T, Dyrness A (1991) A prediction of the expansion of fluidised beds containing tubes. Chem Eng Sci 46(7):1561–1571 3. Johnsson F, Andersson S (1989) Expansion of a freely bubbling fluidized bed - Parts I and II. Report A89–178, Department of Energy Conversion, Chalmers University of Technology, Gothenburg 4. Godard K, Richardson JF (1969) Bubble velocities and bed expansions in freely bubbling fluidized beds. Chem Eng Sci 24:663–676 5. Mazumdar P, Ganguly UP (1985) Effect of aspect ratio on bed expansion in particulate fluidization. Canad J Chem Eng 63:850–852 6. Dora TKD, Mohanty YK, Roy GK, Sarangi B (2014) Prediction of bed fluctuation and expansion ratios for homogeneous ternary mixtures of spherical glass bead particles in a three-phase fluidised bed. Can J Chem Eng 92:536–542 7. Avidan AA, Yerushalmi J (1982) Bed expansion in high velocity fluidization. Powder Technol 32:223–232 8. Al-Zahrani AA, Daous MA (1996) Bed expansion and average bubble rise velocity in a gas-solid fluidized bed. Powder Technol 87:255–257 9. Cranfield RR, Geldart D (1974) Large particle fluidization. Chem Eng Sci 29:935–947 10. Hilligardt K, Werther J (1986) Local bubble gas hold-up and expansion of gas/solid fluidised beds. German Chem Eng 9:215–221 11. Johnson F, Andersson S, Leckner B (1991) Expansion of freely bubbling fluidized bed. Powder Technol 68:117–123 12. Oka SN (2004) Fluidized bed combustion. Marcel Dekker Inc., New York 13. Hepbasli A (1998) Estimation of bed expansions in a freely-bubbling three-dimensional gasfluidized bed. Int J Energy Res 22:1365–1380 14. Miller G, Kolar A, Zakkay V, Hakim S (1981) Bed expansion studies and slugging characteristics in a pressurized fluidized bed of large particles. Am Inst Chem Eng Sympos Series 77(205):166–173 15. Asghar U, Khan WA, Shamshad I Effect of initial bed height and liquid velocity on the minimum fluidization velocity (Umf) and pressure drop for the bed of semolina particles in liquid-solid fluidization. Am J Sci Eng Technol 1(2):201658–62 16. Philippsen CG, Vilela ACF, Zen LD (2015) Fluidised bed modelling applied to the analysis of processes: review and state of the art. J Market Res 4(2):208–216 17. Zhang X, Zhou Y, Zhu J (2020) Enhanced fluidization of group A particles modulated by group C powder. Pow Technol J 684–692 18. Zhou Y, Zhu J (2020) Different bubble behaviours in gas- solid fluidized bed of Geldart group A and group C+ particles. Pow Technol J 59:431–441 19. Nguyen V, Chian K (2021) Multiresolution analysis of pressure fluctuations in a gas–solids fluidized bed: application to glass beads and polyethylene powder systems. Int J Hydrogen Energy 20. Davidson JF, Harrison D (1971) Fluidization. Academic Press, London 21. Thonglimp V, Hiquily N, Laguerie C (1984) Minimum fluidization velocity and expansion of layers of mixtures of particulates solids fluidized by a gas. Powder Technol 39:223–239
Experimental Investigation on the Influence of Bed Height and Bed …
535
22. Dry RJ, Judd MR, Shingles T (1983) Two-phase theory and fine powders. Powder Technol 34:213–223 23. Sabreiro LEL, Monteiro JLF (1982) The effect of pressure on fluidized bed behaviour. Powder Technol 33:95–100 24. Rowe PN, Yacono CXR (1976) The bubbling behaviour of fine powders when fluidized. Chem Eng Sci 31:1179–1192 25. Canada GS, McLaughlin MH, Staub FW (1978) Flow regimes and void fraction distribution in gas fluidization of large particles in beds without tube banks. Am Inst Chem Eng Sympo Series 74(176):14–26 26. Almstedt AE (1987) Distribution of the gas flow in fluidised beds with a slugging behaviour. Chem Eng Sci 42(3):581–590 27. Sahoo P, Sahoo A (2013) Fluidisation and spouting of fine powders: a comparison. Adv Mater Sci Eng 369380/13 28. Hilal N, Gunn DJ (2002) Solids hold up in gas fluidized beds. Chem Eng Process 41:373–379 29. Babu SP, Shah B, Talwalkar A (1973) Fluidization correlation for coal gasification materialsminimum fluidization velocity and fluidized bed expansion ratio. Am Inst Chem Eng Sympos Series 74:176–186 30. Sahoo A (2011) Bed expansion and fluctuation in cylindrical gas solid fluidized beds with stirred promoters. Adv Powder Technol 22:753–760 31. Rowe PN, Santoro L, Yates JG (1978) The division of gas between bubble and interstitial phases in fluidized beds of fine powders. Chem Eng Sci 33:133–140 32. Geldart D (1973) The effect of particle size and size distribution on the behaviour of gas fluidised beds. Powder Technol 6:201–215
Study on Escapes Probability of Gas Bubble in Surge Tank Using Water Model Experiment P. Lijukrishnan, Indranil Banerjee, S. Manikandan, S. Rammohan, V. Vinod, and S. Raghupathy
1 Introduction The presence of argon bubbles in the secondary sodium circuit of SFR can cause various reactor operational difficulties majorly flow fluctuations and associated vibration issues in the sodium flow circuit. The source of these argon bubbles can be from the locked up gas inside the secondary sodium loop during filling. These entrapped bubbles can be transported to different locations in the secondary circuit. Surge tank is one important component in the secondary sodium circuit where argon cover gas plenum is provided above liquid sodium column for absorbing the pressure surges in loop. Once the bubbles reach the surge tank, depending on sodium flow rate, gas flow rate and bubble size, there is a probability of bubbles to escape into the cover gas plenum or carry through the flowing sodium in the system. The mechanism is depending on the buoyancy and inertia forces acting on these bubbles. The surge tank geometry also plays important role in deciding the escape probability. Therefore, if the gas bubble escape probability through surge tank is sufficiently high, then over time, the secondary sodium loop will be free from re-circulating gas bubbles. In this work, it is aimed to quantify the escape probability of gas volume circulation in the secondary sodium circuit through surge tank by experimental methods. The reference design of surge tank is taken from the prototype fast breeder reactor (PFBR), where the surge tank has a nozzle configuration of two bottom inlets and four side outlets [1]. And there are gas entrainment mitigation devices to eliminate direct entrainment of argon gas bubbles from cover gas plenum to sodium [2, 3].
P. Lijukrishnan (B) · I. Banerjee · V. Vinod · S. Raghupathy Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu 603102, India e-mail: [email protected] S. Manikandan · S. Rammohan Fluid Control Research Institute, Kanjikode West, Palakkad, Kerala 678623, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_49
537
538
P. Lijukrishnan et al.
2 Experimental Model Details The study is carried out in a large scale (5/8 scale) model of surge tank using water and air. The transportation of argon bubbles in sodium has been simulated by injecting air bubble to water through inlet lines of surge tank. Since the density difference between sodium and argon is nearby equal to that of water and air, argon–sodium mixture behave similarly with air–water mixture (variation of density difference ratio of mixtures is less than 15%). An arrangement for injection of measured quantity of air to both inlet lines of surge tank is provided in the facility. The schematic of overall arrangement for experimental study is shown in Fig. 1. PT2 T2 T
V3 GAS FLOW METER
COVER GAS VOLUME
FREE SURFACE SURGE TANK MODEL
OUTLET
OUTLET
SURGE TANK INLET
PT1 T1 T
Q1 AIR FLOW METER
AIR COMPRESSOR AIR FLOW
Fig. 1 Schematic of experimental model
SURGE TANK INLET
WATER FLOW
Q2 AIR FLOW METER
Study on Escapes Probability of Gas Bubble in Surge Tank Using Water … AIR PRESSURE REGULATOR
FLOW METER NON RETURN VALVE
539
TO SURGE TANK INLET 2
AIR COMPRESSOR
FLOW METER
AIR PRESSURE REGULATOR
NON RETURN VALVE
SURGE TANK INLET 1
WATER FLOW
AIR INJECTING TUBE
Ø5mm HOLE
Fig. 2 Arrangement of air injection to the inlets of surge tank
The air injection nozzles are connected to a compressor through valves and pressure regulators. The injection of air into the inlets has been measured using calorimetric air flow meters which measures both instant and cumulative flow at standard pressure and temperature. The top end of surge tank has been closed with a cover plate. The top cover plate has a tapping with a valve connected to air flow meter. Any additional air coming to the cover gas plenum is measured by the air flow meter. This air flow meter also measures the cumulative flow from cover gas plenum. The temperature and the pressure of the cover gas plenum also measured for converting the flow in to standard pressure and temperature. Three types of air injection nozzle arrangements are used for air injection in to the surge tank inlets, i.e., nozzles with small (1 mm—26 nos. of holes)-sized, medium (2.5 mm—4 nos. of holes)-sized, and large (5 mm—1 no. of hole)-sized holes. The number of holes in a nozzle for each sized holes has been decided such that each arrangement will have same flow resistance. The arrangement of air injection nozzle on inlets of surge tank is shown in Figs. 2, 3 shows the air injection nozzles/tubes used for studies [4].
3 Experimental Procedure Initially, the valve on the top flange of the surge tank is kept open, and the line is connected to the cover gas flow meter. Required flow of water has been delivered to the surge tank using pumps. The water flow has been measured using averaging pitot tube flow meter, and equal water flow is maintained in each inlet and all four outlets.
540
P. Lijukrishnan et al.
Fig. 3 Nozzles used for air injection
The nominal level of water (2.3 m) is maintained inside surge tank. Once the flow is stabilized, the air injection to both inlets is started. In the air injection line, by throttling the valves, same flow rate of air is injected to both inlets for duration of 5 min, and the same was monitored in air flow meters connected to inlet 1 and inlet 2. A portion of the injected air is escaped to the cover gas plenum and balance is taken away by the flow to the outlets and thereby back to sump. The escaped air bubbles coming out through the cover gas plenum have been measured using air flow meter connected with the cover gas plenum. The air flow meter showing cumulative air flow for duration of 5 min is converted in to instantaneous flow rate. The air flow meter in the inlet side gives air flow rate in standard pressure and temperature. The air outlet (from cover gas plenum) temperature has been measured as 33.6° C and the flow has been normalized in to flow at standard pressure and temperature. From the injected air flow rate and escaped air flow rate, the percentage of air escaped is estimated. The same procedure is followed for different water flow rates (1000–2500 m3 /h), different air injection flow rates (5–20 m3 /h) and different hole sized (1, 2.5, and 5 mm) nozzles, and the results are compared. The gas escape probability is given by Gas escape probability(%) =
Q out × 100 Q in
(1)
where Qin and Qout are the injected and escaped air flow rates normalized to STP conditions. Measurements are taken at least three times to ensure repeatability of measured data. The overall error in the measurement of gas escape probability has been estimated as 1.32% from the error percentage of two numbers of inlet air flow meters (±0.5%), cover gas air flow meter (±0.5%) and pitot tube water flow meter (±1%).
Study on Escapes Probability of Gas Bubble in Surge Tank Using Water …
541
35
5 mm hole 2.5 mm hole 1 mm hole
30
% of gas escape
25
3
Gas injection rate - 10 m /h
20 15 10 5 0 800
1000
1200
1400
1600
1800
2000
2200
2400
2600
3
Water flow (m /h)
Fig. 4 Variation of bubble escape probability inside surge tank with water flow rate
4 Results and Discussion 4.1 Effect of Water Flow Rate in Surge Tank From the experiments, it is found that for a constant rate of bubble injection to the surge tank, escape probability of bubbles reduces from 30 to 5% for an increase of water flow rate from 20 to 50% of nominal flow rate. When the water flow rate to the model is 50% of nominal flow rate and an air injection flow rate is 10 m3 /h, 6% of the bubbles escape to the cover gas plenum. When water flow rate to the surge increases, more bubbles are carried away by the flow to the outlets and the quantity of bubbles escaped to cover gas plenum reduces. By varying the hole sizes in the air injection nozzle, it is found that nozzles with higher hole size gives better escape probability compared to lower sizes for a constant air rate and constant water rate. Lower size bubbles are easily taken by the flow to the outlets. The variation of bubble escape probability inside surge tank with water flow rate is depicted in Fig. 4.
4.2 Effect of Air Injection Rate to the Surge Tank For a constant water flow rate to surge tank, when the air injection rate to the inlets increases, more bubbles are escaped to the cover gas plenum. The bubble escape probability is nearly 6% for a water flow rate of 2400 m3 /h (which is water flow rate
542
P. Lijukrishnan et al. 7
% of gas escape
6
5
4
5 mm hole 2.5 mm hole 1 mm hole
3
Water flow rate of 2400 m3/h
2 8
10
12
14
16
18
20
22
3
Air injection rate (m /h)
Fig. 5 Variation of bubble escape probability inside surge tank with air injection rate
corresponding to 50% of nominal secondary sodium flow rate) and air injection flow rate of 10 m3 /h [4]. The variation of bubble escape probability inside surge tank with air injection rate is depicted in Fig. 5 From the comparison of results with different holes sizes in the air injection nozzle, it can be observed that nozzles with higher hole size gives better escape probability compared to lower sizes. This study shows that PFBR surge tank is capable of releasing the at least 5% of gas bubbles present in the secondary sodium circuit to the cover gas plenum of surge tank in every pass.
5 Conclusions Experiments are carried out in 5/8 scale of surge tank model with water and air to study the escape probability of air bubbles coming from the surge tank inlets to the cover gas plenum. Water is made to flow into the surge tank, and air is injected to both the inlet lines of surge tank. Air injected to the surge tank and air released from the surge tank to cover gas plenum is measured. The study has been carried out by varying air injection rate as well as water flow rate in to the surge tank. All these experiments are repeated with three different nozzles with different hole sizes. During experimentation, the nominal water level in the surge tank is maintained. From the experiments, it is observed that for a constant rate of bubble injection to the surge tank, when the water flow to the surge tank is increased, escape probability of bubbles changes from 30 to 5%. When water flow rate increases the escape probability
Study on Escapes Probability of Gas Bubble in Surge Tank Using Water …
543
of bubbles reduces. When the air injection rate to the inlets increases for a constant water flow rate, more bubbles escape to the cover gas plenum. The bubble escape probability is nearly 6% for a water flow rate corresponding to 50% of nominal secondary sodium flow rate. It is concluded that the PFBR surge tank is capable of releasing at least 5% of gas bubbles present in the secondary sodium circuit to the cover gas plenum of surge tank in every pass.
Nomenclature Q
Air flow rate[m3 /h]
References 1. Ramdasu D, Shivakumar NS, Padmakumar G, Anand Babu G, Vaidyanathan G (2006) Devices to mitigate gas entrainment in the surge tank of PFBR. In 14th international conference on nuclear engineering, pp 55–61 2. IGCAR Report on Proposal for gas entrainment studies in 5/8 scale model of surge tank, EDG/ ETHS/99504/EX/3004/R-B (2003) 3. FCRI Report on Gas entrainment studies in surge tank, FCRI/PROJ/GES/06 (2007) 4. IGCAR Report on Bubble transportation study inside PFBR surge tank using water model experiment, PFBR/33150/EX/1018/RA (2019)
Mathematical Modelling and Optimization of Cylindrical Heat Pipe Dinesh Kumar Jain and A. V. Deshpande
Nomenclature le lad lc leff lt rv ri rw ro reff rn Av Aw d h T k
Evaporation length of heat pipe [m] Adiabatic length of heat pipe [m] Condensation length of heat pipe [m] Effective length of heat pipe [m] Total length of heat pipe [m] Cross-sectional radius of vapour core [m] Inner Container radius [m] Wick radius [m] Outer radius [m] Effective radius of capillary structure [m] Nucleation radius [m] Cross-sectional area of vapour core [m2 ] Cross-sectional area of wick [m2 ] Sphere diameter [m] Width of capillary structure [m] Temperature [ºC] Thermal conductivity [w/mK]
D. K. Jain (B) · A. V. Deshpande Department of Mechanical Engineering, Veermata Jijabai Technological Institute (VJTI), Matunga, Mumbai 400019, India e-mail: [email protected] A. V. Deshpande e-mail: [email protected] Present Address: D. K. Jain Department of Mechanical Engineering, Rizvi College of Engineering (RCOE), Bandra (W), Mumbai 400050, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_50
545
546
D. K. Jain and A. V. Deshpande
Fig. 1 Schematic representation of heat pipe [2]
ψ
Inclination of heat pipe with vertical axisdegree
1 Introduction A Heat Pipe transfers thermal energy of system from high temperature point to the low temperature point efficiently. Latent heat of the vaporized working fluid is utilized by heat pipe. The three components of heat pipe are [1]: 1. Container 2. Wick or capillary structure 3. Working fluid Schematic representation of Cylindrical Heat Pipe is depicted in Fig. 1. The exterior walls of Cylindrical Heat Pipe are splitted into three regions as evaporator region, adiabatic region and condenser region. The heat flux applied at the evaporator region is first conducted through the heat pipe wall and then the wick region. At the interface of the wick and the vapour region, working fluid vaporizes to vapour and flows to the vapour core which escalates pressure of vapour core. The vapour is forced into the condenser by the rising vapour pressure. There, the vapour condenses into liquid and flows back to the wick area, where it releases its latent heat of vaporization. On the other hand, the capillary pressure created by the menisci in the wick structure pumps the condensed liquid through the wick region and into the evaporation region.
2 Literature Review and Objective Using a mathematical model, Nemec et al. [3] developed and validated the heat transport constraints of heat pipes. The operation of a heat pipe depends on the design of the heat pipe container, wick structure, operating temperature range and thermophysical properties of working fluid. Figure 2 depicts a schematic of a heat pipe that is oriented at an angle with respect to the vertical axis. An analytical investigation of the viscous limit was provided by Busse [4]. Objective of this paper is to optimize Cylindrical Heat Pipe for good thermal performance.
Mathematical Modelling and Optimization of Cylindrical Heat Pipe
547
Fig. 2 Schematic of the wick heat pipe [3]
3 Mathematical Modelling 3.1 Cylindrical Heat Pipe Figure 3 shows the two-dimensional physical representation of the Cylindrical Heat Pipe.
Fig. 3 Schematic model of two-dimensional axis-symmetric cylindrical heat pipe [5]
548
D. K. Jain and A. V. Deshpande
Table 1 Thermo-physical properties of water T
ρL
σL × 10–1
kL
μl × 10–2
lv × 103
pv x 103
ρv
μv × 10–6
(°C)
kgm−3
Nm−1
Wm−1 K−1
Nsm−2
J·kg−1
Pa
kgm−3
N.s.m−2
20
998.2
0.728
0.603
0.1
2448
2
0.02
9.6
40
992.3
0.696
0.630
0.065
2402
7
0.05
10.4
60
983.0
0.662
0.649
0.047
2359
20
0.13
11.2
80
972.0
0.626
0.668
0.036
2309
47
0.29
11.9
100
958.0
0.589
0.680
0.028
2258
101
0.60
12.7
120
945.0
0.550
0.682
0.023
2200
202
1.12
13.4
140
928.0
0.506
0.683
0.020
2139
309
1.19
14.1
160
909.0
0.466
0.679
0.017
2074
644
3.27
14.9
180
888.0
0.429
0.669
0.015
2003
1004
5.16
15.7
200
865.0
0.389
0.659
0.014
1967
1619
7.87
16.5
3.2 Thermo-Physical Properties of Water Thermo-physical properties of water for operating temperature range from 20 to 200 ºC are considered for mathematical model and heat transfer limitations [1]. Thermo-physical properties of water are given in Table 1.
3.3 Heat Transfer Limitations of Cylindrical Heat Pipe with Sintered Capillary Structure The thermo-physical properties of the working fluid in the Cylindrical Heat Pipe, the thermal conductivity of the heat pipe material, heat pipe parameters and axial orientations of the heat pipe are required for calculations of heat transfer limitations/limits. Due to viscous, sonic, capillary, entrainment and boiling effects, heat transport in heat pipes is constrained. The working fluid, operating temperature, thermo-physical properties, wick structure and heat pipe parameters, all have a role in heat pipe limitations [1].
3.3.1
Capillary Limit
( ) 2 σl ρl lv K Aw ρl glt cos𝚿 Qc = − μl le f f re f f σl
Mathematical Modelling and Optimization of Cylindrical Heat Pipe
3.3.2
Viscous Limit Qv =
3.3.3
549
πrv 4 lv ρv Pv 12μv le f f
Sonic Limit Q s = 0.474 Av lv (ρv Pv )1/2
3.3.4
Entrainment Limit
( Q e = Av lv
3.3.5
Boiling Limit
ρv σl 2re f f
)0.5
( ) 4πle f f ke f f Tv σl 1 1 Qb = − lv ρv ln rrvi rn re f f
3.4 Parameters of Cylindrical Heat Pipe Parameters of Cylindrical Heat Pipe are given in Table 2 [5]. Table 2 Parameters of cylindrical heat pipe Cross-sectional radius of vapour core
m
rv
5.27E-03
Inner radius of container
m
ri
5.55E-03
Evaporation length
m
le
1.10E-01
Adiabatic length
m
lad
1.00E-01
Condensation length
m
lc
1.60E-01
Total length
m
lt
0.370
Effective length
m
leff
0.235
Cross-sectional area of vapour core
m2
Av
8.725E-05
Cross-sectional area of wick
m2
Aw
9.518E-06
Thermal conductivity of copper
Wm−1 K−1
km
387.6
550 Table 3 Parameters of capillary structure
D. K. Jain and A. V. Deshpande
Sphere diameter of wick particle
m
d
5.00E-05
Porosity of wick
–
ε
0.713
Width of capillary structure
m
h
2.80E-04
Permeability
m2
K
7.3342E-11
Capillary structure effective radius
m
reff
1.05E-05
Nucleation radius
m
rn
2.54E-06
Vapour saturation temperature
K
Tv
363
3.5 Parameters of Capillary Structure Capillary structure parameters of Cylindrical Heat Pipe are given in Table 3 [3].
4 Results and Discussion 4.1 Heat Transfer Limitations of Cylindrical Heat Pipe with Sintered Capillary Structure for 180° Axial Orientation Heat transfer limitations of Cylindrical Heat Pipe with sintered capillary structure for 180° axial orientation are shown in Fig. 4. Capillary limitation and boiling limitation are the main limitations for heat transport capacities of Cylindrical Heat Pipe at which Heat Pipe will function properly. Capillary limitation as a main limitation at temperatures from 20 to 187 ºC and boiling limitation as a main limitation at temperatures from 187 to 200 ºC for Cylindrical Heat Pipe.
4.2 Heat Transfer Limitations of Cylindrical Heat Pipe with Sintered Capillary Structure for 90° Axial Orientation Heat transfer limitations of Cylindrical Heat Pipe with sintered capillary structure for 90° axial orientation are shown in Fig. 5. Capillary limit is the only main limit for heat transport capacities of Cylindrical Heat Pipe at which Heat Pipe will function. Capillary limitation as a main limitation at temperatures from 20 to 200 ºC for Cylindrical Heat Pipe.
Mathematical Modelling and Optimization of Cylindrical Heat Pipe
551
Fig. 4 Heat transfer limitations of cylindrical heat pipe with sintered capillary structure for axial orientation 180° a actual view b enlarged view
4.3 Comparison of Capillary Limit of Cylindrical Heat Pipe on the Basis of Axial Orientation Comparison of Capillary limits of Cylindrical Heat Pipe with spherical diameter of 50 μm on the basis of axial orientations of 180º and 90º is shown in Fig. 6. Cylindrical Heat Pipe with 180º axial orientation has good heat transport capacity and thermal performance at operating temperature range.
4.4 Comparison of Heat Transfer Limitations of Cylindrical Heat Pipe on the Basis of Axial Orientations Comparison of Heat transfer limitations of Cylindrical Heat Pipe with spherical diameter of 50 μm on the basis of axial orientations of 180º and 90º is shown in
552
D. K. Jain and A. V. Deshpande
Fig. 5 Heat transfer limitations of Cylindrical Heat Pipe with sintered capillary structure for axial orientation 90°. a actual view b enlarged view
Fig. 6 Comparison capillary limit of cylindrical heat pipe on the basis of axial orientation (axial orientation 180º and 90º)
Fig. 7. Cylindrical Heat Pipe with 180º axial orientation has good heat transport capacity and thermal performance at operating temperature range.
Mathematical Modelling and Optimization of Cylindrical Heat Pipe
553
Fig. 7 Comparison heat transfer limitation of cylindrical heat pipe on the basis of axial orientation (axial orientation 180º and 90º)
Fig. 8 Comparison capillary limit of cylindrical heat pipe on the basis of axial orientation 90º for different spherical diameters of copper powder (50 and 56 μm)
4.5 Comparison of Capillary Limit of Cylindrical Heat Pipe on the Basis of Spherical Diameter of Copper Powder for 180º Axial Orientation Comparison of Capillary limit of Cylindrical Heat Pipe on the basis of axial orientation 180º for different spherical diameters of copper powder as 50 and 56 μm is shown in Fig. 8. Cylindrical Heat Pipe with 56 μm spherical diameter of copper powder has good heat transport capacity and thermal performance at operating temperature range.
4.6 Comparison of Cylindrical Heat Pipe on the Basis of Spherical Diameter of Copper Powder (56 µm) and Width of Wick Structure for 180º Axial Orientation Comparison of Capillary limit of Cylindrical Heat Pipe on the basis of different widths of wick structure (0.28 and 0.48 mm) for spherical diameter of 56 μm and axial orientation 180º is shown in Fig. 9. Heat Pipe with 0.48 mm width of wick
554
D. K. Jain and A. V. Deshpande
Fig. 9 Comparison of capillary limit of cylindrical heat pipe on the basis of spherical diameter of copper powder and width of wick structure for axial orientation 180º (spherical diameter 56 μm and width of wick structure, 0.28 and 0.48 mm)
structure has good heat transport capacity and thermal performance in operating temperature range.
5 Conclusion In general, the capillary limitation, entrainment limitation and boiling limitation have the biggest effects on heat pipes. Due to the wick’s structure, which impacts capillary pressure, the capillary limitation is the main limitation for Cylindrical Heat Pipes. The primary limitation that must always be taken into account while analysing heat pipe performance is the capillary limitation. From various graphs of heat transfer limitations, it is concluded that heat pipe with 180º axial orientation is more effective to remove heat. Heat pipe with 0.48 mm width of wick structure and 56 μm spherical diameter of copper powder will transport maximum heat from system to atmosphere. Heat pipe with 180º axial orientation, 0.48 mm width of wick structure and 56 μm spherical diameter of copper powder is optimized for good thermal performance.
References 1. Bahman Z (2016) Heat pipe design and technology, Springer Publication 2. Aishwarya SC, Deshpande A, Tendolkar M (2018) Dissertation titled parametric analysis of cylindrical heat pipe performance using CFD, VJTI Mumbai 3. Nemec P, Caja A, Malcho M (2013) Mathematical model for heat transfer limitations of heat pipe. Math Comput Model 57:126–136 4. Busse CA (1973) Theory of the ultimate transfer of cylindrical heat pipes. Int J Heat Mass Transf 16:169–186 5. Mehdi F, Mahdi Abdollahzadeh M, Ahmed A, GuangHan H, Gerardo C, Chen L (2016) Transient analysis of a cylindrical heat pipe considering different wicks structures. In ASME 2016 summer heat transfer conference HT2016
Anomalous Motion of a Sphere upon Impacting a Quiescent Liquid: Influence of Surface Wettability Prasanna Kumar Billa, Tejaswi Josyula, and Pallab Sinha Mahapatra
1 Introduction Understanding the dynamics of a solid impacting on liquids is of importance in broad areas of engineering, sports, and industries, such as drag reduction of swimmers and divers, entry and exit of the oars from the earth, ship slamming, the impact of boat hulls on water, extreme waves, weather on oil platforms, the locomotion of water-walking creatures, ink-jet printing, and spray adhesives. Dredging of solid particles and waste water discharge is two important applications of the solid–liquid interaction. One of the first studies reporting the impact of solids on quiescent liquids is by Worthington [1], where single-spark photography is used to investigate the motion of solid spheres descending vertically into the water. Later, Kuwabara et al. [2] studied the motion of a solid sphere impacting the water for the Reynolds number between 1500 and 40,000. The positions of the steel, glass, and nylon spheres are calculated from the successive photographs taken simultaneously with two cameras. Valladares et al. [3] investigated the motion of a solid sphere travelling through a viscous fluid numerically. To analyse the behaviour of different density materials, such as steel, lead, sand, and concrete, the Verlet algorithm is employed. Further, the terminal velocities of the spheres are computed inside the water and other viscous fluids by using the particle positions. Rubinow and Killer [4] numerically studied a spinning sphere moving in a stationary fluid. The Basset–Boussinesq–Oseen (BBO) equation is used for obtaining transverse forces acting on a sphere in a liquid. Firstorder Stokes approximation is employed to obtain the position of the sphere. Particle P. K. Billa · T. Josyula · P. S. Mahapatra (B) Micro Nano Bio Fluidics Group, Indian Institute of Technology Madras, Chennai 600036, India e-mail: [email protected] T. Josyula Institute for Technical Thermodynamics, Technical University of Darmstadt, 64287 Darmstadt, Germany © 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_51
555
556
P. K. Billa et al.
image velocimetry (PIV) experiments and Lattice–Boltzmann (LB) simulations on a single sphere settling under gravity are investigated by Ten Cate et al. [5]. A sediment sphere (Nylon) is used to study the behaviour under silicon oil of different viscosities. They have studied the sphere trajectories, flow field, and velocity for low Reynolds numbers (Re = 1.5–31.9) and low Stokes numbers (0.2–4). Total forces acting on a sphere and torque are investigated using LB simulations, and the sphere trajectories are observed experimentally with PIV. A complete study of the negatively and positively buoyant rigid spheres are reported by Horowitz et al. [6]. Different release techniques are used depending on the density ratio. The sphere with a density greater than the impacting liquid uses a top-releasing mechanism. A bottomreleasing mechanism is used for spheres with densities lower than the target liquid. An extensive study is conducted on vortex formation in the wake and the behaviour of spheres inside the liquid. According to the motion of the sphere, regime maps are used to visualise the wakes and trajectories of freely falling and rising spheres. For various Re, vorticity measurements and drag coefficients are studied. The major regimes of the sphere motions are as follows: vertical, oblique, intermittent oblique, and zigzag. Wake-induced oscillatory paths of the sphere, cylinder, and disk are investigated by Ern et al. [7]. The transverse displacement oscillations are observed for the various Reynolds numbers. Will and Krug [8] studied the dynamics of freely rising spheres and the effect of the moment of inertia in quiescent and turbulent fluids. A complete analytical solution of the sphere falling through the liquid is given by Guo [9] for various Reynolds numbers for the nonzero initial velocity. They have considered a rectilinear fall of a sphere onto a quiescent fluid. The BBO equation is solved for the acceleration of the spheres inside the viscous liquid. Hydrodynamic interactions of solid particles are studied by Mahapatra et al. [10], Harikrishnan et al. [11], Theofanous et al. [12]. Dredging of solid particles and waste water discharge is investigated by Carr et al. [13]. Using various analytical techniques, Nouri et al. [14] carried out a comprehensive study on the sedimentation of spherical particles in Newtonian fluid media. They presented three simple and exact analytical models for the problem of resolving the nonlinear equation of sedimentation. Lead, copper, and aluminium are impacted onto the water and solve the unsteady motion of the particle with the BBO equation by neglecting the Basset force. It is concluded that the analytical methods and numerical methods are able to predict the positions, velocities, and accelerations of the particles falling through the water. The impact of a solid mass on a transient cavity of air has been explored by Duclaux et al. [15]. For steel spheres, the formation of a transitory cavity for materials with carbon soot on their surfaces is explored. The effect of the cavity on the radius, density, and radius of the releasing sphere and cylinder is thoroughly discussed. Also, they constructed an analytical model to investigate the theoretical radius of the cavity as a function of time and the density of the releasing solid. The authors used potential flow to calculate cavity formation inside the liquid at various intervals. After impacting the quiescent liquid, the solid sphere entrains an air cavity behind the sphere. The formation of pinchoffs and cavity ripples is investigated by Louf et al. [16]. However, the authors have not discussed the dynamics of the sphere and how the surface wettability affects the
Anomalous Motion of a Sphere upon Impacting a Quiescent Liquid …
557
sphere trajectories. It is essential to study the effect of surface wettability modification on the dynamics of the sphere. The main goal of this study is to systematically investigate the effect of impact conditions and effective parameters on the underwater behaviour of a solid sphere. The experimental outcomes reveal that, while a sphere with higher density (ρ/ρ l ≥ 7.926) descends in a straight path, a sphere with lower density (ρ/ρ l ≤ 2.771) deviates from its vertical impact position. Additionally, a polynomial equation is used to investigate distinct stages in the evolution of penetration depth as a sphere moves through the quiescent water. The transition length lt , at which the transition occurs, i.e., the trajectory of the sphere deviates from a straight line path, is delineated for all the cases. It is evident that this lt is greatly influenced by the diameter of the sphere. Lastly, we demonstrate here that the observed anomalous motion of lowdensity spheres can be controlled by modifying the surface wettability of the sphere. It is observed that a superhydrophobic sphere of lower density follows a straight line path, which is attributed to the formation of an air cavity in the wake. Further, a superhydrophobic sphere moves faster than a non-coated sphere in quiescent water.
2 Experimental Setup and Methodology A. Experimental setup A schematic of the experimental setup is shown in Fig. 1a. The spheres are released into a water tank of 200 × 200 × 400 mm, which is made of clear, translucent acrylic material. A vacuum system is used to release the sphere with varying density and diameter, as shown in Fig. 1. This vacuum system is designed to create an impacting condition with zero initial velocity while ensuring no rotation is imparted onto the sphere. This is confirmed by observing the trajectory of the sphere prior to the impact on the liquid–air interface. Figure 1b shows various time instants during the motion of a sphere, with a black mark on the surface after the release from the pipette tip. A constant location of the black mark at different time instants highlights that there is no rotation of the sphere in the air before the impact. Before conducting the experiment, the water tank and the spheres are cleaned using isopropyl alcohol and ethanol. The length and width of the water tank are 20 times that of the diameter of the largest sphere used in the experiment. This results in a negligible influence of the walls of the tank on the trajectory of the spheres. All the experiments are conducted in a closed room with an ambient temperature of 25 ºC. An optical table (HOLMARC, India) is used to support the experimental apparatus, providing mobility and a vibration-free base. The motion of the sphere inside the liquid is recorded using a Phantom HighSpeed camera (VEO E-340L) with a Nikon Micro 60 mm lens. The camera records images at a rate of 1000 frames per second with a resolution of 768 × 1024 pixels and an exposure period of 990 µs. A 15-W recessed panel light is used to illuminate the back side of the tank.
558
P. K. Billa et al.
Fig. 1 (a) Schematic of the experimental setup and (b) Chronophotography of a sphere travelling in the air following the release from a pipette tip. The time step between the consecutive images is 33.33 ms
B. Methodology Preparation of SH spheres: A commercially available coating spray (NeverWet spray) is used to render the surface of the spheres as superhydrophobic. NeverWet spray is a two-step product system designed to create moisture repelling barrier on a variety of substrates. The following is the procedure for modifying the surface wettability of the sphere. The spheres are placed in a clean petri dish and are arranged so that they do not come into contact with each other or the wall. First, the base coat is applied, followed by the application of the top coat. The base coating is sprayed twice, and the spheres are then placed in the petri dish for one minute. A toothpick is then used to disperse the spheres. The process is repeated on the opposite surface of the spheres. The spheres are cured at ambient temperature for 30 min before applying the top coating in the same procedure followed for the base coat. The spheres are then placed in a dry environment for 12 h before doing the experiment. This results in a high water repellent surface. In the present study, polytetrafluroethylene (PTFE), glass, and steel spheres are used, which are procured from local vendors. The mass of a sphere (M) is measured
Anomalous Motion of a Sphere upon Impacting a Quiescent Liquid … Table 1 Properties of the spheres used in the present study
559
S. No.
Material
ρ/ρ l
d (mm)
1
PTFE
2.167
4, 6, 10
2
Glass
2.771
4, 6, 10
3
Steel
7.926
4, 6, 10
using an electronic weighing machine, and by measuring the diameter using a Vernier Calliper, the volume of the sphere (V) is calculated. The density of the spheres (ρ) . The measured densities are 2167, 2771, and 7926 kg/ is then calculated as ρ = M V m3 , PTFE, glass, and steel, respectively. Three different diameters are considered for each material, as given in Table 1. In the present study, three different impact situations are considered by varying the distance above the air–water interface from which the spheres are released. Consequently, the impact Weber numbers (We = ρv 2 d ) range from 68.11 to 204.37. Here, v is the impact velocity of the sphere, d is 2σ the diameter of the sphere, µ is the dynamic viscosity (0.89 mPa) of the water at 25ºC, ρ l is the density of water (1000 kg/m3 ), g is gravitational acceleration (9.81 m/ s2 ), and σ is surface tension of the air–water interface (0.072 N/m). The captured images from the high-speed camera are post-processed using ImageJ and MATLAB. Briefly, the background is subtracted from each image, and then, by thresholding based on the intensity, the location of the sphere is identified. Followed by this, the penetration depth (PD) is extracted by tracking the centre of the sphere.
3 Results and Discussion Figure 2 shows the motion of non-coated rigid spheres of different densities after impacting quiescent water, using the snapshots captured by the high-speed camera at different time instants. The effect of density is highlighted here by presenting the trajectories of spheres varying density, for a fixed sphere diameter and We. First, Fig. 2a depicts the chronophotography of a sphere with ρ/ρ l = 7.926. It is observed that the sphere moves in a straight line (black dashed line) without deviating from its initial impact point. Such a behaviour is expected for a sphere whose density is much larger than the liquid, and is in line with previous reports. This straight line trajectory of the sphere is further depicted in the figure in the last snapshot. Figure 2b depicts the chronophotography of spheres with ρ/ρ l = 2.771. It is observed that the sphere (ρ/ρ l = 2.771) is straying away from a straight line path. The same phenomenon is observed for spheres with ρ/ρ l = 2.167, as shown in Fig. 2c. Due to the high impact Weber number (We > 68), the wake of the sphere forms a wave motion in a plane containing its streamwise axis (Taneda [17]). This wake rotates slowly about that axis, resulting in a change in the path of the sphere from straight to random. The formation of vortices and wave motion for the different Reynolds numbers is investigated by Horowitz et al. [6]. In the present investigation, for current
560
P. K. Billa et al.
Fig. 2 Chronophotography of the impact of non-coated spheres with 10 mm diameter is shown for (a) Steel (ρ/ρ l = 7.926) (b) Glass (ρ/ρ l = 2.771) and, c PTFE (ρ/ρ l = 2.167), at an impact We of 204.37. The time duration between consecutive images is 8 ms for (a), 15 ms for (b), and 30 ms for (c). For the respective cases presented in (a), (b) and (c), (d) show the (PD) of the sphere with time (t). The black dashed line illustrates a straight path from the air–water interface during impact. The red arrow shows the deviation of the sphere from the vertical impact position, and the scatter points show the position of the centre of the sphere
We, lower density spheres deviate from a vertical impact position. The production of an asymmetric wake causes the sphere to deviate from the straight path. The variation of the penetration depth with time for the cases presented in Fig. 2a– c is presented in Fig. 2d. By observing the trajectories and analysing the penetration depth of spheres with varying densities, this deviation in the trajectory is inversely correlated to the density of the sphere. The influence of vortices on a sphere with ρ/ ρ l = 7.926 is less than others due to the relatively higher density than the impacting liquid, which reduces the effect of the vortices in the wake of the sphere so that the penetration depth from the air–water interface increases linearly. For the non-coated spheres ρ/ρ l = 2.167 and 2.771, the transition inside the fluid occurs at a precise point, where the sphere starts moving away from the vertical impact position. The wake of a sphere moves like a progressive wave in a plane where the streamwise axis passes through the middle of the sphere and moves abnormally in a different direction, as demonstrated in Fig. 2. The vortex loops diffuse rapidly at the wake of the sphere in the radial direction (randomly) as the impact velocity increases. As a result, a sphere impacting with high We on a quiescent liquid experiences a side force (oblique). The direction of this oblique force is entirely unpredictable. The sphere moves anomalously in a different direction, as demonstrated in Fig. 2a– c. The penetration depth increases nonlinearly from the air–water interface due to progressive wave motion in the wake of the spheres. It is also found that the time
Anomalous Motion of a Sphere upon Impacting a Quiescent Liquid …
561
required to reach a specific position is lower for the case of ρ/ρ l = 7.926 than the lower density spheres (Fig. 2d). Here, the deviation observed for lower density spheres is quantified in terms of the transition length (lt ). This lt is the location at which the deviation in the path is observed. Taking a clue from Fig. 2d, where the maximum deviation is observed for the lowest density spheres considered, the effect of sphere diameter and We is presented for the case of spheres with the lowest density (ρ/ρ l = 2.167). Overall, the diameter of the sphere has a greater influence on lt , compared to We as shown in Fig. 3. For a fixed We, the transition length increases as the diameter of the sphere increases. The increase in mass of the sphere as diameter increases will help sustain the upward lift force generated due to the difference in densities between the sphere and the liquid. Hence, larger diameter spheres can further continue on a straight path compared to spheres with smaller diameters. Interestingly, the effect of We is negligible on the lt . This can be explained by the fact that in the present study, We is varied by varying the height from which the sphere is released. Therefore, considering a smaller range of change in height employed in the present study, we believe the change in We is not significant to greatly influence the impact dynamics. As mentioned before, one novel aspect of the present study is the detailed analysis of the anomalous motion resulting from a deviation in the path for spheres of low density. To further highlight this outcome, here, the evolution of the penetration depth is thoroughly analysed to understand the patterns in the overall evolution of the trajectories of impacting spheres. Overall, three distinct stages in the evolution of the penetration depth are observed, as shown in Fig. 4. These various stages are identified based on the velocity of the sphere during the descent. For We = 68.11, after impacting the liquid, the sphere accelerates quickly (stage 1). A quadratic polynomial (PD = −1.0874 – 0.9526t + 0.0019t2 ) with a coefficient of determination of R2 = 0.99 is used to fit stage 1 (blue colour) starting from 0 to 122 ms. Fig. 3 For spheres with ρ/ρ l = 2.167, the transition length lt is plotted for different impact We. The effect of the diameter of the sphere is also presented. Evidently, the diameter of the sphere has a greater influence on the transition length when compared to the impact We
562
P. K. Billa et al.
Fig. 4 Evolution of the penetration depth of a 10 mm diameter sphere with ρ/ρ l = 2.167 is plotted for different impact We. Three distinct stages observed are plotted in different colours
When the sphere deviates from the straight path, the velocity of the sphere decreases gradually, as shown in Fig. 4. Stage 2 (red colour), starts at 122 ms and ends at 153 ms, where a quadratic polynomial (PD = −1.6568–1.2438t + 0.0038t2 ) correctly explains the evolution of penetration depth. After deviating from the path, in stage 3 (black colour), the sphere moves with a constant velocity. This results in a linear increase in the penetration depth. This linear variation (PD = −1.9379–1.4932t + 0.0015t2 ) of penetration depth in Stage 3 is further confirmed by conducting experiments with a larger field of view. Figure 4 also shows the effect of impact We. For brevity, the equations that fit the data are omitted for the higher We numbers. Overall, qualitatively, the three stages are observed here also. However, the total amount of time encompassing each of the stages reduces as the impact We increase. This is expected due to the fact that the kinetic energy of the sphere increases with increasing We, which results in a faster movement of the sphere inside the liquid. Similar stages are observed for other lower density (ρ/ρ l = 2.771) spheres also. Lastly, for the cases of steel spheres, where no deviation in path is observed, the penetration depth increases in an almost linear fashion, as can also be seen in Fig. 2 Surface wettability modification is an emerging field of study with various advantages such as drag reduction and low contamination. Such wettability modification is previously studied for impinging spheres (McHale et al. [18], Jetly et al. [19], and Vakarelski et al. [20]). The entry of a surface wettability modified sphere in an air– water mixture is explored using Particle Image Velocimetry by Yuan et al. [21]. Here, we study the impacting scenario of spheres with superhydrophobic coating on their surface. Figure 5 illustrates the differences in how non-coated and SH-coated spheres behave when submerged in water. The trajectory of the path followed by the spheres is also plotted. As explained before, the non-coated sphere shows a deviation from the interface (see Fig. 5a). However, a superhydrophobic sphere follows a straight line path (see Fig. 5b). The black dashed line shows the straight path of the sphere (ρ/ρ l = 2.771). The entrapped air causes an axisymmetric cavity in the wake of the
Anomalous Motion of a Sphere upon Impacting a Quiescent Liquid …
563
Fig. 5 Chronophotography of the impact of (a) non-coated and (b) superhydrophobic coated sphere (ρ/ρ l = 2.771) with d = 10 mm and We = 204.37. The time step between two consecutive images in (a) and (b) is 15 ms. A comparison of the evolution of penetration depth for both cases is shown in (c)
sphere, due to which the SH-coated spheres (regardless of sphere density) follow a straight line path. Entrapped air in the wake of the sphere as it falls through the water prevents the creation of a vortex because of the Cassie-Baxter state, as explored by Aristoff et al. [22]. Also, the volume of the cavity (entrained) at the wake tries to keep the sphere from moving away from the vertical impact position. Further, for the same initial conditions, a superhydrophobic sphere moves with a higher velocity than a non-coated sphere, which is the result of the reduction in drag. The evolution of penetration depth of the non-coated sphere (ρ/ρ l = 2.771) is depicted in Fig. 5c shows that the transition length of the SH-coated sphere is changed. The deviation of the sphere from the vertical impact position is not seen in SH-coated spheres because vortices will not directly act on the sphere. In SH-coated spheres, the threestage behaviour of non-coated spheres changes is not observed in SH-coated sphere for the densities (ρ/ρ l ≥ 1).
4 Conclusions A systematic experimental study is reported here showing the settling behaviour of rigid spheres of various densities. It is found that a deviation in path occurs for spheres with lower density (ρ/ρl ≤ 2.771), where such a deviation is not observed for spheres with high density (ρ/ρl ≤ 7.926). The transition length of a sphere, where the path of the sphere shows a deviation, depends more on the diameter and density than on the impact Weber number. Three distinct stages are observed in the evolution of the penetration depth, which is the distance from the air–water interface. As a future avenue of research, the radial random motion of the sphere can be examined
564
P. K. Billa et al.
and quantified with precision by employing two high-speed cameras, each oriented orthogonally to the water tank. One novel outcome from the present study is the difference in behaviour of descending spheres when the surface is superhydrophobic. The presence of an airentrapped cavity in the wake of a superhydrophobic sphere results in axisymmetry of the flow field, resulting in a straight line path for the trajectory of the sphere. Further, the superhydrophobic spheres move faster than a non-coated sphere, which further affirms the previously reported ability of superhydrophobic surfaces decreasing the drag. Additionally, unlike a non-coated sphere, the trajectory of the superhydrophobic does not exhibit a three-stage behaviour. Acknowledgements The authors acknowledge the funding received as part of the Institution of Eminence scheme of the Ministry of Education, Government of India [Sanction No: 11/9/2019 − U.3(A)]. The authors would also like to thank Dr. Imdad Uddin Chowdhury, Dr. Harikrishnan, and Mr. Saikat Halder for their assistance with experiments and insightful comments.
Nomenclature PD d lt t ρ/ρ l Re We
Penetration depth[mm] Diameter[mm] Transition length[mm] Time[ms] Density ratioReynolds numberWeber number
References 1. Arthur MW (1883) On impact with a liquid surface. Proc Roy Soc London 34:217–230 2. Kuwabara G, Chiba S, Kono K (1983) Anomalous motion of a sphere falling through water. J Phys Soc Jpn 52(10):3373–3381 3. Valladares RM, Goldstein P, Stern C, Calles A (2003) Simulation of the motion of a sphere through a viscous fluid. Rev Mex Fis 49(2):166–174 4. Rubinow SI, Joseph BK (1961) The transverse force on a spinning sphere moving in a viscous fluid. J Fluid Mech 11(3):447–459 5. Ten Cate A, Nieuwstad CH, Derksen JJ, Van den Akker HEA (2002) Particle imaging velocimetry experiments and latticeBotlzmann simulations on a single sphere settling under gravity. Phys Fluids 14(11):4012–4025 6. Horowitz M, Williamson CHK (2010) The effect of Reynolds number on the dynamics and wakes of freely rising and falling spheres. J Fluid Mech 651:251–294 7. Ern P, Risso F, Fabre D, Magnaudet J (2011) Wake-induced oscillatory paths of bodies freely rising or falling in fluids. Annu Rev Fluid Mech 44:97–121 8. Jelle W, Dominik K (2020) Rising and sinking in resonance: probing the critical role of rotational dynamics for buoyancy driven spheres. arXiv preprint arXiv:03643.
Anomalous Motion of a Sphere upon Impacting a Quiescent Liquid …
565
9. Guo J (2011) Motion of spheres falling through fluids. J Hydraul Res 49(1):32–41 10. Mahapatra PS, Manna NK, Ghosh K (2013) Hydrodynamic and thermal interactions of a cluster of solid particles in a pool of liquid of different Prandtl numbers using two-fluid model. Heat Mass Transf 49:1659–1679 11. Harikrishnan S, Pallab SM (2021) Effect of liquid–air interface on particle cloud dynamics in viscous liquids. Phys Fluids 33(6):063306 12. Theofanous TG, Yuen WW, Angelini S (1999) The verification basis of the PM-ALPHA code. Nucl Eng Design 189(1–3):59–102 13. Carr SA, Liu J, Tesoro AG (2016) Transport and fate of microplastic particles in wastewater treatment plants. Water Res 91:174–182 14. Nouri R, Ganji DD, Hatami M (2014) Unsteady sedimentation analysis of spherical particles in Newtonian fluid media using analytical methods. Propul Power Res 3(2):96–105 15. Duclaux V, Caille F, Duez C, Ybert C, Bocquet L, Clanet C (2007) Dynamics of transient cavities. J Fluid Mech 591:1–19 16. Louf J-F, Chang B, Eshraghi J, Mituniewicz A, Vlachos PP, Jung S (2018) Cavity ripple dynamics after pinch-off. J Fluid Mech 850:611–623 17. Taneda S (1978) Visual observations of the flow past a sphere at Reynolds numbers between 104 and 106. J Fluid Mech 85(1):187–192 18. McHale G, Shirtcliffe NJ, Evans CR, Newton MI (2009) Terminal velocity and drag reduction measurements on superhydrophobic spheres. Appl Phys Lett 94(6):1–4 19. Aditya J, Ivan UV, Sigurdur TT (2018) Drag crisis moderation by thin air layers sustained on superhydrophobic spheres falling in water. Soft Matter 14(9):1608–1613 20. Ivan UV, Evert K, Aditya J, Mohammad MM, Andres AA, Derek YC, Sigurdur TT (2017) Self determined shapes and velocities of giant near-zero drag gas cavities. Sci Adv 3(9):1–8 21. Yuan Q, Hong Y, Zhao Z, Gong Z (2022) Water–air two-phase flow during entry of a sphere into water using particle image velocimetry and smoothed particle hydrodynamics. Phys Fluids 34(3):032105 22. Jeffrey MA, John WMB (2009) Water entry of small hydrophobic spheres. J Fluid Mech 619:45–78
Analysis of Fission Gas-Fuel Particle Dispersion in a Voided Triangular Channel Under Sustained PCM Conditions B. Thilak, P. Mangarjuna Rao, and S. Raghupathy
Nomenclature K Q U0 a d m p t u v w1 w2 x,y,z Ω ρ μ ν λ τ
Stokes resistance coefficient [kgm− 1 ] Flow rate [m3 s− 1 ] Fission gas initial velocity [m2 ] Coolant channel side [m] Fuel particle diameter [m] Fuel particle mass [kg] Fission gas pressure [Pa] Time [s] Fission gas velocity [ms− 1 ] Fuel particle velocity [ms− 1 ] Z-component of fission gas velocity [ms− 1 ] Z-component of fuel particle velocity [ms− 1 ] Spatial coordinates [m] Coolant channel wallDensity of air [kg/m3 ] Rotor rotational speed [rad/s] Fission gas kinematic viscosity [m2 s− 1 ] Characteristic time constant [s] Fuel particle relaxation time [s]
B. Thilak · P. Mangarjuna Rao (B) Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India e-mail: [email protected] P. Mangarjuna Rao · S. Raghupathy Reactor Design & Technology Group, Indira Gandhi Center for Atomic Research, Kalpakkam, Tamil Nadu, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_52
567
568
B. Thilak et al.
1 Introduction The sodium fast reactor (SFR) is one of the six technologies identified for generation IV nuclear power plants by Generation IV International Forum [1] to ensure future energy security. A typical SFR with oxide core uses UO2 + PuO2 mixture as fuel and liquid sodium as coolant. In the event of prolonged loss of coolant flow and simultaneous unavailability of shutdown systems, termed Unprotected Loss of Flow (ULOF), the SFR core is exposed to a sustained power coolant mismatch (PCM) condition, where the thermal energy deposition continues without sufficient heat removal. This sustained PCM can result in core material (fuel, fission products and stainless steel) melting and termed as design extended conditions (DEC). Several inherent and engineered features of SFR core [2] make DEC extremely unlikely. However, knowledge about event progression path following the sustained PCM and the postulated release of radioactive material from the damaged core would help in effective mitigation planning for DEC. During ULOF, the continued thermal energy deposition leads to the overheating of the fuel pin. In the early stage, the inside of the fuel pin begins to melt and an internal cavity is formed. The internal cavity is filled with a mixture of molten fuel and fission gas. As the thermal energy deposition continues in the fuel pin, the cladding temperature approaches the melting point and the pressurization of the cavity leads to fuel pin failure. The molten fuel inside the pin is accelerates rapidly toward the pin failure location and ejected into the voided-liquid sodium coolant channel where it is dispersed axially. Figure 1 shows the schematic of a fission gas-fuel particle release and dispersal phenomenon in the liquid sodium coolant channel following fuel pin rupture. The fission gas-fragmented fuel particle axial dispersion process has a time scale of few hundred milliseconds. This fuel dispersal can lead to a large insertion of negative reactivity and neutronic shutdown of the core. Also, the released fuel particles are transported through the sodium pool due to bubble transport and contribute to the in-vessel radiological source term. The motion of fuel and sodium in coolant channels under accident conditions is generally analyzed using mechanistic codes such as SAS4A [3] or SIMMER [4]. Kenichi and Suzuki [5] discuss the dominant aspects of ULOF to be evaluated for SFR using a mechanistic code. A detailed investigation of ULOF with a combined mechanistic and statistical approach is presented in Droin et al. [6]. In the present work, an analytical model for gas-particle flow through a voided triangular coolant channel is developed using Eulerian–Lagrangian approach. The gas-particle flow is considered to take place under the influence of an exponentially decaying pressure gradient with a characteristic fission gas depressurization time. The analysis will be useful to understand the behavior of fuel particle in the voided coolant channels with triangular cross section. This paper gives the model details and parametric analysis results of a typical fuel particle release phenomenon in the voided SFR subassembly coolant channel.
Analysis of Fission Gas-Fuel Particle Dispersion in a Voided Triangular …
569
Fig. 1 Schematic of fission gas—fuel particle flow in liquid sodium coolant channel following fuel pin disruption
2 Equation of Motion for Gas-Particle Mixture Flow The transient flow of gas-particle mixture in the coolant channel involves complex interaction between the gas and particle phases, and significantly influences the dispersal behavior of core materials in the damaged core. A significant reduction in the resistance coefficient of gas-particle flow through a circular channel compared to the clean gas flow has been reported in the literature [7]. This is due to the attenuation of gas turbulence by the heavier particles. Therefore, the gas-particle flow in the voided coolant channel is considered laminar. The equations of motion for an unsteady, incompressible viscous gas-particle mixture under dilute flow regime are given by [8]: 1 μ ∂u + (u.∇u) = − ∇ p + ∇ 2 .u ∂t ρ ρ
(1)
∇.u = 0
(2)
570
B. Thilak et al.
| ∂v + (v.∇v) = K (u − v) m ∂t |
(3)
Here u and v are velocity vectors of fluid (fission gas) and fuel particles, respectively. p is fluid pressure, m is mass of fuel particles and K is Stokes resistance coefficient (3π μd). μ and ρ are gas viscosity and density, respectively. Figure 2 shows the idealized geometry of the voided channel cross section used in the analysis. The gas-particle mixture flows through the channel with an equilateral triangle cross section. The length of each side of the equilateral triangle is denoted by “a”. The channel axis coincides with the z-axis and the velocity components and pressure gradients along x–y plane are considered zero. The velocity distribution of gas and particles is defined as: ux = 0; uy = 0; uz = w1 (x, y, t)
(4)
vx = 0; vy = 0; v3 = w2 (x, y, t)
(5)
( ) ( ) where ux , uy , uz and vx , vy , vz are velocity components of gas and particles, respectively, in Cartesian coordinates (x, y, z). Using Eqs. 1–5 can be expressed as: −
Fig. 2 Geometry of triangular channel
1 ∂p 1 ∂p = 0; − =0 ρ ∂x ρ ∂y
(6)
Analysis of Fission Gas-Fuel Particle Dispersion in a Voided Triangular …
) ( 2 ∂ w1 1 ∂p ∂w1 ∂ 2 w1 =− +ν + ∂t ρ ∂z ∂x2 ∂y2 m
∂w2 = K (w1 − w2 ) ∂t
571
(7) (8)
Here, v is kinematic viscosity of fission gas. The geometry of triangular channel cross section of side “a” with base resting on x-axis (y = 0) is shown in Fig. 3. The pressure gradient is assumed to be exponential [9] and the fluid velocity can be expressed as follow: w1 = f(x, y) e− λ t
(9)
where λ is characteristic time of decaying time constant. The channel walls of the triangular channel, /, where the no-slip wall boundary condition is applied using a weighting function: )( √ ) 18 3 y a −x + √ − a3 3 2 ( )( √ ) 1 3 1 2 1 2 18 3 2 = x y + y + √ ay − a y 3 4 a3 3 (
y a /=y x+ √ − 3 2
)(
(10)
(11)
The first two terms in Eq. (11) constitute a harmonic function with zero Laplacian. Therefore, using Eqs. (7), (9) and (11), the fluid velocity can be expresses as follow: Fig. 3 Equilateral triangle coolant channel cross section
572
B. Thilak et al.
Fig. 4 Contours of equal velocity in triangular cross section with no-slip wall boundary condition
w1 = U0 /e− λ t
(12)
where U0 is the initial gas velocity at the center of the channel, ) ( 1 ∂p 2aν −1 U0 = √ ρ ∂z 3
(13)
Figure 4 shows the contours of equal velocity for the equilateral triangle channel cross section. The contours of equal w1 change from circle near the center to triangle at the walls. The flow rate through the equilateral triangular channel is −λt
Q = 2U0 e
0
√ =
a/ √3( a −x ) 2 2 ∫ ∫ /dxdy
(14)
0
3U0 a2 − t e λ 9
(15)
Rewriting Eq. (3), which defines the particle motion in Lagrangian frame [10] and integrating with respect to time, an expression for particle velocity, w2 is obtained: m
dw2 = K (w1 − w2 ) dt
(16)
t dt dw2 =∫ (w1 − w2 ) 0 τ
(17)
w2
∫ 0
Analysis of Fission Gas-Fuel Particle Dispersion in a Voided Triangular …
( ) t t w2 = U0 /e− λ 1 − e− τ
573
(18)
Here, τ (= m/K ) is fuel particle relaxation time. Equations (12) and (18) are used to evaluate the individual phase velocities of a gas-particle mixture flowing through a triangular channel under exponentially decaying driving pressure gradient.
3 Model Parameters The fuel particle release and dispersion process depend on the effective smear density of fuel material and its decrease due to irradiation. The axial and radial distribution of the fission gas and volatile fission products in the fuel pin before failure and the fuel enthalpy increase rate during the fuel pin failure also plays a dominant role. Based on the above considerations, a parametric analysis is carried out to evaluate the gas-particle mixture flow under an exponentially decaying pressure gradient. Model parameters used in the analysis are given below: (a) (b) (c) (d) (e) (f)
Triangular channel side, a = 5 mm Pressure decay time constant, λ = 0.1 s Viscosity of fission gas, μ = 10–4 Pa s Initial velocity at centroid, U0 = 5 ms−1 Fuel particle density, ρ = 8600 kg m−3 Fuel particle diameter range, d = 50 to 1000 μm
4 Results and discussion Figure 5 shows the unsteady velocity distribution of fission gas along the y-axis (x = 0) for an exponentially decaying pressure gradient with the characteristic time constant ) s. The peak velocity is observed at the centroid which has coordinates ( / of 0.1 √ 0, a 2 3 , where a is the side of the triangle. The momentum transfer from the gas flow to particles due to drag generates particle motion described using Eq. (18). The particle flow dynamics in the gas depends on the relative magnitude between pressure decay time constant, λ and particle relaxation time, τ. When λ > τ , the fuel particle behaves like gas-particle, and heavier particle (i.e., λ < τ ) shows minimum excitement for the driving pressure. Figures 6 and 7 show the fuel particle velocity and displacement in the triangular coolant channel under exponentially decaying pressure gradient with a characteristic time of 0.1 s. The particle diameter range chosen is 50 to 1000 μm. Results show that the peak velocity obtained by the 50 μm particle is 3.35 ms-1 and the total displacement is 0.41 m at 0.3 s from the transient initiation. The peak velocity and total displacement decreases with increase in particle diameter and becomes negligible when the particle diameter reaches 1000 μm. Therefore, particles of diameter greater
574
B. Thilak et al.
Fig. 5 Variation of fission gas velocity distribution along y-axis (x = 0) with time
than 1000 μm are not expected to participate in the fuel particle dispersion under the chosen condition. The drag forces in the analysis are based on steady-state drag law for a single spherical particle and applicable at channel cross sections with low particle concentration (i.e., few centimeters away from the fuel pin rupture location). In the vicinity of rupture location, the fuel particle transport is influenced by interaction between
Fig. 6 Fuel particle velocity in triangular channel following pin rupture (d = fuel particle diameter)
Analysis of Fission Gas-Fuel Particle Dispersion in a Voided Triangular …
575
Fig. 7 Fuel particle displacement in triangular channel following pin rupture. (d = fuel particle diameter)
particles. So the particle transport near the rupture location is generally analyzed with drag laws based on particle concentration [11], which is not in the scope of present analysis. The present axial fuel dispersion model can be integrated with a fission gas-fuel particle release model to evaluate the time-dependent fuel distribution in the voided coolant channel and its reactivity feedback due to the fuel pin rupture. The fuel particle transport in the present model is based on analytical solution and the fuel particle velocities are consistent with the experimental results available in the literature [12]. Hence, the model predictions are useful for the validation of the integrated axial fuel dispersion model.
5 Conclusions The unsteady flow of fission gas-fuel particle flow through a triangular channel under the influence of an exponentially decaying pressure gradient has been investigated. The driving pressure characteristics time is typical of fission gas release due to a fuel pin disruption process under sustained power coolant mismatch. The time-dependent gas and particle velocities are evaluated for a pressure decay time constant of 0.1 s and the fuel particle diameter range from 50 to 1000 μm. Results show that the fuel particle velocity and displacements for particles of diameter larger than 1000 μm are negligible and not expected to participate in the fuel particle dispersion following the sustained PCM resultant pin rupture. Also, dispersion behavior of fuel particles less than 100 μm is identical to that of fission gas. The result brings out the influence of pressure decay time constant and the fuel particle relaxation time on the behavior of
576
B. Thilak et al.
gas-particle mixture flow following fuel pin rupture. It contributes to the definition of initial condition for the estimation of radiological source term, due to refractory materials. The result will be used in the validation of the axial dispersion model developed for the analysis of dispersive characteristics of fragmented fuel particles under sustained PCM conditions in SFR.
References 1. Kelly JE (2014) Generation IV international forum: a decade of progress through international cooperation. Prog Nucl Energy 77:240–246 2. Cagliostro DJ, Florence AL, Abrahamson GR (1974) Characterization of an energy source for modeling hypothetical core disruptive accidents in nuclear reactors. Nucl Eng Des 27:94–105 3. Weber DP, Birgersson G, Bordner GL (1985) SAS4A LMFBR whole core accident analysis code, CONF-850410–23. Argonne National Lab 4. Kondo S, Tobita Y, Morita K, Shirakawa N (1992) SIMMER-III: An advanced computer program for LMFBR severe accident analysis. In: ANP’92 international conference on design and safety of advanced nuclear power plants 5. Kenichi K, Suzuki T (2021) Study on dominant aspects of unprotected loss-of-flow to be evaluated in the initiating phase for a sodium-cooled fast reactor. J Nucl Sci Technol 58(3):347– 360 6. Droin JB, Marie N, Bertrand F, Marrel A, Bachrata A (2020) Design-oriented tool for unprotected loss of flow simulations in a sodium fast reactor: validation and application to stability analyses. Nucl Eng Des 361:110550 7. Gore RA, Crowe CT (1991) Modulation of turbulence by dispersed phase. J. Fluids Engg 113:304–307 8. Gupta RK, Gupta SC (1976) Flow of a dusty gas through a channel with arbitrary time varying pressure gradient. Zeitschrift für angewandte Mathematik und Physik ZAMP 27(1):119–125 9. Wider HU (1987) Fuel modelling in fast reactor whole core accident calculations. Nucl Eng Des 101(3):363–377 10. Crowe C, Sommerfeld M, Tsuji Y (2011) Multiphase flows with droplets and particles. CRC Press, USA 11. Wallis GB (2020) One-dimensional two-phase flow. Courier Dover Publications 12. Shirolkar JS, Coimbra CFM, McQuay MQ (1996) Fundamental aspects of modeling turbulent particle dispersion in dilute flows. Progr Energy Combust Sci 22(4):363–399
Experimental Investigation of Droplet Spreading on Porous Media Anushka, Prashant Narayan Panday, Prasanta Kumar Das, and Aditya Bandopadhyay
1 Introduction Coatings, printing, and painting are all applications for spreading liquids over solid surfaces [1–3]. Most of the literature has considered spreading over smooth homogeneous surfaces [4–6]. Singularity can be removed from three-phase contact lines by the action of surface forces. A large percentage of solid surfaces are either porous or covered with a thin layer of porous material while most solid surfaces are rough to some extent. The differentiating characteristics of porous substrates include their porosity, size of the pore, permeability, and pore morphology. X-ray tomography and confocal microscopy can be utilized to conduct experimental inspections of porous surfaces with interconnected pores, such as paper, rock, or sponge. The spreading parameters are significantly changed when roughness or a porous sublayer is present [7–10]. It is crucial to take into account the system’s physical and chemical properties, as well as the porous medium’s geometrical structure, in order to wet and absorb liquids on porous surfaces. The effects of incorporating a surfactant to enhance a liquid’s spreading capacity are well-known when evaluating wettability. In the sections that follow, we use liquid with various characteristics to examine how these changes impact the kinetics of spreading across a porous substrate. Inks are frequently supplemented with surfactants during printing to control drop growth, spreading, and substrate imbibition [11]. Two spreading behaviours that are distinguished and revised are competitive wetting and particle wetting instances. The difference between cases of complete and partial wetting can be easily distinguished when spreading over non-porous substrates: When a droplet is completely wet, it spreads out completely, but when a droplet is only partially wet, it ceases to expand after the static advancing
Anushka (B) · P. N. Panday · P. K. Das · A. Bandopadhyay 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_53
577
578
Anushka et al.
contact angle is reached. The two spreading regimes over porous substances are sometimes difficult to differentiate. Starov et al. [12] offered recommendations on how to distinguish between complete and partial wetting situations while dispersing droplets across porous material. In the partial wetting instance, there are three different spreading regimes. The contact angle decreases during the first stage from the initial, adequately high contact angle immediately after deposition to the static progressing contact angle. The droplet base remains stationary during the second stage, during which the contact angle steadily decreases until it approaches the static receding contact angle. Stage three of the spreading/imbibition process is when the contact angle stays constant and equal to the static receding contact angle. Contact angle hysteresis is the key attribute of partial wetting, as shown by the spreading and imbibition processes for partial wetting that were previously discussed. In the event of total wetting, there are only two spreading regimes present, and they are distinguished by the absence of contact angle hysteresis. As the contact angle decreases throughout the first stage, the spreading area increases until it reaches its maximum value. However, the consistent behaviour of the contact angle during the second stage is due to hydrodynamic forces rather than contact angle hysteresis, which is missing in the occurrence of complete wetting. The contact angle is essentially constant at this point. The flow within the droplet, the flow within the porous substrate, and the coupling of these two flows through boundary conditions at the interface between the droplet and the porous substrate must all be described in a theoretical model of the concurrently processed spreading and imbibition of droplets over a porous substrate. Equalized velocities and viscous stress are two necessary boundary conditions at the contact between the droplet and the porous substrate. Since the Navier–Stokes equations are employed to describe this flow and their application to the flow inside droplets is well-known, there is no problem with the description of the flow inside a droplet. Now, one of the following models must be taken into consideration in order to determine the flow inside the porous substrate: The substrates are portrayed in one-dimensional models as an array of capillaries with identical radii [13, 14], Bethe lattice models, which take interconnection into account [13], and network mod else, which further encourage the existence of interlinking pores [15], in addition to the benefits of the Bethe lattice model. The Koch curve and random walk [16] methods were used to improve the network model and integrate roughness. Darcy equations are frequently used to describe flow within the porous substrate. The Darcy equations are second order differential equations, whereas the Navier–Stokes equations are of first order. In other words, using Darcy equations, the contact among a droplet and a porous substrate cannot satisfy two boundary requirements. It has been proposed to model the flow inside a porous substrate using Brinkman’s equations, which are second order differential equations. Therefore, Brinkman’s equations can be used to fulfil both boundary conditions at the point where a droplet and a porous substrate come into contact. This is the rationale underlying Brinkman’s Equations [17], which are frequently used to explain fluid flow in porous material. Two processes govern the growth of drop bases: liquid infiltration into the porous layer and expansion over
Experimental Investigation of Droplet Spreading on Porous Media
579
already saturated parts of the porous layer. These opposing processes cause the radius of the drop to reach its maximum over time. To fully understand the phenomenon of droplet spreading, both the path of the drop inside the porous layer and the behaviour of the drop above the porous layer must be described. We must look at the scenario when “big drops” of liquid are placed over “thin porous layers.” The height drop, h, is considered to be significantly greater than the thickness of the porous layer, l. Capillary force takes control since it is believed that the drop created on the porous layer has a gentle slope and limited gravitational pull. The radius of the wetted region of the porous substrate b(t) and the drop base B(t) will change over time. The evolution of the axisymmetric drop profile with time is described by [18]: δh δh γ δ 1 δ 1 δ = u0 − r h3 r δt r δr 3μ δr r δr δr
(1)
where h(t, r) is the profile of the drop; t and r are the time and the radial coordinate, respectively; z > 0 corresponds to the drop and < z < 0 correspond to the porous layer; z = 0 is the drop-porous layer interface as shown in Fig. 1; v and u are the radial and vertical velocity components, respectively; v0 and u0 are velocities at the drop–porous layer interface and can be calculated by matching the flow in the drop with the flow inside the porous layer; μ is the liquid viscosity. The evolution with time of the radius of the drop base, B(t), is defined as dB = dt
1/3 2π mlk p Pc μB 4 V0 − π mlb2 10γ ω 1/10 1 − π μ 3 V0 − π mlb2 ln Bb (t + t0 )9/10 (2)
Fig. 1 Schematic representation of the experimental setup
580
Anushka et al.
The evolution with time of the radius of the wetted region inside the porous layer, b(t), is defined as K P Pc
db μ = b dt b ln B
(3)
The two parameters B(t) and b(t) should therefore lie on two nearly universal curves, which is in excellent agreement with our experimental data as presented in the Results and Discussion section.
2 Experimental Set-Up The circumstances in which spreading experiments are carried out require special consideration. It is important to remember how variables like temperature, humidity, vibrations, and others influence both the outcomes of experiments and the properties of liquids. For these reasons, a special hermetically sealed chamber was created, assembled, and then mounted on a vibration-resistant table. A side view and a top view of the spreading drops need to be monitored. To view a wetted area on the surface of the porous substrate while a drop is spreading, optical glass windows are therefore placed to the chamber sides and ceiling. The chamber was constructed of brass to avoid fluctuations in humidity and temperature. To enable the pumping of a liquid thermostat, many channels were drilled through the chamber walls. The compartment had a fan installed. To gauge the temperature, a thermocouple was employed. In order to syringe the liquid droplet across the substrate, a very small hole is also constructed in the chamber. The spreading phases were captured by a pair of cameras, as seen in the Fig. 1. Side and top views were lighted with various hues of monochromatic light in order to remove erroneous illumination of the photographs. For seeing from above, interferential light filters were added to the optical circuitry. Measurement accuracy was raised since the membrane dispersed the light and reduced the illumination of camera. The lighting was tuned to ensure that the porous substrate and liquid droplets were both clearly visible. An AC power source was used to power these lights. Whatman filter papers were employed in order to comprehend the spreading over a thin porous layer. All membrane samples were collected on a circle plane plate with a radius of 25 mm and a thickness that ranged from 0:0130 to 0:0138 cm. To make a liquid droplet, pure milk and milk with varying amounts of water were both employed. The circular porous substrate for the experiment is set up on a wooden table. For the proper lighting arrangement in the setup, the table must be hollow.
Experimental Investigation of Droplet Spreading on Porous Media
581
2.1 Droplet Creation The liquid droplet was deposited using a syringe pump with a removable capillary. With the use of this functionality, the drop volume might be changed between 1 and 15 ml. In several trials, the distance between the dosator tip and the porous substrate varied from 0.05 to 1 cm. The accuracy of time and spatial measurement of the deformation determines the reproducibility of experimental outcomes. To conduct the spreading test on the droplets, a porous paper sheet was put inside a thermostated, hermetically sealed space with a fixed humidity and temperature. After turning on both lights, the droplet was placed over the Whatman filter paper. Additionally, recording of both cameras was initiated. The time development of the drop base diameter, dynamic contact angle, and the diameter of the wetted zone on the porous substrate surface was tracked in the spreading scenario over dry thin porous substrates. The suitable term for the contact diameter is the drop base diameter on top of porous material. Within the porous medium, the flow front would be in a different location. By observing the dynamics of spreading through contact diameter and height, one can ascertain the rate of imbibition into the porous material. During the early spreading stage, the fluid front within the porous medium moves in front of the superficial spreading front. During the last stage of spreading, the drop base starts to shrink, and eventually the drop disappears entirely. The flow generated by capillary forces or imbibition causes a simultaneous drop and increase in the spreading diameter and the contact diameter, respectively.
2.2 Image Recording and Data Processing The dispersion of the droplets was recorded at a frame rate of 1000 frames per second using a Phantom high-speed camera. To measure the diameter of the drop base and the diameter of the wetted area inside the porous substrate, one camera was positioned immediately on top of the chamber. To capture changes in droplet height or switching, a second camera was mounted on the chamber’s side. The camera was stabilised via a three-dimensionally rotating and translating support. The camera’s focus was changed using this translation stage. Two LED light sources illuminated the work surface. The chamber and all of the optical apparatus were supported by an optical bench with vibration isolation. The entire spreading and imbibition of the droplet were captured on camera. To better comprehend the procedure, these videos were split up into many frames. The photos were edited using the image-processing programme “ImageJ.” In several experimental runs, the processing discretization was 0.04 to 10 s, and the pixel size for each image was 0.02–0.04 mm. The experimental observations were also post-processed using the IMAGE-J programme. MATLAB was used to plot the data that was collected from the image-processing software (Figs. 2, 3, 4 and 5).
582
Anushka et al.
Fig. 2 Image sequence representing the spreading process of small size ink droplet over porous substrate with time
Fig. 3 Image sequence representing the spreading process of large size ink droplet over porous substrate with time
3 Results and Discussion A droplet spreading over porous substrate involves two processes: First, the droplet spreads over porous media that is already wet, resulting in the drop becoming larger. As for the second procedure, it involves absorbing liquid from the drop into the porous layer so that it will reduce the drop diameter. In the porous substrate, the wetted area expands when the imbibition of liquid takes place. Due to these two opposing processes, droplet diameter increases until a maximum value is reached. There are numerous stages involved in the spreading of liquid over porous media. In order for liquid to spread on porous material, there are three distinct phases. A variation in contact diameter, height of the droplet, or contact angle can be used to identify these stages. At the beginning, the drop of liquid enlarges until it reaches
Experimental Investigation of Droplet Spreading on Porous Media
583
Fig. 4 Image sequence representing the spreading process of small size silicon oil droplet over porous substrate with time
Fig. 5 Image sequence representing the spreading process of large size silicon oil droplet over porous substrate with time
its maximum diameter. As a result of the second step, the contact diameter remains unchanged. Towards the end of the process, the drop diameter decreases until the drop disappears. As a result of capillary forces or imbibition, the spreading diameter with in porous substrate increases and the contact diameter of droplet shrinks. The front of fluid spreading inside the porous medium moves ahead of the front of fluid spreading on the surface during the early spreading stage. The diameter of the liquid droplet and the wetted area inside the porous substrate both increase as a result of this fluid front. The drop base begins to contract during the final spreading phase, and eventually the drop completely vanishes. This results in a simultaneous decrease and increase in the contact diameter and the spreading diameter, respectively, and is caused by flow brought on by capillary forces or imbibition. The diameter of the liquid droplet gradually decreases as it is absorbed inside the porous substrate, while
584
Anushka et al.
the diameter of the wetted zone within the substrate gradually grows. Figures 6, 7, 8 show the outcomes of an experiment that involved distributing ink and silicon oil droplets of different sizes over the same filter paper. To start the experiment, a 3 mm droplet of ink is utilized, and then droplet of different size is created using the syringe pump. The same method was used to make the droplets of silicon oil. Figure 6a shows a plot that depicts the spreading of ink droplet of 3 mm diameter (maximum diameter of drop base: 3.573 mm, maximum diameter of wetted circle on outer surface of filter paper: 7.85 mm, maximum contact angle of spreading: 57 degrees). Figure 6b denotes the spreading of ink droplet of 4 mm diameter (maximum
Fig. 6 Diameter as the function of time for (a) 3 mm ink droplet (b) 4 mm ink droplet (c) 5 mm ink droplet (d) 3 mm silicon oil droplet
Fig. 7 Diameter as the function of time for (a) 4 mm silicon oil droplet (b) 5 mm silicon oil droplet. Diameter of wetted region as the function of time for (c) 3 mm ink droplet (d) 4 mm ink droplet
Experimental Investigation of Droplet Spreading on Porous Media
585
Fig. 8 Diameter of wetted region as the function of time for (a) 5 mm ink droplet (b) 3 mm silicon oil droplet (c) 4 mm silicon oil droplet (d) 5 mm silicon oil droplet
droplet base diameter is 5.144 mm, maximum wetted circle diameter on the filter surface is 14.43 mm, and maximum contact angle of the spreading is 62°). Figure 6c represents the spreading of ink droplet 5 mm (maximum droplet base diameter of 5.58 mm, maximum diameter of wetted circle on the filter paper surface of 16.8 mm, and maximum contact angle of spreading of 82°). Figure 6d shows a plot that depicts the spreading of silicon oil droplet of 3.2 mm diameter (maximum diameter of drop base: 3.681 mm, maximum diameter of wetted circle on outer surface of filter paper: 12 mm, maximum contact angle of spreading: 59°). Figure 7a indicates the spreading of silicon oil droplet of 4 mm diameter (maximum droplet base diameter is 4.83 mm, maximum wetted circle diameter on the filter surface is 13.43 mm, and maximum contact angle of the spreading is 62°). Figure 7b shows the spreading of silicon oil droplet of 5 mm diameter (maximum droplet base diameter is 5.92 mm, maximum wetted circle diameter on the filter surface is 17.43 mm, and maximum contact angle of the spreading is 71°). It was demonstrated that for droplets of different fluids with great difference in viscosities, the dynamics of spreading and imbibition vary considerably. The diameter of highly viscous droplets increases in a slow and steady manner and then becomes constant. The second phase of spreading, where contact angle and contact diameter remain constant, stays for a longer duration than the liquid droplet of low viscosities. In the final phase of spreading, the imbibition of the droplet also takes longer than the low viscosity liquid droplet. As the viscosity of the liquid decreases, the slope of the graph increases steeply in the first and the third phases of spreading. While the duration of the second phase of spreading is reduced with decreasing viscosity of the milk. Figure 7c, d and Fig. 8 dictates the variation of the wetted region inside the porous substrate with time. Initially, the fluid front moves at a higher rate which causes fast imbibition of the liquid into the porous substrate and a high rate of spreading of liquid inside the substrate. This results in the large slope of the wetted region curve
586
Anushka et al.
with time for a limited time duration. After some time, when the substrate absorbs a good amount of liquid, the imbibition process slows down until the complete drop disappears. On the other hand, universal curves tend to group all experimental data into similar trends. All these experimental runs demonstrate that, for the right coordinates, the spreading behaviour of drops with varying viscosities and volumes over the same thick porous substrate is equal. In conclusion, it is found that lighter liquid spread more quickly than heavier liquid. This outcome shows that the liquid redistribution within the drop and, subsequently, adsorption kinetics, affect the rate of spreading. Therefore, “spreading” behaviour can come from dynamic causes as well as equilibrium levels of surface and interfacial tension; his behaviour cannot be produced just by high adsorption on the solid/liquid or vapour/liquid interface. We used theoretical analysis from [18] and experimental data from [19] to validate our experimental findings.
4 Conclusions The spreading and penetration of liquids on porous media are encountered in many applications, such as composites processing, printing, painting, coating, adhesives, agriculture, and oil recovery. Kinematics for liquids changes as their density, viscosity, fluidity, surface tension, and temperature change. Spreading is significantly influenced by the porosity of the substrate as well. This study recorded the wetting dynamics of ink and silicon oil droplets of various sizes on porous substrates to understand the effect of various parameters on the spreading phenomenon. To determine the spreading over a thin porous layer, droplets of ink and silicon oil were created using a syringe pump over the Whatman filter papers. By evaluating the contact diameter, and wetted region diameter, the drop-spreading behaviour is investigated. It is found that the overall duration of the spreading process can be broken down into three stages: the first stage, during which the contact diameter grows until it reaches its maximum, and the final stage, during which the drop base contracts to zero. Additionally, we observed an intermediate stage during which the diameter of the contact remained relatively constant. In contrast, the diameter of the wetted region increases monotonically. It is also found that lighter liquid spread more quickly than heavier liquid.
Nomenclature B(t) v u μ Kp
Radius of drop base (m) Radial velocity (m/s) Vertical velocity (m/s) Liquid viscosity (Pa-s) Permeability of the porous layer (m2 )
Experimental Investigation of Droplet Spreading on Porous Media
b(t) h t r Pc γ ω V0
587
Radius of wetted region (m) Profile of drop (m) Time coordinate (s) Radial coordinate (m) Capillary pressure (Pa) Liquid–air interfacial tension (N/s) Effective lubrication parameter (N/s) Initial volume of the drop (m3 )
References 1. Chebbi R (2021) Absorption and spreading of a liquid droplet over a thick porous substrate. ACS Omega 6(7):4649–4655 2. Morais FP, Vieira JC, Mendes AO, Carta AM, Costa AP, Fiadeiro PT, Curto JM, Amaral ME (2022) Characterization of absorbency properties on tissue paper materials with and without “deco” and “micro” embossing patterns. Cellulose 29(1):541–555 3. Sch¨onecker C (2022) Spreading of complex fluids drops 4. Du J, Wang X, Li Y, Min Q, Wu X (2021) Analytical consideration for the maximum spreading factor of liquid droplet impact on a smooth solid surface. Langmuir 37(24):7582–7590 5. Du J, Zhang Y, Min Q (2021) Numerical investigations of the spreading and retraction dynamics of viscous droplets impact on solid surfaces. Colloids Surf, A 609:125649 6. Qin M, Tang C, Tong S, Zhang P, Huang Z (2019) On the role of liquid viscosity in affecting droplet spreading on a smooth solid surface. Int J Multiph Flow 117:53–63 7. Johnson P, Trybala A, Starov V (2019) Kinetics of spreading over porous substrates. Colloids Interfaces 3(1):38 8. Fu F, Li P, Wang K, Wu R (2019) Numerical simulation of sessile droplet spreading and penetration on porous substrates. Langmuir 35(8):2917–2924 9. de Goede TC, Moqaddam AM, Limpens K, Kooij S, Derome D, Carmeliet J, Shahidzadeh N, Bonn D (2021) Droplet impact of Newtonian fluids and blood on simple fabrics: effect of fabric pore size and underlying substrate. Phys Fluids 33(3):033308 10. Das S, Patel H, Milacic E, Deen N, Kuipers J (2018) Droplet spreading and capillary imbibition in a porous medium: a coupled ib-vof method based numerical study. Phys Fluids 30(1):012112 11. Wijshoff H (2018) Drop dynamics in the inkjet printing process. Curr Opin Colloid Interface Sci 36:20–27 12. Starov V, Kostvintsev S, Sobolev V, Velarde M, Zhdanov S (2002) Spreading of liquid drops over dry porous layers: complete wetting case. J Colloid Interface Sci 252(2):397–408 13. Sahimi M (2011) Flow and transport in porous media and fractured rock: from classical methods to modern approaches. Wiley 14. Scheidegger AE (2020) The physics of flow through porous media. In: The physics of flow through porous media (3rd edn). University of Toronto press 15. Sahimi M, Tahmasebi P (2021) Reconstruction, optimization, and design of heterogeneous materials and media: Basic principles, computational algorithms, and applications. Phys Rep 939:1–82 16. Adegbite JO, Belhaj H, Bera A (2021) Investigations on the relationship among the porosity, permeability and pore throat size of transition zone samples in carbonate reservoirs using multiple regression analysis, artificial neural network and adaptive neuro-fuzzy interface system. Petroleum Res 6(4):321–332
588
Anushka et al.
17. Lef‘evre V, Lopez-Pamies O (2022) The effective shear modulus of a random isotropic suspension of monodisperse rigid n-spheres: from the dilute limit to the percolation threshold. Extreme Mech Lett 55:101818 18. Zhdanov VG, Starov V (2002) Calculation of the effective properties of porous and composite materials. Colloid J 64(6):706–715 19. Kumar SM, Deshpande AP (2006) Dynamics of drop spreading on fibrous porous media. Colloids Surf A 277(1–3):157–163
Investigation of the Liquid Sheet Breakup Dynamics in Like-On-Unlike Impinging Injectors Aditi Sharma, Bikash Mahato, P. Ganesh, and K. P. Shanmugadas
1 Introduction Impinging injectors are used in liquid bipropellant rocket engines for the rapid mixing of fuel and oxidizer [1]. The combustible mixture is fed to the combustor through inclined holes under certain injection pressure. The two jets impinge at a point forming a sheet in an orthogonal direction to the impingement plane which subsequently atomizes into ligaments and droplets. The droplet cloud formed from multiple impinging injectors mixes and undergoes combustion in the chamber. The impinging injectors are classified as like-on-like and like-on-unlike based on whether the impinging liquids are of the same nature or not [2]. In gas generators of liquid rocket engines, the propellants used are hypergolic in nature which ensures spontaneous combustion upon impingement. The sheet formation and subsequent atomization in unlike impingement injectors are complicated in nature and involve the interaction of multiple hydrodynamic instabilities. The impingement results in the formation of large-scale convective waves and they propagate radially and axially resulting in the expansion of the sheet. The breakup of the sheet into ligaments results in droplet formation of different size and velocities. The periodic shedding of droplets and ligaments from the sheet surface and their relations with the liquid sheet oscillations is of particular interest. The sheet breakup involves the propagation of different instability waves. The momentum exchange between the jets at the point of impingement results in the formation of waves that are hydrodynamic in origin known as impact waves [1]. The acceleration of liquid sheet in the downstream direction results in the formation of small disturbances on the sheet surface which further grow in amplitude A. Sharma · B. Mahato · K. P. Shanmugadas (B) Department of Mechanical Engineering, IIT Jammu, Jammu 181121, India e-mail: [email protected] P. Ganesh ISRO Propulsion Complex, Mahendragiri 627133, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_54
589
590
A. Sharma et al.
Fig. 1 Schematic diagram of impinging jet atomization
forming the Kelvin–Helmholtz (KH) instability waves. These instability waves drive the sheet disintegration process. These small-scale instability waves form on the liquid surface and propagate in the radial direction. Furthermore, the interaction of unlike impinging jets of different diameters results in the formation of a curved wave that is deflected towards the lower momentum jet side. A schematic representation of the atomization process is shown in Fig. 1. The present work aims to investigate the propagation mechanics of different instability waves on the liquid surface and the correlation of these waves with the ligament shedding process.
2 Literature Review and Objective The investigation on like-on-like impinging injectors has been carried out by many researchers to study the characteristics of the atomization process. Heidmann et al. [3] and Lee et al. [4] characterized the breakup regimes for the like-doublet injectors into four different regimes, namely the closed rim, periodic rim, open rim, and fully developed. Dombrowski [5] and Hooper [5] in their experimental investigations found that the breakup characteristics are influenced by the flow parameters. For laminar cases, the closed rim or periodic drop is formed below a critical jet velocity and impact waves above the critical jet velocity. It is concluded that the breakup length is inversely
Investigation of the Liquid Sheet Breakup Dynamics in Like-On-Unlike …
591
proportional to the two jet velocities, i.e. of both fuel and oxidizer. As the wave number decreases and the sheet thickness grows, larger ligaments are formed, and further downstream of the injection plane secondary atomization occurs. Ibrahim et al. [6] proposed with their experimental finding on the impinging injectors that the sheet thickness is inversely proportional to the distance from the impingement point. Anderson et al. [7] studied the impinging jet atomization mechanism and found that the breakup characteristics are dependent on the flow parameters. The breakup length is dependent on the Weber number in the case of laminar jets and independent of the Weber number in turbulent flows. With an increase in Weber number, the drop size decreases. Ryan et al. [8] found that for the like-doublet injectors, the wavelengths and ligaments are dependent on the orifice diameter and independent of jet velocity. Sweeney et al. [9] found that for the like-doublet injectors, the breakup characteristics of the sheet are dependent on the flow conditions. For the ratio of the breakup length to the impingement distance greater than one, a flat sheet is formed and for the ratio equal to one, an unsteady sheet is formed. Shorter breakup length is observed in the case of jets with larger pre-impingement length, in which the waves are propagating at a much faster rate. Up to a transition point, the breakup length increases with the increase in Weber number, and thereafter it starts to decrease. Jung et al. [10] characterized the like-double injector using cold flow experiments and indicated that an increase in impact force and turbulence strength decreased the breakup length and wavelengths of the liquid sheets and ligaments. Cold flow models are extensively used for understanding the phenomenon of waves, their propagation, and the disintegration of ligaments [11]. Most of the previous investigations focused on the like-on- like impinging injector atomization considering the wide use and simple geometrical and flow arrangements. The atomization processes related to unlike impinging injector are not explored in detail so far. However, unlike impinging atomization is very much important in many propulsion applications, especially in the case of gas generators where hypergolic propellants are used. A detailed understanding of the hydrodynamics instabilities and their propagation mechanics is required to improve the understanding related to the ligament shedding and secondary atomization processes. The present work is carried out to study the atomization process, sheet breakup dynamics, and ligament shedding process related to unlike impinging injectors.
3 Materials and Methods 3.1 Experimental Arrangements Experiments are conducted at atmospheric ambient conditions using water as the test fluid. The experimental Set-up for the investigation of unlike impinging injectors includes a liquid pressurized supply system, injector, injector holder and mounting
592
A. Sharma et al.
stand, collection tank, and other associated instruments. The schematic of the spray test rig is shown in Fig. 2 and the representation of experimental arrangements is shown in Fig. 3. The impinging injector is mounted on a specially designed three-axis nozzle holder. Liquid lines include individual control valves and pressure sensors. The liquid line is pressurized using two pressure vessels which are connected to individual nitrogen cylinders. The spray is collected in a collection tank and pumped back to the storage tank using a centrifugal pump.
Fig. 2 Schematic diagram of the spray test rig
Fig. 3 Experimental arrangements
Investigation of the Liquid Sheet Breakup Dynamics in Like-On-Unlike …
593
Table 1 Test operating conditions Case No.
Fuel (Kg/s)
Oxidizer (Kg/s)
Fuel velocity (m/s)
Oxidizer velocity (m/s)
Average weber No.
1
0.005
0.0125
2.20
2.76
187
2
0.006
0.015
2.64
3.32
269
3
0.008
0.02
3.52
4.42
479
4
0.01
0.025
4.41
5.53
748
5
0.012
0.03
5.29
6.63
1077
6
0.014
0.035
6.17
7.74
1467
7
0.016
0.04
7.05
8.84
1916
8
0.018
0.045
7.92
9.95
2425
9
0.02
0.05
8.81
11.05
2994
3.2 Test Conditions Based on the scaled-down test matrix, the operating conditions for the experiments are selected. For all the operating cases, the momentum ratio is kept at 3.14 and the mixture ratio is maintained at 2.5. The average Weber number is varied from 187 to 2994 and is defined as the mean value of fuel and oxidizer Weber numbers. For different operating conditions, the liquid flow rate for the fuel is varied from 0.005 to 0.02 kg/s and for the oxidizer from 0.0125 to 0.05 kg/s. The velocity of the fuel is varied from 2.20 to 8.81 m/s and for the oxidizer from 2.76 to 11.05 m/s. The test conditions are given in Table 1.
3.3 Diagnostic Techniques Atomization of the liquid sheet is captured using the backlight imaging technique. A high-intensity LED strobe light along with a diffuser sheet is used to illuminate the background. A photron high-speed camera along with a 100 mm Nikon lens is used to capture the images at 15,000 fps. The camera and strobe light are synchronized using an in-house developed synchronizer. The field of view is 32.48 mm × 25.88 mm and the corresponding scale factor is 19.7 pixels/mm. A total of 29,000 images are captured for each case.
3.4 Data Processing Techniques One of the major objectives of the present work is to capture the liquid sheet breakup dynamics. Proper orthogonal decomposition (POD) technique is employed to capture
594
A. Sharma et al.
the major flow structures related to the liquid sheet atomization [12]. POD is a multivariate statistical technique which can be used for feature extraction and order reduction, in complex fluid flows. It reveals the major flow structures present in the flow and estimates the corresponding frequency spectra. POD algorithm identifies a collection of basis vectors and maximizes the energy along each of its basis vectors such that each new basis vector is orthogonal to all preceding ones. An Eigen value is presented for each mode and corresponding dominant frequencies are identified to adequately characterize an image. Thus, by focusing just on the dominating spatial features represented by those modes, the unsteady flow dynamics can be examined [13]. In the present work, POD analysis was performed on an ensemble of backlight images data using an in-house developed MATLAB algorithm. A data matrix is constructed, corresponding to an image ensemble such that each row of matrix represents the intensity values of an instantaneous image. The POD decomposes this matrix into the product of three matrices, viz. coefficient matrix, Eigen matrix, and modal matrix. In a temporal average sense, the POD modes represent the dominating spatial features, wavy flow structures in the present case. Apart from POD, Fast Fourier transform (FFT) analysis is also performed at selected locations. FFT of the total intensity values of selected regions in each image is performed and the frequency spectrum is obtained. A combined POD and FFT analysis helps to identify the physical phenomena in comparison to the projected POD modes.
4 Results and Discussion 4.1 Morphology of the Liquid Sheet The liquid jets impinge at an angle of 2θ = 87° and form an expanding liquid sheet in the perpendicular direction of impingement. Since the momentum ratio is kept constant, there is not much deviation in the sheet angle with respect to the vertical plane as the flow rates are increased from Case 1–9. However, the structure of the sheet changes significantly as the average Weber number increases. Figure 4 shows an instantaneous image of the liquid sheet breakup process. The impingement of two unlike jets generates a hydrodynamic instability wave called impact wave near to the impingement location. The impact waves propagate radially as the sheet expands. The liquid sheet expansion is dictated by the effective liquid momentum of the impinging jet. As the sheet expands, the liquid bulk gets transported radially and forms a rim like structure at the sheet periphery. The expansion of the liquid sheet results in the thinning of the same. The surface tension forces will not be able to hold the sheet region together and perforations are generated on the liquid surface. These perforations propagate radially forming internal rims and rivulets which accelerate the breakup of liquid sheets. Furthermore, the accumulated liquid
Investigation of the Liquid Sheet Breakup Dynamics in Like-On-Unlike …
595
Fig. 4 Instantaneous image showing the liquid sheet atomization (Weavg = 187)
rims on the sheet periphery will undergo further atomization by the formation of capillary instabilities. Multiple jets will originate from the rim periphery, and they extend in length and form ligaments and droplets. These droplets further undergo secondary atomization forming finer droplets. The liquid sheet breakup process varies considerably as the flow rates are increased. Instantaneous images showing the sheet structure at different cases are shown in Fig. 5. Based on the breakup structure, 4 regimes can be identified, namely: open rim breakup, wavy sheet breakup, perforated sheet breakup, and wavy and perforated breakup. In the open rim breakup regime, the impact waves are driving the atomization process and droplets are generated at the sheet tip and sheet periphery. As the Weavg increases, the velocity of the sheet increases, and Kelvin–Helmholtz instability waves start to originate in the downstream regions of the sheet. The increase in Weber number also makes the sheet more unstable since perforations start to originate at multiple locations the sheet. At very high Weber numbers, a combination of a wavy and perforated breakup occurs (regime 4). Fine droplets are generated in the higher Weavg cases as compared to the low Weavg cases. Altogether this generates complex atomization mechanisms which include impact waves, KH instability wave, propagation of perforations, wave breakup, etc. The droplet formation around the sheet periphery and tip depends on the complex interaction of these multiple instabilities and hydrodynamic oscillations. Clustered ligament and droplet shedding around the sheet depends on the various flow processes associated with the liquid sheet. In order to get an understanding of the major flow structures, POD and FFT analyses are conducted, and results are explained in the subsequent sections.
596
A. Sharma et al.
Fig. 5 Instantaneous image showing the liquid sheet atomization for all cases
4.2 Dynamics of the Liquid Sheet Breakup The liquid sheet breakup process is analysed using POD analysis. The centre region of the sheet covering a field of view of 50 mm × 50 mm is selected for the analysis since it includes most of the flow structures. The first four POD modes contain most of the energy structures associated with the flow. The POD mode shapes and corresponding frequencies are shown in Fig. 6. Average POD images are not presented here. Among the 4 modes, the first and second, and third and fourth modes are complementary in nature. The first and second modes show the propagation of impact waves. It is evident that impact waves originate near the impingement location and are purely hydrodynamic in origin. The POD mode shapes also substantiate this finding. Modes
Investigation of the Liquid Sheet Breakup Dynamics in Like-On-Unlike …
597
Fig. 6 POD Modes and corresponding dominant frequencies
3 and 4 represent KH instability and other convective waves. As the fluid velocity increases, towards the downstream direction, the amplitude of KH waves increases. Figure 7 shows the mode shapes 1 and 3 for all cases. The predominance of mode 1 (impact waves) is present more in low Weber numbers. At high Weavg cases (Regime 4), both impact waves and KH waves are present as shown in Fig. 7b. Frequency spectra corresponding to each mode are presented in Fig. 8b. Impact wave frequencies vary from 285.9 to 1211 as the Weavg number is increased from 187 to 2994 (Case 1–9). In the case of modes 3 and 4, KH instability waves with two different dominant frequencies are present which are represented as the bimodal peaks. The propagation mechanics of perforations are not highlighted in the POD analysis. Since the perforations are occurring at multiple locations randomly, no dominant frequencies are associated with this phenomenon. In order to verify the presence of impact waves on the liquid sheet, FFT analysis is performed selecting multiple locations on the sheet as presented in Fig. 8a. The FFT frequency peaks at the top region of the sheet matched well with the mode 1 frequency values. This shows that the impact waves are captured in the POD analysis. As the sheet expands, the wave number decreases in the bottom region of the sheet and the wave frequency values decrease (Fig. 8b). It is also observed that for Cases 4 and 5, the frequency values are not changing with the axial distance.
4.3 Ligament Shedding Process The atomization of the liquid sheet generates ligaments and droplets at multiple locations around the sheet. These ligaments are originated either from the breakup of the liquid rim at the sheet periphery or due to the breakup of the perforated sheet or due to the wave breakup. The process depends on sheet breakup mechanisms discussed earlier. The ligaments originate as clusters and are shed at different frequencies at
598
A. Sharma et al.
Fig. 7 a POD Mode 1 for all cases, b POD Mode 3 for all cases
a POD Mode 1 for all cases.
b POD Mode 3 for all cases.
Investigation of the Liquid Sheet Breakup Dynamics in Like-On-Unlike …
599
a
b Fig. 8 a FFT analysis of sheet oscillations along the axial distance-Instantaneous image showing analysis regions, b FFT analysis of sheet oscillations along the axial distance-Frequency variation with average Weber number
different locations. It is really important to understand the dominant structures related to the sheet breakup and their relation with the ligament and droplet breakup. Droplet clustering and shedding results in non- uniform spray pattern. The ligament shedding process around the sheet periphery and sheet tip is analysed further using FFT analysis. Different regions around the liquid sheet are selected for the analysis as shown in Fig. 9a. The analysis showed that ligaments are shedding from the periphery and from the sheet tip. The ligament shedding frequency at the
600
A. Sharma et al.
a
1400
Liquid sheet FFT Modes 1&2- Impact waves
1200
Frequency (Hz)
Ligament shedding near tip
1000
Modes 3&4- KH waves
800 600 400 200 0 187
269
479
748
1077 1467 1916 2425 2994
Average Weber Number weavg b Fig. 9 a FFT analysis of the ligament shedding process- regions selected for analysis, b FFT analysis of the ligament shedding process- frequency variations with average Weber number
tip region is matching with the mode 4 and 5 frequencies, and not with the impact wave frequencies. Hence, it is clear that the KH waves are dictating the sheet breakup process at the tip region and the droplet shedding is directly correlated with the KH instability frequencies. The droplets are shed from the sheet periphery at different frequencies ranging from 63.57 to 947.14 Hz (Fig. 9b). The shedding frequency
Investigation of the Liquid Sheet Breakup Dynamics in Like-On-Unlike …
601
depends on various time scales related to the transport of impact waves, rim formation, and RT instability waves related to rim breakup. The combined effect results in the droplet shedding at different frequencies other than that of impact waves.
5 Conclusions The liquid sheet breakup dynamics of a like-on-unlike impinging injector is investigated using time resolved backlight imaging technique. The propagation of impact waves and KH instability waves is captured using combined POD and FFT analysis. The breakup modes change significantly with the increase in Weber number. It is observed that near the impingement locations, impact waves dominate the flow field and KH wave motion becomes predominant near the sheet tip regions. The ligament shedding frequencies near the tip regions matches with the KH instability wave frequencies. The rim breakup around the sheet periphery is not dependent on the impact wave propagation alone and hence no specific correlations can be made with impact wave frequency and droplet shedding due to rim breakup. The sheet perforations are occurring randomly at multiple locations on the sheet and no dominant modes are observed related to the propagation of sheet perforations. Acknowledgements The authors would like to thank NCCRD, IIT Madras for providing the test facilities for carrying out the experiments. The authors would also like to acknowledge the technical support and laser diagnostic arrangements. NCCRD is supported by the department of science and technology India.
References 1. Ashgriz N (ed) (2011) Handbook of atomization and sprays. Springer US, Boston, MA. https:// doi.org/10.1007/978-1-4419-7264-4 2. Brinckman KW, Feldman G, Hosangadi A (2015) Impinging fuel injector atomization and combustion modelling. In: Presented at the 51st AIAA/SAE/ASEE joint propulsion conference, Orlando, FL. https://doi.org/10.2514/6.2015-3763 3. Heidmann MF, Priem RJ, Humphrey JC (1957) A study of sprays formed by two impinging jets. UNT Digital Library. https://digital.library.unt.edu/ark:/67531/metadc56084/m1/3/. Accessed 31 Mar 2022 4. Lee SS, Kim WH, Yoon WS (2009) Spray formation by like-doublet impinging jets in low speed cross-flows. J Mech Sci Technol 23(6):1680–1692. https://doi.org/10.1007/s12206-0090419-z 5. Dombrowski ND, Hooper PC (1964) A study of the sprays formed by impinging jets in laminar and turbulent flow. J Fluid Mech 18(03):392. https://doi.org/10.1017/S0022112064000295 6. Ibrahim EA, Przekwas AJ (1991) Impinging jets atomization. Phys Fluids Fluid Dyn 3(12):2981–2987. https://doi.org/10.1063/1.857840 7. Anderson WE, Miller KL, Ryan HM, Pal S, Santoro RJ, Dressler JL (1998) Effects of periodic atomization on combustion instability in liquid-fueled propulsion systems. J Propuls Power 14(5):818–825. https://doi.org/10.2514/2.5345
602
A. Sharma et al.
8. Ryan HM, Anderson WE, Pal S, Santoro RJ (1995) Atomization characteristics of impinging liquid jets. J Propuls Power 11(1):135–145. https://doi.org/10.2514/3.23851 9. Sweeney BA, Frederick RA (2016) Jet breakup length to impingement distance ratio for like doublet injectors. J Propuls Power 32(6):1516–1530. https://doi.org/10.2514/1.B36137 10. Jung K, Khil T, Yoon Y (2006) Effects of orifice internal flow on breakup characteristics of like-doublet injectors. J Propuls Power 22(3):653–660. https://doi.org/10.2514/1.20362 11. Inoue C, Oishi Y, Daimon Y, Fujii G, Kawatsu K (2021) Direct formulation of bipropellant thruster performance for quantitative cold-flow diagnostic. J Propuls Power 37(6):842–849. https://doi.org/10.2514/1.B38310 12. Chatterjee A (200) An introduction to the proper orthogonal decomposition. Curr Sci 78(7):808–817. http://www.jstor.org/stable/24103957 13. Berkooz G, Holmes P, Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Annu Rev Fluid Mech 25(1):539–575. https://doi.org/10.1146/annurev.fl. 25.010193.002543
Numerical Investigation of Mist Flow Characteristics in a Hexagonal Fuel Rod Bundle Ayush Kumar Rao, Shivam Singh, and Harish Pothukuchi
1 Introduction Nuclear power generated around 10% [1] of the world’s electricity in 2019. After hydropower electricity, nuclear energy constitutes almost one-third of all low carbon electricity, some countries are even heavily dependent on it like 70% of electricity in France, and more than 40% in Sweden comes from nuclear power plants. The temperature of nuclear rods can go up to 1850 K [2]. To cool down, these rods different techniques are employed. In the earlier days, heavy water is the most commonly employed coolant. Latter, in the recent generation nuclear reactors, light water is being used. In light water-cooled nuclear reactors, the mist flow heat transfer cooling rate in case of an accident plays an important role in the safety of the nuclear reactor. Mist flow is a flow regime of two-phase gas–liquid flow. The flow core in the mist flow entrains all the liquid as droplets in the gas phase. Nowadays, many researchers are focused on flow behavior and heat transfer characteristics of post-CHF mist flow. This post-CHF mist flow region mainly has three modes of heat transfer, these are wall to vapor, wall to liquid droplets and vapor to liquid droplets. Because of the low heat transfer coefficient of continuously flowing vapor, heat transfer is less efficient after post-CHF, and this may cause damage to heating equipment and raise serious safety concerns. The discussion about safety is incomplete without quoting the serious accidents in the past such as, Chornobyl in Ukraine (1986) and Fukushima in Japan (2011). Although the probability of such accidents is very less, safety concerns of the reactors cannot be ignored to ensure the safe working conditions in nuclear power plants. To this end, the present study focusses on the study of mist flow characteristics in a hexagonal fuel rod bundle of an advanced water reactor (AWR).
A. K. Rao · S. Singh · H. Pothukuchi (B) Department of Mechanical Engineering, IIT Jammu, Jagti 181221, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_55
603
604
A. K. Rao et al.
2 Literature Review and Objective The study of post-CHF conditions is very much essential to preserve the structural integrity of the reactor fuel core. As it is very much challenging and difficult to mimic the CHF or accident scenario, the experimental study of post-CHF characteristics in a fuel rod bundle is not feasible. Therefore, the numerical study is the only alternative for the study and analysis of mist flow characteristics in a fuel rod bundle. John et al. [3] had done an analytical study to observe the vapor heat transfer in the post-dryout region. The momentum transfer analogy in thermodynamic non-equilibrium is used. From the study, they proposed a correlation to predict the vapor convective heat transfer under post-CHF. Barzoni and Martini [4] conducted an experimental study on post-dry-out heat transfer in a vertical tube. They find out the temperature distribution over the length of the tube for different values of thermodynamic quality. Dougall-Rohsenow, Heineman and Hadaller correlation (thermal equilibrium-based prediction method) overpredict the experimental data. Rossakhin and Kabanov [5] analyzed heat transfer in the post-dry-out region and on wetting heated surfaces. They developed computational techniques for when both processes occur simultaneously. Andreani and Yadigaroglu [6] studied various prediction methods for dispersed flow film boiling DFFB. For high pressure and medium to high mass flux, they conclude that the thermal equilibrium-based correlations give accurate results. Doerffer et al. [7] investigate the critical heat flux between annuli and tubes which internally heated. They proposed a correlation based on eccentricity, void migration, and quality. Guo and Mishima [8] developed a model for heat transfer in a post-dry-out regime. They compared the experimental results and the result obtained from the model. From the study, they conclude that their model accurately predicts the wall temperature and the vapor superheat. Hass et al. [9] studied experimentally the critical heat flux at low pressure for flow boiling in vertical annuli. Michael et al. [10] characterized the droplet size distribution and validated the use of a quasi-static Lagrangian subscale trajectory within COBRA-IE in Dispersed Flow Film Boiling (DFFB). Nguyen and Moon [11] studied a developing post-dry-out region and found that there is significant heat transfer enhancement compared to a fully developed post-dry-out region, because of upstream disturbance and droplets present downstream of the CHF. Li et al. [12] developed a three-field CFD model to study both pre and post-dry-out regions and found out that wall-gas heat transfer is maximum compared to direct contact of wall-droplet ( 1), oblate (AR < 1), and spherical (AR = 1). The large bubble entrapment is observed only for the prolate-shaped drops [5–7]. It has also been revealed that large bubble entrapment is a vortex-driven phenomenon [6, 7]. The vortex ring generated near the interface during the impact of a drop onto the pool penetrates the pool and later rolls up the liquid near the interface to form a thin liquid sheet. The thin liquid sheet then moves towards the centreline and later merges at the centreline, entrapping a large bubble. The interaction of the vortex ring with the liquid surface is a critical parameter for the entrapment of the large bubble, e.g. large bubble entrapment is not observed for an oblate-shaped drop, whereas the oblate-shaped drops produce the strongest vortex ring. This is because of the early interaction of the vortex ring with the liquid pool interface (during the expansion stage of the crater), where the expanding motion resists the interface’s rolling motion. The above discussion briefly summarizes the advancement in understanding the large bubble entrapment phenomena through experimental and numerical tools. However, the major setback of the previous numerical simulation is that the drop is initialized with an ellipsoidal shape in all the simulations, where a and b pass through the centreline, i.e. a and b always intersect at their midpoints. However, in reality, the drop shape is more complex. The maximum radius a location may lie above or below the midpoint of b. In this study, we revisit the large bubble entrapment phenomena, considering the exact shape of the drops. The exact shape of the drops is modelled using Lamb’s theory [8], and numerical simulations are performed for different drop shapes. The rest of the paper is arranged as follows: the numerical methodology is discussed in Sect. 2, the results of the numerical simulations are discussed in Sect. 3, and finally, the conclusion of the investigation are summarized in Sect. 4. All the simulations are performed considering water as the liquid phase and air as the gaseous phase.
2 Methodology In this study, we have investigated the large bubble entrapment during drop impact on a liquid pool using numerical simulations. The open source flow solver Basilisk (www.basilisk.fr) is used to perform the numerical simulations. Basilisk solver is a volume-of-fluid method-based fluid flow solver that can handle the interface’s complex topology very efficiently. The adaptive mesh refinement (AMR) technique of the Basilisk solver facilitates an excellent approximation of the interface when it undergoes complex morphological changes, such as coalescence and pinch-off. Several researchers have used the solver to study complex multi-phase flow simulations ranging from the atomization of liquid jets to studying turbulent breaking waves. In this study, the mass and momentum conservation equations are solved with the
Drop Impact on a Deep Pool: A Revisit to the Large Bubble Entrapment …
659
axisymmetric assumption in a cylindrical coordinate system (r; z). The mass and momentum conservation equations are given as
∇ · V = 0,
(1)
⎤ ∂ V + ∇. V V ⎦ = −∇ P + ρ(F) g ρ(F)⎣ ∂t
(2)
⎡
+∇ · [μ(F)(∇ V +(∇ V )T )] + σ κ nδ ˆ s
where V is the velocity field, σ is the surface tension coefficient, κ is the mean curvature of the interface, nˆ is the unit normal vector on the interface boundary and δs is the interface delta function which is zero elsewhere except on the interface. The density ρ (F) and the viscosity μ(F) are volume fraction-based density and volume fraction-based viscosity, respectively. The volume fraction F, defined as F = 1 for the liquid and F = 0 for the gas, separates the liquid and gas phases. The advection equation of F is solved to capture the movement of the interface. ∂F + V ·∇ F = 0 ∂t
(3)
A representative diagram of the computational domain is shown in Fig. 1. The computation domain has width W = 5D and height H = 5D, where D is the equivalent diameter of a spherical drop having the same volume as that of the deformed drop considered in the numerical simulations. The depth of the liquid pool is 3D. The gap between the drop and the pool interface at the start of the simulation is 0.1D. The following boundary conditions are used to perform the simulations: no-slip boundary condition on the bottom boundary, free slip boundary condition on the right boundary, outflow boundary condition on the top boundary, and symmetry boundary condition on the left boundary. The depth of the liquid pool and the side boundary distance is sufficient to eliminate any influence of the boundary in the impact dynamics [9]. The solver is already validated and has been used in our previous investigations on drop impact on a deep pool [10]. More details about the numerical methodology and solution algorithms of the Basilisk solver are available in Refs. [10–13].
3 Results and Discussion Using Lamb’s theory [8], the time dependent outer shape of a drop can be represented as
660
Y. C. Nikhil et al.
Fig. 1 Representative diagram of the computational domain
R(θ, t) = R0 1 +
∞
Cn (t)Pn (cosθ ) .
(4)
n=0
Here, R0 is the equivalent radius of a spherical drop having the same volume as that of the deformed drop, t is time, Θ is the polar angle, and Pn is the associated Legendre polynomial. The coefficient Cn (t) can be calculated as Cn (t) = An sin(2πt/Tn + φn ), where An is the amplitude damping coefficient, Tn is the time period of oscillation, and φn is the phase angle for the n-th mode of oscillation. Here, mode 0 and mode 1 represent the volumetric pulsation and translation of the droplet. Therefore these two modes can be omitted. Mode 2 contributes to symmetric oscillation with respect to the centre of mass (change in the outer shape of the drop between prolate and oblate), and mode 3 contributes to the asymmetric oscillation of the drop surface. The characteristic oscillation of a real drop can be well-approximated by the summation of mode 2 and mode 3 oscillation. Here, we consider modes 2 and 3 together to represent the deformed shape of the drops, which are more realistic in comparison with the symmetric oscillation of drops (only mode 2) considered in Deka et al. [6], Thoraval et al. [7]. Figure 2 shows the shapes of a drop at different time instants reproduced by considering both the second and third modes of oscillations. It is evident in Fig. 2 that mode 3 brings asymmetry to the shape of the drop, for instance, a protruding tail (or top) and a blunt front (bottom) at t = T /5, and a protruding front and a blunt tail at t = 7 T /20 for prolate drops.
Drop Impact on a Deep Pool: A Revisit to the Large Bubble Entrapment …
661
Fig. 2 Shape of a 5 mm diameter drop at different time instants. Here T is the time period of oscillation of the drop and can be calculated using Lamb’s theory [8]
It has been reported in earlier investigations [6, 7] that large bubble entrapment takes place only for prolate-shaped drops. In the earlier investigation, only mode 2 oscillations were considered; therefore, the drop oscillation was symmetrical. The centre of mass was always located at the intersection of the two major axes a and b. In those simulations, large bubble entrapment was observed for all the prolate-shaped drops with 1.1 < AR < 1.4. However, the question is does the asymmetry brought by mode 3 affects the large bubble entrapment process? In this study, we focus on the effect of asymmetry brought by mode 3 on the large bubble entrapment phenomena. We have performed numerical simulations for different drop shapes to understand the effect of asymmetric shapes on the impact process and on the subsequent dynamics. The temporal evolution of the crater profile after the impact of an oblate-shaped drop is shown in Fig. 3. The drop diameter and the impact velocity here are 5 mm and 1 m/s, respectively, which are inside the large bubble entrapment regime reported in Deka et al. [6]. It is evident in Fig. 3 that the crater expands without the entrapment of a large bubble. We have performed simulations for other oblate-shaped drops, and no large bubble entrapment is observed in any simulations. These results complement the conclusion of Deka et al. [6], Thoraval et al. [7] that large bubble entrapment does not take place for oblate-shaped drops. The present simulation further suggests that the asymmetry in shape brought by the third mode does not bring any change to large bubble entrapment during the impact of oblate-shaped drops. The temporal evolution of the crater profile for a prolate-shaped (AR = 1.29) drop is shown in Fig. 4. Here, the location of a is below the midpoint of b. The drop diameter and the impact velocity here are 5 mm and 1 m/s, respectively. It is evident in Fig. 4 that although the drop’s shape is prolate, the diameter and impact velocity
662
Y. C. Nikhil et al.
Fig. 3 Evolution of the crater profile during the impact of an oblate-shaped drop of equivalent diameter 5 mm at an impact velocity of 1 m/s
are within the range of large bubble entrapment given in Deka et al. [6], the large bubble entrapment does not take place. This reveals that a large bubble entrapment does not take place for all the prolate-shaped drops when the third mode asymmetry is included. Indeed this was the reason why Pumphrey and Elmore [3] observed large bubble entrapment only in a small regime in the V-D map. The crater profile for another representative case is shown in Fig. 5 with AR = 1.27. The drop diameter and the impact velocity are the same as that of Fig. 4. In this case also the drop shape is prolate, but the third mode has brought an asymmetry with a protruding bottom and a blunt top. In this case, the vertical location of a is above the midpoint of b. We observe the large bubble entrapment in this case, as evident in Fig. 5. The mechanism of large bubble entrapment is identical to that of the symmetrically deformed drops considered in Deka et al. [6], Thoraval et al. [7], with the vortex ring rolling up the liquid near the interface resulting in the formation of a thin liquid sheet that merges at the centre line entrapping a large. The comparison of Figs. 4 and 5 reveals that a critical criterion for large bubble entrapment is the location of the radial major axis a above or below the midpoint
Drop Impact on a Deep Pool: A Revisit to the Large Bubble Entrapment …
663
Fig. 4 Evolution of the crater profile during the impact of a prolate-shaped drop of equivalent diameter of 5 mm at an impact velocity of 1 m/s. Here, the aspect ratio AR = 1.29 and the location of a is below the midpoint of b
of the axial major axis b. In all the simulations that we have performed so far, large bubble entrapment has never taken place whenever a is located below the midpoint of b, even though the shape of the drop is prolate. One possible reason might be that when a lies below the midpoint of b, it creates a blunt shape of the drop at the bottom, having a local radius higher than the top of the drop. As a result, the vorticity generated during the impact of the drop onto the liquid pool is stronger, similar to that of an oblate drop. It has been reported in Deka et al. [6], Thoraval et al. [7] that a stronger vortex ring interacts with the interface early and cannot penetrate into the pool. As a result, although the vortex ring rolls up the interface liquid, it is not sufficient to merge at the centre line and, thus, prevents the large bubble entrapment.
664
Y. C. Nikhil et al.
Fig. 5 Evolution of the crater profile during the impact of a prolate-shaped drop of equivalent diameter of 5 mm at an impact velocity of 1 m/s. Here, the aspect ratio AR = 1.27 and the location of a is above the midpoint of b
4 Conclusions In this study, we investigate the large bubble entrapment phenomena considering the asymmetric deformation of drops. The asymmetric deformation in the drop is incorporated considering the third mode of oscillation in Lamb’s theory, along with the symmetric deformation of the drop incorporated by the second mode of oscillation. Thus, by considering both the second and third modes of oscillation, we have replicated the exact shape of the drop. We have studied the effect of asymmetric deformation of drops on the large bubble entrapment dynamics during drop impact on a deep pool. It has been observed that large bubble entrapment does not take place for all the prolate-shaped drops. Large bubble entrapment is observed only when the location of the radial major axis a lies above the midpoint of the axial major axis b. Further study is needed to build a deeper understanding of the effect of asymmetric deformation on the large bubble entrapment dynamics.
Drop Impact on a Deep Pool: A Revisit to the Large Bubble Entrapment …
665
Acknowledgements The authors acknowledge the HPC facility of IIT Dharwad for performing the computations. Hiranya Deka acknowledges the financial support from the Seed Grant and Network Fund (SGNF) of IIT Dharwad.
References 1. Thoroddsen ST, Thoraval MJ, Takehara K, Etoh TG (2012) Micro-bubble morphologies following drop impacts onto a pool surface. J Fluid Mech 708:469–479 2. Ray B, Biswas G, Sharma A (2015) Regimes during liquid drop impact on a liquid pool. J Fluid Mech 768:492–523 3. Pumphrey HC, Elmore PA (1990) The entrainment of bubbles by drop impacts. J Fluid Mech 220:539–567 4. Oguz HN, Prosperetti A (1990) Bubble entrainment by the impact of drops on liquid surfaces. J Fluid Mech 219:143–179 5. Wang A-B, Kuan C-C, Tsai P-H (2013) Do we understand the bubble formation by a single drop impacting upon liquid surface? Phys Fluids 25(10):101702 6. Deka H, Ray B, Biswas G, Dalal A, Tsai P-H, Wang A-B (2017) The regime of large bubble entrapment during a single drop impact on a liquid pool. Phys Fluids 29(9):092101 7. Thoraval MJ, Li Y, Thoroddsen ST (2016) Vortex-ring-induced large bubble entrainment during drop impact. Phys Rev E 93(3):033128 8. Lamb SH (1932) Hydrodynamics. Cambridge University Press, New York, USA 9. Deka H, Biswas G, Chakraborty S, Dalal A (2019) Coalescence dynamics of unequal sized drops. Phys Fluids 31(1):012105 10. Deka H, Biswas G, Sahu KC, Kulkarni Y, Dalal A (2019) Coalescence dynamics of a compound drop on a deep liquid pool. J Fluid Mech 866 11. Popinet S (2003) Gerris: a tree-based adaptive´ solver for the incompressible Euler equations in complex geometries. J Comput Phys 190(2):572–600 12. Popinet S (2009) An accurate adaptive solver for´ surface-tension-driven interfacial flows. J Comput Phys 228(16):5838–5866 13. Popinet S (2018) Numerical models of surface tension. Annu Rev Fluid Mech 50:49–75
2D Numerical Simulation of the Electrospraying Process of a Viscoelastic Liquid in an Ambient, Highly Viscous Liquid Vimal Chauhan, Shyam Sunder Yadav, and Venkatesh K. P. Rao
Nomenclature x y t p μ gx D φ γ κ nˆ
Horizontal coordinate (mm) Vertical coordinate (mm) Time (s) Pressure (Pa) Dynamic viscosity (Pa.s) Gravity along axis, −9.81 (m/s2 ) Rate of deformation tensor (1/s) Electric potential (V) Surface tension (N/m) Curvature (1/m) Unit normal vector
1 Introduction Electrospraying is an electrohydrodynamic (EHD) process which produces uniformly sized droplets under the influence of an electric field. The electrospraying setup consists of a nozzle and substrate with an electric potential difference applied between them. A Newtonian or a viscoelastic fluid passes through the nozzle in the form of a continuous stream which is collected by the substrate placed under the nozzle. The electrospray process has an advantage over other techniques of generating droplets as it allows the user to precisely control the droplet size. The process V. Chauhan · S. S. Yadav (B) · V. K. P. Rao Department of Mechanical Engineering, BITS Pilani, Pilani, Rajasthan 333031, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_61
667
668
V. Chauhan et al.
has applications in ink-jet printing [1, 2], in thin film deposition process [3], and in polymer particle production [4]. The electrospray process can exhibit many EHD modes depending on the applied electric potential difference and fluid properties. Some of these modes for a Newtonian fluid are as follows: dripping, micro-dripping, spindle mode, con-jet mode and multi-jet mode [5]. For a viscoelastic fluid, the modes are the dripping mode, the bead-on-a-string structure, cone-jet mode, stick jet mode and unstable mode. The dripping mode for viscoelastic fluids starts at lower electric field strengths, but the drop generation process is very slow compared with the case with a Newtonian fluid. A slight increase in the electric field strength results in the bead-on-a string structure of the viscoelastic jet where beads are formed on the liquid filament coming out of the nozzle. With further increase in the electric potential, the beads on the filament disappear and a cone-jet mode develops which produces a fine viscoelastic jet. Further increment in the electric field results in a very thin jet which sticks to the tip of the nozzle. This EHD mode is known as the stick jet mode. Finally, further increase in electric field produces a non-uniform and irregular jet ejected from the nozzle tip. Simulating an electrospray process is computationally challenging as it involves multiphysics-based models including concepts from constitutive modelling for viscoelastic fluids, electrohydrodynamics and interface capturing methods. The behaviour of viscoelastic fluid is approximated using several models such as the Oldroyd-B model, upper-convected Maxwell model, the Giesekus model, the Phan Thien-Tanner (PTT) model, and the finitely extensible nonlinear elastic (FENE-P, FENE-CR) models. While simulating viscoelastic flows at large values of Weissenberg number (Wi), a numerical instability diverges the solution process. To tackle this instability, several methods have been proposed such as the elastic-viscous stress splitting (EVSS), the both side diffusion approach and the log conformation method (LCM) [6]. Several models have been proposed for the dynamic response of fluids under electric fields. These include the perfect dielectric, the perfect conducting, and the leaky dielectric model introduced by Taylor [7]. This model has been used in a number of studies on drop deformation under external electric fields [8–11]. The leaky dielectric model is also used to study the electrospray process because of its ability to apply the tangential force on the jet interface [12–14]. Liu et al. [15] investigated the atomization process of viscoelastic fluid under the influence of electric field. Another experimental and numerical study was done on electrospray process by using FENE-P model for polyisobutylene-based solution [16, 17]. Panahi et al. [18] numerically investigated the viscoelastic properties of polymeric solutions at different Weissenberg numbers in the electrospray process and compared the results with their own experimental data. The aim of the current study is to explore the electrospraying process of a viscoelastic liquid in an ambient, viscous medium. We use the Oldroyd-B model for the constitutive behaviour of the viscoelastic liquid and the log conformation approach for its numerical solution. The interface between the viscoelastic jet and the surrounding fluid is captured by the volume-of-fluid method. The paper is arranged as follows: problem formulation and governing equations are discussed in Sect. 2. The
2D Numerical Simulation of the Electrospraying Process …
669
results on the behaviour of the viscoelastic jet under electric field are summarized in Sect. 3. Finally, the paper is concluded in Sect. 4.
2 Problem Formulation and Governing Equations For the numerical simulations, we take a computational domain of size 20 mm by 20 mm where the nozzle orifice of radius one mm is placed symmetrically at the top wall as shown in Fig. 1a. Initially, a semi-circular profile is taken for the viscoelastic fluid and rest of the domain is filled with a Newtonian, viscous liquid. A flow rate of 36 ml/h is enforced to the viscoelastic fluid at the orifice. A steady and uniform electric field of strength 7500 V/cm is introduced along the y direction to initiate the electrohydrodynamic phenomenon. Consequently, the electric potential is generated inside the domain which is governed by Gauss law. An interfacial surface tension γ = 0.04 N/m is assumed between the two liquids. The various parameters of the viscoelastic liquid and the viscous liquid are represented with subscript ‘1’ and ‘2’, respectively. Physical properties of the viscoelastic fluid such as density, viscosity, relative permittivity and electrical conductivity are represented by ρ 1 , μ1 , ε1 and σ 1, respectively. The respective properties of the surrounding, viscous liquid are density ρ 2 , viscosity μ2 , relative permittivity ε2 and electrical conductivity σ 2 . An open source fluid solver, Basilisk [19], is used to simulate the behaviour of the viscoelastic fluid under the influence of electric field. The governing equations used for the current simulations are described in the following subsections.
2.1 Fluid Flow and Viscoelastic Constitutive Equation The continuity and the Navier–Stokes equation are used as the governing equations for the incompressible flow of the fluids under externally applied electric field: ∇ · u = 0,
(1)
ρ(∂t u + u · ∇u) = −∇p + ∇ · (τm + τe ) + ρg + γκ∇α.
(2)
In Eqs. 1 and 2, the velocity, density, pressure and surface tension are represented by u, ρ, p and γ respectively. κ and α are respectively the curvature and volume fraction used in the volume-of-fluid method. Here, τ e is the Maxwell stress tensor which gives the electric force per unit area. Similarly, τ m is the total stresses developed in viscoelastic fluid and it is the sum of the solvent stresses τ s and the polymeric stresses τ p ,
670
V. Chauhan et al.
Fig. 1 a Schematic of the 2D domain for the dynamics of a viscoelastic fluid in an immiscible, viscous surrounding fluid under an external electric field, b zoomed view of the viscoelastic fluid ejected from the nozzle
τ m = τ s + τ p.
(3)
The stresses in the solvent can be described by assuming Newtonian behaviour, ( ) τ s = 2μs D = μs ∇u + ∇uT ,
(4)
where μs is the viscosity of solvent and D is deformation rate tensor. In this study, the Oldroyd-B model is used with constant polymeric viscosity, under which the polymeric stress τ p is expressed as follows [20] /
λτ p + τ p = 2μ p D.
(5)
Here, λ and μ p are, respectively, the relaxation time and the polymeric viscosity, /
whereas τ p is the upper-convected time derivative of τ p , /
τp =
∂τ p + u · ∇τ p − (∇u)T · τ p − τ p · ∇u. ∂t
(6)
2D Numerical Simulation of the Electrospraying Process …
671
2.2 Log Conformation Method The polymeric stresses can be expressed in terms of the conformation tensor A as τp =
μp (A − I), λ
(7)
where I is the identity matrix. Using Eqs. (5) and (7), the constitutive equation in terms of the conformation tensor (A) becomes ∂t A + (u · ∇)A − A · ∇u − ∇uT · A =
1 (I − A). λ
(8)
According to Fattal and Kupferman [21], it is better to work with the logarithm of the conformation tensor (y = logA) while dealing with the high Weissenberg number problem (HWNP), otherwise the solution process will be unstable. Under the log conformation approach, the transpose of the velocity gradient is decomposed as follows: (∇u)T = y + B + NA−1 ,
(9)
where u is divergence-free velocity field, N and y are antisymmetric tensors and B is a symmetrical, trace-less tensor. In this method, we advance in time with the logarithmic value of conformation tensor, y = log A. Since A is symmetric, positivedefinite tensor, it can always be diagonalized. Furthermore, the conformation tensor and the log conformation tensor can be described as follows: A = R/R T andy = log A = R log /R T ,
(10)
where R is an orthogonal tensor which is formed by the eigenvectors of A. / is a diagonal tensor which is formed by eigenvalues of A. Now, N, y, and B can be decomposed using the tensor R and its transpose R T as follows: ) 0 n12 RT , N=R −n12 0 ) ( 0 ω12 RT , y=R −ω12 0 ) ( b11 0 RT . B=R 0 b22 (
(11)
(12)
(13)
The velocity gradient transpose (∇u)T is obtained using Eqs. (12) and (13). Similarly, the log conformation constitutive equations are obtained after solving Eqs. (8) and (9). After following the steps discussed by Fattal and Kupferman [21], we get
672
V. Chauhan et al.
the following equation, ∂t y + (u · ∇)y − 2B − (yw − yw) =
) 1( y e −I . λ
(14)
The conformation (A) and log conformation (y) tensors are related to each other through the eigenvectors of the conformation tensor (A), ( ) A = Rexp /y R T
(15)
( ) y = Rlog /A R T ,
(16)
where /y and /A are tensors formed by eigenvalues of log conformation tensor and conformation tensor, respectively.
2.3 Electrostatic Equations To compute the electrical stresses (τ e ) on the interface between two fluids, the Maxwell’s electromagnetic equations need to be simplified [22]. The electric field is irrotational and is curl-less, ∇ × E = 0. The electric field can be described in terms of electric potential (φ) as E = −∇ · φ and its distribution is governed by Gauss law, ∇ ·E=
ρe . ε
(17)
Substituting the value of electric field E in Eq. (17) we get, ∇2φ =
ρe , ε
(18)
where ε and ρe are the electric permittivity and volumetric charge density, respectively. The charge conservation equation is given by ∂ρe + ∇ · J = 0, ∂t
(19)
where J is the current density vector defined as, J = J f + Jb ,
(20)
J = σE + ρe u.
(21)
2D Numerical Simulation of the Electrospraying Process …
673
In Eq. (21), J f and Jb are the free current density and bound current density. In Eq. (21), the first term is the Ohmic charge conduction, while the second is due to the convection of charges. Here, σ, E, ρe and u are the fluid conductivity, electric field, volumetric charge density and charge velocity, respectively. For an electrostatic field, the volumetric electric forces Fe can be obtained from the Maxwell stress tensor which is defined as: ( ) 1 2 τe = ε EE − E I . (22) 2 In Eq. (22), I is the identity tensor. By applying the divergence operator, the total electric force per unit volume becomes, 1 Fe = ∇.τe = ρe E − E2 ∇ε. 2
(23)
In Eq. (23), the term ρe E represents the force due to free charge. The last term is the polarizing force and it appears only at the fluid interface along the normal direction due to electric permittivity gradient. 1 2 E ∇ε 2
2.4 Two-Phase Flow Equations The volume-of-fluid method is used to track the interface between the viscoelastic fluid and the surrounding liquid. In the VoF method, the fraction of the reference fluid (which is the viscoelastic fluid in our case) in a grid cell is denoted by α. The value of α varies between 0 < α < 1. If α = 1, then computational cell is completely filled with the viscoelastic fluid, and if α = 0, then the computational cell is completely filled with the surrounding liquid. The advection of the volume fraction field is performed by ∂α + ∇ · (αu) = 0. ∂t
(24)
The unit normal vector nˆ and the curvature to the interface κ are required to calculate the capillary force term in the Navier–Stokes equation. These are given by ∇α nˆ = |∇α| and κ = ∇ · nˆ respectively. The physical properties of both the fluids are given in Table 1. We used the open source code called Basilisk flow solver for solving the governing equations. Basilisk discretizes the computational domain using a Quadtree-based structured grid made up of square finite volumes. The solver is capable of using automatic adaptive mesh refinement (AMR) as shown in Fig. 1b. AMR captures the local variations in the numerical solution accurately by selectively refining the mesh locally in that region. Before proceeding towards the discussion on the results, it is
674 Table 1 Physical properties of the two liquids
V. Chauhan et al.
Properties of liquids
Droplet
Ambient
Density, ρ (Kg/m3 )
1000
960
Viscosity, μ (kg/m · s)
0.001
0.96
Dielectric constant, e
81
1
Conductivity, σ (s/m)
1 × 10−4
1 × 10−12
Surface tension, γ (N/m)
0.04
Weissenberg no., Wi
1
good to perform a grid independence test so that the results do not depend on the grid size.
2.5 Grid Independence Test The interface between the viscoelastic jet and the surrounding liquid at three different grid sizes is shown in Fig. 2. The numerical solution is performed on three different grid levels, namely Level 7, Level 8 and Level 9 with the respective interface shapes represented by green, orange and purple colours in the figure. The grid is courser at Level 7 which does not capture the location of interface accurately compared with those done by grids with Level 8 and Level 9 refinements. Moreover, the Level 8 and Level 9 refinements capture the same position of the interface between the viscoelastic jet and the surrounding liquid. Therefore, the refinement Level 8 has been used for further calculations to save on the required computational effort. In the following section, we discuss the various results obtained with the help of above-mentioned numerical formulation. The results are calculated based on Weissenberg number (Wi = 1) and at a flow rate of 36 mL/h and under an applied electric potential of 15 kV.
3 Results and Discussion 3.1 Viscoelastic Jet Analysis Figure 3 represents the evolution of the volume fraction (α) of the viscoelastic liquid (Wi = 1) at a flow rate of 36 mL/h and at applied electric potential of 15 kV. The interface between the viscoelastic jet and the surrounding liquid is successfully captured by the VoF method from t = 0.05 to 0.25 ms. As can be seen in the figure, the jet undergoes the dripping mode instability at around half of the height of the domain at t = 0.2 ms due to electric forces.
2D Numerical Simulation of the Electrospraying Process …
675
Fig. 2 Viscoelastic jet interface on different grid refinement levels
Figure 4 shows the distribution of the electric potential (φ) in the domain. The electric potential varies between the top and the bottom walls of the domain which are respectively at φtop = 15,000 V and φbottom = −15,000 V. Due to the higher conductivity of the viscoelastic fluid, most of the jet is at an electric potential which is equal to that applied on the top wall. The volume charge density contours at an applied electric potential of 15 kV are shown in Fig. 5. Initially, at time t = 0.01 ms, the volume charge density at the interface is low. As the process proceeds further, at t = 0.05 ms, the jet profile takes a conical shape due to the accumulation of charged particles at its tip. The volume charge density increases up to 0.24 C/m3 at the interface which results in electric forces overpowering the surface tension forces. With increasing time, the jet is stretched in length and simultaneously it becomes very thin. Finally, Coulombic fission of this thin jet occurs resulting into the formation of small droplets which drain the charge from the viscoelastic jet. The positively charged, detached droplets are finally attracted towards the negatively charged bottom electrode. Figure 6 represents the three components of the polymeric stresses, namely, τ p,x x , τ p,x y and τ p, y y at the interface of the viscoelastic jet. As shown in Fig. 6a and b, the stress components τ p,x x and τ p,x y dominate in the conical portion near the top wall. In contrast to this, the component τ p, y y dominates in the thin viscoelastic jet, which is stretched by the electrostatic forces as can be confirmed from Fig. 6c. In the following section, we conclude the current work.
676
V. Chauhan et al.
Fig. 3 Evolution of the volume fraction of the viscoelastic liquid
4 Conclusions In this manuscript, we investigated the electrospraying process of a viscoelastic fluid in an ambient viscous liquid. The Oldroyd-B model is used for the viscoelastic behaviour of the fluid, which is numerically, simulated using the Log conformation approach. The electric forces are induced in the fluids by applying electric potential to the top and bottom boundaries. It can be concluded from the numerical simulations that the high-volume charge density at the tip of initial interface leads to the viscoelastic jet formation. The electric stresses cause the formation of a Taylor cone at initial stages. Afterwards, due to further charge concentration, the electric forces stretch a very thin jet. When the jet becomes thin enough, the surface tension forces cause fission of the jet into smaller, charged droplets. τ p,x x and τ p, yx components of the polymeric stresses dominate in the conical portion of the jet, while the τ p, y y component dominates in the stretched, thin jet. The outcomes from the current work
2D Numerical Simulation of the Electrospraying Process …
677
Fig. 4 Effect of the electric potential on the interface of the viscoelastic jet
Fig. 5 Volume charge density at the interface of the viscoelastic jet
shed further light on the mechanism of small, charged droplet formation under the electrospraying process.
678
V. Chauhan et al.
Fig. 6 Distribution of the three components of the polymeric stresses: a τ p,x x , b τ p,x y and c τ p,yy , at the interface of the jet
References 1. Shin W-T, Yiacoumi S, Tsouris C (2004) Electric-field effects on interfaces: electrospray and electrocoalescence. Curr Opin Colloid Interface Sci 9(3–4):249–255 2. Cai S, Sun Y, Wang Z, Yang W, Li X, Yu H (2021) Mechanisms, influencing factors, and applications of electrohydrodynamic jet printing. Nanotechnol Rev 10(1):1046–1078 3. Jaworek A (2007) Micro-and nanoparticle production by electrospraying. Powder Technol 176(1):18–35 4. Jung JH, Park SY, Lee JE, Nho CW, Lee BU, Bae GW (2011) Electrohydrodynamic nanospraying of ethanolic natural plant extracts. J Aerosol Sci 42(10):725–736 5. Lopez-Herrera JM, Barrero A, Boucard A, Loscertales IG, Marquez M (2004) An experimental study of the electrospraying of water in air at atmospheric pressure. J Am Soc Mass Spectrom 15(2):253–259 6. Fattal R, Kupferman R (2005) Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log conformation representation. J Nonnewton Fluid Mech 126(1):23–37 7. Taylor GI (1966) Studies in electrohydrodynamics. I. The circulation produced in a drop by an electric field. Proc Roy Soc Lond Ser A Math Phys Sci 291(1425):159–166 8. Lopez-Herrera JM, Popinet S, Herrada MA (2011) A charge conservative approach for simulating electrohydrodynamic two-phase flows using volume-of-fluid. J Comput Phys 230(5):1939–1955 9. Paknemat H, Pishevar AR, Pournaderi P (2012) Numerical simulation of drop deformations and breakup modes caused by direct current electric fields. Phys Fluids 24(10) 10. Lima NC, D’Avila MA (2014) Numerical simulation of electrohydrodynamic flows of Newtonian and viscoelastic droplets. J Nonnewton Fluid Mech 213:1–14 11. Sunder S, Tomar G (2020) Numerical investigation of a conducting drop’s interaction with a conducting liquid pool under an external electric field. Eur J Mech B/Fluids 81:114–123 12. Lim LK, Hua J, Wang C-H, Smith KA (2011) Numerical simulation of cone-jet formation in electrohydrodynamic atomization. AIChE J 57(1):57–78 13. Herrada MA, Lopez-Herrera JM, Gañán-Calvo AM, Vega EJ, Montanero JM, Popinet S (2012) Numerical simulation of electrospray in the cone-jet mode. Phys Rev E 86(2):026305 14. Hashemi AR, Pishevar AR, Valipouri A, Parau EI (2018) Numerical and experimental investigation on static electric charge model at stable cone-jet region. Phys Fluids 30(3):037102
2D Numerical Simulation of the Electrospraying Process …
679
15. Liu L, Liu Y, Lu L (2015) Atomization mechanism of a charged viscoelastic liquid sheet. Chin J Aeronaut 28(2):403–409 16. Carroll CP, Joo YK (2008) Axisymmetric instabilities of electrically driven viscoelastic jets. J non-Newton Fluid Mech 153(2–3):130–148 17. Zhmayev Y, Divvela MJ, Ruo A-C, Huang T, Joo YK (2015) The jetting behavior of viscoelastic boger fluids during centrifugal spinning. Phys Fluids 27(12):123101 18. Panahi A, Pishevar AR, Tavakoli MR (2020) Numerical simulation of jet mode in electrospraying of Newtonian and viscoelastic fluids. Int J Multiphase Flow 129 19. Popinet S. Basilisk flow solver and pde library. http://basilisk.fr/ 20. Lopez-Herrera JM, Popinet S, Castrejón-Pita AJ (2019) An adaptive solver for viscoelastic incompressible two-phase problems applied to the study of the splashing of weakly viscoelastic droplets. J Nonnewton Fluid Mech 264:144–158 21. Fattal R, Kupferman R (2004) Constitutive laws for the matrix logarithm of the conformation tensor. J Nonnewton Fluid Mech 123(2–3):281–285 22. Saville DA (1997) Electrohydrodynamics: the Taylor-Melcher leaky dielectric model. Annu Rev Fluid Mech 29(1962):27–64
Deformation Dynamics During Complete Rebounding During Impact of a Falling Droplet of Varied Surface Tension on a Sessile Drop Pragyan Kumar Sarma and Anup Paul
Nomenclature CMC Di Dih Div Ds h We τ γ η σ ν
Critical micelle concentration (M/L) Falling droplet diameter (mm) Horizontal diameter of the falling droplet (mm) Vertical diameter of the falling droplet (mm) Sessile drop diameter (mm) Sessile drop height (mm) Weber number Non-dimensional time Non-dimensional deformation of horizontal diameter of the falling droplet Non-dimensional deformation of vertical diameter of the falling droplet Non-dimensional deformation of horizontal diameter of the sessile drop Non-dimensional deformation of vertical diameter of the sessile drop
1 Introduction Impact of droplets on both solid surfaces and liquids have a tremendous influence on our daily lives, as well as applications in various industrial fields. Droplet impacts are frequently used in inkjet printing, coating and spray painting, spray cooling of hot surfaces like turbine blades, rolls in rolling operations, annealing processes, quenching of metal alloys, fire suppression sprinklers, combustion in IC engines, P. K. Sarma · A. Paul (B) Department of Mechanical Engineering, National Institute of Technology Arunachal Pradesh, Papum Pare, Arunachal Pradesh 791113, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_62
681
682
P. K. Sarma and A. Paul
criminal forensics, aerial application, plasma spray coating, and the formation of salt crystals in oceans [1]. Rapid prototyping [2], micro manufacturing [3], and electronic packaging [4] are some aspects where analysis of droplet dynamics is useful. When a liquid drop falls on another drop, there occurs various phenomena, such as coalescence, advancing, bouncing, and splashing depending on certain controlled conditions. At low impact velocity a falling droplet impacting on a sessile droplet could lead to bouncing and coalescence. Coalescence is found to happen when two incoming droplets interact with each other and also with the substrate leading to formation of various patterns and forms a single unit of that liquid.
2 Literature Review and Objective Studies from several researchers [5–8] have led to believe that the gas layer between the drop and the target surface is what causes the impact to bounce-off wetted areas. When the impact energy is less, the drop hesitates on the target surface before coalescing, because the contact time between the drop and the surface is longer than the gas layer drain time [9]. Although there are two possible outcomes when the impact energy is rather high: the first is the rupture of the air layer and subsequent direct coalescence, and the second is the deformation of the drop over the surface. The drop is then pushed to rise off the surface by a high pressure that has built up at the contact. Research on impact of a falling droplet on a stationary surface is valuable in various applications, including inkjet printing, the dispersion of raindrop patterns, and the formation of ice on the surface of aerial vehicles, among other areas. Wang et al. [10] observed four varied regimes on impact of an impacting drop on a sessile one, these are: complete rebounding, partial rebounding, coalescence, coalescence with glutination. The development of the surface form during the coalescence of the falling drop with the sessile on a superhydrophobic surface was investigated experimentally and numerically by Farhangi et al. [11]. The impact of various ratios of the sessile drop volume to the falling drop volume on the coalescence dynamics and partial coalescence was studied by Kumar et al. [12]. Recently Abouelsoud and Bai [13] investigated both experimentally and theoretically the dynamics of an impacting drop on a sessile one of the same liquid on a hydrophilic surface and observed different regimes. They calculated the deformation of the falling and the sessile droplet; additionally they determined the restitution coefficient and the contact time and matched their empirical evidences with a theoretical study. It is seen that most studies are based on the droplet impacts on another drop of the same fluid. Very few studies are found pertaining to studies of a falling droplet on a stationary one. We carried out an experimental investigation of a falling droplet of varied surface tension (not changing the viscosity) on a water droplet on a glass substrate. It is observed that complete rebounding of the falling drops of varied surface tension occurs at low velocity of 0.4 m/sec. So we are interested to find out the deformation of the falling droplet as well as the sessile one from the point of contact to the point where the falling droplet bounces of the sessile droplet. The
Deformation Dynamics During Complete Rebounding During Impact …
683
droplet impact of two drops having different surface tension has not been taken up yet. This work is applicable in inkjet printing where droplet bouncing is not a desired result to be had. It is a case for studying the raindrop splashing patterns and growth.
3 Materials and Methods Figure 1 shows the experimental set-up. A droplet dispensing glass micrometre syringe (Gilmont Instruments, 2.0 ml) is installed on a height-adjustable syringe holding fixture (Ramé-hart Instruments, USA). An electronically controlled dispenser system (Ramé-hart instruments, USA) pumps the sessile water droplet out through a glass syringe (50 μL). It is then allowed to fall on a glass slide (76 × 26 × 1, Borosil), which has been thoroughly cleaned with acetone and DI water and blown dried in a hot air oven. The glass slide containing the sessile droplet is placed on a three-axis moveable specimen stage (Ramé-hart instruments, USA) underneath the mounted glass micrometre syringe. A fixed droplet of DI water is employed, and droplets of varying surface tension are allowed to fall on it at a speed of 0.4 m/ sec. The specimen stage, which has an accuracy of 0.001 mm, is moved horizontally to impact and coalesce head-on. A high-speed camera (VEO 340 Phantom, Ametek Materials Analysis Division) with a macro lens (Nikon) was used to record the coalescence dynamics. The macro lens was positioned in line with the sessile droplet on the glass slide, and an LED light was positioned at the opposite end of the camera lens side. Images at a resolution of 1280 × 800 pixels were taken at a rate of 3600 frames per second. With the help of the free source programme FIJI (a distribution of ImageJ), the pictures of the collisional dynamics are post-processed to estimate the non-dimensionalised deformations. Aqueous sodium dodecyl sulphate (Sigma-Aldrich) in DI water was diluted to different concentrations to create test fluids with varying surface tension, without altering viscosity (0.25, 1 of the Critical Micelle Concentration). By using a co-axial cylinder rheometer (Brookfield, USA) and pendant drop analysis (Ramé-hart instruments, USA), respectively, the surface tension was determined by pendent drop analysis (Ramé-hart Instruments, USA) and viscosities were found out by co-axial cylinder rheometer (Brookfield instruments, USA). Table 1 lists the test fluids’ characteristics. The impacting droplet diameter is evaluated by Di = (Di2h × Div ), where Di h and Div are the horizontal diameter and vertical diameter, respectively and the sessile drop axial diameter Ds and height h are obtained from image analysis. The velocity is calculated by virtue of free fall and neglecting the air resistance since the distance from the needle tip to the glass substrate is very small. The head-on collision has been ensured through image analysis and an uncertainty of ± 0.02 mm is observed.
684
P. K. Sarma and A. Paul
Fig. 1 Schematic representation of the experimental set-up a LED light, b diffuser plate, c height adjustable syringe mounting assembly, d glass slide, e micrometre syringe, f falling droplet, g sessile droplet, h horizontally movable specimen stage, i High-speed camera and j Laptop PC
Table 1 Properties of the test fluids at 25 °C Liquid
Density (kg/m3 )
Surface tension (N/m)
Viscosity (mPas)
Di water
0.071
0.071
0.88
SDS solution (0.25 CMC)
987
0.045
0.88
SDS solution (1 CMC)
987
0.020
0.88
4 Results and Discussion Figure 2 shows the effect of the variation of surface tension of the falling droplet resulting in the variation of Weber number as well on the bouncing phenomena when impacted on a sessile water drop. Figure 2a shows the impact of a slight lesser surface tension (0.25 CMC, W e = 9.87) falling droplet on a sessile water droplet on a glass substrate, and Fig. 2b shows a droplet of least surface tension (1 CMC, W e = 25.6) falling on a water droplet resting on a glass surface. It can be observed that both the falling droplets undergo noticeable elastic deformation and shows the bouncing phenomena as it falls on the sessile water droplet impacting head-on with a velocity of 0.4 m/sec. It is wellknown that the bouncing action is owing to the higher pressure of the air film present at the interface between the falling and the sessile drop. After touching the sessile water droplet, the falling droplet of the both surface tension variants oscillates on the surface of the sessile drop with some deformations (1.1–11.11 ms). The falling droplet doesn’t have enough inertial force to break the higher surface tension of sessile water droplet, the inertial force makes the falling droplet to squeeze like a pancake (5 ms) and gain potential energy like that of a spring after the inertial force gets converted into stored energy it then changes to kinetic energy, and part of surface
Deformation Dynamics During Complete Rebounding During Impact … Fig. 2 Snapshots of the effect of surface tension on the bouncing phenomena for the impact of the falling droplet on the sessile water droplet on a glass surface: a 0.25 CMC, We = 9.87 and b 1 CMC, We = 25.60
685
686
P. K. Sarma and A. Paul
energy also contributes to the kinetic energy for the falling droplet to spring back and rebound completely.
4.1 Droplet Deformation During Full Rebounding We are interested in finding the deformation of the impacting droplet as well as the sessile one in both lateral and longitudinal axis. Measurement of the deformations of the impacting drop has been taken when the drop touches the sessile drop for the first time until the impacting drop is about to lift off from the surface of the sessile drop after going through deformations. Figure 3 shows the deformation of the impacting drop along the non-dimensional time τ = tv/Di where v is the velocity of the impacting drop (v = 0.4 m/sec) and t is time in ms. The left axis of the figure denotes dimensionless deformation in the lateral diameter, γ = Di h /Di and the right axis of the figure represents non-dimensional deformation in the longitudinal diameter, η = Di v /Di . From Fig. 3, it can be seen that as the impacting droplet falls on the sessile one while undergoing deformations, the γ decreases slightly at first and then increases up to a maximum value and again decreases and oscillates in between when the impacting droplet is about to rebound (τ = 122–1.4) whereas the η decreases to a minimum value and finally increases. It can be seen from the figure that the impacting drop with lower surface tension (1CMC) reaches the maximum γ in a shorter time than the relatively higher surface tension impacting drop (0.25 CMC). It can be said that the lower surface tension force is responsible to make the deformation faster due to the stronger inertial force. As the impacting drop is rebounded, the sessile water drop also undergoes deformation which can be observed from Fig. 4. In the similar vein as Fig. 3 here, the left axis denotes non-dimensional axial diameter σ = Ds /Di and the right axis is for dimensionless sessile drop height ν = h/D f along non-dimensional time τ. It could be observed from the Fig. 4 that there is very little perturbations of σ with respect to time, it means that the impact energy of the falling droplet of W e = 9.87, 25.6 is absorbed by the higher surface tension of the sessile water droplet. In contrast to σ , ν on the other hand decreases as the impacting drop undergoes deformation, it is obvious as the impact energy of the impacting droplet compresses the sessile droplet which makes the sessile drop also like a pan cake shape. As the rebounding phase starts, the compressed region gets pushed back and returns to its initial position more or less.
Deformation Dynamics During Complete Rebounding During Impact …
687
Fig. 3 Impacting drop deformation along with non-dimensional time for varied surface tension
Fig. 4 Sessile drop deformation along with non-dimensional time for varied surface tension impacting drops
688
P. K. Sarma and A. Paul
4.2 Changes in the Dynamic Contact Angle During Rebounding To know the contact line dynamics of both the droplets, it is imperative to measure the dynamic contact angle at both the left and right edge of the both impacting droplet and the sessile one. Figure 5 shows the dynamic contact angle (°) with respect to time (ms) for the impacting droplet of 0.25 CMC falling on sessile water drop. It can be seen from the Fig. 5 that both the left and right edges of the impacting behaves in a similar manner albeit the left right edge of the impacting droplet has higher dynamics contact angle than the left edge and initially the contact angle of the sessile drop is same, but as the compression starts the left edge gets a higher dynamic contact angle than the left edge. It is to be noted that the impacting droplet is in contact with the air layer sandwiched between the impacting drop and the sessile water droplet; hence, the contact angle is of liquid–air interphase. The elastic deformation of the impacting droplet of 0.25 CMC is very noticeable and the dynamic contact angle also denotes it as it oscillates in a good amount. Somewhere around 5–6 ms the right edge of the impacting droplet reaches a highest dynamic contact angle of 160° this is during the retraction phase. The dynamic contact angle variation with time of the 1 CMC droplet impacting the sessile water droplet is shown in Fig. 6. The deformations of the impacting droplet (1 CMC) are clearly noticeable as the oscillations of the dynamic contact angle of the left and right edge are prominent. A steep rise and fall of the dynamic contact angle of the 1 CMC impacting drop are seen as well as around 4–6 ms, this period is also when the 1 CMC falling droplet is on the rebound phase. A very steep rise of 170° of the right edge is seen of the 1 CMC falling droplet and the left is about 130° as the droplet is about to rise off.
5 Conclusions Experimental observations are carried out for the complete rebounding cases of impacting droplet of varied surface tension (0.25 CMC and 1 CMC) falling on a sessile water drop at a velocity of 0.4 m/sec. The impacting drop goes through remarkable elastic deformation and finally lifts off the surface of the sessile water droplet. The non-dimensionalised deformations are plotted with respect to nondimensionalised time and it is found that the lower surface tension impacting droplet (1 CMC) undergoes deformation a bit faster than the relatively higher surface tension impacting droplet (0.25 CMC). The dynamic contact angle of left and right edges of both the impacting droplet and the sessile one are also plotted against time and a sharp increase and decrease of the dynamic contact angle is found when the impacting droplet is in its retracting stage. It can be also concluded that when at low velocity (< 1 m/sec), droplets of different surface tension when collides show a complete
Deformation Dynamics During Complete Rebounding During Impact …
689
Fig. 5 Dynamic contact angle at the left and right edge of both the sessile drop (water) and impacting drop (0.25 CMC)
Fig. 6 Dynamic contact angle at the left and right edge of both the sessile drop (water) and impacting drop (1 CMC)
690
P. K. Sarma and A. Paul
rebounding phenomenon when collided head-on and the droplets show appreciable deformations as it goes through the processes of deformation.
References 1. Yarin AL (2006) Drop impact dynamics: splashing, spreading, receding, bouncing. Annu Rev Fluid Mech 38:159 2. Zhang YM, Chen Y, Li P, Male AT (2003) Weld deposition-based rapid prototyping: a preliminary study. J Mater Process Technol 135:347 3. Fang M, Chandra S, Park CB (2007) Experiments on remelting and solidification of molten metal droplets deposited in vertical columns. J Manuf Sci Eng 129:311 4. Sirringhaus H, Kawase T, Friend RH (2000) High-resolution inkjet printing of all-polymer transistor circuits. Science 290:2123 5. Terwagne D, Vandewalle N, Dorbolo S (2007) Lifetime of a bouncing droplet. Phys Rev E 76:1 6. Kannangara D, Zhang H, Shen W (2006) Liquid-paper interactions during liquid drop impact and recoil on paper surfaces. Colloids Surf A 280:203 7. Blackett PMS (1964) The coalescence and bouncing of water drops at an air/water interface. Proc Roy Soc Lond Ser A 280:545 8. Hicks PD, Purvis R (2011) Air cushioning in droplet impacts with liquid layers and other droplets. Phys Fluids 23(6):062104 9. Thoroddsen ST, Takehara K (2000) The coalescence cascade of a drop. Phys Fluids 12:1265 10. Wang FC, Feng JT, Zhao YP (2008) The head-on colliding process of binary liquid droplets at low velocity: high-speed photography experiments and modeling. J Colloid Interface Sci 326:196 11. Farhangi MM, Graham PJ, Choudhury NR, Dolatabadi A (2012) Induced detachment of coalescing droplets on superhydrophobic surfaces. Langmuir 28:1290 12. Kumar M, Bhardwaj R, Sahu KC (2020) Coalescence dynamics of a droplet on a sessile droplet. Phys Fluids 32(1):012104 13. Abouelsoud M, Bai B (2021) Bouncing and coalescence dynamics during the impact of a falling drop with a sessile drop. Phys Fluids 33:063309
Real-Time Strengthening of Natural Convection and Dendrite Fragmentation During Binary Mixture Freezing Virkeshwar Kumar, Shyamprasad Karagadde, and Kamal Meena
1 Introduction Solidification is an exothermic process where liquid converts into solids by lowering the temperature below the melting point. When a mixture in containers is cooled through their walls, solid nucleates from the walls or external particulate matter grow in the direction of the cooling [1]. The solid growth occurs typically in an anisotropic manner by forming different microstructural morphologies such as dendrites (treelike, such as snowflakes), cellular, equiaxed, and faceted, depending on the material and composition. Dendritic structures, where branched secondary/side arms grow on the parents, (Fig. 1b) and columnar structures are commonly found in most metals and a few non-metallic solutions [2]. When a region (such as a container) is cooled, the freezing location is identified by following the iso-contour of the melting point in a pure system [3]. In mixtures, a range of freezing temperatures often exists for a given composition, with the liquidus temperature (liquidus line with circle symbol in Fig. 1a) marking the beginning and the solidus (solidus line with squares symbol in Fig. 1a) marking the end of solidification. The presence of solid and liquid phases between the above two temperatures makes the region porous and is termed the mushy zone. (Fig. 1b) [3]. The binary solution consists of two components, A and B, in Fig. 1, where A and B have freezing points T A and T B, respectively. C E is a eutectic composition for V. Kumar (B) Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Uttar Pradesh, Kanpur 208016, India e-mail: [email protected] S. Karagadde Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India K. Meena Indian Oil Corporation Ltd, Bhopal 462011, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_63
691
692
V. Kumar et al.
Fig. 1 a Schematic of an equilibrium binary eutectic phase diagram. Red dots indicate points on the liquidus, depicting a gradual increase of the solute concentration in the liquid phase (for C 0 < C E ), b schematic of bottom-cooled solidification of a binary system where composition lies between A and C E and where the bottom temperature is below the eutectic temperature
A–B binary system where liquid directly solidifies from liquid to solid (two phases) at T E (eutectic temperature). Suppose the liquid composition (amount of B in the mixture) lies between pure A and C E (e.g., C O as the initial composition). In that case, A solidifies as α-phase and rejects B into the liquid. This is due to the limit of solubility of B in the solid phase. However, the growing solid phase consists of some amount of solute B, and the locus of the solidus curve gives this amount. Similarly, for other sections (between C E and pure B, also termed hyper eutectic), the solid occurs as β-phase (with predominant B), and the liquid is enriched with A. However, the growing solid phase consists of some amount of solute B, which is quantified by partition coefficient (k). The partition coefficient is the ratio of solute composition in growing solid and bulk liquid in equilibrium solidification (k = C S / C L ). During solidification, composition moves toward eutectic composition, which has the lowest solidification temperature. Similarly, for other sections (between C E and pure B) where B solidifies as β-phase and A-enriched liquid rejects in the mushy and bulk zone. When a lighter solute is rejected into the liquid, the concentration is expected to increase gradually, which initiates natural convection in the upward direction. Sometimes convective flow forms stabilized plumes that constantly carry the lighter fluid from the bottom (mushy region) to the top of the container. In equilibrium solidification rate or cooling rate is very low, which allows proper diffusion of solute in growing solid as well as liquid. In this case, composition in solid and liquid follows the solidus and liquidus curve, and at any temperature, the composition can be estimated by lever rule [4], e.g., at T0 temperature and C0 composition, the liquid fraction will be (C 0 -C S )/(C L -C S ). In general, the diffusion of the dissolved solute within the solid phase is very low ( 20 min), all regime has solutal convection with low strength. Region P2 and P4 had a low convective velocity but higher than P1. In P3, the convective velocity was similar to P2 and P4 for the initial time instant. As solidification proceeds, the convective velocity in P1 and P4 decreases with time, whereas P2 and P3 region velocity increase with time, indicating that convection was focused on regimes P2 and P3. Region P2–P4 shows an increase in localized convective velocity with time; however, it was highest at P3. The localized strengthened convective regime can be identified as a chimney/plume origination that drags surrounding rejected solute in the mushy zone which reduces the velocity of the surrounding and increases its own velocity. From the experiments, a plume with 0.3–0.4 mm/s convective velocity can drop off the dendrite. A similar velocity scale is observed in numerical work for the plumes in Ga-In alloy system solidification [7, 17, 18]. An increase in solute rejection in the mushy zone led to a shift in composition near the eutectic region. Generally, the growth rate/cooling rate should be uniform in each location of the mushy zone; it is not a reality due to solute rejection and interfacial kinetics. For better understanding, the mushy zone can be divided into two regions: one near the tip region (solid–liquid interface region; horizontal dashed red line in Fig. 4b) and the second region, which is beneath the mushy zone (Fig. 6). Due to solute rejection, most of the solutes are entrapped between the side arms of the dendrite (Fig. 6), which reduces the growth rate of the side arms in the second region. Lower region side arms are bigger in dimension and reject more solute compared with the adjacent small arms. For a given cooling rate, the first region, or dendritic tip, grows faster than the second region (the high-solute region solidifying in the last stage). With this localized variation in growth rate, the tip region grows faster than
Fig. 5 Plot of solutal convective velocity with nondimensional height a For the region P1 and P2 with the time, b for regions P3 and P4. P1, P2, P3, and P4 region is marked in Fig. 4a. 0 and 1 of non-dimensional height indicate the lowest and highest position of the visualization region (Fig. 4)
Real-Time Strengthening of Natural Convection and Dendrite …
697
Fig. 6 Schematic of the a Dendrite, b zoomed view of two side arms of region two, and c reduction in cross-section area at the side arms neck (change in cross-section area is indicated by arrow)
the neck region, leading to remelting kind of situation of the side arms at the neck (Fig. 6c). Finally, the pinch-off of the side arms from the roots occurs. Side branch remelting changes the localized curvature of the primary arm (Fig. 4c). Convection also enhances the remelting of the side and primary arms. The pinched-off location on the primary arm experiences more remelting due to the formation of extra cavities, leading to localized recirculation and weakening of the dendrites. With time, higher solute increases the convective strength, and dendrite D2 starts to break off (a circle in Fig. 4c marks a fragmented solid). The fragmented dendritic part moves with the plume and is remelted in bulk during motion. Remelting is the process of resolving a grown solid into a melt. The main causes of remelting are (a) temperature and (b) composition differences. The top region is at a higher temperature compared with the bottom region (bottom-cooled solidification configuration). The fragmented solid (cool) moves with the convective flow in bulk (hot), which remelts the solid into the bulk. Fragmented solid dendrite is SCN-enriched material (nearly 100% SCN or 0% acetone), whereas bulk liquid has approximately 15 wt.% acetone. The plume or channel contains more acetone (closer to about 20 wt.%) which changes the overall bulk composition (> 15% acetone) and reduces the equilibrium liquidus temperature of the liquid (Fig. 1a). The fragmented solid and bulk liquid composition is not at equilibrium, leading to the remelting of solid dendrite. The remelting of fragmented solids in bulk is the dissolution of solute in solvent, and it is based on solubility. The equilibrium composition is also known as a saturated composition, where a maximum solute composition can dissolve in the liquid, beyond that it cannot dissolve at that temperature, e.g., at room temperature,
698
V. Kumar et al.
Fig. 7 Solidification of water-25 wt.% NH4 Cl salt under bottom-cooled configuration. The image was captured using a mobile camera. Salt solidifies and water rejects in bulk which forms plumes. The naked eye can visualize plumes, and fragmented dendrites can be seen near the plume
26 wt.% NaCl salt can be dissolved in water, and it is called a saturated composition or maximum solubility of salt in water at that temperature. In SCN-acetone mixture, beyond 11 wt.% acetone, there always exists a liquid mixture at room temperature. In the present experiments, the bulk composition has more than 15 wt.% acetone, in which SCN dissolves in bulk and remelt. In the presence of convection, acetone is constantly enriched in bulk leading to the remelting of the solid. With time, plumes flow physically detached from the base (e.g., the D2 dendrite from the base (Fig. 4d–f) and remelts in bulk. Similar plumes and fragmentation can be observed in the water-salt (NH4 Cl) system (Fig. 7) [19, 20]. Freckle defects or A-segregates are plume structures made up of grain boundaries and disoriented crystals [17, 21–23]. This remelting of fragmented dendrites changes the bulk composition and affects the final product composition. In a few cases of fragmentation, the fragmented solid is not able to remelt and act as a nucleation site which may form equiaxed grains or disoriented grains (the direction of the new dendrite may not be similar to the initial dendrite). For example, suppose the bulk composition is around 10 wt.% acetone. In that case, the fragmented solid cannot be remelted (composition will move < 10 wt.% acetone), and it can grow in size in the presence of thermal undercooling.
4 Conclusions The present study shows the effect of natural convection on the growing solid in simple binary alloy solidification. Succinonitrile-15 wt.% acetone was cooled from the bottom where succinonitrile was growing as dendritic morphology, and acetoneenriched less dense liquid was rejected in bulk. Due to localized variation of growth rate and natural convection, dendritic grains were fragmented and carried upward.
Real-Time Strengthening of Natural Convection and Dendrite …
699
Further, dendrites remelt in bulk. The remelting of fragmented dendrites changes the bulk composition, which leads to several defect formations. Acknowledgements The authors warmly acknowledged the Department of Science and Technology (DST), India (Grant No. EMR/2015/001140).
References 1. Soletta I, Branca M (2005) The frozen lake: a physical model using calculator-based laboratory technology. Phys Teacher 43(4):214–217 2. Yasuda H, Morishita K, Nakatsuka N, Nishimura T, Yoshiya M, Sugiyama A, Uesugi K, Takeuchi A (2019) Dendrite fragmentation induced by massive-like δ–γ transformation in Fe–C alloys. Nat Commun 10(1):1–5 3. Velasco S, White JA, Román FL (2010) Cooling of water in a flask: convection currents in a fluid with a density maximum. Phys Teacher 48(1):60–62 4. Porter DA, Easterling KE (1992) Phase transformations in metals and alloys. Springer-Science Business Media 5. Dantzig JAA, Rappaz M (2009) Solidification. EPFL Press 6. Shih YC, Tu SM (2009) PIV study on the development of double-diffusive convection during the solidification effected by lateral cooling for a super-eutectic binary solution. Appl Therm Eng 29(14–15):2773–2782 7. Karagadde S, Yuan L, Shevchenko N, Eckert S, Lee PD (2014) “3-D Microstructural model of freckle formation validated using in situ experiments. Acta Mater 79:168–180 8. Neumann-Heyme H, Eckert K, Beckermann C (2015) Dendrite fragmentation in alloy solidification due to sidearm pinch-off. Phys Rev E Stat Nonlinear Soft Matter Phys 92(6):1–5 9. Hellawell A, Liu S, Lu SZ (1997) Dendrite fragmentation and the effects of fluid flow in castings. JOM 49(3):18–20 10. Pollock TM, Tin S (2006) Nickel-based superalloys for advanced turbine engines: chemistry, microstructure and properties. J Propul Power 22(2):361–374 11. Rad MT, Kotas P, Beckermann C (2013) Rayleigh number criterion for formation of A-segregates in steel castings and ingots. Metall Mater Trans A 44(9):4266–4281 12. Reinhart G, Grange D, Abou-Khalil L, Mangelinck-Noël N, Niane NT, Maguin V, Guillemot G, Gandin C-A, Nguyen-Thi H (2020) Impact of solute flow during directional solidification of a Ni-based alloy: in-situ and real-time X-radiography. Acta Mater 194:68–79 13. Li Q, Beckermann Q (1999) Evolution of the sidebranch structure in free dendritic growth. Acta Mater 47(8):2345–2356 14. Ustün E, Cadirli E, Kaya H (2006) Dendritic solidification and characterization of a succinonitrile-acetone alloy. J Phys Condens Matter Inst Phys J 18(32):7825–7839 15. Kumar V, Abhishek GS, Srivastava A, Karagadde S (2020) On the mechanism responsible for unconventional thermal behaviour during freezing. J Fluid Mech 903(A32):1–29 16. Kurz W, Fisher DJJ (1998) Fundamentals of solidification. Trans Tech Publications 17. Shevchenko N, Roshchupkina O, Eckert S (2015) X-ray observations showing the effect of fluid flow on dendritic solidification in Ga-In alloys. TMS annual meeting, pp 241–248 18. Shevchenko N, Roshchupkina O, Sokolova O, Eckert S (2015) The effect of natural and forced melt convection on dendritic solidification in Ga-In alloys. J Cryst Growth 417:1–8 19. Kumar V, Srivastava A, Karagadde S (2018) Compositional dependency of double-diffusive layers during binary alloy solidification: full-field measurements and quantification. Phys Fluids 30(11):113603 20. Kumar V, Srivastava A, Karagadde S (2020) Characteristics of solidification-driven doublediffusive layers in mixtures. J Flow Vis Image Process 27(4):427–451
700
V. Kumar et al.
21. Pollock TM, Murphy WH (1996) The breakdown of single-crystal solidification in high refractory nickel-base alloys. Metall Mater Trans A 27(4):1081–1094 22. Shevchenko N, Boden S, Gerbeth G, Eckert S (2013) Chimney formation in solidifying Ga25wt Pct in alloys under the influence of thermosolutal melt convection. Metall and Mater Trans A 44(8):3797–3808 23. Felicelli SD, Sung PK, Poirier DR, Heinrich JC (1998) Transport properties and transport phenomena in casting nickel superalloys. Int J Thermophys 19(2–6):1657–1669
Study of Bubble Growth on a Heated Vertical Surface: Influence of Axial Flow Vibration Nikhil Chitnavis, Harish Pothukuchi, and B. S. V. Patnaik
Abbreviations Nomenclature Rb α Cp Dh G qw dw hlv t db S T Tτ u* k Jd Re Y y+
Bubble radius (mm) Thermal diffusivity (m2 /s) Specific heat (kJ/kg-K) Hydraulic diameter (m) Mass flux (kgm− 2 s− 1 ) Wall heat flux (kWm− 2 ) Contact diameter (kk) Latent heat (kJkg− 1 ) Time (s) Bubble departure diameter (mm) Suppression factor Temperature (K) Frictional temperature (K) Frictional velocity (m/s) Thermal conductivity (Wm− 1 K− 1 ) Jacob number Reynolds number Wall distance (m) Nondimensional wall distance
N. Chitnavis (B) · H. Pothukuchi · B. S. V. Patnaik Department of Applied Mechanics, IIT Madras, Chennai 600036, India e-mail: [email protected] H. Pothukuchi Department of Mechanical 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_64
701
702
h g ID OD
N. Chitnavis et al.
Convective heat transfer coefficient (Wm− 2 K− 1 ) Gravitational acceleration (ms− 2 ) Internal diameter (mm) Outer diameter (mm)
Greek Letters α β φ θ ρ μ σ τ
Advancing contact angle Receding contact angle Inclination of bubble Inclination of heated surface Density of fluid (kgm− 3 ) Dynamic viscosity of fluid (kgm− 1 s− 1 ) Surface tension (N/m) Shear stress
Subscripts conv d nb sat sub w
Convection Departure Nucleate boiling Saturation Sub-cooling Wall
1 Introduction Heat transfer enhancement is of interest to a large number of different industrial applications. Coolant phase change is one of the approaches to improve the heat transfer rate from the heated surface to the coolant. Coolant phase change in the forced convection flow is known as flow boiling of the coolant. As the heated surface temperature crosses the coolant saturation temperature, the sufficient wall superheat triggers the nucleation of vapour bubbles at different sites called nucleation sites on the heated surface. The vapour bubbles form at the nucleation sites, grow in size and depart from the heated surface. This cycle is termed as bubble ebullition cycle, and during this process, coolant extracts additional heat from heated surface in the form of latent heat. The improved heat transfer rate significantly depends on the factors, such as subcooling, wall superheat, bubble growth rate, departure diameter and departure
Study of Bubble Growth on a Heated Vertical Surface: Influence of Axial …
703
frequency. Among these parameters, controlled phase change phenomenon can be achieved by varying the subcooling and by decreasing the bubble departure diameter. One such approach is by vibrating the heated surface. The additional forces acting on the bubble result in its early departure from the heated surface. As the bubble departure diameter reduces, the number of bubbles that are generated, i.e., bubble departure frequency increases which results in enhanced heat transfer rate. Therefore, it is important to study the bubble growth rate phenomenon on the vibrating surface. The important aspect in flow boiling is to predict the departure diameter of the bubble. Experiments were conducted for predicting the bubble diameter during its growth and departure for different subcooling and pressure conditions. Several researchers have developed numerical models to accurately predict the bubble dynamics during its life cycle. However, there is no accurate numerical model to predict the bubble diameter. For the calculation of bubble growth rate, heat flux partitioning method is employed. In this method, the applied total wall heat flux (qw = qml + qsl + qc ) is considered to contribute for (i) heat transfer due to the evaporation of microlayer (qml ) that gets formed beneath the bubble and the heated surface, (ii) heat transfer from superheated layer (qsl ) and (iii) heat transfer due to the condensation (qc ) from the top portion of the bubble that is exposed to the subcooled bulk liquid phase. Situ et al. [7] accounted for only superheated layer heat transfer similar to the model proposed by Zuber [9] for the calculation of bubble radius and compared their predictions with the experimental data. A force balance-based approach is often widely used for predicting the bubble departure diameter. However, as described by [1], these model predictions are very sensitive and not robust. [4] developed the analytical model based on force balance approach where bubble departure diameter is calculated with forces acting on it, i.e., surface tension, buoyancy, unsteady drag, static drag, contact pressure force, hydrodynamic force and lift force. These forces are resolved to act on the bubble in both x- and y-directions. If the sum of the forces either in x-direction or in y-direction becomes greater than zero, then the bubble is considered to be departed either by sliding along or by lift-off from the heated surface. The corresponding bubble size is termed as the bubble departure diameter. Based on the brief literature review presented above, it can be inferred that the bubble growth models for the vibrating heated surfaces have not received the necessary attention. To this end, the present study focuses on analyzing the effect of the vibration of heated surface along the flow direction, to study bubble growth rate. The influence of vibration frequency as well as degree of subcooling are also investigated.
2 Methodology In the present study, the flow boiling through a heated channel is studied. The annular test section with ID 38 mm, OD 19 mm, thus the equivalent hydraulic diameter is 19 mm [7]. Conservation of energy for the vapour bubble is used for calculating its growth rate, and a force balance method is used for calculating the bubble departure
704
N. Chitnavis et al.
diameter. The detailed description of these models are presented in the following sub-sections.
2.1 Nucleating Bubble The growth rate of the vapour bubble is calculated by applying conservation of energy principle. The energy balance equation for the nucleating bubble can be written as follows: E˙ stored = E˙ in − E˙ out + E˙ gen
(1)
As no heat is generated by the bubble, so E˙ gen = 0 and E˙ stored is equal to the latent heat of vapourization (h lv ), where the subscripts l and v denote the liquid and vapour phases respectively (Fig. 1). For simplification, the bubble is considered to be spherical in shape. Further, the heat transfer to the bubble from the superheated layer is only considered. Now, the bubble radius can be defined as [7]: d(R) = dt Fig. 1 Schematic of energy balance in the growing bubble
(/ ) 3 Ja ∗ a 0.5 t −0.5 , π
(2)
Study of Bubble Growth on a Heated Vertical Surface: Influence of Axial … Table 1 Models for wall temperature prediction
705
Correlation qw = qconv + qnucleate qconv = h conv (Twall − Tbulk ) qnucleate = h nb (Twall − Tsat )
Chen [2]
h conv = 0.023Rel0.8 Pr 0.4 Dklh F ( 0.79 0.45 0.49 ) k C ρ h nb = 0.00122 0.5l 0.29pl 0.24l 0.24 b σ
μl
h lv ρg
0.24 /P 0.75 S b = /Tsat
F = 1&S =
1 , 1+2.53×10−06 Re1.17 tp
( ) ρ C (T −Tsat ) where J a is Jacob number l pf ρvwall , α is thermal diffusivity and Rb is the h lv bubble radius which is a function of time (t) only.
2.2 Wall Temperature Chen’s correlation [2] is used for the calculation of the wall temperature of the heated surface, and the detailed correlation is given in Table 1. Where qw is the total applied wall heat flux, qconv is the convective heat transfer by single-phase liquid and qnucleate is the heat transfer from nucleate pool boiling. S and F are the suppression and the enhancement factors, respectively, considering F = 1 for the subcooled flow boiling.
2.3 Force Balance Model For the calculation of bubble departure diameter, force balance approach is used. Figure 2 shows the schematic of different forces acting on the bubble, which can and in E y-directions. If the sum of the forces in any be resolved in both x-direction E Fy > 0), then the corresponding bubble direction crosses zero ( Fx > 0 or diameter is called as bubble departure diameter. Surface tension force (Fs ) tries to oppose the detachment of the bubble from the surface which is defined in Eqs. (3, 4) Fsx = −1.25dw σ
π (α − β) (sin α − sin β), − (α − β)2
π2
Fsy = −dw σ
π (cos β − cos α), (α − β)
(3) (4)
706
N. Chitnavis et al.
Fig. 2 Schematic of forces on a growing bubble
where Fsx and Fsy are the surface tension force in x-direction and in y-direction, α and β are the advancing and receding contact angles and dw is the contact width. Shear lift force Fsl , acts perpendicular to the heated surface and supports the detachment of the bubble. This shear lift force was given by [4]. | ( −2 )0.25 | Fsl = 0.5ρl Ul2 π R 2 3.87G 0.5 Re + 0.118G 2s s
(5)
Gs =
dUl R dy Ul
(6)
Re =
ρl Ul db μl
(7)
The fluid velocity defined near the heated wall surface is assumed as single-phase turbulent velocity profile. ⎧ ⎨
y+ ≤ 5 u ∗ (= y)+ , + u = 5 ln y ( −) 3.05, 5 < y + < 30 , ⎩ ∗ u = 2.5 ln y + + 5. 5, y + ≥ 30 ∗
(8)
/ ∗ f ρ v2 where y + = yuϑ , u ∗ = τwall and τwall = l 8l l , fl is the friction factor and vl is the ρl average velocity of the fluid. Unsteady drag force is the force exerted by the growing bubble on the surrounding fluid which is defined as: | | Fdu = −ρl π R 2 R R¨ + 1.5 R˙ 2 ,
(9)
where Rb is the bubble radius, R˙ b is the rate of change of bubble size (radius) with ˙ time and R¨ b = ddtRb . Quasi-static drag force is the force applied on the bubble, considering it as a stationary body as defined by Mei and Klausner [5, 6]. The drag force for the turbulent flow is defined as:
Study of Bubble Growth on a Heated Vertical Surface: Influence of Axial …
Fqs 2 = + 6π μl Ul R 3
|(
12 Re
707
|−1/n
)n + 0.769
n
,
(10)
where n = 0.65 and Rb is the bubble radius. Buoyancy force acts on the bubble due to density difference between the liquid and the vapour phases. This force tries to detach the vapour bubble from the heated surface. The contact pressure force arises due to difference in pressure inside and outside the bubble, whereas the hydrodynamic pressure force acts on the bubble in the direction perpendicular to the heated wall. The above listed forces are defined as: Fb =
4 π R 3 (ρl − ρv )g, 3
(11)
π dw2 2σ , 4rr
(12)
π d2 9 ρl U 2 w , 8 4
(13)
Fcp = Fh =
where ρl and ρg are the densities of surrounding liquid and vapor phases. If the sinusoidal motion is given to the heated surface, the displacement, acceleration and mass flux variation with respect to the sinusoidal motion of the heated surface is defined as: y(t) = ym sin(2π f t),
(14)
a(t) = −4π 2 f 2 ym sin(2π f t),
(15)
where y(t) is the displacement of the heated surface in the vertical direction, a(t) is the acceleration of the plate and ym is the maximum amplitude of the displacement. The gravitational acceleration is modified as: g ' = g − a(t).
(16)
Also due to variation in acceleration with respect to time, velocity of the incoming liquid, mass flux and finally the buoyant force will change, which is defined as: u 'f = u f + /u f ω(t), ) 2π t , ω(t) = sin T (( )) 4π 2 ym /T 2 /u f = 53,000 Re−1.45 u f , g
(17)
(
(18)
(19)
708
N. Chitnavis et al. ' τwall = 0.5C f ρ f u ,2f ,
Fb' =
4 π R 3 (ρl − ρv )g ' , 3
(20) (21)
where g ' , u 'f , f , Fb, are the updated gravitational acceleration, liquid velocity, frequency with which the heated plate vibrates and the updated buoyant forces exerted on the bubble. Rb is the bubble radius. The summation of the forces in both x- and y- directions are written as follows: E E
Fx = Fsx + Fqs + Fdux + Fb + Fa ,
(22)
Fy = Fsy + Fduy + Fsl + Fh + Fcp .
(23)
E E If Fx > 0 then the bubble slides along the heated surface. Whereas, if Fy > 0 then the bubble lifts-off from the heated surface. In both the cases, either the bubble slides or lifts-off from heated surface is considered as departure from the heated surface. The corresponding bubble size is identified as the bubble departure diameter.
3 Results and Discussion This section presents the validation studies for both the static and vibrating heated surface cases. Further, the study is extended to investigate the influence of vibrating frequency and the degree of subcooling on the bubble growth rate and the departure diameter.
3.1 Validation The present model developed based on the energy and force balance for the calculation of bubble growth rate and the departure is validated with the experimental dataset of Situ et al. [7] and Sugrue et al. [8]. The corresponding experimental conditions are mentioned in Table 2. The present model accounts for only conduction heat transfer from the superheated layer and the predictions are compared with the experimental data as shown in Figs. 3, 4, 5, 6 and 7. A good agreement is noticed with a mean percentage error of 30%. Figures 3, 4 present the comparison of bubble departure diameter predictions against the measured data for the static plate subjected to subcooled flow conditions. The same model is extended to predict the bubble growth rate and the departure diameter when the heated plate is subjected to a vibration. When the heated plate is
Study of Bubble Growth on a Heated Vertical Surface: Influence of Axial … Table 2 Experimental data for validation
709
Situ et al. [7]
Sugrue et al. [8]
Fluid
Water
Water
Dh (mm) ) ( G kgm−2 s−1
19.1
16.7
466–900
250–400
/Tsub (◦ C) ( ) q kW m −2
1.5–20
10–20
60.7–206 ( ◦) Vertical 90
50–100
Orientation
◦
0 − 90
◦
Fig. 3 Comparison of predicted bubble departure diameter against the experimental data of Situ et al. [7]
Fig. 4 Comparison of predicted bubble departure diameter with the experimental data of Sugrue et al. [8]
subjected to sinusoidal motion, the vibration results in additional acceleration and hence mass flux also changes as the heated surface is moving along the direction of the incoming liquid flow. As a result, additional forces are exerted on the bubble due to the variation in the acceleration, as defined in [3]. The variation of displacement, acceleration and mass flux are compared with the experimental data of Hong et al. [3] as shown in Fig. 5. The bubble growth rate predictions at a single nucleation site on the heaving heated surface are shown in Fig. 6. A good agreement is observed when compared
710
N. Chitnavis et al.
Fig. 5 Temporal variation of mass flux due to an imposed axial vibration Hong et al. [3]
Fig. 6 Bubble growth rate over six different time periods against the experimental study of Hong et al. [3]
Fig. 7 Bubble departure diameter for static and vibrating plate against the experimental data of Hong et al. [3]
Study of Bubble Growth on a Heated Vertical Surface: Influence of Axial …
711
with the experimental data of Hong et al. [3]. Further, the bubble departure diameter is calculated for the dataset of Hong et al. [3] for the static and heaving heating surfaces. A good agreement with a mean relative error of around 30% is observed as shown in Fig. 7. Further, the mass flux also varies as a sinusoidal function because of the heated surface motion. As a result of the fluctuating mass flux, bubble departure diameter also changes with time. If the mass flux is greater than its mean value for the same heat flux and subcooling, then the bubble departure diameter is observed to be smaller compared with the static case. Whereas, if the mass flux is smaller than its mean value then, the bubble departure diameter is noticed to be larger than the static case. These predictions are in line with the experimental observation of Hong et al. [3].
3.2 Influence of Subcooling and Vibration Frequency Heat transfer enhancement from the heated surface can be achieved by disrupting the thermal boundary layer. Due to the disruption of the thermal boundary layer, the heated surface gets in to direct contact with the lower temperature liquid coolant which results in heat transfer enhancement. There are various ways for disrupting the formation of thermal boundary layer such as (a) providing a small sinusoidal displacement to the heated surface either along the fluid flow or normal to the flow and (b) by the nucleation of vapour bubble on the heated surface. Thermal boundary layer gets disturbed due to the bubble growth and departure from the heated surface. At the time of bubble departure, the bubble takes away the significant amount of heat from the surface. When the bubble leaves the nucleation site, the surrounding cold fluid rushes to that site and thus increases the heat transfer. Therefore, the bubble departure diameter and the frequency have a strong influence on the heat transfer from the heated surface. The bubble departure diameter can be varied to further increase the heat transfer by vibrating the heated surface with a sinusoidal displacement along the flow for subcooled flow boiling conditions. When a vapour bubble forms over the heated surface, during its growth, it extracts an additional amount of heat from the surface in the form of latent heat, and then departs from the nucleation site. In the present study, the heated surface is vibrated with a certain frequency along the coolant flow direction to study the growth rate of the bubble on the vibrating heated surface. The additional forces acting on the bubble due to the vibration of heated surface may result in the early departure of the vapour bubble. As a result, the heat transfer rate enhances compared with the heat transfer rate from the static heated surface. To this end, the influence of subcooling and vibration on the bubble growth rate and departure diameter is studied in the present work. Figure 8 shows that with increase in frequency of vibration of the heated surface, the bubble departure diameter gradually decreases. The effect of subcooling on the bubble departure diameter is studied by considering two different values of subcooling, 20 and 30 K for a heat flux of 300 kWm−2 , mass flux of 250 kgm−2 s−1
712
N. Chitnavis et al.
at 1 bar pressure. This shows that (Fig. 8), with decrease in bubble departure diameter with frequency, decreases the total ebullition time, due to which large amount of bubble will able to depart from the heated surface which in turn increases the heat transfer from the heated plate. Similarly, shown in Fig. 9 is the evolution of bubble with time by varying frequency of the heated plate. Figure 9 also shows that with increase in frequency, the bubble growth time and bubble departure diameter decrease. Fig. 8 Variation of bubble departure diameter with the vibration frequency of the heated surface for two different subcoolings, 20 and 30 K
Fig. 9 Influence of the frequency of vibration of the bubble growth rate
Study of Bubble Growth on a Heated Vertical Surface: Influence of Axial …
713
4 Conclusions Numerical simulations are carried out to determine the bubble growth rate and departure diameter under subcooled flow boiling conditions for both the static and vibrating surfaces. For the calculation of vapour bubble growth rate, only conduction heat transfer from the superheated layer is considered, whereas for calculating the bubble departure diameter force balance model was implemented. The following conclusions were made based on the present study. • The present numerical model is validated against the experimental data for static and vibrating heated surfaces, and a good agreement was noticed. • Due to the vibration of heated surface along the flow direction, the bubble departure diameter decreases compared with the static case. • By increasing the degree of subcooling, the bubble departure diameter is found to decrease, which in turn increases the heat transfer rate from the heated surface.
References 1. Bucci M, Buongiorno J, Bucci M (2021) The not-so-subtle flaws of the force balance approach to predict the departure of bubbles in boiling heat transfer. Phys Fluids 33(1):017110 2. Chen JC (1966) Correlation for boiling heat transfer to saturated fluids in convective flow. Ind Eng Chem Process Des Dev 5(3):322–329 3. Hong G et al (2012) Bubble departure size in forced convective subcooled boiling flow under static and heaving conditions. Nucl Eng Des 247:202–211 4. Klausner JF et al (1993) Vapor bubble departure in forced convection boiling. Int J Heat Mass Transfer 36(3):651–662 5. Mei R, Klausner JF (1992) Unsteady force on a spherical bubble at finite Reynolds number with small fluctuations in the free-stream velocity. Phys Fluids A 4(1):63–70 6. Mei R, Klausner JF (1994) Shear lift force on spherical bubbles. Int J Heat Fluid Flow 15(1):62– 65 7. Situ R et al (2005) Bubble lift-off size in forced convective subcooled boiling flow. Int J Heat Mass Transfer 48(25–26):5536–5548 8. Sugrue R, Buongiorno J, McKrell T (2014) An experimental study of bubble departure diameter in subcooled flow boiling including the effects of orientation angle, subcooling, mass flux, heat flux, and pressure. Nucl Eng Des 279:182–188 9. Zuber N (1961) The dynamics of vapor bubbles in nonuniform temperature fields. Int J Heat Mass Transf 2(1–2):83–98
Comparative Study of Droplet Impact Characteristics with Various Viscous Liquids: A Study of Both Miscible and Immiscible Droplet Impacts Kollati Prudhvi Ravikumar, Abanti Sahoo, and Soumya Sanjeeb Mohapatra
Nomenclature d d1 dh dv g h v μ ρ σ τ RBO VR
Diameter of the impacting droplet (mm) Diameter of the secondary droplet (formed after a jet breakup) (mm) Horizontal diameter of the secondary droplet (mm) Vertical diameter of the secondary droplet (mm) Acceleration due to gravity (m/s2 ) Maximum jet height (mm) Droplet velocity (m/s) Dynamic viscosity (mPa · s) Density (kg/m3 ) Surface tension (N/m) Dimensionless time Rice-bran oil Viscosity ratio of liquid pool viscosity to impinging droplet viscosity
1 Introduction The physics behind the droplet impact on a liquid surface is a complex fluid dynamics problem. Solving such problems helps in creating real applications such as inkjet printing [1], spray cooling [2, 3], fire extinguishing [4], agriculture equipments for spraying [5], emulsion [6], micro fluidics [7] and other industrial applications [8]. The droplet impact on a liquid surface depends on the impact characteristics: K. P. Ravikumar · A. Sahoo · S. S. Mohapatra (B) Department of Chemical 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_65
715
716
K. P. Ravikumar et al.
droplet diameter, droplet velocity and height of the pool liquid and on fluid properties: viscosity, surface tension and density [9]. The impact characteristics vary from miscible to immiscible droplet impact: the determination of immiscible impact was more complex than the miscible droplet impact, as it involves the two different liquids properties [10]. The droplet impingement on the liquid surface results in floating, bouncing, coalescence, crater formation, crown formation, central jet and the central jet breakup [11]. Crater, crown and central jet formation come under the splashing phenomena. After the droplet impingement, the crown formation initially takes place with the dissipation of tiny droplets, simultaneously crater forms under the liquid surface, depending on the impact strength depth of the crater increases. Whenever the buoyancy force overcomes the impact force, the liquid jet ejects above the surface. Later the jet breaks up due to the instability and forms the secondary droplets [12–14]. Several pieces of research explain the irregularities in the splashing phenomena and secondary droplet formation with different mechanisms. Roisman, with his contributors, finally concluded that the liquid rim subjected to the Rayleigh–Taylor and Rayleigh–Plateau instability [15–17]. When the rim is experiencing deceleration, the Rayleigh–Taylor instability dominates and when it is at later times the Rayleigh– Plateau instability dominates [18]. Cossali et al. [19] reported that secondary droplets are produced before the full crown formation for low viscous liquids, called early splash, and after the crown formation for the high viscous liquids, called late splash [10]. Motzkus [20] reported first the splashing thresholds for early and delay (late) splash. The secondary droplet size was large in the late splash, which plays a vital role in the inkjet printing [13]; the secondary droplet size was small in the early splash, which can entrap the air more easily in aerosol production [22] and gas–liquid separation [23].
2 Literature Review and Objective The splashing is an undesirable parameter in pool fire extinguishing. In liquid pool fire burning, the water mist is used for economical fire extinguishing. At the time of fire extinguishing, the water mist sprinkled on the surface of the liquid pool results in splashing, liquid (liquid fuel) splashes to burn the fire more vigorously [25]. In the current research, the splashing characteristics were investigated with miscible and immiscible droplet impacts. Fuel (petrol, kerosene, diesel and rice-bran oil) droplet impact on a fuel pool is considered as miscible droplet impact, whereas the water droplet impact on a fuel pool is considered as a immiscible droplet impact. For every experiment, the height of the liquid droplet varied to attain different velocities. After the liquid droplet impact, the jet height and the secondary droplet characteristics were measured for both miscible and immiscible droplet impact. The non-dimensional numbers characterise all the liquid’s impact characteristics, and Table 1 explains the non-dimensional numbers in the current work. After the droplet
Comparative Study of Droplet Impact Characteristics with Various …
717
Table 1 Brief explanation about the non-dimensional numbers Non-dimensional number
Mathematical equation
Definition
μv σ
Viscous force Surface tension
Ohnesorge number (Oh)
√μ ρσ d
√
Weber number (We)
ρv2 d σ
Inertia force Surface tension
Froude number (Fr)
v2 gd
Inertia force Gravitational force
Capillary number (Ca) (Dimensionless velocity)
Viscous force Inertia force x surface tensionnsion
impact on the liquid surface, the primary droplet combines with the impacted liquid produces the central jet. The central jet liquid is a combination of primary droplet liquid and pool liquid. So, the non-dimensional numbers for the secondary droplet can be calculated with little effort in the miscible droplet impact case, whereas for the immiscible droplet case, it’s a complex phenomenon. To reduce complexity, for both miscible and immiscible droplet impact cases, the dimensional numbers calculated for the secondary droplet are based on the primary droplet but not on the properties of the secondary droplet.
3 Experimentation The liquid droplet was released from three heights (0.04, 0.03 and 0.02 m) to vary the droplet velocity. Five different viscous liquids (water, petrol, kerosene, diesel and rice-bran oil) were used in the current investigation for the liquid droplet and four different liquids for the liquid pool (petrol, kerosene, diesel and rice-bran oil). The liquid pool consists of 0.014 m depth and 0.96 m diameter. VITA19 (Rame-hart, 0.69 mm internal diameter) dispensing needle is used for droplet generation using droplet-generating system (syringe). A high-speed imaging system with 1000fps captured all droplet impact processes at 1080 × 1920 pixels. The photographs were taken with 1/1600 s shutter speed, 11 mm focal length and the optical axis aligned normal to the object. The special resolution occupied by the droplet is about 16.3 pixel/mm. As spatial resolution plays a vital role in calculating diameter and velocity, camera settings should not be altered. The droplet dimensions like diameter, jet height and velocity were calculated using the image processing techniques (Fig. 1). When the droplet was in motion, its shape was not spherical; it was like prolate ellipsoid and oblate ellipsoid shapes. The equivalent spherical droplet diameter was calculated using Eq. 1 [10]. All the experiments were performed at room temperature of 25 °C. 1 d = d v d 2h 3
(1)
718
K. P. Ravikumar et al.
Fig. 1 Schematic diagram of droplet dispensing system with a high-speed camera
For calculation, the liquid properties were considered at room temperature from Table 2—all the performed experimental data for liquid is tabulated in Table 3. The experimental dimensionless numbers of ranges are mentioned in Table 4. The gravity effect was considered insignificant for the Froude number greater than 100. For all the experiments, the Froude number was greater than 100, so the gravitational effect was given the least importance. Table 2 Fluid properties at 25 °C ρ (kg/m3 )
μ (mPa.s)
σ (N/m)
S. No.
Fluid
1
Petrol
680
0.6
0.0216
2
Kerosene
810
1.64
0.025
3
Diesel
850
3
0.028
4
RBO
910
59.26
5
Water
1000
0.89
0.032625 0.072
Table 3 Droplet impact parameters Fluid
d (mm)
v (m/s)
h (mm)
d 1 (mm)
Petrol
2.37–2.41
2.03–3.11
12.64–23.44
1.45–3.55
Kerosene
2.55–2.73
1.92–2.73
20.07–25.69
1.02–3.45
Diesel
2.38–2.88
2.07–2.73
9.30–22.85
1.52–2.34
RBO
2.78–2.94
1.99–2.73
5.56–7.09
0
Water
2.99–3.61
2.03–3.52
NA
NA
Comparative Study of Droplet Impact Characteristics with Various …
719
Table 4 Non-dimensional number ranges Fluid
We
Oh
Fr
Ca
Petrol
313–719
0.003
174–416
0.0564–0.0863
Kerosene
326–614
0.007
137–299
0.126–0.179
Diesel
376–539
0.012
152–319
0.222–0.292
RBO
298–588
0.204–0.210
150–269
3.622–4.956
Water
193–597
0.002
130–364
0.025–0.044
4 Results and Discussion In the current investigation, the liquid droplet impacts on a liquid surface result in the crown, crater, central jet, jet breakup, secondary droplets and secondary jet formation. Figure 2 depicts all the impact characteristics; all the droplets impact characteristics may not be noticed in all the impact phenomena. The impact characteristics cannot be distinguished with a single non-dimensional parameter. It depends on the properties of the droplet: density, viscosity and surface tension; pool properties: density, viscosity, surface tension and depth of the pool and impact parameters: droplet size and velocity. The droplet impact on the liquid surface is considered in two categories, with miscible and immiscible droplet impact.
4.1 Miscible Droplet Impact In the current research, Oh = 0.002–0.032 is treated as low viscous liquid and Oh = 0.204–0.210 is treated as a very high viscous liquid. The low viscous liquids increase crown height and crown base diameter with increasing Ohnesorge number for the same impact height, and it can be observed clearly in the snapshots taken by a highspeed imaging camera in Fig. 3a–c . In addition, the jet breaks up early, and the thickness of the jet is higher for less viscous liquid than the high viscous liquid to the low viscosity range liquids. After the droplet impacts the liquid surface, the crown formation is noticed, and later, the crown ejects into ring shape for low viscous liquids, whereas compound crown for higher viscous liquids in the low viscosity
Fig. 2 Snapshots of a petrol droplet impact on a liquid pool (petrol) at different instants, We = 719 and Oh = 0.032
720
K. P. Ravikumar et al.
Fig. 3 a Petrol, b Kerosene, c Diesel, and d Rice-Bran oil; overview of all the four liquid droplets impact the liquid pool at different instants. Liquid droplet and the pool liquid are maintained with the same liquid
range liquids. The high viscous liquid with the Oh = 0.204 did not produce any crown due to insufficient impact energy (less Weber number) on the liquid surface, as seen in Fig. 3d. All the above phenomena deal with the miscible droplet impacts.
4.2 Comparison of Miscible and Immiscible Droplet Impact (a) Jet Height For all the immiscible droplet impacts, the thickness of the jet is larger than the miscible droplet impact; it is due to the high impact energy for water compared with other test liquids, clearly observed from Figs. 4, 5, 6 and 7. As the impact energy is higher for the water, the immiscible droplet impact dissipates the larger number of tiny droplets during splashing. Therefore, for pool fire extinguishing with water, the impact energy is regulated to control the liquid splashing from the pool. The jet height in the miscible droplet impact was higher than the immiscible droplet impact, as seen in Fig. 8a and b. Failing to control the splashing in liquid pool fire burning results in fire expansion over the liquid pool’s surface.
Fig. 4 a Petrol droplet impact on a petrol pool surface and b water droplet impact on a petrol pool surface from the same height
Comparative Study of Droplet Impact Characteristics with Various …
721
Fig. 5 a Kerosene droplet impact on a kerosene surface and b water droplet impact on a kerosene surface from the same height
Fig. 6 a Diesel droplet impact on a diesel surface and b water droplet impact on a diesel surface from the same height
Fig. 7 a Rice-bran oil droplet impact on a rice-bran oil surface and b water droplet impact on a rice-bran oil surface from the same height
Fig. 8 The Variation of the maximum jet height with Weber number represents various viscous liquids impacts: a Miscible droplet impact, b immiscible droplet impact
722
K. P. Ravikumar et al.
Fig. 9 Variation of the maximum jet height vs non-dimensional time represented for various viscous liquids impacts: a Miscible droplet impact, b immiscible droplet impact
For different viscous liquids, max jet height increases with the Weber number, as shown in Fig. 8. In addition, the time to achieve the maximum jet height also increases with the Weber number, depicted in Figs. 8 and 9. The maximum jet height increases linearly with the Weber number for the Oh = 0.012. Above discussed phenomena noticed similar for both miscible and immiscible droplet impacts. By comparing the Figs. 9a and b, the jet height in immiscible droplet impact was higher than the miscible droplet impact, due to low Oh of water when compared with the other test liquids. The time duration for attaining the maximum jet was larger for low viscous liquids and shorter for high viscous liquids, as depicted in Fig. 9a and b. (b) Secondary Droplet Diameter After droplet impact on the liquid surface, the secondary droplet produced from the central jet breakup. In the current research, the secondary droplet did not produce for the very high viscous liquid for Ohnesorge number greater than 0.204, in case of immiscible droplet impact. As all the experiments were conducted at atmospheric conditions, the secondary droplet undergoes deformation and forms a spheroid shape. The diameter of the spheroid in the horizontal direction was more than the vertical direction due to the effect of buoyancy effect; these phenomena were noticed similarly for miscible and immiscible droplet impact. The variation noticed for the diameter of the droplet in horizontal and vertical direction was very less and justifies with the literature as the Fr > 100(refer Table 4) for all the liquids considered in the current research. Figure 10 shows that the d1 /d ratio increases with the increase in capillary number for low viscous liquids and no secondary droplet formed for higher viscous liquids in case of miscible droplet impact. The maximum secondary droplet declines with the increase in Ca due to increased effect of viscous force and decrease in surface tension, as depicted in Fig. 10. Figure 11 depicts the variation of d1 /d with the Ca for immiscible droplet impact. No secondary droplet was noticed for immiscible droplet impact from the low height in case of higher viscosity ratio (greater than 3.37) due to the poor impact strength
Comparative Study of Droplet Impact Characteristics with Various …
723
Fig. 10 Secondary droplet variation with the Capillary number for different Ohnesorge numbers (miscible droplet impact)
(We) and Ca. So, there is need to identify the critical limit for the secondary droplet formation in case of higher VR, the vertical line in Fig. 11 represents critical Capillary number at 0.02635. Fig. 11 Critical limit for identifying the formation of secondary droplet for VR > 3.37
724
K. P. Ravikumar et al.
5 Conclusions The droplet impact on the liquid surface is considered with five impact liquids and four pool liquids. The characteristics of the miscible and immiscible droplet impact investigated the results compared with the non-dimensional parameters. The critical findings in the current investigation are as follows: • The only single non-dimensional number cannot explain the liquid splashing characteristics for different liquids. The splashing characteristics vary from liquid to liquid. Therefore, it was suggested to use the non-dimensional numbers to express the liquid droplet impact characteristics. • After the liquid droplet impact on the liquid surface, splashing takes place with crown, crater, central jet, jet break up, secondary droplet and secondary jet formation were noticed. • The splashing characteristics increase for all the immiscible droplet impacts compared with the miscible droplet impact for the same droplet impact height. • The jet becomes thin with the increase in Ohnesorge number for lower viscous liquids in case of miscible droplet impact. • The liquid jet breaks up early for lower Ohnesorge number in case of low viscous miscible droplet impacts. • The jet height in immiscible droplet impact was higher than the miscible droplet impact, as the water has the low Oh when compared with the other test liquids. • The time taken to attain the maximum height was higher for high viscous liquids (Oh > 0.204) in case of immiscible droplet impact than the miscible droplet impact. • The critical Ca number for formation of secondary droplet in case of higher VR range was identified at 0.02635.
References 1. Castrejón-Pita JR, Muñoz-Sánchez BN, Hutchings IM, Castrejón-Pita AA (2016) Droplet impact onto moving liquids. J Fluid Mech 809:716–725. https://doi.org/10.1017/jfm.2016.672 2. Fujimoto H, Oku Y, Ogihara T, Takuda H (2010) Hydrodynamics and boiling phenomena of water droplets impinging on hot solid. Int J Multiph Flow 36(8):620–642. https://doi.org/10. 1016/j.ijmultiphaseflow.2010.04.004 3. Li H, Mei S, Wang L, Gao Y, Liu J (2014) Splashing phenomena of room temperature liquid metal droplet striking on the pool of the same liquid under ambient air environment. Int J Heat Fluid Flow 47:1–8. https://doi.org/10.1016/j.ijheatfluidflow.2014.02.002 4. Hsieh TL, Wu YL, Ho MC, Chung KC (2006) Characterisation of water spray on fire suppression. JSME Int J Ser B Fluids Therm Eng 49(2):490–497. https://doi.org/10.1299/jsmeb. 49.490 5. Ashgriz N (2011) Handbook of atomization and sprays. Springer, US, Boston, MA 6. Dudek M et al (2019) Microfluidic method for determining drop-drop coalescence and contact times in flow. Colloids Surf A Physicochem Eng Asp 586:124265. https://doi.org/10.1016/j. colsurfa.2019.124265
Comparative Study of Droplet Impact Characteristics with Various …
725
7. Narhe R, Beysens D, Nikolayev VS (2004) Contact line dynamics in drop coalescence and spreading. Langmuir 20(4):1213–1221. https://doi.org/10.1021/la034991g 8. Ersoy NE, Eslamian M (2020) Phenomenological study and comparison of droplet impact dynamics on a dry surface, thin liquid film, liquid film and shallow pool. Exp Therm Fluid Sci 112:109977. https://doi.org/10.1016/j.expthermflusci.2019.109977 9. Zhang G, Quetzeri-Santiago MA, Stone CA, Botto L, Castrejón-Pita JR (2018) Droplet impact dynamics on textiles. Soft Matter 14(40):8182–8190. https://doi.org/10.1039/C8SM01082J 10. Okawa T, Kubo K, Kawai K, Kitabayashi S (2021) Experiments on splashing thresholds during single-drop impact onto a quiescent liquid film. Exp Therm Fluid Sci 121:110279. https://doi. org/10.1016/j.expthermflusci.2020.110279 11. Rodriguez F, Mesler R (1988) The penetration of drop-formed vortex rings into pools of liquid. J Colloid Interface Sci 121(1):121–129. https://doi.org/10.1016/0021-9797(88)90414-6 12. Engel OG (1966) Crater depth in fluid impacts. J Appl Phys 37(4):1798–1808. https://doi.org/ 10.1063/1.1708605 13. Leng LJ (2001) Splash formation by spherical drops. J Fluid Mech 427:73–105 [Online]. Available: http://journals.cambridge.org/abstract_S0022112000002500 14. Engel OG (1967) Initial pressure, initial flow velocity, and the time dependence of crater depth in fluid impacts. J Appl Phys 38(10):3935–3940. https://doi.org/10.1063/1.1709044 15. Roisman IV, Horvat K, Tropea C (2006) Spray impact: rim transverse instability initiating fingering and splash, and description of a secondary spray. Phys Fluids 18(10). https://doi.org/ 10.1063/1.2364187 16. Roisman IV, Gambaryan-Roisman T, Kyriopoulos O, Stephan P, Tropea C (2007) Breakup and atomisation of a stretching crown. Phys Rev E Stat Nonlinear Soft Matter Phys 76(2):1–9. https://doi.org/10.1103/PhysRevE.76.026302 17. Roisman IV (2010) On the instability of a free viscous rim. J Fluid Mech 661:206–228. https:// doi.org/10.1017/S0022112010002910 18. Agbaglah G, Josserand C, Zaleski S (2013) Longitudinal instability of a liquid rim. Phys Fluids 25(2). https://doi.org/10.1063/1.4789971 19. Cossali GE, Coghe A, Marengo M (1997) The impact of a single drop on a wetted solid surface. Exp Fluids 22(6):463–472. https://doi.org/10.1007/s003480050073 20. Kitabayashi S, Enoki K, Okawa T (2017) Experiments on the splashing limit during drop impact onto a thin liquid film. In: International conference on nuclear engineering proceedings (ICONE), vol 9, no 2, pp 1–5. https://doi.org/10.1115/ICONE25-67021 21. Motzkus C, Gensdarmes F, Géhin E (2011) Study of the coalescence/splash threshold of droplet impact on liquid films and its relevance in assessing airborne particle release. J Colloid Interface Sci 362(2):540–552. https://doi.org/10.1016/j.jcis.2011.06.031 22. Motzkus C, Gensdarmes F, Géhin E (2009) Parameter study of microdroplet formation by impact of millimetre-size droplets onto a liquid film. J Aerosol Sci 40(8):680–692. https://doi. org/10.1016/j.jaerosci.2009.04.001 23. Nakao T, Saito Y, Souma H, Kawasaki T, Aoyama G (1998) Droplet behavior analyses in the BWR dryer and separator. J Nucl Sci Technol 35(4):286–293. https://doi.org/10.1080/188 11248.1998.9733858 24. Shrigondekar H, Chowdhury A, Prabhu SV (2021) Performance by various water mist nozzles in extinguishing liquid pool fires. Fire Technol 57(5):2553–2581. https://doi.org/10.1007/s10 694-021-01130-0 25. Deegan RD, Brunet P, Eggers J (2008) Complexities of splashing. Nonlinearity 21(1):C1–C11. https://doi.org/10.1088/0951-7715/21/1/C01
Parametric Study on Marangoni Instability in Two-Layer Creeping Flow Ankur Agrawal and P. Deepu
Nomenclature Ca Ma Pe Re x y β D1 μ1 σ0
Capillary number Marangoni number Peclet number Reynolds number Dimensionless streamwise direction Dimensionless spanwise direction Interfacial sensitivity parameter (1/mol) Diffusivity of speice in top layer fluid (m2 /s) Dynamic viscosity (kg/m-s) Interfacial tension at reference concentration (kg/s2 )
1 Introduction The Marangoni instability has attracted many researchers because of its relevance to many physio-chemical applications. Such instability occurs in systems involving concentration variation or temperature gradient across an interface. The current study performs linear stability analysis of a fully developed two-layer stratified flow of Newtonian fluids with concentrations that are maintained uniformly at the walls of the two-dimensional channel. The problem setup is representative of system employed in extraction processes [1], catalyst phase transfer reactions [2, 3], etc. Many experimental studies have been carried out over the extraction processes [4–7] in a two-layer A. Agrawal (B) · P. Deepu Department of Mechanical Engineering, IIT Patna, Patna 801106, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_66
727
728
A. Agrawal and P. Deepu
stratified pressure-driven flow and found that species can be transferred efficiently from one fluid to another by adjusting the flow rate of fluids. The history of stability analysis picked up the pace when Yih [8] carried out stability analysis on two-layer stratified flow and identified the interfacial mode of instability. Yiantsios and Higgins [9] carried out the asymptotic study to reveal the instability behavior for long-wave perturbations. Boomkamp [10] studied the stratified flow system and carried out the energy budget analysis which confirms that the instability mode identified by Yih is viscosity-driven. The flow is generally found stable in the creeping flow limit, however, it could become unstable under the influence of soluble surfactants. There are a few studies on the stability analysis related to soluble surfactants in a shear flow system. Some studies carried out Marangoni instability analysis. Marangoni instabilities can originate when a liquid–liquid interface is subjected to a concentration or temperature gradient. Some recent works [11, 12] showed Marangoni instability of drop lay within stable stratified liquids. Soleimani [13] carried out Rayleigh–Taylor instability over non-isothermal two-layered fluids of distinct densities and showed some control over instability within an interval of Marangoni numbers. Some studies [14, 15] carried out linear stability analysis over two-phase flow where Marangoni triggers instability, assuming a non-deformable interface. Deformable interface disturbs the uniformity of species near it, which varies the interfacial tension and thus could lead to Marangoni instability. You [16] carried out an analysis considering the deformed interface but missed to account for its effect on interfacial concentration gradient. Picardo [1] filled this gap by investigating the Marangoni instability for two-layer stratified flow. The present work aims to study the Marangoni instability in creeping flow for a wide range of parameters, viz. Peclet number and diffusivity ratio.
2 Formulation 2.1 Base State Consider two Newtonian fluids flowing in two-layer stratified fashion in between two infinitely long plates. The plates are maintained at two different but constant concentration of species or surfactants. The system is limited to a two-dimensional, fully developed isothermal incompressible flow regime, where flow is considered to be pressure driven. The system of two-layered fluid flows in a positive x-direction from the origin, considered to be placed at the interface of two liquids. On the basis of these assumptions, the continuity equation, momentum equation, and transport equation are scaled with the interface velocity (U0 ), concentration maintained at the top plate (C10 ), and top fluid layer thickness (d1 ). The base state solutions for the considered system are given by:
Parametric Study on Marangoni Instability in Two-Layer Creeping Flow
729
u j = b j y 2 + a j y + 1, v j = 0.
(1)
cj = sj y + tj
(2)
where j denotes the fluid layer; j = 1 (top layer) and j = 2 (bottom layer). u j and c j are the velocity and concentration fields respectively. The constants appearing the solutions are: a1 = m − n 2 /n(n + 1), a2 = a1 /m,b1 = −(m + n)/n(n + 1), b2 = b1 /m, s1 = Dr (1 − γ K )/(Dr + K n), s2 = s1 /Dr , t1 = K (n + Dr γ )/(Dr + K n), t2 = t1 /K . Here, m, n, Dr , γ , K are the viscosity ratio, depth ratio, diffusivity ratio, ratio of concentration at walls, and concentration jump at the interface. For more details, refer to Picardo et al. [1].
2.2 Linearized Eigenvalue Problem for Stability Analysis The base state superposed with infinitesimal perturbations is substituted in to the non-dimensional set of the governing equations and boundary conditions. These perturbations added to the base state parameters, velocity, pressure, concentration, and deforming interface, are further expanded in the normal mode form (eiα(x−ωt) ) for the streamwise wavenumber (α) and complex wave speed (ω). The two velocity components are eliminated in favor of stream function (φ). These linearized set of perturbation equations read 2 i αRe u j − ω D2 − α 2 − 2b j φ j = m j D2 − α 2 φ j ,
(3)
iαPe u j − ω c j − s j φ j = Dr, j D2 − α 2 c j ,
(4)
φ1 (1) = Dφ1 (1) = φ2 (−n) = Dφ2 (−n) = 0 = c1 (1) = c2 (−n),
(5–10)
Dφ1 (0) − Dφ2 (0) + ha1 − ha2 = 0,
(11)
φ1 (0) − φ2 (0) = 0,
(12)
c1 (0) + hs1 − K (c2 (0) + hs2 ) = 0,
(13)
Dc1 (0) = Dr Dc2 (0),
(14)
Ma D2 + α 2 φ1 (0) − m D2 + α 2 φ2 (0) − i α (c1 (0) + hs1 ) = 0, Pe
(15)
730
A. Agrawal and P. Deepu
mD3 φ2 (0) − D3 φ1 (0) − 3α 2 (mDφ2 (0) − Dφ1 (0)) −
i α3 h = 0, Ca
φ1 (0) + h = ωh.
(16) (17)
where, D = d/dy, and h is the vertical distance of the deformed interface from the base state flat interface. Re = ρU0 d1 /μ1 , Pe = U0 d1 /D1 , Ma = σ0 βC10 d1 /D1 μ1 , Ca = μU0 /σ0 are the Reynolds number, Peclet number, Marangoni number, and Capillary number. These dimensionless numbers are discussed in detail by Picardo et al. [1]. Here, the system is defined by governing Eqs. (3, 4), where (3) is the OrrSommerfeld equation and (4) represents the transport equations. The remaining Eqs. (5–17) are boundary conditions.
2.3 Numerical Technique Many researchers used the approach used by Schmid and Henningson [17] to explore the linear stability analysis of the coupled eigenvalue problem for infinitesimal disturbances of any wavenumber. The same technique is used here to solve the momentum equation coupled with transportation equation for species. The eigenvalue problem is cast as below: AX = ωB X
(18)
where, X is the eigenvector [φ1 , φ2 , c1 , c2 , h]T for the corresponding eigen value ω. A and B are the complex matrix (n × n) containing governing equations of the system along with the boundary conditions. Further, this problem (18) is solved by Chebyshev spectral collocation method. In this technique, each Chebyshev polynomial in the column matrix X is truncated to represent each perturbation amplitude. These Chebyshev polynomials are expressed as: φ j (y) =
N n=0
a jn Tn (y), c j (y) =
N
b jn Tn (y)
(19)
n=0
where, φ j and c j are Chebyshev coefficients that need to be evaluated. As liquid layer domains are [−1, 0] (for layer 2), [0, 1] (for layer 1) and Chebyshev polynomials work across the domain [−1, 1], we adopt a linear transformation which shifts these domains into the desired range as follows. For layer 2, y = (z − 1)/2, where z ∈ [−1, 1], and for layer 1, y = (z + 1)/2, where z ∈ [−1, 1]. These Chebyshev polynomials (19) are substituted into the eigenvalue problem (18) and eigenvectors for each Gauss–Lobatto points z i = cos(πi /N ) are evaluated, where i = 0, 1, . . . N . Further, this eigenvalue problem equipped with Chebyshev polynomial is solved using MATLAB subroutine eig. On solving, the set of eigenvalues
Parametric Study on Marangoni Instability in Two-Layer Creeping Flow
731
Fig. 1 a Spectrum showing the eigenvalues for N = 46 nodes; b showing the convergence of the eigenvalue spectrum for different number of nodes, N. The parameters used are Re = 0, Ca = 100, m = 1.5, K = 0.5, n = 1, γ = 0.5, Ma = 13726.52, Dr = 0.5, Pe = 1000
with the corresponding eigenvectors is analyzed to observe whether the system is converging. A set of eigenvalues (ω) is plotted between ωi with ωr and its spectrum of eigenvalues is shown in Fig. 1a. According to Tilton [18], the relative error method can be used to check for convergence. Twenty least stable eigenvalues are used to estimate the relative error of each number of nodes (N ). Figure 1b shows that the relative error (E N = ||ω N +1 − ω N ||2 /||ω N ||2 ) gets stable and shows convergence for N = 40, where ||.||2 is the Euclidean norm. For more details, check Tilton et al. 2008. So, the current study uses the N = 43 to produce results. The least stable eigenvalue with ωi > 0 and its corresponding eigenvector leads the system toward instability.
3 Results In this work, instability under the creeping flow assumption is analyzed. The parametric variations are carried out to analyze the influence of Peclet number and diffusivity ratio on the instability is studied. But before discussing the results, first, it is necessary to get assured that the current numerical result is in good agreement with [1]. Figure 2 shows the dispersion curve (in dots) discussed in Picardo et al. [1] for long-wave instability and the result for the same parameter produced from the current numerical study matches well. The two most unstable modes are named M1 and M2, following Ref. [1].
732
A. Agrawal and P. Deepu
Fig. 2 Dispersion curve showing the validation of the current working result with one of the Picardo et al. [1] results (Fig. 3b). The parameters used are Re = 0, Ca = 1, m = 0.9, k = 1.2, n = 1.3, γ = 0, Ma = 10000, Dr = 0.5, Pe = 1000
3.1 System Stability Under Peclet Number Variation In the study performed by Picardo et al. [1], the Peclet number is limited to [1000, 2500]. In the present study, Peclet number is varied over a wider range and its influence over the onset and growth of instability are examined. Upon varying Pe from 200 to 2500, the different dispersion curves are shown in Fig. 3. Here the instability growth of both the modes seems non-monotonic with increasing the values of Peclet number. When Pe rises from 200 to 500 growth of M1 mode instability increases (cf Fig. 3a-b), and on further increase in Pe value growth to M1 mode shows the stabilization effect. Whereas M1 mode seems to be stable till 500 and found unstable from 1000 (as per the chosen value of Pe). Here, for M2 mode also instability grows with values of Peclet number (cf Fig. 3b-c) and diminishes on further increment in Pe (cf Fig. 3c-d). In order to uncover this non-monotonic variation in instability growth with Pe, neutral curve is needed to have its full picture. Neutral curve, for the same parameters, is shown in Fig. 4a. Here, neutral curve for M1 mode and M2 mode clears the picture for instability growth. From neutral curve, it can be interpreted that on increasing the Peclet number, M2 mode becomes unstable at Pe = 118 and its instability diminishes after Pe = 2806, whereas the M1 mode shows the instability between Pe = 714 to Pe = 3454. This observation is noticed for a Marangoni number of Ma = 13726.52.
3.2 System Stability Under Diffusivity Ratio Variation Here, diffusivity ratio is varied and its influence over the instability growth is examined. Plazl and Plazl 2007 performed the extraction of steroids from the water phase to organic phase. On the basis of their data related to diffusivity, one obtains a diffusivity ratio of Dr = 3.125 if water phase is considered as top layer. This shows
Parametric Study on Marangoni Instability in Two-Layer Creeping Flow
733
Fig. 3 Showing the influence of the instability growth rate (αωi ) due to the Peclet number (Pe) variation. Here Peclet number is varied from 200 to 2500. The parameters used are Re = 0, Ca = 100, m = 1.5, k = 0.5, n = 1, γ = 0.5, Ma = 13726.52, Dr = 0.5, a Pe = 200; b Pe = 500; c Pe = 1000; d Pe = 2500
that diffusivity ratio can have a much wider range as compared to the values tried in [1]. Dispersion curves shown in Fig. 5 represents this instability growth for different values of Dr . For the prescribed parameters, system gets unstable to M2 mode (cf Fig. 5a) for Dr = 0.3 and M1 mode gets unstable before Dr = 0.8. Subsequently, instability growth of M1 and M2 modes decreases with Dr . Up to a threshold value of Dr , the instability grows and further increase in Dr stabilizes the system. This means that system’s instability to M1 and M2 modes exists for some range of Dr value. It is also seen that this range of Dr to the instability shrinks with a decrease in Ma value.
4 Conclusions In the present study, linear stability is carried out for the two-phase stratified flow. Here, Newtonian fluids flow in two layers in between two infinitely long plates. These plates are maintained with the two different but constant concentration of
734
A. Agrawal and P. Deepu
Fig. 4 Neutral curves for M1 and M2 mode, showing the instability lies for the enclosed region. The parameters other than Ma are similar to Fig. 3. a Ma = 13726.52; b Ma = 15000
soluble surfactant. Picardo et al. [1] performed a numerical study on such a system, where detailed parametric variations in terms of Peclet number and diffusivity ratio are not discussed. On varying the Peclet number, non-monotonic instability growth was noticed for both modes. Neutral curves show the system’s instability lies in a range of Peclet number. This region of instability under Peclet number spreads with increasing the Marangoni number. Further, varying the diffusivity ratio, Dr , here instability grows to some limit but on further increase in Dr instability starts diminishing.
Parametric Study on Marangoni Instability in Two-Layer Creeping Flow
735
Fig. 5 Shows the influence of the instability growth rate (αωi ) due to the diffusivity ratio (Dr ) variation. Here Dr is varied from 0.2 to 2.5 to show the possibilities of instability. The parameters used are Re = 0, Ca = 100, m = 1.5, k = 0.5, n = 1, γ = 0.5, Ma = 13726.52, Pe = 2000, a Dr = 0.3; b Dr = 0.8; c Dr = 1.5; d Dr = 2.2
References 1. Picardo JR, Radhakrishna TG, Pushpavanam S (2016) Solutal Marangoni instability in layered two-phase flows. J Fluid Mech 793:280–315 2. Aljbour S, Yamada H, Tagawa T (2010) Sequential reaction-separation in a microchannel reactor for liquid–liquid phase transfer catalysis. Top Catal 53(7):694–699 3. Šinkovec E, Pohar A, Krajnc M (2013) Phase transfer catalyzed esterification: modeling and experimental studies in a microreactor under parallel flow conditions. Microfluid Nanofluid 14(3):489–498 4. Sternling CA, Scriven LE (1959) Interfacial turbulence: hydrodynamic instability and the Marangoni effect. AIChE J 5(4):514–523 5. Javed KH, Thornton JD, Anderson TJ (1989) Surface phenomena and mass transfer rates in liquid-liquid systems: part 2. AIChE J 35(7):1125–1136 6. Tadmouri R, Kovalchuk NM, Pimienta V, Vollhardt D, Micheau JC (2010) Transfer of oxyethylated alcohols through water/heptane interface: transition from non-oscillatory to oscillatory behaviour. Colloids Surf, A 354(1–3):134–142 7. Žnidaršiˇc-Plazl P, Plazl I (2007) Steroid extraction in a microchannel system—mathematical modelling and experiments. Lab Chip 7(7):883–889 8. Yih CS (1967) Instability due to viscosity stratification. J Fluid Mech 27(2):337–352 9. Yiantsios SG, Higgins BG (1988) Linear stability of plane Poiseuille flow of two superposed fluids. Phys Fluids 31(11):3225–3238
736
A. Agrawal and P. Deepu
10. Boomkamp PAM, Boersma BJ, Miesen RHM, Beijnon GV (1997) A Chebyshev collocation method for solving two-phase flow stability problems. J Comput Phys 132(2):191–200 11. Li Y, Meijer JG, Lohse D (2022) Marangoni instabilities of drops of different viscosities in stratified liquids. J Fluid Mech 932 12. Li Y, Diddens C, Prosperetti A, Lohse D (2021) Marangoni instability of a drop in a stably stratified liquid. Phys Rev Lett 126(12):124502 13. Soleimani R, Azaiez J, Zargartalebi M, Gates ID (2022) Analysis of Marangoni effects on the non-isothermal immiscible Rayleigh-Taylor instability. Int J Multiph Flow 156:104231 14. Sun ZF, Fahmy M (2006) Onset of Rayleigh−Bénard−Marangoni convection in gas−liquid mass transfer with two-phase flow: theory. Ind Eng Chem Res 45(9):3293–3302 15. Zaisha MAO, Ping LU, Zhang G, Chao YANG (2008) Numerical simulation of the Marangoni effect with interphase mass transfer between two planar liquid layers. Chin J Chem Eng 16(2):161–170 16. You XY, Zhang LD, Zheng JR (2014) Marangoni instability of immiscible liquid–liquid stratified flow with a planar interface in the presence of interfacial mass transfer. J Taiwan Inst Chem Eng 45(3):772–779 17. Schmid PJ, Henningson DS, Jankowski DF (2002) Stability and transition in shear flows. Appl Math Sci 142. Appl Mech Rev 55(3):B57–B59 18. Tilton N, Cortelezzi L (2008) Linear stability analysis of pressure-driven flows in channels with porous walls. J Fluid Mech 604:411–445. Rakopoulos CD, Giakoumis EG (2006) Second-law analyses applied to internal combustion engine operation. Prog Energy Combust Sci 32(1):2–47
Measurement of Force in Granular Flow Past Cylindrical Models for Various Inclination Angle Aadarsh Kumar, Deepika Chimote, Aqib Khan, Yash Jaiswal, Rakesh Kumar, and Sanjay Kumar
Nomenclature F D f ρ v
Force (N) Diameter of models (mm) Angle of Inclination (°) Density of air (kg/m3 ) Incoming flow velocity (m/s2 )
1 Introduction In recent years, granular flows grabbed a lot of attention due to their easily observing nature in natural processes as well as in many industries. Granular flows past obstacles play an important role in designing the masts of electric power lines [1], and ski lifts [1], deflecting and catching dams against landslides, etc. [2]. Granular flows find huge applications in industries and naturally occurring flows on the planet Earth and beyond. Industries that predominantly use the material in the form of grains include the food and agriculture industry, pharmaceutical industry, mining industry, construction industry, chemical industry and plastic industry. Naturally occurring events such as snow avalanches, landslides and earthquakes involve the transport of granular material in large quantities. Granular flows are not limited to earth, grains and dust abound in space as well, such as planetary rings, granular asteroids, Martian dunes and dispersion of dust on the lunar surface during landing. In many applications complex flow field evolves due to the interaction of flowing grains with solid A. Kumar (B) · D. Chimote · A. Khan · Y. Jaiswal · R. Kumar · S. Kumar Department of Aerospace Engineering, IIT Kanpur, UttarPradesh, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_67
737
738
A. Kumar et al.
structures. Some important examples where granular fluid/solid interaction plays a crucial role is the interaction of landslides with trees, residences and flow deflectors. There are other examples from industries such as mixer blades, closed conveyors, underground tunneling, etc. where the dynamics of granular motion are significantly influenced by the interaction of grains with other structures. Therefore, understanding the interaction of granular flows with solid structures is important from the fundamental aspects as well as for practical utility. There are many studies that have investigated such flows using experiments and numerical techniques. However, due to the complex nature of such flows, these are not well understood. Furthermore, the force characteristics in granular flows past a cylinder are not given due consideration in the previous works. Therefore, the present work is carried out to study the flow characteristics and the force on the cylindrical obstacle placed in a dry granular stream.
2 Objective The objective of the present study is to analyze the force variation and flow behavior past standard geometries. In granular media, the force imparted on the obstacle is a function of model geometry size (D), grain size (d), inclination angle (φ) and mass flux (m). For lab scale experiments, a novel granular flow chute is fabricated to facilitate the force calculation and shock wave visualization for various channel inclinations. Experiments are carried out on cylinders with different diameters. Unlike previous studies reported in the literature, the chute is kept at different inclination angles. The storage hopper is placed on the top of the chute and feeds glass particles into the channel at a specific mass flux. Velocity profiles are estimated using the particle image velocimetry method. The drag force acting on the models is calculated using the strain gauge force measurement system. Studies show that the force acting on the cylinder is independent of the mean flow velocity and that the force increases with cylinder diameter for the vertical channel. So, we have extensively carried out experiments for most possible cases to properly understand the drag force characteristics and shock wave visualization and their correlation.
3 Experimental Setup The schematic of the experimental setup used to perform the experiments is shown in Fig. 1. The setup consists of a rectangular shape acrylic glass sheet of 310 mm in width and 1200 mm in length sliding on the Aluminum metal frame. The metal frame structure is supported by a variable-length stand, allowing it to change its inclination angle (φ). The frame can be inclined to the horizontal at an inclination of 15° to 80°. The reservoir or hopper is fixed to the top of the variable-length stand. An inclinometer with the least count of 1° is used to precisely measure an inclination
Measurement of Force in Granular Flow Past Cylindrical Models …
739
Fig. 1 Schematic diagram of the setup
angle. The bending beam load cell is used to study the drag force measurement. The load cell with the range of 20 mg to 1 kg and an accuracy of ± 20 mg according to manufacturer specification is used for experiments. The fixed end of the load cell is fitted on the L-shaped bar which is attached to the bottom side of the setup (Fig. 2b). The shaft is fitted on the free end of the load cell, and it passes through a small 10 mm size hole made on the glass sheet at a distance of 750 mm from the top of the glass sheet (Fig. 2a). The test model used for the experiments is mounted on the sheet with the help of the shaft. The LED-based light source is used to illuminate the flow field. The granular material used to perform the experiments is solid colorless spherical glass beads made from high-grade soda-lime-silica glass. The nominal size range of glass beads is from 125 ± 25 μm to 925 ± 25 μm. Grain size with a diameter (d) of 200–250 μm is used for the present experiments. The granular material used for experiments has a bulk density (ρ) of 1600 kg, and a solid fraction (νmax) of 0.64, and five different diameter cylindrical test models are used. The models are manufactured from Aluminum by a CNC lathe machine. All the models are 50 mm in height (H) and diameter (D) ranging from 20 to 60 mm. The models have a 10 mm inner threading slot at the center to fit it in the load cell.
3.1 Experiment for Force Measurement on Cylinder The cylindrical models with varying diameters (D) and the same height (H) were mounted on the top of the glass sheet at a location of 750 mm away from the hopper.
740
A. Kumar et al.
Fig. 2 a Top side of the setup, b bottom side of the setup
At the start of each experiment, the load cell reading is set to zero. To perform the experiment, the grains are stored in the hopper or reservoir and released with the help of a gate mechanism. Once the grains are released, it accelerates continuously on the glass sheet due to the gravity effect. The load cell starts to show readings as grains pass through it. As the grains pass through the model, the load cell starts to give more fluctuating value but after achieving steady-state conditions the load cell starts to show consistent value. The value displayed during the steady-state condition is considered for the force measurement. The grains that are moving sideways on the glass surface do not influence the observed load. The load observed in all the experiments is caused only by the material that strikes the model. A residual non-zero load is observed on the test model at the end of every experiment. This residual force results because of material remaining upstream of the model.
4 Results and Discussion 4.1 Drag Force Calculation The force calculation is done with the help of a load cell. The load cell shows the amount of load (W ) acting on the test model in grams. This load is converted to drag force (FD) by using Eq. (1) FD = W ∗ g
(1)
Measurement of Force in Granular Flow Past Cylindrical Models …
741
The drag force measurement is done on 13 different inclination angles (φ) and for three different test model diameters (D). All force measurement experiments are performed on a plexiglass sheet.
4.2 Effect of Inclination Angle on Force From the experiments, it is observed that grains move very slowly for low inclination angles and come to rest after impingement with the test models. After striking with the Stationary, grains present on the models the incoming grains also come to rest. This results in the formation of diffuse shock on the model. The particles affected by the model’s presence stay in the diffuse shock zone and unaffected particles continue to move downstream. The diffuse shock zone is visible till the inclination angle of φ = 16° to 22°, the drag acting on the model between these angles is due to the accumulation of grains on top of the model. As the inclination angle increases the diffuse shock zone move closer to the model due to the strong impingement of grains with a model (Fig. 3). Initially, the drag force is maximum at the starting of the flow, decreases to a minimum value, and then increases continuously with the channel inclination. This observation can be considered as there is a minimum drag force applied on an obstacle in granular flows on a specific inclination. The initial decrement of the drag force is because of the continuous decrement of the accumulation of grains in front of the obstacle which is the cylinder in our case and the increment of the drag force after a minimum value is because of the granular shock formation effect which can be easily seen after the minimum drag force. These phenomena can be explained based on the frictional effect between grains and channel surfaces (Fig. 4). Fig. 3 Variation of F with φ
742
A. Kumar et al.
Fig. 4 High speed image of Shock structure on a 20 mm diameter cylinder
From previous experiments [3] in our laboratory, we know that the angle of repose for 250 ± 50 μm grains (which we have used in our experiments) is 16.1° for wall particle pairs and 26° for particle–particle pairs. So, we can say that from the start of the flow till 26° there is no relative motion between the grains, and the whole mass of grains just slides on the channel as a solid body that’s why the grains just accumulate in front of the cylinder which we say as the diffuse shock resulting in drag force on the cylinder. As we increase the inclination of the channel, this accumulated mass becomes unstable in front of the cylinder which passes through the sides of it resulting in the continuous decrement of the drag on the cylinder till it approximately reaches the angle of repose of particle–particle pairs. Now, from this inclination, the relative motion between grains also starts which makes this granular media flow like a fluid. The interesting part is that now, the flow velocity is greater than the sound velocity in this granular media and we can easily see the shock formation after this. So, it is the direct consequence of the gas dynamics, which also follows in granular media. If we say the inclination for minimum drag as critical inclination, after this critical inclination, we have observed two types of load readings. The first one is at the start of the flow due to the direct impact of the grains on the cylinder and the second one is the steady-state reading after some time. This initial reading due to direct impact is always higher than the steady-state reading and we have focused on the steadystate reading because readings due to impact are abruptly fluctuating which is not gradual. The gradual increase in the drag force after the critical inclination is just because of the increase in the flow velocity because from now onwards the granular flow behaves like a fluid flow. From particle image velocimetry visualization of the shock formation (Fig. 5), we can see that the velocity distribution is approximately the same as shock formation on a cylindrical obstacle in the gaseous medium.
Measurement of Force in Granular Flow Past Cylindrical Models …
743
Fig. 5 PIV of the granular flow field on a 20 mm diameter cylinder
4.3 Effect of Diameter on Drag Force There is a significant increase in the drag force with the increase in diameter of the cylinder at each inclination. From experiments, it is observed that drag force shows the linear dependence on the model diameter (Fig. 6). Fig. 6 Force versus diameter of different size cylinders
744
A. Kumar et al.
5 Conclusions In conclusion, we have got the variation between the Drag coefficient and Froude number in which the Drag coefficient is maximum at the start of the flow, decreases rapidly with increasing the inclination angle, and then approaches a constant value which is much less than the starting Drag coefficient. As we have also seen, there is a unique behavior of the drag force on the obstacles which means there is a minimum optimum drag force applied on the obstacles. So, based on the application point of view, this fundamental result can be applied in areas where usually people face avalanches, landslides, earthquakes, sand dunes, etc. To avoid these disasters an optimum slope should be preferred so that in case of these disasters minimum possible drag occurs on the residences, trees, flow deflectors, etc. Further, we saw that after the critical inclination shock wave is formed which is very sharp and can be seen easily. With the increase in velocity, dragforce increases because of the strong shock formation which shows the direct consequence of the gas dynamics results. So, this granular flow can play a very vital role to understand the gas dynamics results very easily and at less cost. Acknowledgements This research was supported by Science and Engineering Research Board (SERB) through the Grant Numbers CRG/2019/003989 and CRG/2020/000504.
References 1. Hauksson S, Pagliardi M, Barbolini M, Johannesson T (2007) Laboratory measurements of impact forces of supercritical granular flow against mast-like obstacles. Cold Reg Sci Technol 49(1):54–63 2. Cui X, Gray JMNT (2013) Gravity-driven granular free-surface flow around a circular cylinder. J Fluid Mech 720:314–337 3. Khan A, Hankare P, Verma S, Jaiswal Y, Kumar R, Kumar S (2022) Detachment of strong shocks in confined granular flows. J Fluid Mech 935 4. Chehata D, Zenit R, Wassgren CR (2003) Dense granular flow around an immersed cylinder. Phys Fluids 15(6):1622–1631 5. Ding Y, Gravish N, Goldman DI (2011) Drag induced lift in granular media. Phys Rev Lett 106(2):028001 6. Teufelsbauer H, Wang Y, Chiou MC, Wu W (2009) Flow–obstacle interaction in rapid granular avalanches: DEM simulation and comparison with experiment. Granular Matter 11(4):209–220 7. Boudet JF, Kellay H (2010) Drag coefficient for a circular obstacle in a quasi-two-dimensional dilute supersonic granular flow. Phys Rev Lett 105(10):104501
Experimental Interfacial Reconstruction and Mass Transfer Modelling of a Slug Bubble During Co-current Flow in a Millimetric Tube Lokesh Rohilla, Ravi Prakash, Raj Kumar Verma, and Arup Kumar Das
1 Introduction Process intensification in the monolith reactors with gas liquid interactions is of paramount importance in the chemical industries. The applications of gas liquid interactions range from applications ranging from anaerobic reactions in bioreactors, hydrogenation and photochemical reactions to carbonated drinks [1]. Gas–liquid mass transfer process is generally performed in stirred reactors, trickle bed reactors and bubble column reactors. However, monolith reactors provide a distinct advantage of low pressure drop, modular construction and high performance. Monolith reactors generally consist of a metal or ceramic block with multiple millimetric/micrometric passage channels in which two phase flow is established. The walls of these reactors could have specific metal coatings to catalyze the reactions like hydrogenation processes with Nickel or Palladium coating. The performance of these reactors is dependent on the phase distribution, pressure drop, presence of catalyst, fluid thermophysical properties, order of reaction and channel geometry. The geometry of the channel, and superficial velocities of gas, liquid, and flow distribution device dictate the gas holdup. These monolith reactors are generally operated in Slug flow/Taylor flow regime due to its inherent advantages like, known shape of the bubbles, well defined pressure drop, thin film around the gas bubble and liquid recirculation region. L. Rohilla (B) Process Engineering and Instrumentation Department, CSIR-IMMT, Bhubaneshwar 751013, India e-mail: [email protected] R. Prakash Department of Chemical Engineering, IIT Roorkee, Roorkee 247667, India R. K. Verma Department of Chemical Engineering, Chaitanya Bharathi Institute of Technology, Hyderabad 500075, India A. K. Das Department of Mechanical and Industrial Engineering, IIT Roorkee, Roorkee 247667, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_68
745
746
L. Rohilla et al.
The thin annular film and the recirculation liquid slug region are the major source for process intensification in the gas liquid mass transfer process. The design of mass transfer equipment requires the knowledge of liquid side mass transfer coefficient, K l to predict the gas volumetric flux to the liquid in contact. The liquid side mass transfer coefficient is further utilized to compute the overall mass transfer coefficient by considering gas phase resistance. The conventional quantification methods of the mass transfer include various off-line experimental techniques like spectroscopy, titration, etc. However, the online measurement is a necessity to design the closed loop control system which tunes the process parameter like superficial velocities and conduit length. Various online techniques include the amperometric methods which require careful calibration before field deployment. The optical techniques are based upon fluorescence quenching principle. Both amperometric and fluorescence quenching techniques require sophisticated expensive equipment with frequent maintenance. Researchers have also demonstrated Planar Laser Induced Fluorescence with dye Inhibition (PLIF-I) technique at lab scale to identify the mechanism of mass flux from a bubble. In this paper, we have utilized colorimetric technique based upon the oxidation reaction of an oxygen sensitive dye. Such techniques could be a prelude to the development of an online control system which optimizes the mass transfer in the liquid–gas processes.
2 Literature Review and Objective Dietrich et al. [1] utilized the colorimetric technique to investigate the effect of the superficial velocities and bubble length on liquid side mass transfer coefficient in a polymethyl methacrylate (PMMA) square channel. Butler et al. [2] utilized Planar Laser Induced Fluorescence with dye Inhibition (PLIF-I) technique to provide a correlation for evaluating overall mass transfer coefficient, k L for different slug lengths. PLIF-I technique requires the presence of the laser sheet to highlight the dissolved oxygen region. Kováts et al. [3] investigated the effect of the Deans vortices and helical coil geometry on the overall mass transfer coefficient by using resazurin dye. They deduced that the liquid side mass transfer coefficient is approximately six times higher in the helical coil geometry as compared to the straight tubes. This technique does not require the presence of any external laser source to highlight the dissolved oxygen region. Colorimetric technique is a lucrative method to create a closed loop control system for optimizing performance in gas–liquid-related mass transfer interactions. Moreover, this system could be deployed in a resource limited setting also. In this study, we have deployed a similar colorimetric technique to experimentally investigate the mass transfer process by using an Oxygen sensitive dye and a highspeed camera in co-current flow setting.
Experimental Interfacial Reconstruction and Mass Transfer Modelling …
747
3 Materials and Methods 3.1 Co-current Experimental Apparatus The experimental apparatus consists of a reactor with glass tube (ID = 4 mm) treated with piranha solution to make the glass hydrophilic. The schematic of the experimental apparatus has been shown in Fig. 1. The co-current flow is established in the reactor tube with an alkaline solution with an Oxygen sensitive resazurin dye (molecular weight = 229.19 gm/mole) and Oxygen slug bubbles. The pure medical grade Oxygen is injected into the glass tube with the help of the peristaltic pump (make: Precision Pump Co Ltd, Lab KD1 model). The dye solution passes through a filter before entering into the reactor tube. A view box has been built around the round glass tube to eliminate the optical distortion issues during high-speed photography. The Oxygen sensitive dye reacts with the Oxygen present in the injected bubble. The dissolved oxygen in dye solution is highlighted with the shedding of pink colour from the bubble. To prepare the solution, Sodium hydroxide and glucose have been diluted with deionized water to achieve the concentration of 20 gL−1 each (Table 1). The concentration of the resazurin dye has been fixed to 0.1 gL−1 . These optimum concentrations give better intensity of colour and fast reaction kinetics. O2 out
D
1 9
2 3
LS LIS
4
10
8
5
6
7
Fig. 1 Schematic diagram of the co-current experimental apparatus, (1) reactor tube, ID = 4 mm, (2) view box, (3) holding stand, (4) oxygen cylinder, (5) peristaltic pump, (6) resazurin dye solution, (7) oxygenated solution, (8) mass flow metre, (9) high-speed camera and (10) computer. Inset figure shows the slug bubble train (note: image not to scale)
748
L. Rohilla et al.
Table 1 Thermo-physical properties of the test fluids at 25 ± 0.1 ◦ C
Thermo-physical properties
Value
Density, ρ (kg.m−3 )
1004.5
Surface tension, σ (N.m−1 )
0.075
Viscosity, μ (Pa.s)
1.118
Diffusivity of oxygen, DL (m−2 .s)
3 × 108
3.2 Principle of Colorimetric Technique The image processing technique is a powerful technique that can detect any minute change in the grey scale value generated due to the dye colour (refer Sect. 3.3). Therefore, the reaction should be such that the colour is induced in the colourless solution during oxygen slug bubble translation. Resazurin dye is one such chemical which fulfils this pre-requisite condition with fasted reaction kinetics. The reaction kinetics is catalysed in the presence of glucose and Sodium hydroxide. The pink colour from the bubble interface results from the reversible oxidation reaction between resorufin (pink colour) and dihydroresorufin (colourless). The glucose present in the solution reacts with OH− ions from NaOH to form the D-Gluconic acid by the following oxidation reaction (Eq. 1). HOCH2 (CHOH)4 CHO +3OH− → HOCH2 (CHOH)4 CO2 +2H2 O + 2e− )( ) ( )( ) ( (Dextrose)
(1)
(Gluconic acid)
The generation of the pink colour from the oxygen bubble results from the reversible reaction between resorufin (pink) and dihydroresorufin (colourless): O + 2.Dihydroresorufin → 2.Resorufin + 2.H2 O )( ) (2
(2)
Resorufin + Dextrose → Dihydroresorufin + Gluconic acid )( ) (
(3)
(Fast reaction)
(Slow reaction)
The conversion of dihydroresorufin (colourless) to the resorufin (pink) is faster as compared to the convective time scale of the bubble (Eqs. (2) and (3)). This feature of the resazurin allows us to comfortably visualize the mass transfer around the bubble. The equivalent concentration of the Oxygen (mg/L) could be calculated back with the help of the stoichiometry from the above reactions Eq. (4): n O2 ,transfered = n O2 ,reacted = n resazurin /2
(4)
Experimental Interfacial Reconstruction and Mass Transfer Modelling …
749
From stoichiometry (Eq. 2), one mole of dihydroresorufin reacts with two moles of oxygen. Similarly, one mole of resazurin reacts to produce one mole of dihydroresorufin. The concentration of the dissolved oxygen around the bubble could be back calculated. The whole reaction could be summarized in a nutshell as follows:
3.3 Image Processing and Concentration Contours High-speed photography (Model: CHRONOS 2.1, make: Canada) is utilized to capture the flow field around the slug bubble train (Fig. 2a) at 1000 frames per second. The image analysis has been done with the in-house written image analysis codes in MATLAB. The slug bubble train image is subtracted from the background image (Refer Fig. 2a, b). The grey scale image is necessary for mask reconstruction in the concentration detection step. The grey scale image is converted to the black and white image with a threshold. Thresholding is selected to detect the interface of the bubble only. The holes in the image have been removed by processing (Fig. 2e). Thresholding partitions the image into the background with black colour and bubble with white colour (Fig. 2f). The detected interface in the slug bubble train is shown in Fig. 2g. The extracted interface coordinates from the image analysis have been further utilized to generate three-dimensional reconstruction of the bubble (Fig. 3). Figure 4 shows the grey scale image with subtracted background. The grey scale value for the image ranges from 0 to 255. The grey scale bubble has been masked with white colour to remove it from the calculations. The calibration curve (Fig. 5) mapping the grey scale value difference and equivalent oxygen concentration (Eq. 4) is utilized to deduce the concentration plots (Fig. 4). The Oxygen dissolution region could be further measured by performing discrete integration in MATLAB. The measurement of the dissolved concentration around the bubble requires a priori knowledge of the grey scale value at different resazurin concentration. Different resazurin concentration results in different sample colour due to the differential dissolution of the oxygen. The samples ranging from 0 to 0.015 gm/L of resazurin have been prepared under the standard lab conditions with deionized water and were allowed to uptake oxygen from the ambient for 48 h. These samples were filled in the experimental apparatus and the grey scale value of the glass tube region has been taken. It must be noted that the calibration thus performed is only valid for the specific geometry, light intensity and camera settings. A new calibration curve may be deduced with the change in any of the above-mentioned parameters. The difference in the grey scale intensity ΔGV is the absolute difference between the grey value of the deionized water sample and resazurin dye dissolved sample. Figure 5 reports the change in the grey scale intensity, ΔGV and oxygen concentration (mg/L). The inset in Fig. 5 shows the resazurin concentration and the graph shows the corresponding points with the Oxygen concentration. Figure 4 summarizes the implementation of the calibration curve with image analysis. The reconstruction of the tube domain is a crucial step for the computation of the overall mass transfer coefficient. The axisymmetric nature of the bubble makes
750
L. Rohilla et al.
Resazurin O
O
O
OH
GLUCOSE
N
Dihydroresorufin
Resorufin O
O
N
N
O2
(b)
(c)
(d)
H
(COLORLESS)
(PINK)
(BLUE)
OH
GLUCOSE
O
(a)
O
OH
(e)
(f)
(g)
2 mm
Fig. 2 Interface detection technique for bubbles, a image background, b slug bubble train, c grey scale image, d background subtracted grey scale image, e black and white image with threshold, f holes removed image, g detected image interface superimposed over the monochromatic image
the domain reconstruction facile. Figure 6 shows the reconstructed tube domain by axisymmetric rotation around the tube axis in solid colour. The concentration computation is performed on the unit cell which includes the bubble length, L B and liquid slug length, L s . Figure 7 shows the bubble concentration profile at different gas superficial velocities.
4 Results and Discussion 4.1 Superficial Velocity of Gas and Unit Cell Length The superficial velocity of the liquid dye has been kept constant at Ul = 0.0196 m/s. The gas superficial velocity Ug has been varied from 0.21 to 0.31 m/s. Figure 7 shows the concentration profile at different gas superficial velocities. The concentration profile shows the role of the Taylor recirculation vortices in the mass transfer
Experimental Interfacial Reconstruction and Mass Transfer Modelling …
751
700 600 500 400 300 200 100 0 0
Monochromatic Image
100
200
Bubble Coordinates
Axisymmetric Rotation
Interfacial Reconstruction
Fig. 3 Three-dimensional interfacial reconstruction of the slug bubble from the monochromatic
Grey scale image
Concentration plot
Fig. 4 Mass concentration detection algorithm. Grey scale image bubble has been masked to eliminate it from the calculation
L. Rohilla et al.
O2 concentration (mg/L)
752
0.006
B
A
C
C
0.005 0.004
0 gm/L Resazurin
0.003
0.06 gm/L Resazurin
0.15 gm/L Resazurin
B
0.002 0.001
A
0 0
4
8
12
16
20
24
28
ΔGV Fig. 5 Calibration curve between the grey value difference, ΔGV and Oxygen concentration (mg/ L) Fig. 6 Three-dimensional reconstruction of the tube domain. The schematic shows the slug bubble length and liquid slug length LB
LS
Bubble
process. The accumulation of the dissolved Oxygen has been observed near the cap of the bubble. The majority of the dissolved Oxygen is accumulated in the liquid slug due to high convective flux. The periodic injection of the gas into the tube results in the formation of the slug bubble train. The slug bubble train is analysed in the form of the unit cell. The unit cell of the slug bubble train consists of bubble length, L b and liquid slug length, L s . Figure 8 shows the effect of gas superficial velocity on the unit cell. The power law variation in the liquid unit cell variation in the gas superficial velocity has been observed.
Experimental Interfacial Reconstruction and Mass Transfer Modelling …
(a)
(b)
(c)
753
(d)
(e)
Fig. 7 Concentration profile around the bubble. Experiments have been performed at a constant liquid superficial velocity, Ul = 0.019 m/s. Gas superficial velocities are, a Ug = 0.027 m/s, b Ug = 0.024 m/s, c Ug = 0.022 m/s, d Ug = 0.021 m/s and e Ug = 0.018 m/s 0.017
Fig. 8 Length of the unit cell variation with the gas superficial velocity
0.016
LUC
LUC (m)
0.015 0.014 Ls
0.013
LB
0.012 0.011 0.01 0.02
0.022 0.024 0.026 0.028
0.03
0.032 0.034
Ug (m/s)
4.2 Estimation of Liquid Side Mass Transfer Coefficient The equivalent concentration field could be quantified from the image analysis presented in the above Sect. 4.2. The liquid side mass transfer coefficient, k L could
754
L. Rohilla et al.
be deduced from the equivalent ( ) concentration of the Oxygen around the bubble. The volumetric mass flux, φ X ' from the bubble along the channel length could be expressed as Eq. (5). ) ( ) ( φ X ' = Ug × ∂Cmes /∂ X '
(5)
( ) The volumetric flux from the bubble φ X ' is dependent upon the terminal velocity of the bubble, V , measured Oxygen concentration, Cmes around the bubble and the length scale, X ' which is generally( the) distance in the channel from the bubble origin. The mass flux per unit volume φ X ' from the bubble interface could be expressed by Eq. (6). ( ) φ X ' = kL × a × C ∗
(6)
The above expression assumes that the concentration of dissolved Oxygen in the Resazurin solution is zero due to the corresponding chemical reaction which consumes Oxygen. The maximum saturation concertation, C ∗ of Oxygen in the Resazurin solution has been reported by Dietrich et al. [1] as 8.5 mg/L. The liquid side mass transfer coefficient k L and interfacial area of the bubble a are of paramount importance for enhancing the mass flux from the bubble. Equating Eqs. (5) and (6), we get Eq. (7): (
) ∂Cmes /∂ X ' = (k L × a × C ∗ )/Ug
(7)
Cmes = (k L × a × X ' × C ∗ )/Ug
(8)
Performing an integration over the axial length scale, X ' , we get Eq. (8) for the measured Oxygen concentration, Cmes . The interfacial area, a of the bubble in contact with the liquid could be deduced from the image analysis (Fig. 6b, inset Fig. 7). The measured concentration of the oxygen could be estimated from the image analysis (Fig. 5) by performing the numerical integration (Eq. 9). (˚ Cmes =
) ) ( C(x, y, z)dxdydz / d2 × L U C
(9)
where, C(x, y, z) is the concentration in each pixel of projected area, dxdydz. The integration has been performed over the reconstructed axisymmetric domain (Fig. 6). Equationd (8) and (9) could be combined to derive the liquid side mass transfer coefficient (Eq. 10). ( ) ( ) k L = Cmes × V / X ' × C ∗ × a
(10)
The axial length scale X ' is the length of the unit cell. Figure 9 shows the monotonic increase in the liquid side mass transfer coefficient with the gas superficial velocity.
Experimental Interfacial Reconstruction and Mass Transfer Modelling … 0.0024 0.002
Kl (m/s)
Fig. 9 Variation of the liquid side mass transfer coefficient with the gas superficial velocity
755
0.0016 0.0012 0.0008 0.0004 0 0.02
0.022 0.024 0.026 0.028
0.03
0.032 0.034
Ug (m/s)
4.3 Radial Oxygen Concentration Profile in Liquid Slug Equation (10) could be utilized to estimate the order of magnitude of the liquid side mass transfer coefficient. The measured oxygen concentration Cmes results from axisymmetric domain analysis presented in Sect. 3.3. Figure 10 shows the oxygen concentration in the liquid slug at different location for Ug = 0.01 m/s. The peak has been noticed at the tube centre ∼ 0.01 mg/L at the bottom of the leading bubble and the nose of the trailing bubble. The middle section, B between the two bubbles reports the concentration stem with concentration ∼ 0.004 mg/L. A peak in the concentration profile has also been observed which highlights the presence of the recirculating Taylor vortices. These Taylor vortices between the bubbles create a non-uniform distribution of the dissolved Oxygen concentration in the liquid slug. Figure 11 shows the concentration profile for Ug = 0.021 m/s. The peak dissolved oxygen concentration ∼ 0.020 mg/L. More oxygen concentration has been found at the nose of the slug bubble as compared to its tail. Further increment in the superficial velocity of the gas results in the peak dissolved Oxygen concentration ∼ 0.022 mg/ L at Ug = 0.031 m/s (Fig. 12). All the concentration profiles have a peak near the tube wall at 0.018 m in all the cases which represents the non-uniform dissolution of Oxygen into the liquid slug. 0.014
Oxygen concnetration (mg/L)
Fig. 10 Radial Oxygen concentration in the liquid slug at different sections for Ug = 0.018 m/s
A
0.012 0.01 0.008 0.006
B
C
A B C
0.004 0.002 0 -0.002
-0.001
0
0.001
Tube radial dimension, r (mm)
0.002
756 0.025
Oxygee concnetration (mg/L)
Fig. 11 Radial Oxygen concentration in the liquid slug at different sections for Ug = 0.021 m/s
L. Rohilla et al.
0.02
A
A
B
B
0.015
C
C
0.01 0.005 0 -0.002
-0.001
0
0.001
0.002
Tube radial position, r (mm)
0.025
Oxygen concnetration (mg/L)
Fig. 12 Radial Oxygen concentration in the liquid slug at different sections for Ug = 0.031 m/s
0.02
A
A
B B
0.015
C
C
0.01 0.005 0 -0.002
-0.001
0
0.001
0.002
Tube radial dimension, r (mm)
5 Conclusions A colorimetric technique has been utilized to identify the mass transfer region in the slug bubble train at a fixed liquid superficial velocity. An oxidation of Resazurin dye compound has been used to generate colour from the bubble in the dissolved Oxygen region. An in-house code has been written for interface detection, bubble masking, axisymmetric interfacial reconstruction and Oxygen calibration. The liquid side mass transfer coefficient increases with the gas superficial velocity. The radial concentration plots highlighted the non-uniform dissolution of the Oxygen in the liquid slug. Taylor vortices below the slug bubble are the major source of the mass transfer enhancement with the gas superficial velocity. Acknowledgements The authors acknowledge the research grant OLP-121 from Council of Scientific and Industrial Research-IMMT, India.
Experimental Interfacial Reconstruction and Mass Transfer Modelling …
757
Nomenclature d Ul Ug DL
Tube diameter (m) Liquid superficial velocity (m s− 1 ) Gas superficial velocity (m s− 1 ) Diffusivity of Oxygen (m2 s− 1 )
References 1. Dietrich N, Loubiere K, Jimenez M, Hebrard G, Gourdon C (2013) A new direct technique for visualizing and measuring gas–liquid mass transfer around bubbles moving in a straight millimetric square channel. Chem Eng Sci 100:172–182 2. Butler C, Cid E, Billet AM (2016) Modelling of mass transfer in Taylor flow: investigation with the PLIF-I technique. Chem Eng Res Des 115:292–302 3. Kováts P, Pohl D, Thévenin D, Zähringer K (2018) Optical determination of oxygen mass transfer in a helically-coiled pipe compared to a straight horizontal tube. Chem Eng Sci 190:273–285
Novel and Efficient Superhydrophilic Surface for Improved Critical Heat Flux in Heat Pipe Applications Pradyumna Kodancha, Siddhartha Tripathi, and Vadiraj Hemadri
1 Introduction The increase in the functionality, performance, and miniaturization of the electronic components necessitates the need for improved, high-performance cooling systems. In this regard, multiphase cooling techniques are an attractive option due to the high heat transfer coefficients associated with phase change systems. Multiphase cooling systems predominantly involve boiling and condensation mechanism. Boiling is a complex phenomenon where energy is transported through latent heat associated with the phase change process. It is a combination of convection and multiphase flow of liquid–vapor phases. The boiling process can be divided into three major stages: nucleate, transition, and film boiling. One of the limiting factors which dictate the working of heat pipe is the critical heat flux (CHF). It occurs at the end of the nucleation phase of boiling; thereafter heat removal rate decreases as the boiling transitions to the film boiling phase. Copper is generally the material of choice in heat pipes due to its high thermal conductivity and cheap availability. Therefore, any technique that improves the CHF of copper surface has a major impact on the performance limit of heat pipes. In this context, many studies [1–6] focus on altering the surface morphology of copper surface by producing CuO microstructure by sol–gel immersion technique. The CuO microstructure, which is superhydrophilic in nature, helps in superior spreading and thus increases the performance of heat pipes. In addition to the CuO microstructure, the introduction of grooves is shown to be helpful in heat transfer, especially in horizontal heat pipes [3–5]. A few studies also explore the effect of surface wettability gradients on the performance of heat pipes [4, 7]. Leidenfrost point is another critical P. Kodancha (B) · S. Tripathi · V. Hemadri Department of Mechanical Engineering, BITS Pilani K K Birla Goa Campus, Zuarinagar, Sancoale, Goa 403726, India V. Hemadri 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_69
759
760
P. Kodancha et al.
parameter in the design of heat pipes. Studies have shown that superhydrophilic surfaces also delay the Leidenfrost point along with the CHF [8, 9]. From the literature, it can be seen that the change in surface morphology gives superior results than the bare copper surface in heat pipe applications. Sol–gel coating method is a promising technique for industries to produce superhydrophilic surfaces, as it results in uniform and cost-effective coating while being easily scalable for mass production. The present work proposes a novel combination of alkali solution to generate superhydrophilic surface with improved wettability and higher CHF.
2 Materials and Methods 2.1 Contact Angle Determination The wettability of a surface can be determined by measuring the static contact angle. The sessile drop method is used to measure the contact angle of bare copper surfaces. In this method, a single drop of ~2 µL is placed onto the surface by a micropipette. The photograph is captured by a microscopic camera, and the image is processed using the ImageJ software. The arrangement for measuring the contact angle is shown in Fig. 1a. The static contact angles are close to zero for SHP surfaces and hence it is difficult to compare the wettability of different SHP surfaces using this technique. Therefore, we employ a method based on the spreading area of water droplet, to determine the effective contact angle (θ eff ) to compare different SHP surfaces. Infrared camera (FLIR) is used to precisely calculate the spreading area (Fig. 1b). Figure 2a shows the schematic of the behavior of droplets on a solid surface. For bare copper surface and for hydrophilic surface (i.e. for θ > 10°), the traditional
Fig. 1 a Set up for measuring static contact angle (θ). b Set up for determining effective contact angle (θ eff ) using the spreading area
Novel and Efficient Superhydrophilic Surface for Improved Critical …
761
way of determining the static contact angle is by using a goniometer. For superhydrophilic surfaces (i.e. for θ ~ 0°), the contact angles are too small to be measured by goniometer. Hence, to compare different superhydrophilic surfaces, it is customary to define an effective contact angle (θ eff ) based on the area of liquid droplet spreading on the surface (Asp ) [9]. As shown in Fig. 2b, the spreading area (Asp ) is determined by the IR camera which can be used to determine θ eff by employing Eqs. 1 and 2. With the help of known droplet volume (V l ), and solid–liquid contact area (i.e., spreading area, Asp ), the radius of curvature (R) and the effective contact angle (θ eff ) for SHP surfaces are calculated using [9] Vl =
π R3 (2 − 3 cos θ + cos3 θ ) 3
(1)
and Equations 1 and 2 are simultaneous equations. So they should be in consecutive lines. Asp = π R 2 (1 − cos2 θ )
(2)
Fig. 2 a Idealization of the spherical cap of a liquid droplet on a solid surface used to calculate effective contact angle. b (i) static contact angle (θ) (ii) effective contact angle (θ eff )
762
P. Kodancha et al.
Table 1 Alkali solutions used in the present study for chemical oxidation processes Name
KOH
NaOH
(NH4 )2 S2 O8
K2 S2 O8
References
CuO-1
2.5 M
–
–
0.065 M
[2, 6, 11, 12]
CuO-2
–
2.5 M
0.065 M
–
[3–5, 7, 13]
CuO-3
2.5 M
2.5 M
0.065 M
0.065 M
Present work
2.2 Surface Fabrication A copper specimen of 20 × 20 × 1.5 mm is used to as the substrate to investigate the superhydrophilic CuO microstructure formation. In the sol–gel process, the chemical mixture is prepared using different concentrations, as shown in Table 1, and the mixture is stirred in a magnetic stirrer for 10 min. To remove the impurities and oxide layers, the specimen is cleaned ultrasonically in acetone, ethanol, HCl, and H2 SO4, respectively. Further samples are washed with D.I. water for 5 min. The cleaned samples are dipped in hot alkaline solution at 70 °C. The specimen is removed and washed with D.I. water. Table 1 presents information on the combination of chemicals used in the alkaline solution in processes CuO-1, CuO-2, and CuO-3. Of these combinations, CuO-1 and CuO-2 have been extensively used in the literature. CuO-3 is the new combination that is tested and compared in the present work. The obtained CuO samples are dehydrated for 8 hours. at 200 °C on the hotplate. The scanning electron microscopy (SEM) (Quanta 250FEG) and X-ray diffraction (XRD) (BRUKER D8 ADVANCE) are used to study surface morphology and chemical composition, respectively. In-house designed goniometer [10] is used for measuring the static contact angle.
3 Results and Discussion 3.1 Structural Growth of Microstructure In this section, the formation of the microstructure is discussed. The superhydrophilicity of the surface is the outcome of the multi-layered microstructure and the progression of growth of this microstructure provides insights on its effect on the superhydrophilicity. The coated surface exhibits microstructure growth of two major different sizes. To distinguish the different structures, the larger structures (observed at 5 µm zoom depth) are named structure-1 (S1) and the smaller structural forms (observed at 2 µm zoom depth) are termed structure-2 (S2). Figure 3a shows the ready-to-coat copper sample. Figure 3b shows the S1 and S2 structures, and a zoomed view of S1 is shown in Fig. 3c. Figure 3d–f correspond to the growth of microstructure for CuO-1. It is seen that the initial bud-like S1 structures (shown in Fig. 3d) gradually deteriorate to transform into hair-like S2 structure
Novel and Efficient Superhydrophilic Surface for Improved Critical …
763
observed in Fig. 3f. This process is continuous, and as the reaction proceeds, new S1 and S2 develop over the previous layers. This results in the formation of porous copper oxide of microlayer thickness on the copper sample. The superhydrophilic surfaces formed by S1 and S2 are helpful in aiding the spreadability of the liquid, which prominently depends on the porosity formed by the structures and the adhesion chemistry. The balanced chemical reaction for CuO-1 and CuO-2 are discussed in detail in previous works [7, 12] respectively. The combination of these multiscale structures increases the transition boiling range and thus increases the Leidenfrost temperature [8, 14]. A similar transformation is also seen in the case of CuO-2 (Fig. 3g–i). In CuO-2, the critical difference is that S1 will not reduce to pillars/ribbon to form micro hairlike/ribbon structure; instead, S1 structure is reduced to a thin sheet-like structure (Fig. 3g). The subsequent reduction step is the same as the previous one, in which the S1 structure will bloom and spread over the area to form a thin micro petallike structure (S2) shown in Fig. 3i). The copper from the surface is displaced by the persulphate, which spontaneously reacts with the hydroxide and leads to the formation of oxide i.e., the structures mentioned above. As the reaction is chaotic with construction and reduction phenomenon, the final surface is rough and nonuniform, and there is no patterned or layer-by-layer construction of structures (S1 and S2). CuO-3, being a combination of the chemicals used in CuO-1 and CuO-2, a chaotic and random growth of microstructures is observed (Fig. 3j–l). In CuO-3,
Fig. 3 SEM images of a pristine copper at 20 µm, b final coated surface at 20 µm showing S1 and S2, c magnified view of S1 at 5 µm. The subsequent images show the gradual destruction of S1 and formation of S2 for the three cases, d Cuo-1-S1 at 5 µm, e CuO-1-S1 at 5 µm, f CuO-1-S2 at 2 µm, g CuO-2-S1 at 5 µm, h CuO-2-S1 at 5 µm, i CuO-2-S2 at 2 µm, j CuO-3-S1 at 5 µm, k CuO-3-S1 at 5 µm, l CuO-3-S2 at 2 µm
764
P. Kodancha et al.
due to presence of four chemicals, the exact sequence of chemical reaction resulting in the chaotic final structure is hard to identify.
3.2 Surface Morphology The previous section discussed the formation of CuO microstructures in all three cases. It is difficult to comment on the porosity enhancement of the surface by visual observations. Hence spreading parameter can be used as an indicator of the porosity of the microstructure. In order to establish a relationship between the surface wettability and microstructure, the structural dimensions are presented alongside the droplet contact angle (θ ) and the effective contact angle (θ eff ) on the respective surfaces (Fig. 4). By comparing, Fig. 4a with Fig. 4b, d, it is evident that the formation of microstructures is responsible for rendering the copper surface superhydrophilic. Furthermore, it is seen that CuO-1 and CuO-2 exhibit similar values of spreadability as indicated by the values of their respective effective contact angles.
Fig. 4 SEM images and respective contact angles of a pristine copper at 20 µm, b CuO-1 Coating at 2 µm, c CuO-2 Coating at 2 µm, d CuO-3 Coating at 2 µm
Novel and Efficient Superhydrophilic Surface for Improved Critical …
765
Fig. 5 Non-dimensional spreading area (A* ) and effective contact angle for different surfaces. The non-dimensional spreading area (A* ) is obtained by scaling the spreading area (Asp ) with respect to the area of the specimen
CuO-3 shows a relatively lower value of θ eff (Fig. 4d) which can be attributed to the increase in microstructure density. To further illustrate the effect of the microstructure on the spreadability, the spreading area and effective contact angles for all three cases are shown in Fig. 5. It is seen that the CuO-3 case, proposed in the present work, has the highest non-dimensional spreading area (A*) and the lowest value of θ eff . While A* and θ eff for CuO-1 and CuO-2 are comparable, they are considerably better than the values corresponding to bare copper surface. For the CuO-3 case, A* increases by 99.21% and θ eff decreases by 99.88% compared to the bare copper surface.
3.3 Thermal Performance The main objective of this work is to evaluate the SHP surfaces obtained in the present study for their thermal performance, particularly CHF values associated with the surface. In a two-phase thermal application, the critical heat flux (CHF) is the crucial thermal parameter that defines the performance, especially in heat pipe applications where instant high heat transfer is required. From the onset of nucleate boiling till CHF, the initial heat transfer phase takes place, this phase is associated with high heat transfer coefficient. The operational range of the heat pipe is increased by increasing CHF. The superhydrophilic surface decreases the thermal resistance and increases the spreadability thereby increasing the CHF [3–5, 11, 13, 15]. Generally, pool boiling experiments are conducted to correlate the change in surface wettability with the
766
P. Kodancha et al.
Fig. 6 Effect of contact angle on CHF on a horizontal surface
CHF. In this work, we employ a semi-analytical equation provided by Liu et al. [16]. It considers the effect of adhesion, hydraulic pressure, and gravity along with the contact angle and heating surface area on the CHF. As far as the authors know, it is the only predictive relation available in the literature that relates CHF with contact angle for a wide range of values. The CHF was obtained by using [16] plotted in Fig. 6 for all four cases of the present study. It can be seen that the highest CHF is observed in CuO-3 case. In comparison, CuO-1 and CuO-2 have nearly half of the CHF of CuO-3. The reduction in contact angle in CuO-3, compared to CuO-2 is 87.12%, and the CHF of CuO-3 increased by 63.5%, compared to CuO-2. For comparative purposes, the CHF obtained by other studies [15–17] for different contact angles is also plotted in Fig. 6.
4 Conclusions A novel chemical combination is investigated in the present work to obtain superhydrophilic surface from the bare copper surface. The progressive growth of the microstructure is discussed. The surface formed with the new combination (CuO-3) successfully enhanced the wettability performance of the bare copper surface. The enhancement was characterized by the spreading area improvement of 99.21% and a contact angle decrement of 99.88% compared to bare copper. The semi-analytical relation used to determine the CHF predicted an improvement of ~ 200% for CuO-3 compared to the bare copper surface. Acknowledgements Authors would like to acknowledge the support received by SERB-DST CRG grant funded by the Government of India (CRG/2020/003490). Authors acknowledge the support of center for sophisticated instrumentation facility (CSIF) and mechanical workshop, Bits-Pilani K.K. Birla Goa Campus, India.
Novel and Efficient Superhydrophilic Surface for Improved Critical …
767
References 1. Cecere A et al (2018) Visualization of liquid distribution and dry-out in a single-channel heat pipes with different wettability. Exp Therm Fluid Sci 96:234–242. https://doi.org/10.1016/j. expthermflusci.2018.03.012 2. Hao T, Ma X, Lan Z, Li N, Zhao Y, Ma H (2014) Effects of hydrophilic surface on heat transfer performance and oscillating motion for an oscillating heat pipe. Int J Heat Mass Transf 72:50–65. https://doi.org/10.1016/j.ijheatmasstransfer.2014.01.007 3. Xu P, Li Q (2017) Visualization study on the enhancement of heat transfer for the groove flat-plate heat pipe with nanoflower coated CuO layer. Appl Phys Lett 111(14):2017. https:// doi.org/10.1063/1.4986318 4. Cheng J, Wang G, Zhang Y, Pi P, Xu S (2017) Enhancement of capillary and thermal performance of grooved copper heat pipe by gradient wettability surface. Int J Heat Mass Transf 107:586–591. https://doi.org/10.1016/j.ijheatmasstransfer.2016.10.078 5. Hu Y, Cheng J, Zhang W, Shirakashi R, Wang S (2013) Thermal performance enhancement of grooved heat pipes with inner surface treatment. Int J Heat Mass Transf 67:416–419. https:// doi.org/10.1016/j.ijheatmasstransfer.2013.08.035 6. Hao T, Ma X, Lan Z (2018) Effects of hydrophilic and hydrophobic surfaces on start-up performance of an oscillating heat pipe. J Heat Transfer 140(1). https://doi.org/10.1115/1.403 7341 7. Huang Z et al (2012) Preparation and characterization of gradient wettability surface depending on controlling Cu(OH)2 nanoribbon arrays growth on copper substrate. Appl Surf Sci 259:142– 146. https://doi.org/10.1016/j.apsusc.2012.07.006 8. Talari V, Behar P, Lu Y, Haryadi E, and D. Liu, “Leidenfrost drops on micro/nanostructured surfaces. Frontiers in energy, vol 12, no 1. Higher Education Press, pp 22–42. https://doi.org/ 10.1007/s11708-018-0541-7 9. Padilla J, Carey VP (2014) Water droplet vaporization on superhydrophilic nanostructured surfaces at high and low superheat. Available http://proceedings.asmedigitalcollection.asme. org/pdfaccess.ashx?url=/data/conferences/asmep/83091/ 10. Sow PK, Ashwin Y (2020) A design framework for the fabrication of a low-cost goniometer apparatus for contact angle and surface tension measurements. Meas Sci Technol 31(12). https:// doi.org/10.1088/1361-6501/aba78c 11. Hao T, Ma X, Lan Z, Li N, Zhao Y (2014) Effects of superhydrophobic and superhydrophilic surfaces on heat transfer and oscillating motion of an oscillating heat pipe. J Heat Transfer 136(8). https://doi.org/10.1115/1.4027390 12. Ji Y, Xu C, Ma H, Pan X (2013) An experimental investigation of the heat transfer performance of an oscillating heat pipe with copper oxide (CuO) microstructure layer on the inner surface. J Heat Transfer 135(7). https://doi.org/10.1115/1.4023749 13. Zhang FZ, Winholtz RA, Black WJ, Wilson MR, Taub H, Ma HB (2016) Effect of hydrophilic nanostructured cupric oxide surfaces on the heat transport capability of a flat-plate oscillating heat pipe. J Heat Transfer 138(6). https://doi.org/10.1115/1.4032608 14. Agapov RL et al (2014) Asymmetric wettability of nanostructures directs Leidenfrost droplets. ACS Nano 8(1):860–867. https://doi.org/10.1021/nn405585m 15. Ahn HS, Park G, Kim JM, Kim J, Kim MH (2012) The effect of water absorption on critical heat flux enhancement during pool boiling. Exp Therm Fluid Sci 42:187–195. https://doi.org/ 10.1016/j.expthermflusci.2012.05.005 16. Liu Y, Du Y, Xiong S, Zhou T, Zhao Z (2019) An innovative model for critical heat flux prediction in saturated pool boiling based on bubble behaviors on dry hot spots 17. Ahn HS, Jo HJ, Kang SH, Kim MH (2011) Effect of liquid spreading due to nano/ microstructures on the critical heat flux during pool boiling. Appl Phys Lett 98(7). https:// doi.org/10.1063/1.3555430
Effects of Wettability on the Flow Boiling Heat Transfer Enhancement Akash Priy, Israr Ahmad, Manabendra Pathak, and Mohd. Kaleem Khan
1 Introduction With the rapid development of miniaturized high power density devices, there is a constant surge in demand to dissipate high heat flux from electronic circuitries. Thermal management in data centers, rocket nozzles, and spacecraft avionics [1] has posed a critical challenge to dissipating more than 1000 W/cm2 . Conventional cooling techniques such as single-phase air or water cooling have several limitations such as non-uniform surface temperature in the flow direction and large coolant volume requirement due to their lower heat transfer performance. Two-phase cooling systems such as heat pipes, thermosiphon, and immersion cooling provide better heat transfer performance but are also limited to low heat flux. Another limitation is that it fails to perform satisfactorily in space microgravity conditions where buoyancy-driven bubble departure is not possible. Flow boiling through microchannels can provide a promising solution [2, 3]. Microchannel heat sinks consist of an array of microchannels fabricated on the back of the device. Flow boiling through a microchannel heat sink utilizes a combination of liquid inertia generated by a pumped fluid and high latent heat of vaporization of the boiling process. This provides excellent heat transfer performance, better temperature uniformity, and less coolant requirement than other single-phase flow systems.
A. Priy (B) · I. Ahmad · M. Pathak · Mohd. K. Khan Department of Mechanical Engineering, IIT Patna, Patna 801106, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_70
769
770
A. Priy et al.
2 Literature Review and Objective Bubble dynamics play a critical role in the heat transfer mechanism in flow boiling through microchannels. The bubble inside a microchannel undergoes a constrained movement due to the presence of sidewalls and the top cover plate. Hsu [4] presented the basic framework of bubble nucleation from trapped vapor/ gas inside a pool of liquid. They proposed a criterion for a cavity of entrapped vapor/gas to become an active nucleation site for boiling. However, in flow boiling, several other parameters such as flow inertia, heated sidewalls, their geometry, liquid subcooling, surface wettability, etc. must be taken into account. Liu et al. [5] reported that the seed bubbles injected inside the microchannels trigger faster nucleation during flow boiling. Later, Lee et al. [6] created a single artificial cavity in a microchannel and analyzed the flow boiling behavior involving bubble growth, expansion, and detachment from the cavity under different mass flow rates and wall superheat. Jafri et al. [7] investigated the Chan-Hilliard phase field approach to model the flow boiling phenomenon from a single artificial cavity inside the microchannel. As the size of the channel decreases, the surface tension dominates over the inertia force. Therefore, it becomes crucial to analyze the role of surface wettability on the bubble dynamics and heat transfer performance of flow boiling through microchannels heat sink. However, there is, a paucity of work examining the role of surface wettability inside the microchannel heat sink. This might be because creating microscale surfaces with different wettability is challenging, and maintaining the wettability of such surfaces is crucial. Liu et al.[8] prepared three kinds of surfaces namely, surperhydrophillic by growing nanowire array (contact angle (C.A.) nearly 0°), hydrophilic (C.A. 36°) by plasma etching, and hydrophobic by coating with low surface energy (C.A. 103°). They observed that superhydrophillic surfaces exhibit highest bubble generation rate while hydrophobic surface requires greater wall superheat temperature faster bubble growth was observed after bubble incipience. Choi et al. [9] reported that the HTC of hydrophobic surface is greater than hydrophilic surfaces due to increased nucleation site density. Tan et al. [10] presented the effects of surface wettability in flow boiling heat transfer in a microtube and observed that the hydrophilic surfaces exhibit better heat transfer performance than the hydrophobic in the bubbly and confined flow region. Vontas et al. [11] performed flow boiling numerical simulations within microchannels to observe the effects of surface wettability on single nucleation sites (with single and multiple nucleation events) as well as multiple nucleation sites. Only a few numerical studies demonstrate the effects of surface wettability in a flow boiling system. In this work, we have used the phase field method-based approach to simulate the flow boiling heat transfer inside a microchannel from an artificial cavity. The objective of the present work is to investigate the role of surface wettability in subcooled flow boiling through the microchannel heat sink under low mass flow rates. The bubble dynamics and the heat transfer characteristics are analyzed.
Effects of Wettability on the Flow Boiling Heat Transfer Enhancement
771
Fig. 1 Computational domain and boundary conditions for the present study (not to scale)
3 Model Description 3.1 Computational Domain The computational domain of the present work is shown in Fig. 1. The computational domain is fixed based on the experimental investigation of Lee et al. [6] for flow boiling. The fluid flows through a microchannel of width 100 µm and length 64 mm with an inlet mass flow rate of 7.97 kg/m2 s. The bottom wall is heated and is kept at a constant temperature of Tsat + 2 °C. Water is taken as the working fluid. Water flows inside the microchannel with the inlet subcooling of 25 °C. An artificial cavity is created on the bottom-heated wall at a distance of 21.4 mm from the channel inlet. A vapor film is patched to the cavity, which serves as a nucleation site. The outlet is set to be at atmospheric conditions. The top cover plate is insulated. Initially, the contact angle (C.A.) of the heated and the top cover plate is set to 60° to start the simulation. The numerical simulation is performed in 2-D in order to save computational time and cost. The simulations have been carried out in Comsol Multiphysics software which is a finite element-based solver.
3.2 Governing Equations The governing equations such as continuity, momentum, and energy equations are modified in such a way that it incorporates the phase change phenomena. In this study, we employ phase field modeling, which is an effective method for resolving mesoscale interfacial issues, to determine each physical parameter in the two-phase flow system. It helps in better tracking of the surface tension at the interface as compared to other fixed Eulerian grids tracking methods such as the level-set (LS) method and volume of fluid (VOF) method. The phase field variable φ ranging from + 1 to − 1 represents the two fluid components. Governing equations are given below. Continuity Equation
772
A. Priy et al.
∇.u = mδ ˙
1 1 − ρv ρw
(1)
Momentum Equation ρ
∂u + ρ(u.∇)u = ∇. − p I + μ ∇u + (∇u)T + ρg + Fst ∂t
(2)
Phase field equation V f,v V f,w ∂φ = γ ∇2G + u.∇φ − mδ ˙ + ∂t ρv ρw
(3)
Energy Equation ρC p
∂T + ρC p (u.∇)T = −∇.k∇T + Q s ∂t Q s = −L mδ ˙
(4) (5)
The rate of mass transfer during a phase change is given as: m˙ = Cρv m˙ = Cρw
T − Tsat Tsat T − Tsat Tsat
when T < Tsat
(6)
when T ≥ Tsat
(7)
3.3 Grid Independence Study The entire computational domain is discretized with triangular mesh elements. The grid size around the cavity is kept finer at 0.5 µm and the grid size varies between 4 µm-, 10 µm for the rest of the geometry. The Grid independence study was performed and 12,06,125 elements were chosen for the present numerical simulation with the maximum deviation in bubble radius less than 1% (Table 1). Table1 Grid independence study
Grids
Maximum deviation (%)
2,38,950 to 5,75,553
4.17
5,75,553 to 12,06,125
3.5
12,06,125 to 17,90,358
0.96
Effects of Wettability on the Flow Boiling Heat Transfer Enhancement
773
4 Results and Discussion This section presents the role of bubble dynamics and its effects on flow behavior during subcooled flow boiling inside the microchannel. Further, the role of varying surface wettability on bubble dynamics and the heat transfer performance is analyzed in detail.
4.1 Validation of Numerical Model Figure 2 shows the variation of bubble radius with time obtained from the present numerical simulation. The present numerical results are compared with the experimental results of Lee et al. [6]. The standard deviation of the present work as compared to the experimental results of Lee et al. [6] is only 3.13%. The results show very good agreement between the present numerical simulation and existing data.
Fig. 2 Comparison of bubble growth results of present prediction with the experimental data of Lee et al. [6]
774
A. Priy et al.
4.2 Effect of Bubble Dynamics on Pressure, Temperature, and Velocity Distribution of Flow Figure 3 presents the bubble growth and flow parameters at different instants. The bubble nucleates from the cavity of the bottom-heated wall inside the microchannel. The bubble then continues to gain mass from its interface, causing it to enlarge. Then it gets constrained by the walls of microchannels allowing it to grow in longitudinal directions only. The inertia of liquid continuously pushes the vapor bubble, and the necking regime is formed before departure. The bubble finally departs from the cavity due to continuous shear thinning at the cavity. Some portion of the bubble remains in the cavity, which can serve as subsequent nucleation sites. The process continues and a slug flow regime is formed inside the microchannel. The size of the bubble increases along the flow due to the heated bottom wall of the microchannel. Figure 3a shows the pressure distribution inside the microchannel as the bubble nucleates from the cavity. The pressure inside the bubble is high at the time of
Fig. 3 Bubble dynamics inside the microchannel for the validated case of CA = 60°at various time steps and its effects on a pressure distribution b temperature distribution c velocity distribution
Effects of Wettability on the Flow Boiling Heat Transfer Enhancement
775
nucleation and decreases as the bubble size grows. Then bubble is confined to the channel wall creating an abrupt change in pressure, both upstream and downstream of the bubble. As the bubble resists the inertia of the continuous liquid flow, the pressure upstream of the bubble rises. It is significant to observe that while the pressure of the subsequent nucleating bubbles is higher than the previous one, the pressure of the vapor slug decreases as it expands in a longitudinal direction. Figure 3b shows the temperature distribution inside the microchannel during subcooled flow boiling. Time t = 0 ms assumes that the vapor embryo is created and the liquid in the near vicinity of the embryo is in a superheated state. Then the embryo grows, and there are phase change phenomena around the bubble interface as evident from the temperature distribution at t = 0.7 ms. The vapor slug then covers the microchannel and its temperature is increased as shown at t = 10 ms. Figure 3c shows the velocity distribution inside the microchannel at various time steps. It can be observed that as the vapor bubble grows it accelerates the fluid in the downstream direction due to its evaporative momentum. Because of this, liquid in the downstream direction has more velocity as compared to its upstream direction. As the vapor slug grows in a longitudinal direction, its velocity increases. The preceding bubble has more velocity as compared to its succeeding one. The fluid between two bubbles also accelerates during the flow. The fluid downstream of the first bubble has the highest velocity, which results in more convective heat transfer.
4.3 Effect of Surface Wettability on Bubble Dynamics During Flow Boiling Heat Transfer Figure 4 shows the effect of bubble dynamics at different surface wettability conditions of the heating surface, (1) 30° (2) 60° (3) 90° (4) 120° and (5) 150°. The CA of the cover plate is kept constant at 60° for all cases. The snapshot at t = 1 ms depicts the nucleation characteristics for all the cases. It is observed that the bubble size for case (1), i.e., C.A = 30° is the highest and high rewetting is observed for this case. For cases (2–4), the bubble size at t = 1 ms is reduced with an increase in C.A. The bubble for case (5) has a high affinity toward the heating surface and it tries to spread over the cavity. As time increases, the bubble size also increases and it gets confined by the top cover plate. The bubble confinement is the fastest in case (1) at t = 1.2 ms. While the time of bubble confinement for other cases is shown in Fig. 4b. As the C.A. increases, the heating surface tends to pull the vapor bubble toward itself. Hence, the time for bubble confinement is delayed for higher C.A. heating surfaces i.e., for case (3) is 3.5 ms and case (4) is 6.2 ms. The confinement snapshot for case (4) shows that the vapor bubble tries to be attached to the heating plate. Therefore, contact with the cover plate is lesser as compared with the heating plate. And in case (5) no confinement is observed till 7.5 ms. The vapor bubble spreads over the heating surface.
776
A. Priy et al.
Fig. 4 Effect of bubble dynamics at various times with different surface wettability conditions, (1) 30° (2) 60° (3) 90° (4) 120° (5) 150°
After confinement, the vapor bubble tends to form a neck at the cavity. The incoming liquid keeps on pushing the vapor bubble toward the outlet. This leads to the formation of the neck at the cavity that keeps on thinning with time. Then a stage comes, where the surface tension force on the bubble cannot resist the inertia force of the incoming liquid, and the bubble departure takes place. The bubble departure time of the first bubble is shown in Fig. 4c. For case (1), the departure time is at the earliest at 4 ms. The departure time is delayed as the C.A. is increased. The departure time for case (2) is 4.4 ms, case (3) is 6.4 ms, and case (4) is 7 ms. There is no departure for case (5), the bubble keeps on increasing in size, and the vapor film spreads over the heating surface. This can even lead to early critical heat flux (CHF) in low mass flow rate conditions. Figure 4d shows the snapshot of bubble dynamics at t = 7.5 ms for various cases. The bubble generation frequency is highest for case (1). It can be observed that for case (1), two bubbles have already departed and the third bubble is nucleated at t = 7.5 ms. While the bubble generation rate is lesser for higher C.A, There is no subsequent nucleation for case (5) at t = 7.5 ms, and the bubble generation frequency
Effects of Wettability on the Flow Boiling Heat Transfer Enhancement
777
is the minimum. Since the simulation is run over 10 ms, the departure is not observed until then in case (5). However, the inertia of incoming liquid pushes the bubble, and the necking region starts to form at this stage.
4.4 Effect on Local Heat Flux and HTC Along the Channel Length Figure 5 shows the variation of local heat flux, along the length, and snapshots of bubble inside the microchannel at 7.5 ms for CA = 30°. It also presents the local heat flux dissipated from the bottom wall along the channel length. The local heat flux is calculated by considering Fourier’s law of heat conduction equation given by the thermal conductivity of liquid multiplied by its temperature gradient. The cavity is at 21,400 µm where the nucleation takes place. It experiences a spike in dissipated heat flux at that point. After the distance of 21,500 µm, the local heat flux is reduced to its minimum value. This is due to the low thermal conductivity of the vapor film covering the heated surface. The heat flux value suddenly increases at around 21,800 µm, which is because of the formation of a thin liquid film between the heated bottom surface and the vapor bubble. Jafari et al. [7] report similar observations. The heat flux again decreases from 22,250 µm to 23,250 µm, where the vapor bubble again touches the bottom surface. The heat flux value again spikes due to the thin film regime around 23,300 µm. After 23,300 µm, the heat flux value rises slowly, which is due to the high convective heat transfer induced by the accelerating bubble at the front. The bubble generation rate with C.A. = 30° is high as compared to other cases. So the heat transfer rate also increases in this case.
Fig. 5 Variation of local heat flux and HTC with channel length at CA = 30° and time = 7.5 ms
778
A. Priy et al.
The elongated bubble observed in case (1) will eventually occur in other cases too. This results in a reduction in heat transfer. The average heat transfer is a cumulative effect of all three modes of heat transfer namely two-phase convective heat transfer, thin film evaporation, and, single-phase convective heat transfer. The boiling heat transfer dominates above the cavity site and has the maximum value among the other two modes. The thin film evaporation dominates at the interface position where thin liquid forms between the bubble and the heated surface. The single-phase convective heat transfer dominates the downstream of the first nucleating bubble. Here the liquid accelerates due to the movement and expansion of the vapor plug near it.
4.5 Effect on Local Removed Heat Flux and HTC with Time To study the local heat transfer with time an arbitrary point is taken on the heated surface at 21,500 µm close to the cavity. Figure 6 represents the removed heat flux associated with fluid passing through the point with respect to time. Initially, the removed heat flux maximizes as the bubble nucleates from the cavity. As the bubble grows with time, it crosses over the arbitrary point. This creates a sharp rise in heat flux due to the creation of thin liquid film and decreases in heat flux when the vapor bubble completely covers the arbitrary point. It is noted that the bubble1 for C.A. = 120° is more attached to the heated surface as seen by the longer duration of lower removed heat flux (2–6 ms) as compared to C.A. = 30° (1.8–3.8 ms). The value of removed heat flux at the time of nucleation is higher for C.A = 120°. Also, the bubble frequency is less for higher C.A. angles.
5 Conclusions The present work reports a numerical study on the effects of surface wettability on flow boiling heat transfer in microchannel heat sink at low mass flow condition. The influences on pressure, temperature, and velocity are studied during two-phase flow boiling inside a microchannel. The effects of heater plate wettability on bubble nucleation, confinement, and bubble departure are observed in detail. Results reveal that the bubble generation rate for C.A. = 30° is the highest and the generation rate decreases with an increase in contact angle. Better surface rewetting is observed for the case with C.A. = 30°. For the superhydrophobic case (C.A. = 150°), the bubble departure has not occurred as the superhydrophobic surfaces have the highest affinity toward vapors. The vapor bubble spreads over the heating surface and blocks the cavity restricting subsequent nucleation. The heat transfer performance at the time of nucleation is highest and later thin film evaporation is the dominant mode of heat transfer.
Effects of Wettability on the Flow Boiling Heat Transfer Enhancement
779
Fig. 6 Variation of removed local heat flux from the bottom-heated wall at 21.5 mm with the time, CA = 30° and 120°
Abbreviations Nomenclature Cp R T u Vf K m˙ K L P
Specific heat, (J/kg K) Bubble radius, (m) Temperature, (°C) Velocity (m/s) Volume fraction Thermal conductivity (W/cm-K) Mass flow rate (kg/m2 s) Thermal conductivity (W/cm-K) Latent heat (J/kg) Pressure (Pa)
Greek symbols ρ ϕ q
Density, (kg/m3) Phase field variable Heat flux (W/cm2)
780
A. Priy et al.
Abbreviations CHF CA HTC
Critical heat flux Contact angle Heat transfer coefficient
Subscripts v l sat
Vapor Liquid Saturation
References 1. Mudawar I (2011) Two-phase microchannel heat sinks: theory, applications, and limitations. J Electron Packag Trans ASME 133. https://doi.org/10.1115/1.4005300 2. Karayiannis TG, Mahmoud MM (2017) Flow boiling in microchannels: fundamentals and applications. Appl Therm Eng 115:1372–1397. https://doi.org/10.1016/j.applthermaleng.2016. 08.063 3. Priy A, Raj S, Pathak M, Kaleem Khan M (2022) A hydrophobic porous substrate-based vapor venting technique for mitigating flow boiling instabilities in microchannel heat sink. Appl Therm Eng 216:119138. https://doi.org/10.1016/j.applthermaleng.2022.119138 4. Hsu YY (1962) On the size range of active nucleation cavities on a heating surface. J Heat Transf 84:207–213. https://doi.org/10.1115/1.3684339 5. Liu G, Xu J, Yang Y (2010) Seed bubbles trigger boiling heat transfer in silicon microchannels. Microfluid Nanofluid 8:341–359. https://doi.org/10.1007/s10404-009-0465-y 6. Lee JY, Kim MH, Kaviany M, Son SY (2011) Bubble nucleation in microchannel flow boiling using single artificial cavity. Int J Heat Mass Transf 54:5139–5148. https://doi.org/10.1016/j. ijheatmasstransfer.2011.08.042 7. Jafari R, Okutucu-Özyurt T (2016) Numerical simulation of flow boiling from an artificial cavity in a microchannel. Int J Heat Mass Transf 97:270–278. https://doi.org/10.1016/j.ijheat masstransfer.2016.02.028 8. Liu TY, Li PL, Liu CW, Gau C (2011) Boiling flow characteristics in microchannels with very hydrophobic surface to super-hydrophilic surface. Int J Heat Mass Transf 54:126–134. https:// doi.org/10.1016/j.ijheatmasstransfer.2010.09.060 9. Choi C, Shin JS, Yu DI, Kim MH (2011) Flow boiling behaviors in hydrophilic and hydrophobic microchannels. Exp Therm Fluid Sci 35:816–824. https://doi.org/10.1016/j.expthermflusci. 2010.07.003 10. Tan K, Hu Y, He Y (2021) Effect of wettability on flow boiling heat transfer in a microtube. Case Stud Therm Eng 26:101018. https://doi.org/10.1016/j.csite.2021.101018 11. Vontas K, Andredaki M, Georgoulas A, Miché N, Marengo M (2021) The effect of surface wettability on flow boiling characteristics within microchannels. Int J Heat Mass Transf 172:121133. https://doi.org/10.1016/j.ijheatmasstransfer.2021.121133
Fluid-Structure Interaction
Computational Study to Assess the Usage of Asymmetric Canard as Yaw Control Device for a Generic Fighter Aircraft V. Sundara Pandian, R. J. Pathanjali, B. Praveen Kumar, Muralidhar Madhusudan, and Dharmendra Narayan
Nomenclature RANS MRP Cp LE Cy Cmx Cmz Re
Reynolds Average Navier Stokes Moment reference point (m) Coefficient of pressure Leading edge Coefficient of side force Coefficient of rolling moment Coefficient of yawing moment Reynolds number
1 Introduction The requirements of modern combat fighter aircraft encompass a multitude of flow regimes ranging from low speeds to high supersonic Mach numbers. In the low speed flow regimes, the maneuvering capability of the fighter aircraft employing delta wings can be significantly improved by enhancing the high angle of attack aerodynamic stability and control characteristics. Some of the well-known issues associated with delta wings at high angles of attack are the shedding of asymmetric vortices from slender fore-body, vortex break down and separated flow over delta wings, and immersion of vertical fin surface in the separated wing flow leading to the loss of stability and control at high angles of attack. In such scenarios, availability of additional control power would help in overcoming the limitations. Such additional V. Sundara Pandian · R. J. Pathanjali · B. Praveen Kumar (B) · M. Madhusudan · D. Narayan Aerodynamics and Performance Group, Aeronautical Development Agency, Bangalore 560017, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_71
783
784
V. Sundara Pandian et al.
control power would go a big way in gaining significant advantage over adversaries during the time of combat. In normal conditions, the canards in a canard-delta wing fighter aircraft are operated symmetrically at the same deflections for pitch stability and control. This paper explores the possibility of using an asymmetric deflection of canard surface for generating yawing moments on a close coupled canard-delta wing fighter aircraft at subsonic Mach numbers. When canards are deflected asymmetrically, their effects on fuselage pressure distribution differ between starboard and port side fuselage. This differential pressure distribution on the sides of the front fuselage results in a side force in the direction of canard deflecting Leading Edge (LE) down. Since this side force is generated ahead of the Moment Reference Point (MRP), this side force generates a yawing moment in the same direction and tends to rotate the nose of the aircraft in the direction of canard deflecting LE down. Agnew and Hess [1] explored the possibility of using a differential canard deflection to generate substantial levels of direct side force for a three surface configuration. Croom and Nguyen [2] studied the data from wind tunnel experiments and stated that apart from generating yawing moment, the differential canard deflection induces a favorable side wash on the vertical tail which results in further augmentation of the yawing moment generated in the front fuselage. This paper builds on their studies and tries to quantify the side force generated from an asymmetric canard deflection. The geometry used for this study is a generic canard-delta wing tailless fighter aircraft with side mounted air intakes and a single vertical fin as shown in Fig. 1. The air intakes are located under the delta wing with the missiles and the associated launchers carried at the tip of the delta wings. For this study, Asymmetric canard deflections are considered, i.e., Starboard side canard leading edge is deflected down by 10° and the port side canard leading edge is deflected up by 10°, as shown in Fig. 2. Fig. 1 Geometry of fighter aircraft with Canards at neutral position
Computational Study to Assess the Usage of Asymmetric Canard …
785
Fig. 2 Geometry of fighter aircraft with asymmetrically deflected Canards
2 Details of Grid and CFD Computations The surface mesh generated on the computational geometry used for the present study is shown in Fig. 3. Sufficient grid densities have been incorporated to capture the local flow in the suction side of the delta wing and also behind the canards particularly to capture the vortices shed by the up and down deflected canards. Similar grid generation strategy was employed for both the configurations—canards at neutral position and canards deflected asymmetrically. The grid generation was carried out using the ANSYS ICEM CFD grid generation software. The grid generated is of hybrid anisotropic unstructured type consisting of 60 million cells in the volume, with a combination of tetrahedral, prismatic, and pyramidal volumes. Prism layers were generated with desired y+ to capture the viscous effects near the wall. The boundary layer was modeled with approximately 38 prism layers. Additional clustering was provided over the wing and canard region to capture vortical flow structures, observed at medium and high angles of attack. All RANS computations in the present study had been carried out in CFD++ solver. The spherical domain was given Riemann invariant characteristics based far field boundary condition and the aircraft surfaces were treated as adiabatic viscous wall. Turbulent shear stress terms were modeled in the computations using two-equation Shear Stress Transport (k-ω SST) model. Residual convergence of 4th to 6th order was checked and achieved for all the cases simulated.
3 Results and Discussion CFD computations were carried out for the configurations with symmetric canards at neutral position and asymmetrically deflected canards, at a subsonic Mach number of 0.7 at zero side-slip conditions. The experimental results shown in this paper were obtained from carrying out tests of scaled aircraft model in a closed circuit wind
786
V. Sundara Pandian et al.
Fig. 3 Surface mesh of computational geometry
tunnel facility. The free stream conditions in the test facility are M CO = 0.7 and Reynolds number per unit length of 3.25 million.
3.1 Pressure Distribution The first section compares the Cp distribution on the aircraft surfaces at low, medium, and high angles of attack between the configuration with symmetrically deflected canards at 0° and asymmetrically deflected canards. In the asymmetric canard case, the port canard is deflected LE 10° up and the starboard canard is deflected LE 10° down. Figures 4 and 5 show the comparison of Cp contours on the Port side surfaces of the aircraft for the canards at neutral position (0° canard) and asymmetric canard for low, medium and higher angles of attack. In Fig. 4, the presence of canard generates suction on the sides of the fuselage above the canards and compression on the sides of the fuselage below the canards and ahead of the splitter plate. Both the suction and compression on the sides of the fuselage increase with increase in angle of attack. Compared to the fuselage suction observed (above the canard) in the canard neutral case, the suction pressures are much higher in the asymmetric canard case, due to the LE up deflection of the canard. Additionally, higher compressive pressures are observed (below the canard) in the region between LE up canard and the delta wing in the asymmetric canard case. Figures 6 and 7 show the comparison of Cp contours on the Starboard side surfaces of the aircraft for the two configurations. For the canards neutral case, the pressure distribution between the port and starboard side forces are almost identical, as shown in Figs. 4 and 6, thereby leading to zero side force and zero yawing moments, throughout the angle of attack range. In the asymmetric canard case, the starboard canard is deflected LE 10° down. The fuselage suction pressures above the canard in the asymmetric canard case are much lesser, due to the LE down deflected canard. Also the fuselage compressive pressures below the canards are also lower as compared to the neutral canard case. The region between the canard and the delta wing in the starboard side forms a channel where higher suction pressures are observed due to the flow acceleration in the channel.
Computational Study to Assess the Usage of Asymmetric Canard …
787
Fig. 4 Cp distribution on aircraft port side for symmetric canard deflection (Canards at 0°)
Fig. 5 Cp distribution on aircraft port side for asymmetric canard deflection
Figures 8 and 9 show the comparison of Cp distribution on the top surfaces of the aircraft for the two configurations. For the canards neutral case, the suction patterns are near symmetrical between starboard and port wing and canards, for all three ranges of angles of attack. For the asymmetrical canard case, at low angles of attack, the port canard shows the presence of distinct vortex flow since it is deflected up whereas the down deflected canard shows much benign suction pressures. The suction
788
V. Sundara Pandian et al.
Fig. 6 Cp distribution on starboard side for symmetric canard deflection (Canards at 0°)
Fig. 7 Cp distribution on aircraft starboard surfaces for asymmetric canard deflection
regions in the wing are concentrated close to the leading edge of the wing and are roughly symmetrical. At medium angles of attack, the delta wing pressure distribution is completely asymmetrical as the LE up canard in port side induces downwash in the port wing whereas the LE down deflected canard in the starboard side induces upwash in the starboard wing. At high angles of attack, the trend changes completely
Computational Study to Assess the Usage of Asymmetric Canard …
789
and the starboard wing experiences a complete vortex break down whereas the port wing continues to generate lift.
Fig. 8 Cp distribution on aircraft top surfaces for symmetric canard deflection (Canards at 0°)
Fig. 9 Cp distribution on aircraft top surfaces for asymmetric canard deflection
790
V. Sundara Pandian et al.
3.2 Side Force Figure 10 shows the comparison of side force between the results from CFD computations and experiments for M CO = 0.7 for the configuration of canards deflected asymmetrically. The asymmetric canard deflection generates a positive side force (force acting from port toward the starboard side), as shown in Fig. 10 and this acts in the direction of canard going LE down. The magnitude of side force increases with increase in angle of attack. The side force shows a satisfactory comparison between computational and experimental results, from low to medium angles of attack range, and small discrepancies in the higher angles of attack. Figures 11 and 12 show the contours of side force on the port side surfaces of the aircraft for the canard neutral and asymmetric canard cases. The regions of the fuselage where suction was observed, mainly on the sides of the front fuselage, above the canard surfaces, contribute to a negative side force and the regions where compressive pressures were seen contribute to a positive side force. Figures 13 and 14 show the contours of side force on the starboard side surfaces of the aircraft for the canard neutral and asymmetric canard cases. On the starboard side, the regions of the fuselage where suction was observed contribute to a positive side force and vice-versa for the compression regions. At high angles of attack, the accelerated flow in the channel interacts with the top surface vortical flows and creates a larger suction region in the top fuselage, which further adds to positive side force, as seen in Fig. 14. Fig. 10 Comparison of side force coefficient between CFD and wind tunnel test results
Computational Study to Assess the Usage of Asymmetric Canard …
791
Fig. 11 Cy distribution on aircraft port side for symmetric canard deflection (Canards at 0°)
Fig. 12 Cy distribution on aircraft port side for asymmetric canard deflection
3.3 Yawing Moment The pressure and skin friction force distribution of all the aircraft surfaces, obtained from the CFD computations are integrated about the MRP to get the aerodynamic moments. Figure 15 shows the comparison of yawing moments between the results
792
V. Sundara Pandian et al.
Fig. 13 Cy distribution on aircraft starboard side for symmetric canard deflection (Canards at 0°)
Fig. 14 Cy distribution on aircraft starboard side for asymmetric canard deflection
from CFD computations and experiments for M CO = 0.7 for the configuration of canards deflected asymmetrically. Since the canards are located far ahead of the MRP, a positive side force generated by the canards tends to produce a negative yawing moment about the MRP (yawing moment that tends to take nose of aircraft toward port side). From Fig. 15, it can be concluded that the asymmetric deflection
Computational Study to Assess the Usage of Asymmetric Canard …
793
Fig. 15 Comparison of yawing moment coefficient between CFD and wind tunnel test results
of canards generates yawing moment that tends to take the aircraft nose toward the direction of canards deflecting LE down. As shown in Fig. 15, the yawing moment generated by deflecting the canards asymmetrically by ±10° remains constant, starting from low angles of attack to medium angles of attack and suddenly drops to zero at higher angles of attack and changes the direction of moment generated at very high angles of attack. The yawing moments show very good comparison between computational (CFD) and experimental (WT) results, throughout the angles of attack regime. The reasons for the change in trend in yawing moments with the change in angle of attack regime are explained by analyzing the solutions of CFD computations. In the same figure, the data from the experiments for the 40% of maximum rudder deflection is shown. It can be inferred from the data that the yawing moment generated by deflecting the canards asymmetrically by ±10° can generate yawing moment close to that generated by 40% of maximum rudder deflection. Figure 16 shows the sectional distribution of yawing moment coefficient at three different angles of attack. It can be seen that the canard regions in the front fuselage generate the negative yawing moment. It can also be seen from Fig. 16, that the asymmetric canard deflection generates negative yawing moment in the vertical fin region of the aircraft. This is due to the vertical fin influenced from the side wash induced due to the asymmetric canard deflection. At high angles of attack, the yawing moment induced at the vertical fin due to the side wash turns adverse and is thus responsible for the drop in yawing moment observed in higher angles of attack. Figures 17 and 18 show the yawing moment contours of the port side surfaces of the aircraft for the canard neutral and asymmetric canard cases. The regions generating positive side force ahead of the MRP generate positive yawing moment (nose yaws toward port side) and the regions aft of MRP generate negative yawing moment. As seen in Fig. 18, the negative side force in the front fuselage sides, above the canard generates maximum negative yawing moment. In starboard side, as seen
794
V. Sundara Pandian et al.
Fig. 16 Sectional distribution of yawing moment coefficient for asymmetric canard deflection at M CO = 0.7
in Figs. 19 and 20, the positive side force in the front fuselage sides above the canard generates maximum positive yawing moment. The yawing moment distribution on the vertical fin shows differences between the two configurations due to the influence of the side wash induced by the asymmetric canard deflection.
Fig. 17 Contours of yawing moment on aircraft port side for symmetric canard deflection (Canards at 0°)
Computational Study to Assess the Usage of Asymmetric Canard …
795
Fig. 18 Contours of yawing moment on aircraft port side for asymmetric canard deflection
Fig. 19 Contours of yawing moment on aircraft starboard side for symmetric canard deflection (Canards at 0°)
796
V. Sundara Pandian et al.
Fig. 20 Contours of yawing moment on aircraft starboard side for asymmetric canard deflection
4 Conclusions In this paper, the possibility of using an asymmetric deflection of canard for generating yawing moments in comparison to that generated by the rudder has been presented using results from CFD computations and Experimental (WT) tests at a subsonic Mach number. The results from the numerical simulations showed very good agreement with the wind tunnel experimental results. The results from CFD computations show encouraging trends that asymmetric canards can generate sufficient yawing moment throughout the range of angles of attack. Though benign asymmetric canard deflections were considered in this study, this configuration showed great promise by generating yawing moments comparable to that produced by the 40% of maximum rudder deflection. The results from the CFD computations were post processed extensively to identify the sources for generating the yawing moments and presented in detail in this paper. Acknowledgements The authors would like to acknowledge Mr. S. Jawahar, Technology Director (ARD&P), Aeronautical Development Agency, for permitting him to present this work in this conference and his constant encouragement in carrying out this work. The authors graciously acknowledge the technical guidance of Dr. Santhosh P. Koruthu for this present study.
Computational Study to Assess the Usage of Asymmetric Canard …
797
References 1. Agnew JW, Hess JR Jr (1980) Benefits of aerodynamic interaction to the three-surface configuration. J Aircr 17(11):823–827 2. Nguyen LT, Croom MA, Grafton SB (1982) High angle-of-attack characteristics of three-surface fighter aircraft. In: AIAA 20th aerospace sciences meeting, no. AIAA-82-0245
Physiological FSI Study for Phonoangiography-Based Rupture Risk Prediction in Abdominal Aortic Aneurysms Sumant R. Morab, Janani S. Murallidharan, and Atul Sharma
1 Introduction Cardio vascular diseases (CVD) have resulted in increased mortality during past few years and deaths due to CVD are expected to grow to 23.6 million (25%) by 2030 as per recent report [1]. Aneurysms are a class of CVD, which involve permanent arterial dilatation due to weakening of arteries caused by pathological processes [2]. Although this vascular abnormality occurs in many arterial networks, it is more commonly observed in abdominal and cerebral arteries. Based on the shape of the arterial dilatation, the aneurysms are mainly categorized into two types: fusiform and saccular. Fusiform aneurysms have symmetric bulge around the vessel centerline while the saccular aneurysms bulge only on one side. The major risk associated with aneurysm is the blood leakage due to sudden rupture caused by blood pressure. This may lead to insufficient blood supply to other organs which ultimately leads to stroke. Medical practitioners across the globe assess the risk of the rupture of an aneurysm by maximum diameter (Dmax ) of widening; Dmax ≥ 5 cm in abdominal aortic aneurysm (AAA) can lead to rupture. In a mechanics perspective, rupture occurs when mechanical stress on wall approaches maximum artery strength. Many recent studies compared efficacy of using the diameter (Dmax ) and wall stress (WS) as parameter to estimate rupture risk in a clinical setup. Fillinger et al. [3] performed CFD simulations on medical images of aneurysms on elective and emergent repair patients (those without and with surgery requirement) and observed that although there was only 3% difference in diameter, 36% difference in WS was observed. Maier et al. [4] also performed a similar CFD investigation but authors considered a new parameter named rupture potential index (RPI) (calculated as ratio of the WS and ultimate strength). It was observed that just 4% difference in diameter between S. R. Morab (B) · J. S. Murallidharan · A. Sharma Department of Mechanical 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_72
799
800
S. R. Morab et al.
elective and emergent group, leads to 32 and 46% difference in WS and RPI values, respectively, indicating RPI to be most accurate predictor of rupture risk. Several other authors have reviewed the parameters used for rupture risk prediction [5] and highlighted the necessity for consideration of fluid–structure interaction (FSI) analysis in a computational study. Diagnosis of aneurysm is performed using physical examination and medical imaging. A palpable, pulsatile mass is usually sensed by medical practitioner on abdominal area. This is followed by imaging using CT scanning, MR angiography, ultrasonography, etc., and based on the size and shape of aneurysm, treatment strategies are decided. For patients with constricted arteries, abnormal sounds that are heard through stethoscope have been observed in aneurysms also [6]. The sound-based diagnosis of arterial constriction led to the field of phonoangiography. To study sound emitted from these arteries, recently Seo et al. [7] developed numerical models as a diagnostic tool. They indicated that a hemodynamic parameter is also capable of indicating possible acoustic frequency in constricted arteries. From an extensive literature survey, no published work is found for phonoangiography-based study on fusiform AAA’s. However, such studies are found for stenosed artery. Thus, the current study tries to extend the sound-based diagnosis for aneurysms to predict rupture risk. Hence, FSI simulations are performed on several aneurysm geometries to study the correlation between rupture risk and qualitative acoustic indicator. The current article is organized as follows: Sect. 2 presents a numerical methodology and validation study for FSI solver. In Sect. 3, results obtained for a qualitative acoustic indicator along with RPI variation for different geometries are discussed along with reasoning provided through flow visualization. Also, a correlation between rupture risk and the obtained qualitative sound frequencies is presented. Finally, conclusions of the current work and future possibilities are presented.
2 Methodology 2.1 Computational Setup and Parametric Details For an axisymmetric AAA, Fig. 1 shows the computational domain, consisting of subdomains for blood flow and artery; along with the boundary condition for the flow and arterial structure. The geometry for aneurysm is modeled using Gaussian curve [8] as follows: [ r (z) =
)] ( z2 D , + H exp − 2 2W 2
where r is the radial distance, D is the unconstricted diameter, H is the maximum bulge height, and W represents width of the aneurysm bulge.
Physiological FSI Study for Phonoangiography-Based Rupture Risk …
801
Fig. 1 Computational setup for coupled FSI simulation on fusiform AAA (blue region indicates lumen, while the structure is indicated by grayed region)
The heights (H) and widths (W ) used in the current are chosen such that H/D = 0.3, 0.5, 0.7, 1.0 and 1.2 and W/D = 0.5 and 1, which are commonly observed in patients with mild and severe aneurysm. To allow the flow to renormalize at the exit, a domain length of 10D is considered post aneurysm bulge.
2.2 Numerical Model Fluid Flow: Blood is assumed to be Newtonian in nature, since non-Newtonian effects are observed to be negligible in large arteries (≥ 2 mm). For laminar and incompressible flow, axisymmetric conservation equations are solved to obtain the flow variables (U z , U r and P); given [8] as, Mass conservation: ∂Ur Ur ∂Uz + + = 0. ∂z ∂r r Momentum conservation: ] [ ∂Uz ∂Uz ∂Uz + Ur + Uz = ρ ∂t ∂r ∂z ] [ ( ) 1 ∂ ∂Uz ∂ 2 Uz ∂P +μ r + − ∂z r ∂r ∂r ∂z 2 ] [ ∂Ur ∂Ur ∂Ur + Ur + Uz = ρ ∂t ∂r ∂z [ ( ) ] ∂P 1 ∂ ∂Ur ∂ 2 Ur Ur − +μ r + , − ∂r r ∂r ∂r ∂z 2 r2
(1)
(2)
(3)
where U z , U r and P are the axial velocity; radial velocity and pressure, respectively. Further, μ is dynamic viscosity and ρ is density of blood. Structural Deformations: Since the ratio of arterial thickness and internal diameter is usually small (t/D ≤ 0.1), Womersley’s thin structure deformable theory [9]
802
S. R. Morab et al.
is used to model artery displacements as follows: [ 2 ] ] [ ∂ η νs ∂ξ Et ∂ 2η ∂Uz ∂Ur + ρs t 2 = −μ + + ∂t ∂r ∂z 1 − νs2 ∂z 2 R ∂z [ [ ] ] ξ ∂ 2ξ ∂Ur νs ∂η Et ρs t 2 = P − 2μ + − , ∂t ∂r 1 − νs2 R 2 R ∂z
(4)
(5)
where η = L—L 0 represents axial deformation and ξ = R—R0 represents radial deformation. Further, E, ρ s and ν s represent elastic modulus, density and Poisson’s ratio of artery wall, respectively. Arbitrary Lagrangian–Eulerian (ALE) method-based in-house FSI solver is developed, which is used to obtain solutions in the current investigation. The fluid flow equations are solved using semi-implicit pressure projection (SIPP) scheme, while structural deformations are obtained using pseudo-transient time stepping.
2.3 Boundary Conditions and Flow Parameters The boundary conditions used in the current FSI study are as follows: Inlet: In order to mimic physiological pulsatile flow waveform, a time varying axial velocity, as measured by Suh et al. [10] during physical activity, is considered and implemented computationally through Fourier harmonics; given as Q(t) = Re
( N ∑
) Qn e
jnωt
,
(6)
n=0
where Q(t) represents instantaneous flow rate which is expressed as sum of Fourier co-efficients Qn . ω is angular frequency of pulsation, j is the complex index and Re() represents real part. Along the cross section, Womersley’s pulsatile flow solution is used to calculate axial velocity given as [ ] r2 2Q 0 1− 2 Uz (r, t) = π R2 R ⎞ ⎛ [ ( r )] N ∑ J Λ Q 0 n n R [ ] 1− + Re⎝ e jnωt ⎠, 2J1 (Λn ) J (Λ ) 2 0 n n=1 π R 1 − Λn J0 (Λn )
(7)
where Λn = W o j 3/2 n 1/2 and J0 and J1 represent Bessel function of first and second order, respectively. For inlet pressure P, a zero gradient condition is applied. Outlet: A Neumann boundary condition is implemented for velocity at the outlet, and a three-element Windkessel model (3-eWkm) is used for pressure. For the model, the pressure profile is directly determined using flow rate at outlet and resistance
Physiological FSI Study for Phonoangiography-Based Rupture Risk … Table 1 Properties of blood flow and artery wall [12] used in the current study
Parameter
Value
Kinematic viscosity (ν)
3.5 × 10−6 m2 /s
Flow rate ‘Q’ (Min.-Max.)
3 l/min–12 l/min
Womersley number (Wo)
16.5
Diameter (D)
2 cm
Elastic modulus (E)
2 MPa
Poisson’s ratio (ν s )
0.45
Thickness of artery (t)
2 mm
803
values are obtained through arterial properties. The 3-eWkm is 0D model which takes into account compliance of artery and calculates pressure values. It has been found to be the most accurate outlet model as per the studies by Xiao et al. [11]. The outlet pressure can be determined using an equation, given as ( ) P ∂Q R1 ∂P = , Q 1+ +C + C R1 R2 ∂t R2 ∂t
(8)
where R1 , R2 and C represent resistance and capacitance values that are determined through arterial and flow properties as described in Xiao et al. [11]. No-Slip Wall: Since the current study involves coupled FSI, walls are moved at every time-step according to the deformation of structure (artery wall). Hence, the velocity and pressure conditions are specified as follows: Ur =
∂P ∂Ur ∂ξ ∂η , Uz = and =− . ∂t ∂t ∂r ∂t
(9)
Axisymmetry: For the present axisymmetric blood flow, the boundary condition at axis is given as Ur =
∂P ∂Uz = 0, and = 0. ∂r ∂r
(10)
Flow and Arterial Wall Properties: The blood and artery wall properties are obtained from Drewe et al. [12], and are presented in Table 1.
2.4 Validation: Wave Propagation in a Flexible Tube In order to validate the current in-house FSI solver, pressure wave propagation in a straight complaint tube is studied and the results are compared with those presented by Kang et al. [13]. The geometry consists of a straight tube with internal diameter (D) of 2 cm surrounded by flexible (complaint) artery, with elastic modulus (E) of
804
S. R. Morab et al.
Fig. 2 Validation study: comparison of the present results with that reported by Kang et al. [13] for wall pressure during blood flow in straight flexible artery
1Mpa, Poisson’s ratio (ν s ) of 0.3 and density (ρ s ) of 1000 kg/m3 . The thickness of artery is considered as 2 mm and length of domain as 0.1 m. Blood flow inside the tube is modeled to be Newtonian, with kinematic viscosity of 4 × 10−6 m2 / s. At inlet, pressure is ramped up from 0 to 5 kPa while keeping exit pressure at 0 kPa. Symmetry boundary conditions are applied for left and right end of tube wall and hemodynamic load is used to calculate displacements. The simulations are performed with a constant time-step size ∆t = 10−5 s and grid size of 200 × 100 (z and r direction). For comparison of wall pressure at different time instants with literature [13], Fig. 2 shows a good agreement with maximum error of 4.87% that is obtained by the present in-house FSI solver.
3 Results and Discussion For the present problem, our in-house unsteady FSI solver requires a total of six pulsatile flow cycles to achieve periodic steady-state for the flow and arterial deformations. The results presented in this section are from sixth cycle onward for the RPI calculation and from fifth and sixth cycle for IPFR calculation. The simulations are run on Dell Xeon Silver machine in serial, and total time for six cycles is close to 48 h.
3.1 Integrated Pressure Force Rate (IPFR) It has been observed in the previous works ([7, 14]) that the IPFR parameter, which is calculated based on pressure rate on walls of vessels, correlates very well with
Physiological FSI Study for Phonoangiography-Based Rupture Risk …
805
the acoustic velocity fluctuation on the epidermal surface. The rate of velocity fluctuation can be measured by a digital stethoscope, so as to check for irregular sounds. In the current study, we utilize the same parameter to check the feasibility of sound-based diagnosis for AAA. The IPFR value is calculated on artery wall, using nondimensional pressure P, time τ and length L x of domain given as: L x dP dFy = ∫ dx. dτ 0 dτ
(11)
The rate of the IPFR, which correlates with acceleration, is calculated and corresponding spectrum is plotted (not shown here). The spectrum shows presence of higher frequencies (≥ 100 Hz) apart from normal heart sounds (20–100 Hz), which indicates feasibility of sound-based diagnosis. These high frequencies (bruits) exhibit a characteristic frequency, after which intensity of spectrum starts to drop suddenly. This cut-off frequency is calculated as per formulation of Jones and Fronek [15], and presented in Fig. 3 for W /D = 0.5, 1. The figure shows that the cut-off frequency is significantly different from normal heart sounds and increases rapidly with height of aneurysm. Also, it is observed that, width of aneurysm has very less effect since less than 10% increase in frequency is observed for W /D = 0.5 and 1, while almost 100% increase in frequency is observed with increase in H/D from 0.3 to 1.2. This indicates a clear probability of diagnosing the maximum diameter of aneurysm through acoustic signal. In order to investigate the cause behind observed variation, a flow-visualization study is performed as shown in Fig. 4, wherein vorticity contours at three different time instants are presented for four geometries. The three-time instants correspond to early systole (T 1), late systole (T 2), and diastole (T 3) phases where maximum flow changes are observed. For lower height H/D, the figure shows that the vortex shedding first occurs in distal aneurysm end and propagates toward proximal end where secondary vortex appears. This secondary vortex slowly dissipates during diastolic phase. These variations in the flow lead to pressure fluctuations that lead to acoustic signals. In case of larger height H/D, the primary vortex which sheds near Fig. 3 Comparison of cut-off frequency obtained from spectrum of ∂ 2 F y /∂t 2 and those obtained from flow velocity and size of recirculation
300 250 200 150 W/D = 0.5 (IPFR based) W/D = 1 (IPFR based) W/D = 0.5 (Velocity-based) W/D = 1 (velocity-based)
100 50
0.2
0.4
0.6
0.8 H/D
1
1.2
806
S. R. Morab et al. H/D = 0.3, W/D= 0.5
H/D = 1.0, W/D= 0.5
H/D = 0.3, W/D= 1.0
H/D = 1.0, W/D= 1.0
Fig. 4 Variation of vorticity at different time instants of a heart cycle for four geometries. Here, T 1—Early Systole, t/T = 0.54, T 2—Late systole, t/T = 0.74 and T 3—Diastolic phase, t/T = 0.95. Further, the stages correspond to the pulsatile inlet flow in the aneurysm
distal end coalesces with persisting vortex from previous cycle and dissipates along distal end during diastole. Borisyuk [16] presented a flow velocity and cut-off frequency correlation, by hypothesizing that primary vortex dissipation on wall leads to a reduction in the intensity of pressure fluctuation and thus leading to appearance of break frequency. Based on this hypothesis, velocity and vortex core sizes are measured through the flow visualization during the dissipation stage and cut-off frequency is predicted as ratio of the above two parameters. Figure 3 shows the value of these frequencies with actual frequencies obtained from IPFR spectrum. The figure shows that the trend of variation, for both the frequencies (velocity and IPFR-based), is similar. Thus, the presence of higher velocity scale and lower recirculation zones in geometries with large ‘H/D’ contribute to larger frequencies. These results indicate that diagnosis of aneurysm is possible through non-invasive phonoangiography technique.
3.2 Rupture Potential Index (RPI) One of the major risks associated with aneurysm is that of its rupture leading to blood loss from vessel network. Medical practitioners usually use diameter of aneurysm as an indicator for rupture risk in a patient. From a physical perspective, rupture occurs when stress acting on aneurysm due to the pressure of the flow approaches
Physiological FSI Study for Phonoangiography-Based Rupture Risk …
807
0.6
Fig. 5 Variation of rupture potential index (RPI) values for different geometries corresponding to W /D = 0.5 (almost same results are obtained for W /D = 1)
0.5 0.4 0.3 Male Female Critical
0.2 0.1 0.2
0.4
0.6
0.8 H/D
1
1.2
the mechanical strength of artery. The ratio of these two quantities, known as rupture potential index (RPI) was recently observed to predict rupture with greater probability in a recent clinical study [4]. In the current study, wall stress (WS) and mechanical strength of wall are calculated using an empirical equation, given as e0.0123(0.85 p+19.5d) in MPa, t 0.63 = 1132 − 156(NORD − 2.46) + 193 G,
W S = 0.006 σultimate
(12)
where G = 0.5 (male) and − 0.5 (female), NORD = r(z)/r(inlet). These equations are found to be most accurate in clinical setup. The RPI values for male and female are presented in Fig. 5 for W /D = 0.5 (since negligible difference is observed with W /D = 1, it is ignored). It can be observed that similar to cut-off frequencies, RPI increases exponentially with H/D values. RPI ≥ 0.3 is reported [4] to be critical for aneurysm rupture in medical literature and is observed to occur above H/D = 0.8 for a female and 0.9 for a male.
3.3 Correlation of RPI with Frequency Since the main objective of a medical practitioner is to determine the risk of aneurysm rupture so that treatment strategy can be decided efficiently, the indicative cut-off frequency obtained from IPFR is correlated with RPI in the current study. In a clinical setup, cut-off frequency can be obtained through a digital stethoscope connected to computing device. From the correlation presented below, RPI can be directly calculated from cut-off frequency and thus helping practitioners to access the rupture probability. It is observed from Figs. 3 and 5 that variation of cut-off frequency and RPI is nonlinear in nature. Inspired by this, an exponential type of curve-fitting is presented to obtain relation between RPI and frequency, given as:
808
S. R. Morab et al.
Fig. 6 Error between the present simulations-based RPI values and those obtain from curve-fitting with n = 1.604 and k = 4.922 × 10−5 for male and the respective values are 1.669 and k = 4.162 × 10−5 for female
12 10 8 6 4 2 0
Error (Male, W/D = 0.5) Error (Female, W/D = 0.5) 0.2
0.4
0.6
0.8 H/D
RPI = k( f b )n ,
1
1.2
(13)
where the values of ‘k’ and ‘n’ are obtained through least square error method. These values are used to predict RPI, and error is calculated between predicted and actual RPI values as follows: Error =
|RPIsimulation − R P Icurve fit | . RPIsimulation
(14)
Figure 6 shows the error obtained for available data at W/D = 0.5 for a male and a female. The figure shows that the errors are minimum at the extreme geometries and maximum error is ≤ 10% for both cases. Thus, an acceptable value of RPI can be calculated from current physics-based phonoangiography technique.
4 Conclusions Axisymmetric pulsatile blood flow and arterial deformation-based simulations are performed on complaint fusiform aneurysms in abdominal aorta. An indicative acoustic break frequency, obtained from pressure rate spectrum along the wall, is correlated with the rupture potential index (RPI) using an exponential curve-fitting. The conclusions are as follows: (1) The RPI values are observed to cross critical value of 0.3 at H/D of 0.7(0.8) for female (male) which indicates that the larger aneurysms are prone to rupture. (2) A correlation, between the RPI and cut-off frequency, is proposed for a phonoangiography-based clinical assessment of the rupture risk of an abdominal aortic aneurysms. The present work is continued for acoustic signals that generate in the arterial vessel and propagate through tissue surface to reach skin surface for a more realistic phonoangiography study in future.
Physiological FSI Study for Phonoangiography-Based Rupture Risk …
809
Nomenclature d D H p t W WS
Aneurysm maximum diameter (mm) Inlet diameter (cm) Aneurysm height (cm) Pressure (mm Hg) Wall thickness (mm) Aneurysm width (cm) Wall stress (MPa)
References 1. Virani SS, Alonso A, Aparicio HJ, Benjamin EJ, Bittencourt MS, Callaway CW, Carson AP, Chamberlain AM, Cheng S, Delling FN et al (2021) Heart disease and stroke statistics—2021 update: a report from the american heart association. Circulation 143(8):e254–e743 2. Lasheras JC et al (2007) The biomechanics of arterial aneurysms. Ann Rev Fluid Mech 39(1):293–319 3. Fillinger MF, Marra SP, Raghavan ML, Kennedy FE (2003) Prediction of rupture risk in abdominal aortic aneurysm during observation: wall stress versus diameter. J Vasc Surg 37(4):724–732 4. Maier A, Gee MW, Reeps C, Pongratz J, Eckstein H-H, Wall WA (2010) A comparison of diameter, wall stress, and rupture potential index for abdominal aortic aneurysm rupture risk prediction. Ann Biomed Eng 38(10):3124–3134 5. Raut SS, Chandra S, Shum J, Finol EA (2013) The role of geometric and biomechanical factors in abdominal aortic aneurysm rupture risk assessment. Ann Biomed Eng 41(7):1459–1477 6. Krupina NE, Sakovich VP, Suslov SA (1992) Phonoangiography in aneurysms of the cerebral vessels. Zhurnal Voprosy Neirokhirurgii Imeni NN Burdenko 2–3:21–25 7. Seo JH, Mittal R (2012) A coupled flow-acoustic computational study of bruits from a modeled stenosed artery. Med Biol Eng Comput 50(10):1025–1035 8. Gopalakrishnan SS, Pier B, Biesheuvel A (2014) Dynamics of pulsatile flow through model abdominal aortic aneurysms. J Fluid Mech 758:150–179 9. Womersley JR et al (1957) Oscillatory flow in arteries: the constrained elastic tube as a model of arterial flow and pulse transmission. Phys Med Biol 2(2):178–187 10. Suh GY, Les AS, Tenforde AS, Shadden SC, Spilker RL, Yeung JJ, Cheng CP, Herfkens RJ, Dalman RL, Taylor CA (2011) Hemodynamic changes quantified in abdominal aortic aneurysms with increasing exercise intensity using mr exercise imaging and image-based computational fluid dynamics. Ann Biomed Eng 39(8):2186–2202 11. Xiao N, Alastruey J, Alberto Figueroa C (2014) A systematic comparison between 1-d and 3-d hemodynamics in compliant arterial models. Int J Numer Methods Biomed Eng 30(2):204–231 12. Drewe CJ, Parker LP, Kelsey LJ, Norman PE, Powell JT, Doyle BJ (2017) Haemodynamics and stresses in abdominal aortic aneurysms: a fluid-structure interaction study into the effect of proximal neck and iliac bifurcation angle. J Biomech 60:150–156 13. Kang S, Choi HG, Yoo JY (2012) Investigation of fluid–structure interactions using a velocitylinked p2/p1 finite element method and the generalized-α method. Int J Numer Meth Eng 90(12):1529–1548 14. Seo JH, Bakhshaee H, Garreau G, Zhu C, Andreou A, Thompson WR, Mittal R (2017) A method for the computational modeling of the physics of heart murmurs. J Comput Phys 336:546–568
810
S. R. Morab et al.
15. Jones SA, Fronek A (1987) Analysis of break frequencies downstream of a constriction in a cylindrical tube. J Biomech 20(3):319–327 16. Borisyuk AO (2002) Experimental study of noise produced by steady flow through a simulated vascular stenosis. J Sound Vib 256(3):475–498
Impact of Building Configurations on Fluid Flow in an Urban Street Canyon Surendra Singh, Lakhvinder Singh, and S. Jitendra Pal
1 Introduction It has been estimated that air pollution alone causes two million premature deaths in India. As per WHO, the world loses seven million people every year due to stroke, heart disease, lung cancer, chronic obstructive pulmonary diseases, and respiratory infections, including pneumonia all caused by polluted air. It is a matter of great concern to anyone. At present 27% of air pollution is caused by motor vehicles plying on the roads that is in close vicinity with the human habitat. Figure 1 shows a typical configuration of a street canyon. The road is flanked on either side by the high-rise buildings. A common parameter to define the street canyon is aspect ratio (ratio of building height to road width). The unplanned growth and poor planning of the urban space create the problem of pollutant dispersion. The street canyon aspect ratio (AR) and the wind conditions are the dominant factors that influence dispersion. Indeed, dispersion plays a significant role in managing the air quality of a region. The common methods to forecast dispersion are (i) full-scale site testing, (ii) scale model experimentations, and (iii) numerical simulations. Urban planners are now increasingly using numerical methods for predicting pollution dispersion. Wang et al. [1] provided the essential guidelines for performing and validating computational fluid dynamics (CFD) simulations. Various studies [2, 3] have shown that the turbulent kinetic energy (TKE), characterized by components of the velocity fluctuations, guides the dispersion phenomenon. The present study is undertaken in the context of Indian Urban space that has non-uniform and irregular street canyon layout.
S. Singh · S. Jitendra Pal Department of Chemical Engineering, NIT Surathkal, Mangalore 575025, India L. Singh (B) Centre for Essential Skills, IIT Jammu, Jammu 181221, 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_73
811
812
S. Singh et al.
Fig. 1 Schematic diagram of the street canyon
The understanding of flow patterns and turbulence intensity in the street canyon are complex in nature and will offer insights into pollution dispersion.
2 Literature Review and Objective The prediction of pollution dispersion is a complex task due to the direct involvement of several factors such as uneven distribution of pollution sources that changes spatially and temporally, dynamic metrological conditions, and non-uniform obstacle. Ming et al. [4] studied dispersion in a non-uniform urban street canyon under the influence of traffic tidal flow for wind direction perpendicular to the street. The author found that there is a higher concentration of pollutants at the pedestrian breathing height when the intensity of source is greater in the leeward side than windward side. A different street canyon configuration yet becoming increasingly common in metro cities is multi-canopy building array that was investigated by Chan et al. [5]. A numerical study by Singh et al. [6] analyzed pollution levels on pedestrian for a dense city’s street canyon. The study by Sharma et al. focused on improving the fresh air concentration in the street canyon by changing building shapes. Madalozza et al. [7] conducted the numerical study on dispersion in a street canyon while incorporating heat transfer. The author found significant differences in the distribution of pollutant concentration under heated conditions. Salim et al. [8] predicted the airflow and dispersion characteristics using three modelling methods that are Reynolds-averaged Navier–Stokes (RANS), unsteady RANS, and large Eddy simulation (LES). As expected, LES provided a better insight into the turbulent fluctuations and TKE. However, the cost of simulating street canyon using LES is quite expensive compared with RANS. An extensive numerical study on an actual urban setup in the city of Osaka, Japan was carried out by Francesca et al. [2]. The author
Impact of Building Configurations on Fluid Flow in an Urban Street Canyon
813
combined flow analysis and chemical reactions to determine a realistic scenario. For determining accuracy, the boundary conditions were obtained from mesoscale meteorological and air quality models. The detail study confirmed significance of roadside emission and airflow patterns in pollutant dispersion. The atmospheric boundary layer (ABL) is a key concept for the numerical study of street canyon. ABL is the region between the Earth surface and free atmosphere in which the velocity gradients are found. ABL is found to exist till 4 km from the Earth surface and multiple sub regions exist within ABL that are identified by the rate of velocity gradients [9]. The lower region up to 200 m is commonly called as surface layer and is defined by the logarithmic law [10]. The region beyond surface layer till free atmosphere is known as transition region. In here, the velocity gradients diminish from a constant value to zero approaching free atmosphere. The standard practice is to employ power law expression to model ABL. The current study aims to investigate three important aspects of street canyon that affects the fluid flow and transport of pollutant dispersion. In the first part, the building height is investigated for street canyon and wind approach angle being 90°. In the second part, four types of building configurations have different arrangement of buildings along the street. The study is done for wind approaching normal to the road. The third aspect of the study aims at understanding the flow patterns as the approach angle of wind changes. All the three aspects will provide an understanding of potential pollutant traps that form due to the recirculation of fluid pockets.
3 Computational Model and Methodology The pollutant dispersion in a street canyon is a classic problem of fluid dynamics. The interaction of wind with buildings results in formation of number of lowpressure zones (recirculation pockets) that governs the dispersion. To obtain flow field, Reynolds-averaged Navier–Stokes (RANS) for continuity and momentum equations is solved. The turbulence modelling is found to be an effective tool to estimate the behaviour of dispersion. The present study chooses RNG k-ε turbulence model for estimating the turbulence variables and studying the flow patterns in a street canyon. The choice of RNG k-ε turbulence model is based on its better suitability for estimating local recirculation [8, 13]. The computational domain is a scaled model in the ratio of 1:100. Figure 2 shows the computational domain including the dimensions of the building, road, and the flow domain. The edges of the flow domain are placed as follows: (i) the extreme right face (inlet) of the domain is 5H from the building, (ii) the extreme right (outlet) is 25H from the building, (iii) the top plane is 14H, and (iv) the lateral faces are 5H from the building edge. The current street model is classified as avenue type canyon since the aspect ratio (height of building/width of road) is 0.587. Likewise, the street type is classified as medium street canyon as the ratio of length of the road to height of building is 5.494 [11].
814
S. Singh et al.
Fig. 2 Computational domain of the street canyon
3.1 Computational Domain and Grid Setting The structured grid approach is used to discretize the domain. The choice of such a grid over unstructured tetrahedral mesh is firstly the domain is unidirectional domain, secondly for being cost effective the number of elements must be less. Thus, for major portion of the domain, polyhedral mesh is generated as shown in Fig. 3, while inflation layers are generated close to the wall for capturing the sharp gradients. The number of inflation layers are 10, having a growth rate of 1.2 and transition ratio of 0.272. The y + value of 30 is maintained at the walls, an essential condition when using RNG k-ε turbulence model. Further, the grid quality is ensured by keeping the aspect ratio of elements to be less than 50.
Fig. 3 Mesh elements and inflation layers in the computational domain
Impact of Building Configurations on Fluid Flow in an Urban Street Canyon
815
Fig. 4 Grid independence study
The mesh independence study is performed to arrive at a grid that could generate numerical results independent of grid size. Three grids having the nodes counts as (i) Mesh 1 = 0.3 × 106 and (ii) Mesh 2 = 0.6 × 106 , Mesh 3 = 0.9 × 106 are numerically solved. The velocity profiles are extracted at the midline of the road at nose level leeward side and at nose level windward side. Figure 4 shows the comparison of velocity profiles for the three grids. The profiles seem to reasonably match for Mesh 2 and Mesh 3. Similar findings are observed at the other two planes. Thus, going forward Mesh 2 is chosen for the numerical study.
3.2 Boundary Conditions The inlet to the computational domain is well within the region of atmospheric boundary layer. Thus, for the current simulation, a logarithmic velocity profile is applied as inlet boundary condition. Further, profiles for kinetic energy and dissipation rate are used rather than static values. Analytical expressions proposed by Richards and Hoxey [12] are used to obtain the inlet conditions. Equation 1 is the expression used to obtain inlet velocity profile, while Eq. 2 gives the inlet profile for kinetic energy and Eq. 3 gives the profile for inlet dissipation rate. u ref
) ( z1 + z0 U∗ = ln kv z0
(1)
U∗ k=√ Cμ
(2)
U∗ ε= k(z1 + z 0 )
(3)
2
2
816
S. Singh et al.
At the outlet, a pressure outlet boundary condition is applied to mimic atmospheric conditions. The values of all the variables at the outlet boundary condition are extrapolated from the adjacent cells by ensuring zero gradient at the outlet. The top and the side boundaries of the flow domain are given symmetric boundary conditions since no normal gradients exit at these boundaries, while the tangential components of the variables are calculated based on zero gradient conditions. The bottom of the flow domain represents the ground and hence no-slip wall boundary condition is applied. Likewise, all the side of the building are also set as no-slip boundary conditions.
3.3 Solver Settings Governing equations are integrated over the discretized computational domain by finite volume technique (FVM). The SIMPLE scheme is used for the pressure velocity coupling and derives the equation for pressure. To increase the accuracy of the simulation, a second-order upwind scheme is used to find the values of the variables. The scheme uses a larger stencil to calculate the value at each node thus reducing the dependence on the immediate node values and provides better accuracy. The convergence criteria for all variables are set to 1e−6 .
4 Results and Discussion The results are broadly presented in three sections explaining the effect of building height, building configuration, and wind direction on the flow distribution in the vicinity of buildings.
4.1 Effect of Building Height The building height is an important parameter that affects the flow distribution and hence transport of pollutants. Three common street canyon types found in urban cities of India are avenue street canyon (AR = 0.5), simple street canyon (AR = 1), and deep street canyon (AR = 2). They are distinguished based on the aspect ratio (AR). Thus, the three cases that are investigated are AR = 0.5, AR = 1, and AR = 2. The wind direction is assumed normal to the street canyon. The inlet velocity profile shown in Fig. 5 is taken for the investigation. Figure 6 shows the locations where data has been extracted and analyzed. Figure 6 shows the vector plot on the nose level plane for the case when AR = 2. It is observed that with the increase in building height the intensity of recirculation pockets also increases, and higher magnitude of velocities are observed for AR = 2. Further, on investigating the flow patterns for the three cases, it is observed that for
Impact of Building Configurations on Fluid Flow in an Urban Street Canyon
817
Logarithmic Inlet Velocity Profile ABL
Domain Height, m
2.5 2.0 1.5 1.0 0.5 0.0 0
1
2
3
4
5
6
7
8
9
10
Velocity, m/s Fig. 5 Inlet velocity profile as boundary condition representing ABL
Fig. 6 Planes of investigation
AR = 2, deep street canyon, additional pocket of recirculation zone is formed on the street. Thus, the flow patterns for AR = 2 are significantly different from AR = 0.5 and AR = 1. Another interesting phenomenon that was observed at the nose level of the street canyon is the intensity of velocity magnitude. As the building height increases, the velocity at the nose plane tends to decrease. This could be due to the flow separation at the windward side of the building and its ability to affect the nose level. As the building height increases, the influence of the windward conditions
818
S. Singh et al.
Fig. 7 Velocity vector plot highlighting the flow distribution at nose level for AR = 2
on the nose level tends to decrease. In Fig. 7, there is a formation of recirculation pocket on the sides of the building, and it seems to travel inwards. The intensity of such pockets of fluid flows increases with the building height, thereby increasing the tendency to trap pollutants on the street canyon. On analyzing the turbulent kinetic energy (TKE), it is found to be maximum at the top side of the building where the flow separates. A region of high turbulent kinetic energy seems to exist in the proximity of the buildings. In the case of AR = 0.5 and AR = 1, the intensity of TKE is more on the leeward side, whereas it is more in the windward side for AR = 2. Thus, the building height affects the pattern of TKE in the region. On observing the intensity of TKE on the street canyon, it seems to decrease with building height. However, as AR increases the maximum value of TKE also increases and the region of their occurrence shifts downstream.
4.2 Effect of Building Configuration The effect of building configurations on flow patterns is studied by selecting four types of building configurations: (i) street canyon (BSB), (ii) plain road (PL), (iii) upwind side building (UWSB), and (iv) downwind side building (DWSB). Among the four cases, there are no changes in boundary conditions except the changes pertaining to building type. Figure 8 shows the velocity profiles at the middle of the road for the four cases under consideration. In the case of PL, a logarithmic velocity profile is observed much like the inlet velocity profile. It is expected since there are no buildings. Further, the PL is equivalent to the case of flow over a flat plate. A similar velocity profile is observed in the case of DWSB but, near to the road, the velocity gradients are significantly lower when compared with PL. Thus, the flow distribution on the street is impacted due to the downstream building. On the other hand, the velocity profile is significantly different in the case of BSB and UWSB. This is primarily due to the
Impact of Building Configurations on Fluid Flow in an Urban Street Canyon
819
Fig. 8 Velocity profiles for the four building configurations at the mid-section of the road
building position upstream of the street. On inspecting the velocity profile up to H = 0.182 m (the building height), the velocity at the top of the building is less than at the nose level. It is more prominent in the case of UWSB and signified a larger and much stronger recirculation pocket on the leeward side of the building. In the case of BSB, the velocity profile is mainly flat. The study suggests UWSB as the worst-case scenario.
4.3 Effect of Wind Direction Another critical parameter that affects pollutant dispersion is the wind direction. The effect of wind direction is studied by varying the angle between the velocity inlet and buildings. Five cases are studied where the angles are varied as 90°, 60°, 45°, 30°, and 0°. One such configuration is shown in Fig. 9.
Fig. 9 Isometric view of the model when wind angle 30°
820
S. Singh et al.
The flow pattern among all the cases is significantly different from each other. In each of the cases, there is a formation of wake on the leeward side of the building and a tail-like structure is seen that tends to change its size with the wind direction. In the case of wind direction being normal (900) to the building, the wake is symmetric across the building length. As the angle changes to 600, the wake becomes asymmetric and moves towards the side edge of the building. The wake is thicker in size but short in length. When the angle reduces further to 450, the wake becomes thin and long as shown in Fig. 10. On subsequent angles, the wake elongates but the size is much smaller. On the contrary, as the wind tends to align with the building, the flow on the street canyon is largely unaffected in the vicinity of the road as shown in Fig. 11. In the case of 00 wind direction, a condition when the wind is aligned with the street canyon, the velocity profile is quite like the inlet velocity profile. The other extreme case, when the wind direction is normal to the street canyon, the velocity is rather flat and significantly lower velocity magnitude is observed. It indicates a condition where the transport of pollutant is least effective. In the other cases, the velocity at the nose level (a location at 15 mm from the road) is large of the order of 4.5 m/s. It indicated higher transport of pollutants. Along the building height, the velocity tends to decrease but still greater than the case of 90°. On analyzing the production of TKE for the five configurations, it is found that the highest intensity of TKE is found to exist on the top edge of the buildings in all the cases. The intensity seems to increase with angle of incident except in the case when the angle is 60°. Indeed, the maximum intensity is seen at 60°. As discussed earlier, the wake size is much wider but short in length. Another important observation is the intensity of TKE in the street canyon that seems to replicate the pattern of maximum intensity. In the street canyon, TKE increases with angle except of the case when the angle is 60°. Again, the intensity is maximum in this case. Thus, it can be said that the most favourable scenario for maximum transport of pollutant is when the angle of wind approaching the street canyon is 60°.
Fig. 10 Velocity vector at nose level (Y = 15mm) wind angle 45°
Impact of Building Configurations on Fluid Flow in an Urban Street Canyon
821
Fig. 11 Velocity profile for different wind conditions at the mid-section of the street canyon
5 Conclusions A numerical study is carried out on urban street canyon to investigate the pollutant dispersion transported by the fluid flow. The study is performed using RNG k-ε turbulence model. For a realistic scenario, the velocity inlet boundary conditions are adapted to represent atmospheric boundary layer by incorporating logarithmic velocity profile. In the present study, effect of building height, building configuration, and wind direction is studied. As the building height increases, the turbulent kinetic energy increases at the top edge of the buildings identified as the zones of flow detachment. At the same time, turbulent kinetic energy is found to increase at the point of reattachment on the leeward side of the building. On the contrary, the turbulent kinetic energy tends to decrease with building height in the street canyon. At the nose level of the windward side, pockets of recirculation are observed that act as sources for pollution traps. While investigating the building configurations, a uniform velocity profile is observed for plain road and downward side building. However, highly skewed velocity profiles are seen in the case of upward side building and street canyon and lower turbulent kinetic energy values that increase the probability of pollution trapping. The study of wind direction shows that the turbulent kinetic energy is maximum on the top edge of the building as well as in the street canyon when wind approaches the building at an angle of 600. Whereas, for 450 wind angles, the intensity of recirculation pockets formed at the nose level is stronger than other cases, thus increasing the chances of pollution trapping.
822
S. Singh et al.
Nomenclature H W k ε AR U* uref kv z1 z0 Cμ
Height of building (m) Width of road (m) Turbulent kinetic energy (m2 /s2 ) Dissipation rate (m2 /s3 ) Street canyon aspect ratio Free stream velocity (m/s) Friction velocity (m/s) Von Karman’s constant Height of building (m) Surface roughness length (m) Turbulent viscosity constant
References 1. Wang Q, Fang W, de Richter R, Peng C, Ming T (2019) Effect of moving vehicles on pollutant dispersion in street canyon by using dynamic mesh updating method. J Wind Eng Ind Aerodyn 187:15–25 2. Olivardia Gonzalez FG, Zhang Q, Matsuo T, Shimadera H, Kondo A (2019) Analysis of pollutant dispersion in a realistic urban street canyon using coupled CFD and chemical reaction modeling. Atmosphere 10(9):479 3. Longo R, Bellemans A, Derudi M, Parente A (2020) A multi-fidelity framework for the estimation of the turbulent Schmidt number in the simulation of atmospheric dispersion. Build Environ 185:107066 4. Ming T, Fang W, Peng C, Cai C, De Richter R, Ahmadi MH, Wen Y (2018) Impacts of traffic tidal flow on pollutant dispersion in a non-uniform urban street canyon. Atmosphere 9(3):82 5. Chan AT, Au WT, So ES (2003) Strategic guidelines for street canyon geometry to achieve sustainable street air quality—part II: multiple canopies and canyons. Atmos Environ 37(20):2761–2772 6. Singh AP (2013) Vehicular pollutants dispersion modelling analysis in street canyons using computational fluid dynamics (CFD). Int J Eng Res 02(06) 7. Madalozzo DM, Braun AL, Awruch AM (2013) Pollutant dispersion simulation in street canyons using the finite element method and shared memory parallelization. Mecánica Computacional 32(15):1271–1295 8. Salim SM, Ong KC (2013) Performance of RANS, URANS and LES in the prediction of airflow and pollutant dispersion. In: IAENG transactions on engineering technologies. Springer, Dordrecht, pp 263–274 9. Abu-Zidan Y, Mendis P, Gunawardena T (2020) Impact of atmospheric boundary layer inhomogeneity in CFD simulations of tall buildings. Heliyon 6(7):04274 10. Snyder WH (1981) Guideline for fluid modeling of atmospheric diffusion. Environmental Sciences Research Laboratory, US Environmental Protection Agency, 81(9) 11. Afiq WMY, Azwadi CN, Saqr KM (2012) Effects of buildings aspect ratio, wind speed and wind direction on flow structure and pollutant dispersion in symmetric street canyons: a review. Int J Mech Mater Eng 7(2):158–165 12. Richards PJ, Hoxey RP (1993) Appropriate boundary conditions for computational wind engineering models using the k- ∊ turbulence model. J Wind Eng Ind Aerodyn 46:145–153
Impact of Building Configurations on Fluid Flow in an Urban Street Canyon
823
13. Kim J, Baik J (2004) A numerical study of the effects of ambient wind direction on flow and dispersion in urban street canyons using the RNG k–ε turbulence model. Atmos Environ 38(19):3039–3048
Experimental and Theoretical Analysis of Flow-Induced Vibration of Cantilevered Flexible Plate Shubham Giri, V. Kartik, Amit Agrawal, and Rajneesh Bhardwaj
1 Introduction Flow-induced vibration (FIV) of the flexible cantilever plate has served as a fundamental problem to understand flag flapping [1], aircraft wing flutter [2], plant biomechanics [3], biolocomotion [4], etc. Further, the vibration energy can be utilized to harvest kinetic energy of ambient fluid flow, to enhance heat transfer and mixing in a process industry [1]. Fluid flow over the thin flexible plate installed to the lee side of a rigidly fixed cylinder causes flow-induced vibration of the plate. The early focus of the researchers was to find the critical velocity and frequency of flutter instability of the flexible plate, experimentally and mathematical modeling [1, 2, 5, 6]. Mass ratio, Ms = ρρfs hL and bending stiffness, K B = ρ f UE I2 L 3 are the two important parameters to find the instability boundary of the standard and inverted flag or the flag behind the bluff body as summarized by Yu et al. [1], where ρs , ρ f , h, L, EI, and U are solid density, fluid density, plate thickness, plate length, flexural rigidity, and free stream velocity, respectively. Hysteretic behavior [6], several flapping states [7, 8], and the effect of aspect ratios on instability [6, 8] were demonstrated. Shukla et al. [9] investigated the dynamical behavior of the flexible splitter plate (in post-critical regime) attached at the lee side of a cylinder in water flow, experimentally. Kumar et al. [10] conducted an experiment to study the structural and flow dynamics of the flexible plate attached to a streamlined body subjected to fluid flow. A linear or nonlinear beam model is used for the analytical study of the structure [5] and a potential flow or vortex shedding model is considered to model fluid [2, 5, 8]. These models are not very economical and not easy to implement. The effect mass ratio on the dynamics of the post-critical behavior of cantilevered flexible plates is not explored in detail to date to the best of our knowledge. Therefore, we investigate the dynamics of a cantilevered flexible S. Giri · V. Kartik · A. Agrawal · R. Bhardwaj (B) 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_74
825
826
S. Giri et al.
plate attached to the lee side of a cylinder in airflow experimentally, varying mass ratios. Further, we compare the present with previous works having different mass ratios. A new reduced-order model is developed to explain the post-critical dynamics of the plate using a wake oscillator for fluid as used for VIV of cylinder [11] and vibration of the rotating blade [12].
2 Methodology 2.1 Experimental Setup The experiments were carried out in a low-speed wind tunnel at IIT Bombay. Length and cross-section area of the test section of the wind tunnel are 1.2 m and 0.5 × 0.5 m2 , respectively. The wind speed (U) can be varied in the range of 2–30 m/s and turbulence intensity is lesser than 1.5%. Airspeed is measured using a manometer and anemometer (Benetech Inc., GM8903, resolution 0.001 m/s). The front view and the top view of the test section along with the plate installed to the lee side of the cylinder are shown in Fig. 1a, b, respectively. The deflection of the plate is measured at the middle of the plate by a laser displacement sensor (Micro-Epsilon Inc., ILD1420-500, repeatability 20–40 μm). The plates are made of polyvinyl chloride (PVC) sheets of a thickness (h) of 0.2 mm and width (W ) of 5D, where D (26.7 mm) is the diameter of the cylinder as shown in Fig. 1. The density of the sheet is calculated (measuring mass and volume) as 1210 ± 17 kg/m3 . Young modulus of the sheet is measured as 1.105 ± 0.053 GPa by a universal testing machine (Instron, 5 kN load cell). The plates of four different lengths (L) and their mass ratios (Ms ) are shown in Table 1. The experimental first (Exp.-fn1 ) and second (Exp.-fn2 ) natural frequencies are measured by an impact hammer test and theoretical first (Theo.-fn1 ) and second (Theo.-fn2 ) natural frequencies are calculated based on the assumption of the Euler–Bernoulli beam (values are provided in Table 1).
2.2 Wake Oscillator Model We assume a two-dimensional small deflection of the plate. Therefore, the plate is modeled as an Euler–Bernoulli cantilever beam [6, 13]. The governing equation of motion of the plate in the transverse direction (y direction), v ∗ (x ∗ , t ∗ ) is given below. m
∂ 2 v∗ ∂v ∗ ∂ 4v∗ + r + E I = S∗, ∂t ∗2 ∂t ∗ ∂ x ∗4
(1)
where, v ∗ , x ∗ and t ∗ are deflection, x-direction coordinate, and time, respectively. The mass per unit surface area of the plate, m includes mass of the structure/plate
Experimental and Theoretical Analysis of Flow-Induced Vibration …
827
Fig. 1 a Front view and b top view of present setup in test section of the wind tunnel
Table 1 Natural frequencies (in Hz) of plates Plate
L
MS
Exp.-fn1
Theo.-fn1
Exp.-fn2
Theo.-fn2
P1
4D
1.88
2.93 ± 0
2.71
21 ± 0
16.96
P2
5D
1.51
2.20 ± 0.41
1.73
13.43 0.36
10.86
P3
6D
1.26
1.39 ± 0.14
1.20
10.10 ± 0.21
7.54
P4
7D
1.08
0.95 ± 0.10
0.88
6.89 ± 0.11
5.54
(m s ) and the fluid-added mass (m f ), hence, m = m s + m f = ρs t + C M ρ f t. ρs and ρ f are the density of the plate and fluid, respectively. C M refers a fluid-added mass coefficient (considered as 1 [12]). The total damping coefficient is represented by r , r = rs + r f , where rs is the structural damping and r f is the fluid-added damping. The flexural rigidity per span-wise length of the plate is E I . S ∗ = 21 ρ f U 2 C L is fluid force per unit surface area on the plate, where C L is the lift coefficient of the plate. We use the van der Pol oscillator to model the fluctuating lift on the plate as ) ∂q ( ∂ 2q + εΩ f q 2 − 1 ∗ + Ω2f q = F ∗ , ∗2 ∂t ∂t
(2)
where, q(x ∗ , t ∗ ) is the dimensionless wake parameter. It is depicted as q = 2C L /C L0 , where C L0 is the reference lift coefficient. ε refers the damping coefficient of the van der Pol oscillator. The frequency of the lift parameter due to flow is Ω f = 2π St L U/L, where St L is Strouhal number based on the characteristic length L. F is expressed as
828
S. Giri et al.
acceleration coupling as suggested by Facchinetti et al. [11] and Hoskoti et al. [12]. 2 ∗ F ∗ = AL0 ∂∂tv∗2 , where A0 is the coupling parameter. ∗ Nondimensionalizing the Eqs. 1 and 2 considering following variables v = vL , / ∗ ∗ ∗ rs +r f x = xL , t = t ∗ Ω f , ζ = mΩ , ωs0 = mΩE2IL 4 , S = mΩS 2 L , and F = FΩ2L = f f
f
f
A0 ∂ 2 v , Ω2f ∂t 2
two coupled dimensionless equations (plate and vortex equations) needed to be solved, are the following:
1
4 ∂ 2v ∂v 2 ∂ v + ω + ζ = Mq s0 ∂t 2 ∂t ∂x4
(3)
( 2 ) ∂q ∂ 2v ∂ 2q + q = A + ε q − 1 , ∂t 2 ∂t ∂t 2
(4)
ρ U 2C q
f L0 where S = 4 mΩ = Mq, M = 16πC2 StL02 μL , μ = ρ fmL 2 , and A = ΩA20 . 2 fL L f In modal analysis, the deflection of the plate is written as v(x, t) = Σi=N i=1 ηi (t)φi (x), where N is the total number of modes. ηi (t) and φi (x) are the generalized coordinates and eigenfunctions of the plate, respectively, satisfying the following boundary conditions.
v(0, t) = v ' (0, t) = 0 at x = 0; (fixed end) v '' (1, t) = v ''' (1, t) = 0 at x = 1; (fixed end)
(5)
The eigenfunctions of the cantilever beam is obtained as (Hodges and Pierce [14]) i )+cos(αi ) φi (x) = cosh(αi x) − cos(αi x) − cosh(α [sinh(αi x) − sin(αi x)], where αi are sinh(αi )+sin(αi ) the solutions of the equation, cos αi cosh αi + 1 = 0. Similarly, wake parameter q is presented as the sum Σ of generalized coordinates, Hk (t), and the modal functions, ψk (x), as q(x, t) = k=N k=1 Hk (t)ψk (x). The modal function, ψk (x) is considered as ψk (x) = sin(kπx) that satisfies orthogonality condition [12]. Substituting these in Eqs. (3) and (4), and using orthogonal property of the modal functions, the following coupled equations are obtained after integrating along the length of the plate. l=N Σ d2 η j dη j 2 4 + ω + ζ α η = M Hl I jl1 s0 j j dt 2 dt l=1 ⎛ ⎞ p=N q=N s=N ΣΣΣ d2 Hl dH dH s l 2 ⎠ + Hl I pqsl + ε⎝2 H p Hq − dt 2 dt dt p=1 q=1 s=1
= 2A
j=N Σ j=1
d ηj , dt 2 2
I jl1
(6)
(7)
Experimental and Theoretical Analysis of Flow-Induced Vibration … Table 2 Model parameters
829
Parameters
Values
Reference
ρf
1.2 kg/m3
Experiment
ρs
1210 kg/m3
Experiment
D
26.7 mm
Experiment
L
4D, 5D, 6D, 7D
Experiment
W
5D
Experiment
H
0.2 mm
Experiment
E
1.1 GPa
Experiment
U
2–30 m/s
Experiment
St L
0.2
Experiment
C L0
0.6
Reference value
ε
0.3
Ref. [11]
A
12
Ref. [11]
ζ
0.7
Reference value
∫1 ∫1 2 where I jl1 = 0 φ j (x)ψl (x) dx and I pqsl = 0 ψ p (x)ψq (x)ψs (x)ψl (x)dx are the coupling integrals. Further, we consider single-mode approximation. As the plate shows mode II vibration in the present experiment bypassing the mode I (also reported in [10]), single-mode II is used in the model (i.e., all index in the Eqs. (6) and (7) are 2). MATLAB ode45 function that is based on Runge–Kutta method, is used to solve the Eqs. (6) and (7) numerically. Initial conditions to solve coupled equations are given below. η2 (t = 0) = 0.005, H2 (t = 0) = 0,
∂η2 (t = 0) = 0, ∂t ∂ H2 (t = 0) = 0 ∂t
(8)
The value of the model parameters to solve the aforementioned coupled Eqs. (6) and (7) are provided in the Table 2.
3 Results and Discussion Amplitude and frequency response with varying wind velocity for plate, P2 are shown in Fig. 2. Three sets of reading (for 15 s) are taken and the average root mean square amplitude (yrms ) and frequency ( f ) are indicated in Fig. 2. The standard deviation of the yrms and f are shown by an error bar at the corresponding mean value. As per the motion of the plate, three regimes, namely (I) pre-critical, (II) transition, and (III) post-critical, are identified. The motion of the plate in the pre-critical regime is aperiodic. The deflection, fast-Fourier transform (FFT), and phase portrait (for
830
S. Giri et al.
Fig. 2 Variation of RMS of deflection and flutter frequency with velocity for the plate P2. Yellow (I), blue (II), and green (III)-shaded regions indicate pre-critical, transition, and post-critical flutter regimes
data of 4 s) are shown in Fig. 3a–c, respectively. Due to aperiodic motion, many frequencies are found in the FFT (dominating frequency is considered in the Fig. 2); hence, the phase portrait is quite irregular. In this pre-critical regime, highlighted in the yellow shaded region in Fig. 2, the error bar of the frequency plot is found to be high because of different dominating frequencies in different sets of data. The motion of the plate in the transition region, highlighted in the blue-shaded region in Fig. 2, fluctuates between aperiodic and periodic motion as indicated in Fig. 4a. FFT plot in Fig. 4b depicts one dominating frequency along with other small frequencies. Therefore, the error bar of the frequency plot is small and the error bar of the amplitude plot is high in the transition regime as shown in Fig. 2. The phase plane in Fig. 4c is shifting from irregular to ordered shape. In the post-critical region, highlighted in the green-shaded region in Fig. 2, deflection signal is periodic as indicated in Fig. 5a. Therefore, one dominating frequency is found in FFT plot in Fig. 5b. Figure 5c depicts the periodic self-sustained limit cycle oscillation of the plate. Hence, the error bar of amplitude and frequency response in this regime is very small as shown in Fig. 2. Dimensionless amplitude and frequency response of all plates and comparison with previous studies are presented in Fig. 6. yrms and f are nondimensionalized as √ A∗ = 2yrms /L and f / f n2 , respectively. The amplitude and frequency response of plates, P3 and P4, are similar to plate P2, while plate P1 does not reach to the periodic limit cycle oscillation. Plate P1 shows the irregular motion as observed in the pre-critical regime. This is because of the higher bending rigidity due to the smaller length of the plate. Therefore, the amplitude ( A∗ ) is very small for the plate P1 (L = 4D) as shown in Fig. 6a. The dominating frequency for this case (L = 4D) is also small compared with other plates as indicated in Fig. 6b. It is observed that the dimensionless amplitude shows a collapse of the results and the periodic limit cycle oscillation (synchronized vibration) starts in the range of U R = 2.5 − 4 for L = 5D, 6D, and 7D cases as shown in Fig. 6a. The values of A∗ for Kumar et al. [10] are higher than the present experiments. This is because amplitudes are measured
Experimental and Theoretical Analysis of Flow-Induced Vibration …
831
Fig. 3 a Time-varying deflection signal, b FFT of the signal and c phase portrait of the plate, P2 at air velocity of 3.6 m/s (pre-critical regime)
at the tip of the plate by Kumar et al. [10], while for the present cases, amplitude is measured at L laser = 0.5L. There is also a collapse of the frequency response at the starting of the mode II synchronized oscillation for L = 5D, 6D, and 7D cases and Kumar et al. [10] as indicated in the Fig. 6b. However, after U R around 7.4, Kumar et al. observed mode III synchronized oscillation (jump in the frequency response), while the present experiments are in mode II synchronized oscillation. It is observed that for all these cases (L = 5D, 6D, and 7D and Kumar et al. [10]) mode II synchronized oscillation starts around f / f n2 ≈ 1 that is around second mode natural frequency. In the present cases, the plate is at the lee side of the cylinder (L/D ratio varies from 4D−7D), while Kumar et al. attached the plate to streamlined airfoil support. However, the response is quite similar in dimensionless parameters space as discussed above. On the other hand, Shukla et al. [9] installed a plate in the wake of a cylinder (same arrangement as the present study) in a water tunnel and their response is also presented in Fig. 6 for L/D = 5. From the results of Shukla et al. [9], it is inferred that mode II synchronized oscillation started at U R ≈ 0.5 (considering the density of the plate is 1000 kg/m3 ). This is due to the low flexural rigidity (EI, order of 10−6 N-m per unit span) and
832
S. Giri et al.
Fig. 4 a Time-varying deflection signal, b FFT of the signal and c phase portrait of the plate, P2 at air velocity of 6 m/s (transition regime)
mass ratio (order of 10−4 ). The flexural rigidity of the present experiment and Kumar et al. [10] are in the order of 10−4 N-m and 10−3 N-m, respectively. The mass ratio for the present cases is mentioned in Table 1 which is in the same order (Ms = 1.4) as Kumar et al. [10]. Further, the slope of the frequency response for synchronized mode II oscillation is increased with a decrease in a mass ratio for L = 5D, 6D, and 7D and the slope is maximum for Shukla et al. [9] which has a very small mass ratio as indicated in Fig. 6b. It is very interesting to note that a similar increase in the slope of frequency response with a decrease in a mass ratio for the VIV of a cylinder in the lock-in regime (synchronized oscillation) is reported by Govardhan and Williamson [15]. Hence, it can be deduced that the FIV of the plate shows the VIV of cylinder-like characteristics for varying mass ratios. It is noted that there is a small discrepancy as the slope of the frequency response of Kumar et al. [10] looks higher than the present P3 and P4 plates, although the mass ratio (Ms = 1.4) of Kumar et al. [10] is higher than the P3 and P4 plates. This may be because of the nonlinearity in the material as the frequency response of Kumar et al. [10] is quite nonlinear compared with the present experiment.
Experimental and Theoretical Analysis of Flow-Induced Vibration …
833
Fig. 5 a Time-varying deflection signal, b FFT of the signal, and c phase portrait of the plate, P2 at air velocity of 9.7 m/s (post-critical regime)
Present reduced-order wake oscillator model (WOM) was first verified with the WOM of Facchinetti et al. [11] and Hoskoti et al. [12]. We attempted to extend the WOM to the FIV of the plate problem as detailed earlier. Considering singlemode approximation similar to Hoskoti et al. [12] and using the value of the model parameters (in Table 2), WOM is run for the present experiments. Figure 7a, b depicts the comparison of the amplitude and frequency response, respectively between WOM and the experiment for plate P2. The amplitude obtained from the WOM overestimated the experiment and the peak is found at a smaller UR than the experiment in Fig. 7a. This is because the natural frequencies of the Euler–Bernoulli beam model are smaller than the experiment as mentioned in Table 1. Hence, the synchronized frequency (same structure and fluid frequency) of WOM is smaller than the experiment in Fig. 7b. It is noted that to nondimensionalize the frequency for model and experiment in Fig. 7b, the second natural frequency of Euler–Bernoulli beam vibration is considered. Therefore, the lock-in of experiment started around ffn = 1.2 rather than ffn = 1. The discrepancy in the natural frequency of the model can be overcome considering the added stiffness due to prestress and lengthwise tension of the plate. Despite this, it is interesting to
834
S. Giri et al.
Fig. 6 Dimensionless a amplitude and b frequency response with reduced velocity for the present plates and comparison with previous studies (Shukla et al. [9] and Kumar et al. [10])
note that this simple and very economical WOM can predict the qualitative behavior of the FIV of the plate. The mode shape of the plate for different instances of time obtained from the WOM at U R = 9.66 is presented in Fig. 7c. Because of the single-mode II approximation, the bending pure mode II oscillation of the plate is observed. However, Kumar et al. [10] and Tang and Paidoussis [5] found the neck formation unlike the present node formation around 0.8L from the root. It suggests the contribution of the other modes needed to consider. Hence, the consideration of the multi modes in WOM can further improve the prediction of the model. Note that, the WOM model is also run for other plates P3, P4, and Kumar et al. [10] and the same qualitative matching of amplitude and frequency response is observed.
Experimental and Theoretical Analysis of Flow-Induced Vibration …
835
Fig. 7 Comparison of a amplitude and b frequency response between wake oscillator model and experiment for plate P2 c mode shape for different instances of time obtained from WOM at U R = 9.66
4 Conclusions The flow-induced vibrational dynamics of a flexible plate attached to the lee side of a circular cylinder in fluid flow is investigated with experiments and wake oscillator model. Three regimes, i.e., pre-critical, transition, and post-critical regimes are observed in the experiment. In the post-critical regime, we found self-sustained limit cycle oscillation of the plate which is known as the synchronized vibration or lock-in regime. The varying length of the plate, hence varying mass ratio exhibits VIV of the cylinder-like characteristics in the lock-in regime. The present new WOM for the
836
S. Giri et al.
plate based on single-mode approximation can predict the amplitude and frequency response in the post-critical regime. The consideration of multi modes of the plate in the WOM may be a potential future work to improve the model response further. Acknowledgements We gratefully acknowledge the financial support of ‘Prof. Yashoda Apte Research Award’ (YARA) from Dean Alumni & Corporate Relations (Dean ACR) office, Indian Institute of Technology Bombay, Mumbai, India.
Nomenclature L, W, h U ρ f , ρs E D f ni v∗, v t ∗, t x ∗, x q ms , m f m CM rs , r f r C L , C L0 Ωf St L S∗, S F ∗, F M, μ ωs0 A0 , A ε η, H φ, ψ i, j, k, l, p, q, s αi yrms f Ms KB A*
Length, span, thickness of the plate (mm) Free stream velocity (m/s) Density of fluid, solid (kg/m3 ) Young modulus of solid (GPa) Diameter of the cylinder [mm) Ith natural frequency of the plate (Hz) Dimensional, dimensionless deflection (m) Dimensional, dimensionless time (s) Dimensional, dimensionless x-axis (m) Lift parameter Mass of solid, added mass of fluid (kg/m2 ) Sum of solid mass and added mass of fluid (kg/m2 ) Fluid-added mass coefficient Structural damping, fluid-added damping (kg/m2 s) Sum of structural and fluid damping (kg/m2 s) Lift coefficient of the plate, reference lift Frequency of the lift (rad/s] Strouhal number based on lift frequency Dimensional, dimensionless fluid forcing (Pa) Dimensional, dimensionless solid forcing (s−2 ) Mass number, mass parameter (m−1 ) Dimensionless solid frequency parameter Different forms of coupling parameter (m−1 , m−1 s2 ) Damping of van der Pol oscillator (s−1 ) Generalized coordinate of deflection, lift Eigenfunctions of deflection, lift Indices Constant of ith mode R.M.S. of the deflection of the plate (mm) Frequency of plate vibration (Hz) Mass ratio Bending stiffness Dimensionless amplitude
Experimental and Theoretical Analysis of Flow-Induced Vibration …
UR
837
Reduced velocity
References 1. Yu Y, Liu Y, Amandolese X (2019) A review on fluid-induced flag vibrations. Appl Mech Rev 71(1) 2. Tang DM, Yamamoto H, Dowell EH (2003) Flutter and limit cycle oscillations of twodimensional panels in three-dimensional axial flow. J Fluids Struct 17(2):225–242 3. Shelley MJ, Zhang J (2011) Flapping and bending bodies interacting with fluid flows. Annu Rev Fluid Mech 43(1):449–465 4. Pa¨ıdoussis MP (2016) Plates in axial flow. In: Pa¨ıdoussis MP (ed) Fluid-structure interactions, 2nd edn, Chapt 6. Academic Press, Oxford, pp 539–641 5. Tang L, Pa¨ıdoussis MP (2007) On the instability and the post-critical behaviour of twodimensional cantilevered flexible plates in axial flow. J Sound Vibr 305(1–2):97–115 6. Eloy C, Lagrange R, Souilliez C, Schouveiler L (2008) Aeroelastic instability of cantilevered flexible plates in uniform flow. J Fluid Mech 611:97–106 7. Alben S, Shelley MJ (2008) Flapping states of a flag in an inviscid fluid: bistability and the transition to chaos. Phys Rev Lett 100(7):074301 8. Zhao W, Pa¨ıdoussis MP, Tang L, Liu M, Jiang J (2012) Theoretical and experimental investigations of the dynamics of cantilevered flexible plates subjected to axial flow. J Sound Vibr 331(3):575–587 9. Shukla S, Govardhan RN, Arakeri JH (2013) Dynamics of a flexible splitter plate in the wake of a circular cylinder. J Fluids Struct 41:127–134 10. Kumar D, Arekar AN, Poddar K (2021) The dynamics of flow-induced flutter of a thin flexible sheet. Phys Fluids 33(3):034131 11. Facchinetti ML, Langre ED, Biolley F (2004) Coupling of structure and wake oscillators in vortex-induced vibrations. J Fluids Struct 19(2):123–140 12. Hoskoti L, Misra A, Sucheendran MM (2018) Frequency lock-in during vortex induced vibration of a rotating blade. J Fluids Struct 80:145–164 13. Guo CQ, Pa¨ıdoussis MP (2000) Stability of rectangular plates with free side-edges in twodimensional inviscid channel flow. J Appl Mech 67(1):171–176 14. Hodges DH, Pierce GA (2011) Introduction to structural dynamics and aeroelasticity, vol 15. Cambridge university press 15. Govardhan R, Williamson CHK (2000) Modes of vortex formation and frequency response of a freely vibrating cylinder. J Fluid Mech 420:85–130
Flow-Induced Vibration of an Elastically Mounted Cylinder Under the Influence of Downstream Stationary Cylinder Abhishek Thakur, Atul Sharma, and Sandip K. Saha
1 Introduction Flow-induced vibrations (FIV) of structures has been widely studied for suppression of structural vibrations as well as its enhancement for hydro-power extraction from flowing fluids in nature. Distinct flow-induced motion (FIMs)-based energy harvesters, such as vortex-induced vibration (VIV), translation and torsional galloping, buffeting, aerofoil flutter, and auto-rotation along with corresponding benchmark technologies have been consolidated and reviewed by Armandei and Rostami [3]. The flow interference between multiple structures can suppress as well as enhance the FIV. Therefore, fluid multi-structure interaction (FmSI) has been widely exploited to enhance FIV and associated energy extraction from FIV converters [4]. Vortex-induced vibration (VIV) is oscillation of an elastically mounted structure excited by aerodynamic forces which are generated by alternate vortex shedding from the flow past a structure. During VIV, large oscillation amplitude A* prevails when oscillation frequency f y∗ synchronizes with vortex shedding frequency [15]. The phenomenon is called lock-in and associated range of reduced velocity (U ∗ ) is called lock-in range. The mechanism of phase φ jump for VIV of an isolated cylinder has been numerically investigated by Prasanth and Mittal [9] at low Re = 100. The authors found that the phase difference φ between displacement and lift coefficient is solely caused by change of orientation and relative magnitude of three pressure regions: front stagnation, top, and bottom suction region, irrespective of mode of vortex shedding pattern. Bokaian and Geoola [2] experimentally examined the fluid elastic instability of an elastically mounted cylinder in wake of stationary upstream cylinder. They found large amplitude oscillations reminiscent of galloping and called them as wake induced galloping (WIG). Further, they reported that the WIG to be completely different A. Thakur (B) · A. Sharma · S. K. Saha Department of Mechanical and Industrial 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 5, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-6074-3_75
839
840
A. Thakur et al.
from classical galloping of sharp-edged bodies. In general, the galloping amplitude response of the non-axisymmetric bodies consists of large amplitude A∗ , low frequency f y∗ , and in-phase oscillation with a monotonous increase of A∗ with U ∗ . Further, an experimental investigation of FIV of elastically mounted cylinder with an identical stationary cylinder downstream is presented by Bokaian and Geoola [1]. They found a large amplitude vibration for smaller gap spacing (G ∗ ≤ 0.75) and called the vibration response as proximity-induced galloping (PIG). Borazjani and Sotirpolous [5] have numerically studied the FIV of two circular cylinders in tandem arrangement at Re = 200 and G ∗ = 0.5. They found modification of vibration response of two cylinders at onset of flow in the gap between two cylinders. Mittal and Sharma [11] have studied PIG of an elastically mounted cylinder by placing a flexible plate downstream. The authors found rapid-increase plateau type amplitude variation with U ∗ for G ∗ ≤ 0.3. Various parameters involved in the WIG vibration such as Re, G ∗ , U ∗ , and diameter ratio have been extensively studied by numerous authors. The intrinsic nature of flow dynamics responsible for PIG in experimental investigation of Bokaian and Geoola [1] is not found in the literature. In this paper, numerical investigation of FIV of an elastically mounted cylinder with an identical downstream cylinder has been presented in laminar flow regime at Re = 100. The effect of gap spacing G ∗ and reduced velocity U ∗ on vibration characteristics and flow dynamics is discussed.
2 Computational Model 2.1 Problem Description Our computational domain consists of an elastically mounted upstream cylinder with an identical stationary downstream cylinder in a free stream flow, as shown in Fig. 1. The diameter D and free stream velocity at inlet U ∞ are considered as length and velocity scales. The G* is non-dimensional distance between the trailing and leading edge of upstream and downstream cylinder, respectively. The two cylinders are arranged in tandem arrangement. The size of domain and location of outer boundaries w.r.t. to the centre of upstream cylinder are shown in the Fig. 1. The boundary conditions are also presented in the Fig. 1.
2.2 Governing Equation and Numerical Method The current FmSI problem involves modelling of fluid dynamics, transverse oscillation of the elastically mounted cylinder, and fluid–structure interface dynamics. The fluid flow is governed by the Navier–Stokes equations, the cylinder motion is modelled using a linear oscillator equation, and interface boundary condition (BC)
Flow-Induced Vibration of an Elastically Mounted Cylinder Under …
841
Fig. 1 Computational setup for free stream flow across an elastically mounted cylinder in proximity of a downstream stationary cylinder
governs the fluid–structure interface dynamics. The non-dimensional form of the governing equations and the interface BCs are given as ∇.U ∗ = 0
(1)
∂ U ∗ ∗ U ∗ = −∇ P ∗ + 1 ∇ 2 U ∗ , U + ∇. (2) ∂t ∗ Re where U * = (u/u∞ ), P ∗ = p/ρ f u 2∞ , and Re = ρ f U∞ D/μ are the nondimensional velocity, pressure, and Reynolds number, respectively. Here, u, p, ρ f and μ are the velocity, pressure, fluid density, and viscosity, respectively. The linear spring mass-damper oscillator equation for FIV is as follows. y¨ ∗ +
4π ζ ∗ y˙ + U∗
2π U∗
2
y∗ =
2Cl , π m∗
(3)
where y¨ ∗ , y˙ ∗ , and y ∗ represent the non-dimensional acceleration, velocity, and displacement, respectively. The C l = F y /(2ρ f u∞ D) is lift coefficient and t ∗ = tU ∞ / D is the non-dimensional time, where, F y is the transverse force acting on cylinder. At the fluid–structure interface, the solid and fluid velocity and stress tensor should be continuous, given as dd ∗ ˆ = U f and σs .nˆ = σ f .n, dt
(4)
where d * and U f is non-dimensional displacement of structure and fluid velocity at interface. Further, σ f and σ s are Cauchy’s stress tensor in fluid and structure domain at the interface and nˆ is the unit normal vector at interface. The present FmSI problem is solved using an in-house code based on Level-Set function-based Immersed Interface Method (LS-IIM), proposed by Thekkethil and
842
A. Thakur et al.
Sharma [8]. The LS-IIM is a hybrid Lagrangian–Eulerian (HLE) method, based on a block-iterative partitioned approach [13]. It is an immersed boundary method and is capable to solve fluid–structure interactions with minimal computational resources than body-fitted methods. A physical-law and discrete math-based finite-volume method (FVM) is used to solve the fluid solver [10], whereas a finite-difference method (FDM) is used to solve the equation of motion for a cylinder. The advection term is discretized using the Quadratic Upwind Interpolation for Convective Kinetics (QUICK) scheme, while the diffusion term is discretized using a first-order central difference scheme. The details of present LS-IIM in-house FSI code can be found in the work of Thekkethil and Sharma [8, 13]
2.3 Grid Generation and Independence Study The 2-D non-dimensional incompressible Navier–Stokes equation is solved numerically on a non-uniform Cartesian grid. The grid over complete computational domain is shown in Fig. 2. The mesh consists of a uniform fine grid δfine near the structure, a uniform coarse grid near the outer boundaries, and hyperbolic stretching function-based non-uniform grid in the intermediate region is connected by a hyperbolic stretching function. The hyperbolic stretching region is indicated as the region between blue and red lines in Fig. 2. Vibrations with larger amplitude A* and higher flow field gradients are observed for the case of G∗ = 0.5 and U * = 10. Hence, this case is chosen to grid independence study. Grid independence was carried out on four different Cartesian grids M 1 , M 2 , M 3 , and M 4 with δfine of 0.04, 0.03, 0.025, and 0.015, respectively. Table 1 presents the results on the various grids, where the relative deviation of variable from grid of finest δfine decreases as δfine is decreased. The values of output variables of mesh M3 and M4 are in a good agreement; the largest deviation of 1.34% is observed for Fig. 2 Non-uniform Cartesian mesh distribution throughout the computational domain. The inset figure presents enlarged view of mesh around cylinders
Flow-Induced Vibration of an Elastically Mounted Cylinder Under …
843
Table 1 Details of grid independence study conducted at Re = 100, m∗ = 2, ζ = 0.005 for gap spacing of G* = 0.5, and reduced velocity U * = 10 Mesh
dXfine
A∗
f ∗
CL,rms
CD,mean
M1
0.04
0.866
0.795
0.2935
1.405
M2
0.03
0.885
0.791
0.296
1.437
M3
0.025
0.899
0.807
0.302
1.466
M4
0.015
0.906
0.814
0.305
1.486
C D,mean . Therefore, the grid size M3(391 × 270) with the δfine = 0.025, δt ∗ = 0.005 is selected for all simulations in the present study.
2.4 Validation The present (LS-IIM) in-house code has been validated for several benchmark problems of FSI by Thekkethil and Sharma [8] for flexible structures. Thekkethil et al. [14] also validated and used it for study of rigid hydrofoils with prescribed fish-like kinematics. Further, the in-house code has been successfully validated and used for FmSI of a fixed or elastically mounted cylinder with flexible splitter plate in the work of Mittal and Sharma [7, 11]. In present study, the Fig. 3 shows a validation by the present code for VIV of an isolated circular cylinder at Re = 150, m∗ = 2, and ζ = 0. Figure 3 shows the comparison of present amplitude response and root mean square of lift coefficient C L of circular cylinder with those of Bao et al. [16], Borazjani and Sotirpolous [5] and Zhao [17]. The results of our numerical simulation of Ay and C L,rms fall more closer to result of Bao et al. [16]. The results of Borazjani and Sotirpolous [5] show the smallest amplitude. A maximum percentage error of 5.4% occurred for U * = 7. The variation of results with U * is similar for all the studies shown. The lock-in starts at U * = 3, amplitude suddenly jumps to its maximum at U * = 4 followed by a decrease till end of lock-in at U * = 8.
3 Results and Discussion 3.1 Vibration Characteristics The scope of present investigation is to study the influence of proximity of downstream cylinder (G* ) and stiffness of elastic-mounting (U * ) of upstream cylinder on FIV of upstream cylinder and associated flow dynamics in the gap. The respective effects are studied with various non-dimensional gap ratios of G* = 0.1, 0.3, 0.5, 1.0, 1.5, 2.0, 2.5, and reduced velocity U * = 3 − 20. The study is
844
A. Thakur et al.
Fig. 3 Amplitude response Ay a and root mean square of lift coefficient C L b of VIV of an isolated cylinder at Re = 150, m∗ = 2 and ζ = 0
carried out at a constant Reynolds number Re = 100, mass ratio of cylinder m∗ = ms /md = 2.0, and damping ratio ζ = 0.005. Here, ms and md are mass of oscillating cylinder and fluid displaced by cylinder, respectively. In present study, the variation of non-dimensional natural frequency f n * of oscillator leads to variation in reduced velocity U * . Further, the reduced velocity is defined as U * = U ∞ /(D × f n * ). Figure 4 presents vibration amplitude A∗ , non-dimensional frequency f ∗ = f y∗ / f n∗ , and phase φ (between displacement y* and lift coefficient C l ) with increasing reduced velocity U * . For larger gap G* ≥ 1.5, Fig. 4a shows an increasing–decreasing followed by asymptotic valley trend of variation of A* , called as vortex-induced vibration (VIV) type of amplitude response. However, with decreasing G* up to 1.5, the figure shows an increase in both maximum amplitude (at U ∗ = 5) and asymptotic value of A* at larger U * . Thus, for G* ≥ 1.5, the proximity of stationary cylinder leads to larger amplitude-based VIV type of response when compared with isolated cylinder. For the larger G* ≥ 1.5, Fig. 4b shows an increasing trend of variation for frequency f * , and phase jumps from in-phase (0°) to anti phase (180°). The vibration response for G * ≥ 1.5 is called as VIV type of response. At G* = 1.0, Fig. 4a shows that the amplitude A* and f * deviates from the VIV type of response, with an increasing followed by a slightly decreasing A* ; instead of sharply decreasing trend at larger G* ≥ 1.5. For G* = 1.0 as seen in Fig. 4b for frequency, a smaller f * is observed when compared with large G* (> 1.0). Finally, for the intermediate gap G* = 1.0, Fig. 4c shows the in-to-anti phase-jump at larger U * = 8 when compared with that for larger G* ≥ 1.5. It is evident from vibration characteristics in Fig. 4 that the gap ratio G* = 1.0 corresponds to a transition between two distinct vibration responses: VIV and proximity-induced galloping (PIG) presented below. For G* = 0.5, Fig. 4a shows a rapid-increase plateau type of variation for A* , Fig. 4b shows a plateau type of variation of f * , and Fig. 4c shows in-phase (φ) variation for phase between the displacement y∗ and lift coefficient C l . The vibration
Flow-Induced Vibration of an Elastically Mounted Cylinder Under …
845
Fig. 4 Effect of non-dimensional gap G* and reduced velocity U * on a amplitude A* , b frequency f * , and c phase ϕ between displacement y* and lift coefficient C l
response is known as proximity-induced galloping (PIG) [1]. This PIG type of amplitude vibration response of the cylinder was also observed for a downstream flexible plate by Mittal and Sharma [11]. For further smaller values of G* = 0.3 and 0.1, Fig. 4a shows a amplitude A* response similar to the PIG response at smaller U * while, at larger U * , there is a sharp decrease in the A* at a critical reduced velocity Uc∗ (20 for G* = 0.3 and
846
A. Thakur et al.
10 for G* = 0.1) followed by an asymptotic decaying trend for smallest G* = 0.1. For U ∗ ≥ Uc∗ , , a staggered vibration of cylinder is found, called here as proximity pressure-induced staggered vibration (PPISV) presented below. In PPISV, the centre of the cylinder never crosses the neutral position y ∗ = 0 or cylinder oscillates on one side of mean neutral position shown below in Fig. 7. The onset of PPISV response leads to decrease of the amplitude A* , sudden change of f * variation similar to VIV response, and an in-to-anti phase-jump, shown in Fig. 4c. This type of response is caused by increased influence of front stagnation region (proximity pressure) of downstream cylinder, with decreasing G* of downstream cylinder, affecting the fluid dynamics in the gap behind the oscillating cylinder.
3.2 Flow States The FIV characteristics of cylinder shows distinct vibration response: VIV, PIG, and PPISV with decreasing gap G* as discussed above for the vibration characteristics in Fig. 4. This transition of vibration dynamics is caused by proximity effects of the flow dynamics or flow state in the gap for the three distinct vibration response. Figure 5a presents vorticity contours in gap for VIV vibration response for a representative case of G* = 2.0 and U * = 5. In VIV, due to larger gap G* , the vortices are formed completely within the gap before interaction with the downstream cylinder. Figure 5a shows that during lock-in (U * = 5), a larger A* causes complete vortex shedding in the gap from upstream cylinder. Further, the vortex detached from upstream cylinder reattaches on downstream cylinder and C(2S) vortex pattern is formed in the wake. However, for desynchronization region (U * > 8), attributed to smaller amplitude A∗ , the vortex formed on upstream cylinder is distorted by downstream cylinder before detachment from upstream cylinder resulting in oscillating wake flow in the gap [11]. The vortex shedding or oscillating wake flow in gap is called as VIV flow state in the gap. The intermediate gap G* = 1.0 in Fig. 5b represents the case of transition from VIV to PIG flow state, which shows a simultaneous vortex formation and its movement in gap. The flow state in the gap is transitioning from oscillating wake flow to nonoscillating wake flow or gap flow and is discussed as follows. Figures 6 and 7 show flow pattern for the PIG flow state and PPISV flow state, respectively. (1) PIG flow state: Considering PIG flow pattern in the gap shown in Fig. 5c, due to small gap G* , the fluid flows upwards or downwards in the gap and shear layers roll up or vortex formation on upstream cylinder is significantly diminished. The fluid flow in gap and associated flow pattern is called as PIG flow state. Figure 6 shows flow patterns for PIG flow state at five time instant during the motion of cylinder from top most to bottom most position for a representative case of G* = 0.5, U * = 10. Flow dynamics at each time instant are as follows:
Flow-Induced Vibration of an Elastically Mounted Cylinder Under …
847
Fig. 5 Instantaneous vorticity contours of vortex interaction occurring in the gap between two cylinders at certain gap G* and reduced velocity U * : (a) Vortex shedding in gap (VIV), (b) Concurrent vortex formation and movement in gap, (c) Gap Flow (PIG), and (d) Biased-base bleed (BBB) flow pattern in gap for PPISV
(1) At the top extreme position, Fig. 6a1 shows that the rear cylinder encounters free stream flow, generating a strong front stagnation region (FSR) on rear cylinder called as proximity pressure. This proximity pressure marked in Fig. 6b1 , deflects large flow in gap; the strong fluid flow in the gap causes induced separation (IS) of top shear layer of rear cylinder as shown in the vorticity contours in Fig. 6c1 . Further, a counter clock-wise (CCW) vortex is formed in the gap at bottom of the oscillating cylinder which generates a suction region. It leads to local dip of C l as shown in the Fig. 6d. (2) The strong gap flow pushes bottom shear layer in gap (marked as gap push in Fig. 6c2 ), which pairs with top shear layer of downstream cylinder. The pair is enveloped by top shear layer of upstream cylinder, the interaction is classified in literature as vortex pairing and enveloping (VPE) [6, 12]. Time history of C l in Fig. 6d shows a local increase in C l at this time instant. This increment is caused by stretching of CCW vortex (gap push) or suppression of bottom suction region of upstream cylinder, shown in Fig. 6b2 . The suppression of suction region is caused by the proximity pressure of downstream cylinder. (3) Around mean position (y* = 0), the Fig. 6a3 shows that the flow in gap changes its direction from upwards to downwards. The vorticity in Fig. 7c3 shows that the shear layers of upstream cylinder in gap moves down with cylinder and reattaches onto downstream cylinder. Figure 6b3 shows that the shear layer reattachment (SLR) generates a strong suction region at bottom of oscillating cylinder. This suction region switches the direction of C l with steeper gradients as seen in Fig. 6d. A clock-wise (CW) vortex is shed followed by gap push of shear layers in gap. (4) As the cylinder crosses mean position, fluid flow in gap in downward direction increases as shown in Fig. 6a4 . The suction region at bottom of cylinder in Fig. 6b4 leads to peak in C l .
848
A. Thakur et al.
Fig. 6 Flow structures (a1 –a5 ), (b1 –b5 ), and (c1 –c5 ) represent the velocity vectors coloured with the magnitude of resultant velocity, pressure contours, and vorticity contours, respectively. Time histories plot d presents displacement y∗ , total lift coefficients C l , and velocity Vc∗ of the upstream cylinder for a complete oscillation cycle at G* = 0.5 and U * = 10. Time instant of flow structures are marked in time history plot d
Flow-Induced Vibration of an Elastically Mounted Cylinder Under …
849
Fig. 7 Flow dynamics at various time instants in PPISV flow state at gap ratio G* = 0.1 and reduced velocity U * = 14. The flow structure and time history representation is similar to Fig. 6
(5) As the cylinder reaches near bottom extreme, Fig. 6a5 , b5 shows that the gap flow in downward direction increases and the bottom suction region of upstream cylinder weakens. It is observed in Fig. 6c2 that a part of CW top shear layer of upstream cylinder reattaches with CW shear layer on top of downstream cylinder and rest part moves in gap. A CCW vortex is shed followed by SLR. (2) PPISV flow state: Fig. 7 shows the flow dynamics of PPISV flow state for a representative case of G* = 0.1 and U * = 14. The flow patterns appear similar to that for the flow over two stationary staggered cylinders [6, 12]. However, the intensity of gap flow in Fig. 7a1 –a3 and strength of proximity pressure in Fig. 7b1 –b3 decreases slightly as the upstream cylinder oscillates with slow velocity V c * from time instant I to III. For each time instant in Fig. 7c1 –c3 , fluid in the gap is directed towards suction region behind the downstream cylinder, which is similar to biased-base-bleed (BBB) flow pattern observed for two
850
A. Thakur et al.
Fig. 8 Vibrational characteristics at a larger, b intermediate, and c smaller gap G∗
stationary staggered cylinders by Lee et al. [6]. Further, both cylinder shed vortex as a single bluff body. Attributed to staggered vibration and biased direction of gap flow, time history of lift coefficients C l is also biased in positive direction as seen in Fig. 7d.
4 Conclusions Figure 8 presents the summary of proximity effects of stationary downstream cylinder on vibration characteristics of elastic upstream cylinder. As the gap G* between the two cylinders is decreased, VIV (G* ≥ 1.5), PIG (G* ≤ 0.5), and PPISV (G* ≤ 0.3) vibrational response are found. The PPISV vibration occurred only for larger reduced velocity at G* = 0.3 (U * > 18) and 0.1 (U * ≥ 10). In VIV flow state, vortex shedding and oscillating wake flow is observed in gap, resulting in A* , f * response similar to VIV of an isolated cylinder. For the PIG flow state, fluid flow in gap occurs, generating large amplitude in-phase oscillation with a rapid-increase plateau variation for A* and f * . The PPSIV flow state occurs due to enhanced suppression of gap side vortex and associated suction region of upstream cylinder caused by stronger influence of proximity pressure at G* = 0.1 and 0.3. Acknowledgements The first author would like to acknowledge the help of Dr. Charu Mittal in getting started with this work.
References 1. Geoola F, Bokaian A (1984) Proximity-induced galloping of two interfering circular cylinders. J Fluid Mech 146:417–449 2. Geoola F, Bokaian A (1984) Wake-induced galloping of two interfering circular cylinders. J Fluid Mech 146:383–415 3. Armandei M, Rostami AB (2017) Renewable energy harvesting by vortex-induced motions: review and benchmarking of technologies. Renew Sustai Energy Rev 70:193–214 4. Bernitsas MM (2016) Harvesting energy by flow included motions. In: Xiros NI, Dhanak MR (eds) A handbook of ocean energy (.), vol 1. Springer International Publishing, pp 1163–1244
Flow-Induced Vibration of an Elastically Mounted Cylinder Under …
851
5. Sotirpolous F, Borazjani I (2009) Vortex-induced Vibration of two cylinders in tandem arrangement in the proximity-wake interference region. J Fluid Mech 621:321–364 6. Yang K-S, Lee K (2009) Flow patterns past two circular cylinders in proximity. Comput Fluids 38:813–851 7. Mittal C, Sharma A (2021) Flow-induced vibration of a flexible splitter-plate in the wake of a stationary cylinder. Phys Fluids 33:113607 8. Sharma A, Thekkethil N (2019) Level set function–based immersed interface method and benchmark solutions for fluid flexible structure interaction. Int J Numer Meth Fluids 1:1–24 9. Prasanth TK, Mittal S (2008) Vortex-induced vibrations of a circular cylinder at low Reynolds numbers. J Fluid Mech 594:463–491 10. Sharma A (2016) Introduction to computational fluid dynamics: development, application and analysis. Wiley 11. Mittal C, Sharma A (2022) Flow induced coupled-vibrations of an elastically mounted cylinder and a detached flexible plate. J Fluid Mech 12. Sumner D (2010) Two cylinders in cross flow: a review. J Fluid Struct 26:849–899 13. Thekkethil N, Sharma A (2020) Hybrid Lagrangian–Eulerian method-based CFSD development, application, and analysis. Immersed boundary method. Springer, pp 361–394 14. Thekkethil N, Sharma A, Agrawal A (2018) Unified hydrodynamics study for various types of fishes-like undulating rigid hydrofoil in a free stream flow. Phys Fluids 30(7):077107 15. Govardhan R, Williamson CHK (2004) Motion, vortex and mode transition in vortex induced vibration. Annu Rev Fluid Mech 36:413–550 16. Bao Y, Zhou D, Jiahuang T (2011) Flow interference between a stationary cylinder and an elastically mounted cylinder arranged in proximity. J Fluid Struct 27:1425–1466 17. Zhao M (2013) Flow induced vibration of two rigidly coupled circular cylinders in tandem and side-by-side arrangements at a low Reynolds number of 150. Phys Fluids 25:123601