Design and Analysis of Thermal Systems 9780367502546, 9780367503260, 9781003049272

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Design and Analysis of Thermal Systems Thermal systems are essential features of all domestic and industrial applications involving heat and fluid flow. Focusing on the design of thermal systems, this book bridges the gap between the theories of thermal science and design of practical thermal systems. Further, it discusses thermodynamic design principles, mathematical and CFD tools that will enable students as well as professional engineers to quickly analyze and design practical thermal systems. The major emphasis is on practical problems related to contemporary energy- and environment-related thermal systems including discussions on computational fluid dynamics used in thermal system design. Features • Exclusive book integrating thermal sciences and computational approaches • Covers both philosophical concepts related to systems and design, to numerical methods, to design of specific systems, to computational fluid dynamics strategies • Focuses on solving complex real-world thermal system design problems instead of just designing a single component or simple systems • Introduces usage of statistics and machine learning methods to optimize the system • Includes sample PYTHON codes, exercise problems, special projects This book is aimed at senior undergraduate/graduate students and industry professionals in mechanical engineering, thermo-fluids, HVAC, energy engineering, power engineering, chemical engineering, nuclear engineering.

Design and Analysis of Thermal Systems

Malay Kumar Das Pradipta K. Panigrahi

First edition published 2023 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2023 Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright. com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 978-0-367-50254-6 (hbk) ISBN: 978-0-367-50326-0 (pbk) ISBN: 978-1-003-04927-2 (ebk) DOI: 10.1201/9781003049272 Typeset in Nimbus Roman by KnowledgeWorks Global Ltd.

Contents Authors..................................................................................................................... xv Chapter 1

Introduction ..................................................................................... 1 1.1

Definition and Importance ...................................................... 1 1.1.1 Design versus Analysis .............................................. 1 1.1.2 Synthesis for Design .................................................. 3 1.1.3 Selection versus Design ............................................. 4 1.2 Thermal System Design Aspects ............................................ 4 1.2.1 Environmental Aspects .............................................. 4 1.2.2 Safety Issues............................................................... 5 1.3 Reliability, Availability and Maintainability (RAM).............. 5 1.4 Background Information and Data Sources............................ 6 1.5 Workable, Optimal and Nearly Optimal Designs ................... 6 1.6 Stages of the Design Process .................................................. 8 1.6.1 DFX Strategies........................................................... 9 1.6.2 Formulation of the Design Problem......................... 10 1.6.2.1 Requirements........................................... 10 1.6.2.2 Given Parameters..................................... 10 1.6.2.3 Design Variables...................................... 11 1.6.2.4 Constraints and Limitations..................... 11 1.6.2.5 Safety, Environmental and Other Considerations ......................................... 12 1.7 Conceptual Designs .............................................................. 13 1.7.1 Modification in the Design of Existing Systems ..... 14 1.7.2 Steps in the Design Process ..................................... 14 1.7.2.1 Physical Systems ..................................... 14 1.7.2.2 Modeling.................................................. 15 1.7.2.3 Simulations .............................................. 15 1.7.2.4 Acceptable Design Evaluations ............... 17 1.7.2.5 Optimal Designs ...................................... 17 1.7.2.6 Safety Features, Automation and Control .............................................. 18 1.7.2.7 Communicating the Design ..................... 18 1.7.3 Computer-Aided Designs......................................... 19 Problems ........................................................................................ 21

v

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Chapter 2

Contents

Modeling and Simulation Basics................................................... 23 2.1 2.2

2.3

2.4

2.5

2.6

2.7 2.8 2.9

Introduction........................................................................... 23 Types of Models.................................................................... 23 2.2.1 Analog Models......................................................... 23 2.2.2 Mathematical Models............................................... 24 2.2.3 Physical Models ....................................................... 24 2.2.4 Numerical Models.................................................... 24 Mathematical Modeling ....................................................... 25 2.3.1 Transient/Steady State.............................................. 25 2.3.1.1 Case 1: τc →∞ (Large τc ) ......................... 25 2.3.1.2 Case 2: τ c > τ r ..................................... 26 2.3.1.4 Case 4: Periodic Processes ...................... 26 2.3.1.5 Case 5: Transient .....................................26 2.3.2 Number of Spatial Dimensions................................ 27 2.3.3 Lumped Mass Approximation ................................. 28 2.3.4 Simplification of Boundary Conditions ................... 28 2.3.5 Negligible Effects .................................................... 29 2.3.6 Idealizations ............................................................. 29 2.3.7 Material Properties................................................... 30 Sample Mathematical Modeling Examples ..........................30 2.4.1 Storage Tank of Solar Collector............................... 30 2.4.1.1 Lumped Mass Approximation.................31 2.4.1.2 Material Properties .................................. 31 2.4.1.3 Spatial Dimensions..................................31 2.4.1.4 Simplifications......................................... 31 2.4.1.5 Governing Equation................................. 31 2.4.1.6 Dimensionless Parameters....................... 32 2.4.1.7 Dimensionless Initial/Boundary Conditions................................................ 32 2.4.2 An Electric Heat Treatment Furnace .......................32 2.4.2.1 Initial and Boundary Conditions ............. 34 Dimensional Analysis...........................................................35 2.5.1 Example of an Electronic Device............................. 36 2.5.1.1 Non-Dimensionalization.......................... 38 Curve Fitting ......................................................................... 38 2.6.1 Least Square Method ............................................... 39 2.6.2 Two Independent Variable Cases ............................. 40 2.6.2.1 Curve-Fitting Procedure .......................... 41 Numerical Modeling............................................................. 41 2.7.1 Accuracy and Validation .......................................... 43 Importance of Simulation .....................................................43 Different Classes of Numerical Simulation .......................... 44 2.9.1 Dynamic or Steady State.......................................... 44

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2.9.2 Continuous or Discrete ............................................ 44 2.9.3 Deterministic or Stochastic ...................................... 45 2.10 Flow of Information.............................................................. 45 2.11 Block Representation............................................................ 45 2.11.1 Information Flow Diagram ...................................... 46 2.12 Initial Design ........................................................................ 47 2.13 Iterative Redesign for Convergence...................................... 48 2.14 Sample Thermal System Design Examples.......................... 49 2.14.1 A Piping Network Problem...................................... 49 2.14.2 A Gas Turbine Problem ........................................... 51 2.14.3 Fin Design................................................................ 53 2.14.3.1 Multiple Fins ........................................... 55 2.14.3.2 An Example of a Fin Problem ................. 56 Problems ........................................................................................ 56 References...................................................................................... 59 Chapter 3

Exergy for Design.......................................................................... 61 3.1

3.2

3.3

Introduction........................................................................... 61 3.1.1 Definition of Exergy................................................. 61 3.1.2 Environment............................................................. 61 3.1.3 Exergy Components................................................. 62 3.1.3.1 Physical Exergy ....................................... 62 3.1.3.2 Dead States .............................................. 62 3.1.3.3 Chemical Exergy .....................................63 Exergy Balance Equation...................................................... 63 3.2.1 Closed System.......................................................... 63 3.2.1.1 Energy Balance of the Closed System.....63 3.2.1.2 Entropy Balance of the Closed System ... 63 3.2.2 Open System ............................................................ 64 3.2.2.1 Exergy Transfers at Inlets and Outlets.....65 3.2.3 Standard Chemical Exergy of Gases and Gas Mixtures............................................................ 66 3.2.4 Standard Chemical Exergy of Fuels......................... 67 Exergy Destruction and Exergy Loss ...................................69 3.3.1 Exergy Destruction through Heat Transfer and Friction..................................................................... 70 3.3.1.1 Thermodynamic Average Temperature ... 70 3.3.1.2 Overview ................................................. 72 3.3.2 Exergy Destruction and Exergy Loss Ratios ........... 72 3.3.3 Exergetic Efficiency .................................................73 3.3.3.1 How Do We Distinguish between Fuel and Product? .................................... 74 3.3.3.2 Compressor, Pump or Fan ....................... 74 3.3.3.3 Turbine or Expander ................................ 74

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3.3.3.4 3.3.3.5 3.3.3.6 3.3.3.7 3.3.3.8 3.3.3.9 3.3.3.10

Heat Exchanger ....................................... 74 Case 1 ...................................................... 74 Case 2 ...................................................... 76 Mixing Unit ............................................. 76 Gasifier or Combustion Chamber ............ 76 Boiler ....................................................... 76 Guidelines for Defining Exergetic Efficiency................................................. 77 3.3.3.11 Subsystem A............................................ 77 3.3.3.12 Subsystem B ............................................ 78 3.4 Exergy Analysis of a Gas Turbine-Based Power Plant........ 78 3.4.1 Guidelines for Evaluating and Improving Thermodynamic Effectiveness................................. 81 3.4.2 Additional Guidelines .............................................. 82 3.5 Exergy Analysis of a Heat Exchanger .................................. 82 3.5.1 Area Constraint ........................................................ 85 3.5.2 Volume Constraint ...................................................86 3.5.3 Combined Area and Volume Constraint .................. 87 3.5.4 Unbalanced Heat Exchanger.................................... 87 3.5.5 Counter-Flow Heat Exchanger................................. 87 3.5.6 Parallel Flow Heat Exchanger.................................. 88 3.6 Exergy Analysis of a Refrigeration System.......................... 88 3.7 Exergy Storage System......................................................... 91 3.8 Solar Air Collector................................................................ 94 3.8.1 Heat Transfer Coefficient......................................... 96 3.8.2 Air Mass Flow Rate .................................................98 3.8.2.1 Energy and Exergy Efficiency ................. 99 3.8.2.2 Parametric Study ..................................... 99 3.9 Ocean Thermal Energy Conversion.................................... 101 3.9.1 Hydrogen Production Using OTEC ....................... 104 3.9.1.1 Energy Analysis..................................... 106 3.9.1.2 Flat Plate Solar Collector....................... 106 3.9.1.3 Organic Rankine Cycle.......................... 107 3.9.1.4 PEM Electrolyzer .................................. 108 3.9.1.5 Energy Efficiency .................................. 110 3.9.1.6 Exergy Efficiency .................................. 111 3.9.2 Simulation Results ................................................. 111 Problems ...................................................................................... 113 References.................................................................................... 115 Chapter 4

Material Selection........................................................................ 117 4.1 4.2 4.3

Material Properties.............................................................. 117 Software.............................................................................. 117 Material Attributes.............................................................. 118

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4.4

Selection Strategies............................................................. 119 4.4.1 Material Indices ..................................................... 120 4.5 Case Studies........................................................................ 121 4.5.1 Case 1: Heat Sink Material .................................... 121 4.5.2 Case 2: Material for Sensible Thermal Energy Storage ................................................................... 122 4.5.3 Case 3: Phase Change Material for Cold Thermal Energy Storage ........................................ 125 4.5.3.1 Thermophysical Properties .................... 125 4.5.3.2 Kinetic Properties .................................. 125 4.5.3.3 Chemical Properties............................... 125 4.5.3.4 Economics ............................................. 126 4.5.4 Case 4: Selection of Insulation Material................ 127 4.5.5 Case 5: Heat Transfer Fluids for Solar Power Systems .................................................................. 127 4.6 Summary............................................................................. 130 Problems ...................................................................................... 130 References.................................................................................... 130 Chapter 5

Heat Exchangers.......................................................................... 133 5.1 5.2 5.3 5.4 5.5 5.6 5.7

5.8 5.9

5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18

Introduction......................................................................... 133 Classification of Heat Exchanger........................................ 133 Overall Heat Transfer Coefficient.......................................136 Log Mean Temperature Difference (LMTD)...................... 140 The ε–NTU Method ........................................................... 147 Variable Overall Heat Transfer Coefficient ........................ 147 Heat Exchanger Thermal Design........................................ 149 5.7.1 Rating Problem ...................................................... 149 5.7.2 Sizing Problem....................................................... 150 Forced Convection Correlation for Single–Phase Side of a Heat Exchanger................................................................ 150 Effect of Variable Properties............................................... 153 5.9.1 For Liquids............................................................. 154 5.9.2 For Gases ............................................................... 154 Flow in Smooth Straight Non-Circular Ducts .................... 155 Heat Transfer from Smooth Tube Bundles ......................... 156 Pressure Drop in Tube Bundles in Cross-Flow................... 158 Shell and Tube Heat Exchangers ........................................ 159 Tube Passes......................................................................... 160 Tube Layout ........................................................................ 160 Baffle Type.......................................................................... 161 Tube-Side Pressure Drop .................................................... 161 Bell–Delaware Method ....................................................... 161 5.18.1 Shell-Side Heat Transfer Coefficient ..................... 164

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5.18.2 Shell-Side Pressure Drop ....................................... 166 5.19 Kern Method ....................................................................... 168 5.19.1 Shell-Side Heat Transfer Coefficient ..................... 168 5.19.2 Shell-Side Pressure Drop ....................................... 169 5.20 Basic Design Process.......................................................... 169 5.21 Preliminary Design Estimation........................................... 169 5.22 Compact Heat Exchanger Design....................................... 172 5.22.1 Heat Transfer and Pressure Drop ........................... 172 5.22.2 Pressure Drop for Finned-Tube Exchangers .......... 175 5.22.3 Pressure Drop for Plate-Fin Exchangers................ 176 5.23 Optimization of Heat Exchangers....................................... 177 Problems ...................................................................................... 187 References.................................................................................... 188 Chapter 6

Piping Flow.................................................................................. 189 6.1 6.2

Introduction......................................................................... 189 Energy Equations................................................................ 189 6.2.1 Minor Losses.......................................................... 190 6.2.2 Graphics Symbol Conventions............................... 191 6.2.3 General Considerations.......................................... 191 6.2.4 Resistance Analogy................................................ 191 6.2.5 Classification of Pumps ......................................... 192 6.2.6 Pump Selection ...................................................... 192 6.3 Pump Performance Using Dimensional Analysis............... 195 6.3.1 Dimensional Analysis ............................................ 195 6.3.2 Specific Speed........................................................ 196 6.4 Pump Curve for Viscous Fluid ........................................... 197 6.4.1 Procedure to Obtain the Correction Factor and Pump Curve for Viscous Fluid............................... 197 6.5 Effective Pump Performance Curve ................................... 198 6.5.1 Computer Implementation ..................................... 200 6.5.1.1 Pumps in Series ..................................... 200 6.5.1.2 Pumps in Parallel ................................... 201 6.6 System Characteristics........................................................ 201 6.7 Pump Placement ................................................................. 202 6.7.1 Cavitation ............................................................... 202 6.7.2 Net Positive Suction Head ..................................... 202 6.7.3 Recirculation Problem ........................................... 203 6.8 Suction-Specific Speed ....................................................... 203 6.9 Net Positive Suction Head Available .................................. 204 6.10 Uncertainty Effect on Pump Selection................................ 204 6.11 Uncertainty Analysis Procedure ......................................... 206 6.11.1 Piping Network Design.......................................... 206 6.12 Piping System Design......................................................... 207

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Contents

6.12.1 Hardy Cross Method .............................................. 208 6.12.2 Hazen–Williams Coefficient .................................. 208 6.12.3 Basic Idea............................................................... 209 6.12.4 Correction Factor ................................................... 209 6.12.5 Implementation Procedure ..................................... 211 6.13 Generalized Hardy Cross Analysis..................................... 214 6.13.1 Block Diagram ....................................................... 216 Problems ...................................................................................... 220 References.................................................................................... 221 Chapter 7

Artificial Intelligence for Thermal Systems ................................ 223 7.1 7.2

Introduction......................................................................... 223 Expert System..................................................................... 223 7.2.1 Advantages of Expert Systems .............................. 224 7.2.2 Disadvantages of Expert Systems .......................... 224 7.2.3 Structure of Expert Systems................................... 224 7.2.4 An Example for Feed Water Pump Selection ........ 226 7.3 Artificial Neural Network (ANN) Overview ...................... 228 7.3.1 Structure of ANNs ................................................. 228 7.3.2 Training of ANNs .................................................. 230 7.4 ANNs for Heat Exchanger Analysis................................... 234 7.5 ANNs for a Thermophysical Property Database ................ 236 7.6 Physics Informed ANNs ..................................................... 240 7.7 ANNs for Dynamic Thermal Systems................................ 241 7.8 Summary............................................................................. 245 References.................................................................................... 246

Chapter 8

Numerical Linear Algebra........................................................... 249 8.1

Bisection Method................................................................ 249 8.1.1 Convergence of Bisection Method......................... 250 8.2 Newton–Raphson Method ..................................................251 8.3 Eigenvalues and Eigenvectors............................................. 253 8.4 Power Iterations .................................................................. 254 8.5 Convergence ....................................................................... 255 8.6 Inverse Power Iterations...................................................... 257 8.7 Curve Fitting ....................................................................... 258 8.8 Fitting of a Straight Line..................................................... 259 8.9 Fitting of a Polynomial ....................................................... 260 8.10 Error Estimation.................................................................. 261 8.11 Solution of Algebraic Equations......................................... 262 8.12 Gaussian Elimination.......................................................... 263 8.12.1 Forward Elimination .............................................. 264 8.12.2 Back Substitution ................................................... 265 8.12.3 How to Improve the Solution................................. 265

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8.13 Jacobi and Gauss–Seidel Iterations .................................... 266 8.13.1 Vector and Matrix Norms....................................... 267 8.13.2 Convergence of the Jacobi Iteration....................... 268 8.13.3 Gauss–Seidel Iteration ........................................... 269 8.14 Extension to Nonlinear Systems......................................... 270 Chapter 9

Ordinary Differential Equations .................................................. 273 9.1 9.2 9.3 9.4 9.5 9.6

Chapter 10

Numerical Differentiation and Integration .................................. 281 10.1 10.2 10.3 10.4 10.5

Chapter 11

Introduction......................................................................... 273 Euler Method ...................................................................... 273 Runge–Kutta Method.......................................................... 274 Higher-Order IVP ............................................................... 276 Boundary Value Problems: Shooting Method .................... 277 Boundary Value Problems: Finite Difference Method ....... 278

Introduction......................................................................... 281 Numerical Differentiation................................................... 281 Nonuniform Grid ................................................................ 282 Double Derivative ............................................................... 283 Numerical Integration: Newton–Cotes Formulas ...............284 10.5.1 Trapezoidal Rule .................................................... 284 10.5.2 Simpson’s One-Third Rule .................................... 286

Partial Differential Equations ...................................................... 289 11.1 Introduction......................................................................... 289 11.2 Classification....................................................................... 290 11.2.1 Marching Problem ................................................. 290 11.2.2 Equilibrium Problem.............................................. 290 11.2.3 Eigenvalue Problem ............................................... 290 11.3 Second-Order Linear PDE.................................................. 291 11.3.1 Parabolic Problem.................................................. 291 11.3.2 Hyperbolic Problem ............................................... 291 11.3.3 Elliptic Problem ..................................................... 292 11.4 One-Dimensional Transient Diffusion................................ 292 11.5 Numerical Schemes ............................................................ 293 11.5.1 Explicit Scheme ..................................................... 293 11.5.2 Implicit Scheme ..................................................... 294 11.5.3 Crank–Nicolson Scheme........................................ 295 11.6 Stability and Consistency ................................................... 296 11.6.1 Round-Off Error..................................................... 297 11.6.2 Truncation Error..................................................... 298 11.6.3 Consistency ............................................................ 298 11.6.4 Stability .................................................................. 302

Contents

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11.7 Two-Dimensional Transient Diffusion ...............................305 11.7.1 Explicit Scheme ..................................................... 305 11.7.2 Implicit and Crank–Nicolson Schemes.................. 306 11.8 Elliptic Equations................................................................ 308 11.8.1 Discretization ......................................................... 309 11.8.2 Solution Procedure................................................. 309 11.8.3 Pseudo-Transient Approach................................... 310 Chapter 12

Computational Fluid Dynamics................................................... 313 12.1 Introduction......................................................................... 313 12.1.1 Non-Dimensionalization ........................................ 314 12.2 Stream Function, Vorticity (ψ − ω) Formulation............... 314 12.2.1 Stream Function ..................................................... 315 12.2.2 Vorticity.................................................................. 316 12.2.3 Vorticity Transport Equation.................................. 317 12.2.4 Solution Strategy.................................................... 318 12.3 Primitive Variable Formulation........................................... 320

Chapter 13

Electrochemical Systems............................................................. 323 13.1 Introduction......................................................................... 323 13.1.1 Fuel Cells ............................................................... 323 13.1.2 Batteries and Fuel Cells ......................................... 323 13.2 Fuel Cell Thermodynamics................................................. 324 13.2.1 Reversible Voltage ................................................. 326 13.2.2 Reversible Efficiency ............................................. 326 13.3 Classifications ..................................................................... 327 13.3.1 PEMFC .................................................................. 328 13.3.2 SOFC ..................................................................... 328 13.4 Losses in Fuel Cells ............................................................ 328 References.................................................................................... 329

Chapter 14

Inverse Problems ......................................................................... 331 14.1 Introduction......................................................................... 331 14.2 Inverse Heat Conduction: Conjugate-Gradient Approach..332 14.2.1 Sensitivity Problem ................................................333 14.2.2 Adjoint Problem..................................................... 334 14.2.3 Descent Direction and Step Size............................ 335 14.3 Regularization and Stopping Criterion ............................... 336 14.3.1 Discrepancy Principle ............................................ 336 14.3.2 Additional Measurement Approach....................... 337 14.3.3 Smoothing of Experimental Data........................... 337 14.4 Complete Algorithm ........................................................... 337 References.................................................................................... 338

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Contents

Appendix A

Thermophysical Properties (Working Fluids) ............................. 339

Appendix B

Thermophysical Properties (Exergy Calculation) ....................... 373

Appendix C

Thermophysical Properties (Emissivity) ..................................... 377

Appendix D

Standard Pipe Dimension ............................................................ 379

Appendix E

Pump Performance Curve............................................................ 387

Appendix F

Minor Loss Coefficient................................................................397

Appendix G

Sample Project Topics ................................................................. 399

Index...................................................................................................................... 403

Authors Malay Kumar Das received a BE degree in mechanical engineering (1989) from Bengal Engineering College, Shibpur, India; MTech. in mechanical engineering (2003) from IIT Kanpur, India and PhD in mechanical engineering (2008) from the Pennsylvania State University, USA. He has been a faculty member in IIT Kanpur since 2008. He has authored approximately 60 technical papers, 3 patents and 1 edited book. Pradipta K. Panigrahi received BTech. degree in mechanical engineering (1987) from UCE Burla, Sambalpur, India; MTech. in mechanical engineering (1992), MS in system (computer) science (1997) and PhD in mechanical engineering (1997) from Louisiana State University, Baton Rouge, USA. He has been a faculty member in IIT Kanpur since 1998. He was Head of the Laser Technology Program from 2010– 2013 and Head of Mechanical Engineering Department from 2014–2017. Presently, he is the Head of Photonics Science and Engineering Program, Center for Lasers and Photonics at IIT Kanpur. He has authored more than 200 technical papers, 3 patents, 1 text book, 2 edited books and 2 research monographs.

xv

1 Introduction 1.1

DEFINITION AND IMPORTANCE

Design is defined as a creative process by which new methods, devices, processes and techniques are developed to solve new or existing problems. Nowadays, most industries are interested in producing new and high-quality products at minimal cost while satisfying the increasing concerns about environmental impact and safety. It is no longer adequate just to develop a product that performs the desired task due to increasing worldwide competition. It is important to optimize the process or system such that a chosen quantity, known as the objective function, is maximized or minimized. For a given system, the profit, productivity, product quality etc. may be maximized or the cost per item, investment, energy input etc. may be minimized. Several classical industries e.g. the steel industry have become less important in recent years due to the advent of many new materials, such as composites and ceramics, and new manufacturing processes. Therefore, it is important to keep abreast of new developments and use new techniques for product improvement and cost reduction. The design and optimization of new processes/systems and optimization of existing ones are closely linked to the prosperity of a given company. Design and optimization methods have been traditionally applied to mechanical systems such as those involved with transmission, vibration, control and robotics. In recent years, there has been a tremendous growth in development and use of thermal systems in which fluid flow and transport of energy play a dominant role. Some examples of thermal systems are: (a) Manufacturing Systems: Continuous casting, plastic screw extrusion, optical fiber drawing, hot rolling (b) Power Systems: Solar energy, nuclear energy, coal power plant, gas-fired power plant, wind power plant (c) Cooling Systems: Electronic equipments, gas turbine blades, gas turbine combustion chamber, nuclear power plants (d) Transportation Systems: Aircraft propulsion (turbojet engine), IC engines (e) Fluid Distribution Systems: Industrial system, residential system This book focuses on the design and optimization methods related to thermal systems. 1.1.1

DESIGN VERSUS ANALYSIS

We are quite familiar with the analysis of engineering problems using information derived in the basic areas i.e. statics, dynamics, thermodynamics, fluid mechanics and heat transfer. There is little interaction between different disciplines during analysis. All inputs needed for the problem are usually given, and the results are generally DOI: 10.1201/9781003049272-1

1

2

Design and Analysis of Thermal Systems

Figure 1.1 An arrangement for electronic cooling applications. unique and well defined during analysis. The solution to a given problem can be carried out to completion during analysis. Such problems are termed as close-ended. Some examples of problems for analysis are: 1. 2. 3. 4.

Parabolic fully developed flow through pipe/channels One-dimensional conduction through a flat wall Fluid flow and heat transfer over a flat plate Isentropic work done during a compression process

The design process is open-ended; i.e. the results are not well known or well defined at the onset. The inputs may also be vague or incomplete for design. Therefore, it is necessary to seek additional information or to employ approximation and assumptions. There is usually considerable interaction between various disciplines i.e. technical areas and those concerned with cost, safety and the environment during design. A unique solution is not generally obtained by the design. One may have to choose from a range of acceptable solutions. A solution that satisfies all the requirements may not be obtained by the design. Therefore, the designer may have to relax some of the requirements to obtain an acceptable solution. Trade-off generally forms a necessary part of the design, since certain characteristics of the system may have to be given up in order to achieve other goals such as lower cost or lower environmental impact. A design solution is not unique. It depends on a variety of factors and many of them are non technical. The following electronics cooling problem can be used to demonstrate the difference between design and analysis. The electronics cooling problem in Figure 1.1 can be either close-ended or open-ended depending on the problem statement. It is termed as close-ended when the information on energy dissipated by the integrated chip, geometry, material properties of the circuit board and forced flow conditions are available. We have to solve for the temperature distribution in the component and circuit board, which is an example of close-ended problem. The solution methods for this problem may be either analytical or numerical. In this case, the solution is well defined and unique. There are no trade-offs, and no additional considerations are required. This problem is termed as an open-ended problem when the problem statement is to find the appropriate materials, geometry and dimensions so that the temperature

Introduction

3

Figure 1.2 Schematic of a typical design procedure. (Tc ) of integrated chip is maintained below a certain value (Tmax ). There is no unique answer for this problem because many combinations of materials, dimensions, geometry, fan capacity etc. may satisfy the Tc < Tmax requirement. It is also possible that a satisfactory solution cannot be found for the given conditions. An additional cooling method such as heat pipe or spray may have to be included in the system. Different approaches, often known as conceptual designs, may be considered for satisfying the given requirements. The design is often much more involved than analysis because of the open-ended nature of design problems. Extensive information on analysis of various thermal processes is available in literature. However, the corresponding design problems have received much less attention. The design and analysis are very different in their objectives and goals. Analysis usually forms the basis for design process. 1.1.2

SYNTHESIS FOR DESIGN

Several components and their corresponding analyses are brought together in design to yield the characteristics of the overall system. This is known as synthesis. The designer cannot consider only the heat transfer aspects while ignoring the strength of materials and manufacturing aspects. The design process incorporates information and outputs from different types of models, including experimental and numerical results from existing systems. Figure 1.2 shows the flow chart of a design process. Here, the details of the component information, material characteristics and

4

Design and Analysis of Thermal Systems

experimental data are used. The cost, properties and characteristics of materials are also provided as input during the design process. Additional effects such as safety, legal, regulatory and environment considerations are also synthesized in order to obtain satisfactory design. 1.1.3

SELECTION VERSUS DESIGN

Selection and design are frequently employed together in the development of a system. Selection involves determining the specifications of items for the given task that are easily available in market over the ranges of interest. A choice is made from the various types of items available with different ratings on the basis of these specifications. Standard items that can be selected from the catalog are valves, control sensors, heaters, flow meters, storage tanks, pumps, compressors, fans, condensers and so on. Design is involved in the development of these components. However, for a given system, the design of these individual components may be avoided in the interest of cost, time and convenience.

1.2

THERMAL SYSTEM DESIGN ASPECTS

Thermal systems involve several issues that are unique compared to other systems. Some key design aspects within the context of thermal systems are discussed in the following sections. 1.2.1

ENVIRONMENTAL ASPECTS

There are several types of pollution sources of a thermal system: (1) air, (2) water, (3) thermal, (4) solid waste and (5) noise pollution. The design engineer has to implement different pollution control strategies based on the types of pollution. Air pollution treatment equipment falls into two general types: (1) particulate removal by mechanical means, such as cyclones, filters and scrubbers etc. and (2) gas component removal by absorption, condensation and incineration etc. Physical, chemical and biological waste treatment measures can be used for liquid wastes effluents. It is advisable to consider the recovery of valuable liquid-borne products to avoid costly waste treatment measures prior to waste treatment. Thermal pollution resulting from the direct discharge of warm water into lakes, rivers and streams can be ameliorated by cooling towers. Solid waste can be handled by incineration and pyrolysis etc. Coupling waste incineration with steam or hot water generation may provide economic benefit. It is necessary to understand the individual noise sources, their acoustic properties and how they interact to cause the overall noise pollution for effective noise control. Traditionally, an end-of-the-pipe approach that mainly addresses the pollutants emitted from the system is used. In this approach, the procedure for treatment of pollutants is considered. This is not the best approach. Rather the environmental aspects should be considered during the conceptual stage of a design, which is known as design for environment (DFE). In DFE, the environmentally preferred aspects of

Introduction

5

a system are treated as design objectives not as constraints. Designers are required to anticipate negative environmental aspects throughout the life cycle and engineer them out. Efforts need to be directed for reducing the creation of waste i.e. changing the process technology and/or plant operation and replacing input materials which are the sources of toxic waste with more benign materials. Compliance with environmental regulations should be considered throughout the design process. It should not be deferred to the end when options might be foreclosed due to earlier decisions. Addressing such regulations early may result in fundamentally better process choices that reduce the size of required clean-up. However, some end-of-the-pipe clean-up might be required to meet the government regulations. Costs to control pollution are generally much higher if left for resolution after the facility has begun operation. Thus, the appropriate pollution control measures i.e. the type of pollutant being controlled and the features of the available control equipment need to be specified by the design team. It should be noted that when reporting a project, it is imperative for the design engineer to submit environmental impact statement. 1.2.2

SAFETY ISSUES

The service life of a thermal system will not always be trouble-free. Occasional failure of some piece of equipment is likely to happen. Safety should be one of the quality factors at the beginning of the life cycle-design process. The design team should take note of the following points: 1. The design team is responsible for anticipating unsafe events and designing the system so that a local failure cannot amplify to an overall system failure or disaster. 2. One approach for safety analysis is to test the response of each component via computer simulation at extreme conditions that are not part of the normal operating plan. 3. Safety studies should be undertaken throughout the design process. It is not wise to defer the safety issues to the end of design process because decisions made earlier during design process may foreclose effective alternatives. 4. Exposure to toxic materials should be prevented or minimized during the operation of a thermal system. Machinery must be guarded with some protective devices. It is important for the design engineer to consider safety aspects. However, an indiscriminate application of safety factors is not a good practice because it may lead to over-design and the system can become uneconomical.

1.3

RELIABILITY, AVAILABILITY AND MAINTAINABILITY (RAM)

Reliability, availability and maintainability (RAM) are three important design attributes. Reliability is defined as probability that a system will successfully perform specified functions for specified environmental conditions over a prescribed operating life. Maintainability is defined as probability that the system can be repaired and maintained easily and economically within a specified time period. Availability is

6

Design and Analysis of Thermal Systems

defined as the measure of how often the system can be available (operational) when needed.

1.4

BACKGROUND INFORMATION AND DATA SOURCES

Design engineers should be aware of current advances in their fields and related fields. They should read technical literature, attend industrial expositions and professional society meetings and develop a network of professional contacts. A design engineer should have both private and public sources of information/data to support a design project. Private sources of information consist of guarded proprietary information accumulated by individual companies, which is usually not available to outsiders. Public sources of information include open technical literature. There is a rapid growth of literature in recent times. Therefore, it is becoming increasingly difficult to search effectively for specific types of information. Online databases have helped to facilitate these searches. Handbooks and review articles describing current technology are useful resources. The Thomas Registrar also provides an exhaustive listing of manufactured items. The patent literature is one of the useful public sources of information. It can provide ideas that can assist in achieving a design solution. It can also help in avoiding approaches that will not work. Codes and standards provide information in the form of allowable limits on the performance of various systems. It can be helpful in shortening the design time and reducing uncertainty in performance of the system. It can also improve product quality and reliability. Court of law normally considers use of codes and standards as a sign of good engineering practice even if there is no legal compulsion to do so.

1.5

WORKABLE, OPTIMAL AND NEARLY OPTIMAL DESIGNS

The initial step in the design process is to specify the requirements of a system quantitatively i.e. formulate the design specifications. A workable design is the one that meets all the design specifications. The optimal design is the one that is best among several workable designs. Several alternative criteria for specifying the best can be applied depending on the type of application i.e. lowest cost, lowest weight, maximum reliability etc. A true optimum is generally impossible to determine due to system complexity and uncertainties associated with a thermal system. Therefore, the designer often accepts a design that is close to optimal and is known as nearly optimal design. Let us consider Bejan et al.’s [1996] example of a counter-flow heat exchanger design. One of the key design variables is the minimum temperature difference between the two streams (∆T )min . The temperature difference between the two streams of a heat exchanger is a measure of nonideality (irreversibility) according to thermodynamics. The heat exchangers approach ideality as the temperature difference between two streams approaches zero. This source of nonideality exerts a penalty on the fuel supplied to the overall plant, as a part of the fuel is used to feed the source of irreversibility. Therefore, we would prefer to reduce the stream-to-stream

Introduction

7

Figure 1.3 An example showing optimal and nearly optimal design of a counterflow heat exchanger. temperature difference for reducing operating costs. However, as the temperature difference narrows, more heat transfer area is required for the same rate of energy exchange from heat transfer perspective. More heat transfer area corresponds to higher materials requirement and therefore higher cost of heat exchanger. Figure 1.3 compares different costs involved in acquisition and operation of a counter-flow heat exchanger. The capital cost includes material cost and fabrication cost, which reduces with increase in temperature difference between the streams. The fuel cost increases with increase in temperature difference due to higher irreversibility. The total cost is summation of fuel cost and capital cost. The point labeled a corresponds to the design with minimum total annual cost. Between location a0 and a00 , there is no significant change in annual cost. The (∆T )min between a0 and a00 is termed as nearly optimal design. Optimal design offers more options for selection of final design. Specifying (∆T )min outside the nearly optimal range would be a design error. Notes: 1. The above example of a heat exchanger design involves just one design variable i.e. temperature difference between streams. Several design variables may be considered and optimized simultaneously. 2. Thermal systems typically involve several components that interact with one another in complicated manner. Because of component inter-dependency, a unit of fuel saved by reducing the nonideality of one component (heat exchanger) may lead to waste of one or more units of fuel elsewhere in the system. As a result, there will be an increase in fuel consumption of the overall system. Therefore,

8

Design and Analysis of Thermal Systems

the objective of a design is to optimize the overall system consisting of several components. Optimization of one component does not guarantee that the overall system is working in an optimum condition.

1.6

STAGES OF THE DESIGN PROCESS

The flow chart of a design’s life cycle is presented in Figure 1.4. The broad design process is considered and the role of the design team has been discussed. This design process is not the only possible choice, and alternative strategies can be employed. There are three distinct stages of the design as follows: 1. Understanding the problem 2. Concept development 3. Detailed design The first step is to explore the viability of the overall design process when the requirement of the thermal system is specified. The design project may begin with an idea that something is worth doing. The design teams has to be formed for evaluating the idea and defining the specification of the design. Different concepts for achieving the design objective can be developed subsequently. The detailed design is carried out after finalizing the concept. Figure 1.4 shows the flow chart with details of different design stages. Iteration, concurrent design reviews and pilot plants are important steps at all stages of the design process. Iteration: The iteration identifies and corrects any problem as early as possible during design stages. Through iteration, more knowledge is obtained and utilized even though iteration involves some cost. Concurrent Design: The design process involves a wide range of skills and experience. The complete system design is well beyond the capabilities of a single individual and a group effort is essential. All the departments of a company are involved in the concurrent design process. Therefore, decisions can be made earlier with better knowledge, avoiding delay and errors and shortening the design process. Concurrent design combines the efforts of process engineering, manufacturing, maintenance, cost accounting, environmental engineering, production engineering, marketing and so on. Concurrent design overcomes the weakness inherent in traditional design approaches where evolving designs are passed from one department to another with minimal communication. There is little appreciation on the significance of design decisions made earlier by another department during traditional design process. As a result, there is a poor understanding of the impact of subsequent changes on such decisions. Design Reviews: Design reviews are carried out in the form of presentations to other design team members. This is a key element of concurrent design approach. It provides formal opportunities to share information and solicit inputs for improving the design. The design team members may challenge one another by frequently asking “why” for thinking more deeply about underlying assumptions and the viability of proposed approaches to problems arising during the design process.

Introduction

9

Figure 1.4 Different stages of the design process. Pilot Plants: Laboratory testing, prototyping or pilot plant tests may be necessary to finalize the design when there is lack of accurate design data. 1.6.1

DFX STRATEGIES

DFX strategies are another concept of the design process where, DF denotes “design for” and X identifies a characteristic to be optimized. For example, DFA stands for “Design for Assembly”, DFM stands for “Design for Manufacturing” and DFE stands for “Design for Environment”. All the critical factors that influence Xs should be identified at the concept development stage and considered throughout the life cycle of a design. Different stages of designs are elaborated in the following sections.

10

1.6.2

Design and Analysis of Thermal Systems

FORMULATION OF THE DESIGN PROBLEM

The formulation of the design problem involves specification of the following: (1) requirements, (2) given parameters, (3) design variables, (4) limitations and constraints and (5) safety, environmental and other considerations [Jaluria, 2007]. 1.6.2.1

Requirements

The requirements form the basis for design and evaluation of different designs. It is necessary to express the design requirements quantitatively and the permitted variation or tolerance level of these quantities. Let a water flow system is needed to obtain a specific volume flow rate of a system. There may be variations in the operating conditions leading to change in the flow rate (Q). Therefore, it is essential to determine the possible increase or decrease in the flow rate that can be tolerated by the system. We may have to design the system for delivering the desired flow rate Q0 with a possible maximum flow variation of ∆Q. This may be expressed quantitatively as Q0 − ∆Q ≤ Q ≤ Q0 + ∆Q. Suppose, we have to design a water cooler. In this case, the flow rate Q0 and the desired temperature T0 at the outflow are the requirements. In addition to the flow rate, Q expressed as before, the temperature is specified as T0 − ∆T ≤ T ≤ T0 + ∆T , where ±∆T is the acceptable variation in the outflow temperature. Note: It is critical to determine the main requirements of the system and to focus on satisfying these requirements because it is often difficult to meet all the desired characteristics of the system. Therefore, the requirements that are not particularly important for the chosen application may have to be ignored. It is recommended to satisfy the most essential requirements. Subsequently, attempts can be made to satisfy other less important requirements by varying the design within the specified constraints and limitations. Example: For a refrigeration system, the design should be completed first for the specified temperature and heat removal rate. Subsequent effort can be to find a substitute for the refrigerants R-11 and R-12, or to replace the compressor with one that is more efficient, or to vary the dimensions of the freezer or to improve the temperature control arrangement. 1.6.2.2

Given Parameters

There are parameters which are given or fixed i.e. materials, dimensions, geometry and basic concept or method during the formulation of the design problem. The other possibility is that some of the materials and dimensions are given and others are to be determined as part of the design. If the basic concept is not fixed or given, different concepts may be considered, resulting in considerable flexibility in design. If most of the parameters are fixed for a particular system, the design problem becomes relatively simple because only a small number of variables are to be determined. For example, for the design of an electronic cooling system, the electronic component

Introduction

11

size, the geometry and dimensions of the board, the number of electronic components on each board and the distance between two boards may be given. In a solar energy system, sensible heat storage in water may be chosen as the concept, with the dimensions, geometry etc. of the tank as the design variables. 1.6.2.3

Design Variables

The design variables are the quantities that may be varied during design in order to satisfy the given requirements. Attention is focused on these parameters during the design process. The design variables are varied to determine the behavior of thermal system and then chosen so that the system meets the given requirements. It is important to isolate the main design variables of the problem, since the complexity of design procedures is a strong function of the number of variables. The variables in the design problem may be classified as (1) hardware, and (2) operating conditions. Hardware can be components of a system, dimensions, materials, geometrical configuration and other quantities that constitute the structure of system. Varying these parameters generally results in changes in the fabrication process and assembly of system. Changes in the hardware are not easy to implement if existing systems are to be modified for a new design or for optimization. Operating conditions are the parameters that can be varied relatively easily, over specified ranges, without changing the hardware of a given system, such as temperature, flow rate, pressure, speed, power input etc. The design process yields the ranges for such parameters, with optimization indicating the values at which the performance is optimal. 1.6.2.4

Constraints and Limitations

The design constraints may arise due to material, weight, cost, availability and space limitations. The maximum pressure and temperature that a given component can be subjected to is limited by the properties of its material. For example, the semiconductor devices are very sensitive to temperature. Therefore, the temperatures of electronic equipment are constrained to values about 100 ◦ C. These constraints may be written as T ≤ Tmax , P ≤ Pmax . There may be weight restrictions in the design of portable computers, airplanes, automobiles and rocket systems. Similarly, volume constraints may be important in room air conditioners, household refrigerators etc. Constraints also arise due to conservation principles. The energy rejected from a power plant to a cooling pond is mC ˙ p ∆T , where m˙ is the mass flow rate of cooling water and ∆T is its temperature rise going through the condenser. This energy must be rejected to the environment through heat loss at the water surface. We should have: mC ˙ p ∆T = hAsur f ace (Tnew − Told ) where (Tnew − Told ) is the rise in the average surface temperature. A limitation on this temperature rise is specified by regulations. As a result, we have restriction on temperature rise in the condensers as well as on the total flow rate.

12

Design and Analysis of Thermal Systems

Figure 1.5 An example of the problem statement for air conditioning a house. 1.6.2.5

Safety, Environmental and Other Considerations

Disposal of waste, particularly hazardous waste from chemical plants and radioactive waste from nuclear facilities, is an important consideration that substantially affects the design of a thermal system. For example, safety concerns of nuclear facilities demand that adequate safety features be built into the system. To illustrate, the system must shut down if the temperature or heat flux levels exceed safe values. A safety feature would not allow the heater to be turned on if the fluid level is too low in a boiler. This will avoid any damage to the heaters and keep the operation safe. Similarly, the energy source may be changed from gas to electricity because of safety concerns in an industrial system. An oil furnace may be developed instead of a gas furnace for the same reason. Use of proper refrigerant for an air conditioner may be essential to minimize ozone layer depletion due to environmental considerations. 1.6.2.5.1 Problem Statement of an Air-Conditioning System Let’s consider the design of a building air-conditioning system, where the interior of the building is to be maintained at a temperature of 23 ± 7 ◦ C. The ambient temperature can go as high as 48 ◦ C. The rate of heat dissipated from the house is given as 5.0 kW. The geometry and dimension of the building is given. The concept for operation of the building air-conditioning system is shown in Figure 1.5. We can systematically describe the above design problem as follows: Requirements: Temperature inside the building must be maintained within 16 and 30 ◦ C. Note: In typical cases, the rate of cooling or response time ττ is also a requirement. We can write the energy balance relationship for the building as: mC p

∂T = Q − Qr ∂τ

Introduction

13

where, mC p is the thermal capacity of the building air, Q is the thermal load and Qr is the heat removal rate or loss from the building. Given: The building geometry, location and dimensions are provided. Maximum ambient temperature is 48 ◦ C. Rate of heat dissipation from the house is 5.0 kW. The time needed (ττ ) to cool the building to 67% of its initial temperature difference from the ambient is defined here as the characteristic response time. Constraints: Limitation on size, volume and weight of air conditioner are possible constraints. Maximum air flow rate circulating in the house can also be a possible constraint. Design Variables: System parts i.e. condenser, evaporator, compressor, valve and the type of refrigerant are possible design variables.

1.7

CONCEPTUAL DESIGNS

The design effort starts with the selection of a conceptual design. The design concept is initially expressed in fairly vague terms as a method that might satisfy the given requirement and constraints. Conceptual design can be either an invention of a new approach not employed before or modifications of the existing systems. Creativity, originality, experience, knowledge of existing systems and information on current technology play a key role in development of conceptual design. Innovative Conceptual Design: Brainstorming, where a group of people collectively try to generate a variety of ideas to solve a given problem is one of the techniques for innovative conceptual design. Design contests and awards to employees with the best ideas can also promote the generation of innovative concepts. Various ideas brought forth must be examined before they are discarded. Selection from Available Concepts: In an industry, the ideas that have been tried in the past to solve problems similar to the one under consideration should be considered. Existing literature can be used to generate additional information on various concepts and solutions that have been previously employed. The conceptual design for the present problem may then be selected from the list of earlier concepts and modified based on this information. In this case, the basic concept is similar to the earlier concepts and the system design may be quite different. Even those ideas that did not yield satisfactory designs earlier must be considered because of changes in the problem statement and availability of new technology. Different concepts can also be combined to yield the conceptual design for a given problem. For example, both forced-air cooling and liquid-immersion cooling may be employed for different parts of an electronic system due to different heat input levels. Example: Let us consider the task of transporting iron ore from the loading dock to the blast furnace of a steel plant. This can be achieved by trucks, trains, conveyor belts and carts. Each of these represents different concepts for the transportation system. The final choice is guided by several factors i.e. the distance over which the material is to be transported, the size and form in which the iron ore is available, and the rate at which the material is to be fed. For small plants, individual carts and

14

Design and Analysis of Thermal Systems

Figure 1.6 Different steps of the detailed design process. trucks driven by the workers may be adequate. Trains may be the most appropriate method for large plants. 1.7.1

MODIFICATION IN THE DESIGN OF EXISTING SYSTEMS

The simplest approach for obtaining a conceptual design for a given problem is the modification in the design of existing systems. The conceptual design provides the design of existing system along with the possible modifications needed to meet the requirements of a given problem. The overall configuration of the system is kept largely unchanged, and only a few relevant components or subsystems are varied. Many thermal systems in use today have evolved through modifications of existing system over the years. Example: Rankine cycle is the basic thermodynamic cycle used for steam power plants. Some of the conceptual designs that have been modified are super-heating of vapour leaving the boiler, reheating of steam passing through the turbine, and regenerative heating of the working fluid using stored energy from an earlier process in the system. 1.7.2

STEPS IN THE DESIGN PROCESS

In this section, we present the detailed steps of a design process. The flow chart showing different steps during the design process is shown in Figure 1.6. The main steps of the detailed design process are (1) initial physical system, (2) modeling of system, (3) simulation of system, (4) evaluation of different designs, (5) optimal design, (6) safety features, automation and control and (7) communicating the design. 1.7.2.1

Physical Systems

The starting point in the design process is the details of physical system obtained from conceptual design. The physical system is well defined in terms of the following details: (1) overall geometry and configuration of the system, (2) different

15

Introduction

components or subsystems that constitute the system, (3) interaction between various components, (4) given or fixed quantities of the system and (5) initial values of the design variables. 1.7.2.2

Modeling

Most practical thermal systems are fairly complex. Therefore, it is necessary to focus on the dominant aspects of the system and neglect relatively small effects. The objective is to simplify a given problem and make it possible to investigate its characteristics and behavior for a variety of operating conditions. Idealization and approximation of processes that govern the system are used to simplify analysis. This is known as modeling. Modeling of a thermal system yields a set of algebraic, differential or integral equations that describes the behavior of the actual system. The system is described as Fi (x1 , x2 , x3 , . . . , xn ) = 0, for i = 1, 2, 3, . . . , n. where, xi and Fi represent, respectively, the physical variables and the equations that govern the system. In thermal systems, the nonlinear ordinary and partial differential equations are often encountered. Discretized equations are derived on the basis of numerical techniques such as finite difference and finite element methods giving rise to a numerical model. The numerical/analytical results must be validated preferably by comparisons with available experimental data. Modeling of a thermal system also makes use of dimensional analysis. The governing dimensionless groups are determined for a thermal system using the principle of dimensional analysis. This simplifies the analysis of experimental and simulation results by reducing the number of parameter that needs to be varied for characterization of a given process/system. Modeling not only simplifies the problem but also eliminates relatively minor effects that only serve to confuse the main issues. It also allows satisfactory inclusion of experimental results into the overall model. Modeling is generally first applied to individual components, parts or subsystems of a thermal system under consideration. The individual models are subsequently combined considering the interactions between various components. Different submodels are linked to each other through boundary conditions and the flow of mass, momentum and energy. When these individual models are coupled with each other, the overall model for the thermal system is obtained. The main subsystems of an air-conditioning system are shown in Figure 1.7. The model of the subsystems are coupled through the fluid and energy transport i.e. the outlet from the evaporator is the inlet to the compressor whose outlet is inlet to the condenser. 1.7.2.3

Simulations

Simulation is the process of subjecting the model of a thermal system to various inputs. The behavior of the system at various operating conditions helps in predicting

16

Design and Analysis of Thermal Systems

Figure 1.7 The main subsystems that combine to make an air-conditioning system. the characteristics of the actual physical system. Both hardware and operating conditions are varied to study the system characteristics. A very important question that must be answered in any numerical simulation is how accurately it represents the actual or real world system. This involves ascertaining the validity of the various approximations used during modeling as well as estimating the accuracy of numerical algorithm. Sometimes, simpler or similar systems for which experimental results are available may be simulated to validate the models. A common approach for simulating thermal system is to fix the hardware and vary the operating conditions over the desired ranges. The hardware is then changed to give a different design and the process is repeated. The simulation of the system is carried out with different design variables until an acceptable design or a range of acceptable designs is obtained. Example: The output of the heat exchanger can be expressed as Q = F(D1 , D2 , L,t1 ,t2 , m˙ 1 , m˙ 2 , T1i , T2i ) T20 = G(D1 , D2 , L,t1 ,t2 , m˙ 1 , m˙ 2 , T1i , T2i ) Here, the heat transfer rate, Q and outlet temperature of the outer fluid, T20 are the outputs from the model of heat exchanger. Mass flow rate (m), ˙ length (L), thickness (t) and temperature (T ) are the inputs to the model, where subscript 1 and 2 correspond to the inner and outer fluid stream, respectively. Subscript i corresponds to the inner fluid. The diameters and the length may be chosen so that the constraints due to size or space limitations are not violated. Tube diameter (D1 , D2 ) and thickness (t1 , t2 ) choices may be restricted by the availability with the manufacturer to reduce costs. Each combination of the three design variables D1 , D2 and L represents a system design, which can be obtained by keeping two of them fixed while varying the other one. Each system design is subjected to different flow rates and temperatures, which represent the operating conditions.

17

Introduction

1.7.2.4

Acceptable Design Evaluations

An acceptable design is one that satisfies the given requirements for the system without violating the given constraints (safety, environmental regulation and financial constraints). In almost all practical cases, there are many possible solutions to a given design problem. The acceptable design obtained is by no means unique. The best design may be chosen on the basis of a given criterion such as minimum cost or highest efficiency. For the counter-flow heat exchanger, the requirements and constraints may be written as: Requirements: Q = Q0 ± ∆Q, T20 = T0 ± ∆T Constraints: (D1 )min < D1 < D2 − 2t2 D2 < (D2 )max , L < Lmax Operating conditions: m˙ 1 , T1i Fixed quantities: m˙ 2 = (m˙ 2 )0 ± ∆m˙ 2 , T2i = (T2i )0 ± ∆T2i 1.7.2.5

Optimal Designs

Optimization is a systematic approach used to minimize a chosen quantity or function applied to a number of acceptable designs. The optimization problem can be expressed in the following manner: Objective function: U(x1 , x2 , x3 , . . . , xn ) → minimum/maximum Equality constraint: Gi (x1 , x2 , x3 , . . . , xn ) = 0 Inequality constraint: H j (x1 , x2 , x3 , . . . , xm ) ≤ C j where, U is the objective function, x1 , x2 , . . . , xn are the design variables, Gi is the equality constraint, H j is the inequality constraint and j = 1, 2, 3, . . . , m are the number of inequality constraints. Example: Let us consider the example of a heat exchanger. The objective is to minimize the cost of equipment. Minimizing the total material used for manufacturing will help in reducing the total cost. Therefore, the cost can be represented by the volume of the material, which can be written in terms of design variables as V = πD1 Lt1 + πD2 Lt2 Design variables D1 , L, t1 , D2 , t2 are to be varied in the domain of acceptable designs in order to minimize objective function, V . The market availability of different tube sizes may also be included in this process to employ dimensions that are easily obtainable without significantly affecting the optimum. Once the optimal design is obtained, the operating conditions m1 and T1i may also be varied to determine if the costs could be further minimized by adjusting the flow rate and inlet temperature at the fluid stream while keeping the other fluid stream fixed. Thus, the overall costs may be minimized.

18

Design and Analysis of Thermal Systems

Figure 1.8 A schematic of a safety arrangement in a thermal system. (Jaluria [2007].) 1.7.2.6

Safety Features, Automation and Control

An important ingredient of a thermal system for safe and successful operation is the control scheme, which not only ensures safety of the system and the operator but also maintains the operating conditions within specified limits for satisfactory output. Sensors that monitor the temperatures, pressures, flow rates and other physical quantities in the system are employed to turn off the system or the inflow of material and energy into the system if the safety of people working on it is threatened. Figure 1.8 shows the typical safety arrangement of a thermal system. The possible sensors used for sensing the behavior of the physical system can be thermistors, thermocouples, flow meters, pressure transducers and so on. The actuators reduce the flow rate, increase/decrease the heat input, turn off power supply etc. as a control measure. 1.7.2.7

Communicating the Design

Communicating the final design to the client or customer and to those who will implement the design i.e. fabrication section is an important step for the overall success of a project. It is necessary to highlight the salient features of the design and justify how it meets the design requirements. The basic approach adopted in the development of a design, including information on modeling and simulation must also be presented. The accuracy and validity of the results should be discussed. Design team consists of individuals who have diverse backgrounds to study different aspects of a design problem. Therefore, good communication is very important. The communication depends on the target audience. The mode of communication by the design group depends on the target groups. When communicating with the fabrication section, the design group needs to communicate through detailed engineering drawings, parts list and materials list. When communicating with the group involved in prototype testing, the computer programs and numerical simulation results need to be supplied by the design group. When communicating with the customer, the working models and important results of the system under different operating conditions must be provided. The design team needs to communicate with the management through reports and presentations. The design team may also have to communicate with

Introduction

19

Figure 1.9 Various elements of a typical CAD system. (Jaluria [2007].) the patents and copyright group of the organization for protecting the design from duplication by the competitors. The design team needs to communicate with the sales and marketing team for feedback and improvement of the existing design. 1.7.3

COMPUTER-AIDED DESIGNS

The term computer-aided design largely refers to an independent or stand-alone system such as single or multiple work stations or personal computers. The interactive use of the computer system considers various design options and obtains an acceptable or optimal design. A CAD system involves several aspects that help in the iterative design process. A representative CAD system and its components are shown in Figure 1.9. Some of the important features of a CAD system are 1. 2. 3. 4. 5. 6. 7.

Interactive application of the computer system Graphical display of result Graphical input of geometry and variables Available software for analysis Available database for considering different options of material and components Knowledge base from current engineering practice Storage and information from earlier designs

Computer software codes for analysis are often based on finite element or finite volume or finite difference method like Ideas, Nastran, Fluent etc. Additional codes for curve fitting, numerical solutions of ordinary algebraic equations, differential equations, integration, differentiation, matrix inversions like Maple, Mathematica, MathCAD, MATLAB etc. are also used.

20

Design and Analysis of Thermal Systems

Notes: 1. The modeling aspect is the most involved part of thermal systems compared to other CAD systems. The remaining aspects are similar to CAD systems for other engineering fields. 2. Most of the CAD systems are devoted to the design of mechanical systems and components such as gears, springs and vibrating devices. It is not easy to develop similar CAD systems for thermal processes due to complexity of thermal systems. Some CAD systems are currently available for simple systems such as heat exchangers, air-conditioners, heating systems and refrigerators. Interactive design is generally not possible for more elaborate thermal systems, since numerical simulation might involve considerable CPU time and memory requirements. Parallel computer systems hold promise for the CAD of practical thermal systems. 3. The extent to which CAD of thermal systems can be applied is limited by the availability of property data in suitable forms. 4. CAD relies heavily on suitable process equipment design programs. Libraries of programs are available for designing or rating one of the most common thermal system components i.e. heat exchangers. Software for other types of equipment, including piping networks, are also available. 5. Common software allow the engineer to model the behavior of a system or system components under specified conditions and do thermal analysis, sizing, costing and optimization required for concept development. They are developed along two lines i.e. a. Sequential-modular approach b. Equation-solving approach In the sequential-modular approach, library models associated with the various components are called in sequence by an executive program. Here, the output stream data for each component is the input for the next component. In the equation-solving approach, all the equations representing individual components are assembled as a set of equations for simultaneous solution. 6. Optimization is an important component of CAD systems. Complex thermal systems are described in terms of a large number of equations, including nonlinear equations and non-explicit variable relationships. Therefore, the term optimization implies improvement rather than calculation of a global mathematical optimum. Conventional optimization procedure may be sufficient for relatively simple thermal systems. However, in thermal systems, costs and performance data are seldom in the form required for optimization. The conventional optimization methods can become unwisely time-consuming and costly with increasing system complexity. The method of thermo-economics is a better approach as it can assist in improving system efficiency and reducing the product costs by pinpointing the required changes in structure and parameter values of a thermal system.

Introduction

21

PROBLEMS 1. You have to design a water cooler for drinking water. The water intake on a summer day is at 30 ◦ C and the cooler must supply drinking water in the range of 12–24 ◦ C at a maximum flow rate of 4.5 × 10−3 m3 /min. Give the requirements for the design. Develop an appropriate conceptual design and suggest the relevant design variables and constraints. 2. You have to design a cooling tower for heat rejection from a power plant. The rate of heat rejection to a single tower is given as 100 MW. Ambient air at temperature of 25 ◦ C and relative humidity of 0.8 is to be used for removal of heat from the hot water coming from the condensers of the power plant. The temperature of the hot water is 15 ◦ C above the ambient temperature. Give the formulation of the design problem in terms of the fixed quantities, requirements, constraints and the design variables. 3. The condensers of a 300 MW power plant operating at a thermal efficiency of 35% are to be cooled by the water from a nearby pond. The intake water is available at 22 ◦ C. The temperature of the water discharged back into the lake must be less than 30 ◦ C. Quantify the design problem for the cooling system. 4. You have to design a thermal system for the storage of thermal energy using an underground tank of water. The tank is buried with its top surface at a depth of 2.5 m. It is a cube of 1.5 m on each side. The water in the tank is heated by circulating water through a solar energy collection system. The heat input to the water is due to the solar energy flux. Characterize the design problem in terms of the fixed quantities and design variables. 5. Formulate the design problem for a hot rolling manufacturing process. The steel plate achieves reduction of thickness from 2.5 cm to 1.5 cm at a feed rate of 1.5 m/s during hot rolling. 6. Formulate the design problem of a water supply system to a water treatment plant. The water needs to be supplied from a river to the water treatment plant at 0.2 m3 /min flow rate and a pressure of 4 atm. 7. Discuss the nature, type and possible locations of sensors that need to be used for safety as well as control of a thermal system, which heat short metal rods in a gas furnace and then bend them into desired shapes in a metal forming process. 8. Select the ideal liquid for immersion cooling of an electronic system. 9. You are designing the tank of a water cooler. You have to decide on the location of inlet and outlet port of water from the tank. What is the ideal location for the water tap? Discuss with reasoning.

22

Design and Analysis of Thermal Systems

10. As an engineer at Tata Motors, you are asked to design an engine cooling system. The system should be capable of removing 20 kW of energy from the engine of a car at a speed of 75 km/h and ambient temperature of 35 ◦ C. The system consists of radiator, fan and flow arrangement. The dimensions of the engine are given. The distance between the engine of the car and the radiator must not exceed 2.0 m and the dimensions of the radiator must not exceed 0.45 × 0.45 × 0.15 m3 .

REFERENCES A. Bejan, G. Tsatsaronis, and M. Moran. Thermal Design and Optimization. John Wiley & Sons Inc, 1996. Y. Jaluria. Design and Optimization of Thermal Systems. CRC Press, 2007.

and Simulation 2 Modeling Basics 2.1

INTRODUCTION

Modeling is one of the crucial steps in design of thermal systems. It is defined as the process of simplifying a given problem, so that it can be represented in terms of a system of equations for analysis or a physical arrangement for experimentation. Simulation is the process of obtaining information on the behavior and characteristics of the real system by analyzing, studying or observing a model of the system. In this chapter, various issues related to modeling and simulation of a thermal system are discussed.

2.2

TYPES OF MODELS

There are two types of models: (1) descriptive and (2) predictive (Jaluria 2007). Descriptive model is a working model of compressor, turbine or IC engines, which may be used to explain how the device works. Model made of plastics may have a cutaway section to show the internal mechanism. Predictive models can be used to predict the performance of a given system. For example, the equation describing the cooling of a hot iron rod immersed in a cold water pond represents a predictive model. It allows one to obtain the temperature variation with time, and also to determine the dependence of physical variables such as initial temperature of the rod, water temperature and material properties. The predictive model can also be classified as: (1) analog model, (2) mathematical model, (3) physical model and (4) numerical model. 2.2.1

ANALOG MODELS

An analog model is based on the analogy or similarity between different physical phenomena [Jaluria, 2007]. This type of model helps to use solution and results from a familiar problem to obtain the solutions for a different unsolved problem. Example: The conduction heat transfer through a composite wall of a cold storage wall can be solved as an analogous electrical circuit. The following analogy is applicable based on the principle of heat transfer. • • •

Thermal resistance ≡ Electrical resistance Heat flux ≡ Electrical current Temperature difference ≡ Electrical voltage

It may be noted that the Ohm’s law can be employed to compute the total thermal resistance. The heat flux for a given temperature difference can be calculated using DOI: 10.1201/9781003049272-2

23

24

Design and Analysis of Thermal Systems

the total thermal resistance. Another example of analog model is the use of Reynolds analogy,

where,

2 Cf = StPr 3 2 h St = ρVC p

Cf =

τw 0.5ρV 2

Here, St is the Stanton number, C f is the skin friction coefficient, h is the heat transfer coefficient, ρ is the density, V is the velocity and τw is the wall shear stress. This equation is valid for Prandtl number (Pr) between 0.6 and 60. The skin friction data can be used for calculation/estimation of heat transfer while using the Reynolds analogy. 2.2.2

MATHEMATICAL MODELS

A mathematical model is one that represents the performance and behavior of a given system in terms of mathematical equations. There are two types of mathematical models: (1) theoretical and (2) empirical. Theoretical models use physical principles such as conservation laws, e.g. conservation of mass, momentum and energy, to derive the governing equations. Empirical models use curve fitting of experimental or simulation data to obtain mathematical representations of the thermal system behavior. 2.2.3

PHYSICAL MODELS

A physical model is one that resembles an actual system and is generally used to obtain experimental results on the behavior of systems. Example: A scaled-down model of a truck positioned inside a wind tunnel can be used to study the drag force acting on the truck. Similarly, water channels are used to investigate the forces acting on ships and submarines. The physical model may be a scaled-down version of the actual system or a full-scale experimental model to study the basic characteristics of the system or a prototype, which is essentially the first complete system to be checked in detail before the start of production. 2.2.4

NUMERICAL MODELS

A numerical model facilitates the solution of mathematical model using computational platform. This type of model helps to obtain quantitative results on the behavior of a system for different operating conditions and design parameters using a computer. This model involves selecting the appropriate method for solution e.g. finite difference or finite element method. The mathematical equations are discretized to put them in a form suitable for digital computation, appropriate numerical parameters such as grid size, time step etc. are chosen and numerical solution is obtained by executing the numerical code.

Modeling and Simulation Basics

2.3

25

MATHEMATICAL MODELING

A general step-by-step procedure is outlined for mathematical modeling of thermal systems on the basis of different considerations that arise in these systems. Generally, there is no unique model for a typical thermal system. Some guidelines that may be useful for developing an appropriate model are discussed here. 2.3.1

TRANSIENT/STEADY STATE

Time brings in an additional independent variable of a mathematical model, which increases the complexity of problem. Therefore, it is important to determine whether time variation effects can be neglected. There are two main characteristic time scales that need to be considered. The first time scale, τr , refers to the response time of the material or body under consideration. The second time scale, τc , refers to the characteristic time of variation in ambient operating conditions. Example: The response time, τr , for a lumped body at uniform temperature subjected to a step change in ambient temperature with convective cooling or heating is given by the expression ρCV τr = hA where, ρ is the density, C is the specific heat, V is the volume of the body, A is the surface area and h is the convective heat transfer coefficient. Let’s consider different probable cases based on the relative magnitude of τr and τc . 2.3.1.1

Case 1: τ c → ∞ (Large τ c )

In this case, the environmental conditions don’t change with time and the system may be treated as steady state. At the start of the process, the conditions change sharply over a short time and transient effects are important. Steady-state conditions are attained with increase in time. Example: Let us consider an unheated electronic chip, which is heated by an electric current. The chip temperature increases with time and after some overshoot in temperature, it attains steady state due to the balance between heat loss to the environment and the heat input (see Figure 2.1). 2.3.1.2

Case 2: τ c > τ r

This refers to the case where the material or body responds very quickly but the operating or boundary conditions change very slowly. Example: The solar flux has a slow variation with time on a sunny day while the solar collector has a rapid response. In such cases, the system may be modeled as quasi-steady, with the steady problem being used at different time instants (see Figure 2.2). This implies that the system goes through a sequence of steady states, each characterized by constant operating or environmental conditions. However, as time elapses, the steady-state results do vary because of the changes in environmental conditions. This situation is modeled as quasi-steady modeling, where the ambient temperature variations are replaced with finite number of steps, with the temperature being constant over each step (Figure 2.2). 2.3.1.4

Case 4: Periodic Processes

The behavior of a thermal system repeats over a given time period, τ p , in a periodic process. The main advantage of modeling a system as periodic is that steady-state results need to be obtained only over the time period of the cycle (see Figure 2.3) leading to considerable saving in computation time. 2.3.1.5

Case 5: Transient

When none of the above approximations is applicable, the system has to be modeled as a general time-dependent problem, with the transient terms i.e. time varying terms included in the model. This is the most complicated circumstance for simulating the system behavior.

27

Modeling and Simulation Basics

Figure 2.3 An example of the periodic process, where the body is subjected suddenly to a periodic variation in the heat input. 2.3.2

NUMBER OF SPATIAL DIMENSIONS

The spatial dimension required for modeling a thermal system refers to the number of spatial dimensions needed to describe a given system. Though all practical systems are three-dimensional (3-D), they can often be approximated as two-dimensional (2D) or one-dimensional (1-D). 3-D modeling is generally avoided unless absolutely essential because of additional complexity in obtaining a solution of the governing equations. The results from a 3-D model are also not easy to interpret. The simplification from a 3-D model to a 1-D/2-D model is based on the geometry of the system under consideration and on the boundary conditions. Example: Let us consider a 3-D conduction problem of a solid block of dimension (L × H ×W ) in x, y and z directions, respectively. For constant thermal conductivity (k) and no heat source in the material, the governing equation is expressed as ∂ 2T ∂ 2T ∂ 2T + 2 + 2 =0 ∂ x2 ∂y ∂z Using dimensionless variables, X = Lx ,Y = Hy , Z = Wz , φ = We have L2 ∂ 2 φ L2 ∂ 2 φ ∂ 2φ + 2 + 2 =0 2 2 ∂X H ∂Y W ∂ Z2

T Tre f

Since X, Y and Z all vary from 0 to 1, all the second derivative terms in this equation can be of the same order of magnitude i.e. from 0 to 1. Hence, the overall magnitude of each term in this equation is determined by magnitude of the coefficients i.e. if if

L2 120 ◦ C, we would have to either increase A or U. Th,o = Th,i −

q 35, 000 = 114.29 − = 107.46 ◦ C Ch 5, 129.6

Example: Find the surface area of a 1-2 TEMA E-shell and tube heat exchanger using the following data. Water enters the shell at 20 ◦ C at a rate of 1.4 kg/s. Engine oil flows at a rate of 1.0 kg/s with inlet and outlet temperatures equal to 150 ◦ C and 90 ◦ C, respectively. Assume the overall heat transfer coefficient, U = 225 W/m2 –K. The specific heat of water and oil is equal to 4.19 and 1.67 J/g–K, respectively.

181

Heat Exchangers

SOLUTION The heat capacity of shell liquid (water) is Cs = (mc ˙ p )s = 1.4 × 4.19 × 103 = 5866 W/K The heat capacity of tube fluid (oil) is Ct = (mc ˙ p )t = 1.0 × 1.67 × 103 = 1670 W/K The heat transfer rate from oil is q = Ct (Tt,i − Tt,o ) = 1670(150 − 90) = 100.2 × 103 W Using the energy balance equation, we write q 100.2 × 103 = 20 + = 37.1◦ C Cs 5866 C[t] 1670 C∗ = = 0.2867 = Cs 5866 Tt,i − Tt,o 150 − 90 ε= = = 0.4615 Tt,i − Ts,i 150 − 20    p ∗− ∗2 2 − ε 1 +C 1 +C 1    ln  NTU = p p  2 2 ∗ ∗ 1 +C∗ 2 − ε 1 +C + 1 +C Ts,o = Ts,i +

Cmin 1670 × NTU = × 0.6828 = 5.068 m2 U 225 Example: Calculate the shell-side pressure drop of a shell and tube heat exchanger using the following data: A=

Cross-flow area near the shell centerline (As ): 0.0444 m2 Flow area through the window zone (Aw ): 0.01261 m2 Number of effective tube row baffle section (Nc ): 9 Number of effective rows crossed in one window zone (Ncw ):3.868 Number of baffles (Nb ): 14 Oil flow rate (m˙ s ): 36.3 kg/s Ideal tube bank friction factor ( fc ): 0.23 Oil density (ρs ): 849 kg/m3 Correction factor for baffle-to-shell and tube-to-baffle leakage streams (Rl ): 0.59 Correction factor for baffle to shell bypass streams (Rb ):0.69 Correction factor for unequal baffle spacing on inlet and exit baffle section (Rs ): 0.81 Shell-side Reynolds number (Res ): 242 Tube outside diameter (do ): 19 mm Tube wall thickness (t): 1.2 mm Tube pitch (Pt ): 25 mm (square layout) Tube shell-side lubricating oil is cooled from 70 ◦ C to 65 ◦ C

182

Design and Analysis of Thermal Systems

SOLUTION     Ncw ∆ps = Rl (Nb − 1)∆pbi + Nb ∆pwi + 2∆pbi 1 + Rb Rs Nc   G2 µs,w 0.14 × Nc ∆pbi = 4 fi s 2ρs µs m˙ 36.3 Gs = = = 817.56 kg/m2 –s As 0.0444 µ

' 1 small (because of the small temperature difference). Pressure in Assuming µs,w s the tube bank in one baffle (∆pbi ) and pressure drop in one window section (∆pwi ) are calculated as 817.562 × 10.14 × 9 = 3259.35 Pa 2 × 849 m˙ 2 (2 + 0.6Ncw ) ∆pwi = s 2ρs As Aw 36.32 × (2 + 0.6 × 3.868) = = 5784.42 Pa 2 × 849 × 0.0444 × 0.01261   ∆ps = 0.59 (14 − 1)3259.35 + 14 × 5784.42   3.868 + 2 × 3259.35 1 + × 0.69 × 0.81 9 ∆pbi = 4 × 0.23 ×

= 17249 + 47779.3 + 5209.11 = 70237.41 Pa Example: Consider a single shell and single-tube pass heat exchanger with the following data: Outside diameter of the tube (do ): 19 mm Inside diameter of the tube (di ): 16 mm Square pitch of the tube (Pt ): 0.0254 m Baffle spacing (B): 0.5 m Shell fluid: Distilled water Shell flow rate (m˙ s ): 50 kg/s Shell fluid temperature at the inlet (Th,1 ): 32 ◦ C Shell fluid temperature at the outlet (Th,2 ): 25 ◦ C Tube fluid: Raw water Tube flow rate (m˙ t ): 150 kg/s Tube fluid temperature at inlet (Tc,1 ): 20 ◦ C Thermal conductivity of tube material (k): 42.3 W/m–K Total fouling resistance (R f ): 0.000176 m2 –K/W Flow velocity through each tube (um ): 2 m/s

183

Heat Exchangers

Properties of tube-side fluid c p,t = 4182 J/kg–K

c p,s = 4179 J/kg–K

2

ks = 0.612 W/m2 –K Prs = 5.75

kt = 0.598 W/m –K Prt = 7.01

(a) (b) (c) (d) (e) (f) (g) (h)

Properties of shell-side fluid

ρt = 998.2 kg/m3

ρs = 995.9 kg/m3

µt = 10.02 × 10−4 N-s/m2

µs = 8.15 × 10−4 N-s/m2

Calculate the shell-side heat transfer coefficient by the Kern method. Calculate the shell-side heat transfer coefficient by the Bell–Delaware method. Calculate the tube-side heat transfer coefficient. Calculate the overall heat transfer coefficient of the clean surface. Calculate the overall heat transfer coefficient of the fouled surface. Calculate the net heat transfer requirement. Calculate the net heat transfer area required considering the clean heat exchanger. Calculate the net heat transfer area required considering the fouled heat exchanger.

SOLUTION Total mass flow rate through a tube (m˙ t ) = Nt ρt µm Ac Thus, Number of tubes, Nt = =

m˙ t ρt um Ac 4 × 150 = 373.88 ' 374 998.2 × π × (0.016)2 × 2 "

Nt (CL )(PR)2 do2 Shell diameter, Ds = 0.785(CT P)

#1/2

CTP = 0.93 CL = 1.0 Pt 0.0254 = = 1.336 do 0.019 " #1/2 374 × 1 × 1.3362 × 0.0192 Ds = 0.785 × 0.93

PR =

= 0.5745 m = 575 mm

184

Design and Analysis of Thermal Systems

(a) Kern method Cross-flow area at the shell diameter, As = B(Ds − Ntc do ) where, Ntc =

Ds Pt

=

575 25.4

= 22.637 As = 0.5(0.575 − 22.637 × 0.019) = 0.0724 m2

Equivalent diameter, De =

Pt2 − πdo2 /4 πdo /4

0.02542 − π × 0.0192 /4 Π × 0.019/4 = 0.0242 m m˙ s De 50 × 0.0242 Re = = As µs 0.0724 × 8.15 × 10−4 = 20506.38  0.14 ho De 0.55 1/3 µb = 0.36Re Pr k µw =

Assuming µb = µw due to a small temperature difference. = 0.36 × 20506.380.55 × 5.751/3 = 0.36 × 235.25 × 1.7904 = 151.628 151.628 × 0.612 ho = = 3834.55 W/m2 –K 0.0242 (b) Bell-Delaware method Let’s assume the combined effect of different correction factors i.e. jc , jl , jb , js , jr is 60%. Hence, ho = 0.6 × hid  where, hid = jiC ps ji = a1 Reas 2 a=

m˙ s As 



1.3 Pt /do

a3 1 + 0.14Reas 4

ks C ps µs a

!2/3

µs µs,w

!0.14

185

Heat Exchangers

From Table 5.11, a1 = 0.37, a2 = −0.395, a3 = 1.187, a4 = 0.37 m˙ s do 0.019 × 50 = 16100 = As µs 0.0724 × 8.15 × 10−4 1.187 a= = 0.196 1 + 0.14(16100)0.37 1.3do 1.3 × 0.019 = = 0.9724 Pt 0.0254  a 1.3do = 0.97240.196 = 0.994 ' 1 Pt Res =

Hence, j1 = 0.37 × Re−0.395 = 0.37 × 16100−0.395 = 0.0081 s  2/3  0.612 50 hid = 0.0081 × 4179 0.0724 4179 × 8.15 × 10−4 = 7443.9 W/m2 –K ho = 0.6 × hid = 0.6 × 7443.9 = 4466.3 W/m2 − K (c)

Nut =

( f /2)Reb Prb 2/3

1.07 + 12.7( f /2)1/2 (Prb − 1) ρt µm dt 998.2 × 2 × 0.016 Ret = = = 31878.6 µt 10.02 × 10−4 f = (1.58 × ln(Reb ) − 3.28)−2

= (1.58 × ln(31878.6) − 3.28)−2 = 0.0058 f /2 = 0.0058/2 = 0.0029 0.0029 × 31878.6 × 7.01 Nut = 1.07 + 12.7 × 0.00291/2 (7.012/3 − 1) = 224.16 Nut k 224.16 × 0.598 ht = = = 8377.98 W/m2 –K di 0.016 (d) Overall heat transfer coefficient for a clean surface is 1 1 1 do ro ln(ro /ri ) = + + Uc ho hi di k 1 1 0.019 9.5 × 10−3 × ln(19/16) = + + 3835.55 8378 0.016 42.3 Uc = 2445.5 W/m2 –K

186

Design and Analysis of Thermal Systems

(e) Overall heat transfer coefficient of the fouled surface is 1 1 1 = +Rf = + 0.000176 Uf Uc 2445.5 U f = 1709.7 W/m2 –K

Q = (mc ˙ p )h (Th,1 − Th,2 ) = 50 × 4179 × (32 − 25) = 1462650 W Q Tc,2 = + Tc,1 (mc ˙ p )c 1462650 + 20 = 22.33◦ C = 150 × 4182 (32 − 22.3) − (25 − 20)   = 7.09 ◦ C LMTD = ln 32−22.3 25.20

(f)

(g) Heat transfer area for a clean surface is Ac =

Q 1462650 = 84.36 m2 = Uc ∆Tm 2445.5 × 7.09

(h) Heat transfer area for a fouled surface is As =

Q 1462650 = = 120.66 m2 U f ∆Tm 1709.7 × 7.09

Example: Consider a compact heat exchanger using matrix surface type 8.0 − 3/8 T. The matrix is 0.6 m long, and air at 1 atm and 400 K flows across the matrix at a velocity of 24.8 m/s. Calculate the heat transfer coefficient and friction pressure drop for the air side. Use the following properties of air at 1 atm and 400 K. ρ = 0.8825 kg/m3 ,

µ = 2.29 × 10−5 kg/m–s,

c p = 1013 J/kg–K,

SOLUTION From Figure 5.35 for this surface, Amin = σ = 0.534 Afr Dh = 0.3633 cm ρu∞ A f r ρu∞ 0.8829 × 24.8 = = == 40.98 kg/m2 –s Amin σ 0.534 GDh 40.98 × 0.3633 × 10−2 Re = = = 6501.3 µ 2.29 × 10−5

G=

Pr = 0.719

187

Heat Exchangers

From Figure 5.35 for Re = 6501.3, h Pr2/3 = 0.005 Gc p 0.005 × 40.98 × 1013 = 258.6 W/m2 –K h= (0.719)2/3 For Re = 6501.3, f = 0.02 (from Figure 5.35) At 4L 4 × 0.6 = 660.6 = = Amin Dh 0.3633 × 10−2 G2 At 40.982 ∆p f = f = 0.02 × × 660.6 2pa Amin 2 × 0.8825 = 12570.94 Pa

PROBLEMS 1. A shell and tube heat exchanger with one-shell pass and multiples of two-tube passes is constructed from 0.0254 m OD tube to cool 693 kg/s of a 95% ethyl alcohol (C p = 3810 J/kg-K)from 66 ◦ C to 42 ◦ C, using 6.3 kg/s of water available at 10 ◦ C (C p = 4187 J/kg-K). ln the heat exchanger, 72 tubes will be used. Assume that the overall heat transfer coefficient based on the outer-tube area is 568 W/m2 -K. Calculate the surface area and the length of the heat exchanger. 2. A two-pass tube baffled single-pass shell; shell and tube heat exchanger is used as an oil cooler. Cooling water flows through the tubes at 20 ◦ C at a flow rate of 4.082 kg/s. Engine oil enters the shell side at a flow rate of 10 kg/s. The inlet and outlet temperatures of oil are 90 ◦ C and 60 ◦ C, respectively. Determine the surface area of the heat exchanger by the e–NTU method if the overall heat transfer coefficient based on the outside tube area is 262 W/m2 -K. The specific heats of water and oil are 4179 J/kg-K and 2118 J/kg-K, respectively. 3. A heat exchanger is to be designed to heat raw water by the use of condensed water at 67 ◦ C and 0.2 bar, which will flow in the shell side with a mass flow rate of 50,000 kg/h. The heat will be transferred to 30,000 kg/h of city water coming from a supply at 17 ◦ C (C p : 4184 J/kg-K). The fouling resistance equal to 0.000176 m2 K/W is assumed. The maximum pressure drop on the shell side is 5.0 psi. The water outlet temperature should not be less than 40 ◦ C. Calculate the heat exchanger length and pressure drops in both streams. The TEMA standard shell and tube heat exchanger with the following geometrical parameters is selected. Shell internal diameter: 0.39 m Number of tubes: 124

188

Design and Analysis of Thermal Systems

Outer tube diameter: 19 mm Inner tube diameter: 16 mm Thermal conductivity of tube material: 60 W/m2 -K Pitch size: 0.024 m Number of tube passes: 2 4. A single-pass shell and tube heat exchanger (condenser) heats 946 m3 /h of water from 10 ◦ C to 38 ◦ C . The heat exchanger uses a plain steel tube (k: 45 W/m-◦ C) with an internal diameter 0.0266 m and an outside diameter 0.0333 m. Steam is supplied at 107 ◦ C with the steam-side heat transfer coefficient equal to 68143 W/m2 -◦ C. The water-side heat transfer coefficient is equal to 1937 W/m. The mass velocity through the tube is equal to 8258.3 kg/m2 -s. Calculate the number of tubes required and the length of the tube. 5. Air at 2 atm and 500 K with a velocity of U = 20 m/s flows across a compact heat exchanger matrix having surface type I1.32-0737-S-R. The length of the matrix is 0.8 m. Calculate the heat transfer coefficient and the frictional pressure drop. Use the following properties of air at 500 K and 2 atm: ρ = 1.41 kg/m3 ; C p = 1030 J/kg-K; µ = 2.69x10−5 kg/m-s; Pr = 0.718. Air at 2 atm and 500 K with a velocity of U = 20 m/s flows across a compact heat exchanger matrix having surface type 9.29–0.737–S–R. Calculate the heat transfer coefficient and the frictional pressure drop. The length of the matrix is 0.8 m.

REFERENCES K. J. Bell. Heat Transfer Equipment Design, chapter Delaware method for shell design, pages 145–166. Hemisphere Publishing, 1988a. K. J. Bell. Heat Transfer Equipment Design, chapter Overall design methodology for shell and tube heat exchangers, pages 131–144. Hemisphere Publishing, 1988b. S. Kakac, H. Liu, and A. Pramuanjaroenkij. Heat Exchangers: Selection, Rating, and Thermal Design, Third Edition. CRC Press, 2012. ISBN 978-1-4398-4991-0. W. M. Kays and K. L. London. Compact Heat Exchangers. Mc-Graw Hill, third edition, 1984. R. K. Shah and D. P. Sekulic. Handbook of Heat Transfer Applications, chapter Heat exchangers, pages 17.1–17.169. McGraw-Hill, 1985. J. Taborek. Handbook of Heat Exchanger Design, chapter Shell and tube heat exchangers: single phase flow, pages 3.3.3–1–3.3.11–5. Begell House, 1991. TEMA. Standards of the Tubular Exchanger Manufacturers Association, ninth edition. Tubular Exchanger Manufacturers Association, Inc., 2007.

6 Piping Flow 6.1

INTRODUCTION

A piping network is an integral part of the majority of energy systems. It consists of pumps, turbines, heat exchangers, pipes, valves and other auxiliary devices. There are two aspects of a piping system design: (1) to obtain the performance characteristics of a given piping network and (2) to decide the sizing of the system i.e. the diameter, length, arrangement, material of the pipes etc. Most of the devices present in a piping system are described by the manufacturers’ performance curves for specified fluids. The procedure for computer implementation of piping network design is presented in the following sections.

6.2

ENERGY EQUATIONS

The energy equation forms the basis of piping system design, which is a modified Bernouli’s equation considering the energy losses, energy consumption and energy output of constituent devices. The general equation based on the conservation of energy is given as L I J K Ws j P2 U22 Wsi P1 U12 + + Z1 = + + Z2 + ∑ −∑ + ∑ h fk + ∑ h fl ρg 2g ρg 2g i=1 g j=1 g l=1 k=1

Here, Wsi is the work output per unit mass for turbine, Ws j is the work per unit mass for pumps, h fk is the minor loss due to valves, fittings etc. and h fl is the major loss due to friction in pipes. The head loss due to friction of fluid in a pipe is expressed as hf =

fD−W LV 2 2gD

(Darcy–Weisback)

The friction factor for laminar flow ReD < 2300 of circular pipe is given as f= hf =

4 fF LV 2 2gD

64 ReD (Fanning)

where, fD−W is the Darcy friction factor, fF is the Fanning friction factor, L is the length of pipe and D is the diameter of pipe. It can be noted that 4 fF = fD−W and fD−W can be obtained from the Moody diagram. However, it is not comfortable to implement Moody diagram in computer. Therefore, different empirical expressions are used. Some sample expressions are presented below.

DOI: 10.1201/9781003049272-6

189

190

Design and Analysis of Thermal Systems

The implicit algebraic equation by Celebrook is  −2 1 ε 2.51 √ √ = ln + 3.7D ReD fD−W fD−W where ε is absolute wall roughness and ReD is Reynolds number based on the pipe diameter (ρUD/µ). Here, ρ is density, U is velocity and µ is dynamic viscosity. The explicit algebraic equation by Haaland is given as 0.3086 fD−W = (  ) 1.11  2  6.9 ε ln ReD + 3.7D The equation by Churchill–Churchill is valid for laminar, transition and turbulent flow, as given below: " #1/12  8 12 1 fD−W = 8 + ReD (A + B)1.5 where, ( A=

6.2.1



1 2.457 ln (7/ReD )0.9 + (0.27 ε/D)

)16

 ,

B=

37, 530 ReD

16

MINOR LOSSES

The head loss in valves and fittings is known as minor loss in piping systems, which takes place due to obstructions to flow, changes in flow path, changes in cross section and shape of the flow path etc. One method to characterize the head loss is using equivalent length ratio, L/D, which will cause the same pressure drop as the obstruction under the same flow condition in the pipeline. The equivalent length of pipe is added to the actual length of straight pipe, and the head loss in straight pipe is calculated using f LV 2 2gD Another way to calculate the head loss due to valve and fittings is using dimensionless coefficient, K. The head loss due to valve and fittings is then given as hL =

hf =

KU 2 2g

where K is the minor loss coefficient. Minor loss is attributed to the loss due to the presence of bends, tees and other fittings. The values of the head loss coefficient for various types of fittings, valves and bends can be found in references Cengel et al. (2010) and White et al. (2022).

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Piping Flow

6.2.2

GRAPHICS SYMBOL CONVENTIONS

Different graphics symbol conventions for heating, ventilating and air conditioning purposes are used in water distribution systems, chilled water lines, solar heating, natural gas lines, compressed air lines, heat pump systems, cryogenic applications etc for graphical representation of different components in a piping circuit. 6.2.3

GENERAL CONSIDERATIONS

A simple piping system can be classified into three categories. Category

Given Conditions

Solution

Comment

I II III

Q, L, D, ρ, ε, µ, k h f , L, D, ρ, ε, µ, k h f , Q, L, ρ, ε, µ, k

hf Q D

Direct Iterative Iterative

Among these, category III shows design problems, since Q, h f , L, ρ, ε, µ, k are given and the diameter of the pipe is solution variable. 6.2.4

RESISTANCE ANALOGY

The flow in a pipe leads to a pressure drop ∆P. Pressure drop ∆P can be related to flow rate Q by resistance analogy given as: ∆P = Rhyd Q where, Rhyd is the hydraulic resistance of the pipe. For example, the hydraulic resistance for laminar flow in a pipe based on the Poiseuille flow assumption is given as 128µL Rhyd = πD4 Similarly, the hydraulic resistance value can also be calculated for turbulent flow in the piping circuit. The piping network can be arranged in either series or parallel configuration (Figure 6.1). Using the additivity of pressure drop in a series coupling, the equivalent resistance R can be written as R = R1 + R2 + R3 + R4

Figure 6.1 A typical series or parallel configuration of piping systems.

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Design and Analysis of Thermal Systems

Similarly, in a parallel coupling, the law of additivity of inverse hydraulic resistance is applicable i.e.   1 1 1 1 −1 1 + + + = R R1 R2 R3 R4 These laws are applicable to N i.e. number of pipes in either series or parallel configuration. The procedure for complex piping system design is presented in the following section. 6.2.5

CLASSIFICATION OF PUMPS

The basic system of pump classification is based on the principle by which energy is added to the fluid. On this basis, pumps can be classified into two major categories: (1) dynamic and (2) displacement. In the dynamic category, energy is continuously added to increase the fluid velocities such that the subsequent velocity reduction within the pump produces a pressure increase. The dynamic pump can be further subdivided as a centrifugal pump and other special effect pumps such as a jet pump, electromagnetic pump etc. In a displacement pump, energy is periodically added by the application of force to one or more than one movable boundaries resulting in a direct increase of pressure. They are broadly divided into reciprocating and rotary types depending on the nature of the movement of the pressure-producing members. 6.2.6

PUMP SELECTION

Figure 6.2 shows a pump capacity chart, which can help in preliminary selection by reviewing the wide range of pump casing sizes for a specific impeller speed. This chart helps to narrow down the choice of pumps that can satisfy the system requirements. Figure 6.3 shows a typical pump performance chart for a given model, casing size and impeller rotational speed. The performance curve is a plot of the total head versus flow rate for a specific impeller diameter and rotational speed. The plot starts at zero flow and the corresponding head point is known as the “shut-off head” of the pump. The head decreases starting from this point till a minimum head, which is called the “run-out point” representing the maximum flow of the pump. Beyond this point, the pump cannot operate. The pump’s efficiency varies throughout the operating range of a pump, which is required to calculate the motor power. The best efficiency point, BEP is the highest efficiency point of the pump. The pumping efficiency should be optimized while selecting a pump. Several performance charts at different speeds should be examined such that one model satisfies the efficiency requirement compared to other models. The lowest pump speed should be preferred from multiple available options, as a lower speed will reduce the wear and tear of the rotating parts. Consequently, the pump will have a longer life. The horsepower curve indicated in the chart gives the power required to operate the pump within a certain range. For example, all points to the left of 1.1 kW curve will be attainable with a 1.5 kW motor (see Figure 6.3). It may be noted that the

Piping Flow

193

Figure 6.2 A typical pump capacity chart. (Adapted from Goulds Pumps.)

horsepower curve shown in the performance curves are valid for water only. The horsepower can be calculated using the total head, flow and efficiency at the operating point. The Net Positive Suction Head (NPSH) requirement shown in the pump curve specifies the minimum requirement of suction head for the pump to operate at its design capacity. The NPSH requirement becomes higher as the flow rate increases,

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Design and Analysis of Thermal Systems

Figure 6.3 Typical pump performance curve. (Adapted from Goulds Pumps, STX 1 × 1 12 − 8 AA at 1450 RPM.) and lower as the flow rate decreases. This essentially means that more pressure head is required at the pump suction for high flows than low flows. Sometimes the operating point may be located between two curves on the performance chart. In this situation, the impeller size required can be calculated by linear interpolation. Suppose the operating point is located between 165 mm and 178 mm impeller diameter. The correct size of the impeller for this situation can be calculated using  
178 − 165 Dop = 165 + ∆Hop − ∆H165 ∆H178 − ∆H165 where, Dop is the required impeller diameter, ∆Hop is the pump total head at the operating point, ∆H165 is the pump total head at the intersection of 165 mm impeller curve and flow rate and ∆H178 is the pump total head at the intersection of 178 mm impeller curve and flow rate. Whenever possible, it is a good practice to select a pump with an impeller that can be either increased or decreased in size permitting a future requirement of change in head and capacity. As a guide, one should select a pump with an impeller size no greater than between 1/3 and 2/3 of the impeller range for that casing with an

195

Piping Flow

High-efficiency area

Best efficiency points

Figure 6.4 Typical pump performance curve. (Adapted from Goulds Pumps, STX 1 × 1 12 − 8 AA at 1450 RPM.) operating point in the high-efficiency area (see Figure 6.4). It is also important not to opt for too far right or left from the BEPs. The general guideline is to locate the operating point between 110% and 80% of the BEP flow rate with an operating point in the desirable impeller selection area.

6.3

PUMP PERFORMANCE USING DIMENSIONAL ANALYSIS

Generally, the pump performance characteristics are available for water at some specific speed and impeller diameter. However, water is not always the working fluid and the operational speed is different than that given in the pump curve for many applications. Three approaches allow for obtaining information from the manufacturer performance curve at different operating conditions: (1) dimensional analysis, (2) correction factors for very viscous fluids and (3) curve fitting. 6.3.1

DIMENSIONAL ANALYSIS

Based on the dimensional analysis, a turbomachine or pump can be described using the following important non-dimensional parameters: Π1 =

H P µ Q , Π2 = 2 2 , Π3 = , Π4 = , Π5 = η 3 3 5 ND N D ρN D ρND2

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Design and Analysis of Thermal Systems

Here, Q is the flow rate, N is the rotational speed of impeller D is the diameter of impeller,, H is the developed head, η is the efficiency, µ is the viscosity, ρ is the density of the working fluid and P is the power. The following points should be kept in mind while using the non-dimensional numbers for predicting the pump performance. 1. Π5 is redundant as Π1 , Π2 and Π3 can be combined to determine Π5 i.e. Π5 = Π1 Π2 Π3 . 2. Straightforward application of similarity concept should be avoided. Example: Using the expression of Π1 , we expect Q (flow rate) to be proportional to D3 , and from Π3 , we expect power to be proportional to D5 . However, it is not true. In Π1 and Π3 , D3 and D5 are really AD and AD3 , where A is the cross-sectional area at pump inlet and outlet. Hence, the correct relationship when the impeller diameter changes in the same casing is D2 Q2 = Q1 D1 Similarly, the change in head can be related as  2 H2 D2 = H1 D1 Hence, the change in power with respect to the change in impeller diameter can be expressed as  3 P2 D2 = P1 D1 Therefore, the blind use of non-dimensional numbers relating power to D is most correct. If only speed changes, for a particular pump, we have  3  2 Q2 N2 H2 P2 N2 N2 = , = , and = Q1 N1 H 1 N1 P1 N1 It may be noted that the above dimensionless rule assumes that the two operating points that are being compared are at the same efficiency. 6.3.2

SPECIFIC SPEED

Specific speed is another important non-dimensional number, which is defined as the speed of an ideal pump geometrically similar to the actual pump, which when running at this speed will raise a unit of volume in a unit of time through a unit of the head. It can be determined from the above non-dimensional numbers as follows:   Q 1/3 Q Π1 = ⇒ D= ND3 Π1 N

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Piping Flow

Π2 = ⇒

H 2 N D2

Π5 = ⇒

⇒

Π2 =

H N2



Π1 N Q

2/3

H(N)2/3 H Π2 ≡ 2 2/3 ⇒ Π5 = 4/3 2/3 2/3 (Π1 ) N Q N Q N 4/3 =

H H 3/4 ⇒ N= 2/3 Π5 Q Π5 Q1/2

3/4

Note that we have written Π5 as Π5 in this equation as it is a non-dimensional number and its power is not going to influence the general relationship. We can rewrite the new non-dimensional number, Π5 as √ N Q 0 ⇒ Π5 = 3/4 H Here, Π05 is known as the specific speed.

6.4

PUMP CURVE FOR VISCOUS FLUID

In general, the similarity principle is applied for relatively low-viscosity fluid. Hence, Π4 is relatively small and the remaining Π groups are nearly invariant with respect to Π4 . If the viscosity is large, the similarity principle cannot be applied to extend the manufacturer’s data. The performance characteristics of a large viscous fluid for a pump can be known from that of water by applying suitable correction as below (Gulich (1999a, 1999b). Qvis = CQ QW ,

Hvis = CH HW ,

nvis = Cn nW

Here, subscript W indicates water, and subscript vis indicates viscous fluid. The coefficients, CQ , CH and Cn are correction factors. The correction factors are published in the form of curves by the Hydraulic Institute (www.pumps.org). The correction factor charts are also available on www.fluidedesign.com. Fluids that are viscous can significantly affect the performance of centrifugal pumps. Hence, these correction factors can be used as approximations only. The exact pump performance for a viscous fluid should be obtained by actual tests on the pump with the viscous fluid. 6.4.1

PROCEDURE TO OBTAIN THE CORRECTION FACTOR AND PUMP CURVE FOR VISCOUS FLUID

The following procedure can be followed to obtain the correction factor for the pump curve of viscous fluid from that of water. 1. Locate the point of maximum efficiency on the H − Q curve for water (see Figure 6.5). If this is QMW , then obtain 0.6 × QMW , 0.8 × QMW and 1.2 QMW .

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Design and Analysis of Thermal Systems

Figure 6.5 A representative pump performance curve for water used to extend for very viscous fluid.

Figure 6.6 A typical correction factor curve for obtaining the pump curve of viscous fluid from that of water. 2. Go to the correction factor curve (Figure 6.6) at capacity corresponding to nmax and go forward to the desired head developed and then horizontal to the desired viscosity. From that point, proceed upward to obtain CH , CQ and Cn . 3. Read CQ , CH and Cn for all four capacities. Note that CQ and Cn are determined once at maximum efficiency point only. 4. Multiply HW by CH to get Hvis . 5. Multiply ηW by Cn to get nvis . 6. Multiply QW by CQ to get Qvis . 7. The power required can be calculated using Qvis , Hvis and nvis . 8. Plot the corrected values and draw a smooth curve through them.

6.5

EFFECTIVE PUMP PERFORMANCE CURVE

Pumps are operated either in series or parallel for added head or flow rate requirements, respectively. Check valves are usually provided on the suction (inlet) side of the pump to prevent back flow and shut-off valves are provided in the discharge (outlet) lines for complete isolation when needed. The losses due to these valves are usually neglected for pump selection.

Piping Flow

199

Figure 6.7 A typical pump system arranged in series (a) and its equivalent pump representation (b). Analysis of pumps in series or parallel arrangement is facilitated by an effective pump performance curve. This effective pump performance curve provides a single head-capacity relationship equivalent to that of all pumps in the network. Figures 6.7 and 6.8 show typical equivalent pump representation in series and parallel arrangement, respectively. Figure 6.9 shows the procedure for the generation of an effective pump performance curve when the pumps are arranged either in series or parallel. For pumps in series, heads are added at a constant flow rate. For pumps in parallel, flow rates are added at a constant head. For a dissimilar pump in parallel, the pump with the lower shut-off head cannot be brought into operation until the head of the larger pump is decreased below this lower shut-off head. Shut-off head is the maximum head generated by a pump with zero flow. Otherwise, the more powerful pump will block the output of the lower shut-off head pump. Figure 6.9 shows the equivalent pump curve for similar pumps (A) and dissimilar pumps (A and B) in both series and parallel arrangements. For series arrangement, H for each pump is added at a fixed value of Q. For parallel arrangement, Q is added for a fixed value of H. For dissimilar pumps in parallel, the head of the lower shut-off head is only added for a flow rate greater than its shut-off head.

Figure 6.8 A typical pumping system arranged in parallel (a) with its equivalent pump representation (b).

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Design and Analysis of Thermal Systems

Figure 6.9 Construction of an effective pump performance curve for pumps in series and parallel. 6.5.1

COMPUTER IMPLEMENTATION

The pump performance curve in suitable form can also be used for design purpose using a computer. The procedure for computer implementation of pump performance curve is outlined below. 6.5.1.1

Pumps in Series

For pumps in series, a pump curve for each pump can be obtained by curve fitting as Hi = A1i + A2i Q + A3i Q2

(Pump curve)

Subsequently, for the equivalent pump curve, the head developed by each pump is added as follows: H = H1 + H2 + H3 + · · ·

201

Piping Flow

Here, H1 , H2 and H3 are the pump curves for pumps 1, 2 and 3, respectively. Thus, we get H = (A11 + A12 + · · · ) + (A21 + A22 + · · · )Q + (A31 + A32 + · · · )Q2 In a generalized form, for pumps in series, we can write H=

∑

Ai j Qi

i=0,1,...,M j=0,1,...,N

Here, i indicates the order and j indicates the pump number for coefficient Ai j . 6.5.1.2

Pumps in Parallel

For pumps in parallel, the flow rate of each pump is expressed as a function of the head developed. Qi = B1i + B2i H + B3i H 2 The flow rate developed by each pump is added for pumps in parallel. Q = Q1 + Q2 + · · · Here, Q1 , Q2 , . . ., corresponds to flow rate of pump 1 and pump 2, respectively. Q = (B11 + B12 + · · · ) + (B12 + B22 + · · · )H + (B31 + B32 + · · · )H 2 In a generalized form, we can write the equivalent pump performance as Q=

∑

Bi j H i

i=0,1,...,M j=0,1,...,N

Here, Bi j is the coefficients of the equivalent pump curve of pumps in parallel, with i indicating the order and j indicating the pump number.

6.6

SYSTEM CHARACTERISTICS

For a given piping system, the head versus flow rate curve are known as system characteristics. This curve can be determined from the basic energy equation. The system characteristics have two uses: (1) pump selection or specification for a desired system flow rate and (2) determination of the operating point for a given pumpsystem combination. The first objective is accomplished with a pump curve by selecting the appropriate pump for a specified flow rate. The second objective i.e. the determination of the operating point is illustrated in Figure 6.10. The intersection of the system characteristic and pump curve indicates the operating point. Figure 6.10 shows the H−Q for two piping systems (A and B). The system curves A and B intersect the pump curve at point C and D, receptively. Here, points C and D are the operating points for systems A and B, receptively.

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Design and Analysis of Thermal Systems

Curve B Curve A

Figure 6.10 Operating point for two arrangements i.e. system characteristic. System characteristic A corresponds to a pipe network between two reservoirs at the same surface elevation. System characteristic B corresponds to pipe networks between two reservoirs at different surface elevations.

6.7

PUMP PLACEMENT

Pump placement is an important factor in the design of piping systems. The issue of cavitation and recirculation needs to be considered for the proper placement of pumps. 6.7.1

CAVITATION

When the static pressure of the fluid is reduced below its vapor pressure, pockets of vapor form and cavitation is said to take place. The growth and collapse of these pockets can cause pressure fluctuation leading to mechanical damage of the pump’s components. Therefore, the pump placement in a piping circuit should be appropriately decided to avoid cavitation. 6.7.2

NET POSITIVE SUCTION HEAD

It is the difference between the total pressure at the pump suction (inlet) and the vapor pressure of the fluid (Ps − Pv ). The required NPSH (NPSHR ) is determined on the basis of the head across the pump for a given speed and flow rate. A decrease of 3% in the head across the pump as the suction pressure is decreased is considered the evidence of cavitation. It is assumed that mechanical efficiency, head increase and power required are essentially constant under the variable suction condition as long as the cavitation is not present. The cavitation problem can be avoided by locating the pump where the available NPSH (NPSHA ) is greater than the required NPSHR specified by the pump performance curve. A typical pump arrangement is shown in Figure 6.11. The suction pressure at the inlet can be expressed as Ps = Patm + ρHg − Pf , where Pf is the pressure loss due to

203

Piping Flow

Shut-off valve

Figure 6.11 A typical pump arrangement and calculation of NPSHA . friction. The available suction pressure is given as NPSPA = Patm + ρHg − Pf − Pv where, Pv is the vapor pressure of the working fluid. Thus, the net positive suction head is expressed as NPSHA =

Pf Patm Pv +H − − ρg ρg ρg

It may be noted that the lower the pump placement, the higher is the NPSHA . Therefore, it is preferred to place the pump at the lowest elevation to meet the cavitation requirement. 6.7.3

RECIRCULATION PROBLEM

Recirculation within the pump inlet and at rotor discharge may be observed at flow rates below the BEP. At the BEP, velocities are high enough to preclude or minimize recirculation. However, as the flow rate and hence the fluid velocities decrease recirculating regions develop (see Figure 6.12). Suction-specific speed is calculated to predict the recirculation.

6.8

SUCTION-SPECIFIC SPEED

Suction specific speed (Nss ) is a parameter that identifies the recirculation characteristic. It is constant for a given pump and function of suction inlet design. √ N Q Nss = (nmax ) (NPSHR )3/4 The recirculation problem is minimized by selecting a pump with Nss less than 9000 to 11000. This value is dependent on the fluid type (9000 for water and 11000 for hydrocarbon). The recirculation problem is more significant for large pumps than small pumps. If a pump operates near BEP, recirculation may not be a problem.

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Design and Analysis of Thermal Systems

Figure 6.12 rate.

Schematic showing suction and discharge recirculation at low flow

It may be noted that (NPSH)R should be low to avoid cavitation. However, low (NPSH)R leads to high Nss and may lead to recirculation problem Karassik et al. (2001). Hence, the requirement of NPSH have a conflicting requirement to avoid cavitation and recirculation problem.

6.9

NET POSITIVE SUCTION HEAD AVAILABLE

The Net Positive Suction Head Available (NPSHA ) is the total energy per unit weight or head at the suction flange of the pump less the vapor pressure of the fluid. It is required to avoid cavitation and can be determined from the pressure head at the suction flange of the pump. The pressure at which a liquid boils is called vapor pressure, which is associated with a specific temperature. The boiling temperature drops with the decrease in environment pressure. The pressure near the impeller eye is lower than the pressure at the pump suction flange. The point of lowest pressure is near the eye of the impeller on the underside of the vane (see Figure 6.25), where bubbles can form. These bubbles get rapidly compressed while traveling from the start of the impeller vane to its tip. The rapid compression of bubbles can cause small pieces of metal to be dislodged from the surface. The collapse of the bubbles near the tip of the vane also causes noise and vibration. The decrease in pressure from points A to D is attributed to friction loss, turbulence and entrance loss due to the right angle turn of horizontal flow to outward radial flow in the impeller. The usual recommendation is to ensure that NPSHA is equal to 15% or more above NPSHR .

6.10

UNCERTAINTY EFFECT ON PUMP SELECTION

Thermal system design specifications have uncertainty due to either use of a standard design database (pump curves) containing its own uncertainty or the use of dimensional analysis. Therefore, there is a need to determine the uncertainty limits of the system/component parameters due to the uncertainty of the design database. This is important for the proper operation of the pumping systems.

Piping Flow

205

Figure 6.13 The procedure for specification of excessive pump capacity to provide factor of safety. However, the uncertainty qualification in thermal system design is not well developed. Therefore, thermal system design procedure is generally based on adhoc safety factor. This approach leads to overspecification and adverse consequences. The procedure for overspecification of a pumping system is demonstrated in Figure 6.13. Before finalizing the impeller size, one may have to apply an extra capacity/flow factor depending on the requirement. Figure 6.13 demonstrates the systematic procedure for providing the safety factor. Let point 1 on the impeller curve A is the operating point before applying the capacity factor. For safety margin on the total head, if we apply a head factor of 10%, we will have to select impeller C. The new operating point for this curve will be point 2. This means that the new system curve will be c instead of a. If the pressure drop calculation is correct and we are operating with curve a, the operating point will shift from point 2 to point 3 on impeller curve C. If we need to get back to the flow corresponding to point 1 for process reason, then throttling a valve at the pump discharge will be required. This will change the operating curve to match curve c and bring the operating point back to point 2. We have to select impeller curve, B if we apply a capacity factor for safety margin on flow rate say 10%. The system curve will be d in this situation. If the original flow estimate is correct, the operating point will shift from point 4 to 5. We have to throttle back so that we shift to point 6 to get back the original flow. Now we will operate on a new system curve b. The overspecification of the pump has the following drawbacks: 1. It consumes an excess amount of energy. 2. Operating point will be lower than the best efficiency point. 3. Flow control is difficult due to the excessive nonlinearity of the valve in the near closed position. It may also lead to flow-induced vibration. To alleviate this problem, rotor diameter is reduced in industry, which can lead to maintenance problem. It may be noted that the pump overspecification is in response to uncertainties in the pump selection process.

206

6.11

Design and Analysis of Thermal Systems

UNCERTAINTY ANALYSIS PROCEDURE

Experimentalists use the procedure for calculation of uncertainty of derived quantities from the experiment. The above procedure can be modified for the calculation of uncertainty in piping system design parameters. Let’s assume that the following data reduction equation is used to report the experimental results. R = f (X1 , X2 , . . . , X j ) where, R represents experimental results and X j are measured quantities. The uncertainties of experimented results are calculated using the following expression:  2  2  ∂R ∂R UX1 + UX2 + · · · + UR =  ∂ X1 ∂ X2

∂R UX ∂ Xj j

!2 1/2 

Here, UR is the uncertainty in result, ∂∂XRj is the sensitivity of the results to the change in measured variable and UX j is the error or uncertainty of the measured variable. We can relate the uncertainty calculation for experiments to design of the piping system. The above uncertainty equation can also be rewritten as !2  2           UR X1 ∂ R 2 UX1 2 X2 ∂ R 2 UX2 2 XJ ∂ R 2 UX j = + +···+ R R ∂ X1 X1 R ∂ X2 X2 R ∂ XJ Xj   X Here, Rj ∂∂XRj is the normalized sensitivity coefficient. This parameter identifies the input parameter to which the computed parameter is most sensitive. We can identify which design inputs need to be known with the most fidelity from the calculation of this parameter. Uncertainty contribution of each design inputs is also given by  ,  ∂R 2 Relative contribution: UXi UR2 ∂ Xi 6.11.1

PIPING NETWORK DESIGN

The previous uncertainty equation can be written for the calculation of uncertainty in piping network design as 

UQ2 j

=

Pipes  

∑

i=1

∂Qj  ∂ Di  +

∂Qj ∂ fi

2

2

UD2 i



2



2

∂Qj + ∂ Li

U f2i +

∂Qj ∂ εi

UL2i



2



2

∂Qj + ∂ Ki

Uε2i +

∂Qj ∂µ

UK2i



∂Qj + ∂ m˙ i

Uµ2 +



∂Qj ∂ρ

2

2

Um2˙ i

Uρ2

1/2  

Piping Flow

207

Figure 6.14 The uncertainty band of a pump curve and system curve. Here, UDi , ULi , UKi , Um˙ i , U f i , Uεi , Uµ and Uρ correspond to the error or uncertainty in diameter, length, minor loss coefficient, flow rate, friction factor, roughness value, viscosity and density, respectively. Here, Q j is the flow rate in each line j of the piping network. Uncertainties in the specifications of each pipeline influence the flow rate uncertainty in other pipelines. Figure 6.14 shows the uncertainty band of a typical pump curve and system characteristic curve. In Figure 6.14, the nominal operating flow rate is 1600 m3 /s. If all the uncertainties combine to degrade the system performance, the flow rate will be 1400 m3 /s. If all the uncertainties combine to enhance the system performance, the flow rate is 1800 m3 /s. If a pump is required to guarantee 1400 m3 /s flow rate, its entire operating region must be to the right of 1400 m3 /s. This way the excess capacity of pump selection can be tolerated. The sensitivity coefficients can be determined by finite-difference schemes if the analytical partial derivative calculation is cumbersome. Figure 6.15 shows typical normalized sensitivity coefficients and relative contributions for a piping circuit problem. The largest normalized sensitivity coefficient is associated with the diameter. This may be related to the power law relation of the flow rate with respect to the pipe diameter. The next contribution is due to the viscosity, which may be related to a large error in viscosity due to temperature fluctuation etc.

6.12

PIPING SYSTEM DESIGN

Most energy systems are composed of piping, pumps, compressor, turbine (primary movers) and heat exchangers. In this section, we discuss the systematic approach for the design of complex series-parallel piping networks.

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Design and Analysis of Thermal Systems

Figure 6.15 A histogram showing the uncertainty estimate of a typical piping system. 6.12.1

HARDY CROSS METHOD

The Hardy Cross method can readily be adapted to computers, and number of software firms have generalized Hardy Cross programs for sale. The basis for the Hardy Cross analysis technique is: (1) conservation of mass at a node and (2) uniqueness of pressure at a given point in the loop. A loop is defined as a series of pipes forming a closed path. A node is defined as a point where two or more lines are joined (see Figure 6.16). (1) Conservation of mass at a node says that the node cannot accumulate mass. For node α, based on mass conservation, we have

∑ Qαβ = 0

β =1

where, β is the no of lines connecting the node, α. (2) The pressure at a node must be single valued i.e. the sum of the pressure drops around a loop must be zero. For the ith loop, we can write

∑ h fi j = 0

j=1

Here, j is the number of lines in the loop. 6.12.2

HAZEN–WILLIAMS COEFFICIENT

Since flow rate rather than velocity is usually of primary interest, the head loss expression can be written as hf =

f LV 2 2gD

(Darcy–Weisback)

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Piping Flow

Figure 6.16 Representation of a piping system.

hf =

16 f LQ2 f L Q2 = 2 2gD A 2Π2 gD5

(For circular pipes)

f LQ2 (where K1 = 8/Π2 g) D5 Hazen and Williams suggested that for a general fluid the head loss can be written as = K1

h f = KQn =

k1 L Q1.852 C1.852 D4.8704

where, C is a dimensionless number that is indicative of the roughness of the pipe. Hazen and Williams coefficient k1 is a constant depending on the dimensions of Q i.e. if Q =(ft3 /s), k1 = 4.727 and if Q =(m3 /s), k1 = 10.466. Different values of C are given as follows: Extremely smooth and straight pipes: C = 140 Cast iron pipes with some years of service: C = 140 Cast iron pipes in bad condition: C = 80 6.12.3

BASIC IDEA

First, the conservation of mass at each node is established without consideration of uniqueness of pressure. Then, the uniqueness of pressure is used to calculate the correction factor for each loop. 6.12.4

CORRECTION FACTOR

Figure 6.16 shows a typical representation of a piping system. The piping network is divided into two loops, and each pipe in each loop is assigned a number. The flow rate in each pipe is then identified as Q(i j) , where i is the loop number and j is the pipe number.

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Design and Analysis of Thermal Systems

Note: The pipes that are common between the two loops have flow rates which are related as for example Q11 = −Q23 i.e. when the flow is in the counter clockwise direction around a loop, that flow toward the node is positive. We know from the consistency that a negative Qi j yields a negative h fi j and similarly, from sign convection, Qi j can be either negative or positive. Therefore, we can write ( KQn Q≥0 hf = −K(−Q)n Q < 0 Taking a derivative of this expression, we can write ( dh f nKQn−1 Q≥0 = nK(−Q)n−1 Q < 0 dQ Using the Taylor series to expand the head loss about Q, we have h f (Q + ∆Q) = h f (Q) +

d 2 h f ∆Q2 dh f ∆Q + +··· dQ dQ2 2

Neglecting higher-order terms,  i  K[Qn + nQn−1 ∆Q Q≥0 h f (Q + ∆Q) =  −K[(−Q)n − n(−Q)n−1 ∆Q] Q < 0 With reference to Figure 6.16 in loop 1, Q011 > 0, Q013 > 0 and Q012 < 0. Here, the superscript “0” denotes the first guess. Using the initial guesses Q0i j for loop 1, we have h0f11 + h0f13 − h0f12 6= 0 But, it is desired that ±h1f11 ± h1f13 ± h1f12 = 0 Here, “1” stands for the next iteration. Substituting the expression of h f (Q + ∆Q), we have h i h i K11 (Q011 )n + n(Q011 )n−1 ∆Q1 + K13 (Q013 )n + n(Q013 )n−1 ∆Q1 i h −K12 (−Q012 )n − n(−Q012 )n−1 ∆Q1 = 0 Solving for ∆Q1 , we obtain ∆Q1 = −

K11 (Q011 )n + K13 (Q013 )n − K12 (−Q012 )n K11 n(Q011 )n−1 + K13 n(Q013 )n−1 + K12 n(−Q012 )n−1

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or using absolute values, ∆Q1 = −

K11 Q011 |Q011 |n−1 + K12 Q012 |Q012 |n−1 + K13 Q013 |Q013 |n−1 n(K11 |Q011 |n−1 + K12 |Q012 |n−1 + K13 |Q013 |n−1 )

Note: Use of the absolute value signs allows the correct sense of the sign of h fi j to be maintained. Using the summation notation for loop 1, we can write ∆Q1 =

− ∑3j=1 K1 j Q01 j |Q01 j |n−1 n ∑3j=1 K1 j |Q01 j |n−1

For any loop i with J lines, we can write the expression for correction as below: ∆Qi = −

6.12.5

∑Jj=1 Ki j Q0i j |Q0i j |n−1 n ∑Jj=1 Ki j |Q0i j |n−1

IMPLEMENTATION PROCEDURE

The following procedure can be adopted for the implementation of Hardy Cross method in a piping circuit. 1. Subdivide the network into a number of loops. Be sure that all pipes are included in at least one loop. 2. Determine the zeroth estimate for the flow rate Q0αβ . If s be the number of nodes and r be the total number of lines, write a node equation for each node. As r > s, there will be more unknowns than the equation. If we assume (r − s) values of Q0αβ , then the system should reduce to s linear algebraic equations with s unknowns. If the resulting set is linearly independent, then a solution for all other Q0αβ can be obtained. Linear independence is tested by taking the determinant of the coefficient matrix. If the system of equations turns out to be linearly dependent, then one additional Q0αβ value must be assumed before a set of values {Q0αβ } can be established. 3. Determine the correction factor ∆Q1 for each loop. 4. Obtain a new value for the flow rate in each line QIi j = Q0i j + ∆Qi . 5. Repeat step 3 and step 4 till all the corrections are equal to zero. Example: Obtain the flow rates in each of the lines of the network (Figure 6.17). Assume the Hazen–Williams coefficient, C = 130. Solution Step 1: Figure 6.18 shows the numbering of nodes, loops and lines.

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Figure 6.17 A piping network. Length in meter and diameter in centimeter along with the inlet and outlet flow rates are provided in the sketch.

Figure 6.18 Schematic showing different steps of implementation. Step 1: (a) the numbering of nodes, loops and lines. Step 2: (b) application of node equation at nodes B, C and D and (c) application of node equations at nodes D, E, F and A. Step 2: Determine the zeroth estimate. Here, S = 6 and r = 7 We have r − s = 1. Therefore, minimum one number of Q0αβ should be assumed. In fact, it is suggested that (r − s + 1) values of Q0αβ must be assumed in general i.e. we have to assume two values of Q0αβ . To start the process, let’s assume Q21 = 0.0283 m3 /s. Figure 6.18b shows the application of node equation for nodes

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Piping Flow

Figure 6.19 Converged solution. B, C and D. For the application of the node equation of D, we need a second assumption. Now, we make second assumption: Q11 = 0.0226 m3 /s

Q24 = −0.02266 m3 /s

or

Thus, we have all the flow rates for node D. The applications of node equations for nodes D, E, F and A are shown in Figure 6.18c. Steps 3 and 4: Determine the correction factor ∆Qi for each loop. ∆Q1 = −

∑4j=1 Ki j Q0i j |Q0i j |(1.852−1) 1.852 ∑4j=1 Ki j |Q0i j |(1.852−1)

= 0.0179 m3 /s

where, K, L C1.852 ∆4.8704 ∆Q2 = 0.004457 m3 /s K=

∴

Q113 = −0.034 + 0.0179 = −0.0161 m3 /s

Flow rates for lines that are contained in more than a single loop are corrected by considering the correction factors for common loops. Q111 = Q011 + ∆Q1 − ∆Q2 = 0.0226 + 0.0179 − (−0.004457) = 0.045 m3 /s Q112 = 0.0123 m3 /s,

Q114 = −0.0161 m3 /s

Similarly, Q121 = 0.0239 m3 /s, Q123 = −0.0328 m3 /s,

Q122 = 0.0239 m3 /s Q124 = −0.045 m3 /s

Q124 = Q124 + ∆Q2 − ∆Q1 = −0.0226 − 0.00446 − 0.0179 = −0.045 m3 /s The converged solution will be obtained after the fifth iteration, which is shown in Figure 6.19.

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6.13

Design and Analysis of Thermal Systems

GENERALIZED HARDY CROSS ANALYSIS

The Hardy Cross approach can be generalized for including minor loss and lines containing devices that result either in additional pressure drop i.e. heat exchanger or turbine (see Figure 6.20). The head loss (turbine or heat exchanger) or pressure increase in a pump can be expressed as a function of flow rate i.e. h fDi j = gi j (Q) In general, the change in the head across the device will depend on the flow rate. This expression of the head loss or increase can be represented as M

h fDi j = Ai j + ∑ Bi jm Qm m

where, m represents the degree of polynomial. Example: The head loss through a pipe fitting is typically described as h fDi j = Ci j

Vi2j 2g

= Ci j

Q2i j 2gA2

Hence, the coefficients of the equivalent polynomial expression are M = 2, Ai j = 1 0, Bi j1=0 , and Bi j2 = Ci j 2gA 2 . For a centrifugal pump, the head versus flow rate can be represented as h fDi j = −(Ai j − Bi j1 Q) Note, h fDi j is less than zero because a pump represents a negative head loss i.e. increase in head. Ai j and Bi jm are obtained from the curve fitting of experimental data i.e. pump performance curve. Incorporation of this polynomial representation in the Hardy Cross method discussed in the previous section can be modified as h i m−1 B Q |Q | ∑Jj=1 Ki j Qi j |Qi j |n−1 + SGN(Qi j )Ai j + ∑M i jm i j i j m=1   ∆Qi = − J M n−1 + ∑m=1 Bi jm m|Qi j |m−1 ∑ j=1 Ki j n|Qi j | where, SGN(Qi j ) = 1 when Qi j > 0 SGN(Qi j ) = −1 when Qi j < 0 Example: Let’s investigate the effect of adding the following device in line 2 of loop 2 of Example 6.1. (a) A pump with h fD = −(15.235 − 4.24Q) m (b) A heat exchanger with h fD = (19022Q2 ) m (c) A very large pump with h fD = −304.79 m

Piping Flow

215

Figure 6.20 A sketch showing the minor loss and device representation for implementation of the Hardy Cross method.

Figure 6.21 Converged solution when a pump is added to the piping circuit. Solution (a) Here, A22 = −15.235, B22 1 = 4.24 and B22 2 = 0.0. Using the same initial estimate as in the previous example, we get the solution shown in Figure 6.21. Note, the addition of the pump results in an increase in the flow rate through lines 1 and 2 of loop 2. Loop 1 is affected very little. (b) h fD = 19022Q2 Here, A22 = 0, B22 1 = 0 and B22 2 = 19022 The converged solution is given in Figure 6.22. Note: The addition of the heat exchanger reduces the flow rate in lines 1 and 2 of loop 2. Loop 1 is not affected much due to the addition of heat exchanger. (c) h fD = −304.79m Here, A22 = −304.79, B22 1 = B22 2 = 0 The converged solution is shown in Figure 6.23. Note that this large pump completely changes the flow rate distributions. The size of the head increase dominates loop 2 and forces a useless circulation in the loop, i.e. Pipe (11) or Pipe (24) now flows in a direction opposite from that of the previous cases. Note: When specific Hazen–Williams coefficients are not known or when a different fluid is used, we have to use the Moody diagram and compute the head loss for a

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Design and Analysis of Thermal Systems

Figure 6.22 Converged solution when a heat exchanger is added to the flow circuit.

Figure 6.23 Converged solution when a very large pump is added. variety of flow rates. A curve fit is used to obtain the K and n values. For fluids in process networks, the exponent n may very from pipe to pipe. These circumstances lead to the most general Hazen–Williams Hardy Cross expression as h im−1 B Q |Q | ∑Jj=1 Ki j Qi j |Qi j |ni j −1 + SGN(Qi j )Ai j + ∑M i jm i j i j m=1   ∆Qi = − J M n −1 ∑ j=1 Ki j ni j |Qi j | i j + ∑m=1 Bi jm m|Qi j |m−1 6.13.1

BLOCK DIAGRAM

A block diagram or flow chart showing the implementation of generalized Hardy Cross method is shown in Figure 6.24. In the first step, the parameters associated with the number of loops, number of lines in each loop, the length, diameter and nature of line/pipe materials in each loop are provided as input to the program. The specifications of device i.e. Ai j and Bi j in a line are also provided. If the device is common to more than one loop, then the device coefficients for each line of each loop are provided. The initial flow rate for each line is estimated. The corrections to the

217

Piping Flow

Figure 6.24 A block diagram showing the implementation of the Hardy Cross method for piping circuit. flow rate is calculated for each loop, and the new flow rate for the line is calculated. If a line is common to two loops, the correction sign is properly taken care of. The intermediate point of output is provided. The iteration is continued till convergence. The convergence is attained when the absolute value of all flow rate corrections is less than a specified tolerance. Example: A piping circuit with a pump is shown in Figure 6.26. The pump characteristic used in circuit is given as h f D = −A + 0.4Q

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Design and Analysis of Thermal Systems

Figure 6.25 Pressure variation within a pump. (https://www. pumpfundamentals.com/images/cavitation.jpg, accessed on 30 Nov. 2020.)

Figure 6.26 Schematic of a sample piping circuit.

219

Piping Flow

where, A is a constant and equal to the shut-off head of the pump and Q is the flow rate. What is the value of constant A? Use the following data for the piping circuit: Q1 = 0.963 m3 /s Q2 = 0.792 m3 /s Q3 = 0.208 m3 /s Q4 = 0.208 m3 /s Q5 = 1.828 m3 /s Q6 = 1.828 m3 /s Q7 = 0.172 m3 /s Q8 = 3.0 m3 /s Qout,1 = 1.0 m3 /s Qout,2 = 2.0 m3 /s K1 = 1.148 K2 = 8.269 K3 = 50.36 K4 = 67.146 K5 = 4.134 K6 = 12.403 K7 = 8.269 K8 = 1.148 Solution The head loss with a loop having a pump should be equal to zero.

∑ h f D (Loop 2) = 0 k5 Q1.852 − A + 0.4Q6 + k6 Q1.852 − k7 Q1.852 − k1 Q1.852 7 1 5 6

=0

A = k5 Q1.852 + 0.4Q6 + k6 Q1.852 − k7 Q1.852 − k1 Q1.852 7 1 5 6 = 4.134(1.828)1.852 + 0.4 × 1.828 + 12.403(1.828)1.852 − 8.269(0.172)1.852 − 1.148(0.963)1.852 = 12.634 + 0.731 + 37.905 − 0.317 − 1.07 = 49.883. Example: A pump and pipe system moves water from a reservoir to a water tower. The water level in the tower is 30 m higher than the reservoir. The pipe system uses a 200 mm diameter pipe. The pipe running from the reservoir to the pump is 5 m. The pipe running from the pump to the water tower is 250 m long. For your calculations, assume that the friction factor in the pipe is a constant, f = 0.024 (note that you would need to check this assumption later in actual applications). The only minor loss is at the entrance to the pipe system (K = 20). The pump characteristic curve is given by h p = 100 − 750Q2 where, h p is the head added by the pump and Q is the flow rate in m3 /s. Compute the flow rate in the pump/pipe system and the head added by the pump (i.e. its operating point). Solution For the piping system,   L2 K 8 L1 + f + Q2 D5 D5 D4 gπ 2    5 250 2 8 = 30 + 0.024 5 + 0.024 5 + Q2 0.2 0.2 0.24 gπ 2   8 = 30 + (375 + 18750 + 1250) Q2 gπ 2 

(zB − zA ) + hL (Q) = 30 +

f

= 30 + 1683.5Q2

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Design and Analysis of Thermal Systems

For the pump, h p (Q) = 100 − 750Q2 . The operating point occurs at 100 − 750Q2 = 30 + 1683.5Q2 70 = 2433.5Q2 r 70 Q= = 0.170 m3 /s 2433.5 The head added by the pump is h p = 100 − 750(0.170)2 = 78.33 m. Example: A pump delivers 5.66 m3 /s against a head of 122 m at 4500 RPM. The impeller diameter is 2.1 m. The positive suction head including velocity head is equal to 3 m. The pump is too large for a laboratory testing. Therefore, a small model of the pump needs to be tested for predicting its operational characteristics. The model pump has an impeller diameter equal to 0.46 m. It is to be tested at a reduced head of 97.5 m. At what speed, capacity and suction head should the test be conducted? Solution r  r D H1 2.1 97.5 N1 = N = 450 = 1836.5 rpm. D1 H 0.46 122 r r  2   D1 H1 0.46 2 97.5 Q1 = Q = 5.66 = 0.2427 m3 /s. D H 2.1 122 Hb − Hs 10 − 3 Cavitation factor, σ = = = 0.057 H 122 Hence, Hs1 = Hb − σ H = 10 − 0.057 × 97.5 = 4.44 m Hence, the model should be tested with a positive suction head of 4.44 m.

PROBLEMS 1. The piping network shown below uses smooth cast iron pipes (C:130). The diameter and length of pipes in the network are pipe 1: dia. = 0.31 m, length = 609.6 m; pipe 2: dia. = 0.203 m, length = 609.6 m; pipe 3: dia. = 0.1524 m, length = 914.4 m; pipe 4: dia. = 0.1524 m, length = 1219.2 m; pipe 5: dia. = 0.203 m, length = 304.8 m; pipe 6: dia. = 0.203 m, length = 914.4 m; pipe 7: dia. = 0.203 m, length = 609.6 m. Perform the Hardy Cross analysis and calculate the flow rate in each line after one iteration. Note that K1 = 4.721 when Q is in cfs. Use the following initial flow rates: Q1 (top loop) = 1.811 cfs, Q1 (bottom loop) = −1.811 cfs, Q2 = 0.686 cfs, Q3 = −0.314 cfs, Q4 = −0.314 cfs, Q5 cfs, Q6 = 0.875 cfs, Q7 = −1.125 cfs. Take 1 cfs = 0.02832 m3 /s. 2. Find out the flow rate in each line of the following piping circuit. Here, K is the Hazen–Williams coefficient with n = 2.

Piping Flow

221

3. A piping circuit with a pump is shown in Figure 6.26. The pump characteristic used in the circuit is given as h f D = 0.4Q − A where, A is a constant equal to the shut-off head of the pump and Q is the flow rate. Calculate the value of constant A using the following data: Q1 Q2 Q3 Q4 Q5 Q6 Q7 Qin Qout,1 Qout,2 0.963 0.792 0.208 0.0208 1.828 1.828 0.172 3.0 1.0 2.0 K1 K2 K3 K4 K5 K6 K7 K8 1.148 8.269 50.36 67.146 4.134 12.403 8.269 1.148 Note: All the flow rates (Q) are in m3 /s.

REFERENCES Cengel, Y. and Cimbala, J., Fluid Mechanics: Theory and Applications, McGrawHill Education, New York (2010). Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane Company. Technical paper. Crane Company, 2013. ISBN 9781400527120. https:// books.google.co.in/books?id=8kDYAAAAMAAJ. Frank M. White, H Xue, Fluid Mechanics, McGraw-Hill (2022) J. G¨ulich. Pumping highly viscous fluids with centrifugal pumps — Part 1. World Pumps, 1999(395):30–34, 1999a. doi: 10.1016/s0262-1762(00)87528-8. J. G¨ulich. Pumping highly viscous fluids with centrifugal pumps — Part 2. World Pumps, 1999(396):39–42, 1999b. doi: 10.1016/s0262-1762(00)87492-1. I. Karassik, J. Messina, P. Cooper, and C. Heald. Pump Handbook. McGraw-Hill Education, 2001. ISBN 9780071500111. https://books.google.co.in/ books?id=d29fFd0kiSgC.

Intelligence for 7 Artificial Thermal Systems 7.1

INTRODUCTION

Artificial intelligence (AI) is defined as a branch of computer science which investigates symbolic and non-algorithmic reasoning processes for use in machine inference. It has two general common traits i.e. it consists of several computer methods and can also reproduce a non-quantitative human thought process. AI consists of several branches, namely, expert systems (ESs), artificial neural networks (ANNs), genetic algorithms (GAs), fuzzy logic (FL), natural language processing (NLP), computer vision (CV) etc. These computer-based algorithms are also widely classified in the domain of soft computing. Algorithms in soft computing are mostly based on simplistic models of human intelligence and evolutionary experience. They generally have very simple computational steps, often accompanied by a large number of repeated computational cycles. This is very much in contrast to hard computing, which generally deals with numerical solutions to differential equations such as conservation laws. Up to the very recent past, thermal science problems have largely been treated by traditional hard-computing approaches, along with experiments carried out for the purpose of validation or development of correlations. However, thermal problems are becoming increasingly more complex, and there is a need for dealing with process dynamics, optimization and control. Unfortunately, the traditional approaches are simply not robust enough to handle such increased complexity, and new methodologies are definitely needed for this purpose. This chapter introduces the concept of AI for thermal system design and analysis. Specific emphasis is given on the discussion of ANN technique, followed by sample examples.

7.2

EXPERT SYSTEM

An ES is an AI application aimed at the resolution of a specific class of problems. It is a well-organized and well-cross-referenced task list with computer as a work tool. An ES is an application consisting of a series of procedures aimed at the solution of a specific class of problems built around the general expert knowledge that can be collected for these problems. ESs allow computers to “make decisions” by interpreting data and selecting from a list of alternatives. ESs take computers a step beyond straightforward programming. It is based on a technique called rule-based inference, in which pre-established rule systems are used to process the data. It associates with a rule for every suggestion or piece of advice a specialist expert would give on how to solve the problem. The ES is better described as knowledge-based system, which is a computer program that emulates the decision-making ability of a human expert. DOI: 10.1201/9781003049272-7

223

224

Design and Analysis of Thermal Systems

An ES makes extensive use of specialized knowledge to solve problems at the level of a human expert. It is an intelligent computer program that uses knowledge and inference procedures to solve problems that are difficult enough to require significant human expertise for the solution. ESs are capable of computational, qualitative, descriptive and explanatory functions. The main idea is to create programs where knowledge and reasoning techniques are introduced, such that it can generate answers similar to those that would be provided by a highly experienced human being. In effect, a user can access the human experts knowledge and experience through the user interface of the computer. The user of an ES asks questions and receives answers and explanations presented in various forms, such as text, video, sound, photo, figure etc. 7.2.1

ADVANTAGES OF EXPERT SYSTEMS

The salient features and advantages of an ES can be summarized as follows: 1. Increased Availability: An ES can be considered as a mass production of expertise because it can be made available on any suitable computer hardware. 2. Reduced Cost: The cost per user of an ES for expertise is less. 3. Life Span: The human experts can retire, quit or die. However, an ES is permanent 4. Multiple Expertise: The knowledge of multiple experts is available in an ES at the same time. An ES can provide the second opinion to a human expert or break a tie in case of disagreements between multiple human experts. An ES can have fast response and is more readily available than a human expert. 5. Intelligent Database: An ES can access a database in an intelligent manner. 7.2.2

DISADVANTAGES OF EXPERT SYSTEMS

An ES can have the following disadvantages: 1. Incorrect Answer: An ES can make an error which can be very costly. 2. Answer outside Its Domain: An ES may provide a solution outside its field of expertise. It may have limited knowledge in a particular domain. As a result, it may give misleading or incorrect answers. A human expert, in contrast, will know the limits of his/her abilities and may not try to solve problems outside his/her expertise. 3. Common Sense Knowledge: It can be difficult to represent common sense knowledge in an ES.

7.2.3

STRUCTURE OF EXPERT SYSTEMS

Figure 7.1 shows a representative architecture of an ES. Knowledge base, inference mechanism and explanation mechanism are the main components of an ES. The ES developer receives knowledge from the experts and prepares it in the form of rules and charts, which are stored in a knowledge base. The inference mechanism receives

225

Artificial Intelligence for Thermal Systems

User

User Interface

Inference Mechanism

Database Variables

KS #1 KS #1

Editor Knowledge Base

KS #n

Knowledge Base KS: Knowledge Source

Explanation Mechanism

Skeleton eleton of the expert system

Experts Domain

Sharing knowledge

ES Developer

Knowledge analysis Construction of rules

Figure 7.1 Architecture of a knowledge-based expert system with its components. (Gonciarz [2014].) input from the user interface, processes the input using a knowledge base and explains it to the user. Several terminologies used in the description and operation of an ES are as follows: 1. Knowledge Base: A knowledge base contains facts and rules that are necessary to solve problems related to a specific field. It contains description of all elements under consideration with a list of their mutual relations, and a list of rules (including mathematical formulae) according to which these elements operate. 2. Editor Knowledge Base: The editor can modify the knowledge contained in an ES. This helps in extension or improvement of the ES with time. 3. Inference Engine: An inference engine is the software that uses the data represented in the knowledge base to reach a conclusion for particular cases. It describes the strategy to be used in solving a problem i.e. it guides from query to solution. 4. Rules: A rule is a way of formalizing declarative knowledge. A rule is a statement of relationships and not a series of instructions. For example, a turbine can be declared as follows: IF a machine is rotating AND IF it has blades AND IF a fluid expands in it, THEN the machine is a turbine. 5. Facts: Facts are specific expressions that describe a particular situation. These expressions can be numerical, logical, symbolic or probabilistic. For example, IF (pressure high-frequency oscillations) is TRUE AND (shaft high-frequency vibrations) is TRUE, THEN probability (compressor stall) = 75 percent.

226

Design and Analysis of Thermal Systems

6. Induction: Induction is a logical procedure that proceeds from effects backward to their causes. For example, high-frequency vibrations detected on a gas turbine shaft may be caused by compressor stall, rotor unbalance in the compressor or turbine, fluid dynamic instabilities in the flow, irregular combustion, mechanical misalignment of the shaft, failure in the lubrication system or bearings wear. Pure induction does not always produce a solution. Other data need to be available that can be included in the inductive process to eliminate some of the probable causes from the list. 7. Backward Chaining: Backward chaining is a procedure that attempts to validate or deny a proposition (goal) by searching through a list of conditional rules and facts to see if they univocally determine the possibility of reaching the goal. Let’s consider an example to ascertain if the machine under consideration is a Francis turbine. Here are the rules: Rule 1: “Francis” IF “turbine” AND “hydraulic” AND “specific speed lower than 2 and higher than 0.3” Rule 2: “turbine” IF “rotating” AND “vaned” Rule 3: “hydraulic”IF “fluid = water” 0.5

f lowrate) Fact 1: “specific speed = rpm” * (volume (g∗total head)3/4 Here are the facts: Fact 2: “rpm = 300” Fact 3: “volume flowrate = 30 m3 /s” Fact 4: “total head = 500 m” Fact 5: “fluid = water” Fact 6: “number of blades = 13”

Goal: “given the knowledge base, determine if the machine is a Francis” Result: “it is not”. Because the specific speed (2.805) criteria is not satisfied. 7.2.4

AN EXAMPLE FOR FEED WATER PUMP SELECTION

Here, we consider an example of feed water pump selection for a power plant. The objective is to devise an automatic procedure for the choice and specification of a feed water pump. The procedure should be able to handle any kind of pump geometry (axial, mixed, centrifugal) and cover the entire range of possible operations. The following tasks list may be constructed by consulting a pump design textbook or by asking an expert: • • •

Read the design data: Flow rate (Q) and Head (H). Read constraints (if applicable): Net Positive Suction Head (NPSH), speed, etc. Is rpm given? if Yes: Then compute: Specific speed, Ns . if No: Then is NPSH given?

227

Artificial Intelligence for Thermal Systems Given H, Q N given ? Yes Compute, Ns

No Ns

No. of stages given ?

Call CAVIT for cavitation check No

Yes Hs = H/nstages

Choose pump type

Design: choose pump geometry, diameter, length etc.

Need drawing ?

Database

Yes

Call AUTOCAD for mesh geometry

Drawing

No Need fluid dynamics calculation

Yes

No Exit

Call ANSYS/COMSOL for CFD simulation

Flow field

Figure 7.2 Decision tree for selection and design of a feed water pump. (Paoletti and Sciubba [1997].)

• • • • •

if Yes: Compute iteratively suction head and derive maximum rpm. if No: Assume rpm, compute NPSH, check Ns . Iterate. Check if multi-staging is required. Check if multiple pumps in parallel are required. Compute overall dimensions of the pump. Check component library for the nearest defined pump type from the database. Produce a technical specification sheet.

The decision tree for the selection and design of the feed water pump is shown in Figure 7.2. Here, CAVIT, AUTOCAD and ANSYS/COMSOL are codes/routines of the knowledge base for cavitation check, meshing and computational fluid dynamics (CFD) calculation, respectively. The CAVIT code is specific to the pump problem. However, AUTOCAD and ANSYS/COMSOL are useful for different types of thermal design problems.

228

7.3

Design and Analysis of Thermal Systems

ARTIFICIAL NEURAL NETWORK (ANN) OVERVIEW

ANN is the leading methodology for the solution of general thermal problems. There are several reasons for this. ANN has the ability to recognize accurately the inherent relationship between any set of input and output without a physical model. This ability is essentially independent of the complexity of the underlying relation such as nonlinearity, multiple variables, noise, certain input and output data. ANN is also inherently fault tolerant due to a large number of processing units in the network. The learning ability of ANN also allows it to adapt to changes in its parameters, which enables the ANN to deal with time-dependent dynamic modeling and adaptive control. The ANN also has ability to incorporate elements of other soft-computing methodologies such as FL and GA to further improve its capability for dealing with additional complexity in thermal problems. One of the limitations of ANN is the requirement of input-output data sets in the learning process to train neural networks. However, it is not a serious shortcoming as a large amount of experimental data sets are available for various thermal systems and device performances. In addition, experimental data obtained under specific dynamic conditions can also be used to train dynamic ANNs. The neural network can be trained in real time when the experimental data are being acquired. This feature is useful in the development of dynamic adaptive-control schemes. 7.3.1

STRUCTURE OF ANNS

The ANN is an electrical analog of biological neural networks. Biological nerve cells, called neurons, receive signals from the neighboring neurons or receptors through dendrites. These neurons process the received electrical pulses at the cell body and transmit signals through a large and thick nerve fiber, called as axon. The electrical model of a typical biological neuron consists of a linear activator, followed by a nonlinear inhibiting function. The linear activation function yields the sum of the weighted input excitation. The nonlinear inhibiting function attempts to capture the signal levels of the sum. An ANN is a collection of such electrical neurons connected in different topologies. The schematic explaining the operation of a neuron is shown in Figure 7.3. Here, xi , y, wi and θ are known as input, output, weight and bias, respectively. The neuron consists of two parts i.e. net function (Σ) and activation function ( f ). The net function determines how inputs are combined inside the neuron. The output of the neuron is determined by the activation function. A fully interconnected ANN consists of a number of processing units known as nodes or artificial neurons, organized in layers. There are three groups of node layers i.e. the input layer, one or more hidden layers and an output layer. Each layer is occupied by a number of nodes. All the nodes of each hidden layer are connected to all the nodes of the previous and following layers by means of inter-node synaptic connectors or simply connectors. Each of the connectors, which mimic the biological neural synapsis, is characterized by a synaptic weight. The nodes of the input layer are used to designate the parameter space of the problem under consideration. The

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Artificial Intelligence for Thermal Systems Bias

x1 Input

θ

Weight

w1

x1

wi

xi

wj

Σ

z

f

y

Output

Σ: Net function f : Activation function

Figure 7.3 Schematic representation of a basic neuron. Neurons

Node number

2,1 w1,1

j=1

Synaptic

Connectors

2,2 w1,1

j=2

2,3 w1,1

j=3

j = Ji Layer number

i=1 Input layer

i=2

i=I Hidden layer

Output layer

Figure 7.4 Schematic of a fully connected multilayer ANN. output layer nodes correspond to the unknowns of the problem. The parameters in the input layer and output layer need not be all independent. At each hidden-layer node, the node input consists of a sum of all the node outputs from the nodes in the previous layer modified by the individual interconnector weights (w) and a local node bias (θ ). The bias represents the propensity of the combined input to trigger a response at the node. The weights are simply weighting functions that determine the relative importance of the signals from all the nodes in the previous layer. At each hidden node, the node output is determined by an activation function, which determines whether the particular node is to activate or not. Information, which starts at the input layer moves forward toward the output layer by the connector and node operations. Such a network is known as a fully connected feed-forward network. Figure 7.4 shows the structure or configuration of a network, consisting of the input, hidden and output layers with node and layer designations. Here, i refers to the layer number with i = 1 is the input layer, and i = I is the output layer with I being the total number of layers. The node number in any layer is denoted as j. Since node numbers are likely to vary from layer to layer, the maximum j value is designated by Ji , depending on the layer number, and JI is thus the number of unknowns in the

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output layer. Each node is designated by (i, j). A somewhat different designation is used for all the connectors since there are two nodes involved. The node on the left is designated by subscripts, while the right node in the forward direction is designated by superscripts. The synaptic weight w2,3 1,1 refers to the connector from Node (1, 1) to Node (2, 3). The nodal input to node (i, j) is written as ji −1

xi, j) = θi, j +

j yi−1,k ∑ wi,i−1,k

(7.1)

k=1

where, θi, j is the nodal bias at (i, j), and yi−1,k is the nodal output at (i − 1, k) of the previous layer. This equation indicates that each signal coming from the previous layer is tampered by the weight in the same connector before they are added, and modified by the local node bias to form the input to the local node (i, j). The information to be processed represents the combined influence of all nodes from the previous layer. The node output, yi, j , is driven by the input, xi, j , through the activation function or threshold function given as yi, j = φi, j (xi, j )

(7.2)

This expression plays the role of the biological neuron i.e. whether it should fire or not on the basis of the strength of the input signal. When the input signal is weak, the artificial neuron simply produces a small output. On the other hand, when the input signal exceeds a certain threshold, the artificial neuron fires and then sends a strong signal to all the connectors and then to all the nodes in the next layer. Several relevant activation functions have been proposed i.e. the step function, the logistic sigmoid function, the hyperbolic tangent, the Gaussian, the wavelet etc. The activation function can also be changed from one hidden layer to another. The most popular and preferred activation function is the continuous version of the step function, known as the logistic sigmoid function. This function possesses continuous derivatives to avoid computational difficulties. It is also highly nonlinear, which is beneficial in dealing with highly nonlinear input-output relations. It is generally written as  −1 φi, j (ξ ) = 1 − eξ /c , = ξ,

i=1

i>1

(7.3) (7.4)

Here, the constant c determines the steepness of the function. It may be noted that the node output yi, j represented by the sigmoid function always lies between 0 and 1 for all xi, j . Therefore, it is desirable to normalize the network input and output data with the largest and the smallest of each of the data sets used in the ANN analysis. 7.3.2

TRAINING OF ANNs

Training is the most important step of ANNs for modeling a thermal system. This section describes the training process of ANNs. When the information reaches from

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input to the output layer, errors can be determined by comparing the calculated feedforward data with the output data to determine the error at each of the output nodes. These errors are then used to adjust all the node biases and connector weights in the entire network to minimize the errors by means of a learning or training procedure. Figure 7.5 shows the flow chart of ANN training process. The most popular training procedure for fully connected feed-forward networks is known as the supervised backpropagation learning scheme based on the steepest-gradient error correction process. The weights and biases are adjusted layer by layer from the output layer toward the input layer during training. The whole process of feeding forward with backward learning is then repeated until a satisfactory error level is reached or becomes stationary. Levenberg–Marquardt and scaled conjugate gradient etc. are the other learning algorithms. For a given chosen network architecture of layers and nodes, the very first step in the training process is to assign initial values to all the synaptic weights and biases in the network. The values may be either positive or negative and are usually taken to be less than unity in absolute values. The second step is to complete all the node input and output calculations based on the equations presented in the previous section. The backpropagation procedure starts with an error function expressed as δI, j = (tI, j − yI, j )yI, j (1 − yI, j )

(7.5)

where, tI, j is the normalized output target for the j-th node of the last output layer. This equation is simply a finite-difference approximation of the derivative of the sigmoid function. Once all δI, j are calculated, the computation moves back to the previous layer, I − 1 subsequently. The target outputs for this layer do not exist. Therefore, a surrogate error is utilized and calculated instead for the hidden layer I − 1, which is given by ji

j δI−1,k = yI−1,k (1 − yI−1,k ) ∑ δI, j wI,I−1,k

(7.6)

j=1

Similar calculations are continued from layer to layer in the backward direction until Layer 2. After all the errors (δi, j ) are known, the changes in the weights and biases can be determined by the generalized delta rule as j ∆wi,i−1,k = λ δi, j yi−1,k

(7.7)

for all i < I, from which all the adjustments in the weights and biases can be determined. The quantity λ is known as the learning rate that is used to scale down the degree of change made to the connectors and nodes. The larger the learning or training rate, the faster the network learns. However, there may be a chance that the ANN may not reach the desired outcome due to oscillatory error behaviors. Its value is normally determined by numerical experimentation, and a commonly arrived value is in the range of 0.4–0.5. The error-correction rate is also modulated by addition of a momentum term based on the old weight and bias changes in the previous learning

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Figure 7.5 Flow chart for the ANN training process. (Mohanraj et al. [2015].)

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iteration, which can be written as j j ∆wi,i−1,k (l) = λ δi, j yi−1,k + β ∆wi,i−1,k (l − 1)

(7.8)

i, j ∆θi−1,k (l)

(7.9)

i, j = λ δi, j + β ∆θi−1,k (l − 1)

where, β is the momentum factor and l is the iteration counter. A cycle of training consists of computing a new set of weights and biases successively for all the experimental runs in the training data. The calculations are then repeated over many cycles while recording an overall error quantity for a specific run within each cycle, given by Er =

1 2

JI

∑

tI, j − yI, j


2

(7.10)

j=1

The weights and biases are continually updated throughout the runs and cycles. The training is terminated when the last cycle error falls below a prescribed threshold or becomes stationary. The final sets of weights and biases can now be used for prediction purposes, and the corresponding ANN becomes a model of the inputoutput relation of the thermal system. The overall ANN analysis involves few deterministic and algebraic steps repeated many times on the computer. It involves a relatively large number of free parameters and choices i.e. the number of hidden layers, the number of nodes in each layer, the initial weights and biases, the learning rate, the minimum number of training data sets and the choice of input parameters. The computational steps are simple. However, overall effort depends on the total number of nodes in the network, as a large number of nodes tends to slow down the training process. One flexibility of the ANN methodology is that both numbers of the hidden layer and the corresponding nodes can be increased at will from one training cycle to another training cycle if the cycle errors do not decrease as expected. On the other hand, it may be noted that too many nodes may suffer from the localizing effect of specific data points similar to selecting the degree of polynomial during curve fitting. The issue of assigning initial weights and biases is also difficult in a new application. Without past information or data, the current practice is simply to generate a set of initial data from a random number generator of bounded numbers. The choice of the training rate can be between 0.4 and 0.5 during the starting point. The sigmoid activation function possesses asymptotic limits of 0 and 1 and may cause difficulties when these limits are approached. Therefore, the usual practice is to normalize all physical variables in an arbitrarily restricted range such as 0.15–0.85 to limit the computational efforts. It is desirable to include as many training data sets as possible as the experimental data set can have error and uncertainty. It is also important to set aside about one-quarter of the entire data set to serve as the testing data set to evaluate the accuracy of the ANN results.

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Figure 7.6 Schematic of a multi-column fin tube heat exchanger. (Pacheco Vega [2001].)

7.4

ANNs FOR HEAT EXCHANGER ANALYSIS

Heat exchangers involve significant geometrical complexity and operating conditions involving complex physics. In this section, we will evaluate the capability of ANN in modeling a heat exchanger. The compact multi-row, multi-column, fintube heat exchanger shown in Figure 7.6 is considered a sample. This heat exchanger utilizes chilled water flow inside the tubes for air-cooling purpose. Mcquiston [1978a,b] studied this heat exchanger in great detail by extensive careful experimental measurements and developed correlations in terms of the Colburn j-factors. The chilled water temperature can cause the air temperature to fall below its dew point leading to condensation on the fin surfaces. The data set contained all three fin-surface conditions i.e. dry surface, surface with dropwise condensation and surface with film condensation. Dropwise and film condensation cases were

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differentiated by subjective visualization. The film spacing was an added important parameter because of the condensation phenomena, which was also treated as an input parameter. Only high Reynolds number turbulent flow conditions were considered on the water side. The following correlations were proposed: js

= 0.0014 + 0.2618Re−0.4 D



A Atb

jt

= 0.0014 + 0.2618Re−0.4 D



A Atb

−0.15 −0.15

fs

(7.11)

ft

(7.12)

Here, A is the total air-side heat transfer area, Atb is the surface area of tubes without fins, fs and ft are the functions of the fin geometry and ReD is the Reynolds number based on the tube diameter. For dry surface, fs ft

= 1.0 = 1.0

(7.13) (7.14)

For dropwise condensation,   −1 δ fs = 0.90 + 4.3 × 10−5 Re1.25 D δ −t   4 δ ft = 0.80 + 4.0 × 10−5 Re1.25 D δ −t

(7.15) (7.16)

where, δ and t are the fin spacing and thickness, respectively. For filmwise condensation, fs ft

= 0.84 + 4.0 × 10−5 Re1.25 D 2   δ −5 1.25 = 0.95 + 4.0 × 10 ReD δ −t

(7.17) (7.18)

The Colburn factors are defined as follows: js =

ha 2/3 Pra ; Gcc pa

jt =

hta 2/3 Sca Gc

GcD Gcδ A 4 xa xb ; Reδ = ; = σf µa µa Atb π Dh D Here, Gc is the air mass velocity based on the free flow area, h is the heat transfer coefficient, ht is the total heat transfer coefficient, σ f is the ratio of the free flow cross-sectional area to the frontal area, Reδ is the Reynolds number based on fin spacing, js is the Colburn j-factor for the sensible heat and jt is for the total heat. Figure 7.6 shows the geometrical arrangement of the tube denoted by xa and xb . Later, Gray and Webb [1986] proposed a new correlation using data from additional sources. For a dry surface,   −0.502  δ − t 0.0312 −0.328 xb js = 0.14ReD (7.19) xa D ReD =

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Design and Analysis of Thermal Systems

W

a,db Tin

θ js

a,wb Tin

ji

w Tin

Q˙

ReD δ Input layer

Hidden layer

Output layer

Figure 7.7 A 5-5-3-3 ANN for analysis of a multi-column fin tube heat exchanger. (Pacheco Vega [2001].) Pacheco-Vega et al. [2000] carried out an ANN analysis of a fin tube heat exchanger and compared it with that of the correlations. They used a fully connected feedforward network of 5-5-3-3 configuration (Figure 7.7) and backpropagation learning algorithm. The five input nodes correspond to the air-inlet, dry-bulb and wet-bulb temperatures, the chilled water inlet temperature, the airflow Reynolds number and the fin spacing. The three output nodes correspond to js for sensible heat transfer jt for total heat transfer and Q for the total heat-transfer rate. They used 91 data sets for dry-surface conditions, 117 data sets for dropwise condensation and 119 data sets for film condensations. They also trained entire 327 data sets for training to compare the ANNs trained with separate data sets involving different physics with the ANN trained with the complete data set. Table 7.1 shows the results in the rms percentage deviations of the prediction by ANNs and correlations. A low level of error in total heat transfer is observed which is comparable to the expected experimental uncertainties. The ANNs give better predictions for dry surfaces than those for wet surfaces, possibly due to the complex physics of wet surfaces. When the ANNs are trained with the entire data sets by disregarding surface conditions, all deviations tend to increase. This may be attributed to more complexity and variability in physics involved.

7.5

ANNs FOR A THERMOPHYSICAL PROPERTY DATABASE

As discussed earlier in Chapter 4, the selection of fluid is an important component of several thermal system designs. There are a large number of fluid types for selection, and also many more types of fluid are also being added to the database continuously. In addition, the properties of fluid are functions of the physical state i.e. temperature, pressure, composition etc. Several correlations are available for predicting the physical properties of fluid as a function of the operational parameters. However, errors in using these correlations are high. Therefore, it’s a challenge for a design engineer to

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Table 7.1 Comparison of Percentage Errors in Predictions between the ANN and Standard Power Law Correlations of Heat Exchangers Surface

Dry

Dropwise

Filmwise Combined

Method

js

jt

Qi

McQuiston Gray and Webb ANN McQuiston Gray and Webb ANN McQuiston Gray and Webb ANN ANN

14.57 11.62 1.002 8.5 – 3.32 9.01 – 2.58 4.58

14.57 11.62 1.002 7.5 – 3.87 14.98 – 3.15 5.05

6.07 4.95 0.928 – – 1.446 – – 1.96 2.69

Source: Yang et al. [2008].

select the best working fluid. The availability of several soft computing techniques has opened a new tool to easily accommodate large data sets of working fluids at various operating conditions for design. For example, nanofluid of several types is being considered a promising working fluid for several applications. The prediction of nanofluid properties is discussed in this section. Nanofluids are suspension of small diameter particles (less than 100 nm) i.e. TiO2 , Al2 O3 , SiC, SiO2 , CuO, etc. dispersed in base fluid i.e. water, deionized water, ethylene glycol (EG), ethanol, polyalphaolefin, transformer oil, mixture of EG and water, mixture of propylene glycol and water, R11 refrigerant and so on. The thermal conductivity of nanofluids has been observed to be higher than base fluid even at very low concentrations. Therefore, nanofluid is actively considered as a coolant in several applications. Accurate prediction of properties e.g. viscosity and thermal conductivity is difficult because of complexities due to hydrodynamic and particle–particle interactions of nanoparticles in dispersion. Several empirical models have been proposed in literature. However, the limitation of these models is the requirement of a sufficient amount of data for calibration and validation purposes that makes these models computationally incompetent. The advantages of ANN methodology compared to conceptual models are its high speed, simplicity and large capacity with a lower computational requirement. Let’s consider one example of nanofluid property modeling using ANN. Heidari et al. [2016] developed a feed-forward backpropagation multilayer perceptron ANN for predicting nanofluid viscosity in broad ranges of operating parameters. They evaluated several configurations and the network performance was optimized by changing the number of hidden layers, number of neurons in the hidden layer and network training algorithm in order to obtain the best network for

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Design and Analysis of Thermal Systems Input layer

Hidden layer

Temperature (T )

1

1

Particle size (S)

2

2

Volume fraction (φ)

3

3

Particle density (D)

4

Fluid viscosity (F )

5

Output layer 1 2 1 14

Relative viscosity

wil3

2 wji

25 1 wij

Figure 7.8 Structure of ANN network for modeling the property of nanofluid. (Heidari et al. [2016].) the prediction of nanofluid relative viscosity. The schematic of the ANN arcitechture used is shown in Figure 7.8. The model input parameters are temperature (T ), nanoparticle size (S), density (D), volume fraction (Φ) and base fluid viscosity (F). They used 1490 data points for different nanofluids with different base fluids (water, ethanol, EG, transformer oil, R-11 refrigerant, toluene etc.) and nanoparticles (TiO2 , Al2 O3 , SiC,CUo, Fe3 O4 etc.) for a wide range of different parameters. The ANN network was compared with the correlation developed by Meybodi et al. [2015] based on the data set of viscosity of water-based Al2 O3 , TiO2 , SiO2 and CuO nanofluids. The mathematical form of the correlation is Relative viscosity = 135.54064976 − 343.82413843(eφ /s ) + 290.11804759(eφ /s )2 − 78.993120761(eφ /s )3 0.91161630781 + 32.33014233

Ln(s) T

− 11.732514460

(Ln(s))2 T

(7.20)

where, S, φ and T are size of nanoparticles in nm, volumetric concentration of nanoparticle in percent and temperature of the system in Kelvin, respectively. Several statistical criteria used to evaluate the performance of ANN i.e. average absolute relative deviation (AARD, %), coefficient of determination (R2 ), mean square error (MSE) are defined as follows: 1 N yi − y p AARD(%) = ∑ × 100 (7.21) N i=1 yi
2 ∑N yi − y p R2 = 1 − i=1 (7.22) ¯2 ∑Ni=1 (yi − y) MSE =

1 N ∑ (yi − y p )2 N i=1

(7.23)

where, yi , y p , y¯ and N are experimental data, predicted data, average value of experimental data and number of data points, respectively.

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Table 7.2 ANN Network Performance for Different Input Type Combinations

No. of Test

Combination

2

R of All Data

1 2 3 4 5

Group of One Variable T 0.5123 S 0.5206 V 0.8766 D 0.2875 F 0.3904

6 7 8 9 10 11 12 13 14 15

Group of Two Variables T+S 0.6368 T+V 0.9036 T+D 0.5926 T+F 0.5312 S+V 0.9895 S+D 0.6848 S+F 0.6111 V+D 0.9782 V+F 0.8958 D+F 0.4883

No. of Test

Combination

2

R of All Data

16 17 18 19 20 21 22 23 24 25

Group of Three Variables T+S+V 0.9999 T+S+D 0.6656 T+S+F 0.6593 T+V+D 0.9953 T+V+F 0.9164 S+V+D 0.9895 S+V+F 0.9998 V+D+F 0.9947 S+D+F 0.6087 T+D+F 0.6302

26 27 28 29 30

Group of Four Variables T+S+V+D 0.9999 T+S+V+F 0.9999 S+V+D+F 0.9998 T+V+D+F 0.9963 T+S+D+F 0.6606

31

Group of Five Variables T+S+V+D+F 1

Source: Heidari et al. [2016]. Note: T: temperature, S: particle size, V: particle volume, D: particle density, F: fluid viscosity.

Heidari et al. [2016] reported the AARD for water-based nanofluid from the ANN model to be equal to 0.47, which is much lower than 11.2 obtained from correlation by Meybodi et al. [2015]. This observation indicates the superior performance of ANN in the prediction of nanofluid properties. A sensitivity analysis was also carried out to determine the relative importance of different input variables (T, S, Φ, D, F) used by the ANN model. The performance evaluations of various possible interactions of input variables i.e. one, two, three, four and five variables were investigated. Table 7.2 compares the network performance for different input types i.e. group of input variables. The volume fraction of nanoparticles is observed to be the most effective variable in the group of one variable models due to its higher R2 value (0.87662) among all parameters. The highest

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Figure 7.9 Schematic of physics-informed neural network (PINN). A fully connected neural network is used to approximate the solution u(x, t), which is then applied to construct the residual loss Lr , boundary conditions loss Lb and initial conditions loss Lo . (Cai et al. [2021].) R2 value observed for 2, 3, 4 and 5 input parameters are equal to 0.98946, 0.99989, 0.99994 and 0.99998, respectively. Overall, this study demonstrated the capability of the ANN technique for accurate prediction of nanofluid viscosity over wide ranges of operating parameters.

7.6

PHYSICS INFORMED ANNs

Both hard computing and soft computing techniques are useful for the design of thermal systems. The shortcomings of the hard computing approach are absence of accurate numerical model for complex problems, non-specific boundary conditions and large computational requirements. The shortcomings of the soft computing technique are the exclusion of the underlying physics of the process and the large resource requirement for the generation of the training data for ANN. The physics informed neural network (PINN) technique can accommodate these shortcomings and will be discussed in this section. Cai et al. [2021] reported the application of PINNs to various prototype heat transfer problems with realistic conditions not readily tackled by traditional computational methods. A schematic of the PINN framework is demonstrated in Figure 7.9, in which a simple heat equation (ut = uxx ) is used to solve a heat transfer problem. The fully connected neural network is used to predict the solution u(x,t) inside the domain, where x and t denotes space and time variable, respectively. The solution u(x,t) is used to calculate different partial derivative terms of the governing equation. Subsequently, the residual loss terms of governing equation are calculated as 2 1 Nr ∂ u i i ∂ 2 u i i Lr = (7.24) ∑ ∂t (x t ) − α ∂ x2 (x t ) Nr i=1

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241

where, Nb is the number of data points inside the domain. Residual loss term of the boundary conditions is calculated as Lb =

2 1 Nb i i u(x ,t ) − gi ∑ Nb i=1

(7.25)

where, Nr is the number of data points in boundary condition and gi is the boundary condition. Residual loss terms of initial conditions are given as Lo =

2 1 No i i u(x t ) − hi ∑ No i=1

(7.26)

where, No is the number of data points representing the initial condition and hi is the initial condition. The backpropagation training of the ANN is carried out for minimizing the total loss. The sensor locations, where the data sets are collected, also affect the inferred results. The best sensor locations are generally decided by trial-and-error, which is a costly process. Cai et al. [2021] also proposed an iterative method for optimizing the sensor location. The residual of the heat transfer equation is considered the criterion for selecting the sensor location. They compared the solution from the ANN with an independent reference CFD solution and reported the difference using L2 error defined as  εv = |V p −V ∗ ||2 ||V ∗ ||2 (7.27) where, V p represents one of the predicted quantities and V ∗ is the corresponding reference solution. Several different types of problems were solved using PINN i.e. (1) forced and mixed convection with unknown thermal boundary conditions on the heated surfaces, (2) Stefan problem for two-phase flow and (3) industrial applications related to power electronics. The results presented for different sample problems demonstrated that PINN can solve ill-posed problems, which are beyond the reach of traditional computational methods. PINN also helps to bridge the gap between computational and experimental heat transfer.

7.7

ANNs FOR DYNAMIC THERMAL SYSTEMS

Most of the thermal systems operate in dynamic conditions responding to changes in the operating parameters and boundary conditions. The performance of a device in a thermal system is also affected by other devices and components that are directly or indirectly connected to it. Therefore, there is a need for dynamic models for prediction of performance under dynamic conditions. There is also a more critical role for such dynamic models in adaptive control systems. Many known control schemes that are accurate require dynamic plant models for their implementation. The traditional available techniques cannot develop accurate dynamic models even for simple thermal devices. However, ANN-based techniques offer a viable alternative. The central scheme in the dynamic modeling by ANN is the addition of

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Figure 7.10 The behavior of the test cell temperature for ANN- and PID-based control with air flow actuation for four changes at the set point. (Varshney and Panigrahi [2005].) time as a variable for both training and prediction. Another scheme is to provide variables at time t as inputs, and the values of the same variables at later times, t + ∆t as the outputs. Varshney and Panigrahi [2005] demonstrated the dynamic modeling and control capability of ANN for a heat exchanger. The air flow rate over a fin tube heat exchanger was controlled by varying the voltage supply to the motor driving the blower. The water flow rate inside the heat exchanger tube was controlled by varying the supply voltage to the motor driving the pump. A thermocouple rack was used to monitor the air flow temperature downstream of the heat exchanger. The water temperature to the heat exchanger was maintained constant using a solid-state relay-based temperature controller. Figure 7.11 shows the general control structure of the neural network-based control. The neural network model and the inverse neural network model are the two important components of the control methodology. The neural network model uses the future process variable i.e. outlet air temperature at later time as output. The process variable i.e. air temperature at previous time step and the corresponding actuator output i.e. supply voltage are used as input. The future actuator output i.e. voltage supply to the motor is the output of the inverse neural network model. The process variables i.e. air temperature and actuator outputs i.e.

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243

Figure 7.11 Schematic of the control structure of the ANN-based control of a heat exchanger. (Varshney and Panigrahi [2005].) supply voltage of motor are the input to the inverse neural network model. Both the neural network model and the inverse neural model control use five inputs and one output for single actuation control. The five inputs for the inverse neural network model are the present temperature, the control voltage and temperature at two previous time steps, while the present actuation voltage is the output of the inverse neural network model. The output of the neural network model is the present temperature, while the five inputs are the temperature and voltage at two previous time steps and the present voltage. The robustness filter with a single tuning parameter is implemented to eliminate the steady-state offset due to the mismatch between the plant and the neural network model. Figure 7.12 shows a typical feed-forward multilayer perceptron neural network used for modeling and control of the heat exchanger. The training data for the ANN were collected using the Labview program supplied by the National Instruments. The voltage to the motor controller from the DAQ card was varied in the range of 0–10 V in a step of 0.2 V and brought back from 10 V to 0 V in the same step size. The time interval was set equal to 60 s. This procedure was repeated with lower step sizes i.e. 0.1 V, 0.05 V and 0.02 V. The data file for the training of single actuation ANN control model was arranged in six columns; the first and second columns show current voltage and temperature data, the third and fourth columns show voltage and temperature data after 60 s and the last two columns show voltage and temperature data after 120 s. The training was carried out with the first five columns as input to the ANN and the last column as the output of the ANN model. The testing data file was generated using the same procedure at a different time i.e. one or two days after the acquisition of the training data. The bias and weight obtained from the training of ANN are used during the testing of the ANN model. The maximum and the rms value of the difference between the output from the ANN model and the testing data are considered as a performance indicator. When using the dual control, both air

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History of past actuators values

xi (k) xi (k − 1) xi (k − 2) y j (k + 1)

Process values at time k + 1

History of process values

y i (k) y i (k − 1)

Input Layer

Hidden Layer

Output Layer

Figure 7.12 Schematic of the multilayer perceptron neural network. (Varshney and Panigrahi [2005].)

Table 7.3 Rise Time Comparison between the ANN- and PID-Based Controls at Different Set Points of Temperature Rise Time (s) Method

Set Point

Air Flow

Water Flow

Dual

Control

Temperature (◦ C)

Actuation

Actuation

Actuation

45 45 40 40

150 240 125 210

140 170 130 150

75 110 70 105

ANN PID ANN PID

Source: Varshney and Panigrahi [2005].

flow and water flow actuator voltages were varied simultaneously and arranged in nine columns of a data file for the training of the ANN, out of which the last two columns are the output of the ANN. The learning rate was set equal to 0.4. The momentum factor was set equal to 0.9. Table 7.3 compares the rise time between proportional-integral-derivative (PID) and ANN controls for different set points of temperature and actuation conditions. The rise time is observed to be significantly lower for ANN control compared to PID control at all actuation conditions. Similarly, the steady-state rms error was also observed to be lower for ANN than PID control.

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245

The ANN- and PID-based control with air flow actuation for multiple changes in set points (40 ◦ C, 45 ◦ C, 47 ◦ C and 49 ◦ C) are presented in Figure 7.10 for both increase and decrease in the set point temperature. Both the ANN- and PID-based control are successful in maintaining multiple changes in set points. The ANNbased control shows a faster response with less overshoot in comparison to the PID-based control for all changes in set points. Overall, this result demonstrates the fast capability of ANN for modeling and control of a dynamic complex thermal system.

7.8

SUMMARY

This chapter introduces the concepts of AI for the design of thermal systems. The concept of an ES was introduced. Sample examples of an ES are discussed to illustrate its role in the design of thermal systems. Subsequently, the soft computing technique was introduced for the design of thermal systems with an emphasis on the ANN technique. The implementation procedure i.e. structure and training of ANN was presented. The capability of ANN for the sample examples i.e. modeling of heat exchanger, representation of physical property database of fluid, physics informed ANN and dynamics of thermal system was also presented. The other soft computing techniques i.e. FL and GA are also useful for thermal systems depending on the nature of applications. The logic of fuzzy sets is useful in approximate reasoning in ESs. The FL deals with fuzzy sets and logical statements for modeling human-like reasoning problems of the real world. A fuzzy set includes all elements of the universal set of the domain with varying membership values in the interval [0,1]. GA is a stochastic algorithm that mimics the natural process of biological evolution. It is inspired by the way living organisms are adapted to the harsh realities of a hostile world i.e. by evolution and inheritance. The algorithm imitates the evolution of the population by selecting only fit individuals for reproduction. It is an optimum search technique based on the concepts of natural selection and survival of the fittest. It works with a fixed size population of possible solutions to a problem, called individuals, which evolves in time. GAs are denoted by chromosomes, which are usually represented by binary strings. A GA utilizes three principal genetic operators: selection, crossover and mutation. Hybrid systems can combine different soft computing techniques to enhance overall effectiveness. The hybrid systems can overcome the limitations of single methods, enhance the prediction performance and have a faster response with lower error. Yang [2008] discussed some of the hybrid ANN technologies. Bahiraei et al. [2019] presented several hybrid AI techniques for different applications related to nanofluids. Bakhtiyari et al. [2021] reported the implementation of a hybrid technique for modeling and optimization of laser beam machining. These references and other resources can be reviewed for more details of improved design of thermal systems.

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REFERENCES M. Bahiraei, S. Heshmatian, and H. Moayedi. Artificial intelligence in the field of nanofluids: A review on applications and potential future directions. Powder Technology, 353:276–301, Jul 2019. doi: 10.1016/j.powtec.2019.05.034. A. N. Bakhtiyari, Z. Wang, L. Wang, and H. Zheng. A review on applications of artificial intelligence in modeling and optimization of laser beam machining. Optics & Laser Technology, 135:106721, 2021. ISSN 0030-3992. doi: 10. 1016/j.optlastec.2020.106721. S. Cai, Z. Wang, S. Wang, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks for heat transfer problems. Journal of Heat Transfer, 143(6), Apr 2021. ISSN 0022-1481. doi: 10.1115/1.4050542. 060801. T. Gonciarz. An expert system for supporting the design and selection of mechanical equipment for recreational crafts. TransNav, the International Journal on Marine Navigation and Safety of Sea Transportation, 8(2):275–280, 2014. ISSN 2083-6473. doi: 10.12716/1001.08.02.13. D. L. Gray and R. L. Webb. Heat transfer and friction correlations for plate finnedtube heat exchangers having plain fins. In Proceeding of International Heat Transfer Conference 8. Begellhouse, 1986. doi: 10.1615/ihtc8.1200. E. Heidari, M. A. Sobati, and S. Movahedirad. Accurate prediction of nanofluid viscosity using a multilayer perceptron artificial neural network (MLPANN). Chemometrics and Intelligent Laboratory Systems, 155:73–85, 2016. ISSN 0169-7439. doi: 10.1016/j.chemolab.2016.03.031. https://www. sciencedirect.com/science/article/pii/S0169743916300740. F. C. Mcquiston. Correlation of heat, mass, and momentum trans-port coefficients for plate-fin-tube heat transfer surfaces with staggered tubes. ASHRAE Trans., 84:294–309, 1978a. F. C. Mcquiston. Heat, mass and momentum transfer data for five plate-fin-tube heat transfer surfaces. ASHRAE Trans., 84:266–293, 1978b. M. K. Meybodi, S. Naseri, A. Shokrollahi, and A. Daryasafar. Prediction of viscosity of water-based Al2O3, TiO2, SiO2, and CuO nanofluids using a reliable approach. Chemometrics and Intelligent Laboratory Systems, 149(Part A):60– 69, 2015. doi: 10.1016/j.chemolab.2015.10.001. M. Mohanraj, S. Jayaraj, and C. Muraleedharan. Applications of artificial neural networks for thermal analysis of heat exchangers – A review. International Journal of Thermal Sciences, 90:150–172, 2015. ISSN 1290-0729. doi: 10. 1016/j.ijthermalsci.2014.11.030. A. Pacheco-Vega, G. Diaz, M. Sen, K. T. Yang, and R. L. McClain. Heat rate predictions in humid air-water heat exchangers using correlations and neural networks. Journal of Heat Transfer, 123(2):348–354, Oct 2001. ISSN 0022-1481. doi: 10.1115/1.1351167. B. Paoletti and E. Sciubba. Artificial Intelligence in Thermal Systems Design: Concepts and Applications, page 234–278. Cambridge University Press, 1997. doi: 10.1017/CBO9780511529528.011.

Artificial Intelligence for Thermal Systems

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K. Varshney and P. Panigrahi. Artificial neural network control of a heat exchanger in a closed flow air circuit. Applied Soft Computing, 5(4):441–465, 2005. ISSN 1568-4946. doi: 10.1016/j.asoc.2004.10.004. https://www. sciencedirect.com/science/article/pii/S156849460400105X. K.-T. Yang. Artificial neural networks (ANNs): A new paradigm for thermal science and engineering. Journal of Heat Transfer, 130(9), Jul 2008. ISSN 0022-1481. doi: 10.1115/1.2944238. 093001.

8 Numerical Linear Algebra In this chapter, we focus on the numerical solution of equation involving one unknown. The general form of such equation is written as f (x) = 0

(8.1)

This equation may have many real and complex roots. For brevity, however, here we focus only on finding a single real root using two methods: bisection method and Newton–Raphson method. Being iterative in nature, both methods start from an arbitrary initial guess and are likely to converge to a solution that is close to the initial guess. Let us assume x = x1 to be a solution of equation 8.1. We can test our assumption very easily by evaluating f (x = x1 ). If we find f (x = x1 ) ≈ 0, we have just solved the problem. Unfortunately, one has to be very lucky to find the solution just from the first guess. In general, we are likely to find f (x = x1 ) 6= 0.

8.1

BISECTION METHOD

In the bisection method, our initial goal is to find x1 and x2 such that f (x = x1 ) f (x = x2 ) < 0

(8.2)

Let us assume f (x = x1 ) < 0 and f (x = x2 ) > 0. If f (x) is continuous in [x1 , x2 ], there must be a root x = x0 , such that x1 < x0 < x2 . The goal now is to reduce the interval [x1 , x2 ] while maintaining f (x = x1 ) f (x = x2 ) < 0, terminating the procedure when f (x = x1 ) or f (x = x2 ) is sufficiently close to zero. The procedure replaces either x1 or x2 with x0 = (x1 + x2 ) /2. If f (x = x0 ) < 0, we set x1 = x0 , while f (x = x0 ) > 0, we set x2 = x0 . The algorithm of the bisection method may be written as follows. Example: Find one root of the following equation: f (x) = x2 − 7x + 10 = 0

(8.3)

f (x = 0) = 10 and f (x = 3) = −2 ⇒ f (x = 0) f (x = 3) < 0

(8.4)

We note that

Clearly equation 8.3 has a root between x = 0 and x = 3. If we now follow the bisection method, we iterate to generate Table 8.1. It is evident from the table that we are approaching toward x = 2, which is a solution of equation 8.3. The computer program for the above problem is added in the course website.

DOI: 10.1201/9781003049272-8

249

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Design and Analysis of Thermal Systems

Algorithm 8.1: Bisection Method Data: f (x) , x1 , x2 where f (x = x1 ) f (x = x2 ) < 0, maximumError Result: x0 where f (x = x0 ) < maximumError y1 = f (x = x1 ) y2 = f (x = x2 ) if (y1 ∗ y2 ) < 0 then error = min (|y1 | ,|y2 |) while error > maximumError do x0 = (x1 + x2)/2; y1, y0 = y(x1), y(x0) error = min (|y1 | ,|y0 |) if error < maximumError then if |y1| < maximumError then x0 = x1 end else if (y1 ∗ y0) > 0 then x1 = x0 else x2 = x0 print(error, x0, x1, x2) end end else print(y1, y2);print(“invalid input”) end

8.1.1

CONVERGENCE OF BISECTION METHOD

Consider a function f (x) is continuous in [x1 , x2 ], where f (x1 ) f (x2 ) < 0. In the bisection method, described above, in every iteration, we update x1 and x2 . We denote (n) (0) (0) x1 as the value of x1 after the nth iteration. Thus, x1 and x2 are the initial values of x1 and x2 . Following the iterative procedure described above, we know that after the first iteration (0) (0) x + x2 (1) (0) (1) (8.5) either x1 = x1 , x2 = 1 2 (0) (0) x + x2 (1) (1) (0) or x1 = 1 , x2 = x2 (8.6) 2 From these equations, we can write (1)

(1)

x1 − x2 =

(0)

(0)

x1 − x2 2

(8.7)

Let us proceed to another iteration to find that (2)

(2)

x1 − x2 =

(1)

(1)

(0)

(0)

x1 − x2 x −x = 1 2 2 2 2

(8.8)

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Numerical Linear Algebra

Table 8.1 Bisection Method Iterations Iteration

x1

x2

x0 = (x1 + x2 )/2

f (x0 )

0 1 2 3 4 5 6 7 8 9 10

0 1.5 1.5 1.875 1.875 1.9688 1.9688 1.9921 1.9921 1.9980 1.9980

3 3 2.25 2.25 2.0625 2.0625 2.0156 2.0156 2.0039 2.0039 2.0009

1.5 2.25 1.875 2.0625 1.9688 2.0156 1.9921 2.0039 1.9980 2.0009 1.9995

1.75 0.6875 0.3906 0.1836 0.0947 0.0466 0.0235 0.0117 0.0059 0.0029 0.0015

Generalizing this procedure, we find that after the n-th iteration (n)

(n)

x1 − x2 = We therefore find that

(0)

(0)

x1 − x2 2n

h i (n) (n) lim x1 − x2 = 0

n→∞

(8.9)

(8.10)

Since f (x1 ) f (x1 ) < 0, equation 8.10 indicates that the bisection algorithm always ensures convergence to the acceptable solution.

8.2

NEWTON–RAPHSON METHOD

The Newton–Raphson method presents a powerful alternative to the bisection method. The method is generally much faster than the bisection method. Further, as opposed to the closed bisection method that requires two initial guesses, the open Newton–Raphson method starts with one initial guess. While solving the equation f (x) = 0, the Newton–Raphson method relies on linearization of f (x). The linearization is usually achieved by expanding the function via the Taylor series expansion of f (x). Neglecting higher-order terms, a function f (x) may be expanded in the Taylor series as f (x + ∆x) = f (x) + ∆x f 0 (x) + · · · (8.11) For f (x + ∆x) = 0, we have ∆x = −

f (x) f 0 (x)

(8.12)

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Design and Analysis of Thermal Systems

Figure 8.1 Geometric interpretation of the Newton–Raphson method. The procedure for the Newton–Raphson method, therefore, proceeds as follows: 1. 2. 3. 4.

Start iteration from an initial guess x = x0 Set ∆x = − ff0(x) (x) Calculate x = x0 + ∆x Go to step 2 and continue till f (x) ≈ 0

An algorithm (pseudocode) of the Newton–Raphson method is shown below. Algorithm 8.2: Newton–Raphson Method Data: f (x) , f 0 (x) , x0 , maximumError >0 Result: x0 where f (x0 ) < maximumError y1 = f (x0 ) while |y1 | ≥ maximumError do y2 = f 0 (x0 ) ∆x = − yy21 x0 = x0 + ∆x y1 = f (x0 ) end print (x0 ); print(“solution converged”) The geometric interpretation of the Newton–Raphson method is described in Figure 8.1. Here, we intend to find the solution of the equation f (x) = 0. The
initial guess, of the solution, is x0 . If we draw a tangent at the point x0 , f (x0 ) , the tangent cuts the x-axis at the point x1 . x1 is the value of the solution after one iteration. In the same manner, we have the value of the solution after two iterations is x2 . If we proceed this way, we will eventually reach the converged solution with acceptable accuracy. It should be noted here that, unlike the bisection method, the convergence of the Newton–Raphson method is not guaranteed. A detailed discussion of the Newton–Raphson method is beyond the scope of this book. √ Example: For an example, let us try to evaluate 2 using √ the Newton–Raphson method. As seen before, one way to find the value of 2 would be to solve the following equation and pick the positive root f (x) = x2 − 2 = 0

(8.13)

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Numerical Linear Algebra

Table 8.2 Newton–Raphson Method Iterations Iteration

x0

∆x

x1

0 1 2 3

2 1.5 1.4167 1.414215

−0.5 −0.0833 −0.00245 −2.1239 × 10−6

1.5 1.4167 1.414215 1.414213

SOLUTION We note that f 0 (x) = 2x

(8.14)

and thus

f (x) x2 − 2 1 x = − = − (8.15) f 0 (x) 2x x 2 For an initial guess of x0 , the solution after the first iteration x1 would be ∆x = −

x1 = x0 + ∆x (x = x0 ) = x0 +

1 x0 1 x0 − = + x0 2 x0 2

(8.16)

If we now follow the procedure suggested by the Newton–Raphson algorithm, we iterate to√generate Table 8.2. It is evident from the table that we are approaching toward x = 2 which again is a solution of equation 8.13 The computer program for the above problem is added in the course website.

PROBLEMS

√ 1. Modify the PYTHON program, shown in the book, to find the value of 2 using the bisection method. The value should be correct up to four decimal places. 2. Modify the PYTHON program, shown in the book, to find a root of x2 − 7x + 10 using the Newton–Raphson method. The value should be correct up to four decimal places.

8.3

EIGENVALUES AND EIGENVECTORS

Any matrix A, with eigenvalue λ and the corresponding eigenvector x, must satisfy Av = λ v

(8.17)

Thus, for a matrix A, eigenvalues are the roots λ of the polynomial p (λ ), where p (λ ) = det (A − λ I)

(8.18)

where, I is the identity matrix. The polynomial p (λ ) is called the characteristic polynomial of matrix A.

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Design and Analysis of Thermal Systems

For small matrices, we may solve the equation p (λ ) = 0 to calculate the eigenvalues λ . Once we know λ , we find the eigenvector x by solving the equation Ax = λ x. Unfortunately, the same methods of finding eigenvalues and eigenvectors cannot be used for large matrices. For large matrices, the calculation of the determinant is a daunting task and so is finding the solution of p (λ ) = 0.

8.4

POWER ITERATIONS

The power iteration, also called the power method, is designed to find dominant eigenvalue of a diagonlizable matrix. Consider an n × n diagonalizable matrix A. Since the matrix A is diagonalizable, it has linearly independent eigenvectors v1 , v2 , . . . , vn , where the corresponding eignevalues are λ1 , λ2 , . . . , λn . The existence of a dominant eigenvale λ1 suggests that |λ1 | > |λi | where i = 2, 3, . . . , n. Power iteration numerically finds λ1 and the corresponding eigenvector v1 using the following algorithm. Algorithm 8.3: Power Iteration Data: Diagonalizable n × n matrix A; a randomly generated n-dimensional vector v such that, kvk 6= 0; maximumError > 0 Result: Dominant eigenvalue λ , corrsponding eigenvector v v v = kvk error = 10 /* value of error is assigned arbitrarily, such that error > maximumError */

while error > maximumError do Av u = kAvk error = ku − vk v=u end λ = kAvk Example: Find the dominant eigenvalue and the corresponding normalized eigenvector for the following matrix: " # 7 3 A= (8.19) 6 4 SOLUTION Analytically, we may find the eigenvalues as follows: det (A − λ I) = 0 ⇒ λ = 10, 1 The normalized eigenvector corresponding to the eigenvalue λ = 10 is " # " # 1 1 0.7071 v= √ = 0.7071 2 1

(8.20)

(8.21)

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Numerical Linear Algebra

Table 8.3 Power Iteration Iteration

0 1 2 3 4 5 6 6

vT

Error

1, 0 0.7592566, 0.65079137 0.71240324, 0.70177035 0.70763704, 0.70657612 0.70715981, 0.70705374 0.70711208, 0.70710148 0.70710731, 0.70710625 0.70710683, 0.70710673

− 0.693892 0.0692394 0.00676845 0.0006751855 6.7502 × 10−5 6.75 × 10−6 6.75 × 10−7

For the initial guess of the eigenvector, we assume " # 1 (0) v = 0

(8.22)

Thus, (0)

Av

" 7 = 6

#" # " #

q
3 1 7

72 + 62 = 9.22 = ⇒ Av(0) = 4 0 6

(8.23)

Therefore, (1)

v

" # Av(0) 0.76

=

(0) = 0.65

Av

We can calculate the error as follows:



error = v(1) − v(0) = 0.694

(8.24)

(8.25)

The results of the iteration are given in Table 8.3. The computer program of the above problem is added in the course website. Clearly, the results match the analytical solution, shown in equation 8.21.

8.5

CONVERGENCE

For a diagonalizable n × n matrix A, the eigenvectors v1 , v2 , · · · , vn are linearly independent. Thus, any n-dimensional vector x may be represented as n
x = ∑ a j v j where a j are scalars (8.26) j=1

Thus, we can write

n

Ax =

∑

j=1

a j Av j




(8.27)

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Design and Analysis of Thermal Systems

We know Av j = λ j v j

(8.28)

Using equations 8.27 and 8.28, we have n

Ax =

∑

a jλ jv j




(8.29)

j=1

Similarly, we can show n

A2 x =

  2 a λ v j j ∑ j

(8.30)

j=1

Continuing in the same manner, we write for a positive integer k ! k  n  n λ j av Ak x = ∑ a j λ jk v j = λ1k ∑ k j j j=1 j=1 λ1

(8.31)

We now assume that |λ1 | > |λi | where i = 2, 3, . . . , n

(8.32)

The above assumption suggests that lim

k→∞

λ jk λ1k

= 0 for j = 2, 3, . . . , n

(8.33)

Using equations 8.31 and 8.33, n

lim Ak x =

k→∞

Furthermore,

∑

  a j λ jk v j = λ1k a1 v1

(8.34)

j=1



lim Ak x = λ1k a1 kv1 k

k→∞

(8.35)

Using equations 8.34 and 8.35, we have the unit eigenvector vˆ 1 Ak x v

= 1 = vˆ 1 lim

k→∞

Ak x kv1 k

(8.36)

Aˆv1 = λ1 vˆ 1 ⇒ kAˆv1 k = λ1 kˆv1 k = λ1

(8.37)

We also know Power iteration thus converges for all diagonalizable matrices that contain a dominant eigenvalue λ1 . The rate of converge, howvever, depends on the ratios λi /λ1 for i > 1. High values of the ratios λi /λ1 will lead to faster convergence. While the convergence of power method does not depend on the initial guess, the above discussion indicates that if the initial guess of eigenvector exactly matches with an eigenvector vˆ i , then the power method converges to λi and vˆ i even if the eigenvalue λi is not the dominant one.

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Numerical Linear Algebra

8.6

INVERSE POWER ITERATIONS

Recall the classical eigenvalue problem for an n × n matrix A with eigenvalues λi and the corresponding eigenvectors vi for i = 1, 2, . . . , n We know Av = λ v (8.38) Consider a real number q such that q 6= λi for i = 1, 2, . . . , n

(8.39)

From equations 8.38 and 8.39, (A − qI) v = (λ − q) v ⇒ (A − qI)−1 v =

1 v λ −q

(8.40)

From equations 8.38 and 8.40, we find that the eigenvectors of the matrices A and A − qI are the same. We also see that for the same eigenvector vi , if the eigenvalue of the matrix A is λi , the eigenvalue of A − qI is µ = 1/ (λ − q). Now with careful selection of q, we may be able to find all the eigenvalues of λ . For instance, if we apply the power method to find the eigenvalue of the matrix (A − qI)−1 for q = 0, we obtain maximum µ and thus the minimum λ . Similarly, if we select q in such a way that |q − λi | < q − λ j where j 6= i and j = 1, 2, · · · , n, applying the power method on the matrix (A − qI)−1 , we may find the eigenvalue λi . Overall, the inverse power method provides all the eigenvalues of matrix A.

PROBLEMS 1. Manually calculate the dominant eigenvalue of the following matrix:   2 1 1   A = 1 2 1 1 1 2 2. Write a PYTHON program to find the dominant eigenvalue of the following matrix:   2 1 1   A = 1 2 1 1 1 2 3. Write a PYTHON program for the inverse power method and thus find the minimum eigenvalue of the following matrix: " # 7 3 A= 6 4

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Design and Analysis of Thermal Systems

4. Write a PYTHON program for the inverse power method and thus find the all the eigenvalues of the following matrix:   2 1 1   A = 1 2 1 1 1 2 5. Write a PYTHON program for the inverse power method and thus find the all the eigenvalues of the following matrix:   1 1 1   A = 1 1 0 1 0 1

8.7

CURVE FITTING

Consider an experiment where we are varying a certain variable x and measuring another variable y. At the end of the experiment, we have generated a discrete data set (xi , yi ) for i = 1, 2, . . . , n where n is a positive integer. No matter how carefully we conduct the experiments, all experimental data contain some noise. Since the data are noisy, fitting all the data exactly through a smooth curve may not provide a physically meaningful trend. In curve fitting, we do not intend to fit all the data exactly, rather we try to fit the noisy data, as closely as possible, through a curve Y = f (x, a0 , a1 , . . . , am ) that is physically meaningful. Here, ai , i = 1, 2, . . . , m are adjustable parameters. Adjusting the values of ai provides the best fit. While there are many options to define and obtain the best fit, the most common technique is least square fitting, where the best fit is obtained by minimizing the following objective function: n

n

i=1

i=1

q (a0 , a1 , . . . , am ) = ∑ (ri )2 = ∑ (yi −Yi )2

(8.41)

where, ri are the residuals, quantify the differences between fit and the measured data, defined as ri = yi −Yi for i = 1, 2, . . . , n (8.42) To minimize the objective function, we enforce ∂q = 0 for i = 1, 2, . . . , m ∂ ai

(8.43)

The above m + 1 equations, in principle, may be solved to obtain m + 1 unknowns a0 , a1 , . . . , am . However, to deduce a mathematical expression of ∂∂aqi , we must first specify a physically meaningful form of Y = f (x). Finally, equation 8.43 may be linear or nonlinear depending on the choice of the function Y = f (x). In this chapter, we will discuss linear and polynomial curve fittings, both of which are in the domain of linear least square problems.

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Numerical Linear Algebra

8.8

FITTING OF A STRAIGHT LINE

Let us assume a discrete data set (xi , yi ) where xi are the independent variables. We wish to fit the above data in a straight line Y = a0 + a1 x

(8.44)

The residuals ri quantify the differences between fit and the measured data ri = yi − a0 − a1 xi for i = 1, 2, . . . , n

(8.45)

Thus, the objective function q is defined as n

n

i=1

i=1

q (a0 , a1 ) = ∑ (ri )2 = ∑ (yi − a0 − a1 xi )2

(8.46)

Therefore, n ∂q = −2 ∑ (yi − a0 − a1 xi ) = 0 ∂ a0 i=1

(8.47)

n   ∂q = −2 ∑ xi (yi − a0 − a1 xi ) = 0 ∂ a1 i=1

(8.48)

Using these equations, we have n

∑ (yi − a0 − a1 xi ) = 0

(8.49)

  xi (yi − a0 − a1 xi ) = 0

(8.50)

i=1 n

∑

i=1

Rearranging   na0 + ∑ xi a1 = ∑ yi     ∑ xi a0 + ∑ xi2 a1 = ∑ (xi yi )

(8.51) (8.52)

Solving the above two equations, we obtain a0 = a1 =

∑ xi2 ∑ yi − ∑ xi ∑ (xi yi ) n ∑ Xi2 − (∑ xi )2 n ∑ (xi yi ) − ∑ xi ∑ yi n ∑ Xi2 − (∑ xi )2

(8.53) (8.54)

260

8.9

Design and Analysis of Thermal Systems

FITTING OF A POLYNOMIAL

Let us once again use the same discrete data set (xi , yi ) for i = 1, 2, . . . , n where xi are the independent variables. We wish to fit the above data with a polynomial of degree m, where m < n. m

Y=

∑

  a jx j

(8.55)

j=0

The residuals ri quantify the differences between fit and the measured data  m  ri = yi − ∑ a j xij for i = 1, 2, . . . , n

(8.56)

j=0

Thus, the objective function q is defined as n

q (a0 , a1 , . . . , am ) = ∑ (ri )2

(8.57)

i=1

Therefore,  n  ∂q ∂ ri = ∑ 2ri =0 ∂ a0 i=1 ∂ a0  n  ∂q ∂ ri = ∑ 2ri =0 ∂ a1 i=1 ∂ a1

(8.58)

··················  n  ∂q ∂ ri =0 = ∑ 2ri ∂ am i=1 ∂ am Simplifying this equation, we obtain the following set of m + 1 linear equations: !  n  m n j+k k = x y a x (8.59) i j ∑ i for k = 0, 1, 2, . . . , m ∑ ∑ i j=0

i=1

i=1

We may expand the above equations to have the following set of equations:  n n  n n n (8.60) a0 ∑ xi0 + a1 ∑ xi1 + a2 ∑ xi2 + · · · + am ∑ xim = ∑ xi0 yi i=1

i=1

i=1

n

n

n

i=1 n 

i=1

n

a0 ∑ xi1 + a1 ∑ xi2 + a2 ∑ xi3 + · · · + am ∑ xim+1 = ∑ xi1 yi i=1

i=1

i=1

i=1



i=1

.. . n

n

n

n

n

a0 ∑ xim + a1 ∑ xim+1 + a2 ∑ xim+2 + . . . + am ∑ xi2m = ∑ xim yi i=1

i=1

i=1

i=1

i=1

Solving the above m + 1 equations, we evaluate a0 , a1 , · · · , am .




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Numerical Linear Algebra

Example: Fit a straight line and a second-degree polynomial through the data points shown in Table 8.4.

Table 8.4 Data Points in the Example in Section 8.4 xi yi

0 5

1 4

2 4

3 7

4 12

5 8

6 13

7 16

8 18

SOLUTION To fit a straight line, we use equation 8.53 to find the parameters a0 and a1 . For the second-order polynomial fitting, we use equation 8.59 to obtain a0 , a1 and a2 . The computer program of the above problem is given in the course website. The result is shown in Figure 8.2.

Figure 8.2 Fitting of arbitrary data using a straight line and a second degree polynomial.

8.10

ERROR ESTIMATION

Assuming that all the fitting parameters have been computed with high accuracy, there are multiple ways to look at the errors in curve fitting. First we would like to

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quantify goodness of fit that can be defined simply as the objective function q itself. A small value of q indicates the fitted curve closely follows the discrete points that are used for the curve fitting. In reality, the discrete points, used in the curve fitting, are coming from experiments and thus the discrete data set (xi , yi ) for i = 1, 2, . . . , n contains noise. A small q, therefore, may lead to creeping of noise in our fitting. So, while a small value of q indicates the closeness of the fitted curve with the discrete points, a small value of q does not necessarily indicate physically meaningful results. Another important discussion point in curve fitting is the extent of dependence of the variable y on the independent variable x. Most promising way to quantify such dependence is to evaluate the correlation coefficient, as defined below
covariance (x, y) ∑ni=1 dx,i dy,i p r =  r=   variance (x) × variance (y) 2 2 ∑ni=1 dx,i ∑ni=1 dy,i

(8.61)

where,

∑n yi ∑ni=1 xi ; dy,i = yi − i=1 (8.62) n n The value of the correlation coefficient is limited to −1 ≤ r ≤ 1. Small values of |r| indicate little dependence of y on x. On the other hand, the higher value of |r|, closes to 1, indicates significant dependence of x on y. For Example 8.9, the value of r is found to be 0.927, indicating a good choice of variables for the curve fitting. dx,i = xi −

PROBLEMS 1. Write a PYTHON program to fit a straight line and a second-degree polynomial through the data points shown in Table 8.4. xi yi

1 1.3

2 3.5

3 4.2

4 5

5 7

6 8.8

7 10.1

8 12.5

9 13

10 15.6

2. For the data set shown in the previous problem, fit the following curve: Y = a0 exp (a1 x) where a0 and a1 are the fitting parameter, to be evaluated. 3. Comparing the above three curve fittings, from the previous two problems, compare their goodness of fit.

8.11

SOLUTION OF ALGEBRAIC EQUATIONS

In this chapter, we focus on the numerical solution of a set of linear equation. The general form of such equation is written as Ax = b

(8.63)

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Here, in equation 8.63, A is an n × n square matrix while x and b are column vectors of size n. To begin with, we assume that the elements of the matrix A and the vector b are real constants. Such assumptions create a system of linear equations where the components of the vector x are unknowns. We will first see direct and iterative methods for the solution of such systems of linear equations. Later in this chapter, we will extend our understanding to the system of nonlinear equations. For an Ax = b system, we may have unique solution, no solution and many solutions. For brevity, in this chapter, we will only discuss the cases of unique solution.

8.12

GAUSSIAN ELIMINATION

Gaussian elimination intends to solve the system of linear equations. Let us first see the idea behind Gaussian elimination by taking a system with two equations and two unknowns. 3x + 2y = 5 4x + 5y = 9

(8.64) (8.65)

The goal now is to eliminate the coefficients below the diagonal elements, a step known as forward elimination. Therefore, in this case, the first step is no change in first row 4 row2 (new)=row2 (old)-row1× 3

3x + 2y = 5 7 7 y= 3 3

(8.66) (8.67)

Now the second step; from equation 8.67, y=1

(8.68)

and now the third and final step, known as back substitution; using equations 8.66 and 8.68 5 − 2y =1 (8.69) x= 3 Let’s extend our understanding to a 3 × 3 system x1 + x2 − x3 = 1 3x1 + x2 + x3 = 9 x1 − x2 + 4x3 = 8

(8.70)

Now after the first step of forward elimination, we have x1 + x2 − x3 = 1 −2x2 + 4x3 = 6 −2x2 + 5x3 = 7

(8.71)

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Elimination one more time and we have x1 + x2 − x3 = 1 −2x2 + 4x3 = 6 x3 = 1

(8.72)

Now the back substitution provides x1 = 3 x2 = −1 x3 = 1

(8.73)

We argue that the systems 8.70–8.73are equivalent to each other. Two linear systems are equivalent to each other if one can be deduced from the other through elementary operations. Elementary operations involve: multiplication of an equation by a constant, exchange of two equations of the same system and replacing an equation with a linear combination of the equation with other equations of the system. We will revisit the simple linear systems once again. But for now, let’s see how the Gaussian elimination works for the most general system of equations a11 x1 + a12 x2 + · · · + a1n xn = b1 a21 x1 + a22 x2 + · · · + a2n xn = b2 .. . an1 x1 + an2 x2 + · · · + ann xn = bn

(8.74)

These equations may be written in a simpler way, known as an index notation or Einstein notation: n

ai j x j = bi

where ai j x j =

∑

ai j x j




for i = 1, 2, . . . , n

(8.75)

j=1

8.12.1

FORWARD ELIMINATION

In the forward elimination step, our goal is to do something to make ak+1,k = 0, where k = 1, 2, . . . , n. To do so, the first step is to set all elements of column 1, except the a11 , to zero, as follows: a21 a11 a31 row3(new) = row3(old) − row1 × a11 .. . an1 rown(new) = rown(old) − row1 × a11 row2(new) = row2(old) − row1 ×

(8.76)

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Repeating the same procedure, we can make all the elements below diagonal to be zero. The above step will change the A matrix and b vector as follows: a11 x1 + a12 x2 + · · · + a1n xn = b1 a022 x2 + · · · + a02n xn = b02 .. . a0nn xn = b0n 8.12.2

(8.77)

BACK SUBSTITUTION

Back substitution starts from the last row: xn =

b0n a0nn

(8.78)

The (n − 1)th row now provides the value of xn−1 : xn−1 =

b0n−1 − a0n−1,n xn a0n−1,n−1

In general, xi for i = n − 1, n − 2, . . . , 1 may be obtained from   b0i − ∑nj=i+1 a0i j x j xi = a0ii

(8.79)

(8.80)

The solution procedure, described above, may be summarized in the algorithm below. The procedure is called naive since it is a very basic form of Gaussian elimination and suffers from several shortcomings. 8.12.3

HOW TO IMPROVE THE SOLUTION

The errors associated with Gaussian elimination stem from what we call round-off error. Any realistic calculation is finite precision in nature. Finite-precision calculations, while dealing with real numbers, can take only few digits after the decimal leading to round-off errors in the calculation. Round-off error is common in all numerical calculations. Solution algorithm must be carefully crafted to control such errors. In Gaussian elimination, we can reduce round-off errors using the following methods. Techniques for improving Gaussian elimination 1. We can simply increase the number of significant digits used in the calculation. As the number of significant digits increases, the solution calls for more computation time and storage space. Resource limitations, therefore, prohibits the increase in the number of significant digits beyond some optimum value.

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Algorithm 8.4: Naive Gaussian Elimination Data: ai j ∈ R where aii 6= 0 and bi ∈ R for i = 1, 2, . . . , n and j = 1, 2, . . . , n Result: xi for i = 1, 2, . . . , n such that ∑nj=1 ai j x j = bi /* Forward elimination starts

*/

for k = 1, n − 1 do for i = k + 1, n do f actor = aik /akk for j = k + 1, n do ai j = ai j − f actor ∗ ak j end bi = bi − f actor ∗ bk end end /* Forward elimination ends, back substitution starts

*/

xn = bn /ann for i = n − 1, 1, −1 do sum = bi for j = i + 1, n do sum = sum − ai j ∗ x j end xi = sum/aii end /* Back substitution ends, program ends

*/

2. Partial Pivoting: The goal of partial pivoting is to rearrange the equations in a way such that |aii | ≥ ai j where i, j = 1, 2, . . . n, and j > i. 3. Scaling: Partial pivoting requires a comparison of |aii | with ai j for j > i. Such a comparison may break down if we multiply an equation with a large number. Therefore, before we compare |aii | with ai j , we make max ai j = 1 before partial pivoting. Such an operation is called scaling.

8.13

JACOBI AND GAUSS–SEIDEL ITERATIONS

We wish to solve for x in the following system of linear equations, where A is an n × n matrix with elements ai j and b is an n-dimensional column vector of elements bi n
Ax = b ⇒ ∑ ai j x j = bi for i = 1, 2, . . . , n (8.81) j=1

The Jacobi iteration uses the following algorithm to solve the above problem: h i (k−1) bi − ∑nj=1, j6=i ai j x j (k) xi = (8.82) aii

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where, k is the iteration number and k = 0 indicates initial guess of the unknown vector x. In matrix notation, the above algorithm may be written as x(k) = Mx(k−1) + C

(8.83) −1

−1

where A = D + L + U and M = −D (L + U) , C = D b D : diagonal matrix L : strictly (zero-diagonal) lower-triangular matrix, U : strictly (zero-diagonal) upper-triangular matrix The exact solution of equation 8.81 is given by x = x(∗) , where x(∗) = Mx(∗) + C

(8.84)

The error vector after the kth iteration is given by   e(k) = x(k) −x(∗) = M x(k−1) − x(∗) = Me(k−1) = M2 e(k−2) = · · · = Mk e(0) (8.85) The problems of the Jacobi iteration is to understand where to stop iteration and how to confirm that the solution converges. To inquire these questions, we need to revisit some basic concepts of vector and matrix norms. 8.13.1

VECTOR AND MATRIX NORMS

Definition 8.13.1 An inner product space is a vector space where the inner product between two vectors u and v, expressed as u · v, is defined as follows: u·v = v·u u · u = 0 for u = 0 > 0 otherwise (au + bv) · w = a (u · w) + b (v · w)

where a, b are scalars

Definition 8.13.2 Inner product space also includes the norm of a vector u, expressed as kuk as follows: kauk = akuk kuk = 0 for u = 0 > 0 otherwise ku + vk ≤ kuk +kvk For an n-dimensional vector u with components u1 , u2 , . . . , un , the l p norm, for p being a positive integer, is defined as " kuk p =

n

∑ |ui |

i=1

#1

p

p

where p = 1, 2, . . . , ∞

(8.86)

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Design and Analysis of Thermal Systems

The commonly used norms are as follows: kuk1 = |u1 | +|u1 | + · · · +|un | q kuk2 = u21 + u21 + · · · + u2n kuk∞ = max |ui |

(8.87)

1≤i≤n

Definition 8.13.3 The l p norm of a matrix A, denoted by kAk p , is defined as kAk p = max kAxk p = max kxk p =1

kxk p 6=0

kAxk p kxk p

(8.88)

The above definition of matrix norm suggests kAxk p ≤ kAk p kxk p for all kxk p 6= 0

(8.89)

Thus, for all kxk p 6= 0, for two matrices A and B, we have kABxk p ≤ kAk p kBxk p ≤ kAk p kBk p kxk p

(8.90)

kABxk p ≤ kAk p kBk p

(8.91)

For all kxk p = 1, we Using equations 8.88 and 8.91, we now find kABk p ≤ kAk p kBk p

(8.92)



2 2

A ≤ kAk p

(8.93)

Setting A = B, we have p

In general, using equation 8.93 successively, we can show that for k being a positive integer



k k (8.94)

A ≤ kAk p p

Further, we can prove (left as an exercise) that   n n kAk∞ = max  max ∑ ui j x j  = max ∑ ui j 1≤i≤n |x j |≤1 j=1 1≤i≤n j=1 8.13.2

(8.95)

CONVERGENCE OF THE JACOBI ITERATION

The Jacobi method generates a sequence of vectors x(k) . The sequence converges to the exact solution x(∗) if for any positive real number ε, we find an integer N (ε), such that

(k)

(8.96)

x − x(∗) < ε for k > N p

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Numerical Linear Algebra

Since, in general, we don’t know the exact solution, we take resort to equation 8.84 or 8.81 to use any one of the following stopping criteria:



(k)

x − x(k−1)

(k)

p < ε or − b

< ε, where ε is a positive real number

Ax

x(k−1) p (8.97) Now we focus on the condition that will enable the Jacobi iteration to provide a convergent solution. Looking back at equation 8.85, we suggest that the condition for convergence is given by



(8.98) lim e(k) = lim Mk e(0) = 0 k→∞

k→∞

∞

∞

Using equations 8.89 and 8.94, we can write the sufficient condition for convergence as lim kMkk∞ = 0 ⇒ kMk∞ < 1 (8.99) k→∞

From equations 8.83, and 8.95, we can, therefore, write the convergence condition as n ai j (8.100) ∑ aii < 1 for all i j=1 j6=i

The Jacobi iteration, therefore, converges if the matrix A has the property of diagonal dominance, as shown below: n

|aii | >





∑ ai j for all i

(8.101)

j=1 j6=i

Please note that the above condition is sufficient, but may not be necessary. 8.13.3

GAUSS–SEIDEL ITERATION

The Gauss–Seidel iteration is almost similar to the Jaobi method with minor modifications to improve the rate of convergence. Here again, we wish to solve for x in the following system of linear equations Ax = b, where A is an n × n matrix with elements ai j while b is an n-dimensional column vector of elements bi n

Ax = b ⇒

∑


ai j x j = bi for i = 1, 2, . . . , n

(8.102)

j=1

The Jacobi iteration uses the following algorithm to solve the above problem: h i h i (k) (k−1) n bi − ∑i−1 a x − a x ∑ ij j ij j j=i+1 j=1 (k) (8.103) xi = aii

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Design and Analysis of Thermal Systems

where, k is the iteration number and k = 0 indicates the initial guess of the unknown vector x. The distinction of the Gauss–Seidel from the Jacobi iteration is now quite clear. While the Jacobi iteration uses previous iteration values of the vector x to find the current x, in the Gauss–Seidel iteration, we always use the latest values of x. The advantages of the Gauss–Seidel algorithm over the Jacobi method are that Gauss– Seidel requires fewer iterations and fewer storage space than the latter. In matrix notation, the Gauss–Seidel algorithm may be written as x(k) = Mx(k−1) + C

(8.104) (8.105)

where A = D + L + U and M = − (D + L)−1 U, C = (D + L)−1 b Here, D is diagonal matrix, L is strictly (zero-diagonal) lower-triangular matrix and U is strictly (zero-diagonal) upper-triangular matrix. The exact solution of equation 8.102 is given by x = x(∗) where x(∗) = Mx(∗) + C

(8.106)

The error vector after the kth iteration is given by   e(k) = x(k) − x(∗) = M x(k−1) − x(∗) = Me(k−1) = M2 e(k−2) = · · · = Mk e(0) (8.107) The Gauss–Seidel algorithm may be written as follows. Algorithm 8.5: Gauss–Seidel Method Data: ai j ∈ R where aii 6= 0 and xi , bi , maximumError ∈ R, error > maximumError for i, j = 1, 2 . . . n Result: xi for i = 1, 2 . . . n such that ∑nj=1 ai j x j = bi while error > maximumError do error = 0 for i = 1, n do res = bi for j = 1, n do res = res − ai j /x j end error = error + res ∗ res xi = xi + res/aii end end

8.14

EXTENSION TO NONLINEAR SYSTEMS

So far, we have learned to solve single nonlinear equations using the Newton– Raphson method and a system of linear equations using Gaussian elimination. If

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Numerical Linear Algebra

we combine these two techniques, we can solve the system of nonlinear equations. Consider the following system of nonlinear equations: f1 (x1 , x2 , . . . , xn ) = 0 f2 (x1 , x2 , . . . , xn ) = 0 .. . fn (x1 , x2 , . . . , xn ) = 0

(8.108)

(0)

Consider the initial guess of the solution is given by xi , where i = 1, 2, . . . , n and the superscript indicates the iteration counter. Thus, after the kth iteration, the (k) solution is xi . We now assume (k+1)

xi

(k)

= xi + ∆xi

f1 (x1 + ∆x1 , x2 + ∆x2 , · · · , xn + ∆xn ) = f1 (x1 , x2 , . . . , xn ) + f1,1 ∆x1 + f1,2 ∆x2 + · · · + f1,n ∆xn + · · ·

(8.109)

(8.110)

where, f1,1 =

∂ f1 ∂ f1 ; f1,2 = etc. ∂ x1 ∂ x2

(8.111)

Assuming f1 (x1 + ∆x1 , x2 + ∆x2 , · · · , xn + ∆xn ) = 0, we have f1 (x1 , x2 , . . . , xn ) + f1,1 ∆x1 + f1,2 ∆x2 + · · · + f1,n ∆xn = 0

(8.112)

Leading to f1,1 ∆x1 + f1,2 ∆x2 + · · · + f1,n ∆xn = − f1

(8.113)

In general, fi,1 ∆x1 + fi,2 ∆x2 + · · · + fi,n ∆xn = − fi where i = 1, 2, . . . , n

(8.114)

The system of equations is thus given by f1,1 ∆x1 + f1,2 ∆x2 + · · · + f1,n ∆xn = f1 f2,1 ∆x1 + f2,2 ∆x2 + · · · + f2,n ∆xn = f2 .. . fn,1 ∆x1 + fn,2 ∆x2 + · · · + fn,n ∆xn = fn

(8.115)

In this system of equations, we evaluate fi, j and fi using the previous iteration (k) values xi . The above system of equation is, therefore, a system of linear equations (k+1) that can be solved to evaluate ∆xi . Once we have the ∆xi , we can evaluate the xi using equation 8.109. The procedure proceeds till convergence.

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PROBLEMS 1. Write a PYTHON program to solve the following system of equations: 3x + 2y + z = 6 4x + 6y + 5z = 15 7x + 8y + 9z = 24 The computer program is provided with the companion, please check your program and the solution. 2. Write a PYTHON program to solve the above system of equations using the Gauss–Seidel iteration.

Differential 9 Ordinary Equations 9.1

INTRODUCTION

In general, an n-th order ordinary differential equation (ODE) is expressed as ! dy d 2 y dny F x, y (x) , , 2 , · · · , n = 0 (9.1) dx dx dx Let us begin with the following first-order ODE: y0 =


dy = f x, y (x) ; y (x = x0 ) = y0 dx

(9.2)

This ODE solves for the unknown y (x) for known values of x > x0 , where the value of (x = x0 ) = y0 is known as the initial condition (IC). The above ODE is thus called an initial value problem (IVP). In many physical problems, x represents time. For simplicity, we assume both x and y to be real-valued scalar variables. For brevity, in this chapter, we only discuss the ODEs where the existance and uniqueness of solutions are guaranteed.

9.2

EULER METHOD

Numerical differentiation is the primary tool for numerical solution of an ODE. Let us now try to solve the ODE shown in equation 9.2. The most primitive method for solving this equation is known as the Euler method. In the Euler method, we first discretize x, preferably with uniform grid size (also called stepsize) of ∆x = h. The Euler method is then applied, with the first-order forward difference approximation of y0 , as yi+1 − yi + O (h) = f (xi , yi ) ⇒ yi+1 = yi + h f (xi , yi ) (9.3) h for i = 0, 2, . . . , n − 1 where y0 = y (x = x0 ) is known from the initial condition (IC). Equation 9.3, derived using the finite difference method, is the difference equation or the discretized form of equation 9.2. While in equation 9.2, unknown y appears more than once, in the discretized equation 9.3, only y0 uses yi+1 , all other terms involving y are discretized with yi . Such a discretization method is called explicit discretization. It is now quite clear that the solution of yi+1 for i = 0, 1, · · · , n − 1 requires sequential calculations as shown in the following algorithm: DOI: 10.1201/9781003049272-9

273

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Design and Analysis of Thermal Systems

Algorithm 9.1: Solution of ODE Using the Euler Method Data: f (x, y) , x0 , y0 , h, n Result: y1, y2, · · · , yn i=0 while i < n do yi+1 = yi + h ∗ f (xi , yi ) xi+1 = xi + h i = i+1 end While the Euler method is very simple to implement, the truncation error O (h), in the Euler method, may grow rapidly. We will learn a group of methods, known as the Runge–Kutta (RK) methods. RK methods fall under the umbrella of predictorcorrector method, where we first predict a reasonable value of yi+1 and then correct the value.

9.3

RUNGE–KUTTA METHOD

Consider the second-order RK method, also known as the Huen method. Now, we once again try to solve equation 9.2 i.e. y0 = f (x, y). Here, the initial prediction from the Euler method is corrected. The second-order RK method suggests the following difference equation:   h (9.4) yi+1 = yi + (k1 + k2 ) + O h2 2 where, k1 = f (xi , yi ) k2 = f (xi + h, yi + k1 h) Let us now see how we reach the above difference equation. Consider the following general form of difference equation: yi+1 = yi + h (ak1 + bk2 ) + O (hm ) (9.5) where, k1 = f (xi , yi ) k2 = f (xi + ph, yi + qk1 h) Our goal is to select the constants a, b, p, and q, such that m = 2. We can clearly see that for a = 1 and b = 0, the above algorithm reverts back to the first-order accurate Euler method. This is the predictor part of the algorithm. The corrector part, then adds the k2 , to improve the order of accuracy of the technique. yi+1 = yi + h (ak1 + bk2 ) predictor: k1 = f (xi , yi ) predicts from the Euler method corrector: k2 = f (xi + ph, yi + qk1 h)

(9.6)

Using the Taylor series expansion of the corrector part and neglecting the higherorder terms, k2 = f (xi + ph, yi + qk1 h) = k1 + ph

∂f ∂f + qk1 h ∂x ∂y

(9.7)

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Ordinary Differential Equations

Using equations 9.6 and 9.7, yi+1 = yi + (a + b) h f + bph2

∂f ∂f + bqk1 h2 ∂x ∂x

(9.8)

The Taylor series expansion of y provides yi+1 = yi + hy0i +

  h2 00 yi + O h3 2

(9.9)

We know

df ∂f ∂f ∂f ∂f = + y0 = +f dx ∂x ∂y ∂x ∂y Combining equations 9.9 and 9.10, we have y0 = f (x, y) ⇒ y00 =

yi+1 = yi + h f +

(9.10)

h2 ∂ f h2 f ∂ f + 2 ∂x 2 ∂y

(9.11)

Comparing equations 9.8 and 9.11, we have a + b = 1, bph2 =

h2 h2 f 1 , bqk1 h2 = ⇒ p = q = ,a = 1−b 2 2 2b

(9.12)

Setting an arbitrary value of b, we get the values of a, p, q. The family of method thus obtained is known as the second-order RK method. For instance b = 1/2, we have a second-order accurate RK method, popularly known as the Huen method, which is shown in equation 9.4. Extending the above procedure, we can device RK method of various orders. The generalized RK method is given by n

yi+1 = yi + h ∑ a j k j




(9.13)

j=1

k1 = f (xi , yi ) k j = f xi + p j−1 h, yi + q j−1,1 k1 h, . . . , yi + q j−1, j−1 k j−1 h for j = 2, 3, . . . , n




The most common of them is the fourth-order RK method. The difference equation of fourth-order RK method is described as follows: h yi+1 = yi + (k1 + 2k2 + 2k3 + k4 ) 6 k1 = f (xi , yi )   h k1 h k2 = f xi + , yi + 2 2   k2 h h k3 = f xi + , yi + 2 2 k4 = f (xi + h, yi + k3 h)

(9.14)

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Example: Solve the following IVP using the fourth-order RK method in the interval [0,2]. Use ten (10) uniform grids y0 = x + y; y (x = 0) = 0

(9.15)

SOLUTION The exact solution of the problem may be shown as y (x) = exp (x) − x − 1. Results are plotted in Figure 9.1. The PYTHON program is given in the course website.

Figure 9.1 Numerical versus exact solutions of y0 = x + y.

9.4

HIGHER-ORDER IVP

So far we have discussed the solution techniques for the IVP y0 = f (x, y) assuming y to be a real scalar. The solution procedures, described so far, are also applicable if y is a vector. As such, a higher-order IVP may be soled as a first-order IVP, if we allow y to be a vector. Consider the following second-order IVP: u00 + au0 + bu = f (x, u) , u (x = 0) = p, u0 (x = 0) = q

(9.16)

This IVP may be written as u0 = v with the initial condition u (x = 0) = p v0 = f − av − bu with the initial condition v (x = 0) = q

(9.17)

In vector form, " # " # " # d u u p 0 y = F where y = ⇒y = and y (x = 0) = v q dx v 0

(9.18)

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Ordinary Differential Equations

and

" # v F= where g = g (x, u, v) = f − av − bu g

(9.19)

The difference equation of the fourth-order RK method, for equation 9.16, is described as follows: h h ui+1 = ui + (k1 + 2k2 + 2k3 + k4 ) ; vi+1 = vi + (s1 + 2s2 + 2s3 + s4 ) 6 6 k1 = vi ; s1 = g (xi , ui , vi )   s1 h h k1 h s1 h k2 = vi + ; s2 = g xi + , ui + , vi + 2 2 2 2   s2 h h k2 h s2 h k3 = vi + ; s3 = g xi + , ui + , vi + 2 2 2 2

(9.20)

k4 = f (vi + s3 h) ; s4 = g (xi + h, ui + k3 h, vi + s3 h) In general, any higher-order IVP may be converted into a set of first-order IVPs. The basic concept of the solution techniques, shown for the above second-order IVP, may be extended to any higher-order IVP.

9.5

BOUNDARY VALUE PROBLEMS: SHOOTING METHOD

So far, we have solved IVPs to find the unkown variable y (x) in the interval x0 < x ≤ xn , where the initial condition are given at x = x0 . In case of boundary value problems (BVPs), the boundary conditions are available at both ends of the independent variable i.e. at x = x0 and x = xn . The solution methodologies, applicable for the IVPs, cannot be readily used for the BVPs. One option is to pose the BVP as an IVP with a reasonable assumption for the missing initial condition. The assumed initial condition is then iteratively updated to match the given boundary condition. Such a solution technique is known as the shooting method. Let us consider the following example. Example: Solve the following ODE, famously known as the Blasius equation, using the shooting method. 1 y000 + yy00 = 0 2 y (x = 0) = 0 y0 (x = 0) = 0 y0 (x → ∞) = 1

(9.21)

SOLUTION This problem is clearly a BVP since all the conditions are not available at x = 0. One way to solve the above BVP is to convert the problem in to an IVP by assuming

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Design and Analysis of Thermal Systems

a reasonable value of y00 (x = 0). The goal here is to iteratively update the value of y00 (x = 0), so that the boundary condition (BC) y0 (x → ∞) = 1 is satisfied. The first step is to pose the above problem as a set of first-order IVPs as follows: y0 = y1 , y (x = 0) = 0 0 y1 = y2 , y1 (x = 0) = 0 1 y02 = − yy2 , y2 (x = 0) = y21 2

(9.22) (9.23) (9.24)

Here, y21 is the value assumed for y2 (x = 0) that has to be iteratively updated to match the boundary condition y0 (x → ∞) = 1. The procedure for updating the value of y2 (x = 0) is as follows: 1. We assume y2 (x = 0) = y21 and solve the above system of IVPs to have y0 (x → ∞) = y21n . In general, y21n 6= 1. 2. We assume another value of the initial condition y2 (x = 0) = y22 and solve the above system of IVPs to have y0 (x → ∞) = y22n . In general, y22n 6= 1. 3. The values of the initial condition y21 and y21 are chosen such that (1 − y21n ) (1 − y21n ) < 0. This condition ensures that if y21n < 1, then y22n > 1 or vice versa. 4. We now solve the above system of IVPs using y2 (x = 0) = y23 = (y21 + y22 ) /2 to have y0 (x → ∞) = y23n . If (1 − y21n ) (1 − y23n ) > 0, we set y21 = y23 . Conversely, if (1 − y22n ) (1 − y23n ) > 0, we set y22 = y23 . 5. We follow the above bisection-type procedure until y0 (x → ∞) ≈ 1. The plots of the numerical solution of the Blasius equation are shown in Figure 9.2. The PYTHON program of the solution is given in course website.

9.6

BOUNDARY VALUE PROBLEMS: FINITE DIFFERENCE METHOD

As an alternative to the shooting method, we can discretize the BVP using finite differences to generate a system of linear equations. The system of equation may then be solved using a standard linear equations solver, described before. Let us solve the following equation using the finite difference method. y00 + ay0 + by = f (x) , y (x = 0) = p, y0 (x = 1) = q

(9.25)

Discretization of this equation leads to yi−1 − 2yi + yi+1 yi+1 − yi−1 +a + byi = fi 2 h     2h  1 a 2 1 a − yi−1 + − 2 yi + 2 + yi+1 = fi h2 2h h h 2h

(9.26)

For i = 2, 3, . . . , n−1, we have now n−2 equations. The boundary conditions provide two more equations. Thus, we have an Ax = b system containing n simultaneous

279

Ordinary Differential Equations

Figure 9.2 Numerical solution of the BVP using the shooting method. linear equations with n unknowns. The system may then be solved using standard linear solvers, developed before. Depending on the value of n, matrix A, could be of large size. Matrix A, however, contains mostly zeroes and thus is called a sparse matrix. The matrix inversion is conducted in a way that the sparsity of matrix A is preserved.

PROBLEMS 1. Write a computer program to solve the following ODE using the fourth-order RK method: y0 = xy2 , y (x = 0) = 1 (9.27) 2. Write a computer program to solve the following ODE using the fourth-order RK and shooting method: y00 = xy2 ,

y (x = 0) = 1,

y0 (x = 0) = 1

(9.28)

3. Write a computer program to solve the following ODE using the fourth-order RK method: y0 = x − y, y (x = 0) = 1 (9.29)

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Design and Analysis of Thermal Systems

4. Write a computer program to solve the following ODE using the finite difference method: y00 = x − y, y (x = 0) = 1, y0 (x = 0) = 1 (9.30) 5. Write a computer program to solve the following ODE using the fourth-order RK method: y0 + y tan x = sin (2x) , y (x = 0) = 1 (9.31)

Differentiation 10 Numerical and Integration 10.1

INTRODUCTION

The idea of continuity, as we learned in calculus, does not readily apply in numerical analysis. Numerical methods deal with discrete numbers and thus a function y = f (x) provides certain discrete values of y for certain discrete inputs of x. In other sense, the independent variable x can attain only some discrete values. Once we get familiar with this idea, the numerical differentiation, of any order, becomes very easy.

10.2

NUMERICAL DIFFERENTIATION

In this section, we study the procedure for numerical differentiation of y = f (x) in some interval [a, b] using the finite difference method. When we analytically evaluate f 0 , we get f 0 as a function of x. For any given x within the interval [a, b], we can then evaluate f 0 . When we evaluate f 0 numerically, we can calculate f 0 only at few discrete points. The procedure of selecting these discrete x is called discretization. To begin with, we assume x is dicretized in n points xi , i = 1, 2, . . . , n, where xi+1 − xi = ∆x for i = 1, 2, . . . , n − 1. The discrete divisions in x form a grid structure, and ∆x is called grid size. When ∆x is independent of x, we call the grid to be uniform, while in case of a nonuniform grid, ∆x varies with x. To find the derivative in the discretized domain, we start by expanding f (x + ∆x) and f (x − ∆x) in the Taylor series as follows: (∆x)2 00 f (x) + · · · = fi + ∆x fi0 + . . . 2! (∆x)2 00 fi−1 = f (x − ∆x) = f (x) − ∆x f 0 (x) + f (x) − · · · = fi − ∆x fi0 + . . . (10.1) 2! fi+1 = f (x + ∆x) = f (x) + ∆x f 0 (x) +

From equation 10.1, we can evaluate f 0 as follows: fi+1 − fi + O (∆x) ∆x fi − fi−1 Backward difference: fi0 = + O (∆x) ∆x   fi+1 − fi−1 Central difference: fi0 = + O ∆x2 2∆x Forward difference: fi0 =

(10.2)

The errors that appear in the evaluation of derivatives due to the truncation of the Taylor series are known as truncation errors. Equation 10.2 indicates that truncation errors in forward and backward differences are of the orders of ∆x, while the central DOI: 10.1201/9781003049272-10

281

282

Design and Analysis of Thermal Systems

difference leads to a truncation error of the order of ∆x2 . The forward and backward differences are thus called first-order accurate numerical schemes, while the central difference is second-order accurate. For a first-order accurate scheme, the truncation error reduces linearly as we reduce ∆x, while for second-order accurate scheme, the same reduces quadratically. Please note that we can specify only the orders of truncation error but not the absolute value of the truncation error. The following example illustrates how we can create higher-order numerical schemes. Example: Express fi0 in terms of fi , fi+1 , fi+2 . Solution: ∆x2 00 ∆x3 000 ∆x4 0000 f + f + f +... 2 i 6 i 24 i (2∆x)2 00 (2∆x)3 000 (2∆x)4 0000 fi+2 = fi + 2∆x fi0 + fi + fi + f +... 2 6 24 i Assuming fi0 = a fi + b fi+1 + c fi+2 , where a, b, c are constants, we have fi+1 = fi + ∆x fi0 +

(10.3)

a+b+c = 0 b∆x + 2c∆x = 1 ∆x2 ∆x2 + 4c =0 2 2 Solving the above set of equations, we have

(10.4)

b

a=−

3 2∆x

b=

2 ∆x

c=−

1 2∆x

(10.5)

Thus, the fi0 may be written as ∆x3 000 (2∆x)3 000 fi − c fi 6 6   + O ∆x2

fi0 = a fi + b fi+1 + c fi+2 − b =

10.3

−3 fi + 4 fi+1 − fi+2 2∆x

(10.6)

NONUNIFORM GRID

For a nonuniform grid, we assume x is dicretized in n points xi , i = 1, 2, . . . , n, where xi+1 − xi = ∆xi for i = 1, 2, . . . , n − 1, and in general, ∆xi 6= ∆x j for i 6= j. While the usual expressions of forward differences, as shown in equation 10.1, do not change, complication appears for higher order accurate schemes as shown in the example in Section 10.2. SOLUTION Example: Express fi0 , in a nonuniform grid, in terms of fi , fi+1 , fi+2 . ∆xi2 00 ∆xi3 000 f + f +... 2 i 6 i (∆xi + ∆xi+1 )2 00 (∆xi + ∆xi+1 )3 000 fi+2 = fi + (∆xi + ∆xi+1 ) fi0 + fi + fi + . . . (10.7) 2 6 fi+1 = fi + ∆xi fi0 +

283

Numerical Differentiation and Integration

Assuming fi0 = a fi + b fi+1 + c fi+2 , where a, b, c are constants, we have a+b+c = 0 b∆xi + c (∆xi + ∆xi+1 ) = 1 ∆xi2 (∆xi + ∆xi+1 )2 +c =0 2 2 Solving this set of equations, we have b

2∆xi + ∆xi+1 ∆xi (∆xi + ∆xi+1 ) ∆xi + ∆xi+1 b= ∆xi ∆xi+1 ∆xi c=− ∆xi+1 (∆xi + ∆xi+1 )

(10.8)

a=−

(10.9)

To obtain the order of accuracy, we evaluate −

 b∆xi3 c (∆xi + ∆xi+1 )3 1 2 ∆xi + ∆xi ∆xi+1 − = 6 6 6

(10.10)

Thus, fi0 may be written as 2∆xi + ∆xi+1 ∆xi + ∆xi+1 ∆xi fi0 = − fi+1 − fi + fi+2 ∆xi (∆xi + ∆xi+1 ) ∆xi ∆xi+1 ∆xi+1 (∆xi + ∆xi+1 )   + O ∆xi2 + ∆xi ∆xi+1 (10.11) This example indicates that three-point derivative in a nonuniform grid may not be second-order accurate. In fact, if ∆xi 0, ν is given by R 1 h 0(n) i2 dz J z=0

ν (n) = R 1   0(n−1) 2 dz z=0 J

(14.24)

To find the optimum value of β , we set 2 Z 1  h i ∂ J (n+1) (n) (n) = 0 for J = θ k −Y δ (z − zi ) dz (14.25) ∂ β (n) z=0 h i Here, θ k(n) indicates θ evaluated with the n-th iteration value of the conductivity k. Now, extending equation 14.25 J

(n+1)

=

Z 1  h

θ k

(n+1)

i

2 δ (z − zi ) dz

−Y

z=0

(14.26)

Combining equations 14.9 and 14.26, J

(n+1)

=

Z 1  h

θ k

(n)

−β

(n) (n)

P

i

2 −Y

z=0

δ (z − zi ) dz

(14.27)

Setting ∆k = P(n) and linearizing using a Taylor series expansion, J (n+1) =

Z 1 

θ − β (n) ∆θ −Y

z=0

2

δ (z − zi ) dz

(14.28)

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Design and Analysis of Thermal Systems

Differentiating both sides, ∂ J (n+1) = 0 = −2 ∂ β (n)

Z 1

  ∆θ θ − β (n) ∆θ −Y δ (z − zi ) dz

z=0

(14.29)

Simplifying, β

14.3

(n)

R1

=

z=0 ∆θ (θ −Y ) δ (z − zi ) dz R1 2 z=0 (∆θ ) δ (z − zi ) dz

(14.30)

REGULARIZATION AND STOPPING CRITERION

The solution of inverse problems requires minimization of the functional J as follows: Z 1

J (z) =

z=0

(θ −Y )2 δ (z − zi ) dz < ε

(14.31)

where, ε is a small, positive, real number, such that ε ≈ 0. J, as shown in equation 14.31, uses experimental data Y . Accuracy of experimental results, therefore, has significant influences on the estimated parameter k. Experimental data always involve random noise. As the inverse algorithm attempts to minimize J, along with the information contained in the data, the noise also creeps in the estimated parameter, and thus, the noise in experimental data may lead to erroneous estimation of parameter. To avoid such a situation, we impose a technique called regularization. Regularization of algorithm is usually achieved using one of the following two techniques: (a) modification of J, or (b) premature stopping of the minimization of J. Some of these techniques are discussed here. 14.3.1

DISCREPANCY PRINCIPLE

Discrepancy principle, initially suggested by Tikhonov [1963], assumes that at z = zi , the difference between the computed and the measured values of temperature is given by |θi −Yi | = wσ (14.32) Here, σ is the standard deviation, calculated based on repeated measurements, while the weight w is a real positive number indicating the confidence interval of measurements. The discrepancy principle may be implemented in two ways. First, the minimization of the functional J may be minimized, as shown in equation 14.31, such that ε ≈ wσ

(14.33)

Thus, iteration stops as soon as the functional J reaches the value wσ . Alternately, setting ε ≈ 0, the functional J may be modified as J (z) =

Z 1 z=0

(θ −Y )2 δ (z − zi ) dz + wσ

(14.34)

Inverse Problems

337

In both these cases, we assume that the standard deviation σ remains independent of location. 14.3.2

ADDITIONAL MEASUREMENT APPROACH

¨ Additional measurement approach, proposed by Ozisik and Orlande [2021], constitutes two functionals J1 and J2 while attempting to minimize J1 only. At some point, during the minimization process, J2 starts to increase, indicating the onset of instability. The iteration stops at this point. The success of additional measurement technique depends on the amount of experimental data. It is difficult to constitute two functionals, if the amount of data is inadequate. Furthermore, in this method, we assume that arbitrary grouping of measured data in J1 and J2 will lead to the same estimated value of the unknown parameter. Such an assumption may not be always justified. 14.3.3

SMOOTHING OF EXPERIMENTAL DATA

If we investigate experimental data in frequency domain, the high-frequency part usually contains noise. If we can remove part of the high-frequency component, using suitable filters, the remaining data may be considered clean enough that does not require any further regularization. This concept was applied successfully by AlKhalidy [1998] for inverse heat conduction problems.

14.4

COMPLETE ALGORITHM

The complete algorithm for inverse estimation is given here. This algorithm may be used for both parameter estimation and function estimation problems. 1. Based on the physical problem, assume a reasonable value of the unknown parameter k. 2. Solve the direct problem, shown in equations 14.6 and 14.7 to calculate θ . 3. Compute the functional J. 4. Check the stopping criterion J < ε. Stop iteration if the stopping criterion is satisfied; otherwise follow the steps below. 5. Solve the adjoint problem, shown in equations 14.18 and 14.19. 6. Compute J 0 using equation 14.21. 7. Compute the descent direction using equations 14.23 and 14.24. 8. Solve the sensitivity problem, using equations 14.12 and 14.13 while setting ∆k = P(n) . 9. Compute step size β using equation 14.30. 10. Update the unknown parameter k using equation 14.9. 11. Go to step 2 and continue till convergence.

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Design and Analysis of Thermal Systems

PROBLEMS Consider steady, 1-D heat conduction problems where we wish to use inverse technique to estimate the unknown thermal conductivity using measured temperatures at few discrete locations. In each of the following cases, formulate the inverse problem, and write the complete algorithm to estimate the unknown thermal conductivity: 1. Thermal conductivity is a second-degree polynomial function of temperature. Assume known temperatures at the boundaries. 2. Thermal conductivity is a unknown function of temperature. Assume known temperatures at the boundaries. 3. Thermal conductivity is a second-degree polynomial function of temperature. Assume known temperature at one boundary and known heat flux at the other. 4. Thermal conductivity is a unknown function of temperature. Assume known temperature at one boundary and known heat flux at the other.

REFERENCES N. Al-Khalidy. A general space marching algorithm for the solution of twodimensional boundary inverse heat conduction problems. Numerical Heat Transfer, Part B, 34(3):339–360, 1998. O. M. Alifanov. Inverse Heat Transfer Problems. Springer Science & Business Media, 2012. J. Hadamard. La notion de diff´erentielle dans l’enseignement. Hebrew University, 1923. ¨ M. N. Ozisik and H. R. Orlande. Inverse Heat Transfer: Fundamentals and Applications. CRC Press, 2021. A. N. Tikhonov. On the solution of ill-posed problems and the method of regularization. In Doklady Akademii Nauk, Volume 151, pages 501–504. Russian Academy of Sciences, 1963.

Properties A Thermophysical (Working Fluids) The thermophysical properties presented below have been adapted from the following reference:

Steven G. Penocello. Thermal Energy Systems Design and Analysis. CRC Press, 2019.

DOI: 10.1201/9781003049272-A

339

340

Appendix A

460

Appendix B

Table A.1 B.1 APPENDIX Thermophysical Properties of Saturated Water (SI Units) Thermophysical Properties of Saturated Water (SI Units) T °C 0.01 10 20 30 40 50 60 70 80 90 99.974 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 373.946

P kPa

ρf kg/m3

vg m3/kg

uf kJ/kg

ug kJ/kg

hf kJ/kg

hg kJ/kg

sf sg kJ/kg-K kJ/kg-K

0.61165 1.2282 2.3393 4.2470 7.3849 12.352 19.946 31.201 47.414 70.182 101.325 101.42 143.38 198.67 270.28 361.54 476.16 618.23 792.19 1002.8 1255.2 1554.9 1907.7 2319.6 2797.1 3346.9 3976.2 4692.3 5503.0 6416.6 7441.8 8587.9 9865.1 11284. 12858. 14601. 16529. 18666. 21044. 22064.

999.79 999.65 998.16 995.61 992.18 988.00 983.16 977.73 971.77 965.30 958.37 958.35 950.95 943.11 934.83 926.13 917.01 907.45 897.45 887.00 876.08 864.66 852.72 840.22 827.12 813.37 798.89 783.63 767.46 750.28 731.91 712.14 690.67 667.09 640.77 610.67 574.71 527.59 451.43 322.00

205.99 106.30 57.757 32.878 19.515 12.027 7.6672 5.0395 3.4052 2.3591 1.6732 1.6718 1.2093 0.89121 0.66800 0.50845 0.39245 0.30678 0.24259 0.19384 0.15636 0.12721 0.10429 0.086092 0.071503 0.059705 0.050083 0.042173 0.035621 0.030153 0.025555 0.021660 0.018335 0.015471 0.012979 0.010781 0.0088024 0.0069493 0.0049544 0.0031056

0.0 42.020 83.912 125.73 167.53 209.33 251.16 293.03 334.96 376.97 418.95 419.06 461.26 503.60 546.09 588.77 631.66 674.79 718.20 761.92 806.00 850.47 895.39 940.82 986.81 1033.4 1080.8 1129.0 1178.1 1228.3 1279.9 1332.9 1387.9 1445.3 1505.8 1570.6 1642.1 1726.3 1844.1 2015.7

2374.9 2388.6 2402.3 2415.9 2429.4 2442.7 2455.9 2468.9 2481.6 2494.0 2506.0 2506.0 2517.7 2528.9 2539.5 2549.6 2559.1 2567.8 2575.7 2582.8 2589.0 2594.2 2598.3 2601.2 2602.9 2603.1 2601.8 2598.7 2593.7 2586.4 2576.5 2563.6 2547.1 2526.0 2499.2 2464.4 2418.1 2351.8 2230.3 2015.7

0.00061178 42.021 83.914 125.73 167.53 209.34 251.18 293.07 335.01 377.04 419.06 419.17 461.42 503.81 546.38 589.16 632.18 675.47 719.08 763.05 807.43 852.27 897.63 943.58 990.19 1037.6 1085.8 1135.0 1185.3 1236.9 1290.0 1345.0 1402.2 1462.2 1525.9 1594.5 1670.9 1761.7 1890.7 2084.3

2500.9 2519.2 2537.4 2555.5 2573.5 2591.3 2608.8 2626.1 2643.0 2659.5 2675.5 2675.6 2691.1 2705.9 2720.1 2733.4 2745.9 2757.4 2767.9 2777.2 2785.3 2792.0 2797.3 2800.9 2802.9 2803.0 2800.9 2796.6 2789.7 2779.9 2766.7 2749.6 2727.9 2700.6 2666.0 2621.8 2563.6 2481.5 2334.5 2084.3

0.0 0.15109 0.29648 0.43675 0.57240 0.70381 0.83129 0.95513 1.0756 1.1929 1.3069 1.3072 1.4188 1.5279 1.6346 1.7392 1.8418 1.9426 2.0417 2.1392 2.2355 2.3305 2.4245 2.5177 2.6101 2.7020 2.7935 2.8849 2.9765 3.0685 3.1612 3.2552 3.3510 3.4494 3.5518 3.6601 3.7784 3.9167 4.1112 4.4070

9.1555 8.8998 8.6660 8.4520 8.2555 8.0748 7.9081 7.7540 7.6111 7.4781 7.3544 7.3541 7.2381 7.1291 7.0264 6.9293 6.8371 6.7491 6.6650 6.5840 6.5059 6.4302 6.3563 6.2840 6.2128 6.1423 6.0721 6.0016 5.9304 5.8579 5.7834 5.7059 5.6244 5.5372 5.4422 5.3356 5.2110 5.0536 4.8012 4.4070

T °C 0.01 10 20 30 40 50 60 70 80 90 99.974 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 373.946

341

Appendix A 464

Appendix B

Table A.2 B.2 APPENDIX Thermophysical Properties Single-Phase Water (SI Units) Thermophysical Properties of SingleofPhase Water (SI Units) T °C

ρ kg/m3

v m3/kg

u kJ/kg

h kJ/kg

50 100 150 200 250 300 350 400 450 500

0.0067072 0.0058075 0.0051210 0.0045797 0.0041419 0.0037805 0.0034772 0.0032189 0.0029963 0.0028025

149.09 172.19 195.27 218.35 241.44 264.52 287.59 310.67 333.74 356.82

2445.3 2516.4 2588.4 2661.7 2736.3 2812.4 2890.1 2969.4 3050.4 3133.0

2594.4 2688.6 2783.7 2880.0 2977.7 3077.0 3177.7 3280.1 3384.1 3489.8

9.2430 9.5139 9.7531 9.9682 10.165 10.346 10.514 10.672 10.821 10.963

600 700 800 900 1000 1100 1200 1400 1600

0.0024816 0.0022266 0.0020191 0.0018470 0.0017019 0.0015780 0.0014708 0.0012950 0.0011567

402.97 449.12 495.27 541.42 587.58 633.71 679.90 772.20 864.53

3303.4 3480.8 3665.4 3856.9 4055.3 4260.0 4470.9 4909.1 5366.6

3706.3 3930.0 4160.7 4398.4 4642.8 4893.8 5150.8 5681.3 6231.1

11.226 11.468 11.694 11.906 12.106 12.295 12.476 12.813 13.124

50 100 150 200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 1400 1600

988.03 0.58967 0.51636 0.46031 0.41560 0.37895 0.34832 0.32230 0.29992 0.28046 0.24827 0.22272 0.20194 0.18472 0.17020 0.15780 0.14708 0.12950 0.11567

0.0010121 1.6959 1.9366 2.1724 2.4062 2.6389 2.8709 3.1027 3.3342 3.5656 4.0279 4.4899 4.9520 5.4136 5.8754 6.3371 6.7990 7.7220 8.6453

209.32 2506.2 2582.9 2658.2 2733.9 2810.6 2888.7 2968.3 3049.4 3132.2 3302.8 3480.4 3665.0 3856.6 4055.0 4259.8 4470.7 4908.9 5366.5

s kJ/kg-K

cp kJ/kg-K

μ mPa-s

k W/m-K

Pr

T °C

1.8761 1.8914 1.9139 1.9403 1.9692 1.9996 2.0312 2.0636 2.0969 2.1309

0.010537 0.012336 0.014252 0.016240 0.018270 0.020325 0.022390 0.024456 0.026515 0.028562

0.020232 0.024160 0.028482 0.033149 0.038120 0.043361 0.048845 0.054544 0.060439 0.066508

0.97716 0.96574 0.95766 0.95056 0.94381 0.93729 0.93107 0.92525 0.91991 0.91510

50 100 150 200 250 300 350 400 450 500

2.2006 2.2716 2.3424 2.4114 2.4776 2.5403 2.5989 2.7035 2.7922

0.032605 0.036564 0.040428 0.044194 0.047860 0.051428 0.054901 0.061575 0.067916

0.079104 0.092212 0.10573 0.11959 0.13371 0.14803 0.16252 0.19183 0.22138

0.90704 0.90074 0.89562 0.89114 0.88686 0.88250 0.87790 0.86779 0.85658

600 700 800 900 1000 1100 1200 1400 1600

0.54652 0.012234 0.014192 0.016204 0.018249 0.020313 0.022384 0.024453 0.026515 0.028564 0.032608 0.036568 0.040433 0.044198 0.047864 0.051432 0.054904 0.061578 0.067918

0.64062 0.024564 0.028843 0.033436 0.038340 0.043530 0.048975 0.054649 0.060527 0.066586 0.079173 0.092282 0.10581 0.11967 0.13380 0.14813 0.16262 0.19194 0.22149

3.5671 1.0342 0.97655 0.95735 0.94690 0.93908 0.93230 0.92617 0.92061 0.91562 0.90727 0.90076 0.89549 0.89091 0.88656 0.88216 0.87754 0.86742 0.85623

50 100 150 200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 1400 1600

P = l kPa (0.001 MPa)

P = 100 kPa (0.01 MPa) 209.42 2675.8 2776.6 2875.5 2974.5 3074.5 3175.8 3278.6 3382.8 3488.7 3705.6 3929.4 4160.2 4398.0 4642.6 4893.5 5150.6 5681.2 6231.0

0.70377 7.3610 7.6148 7.8356 8.0346 8.2172 8.3866 8.5452 8.6946 8.8361 9.0998 9.3424 9.5681 9.7800 9.9800 10.170 10.350 10.688 10.998

4.1813 2.0766 1.9846 1.9754 1.9893 2.0124 2.0399 2.0698 2.1015 2.1344 2.2029 2.2731 2.3434 2.4122 2.4782 2.5407 2.5993 2.7038 2.7923

342

Appendix A 465

Appendix B

Table A.3 B.2 (CONTINUED) APPENDIX Thermophysical Properties Single-Phase Water (SI Units) Thermophysical Properties of SingleofPhase Water (SI Units) T °C

ρ kg/m3

v m3/kg

u kJ/kg

h kJ/kg

50 100 150 200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 1400 1600

988.21 958.54 917.02 2.3528 2.1078 1.9135 1.7539 1.6199 1.5056 1.4066 1.2436 1.1149 1.0104 0.92399 0.85121 0.78910 0.73545 0.64745 0.57828

0.0010119 0.0010433 0.0010905 0.42503 0.47443 0.52260 0.57016 0.61732 0.66419 0.71093 0.80412 0.89694 0.98971 1.0823 1.1748 1.2673 1.3597 1.5445 1.7293

209.26 418.94 631.65 2643.3 2723.8 2803.2 2883.0 2963.7 3045.6 3129.0 3300.4 3478.5 3663.6 3855.4 4054.0 4259.0 4470.0 4908.4 5366.1

209.76 419.47 632.19 2855.8 2961.0 3064.6 3168.1 3272.3 3377.7 3484.5 3702.5 3927.0 4158.4 4396.6 4641.4 4892.6 5149.8 5680.6 6230.7

50 100 150 200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 1400 1600

988.43 958.77 917.31 4.8539 4.2965 3.8762 3.5398 3.2615 3.0262 2.8240 2.4931 2.2330 2.0227 1.8490 1.7030 1.5784 1.4710 1.2948 1.1564

0.0010117 0.0010430 0.0010901 0.20602 0.23275 0.25798 0.28250 0.30661 0.33045 0.35411 0.40111 0.44783 0.49439 0.54083 0.58720 0.63355 0.67981 0.77232 0.86475

209.18 418.80 631.41 2622.2 2710.4 2793.6 2875.7 2957.9 3040.9 3125.0 3297.5 3476.2 3661.7 3853.9 4052.7 4257.9 4469.0 4907.7 5365.5

210.19 419.84 632.50 2828.3 2943.1 3051.6 3158.2 3264.5 3371.3 3479.1 3698.6 3924.1 4156.1 4394.8 4639.9 4891.4 5148.9 5680.0 6230.3

s kJ/kg-K

cp kJ/kg-K

μ mPa-s

k W/m-K

Pr

T °C

0.54660 0.28169 0.18262 0.016060 0.018164 0.020265 0.022359 0.024444 0.026516 0.028571 0.032623 0.036585 0.040450 0.044215 0.047880 0.051447 0.054918 0.061590 0.067928

0.64083 0.67744 0.68103 0.034648 0.039254 0.044223 0.049512 0.055079 0.060886 0.066902 0.079455 0.092568 0.10612 0.12001 0.13416 0.14852 0.16304 0.19238 0.22195

3.5657 1.7526 1.1549 0.99325 0.96191 0.94720 0.93764 0.93005 0.92352 0.91776 0.90823 0.90085 0.89497 0.88997 0.88535 0.88079 0.87608 0.86593 0.85481

50 100 150 200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 1400 1600

0.54670 0.28183 0.18274 0.015876 0.018058 0.020205 0.022329 0.024433 0.026518 0.028581 0.032642 0.036607 0.040473 0.044237 0.047901 0.051466 0.054936 0.061605 0.067940

0.64109 0.67772 0.68137 0.036312 0.040464 0.045122 0.050201 0.055627 0.061343 0.067303 0.079811 0.092929 0.10650 0.12043 0.13462 0.14901 0.16355 0.19294 0.22253

3.5639 1.7522 1.1547 1.0616 0.98657 0.95939 0.94512 0.93526 0.92732 0.92052 0.90945 0.90098 0.89433 0.88881 0.88385 0.87908 0.87427 0.86408 0.85305

50 100 150 200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 1400 1600

P = 500 kPa (0.05 MPa) 0.70358 1.3069 1.8418 7.0610 7.2724 7.4614 7.6346 7.7955 7.9465 8.0892 8.3543 8.5977 8.8240 9.0362 9.2364 9.4263 9.6071 9.9448 10.255

4.1804 4.2148 4.3070 2.1429 2.0788 2.0670 2.0763 2.0957 2.1206 2.1490 2.2121 2.2793 2.3479 2.4155 2.4807 2.5427 2.6008 2.7048 2.7931

P = l000 kPa (1 MPa) 0.70335 1.3065 1.8412 6.6955 6.9265 7.1246 7.3029 7.4669 7.6200 7.7641 8.0310 8.2755 8.5024 8.7150 8.9155 9.1056 9.2866 9.6245 9.9351

4.1793 4.2136 4.3054 2.4281 2.2106 2.1425 2.1248 2.1293 2.1452 2.1677 2.2237 2.2871 2.3534 2.4196 2.4839 2.5452 2.6028 2.7062 2.7940

343

Appendix A 466

Appendix B

Table A.4 B.2 (CONTINUED) APPENDIX Thermophysical Properties Single-Phase Water (SI Units) Thermophysical Properties of SingleofPhase Water (SI Units) T °C

ρ kg/m3

v m3/kg

u kJ/kg

h kJ/kg

50 100 150 200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 1400 1600

990.16 960.63 919.56 867.26 800.09 22.053 19.242 17.290 15.792 14.581 12.706 11.297 10.188 9.2861 8.5364 7.9018 7.3571 6.4687 5.7739

0.0010099 0.0010410 0.0010875 0.0011531 0.0012499 0.045345 0.051970 0.057837 0.063323 0.068582 0.078703 0.088519 0.098155 0.10769 0.11715 0.12655 0.13592 0.15459 0.17319

208.59 417.64 629.55 847.91 1079.5 2699.0 2809.5 2907.5 3000.6 3091.7 3273.3 3457.7 3646.9 3841.8 4042.6 4249.3 4461.6 4902.0 5361.1

213.64 422.85 634.98 853.68 1085.7 2925.7 3069.3 3196.7 3317.2 3434.7 3666.8 3900.3 4137.7 4380.2 4628.3 4882.0 5141.2 5675.0 6227.1

50 100 150 200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 1400 1600

992.31 962.93 922.32 870.94 805.70 715.29 44.564 37.827 33.578 30.478 26.057 22.937 20.564 18.675 17.126 15.827 14.719 12.924 11.528

0.0010077 0.0010385 0.0010842 0.0011482 0.0012412 0.0013980 0.022440 0.026436 0.029781 0.032811 0.038377 0.043598 0.048629 0.053548 0.058391 0.063183 0.067939 0.077375 0.086745

207.86 416.23 627.27 844.31 1073.4 1329.4 2699.6 2833.1 2944.5 3047.0 3242.0 3434.0 3628.2 3826.5 4029.9 4238.5 4452.3 4895.0 5355.6

s kJ/kg-K

cp kJ/kg-K

μ mPa-s

k W/m-K

Pr

T °C

0.54751 0.28290 0.18376 0.13546 0.10658 0.019794 0.022157 0.024406 0.026582 0.028703 0.032821 0.036804 0.040669 0.044423 0.048074 0.051625 0.055080 0.061724 0.068037

0.64317 0.67999 0.68408 0.66287 0.61802 0.054298 0.056664 0.060565 0.065367 0.070782 0.082832 0.095933 0.10969 0.12386 0.13832 0.15295 0.16770 0.19738 0.22712

3.5500 1.7493 1.1531 0.91469 0.83747 1.1564 1.0404 0.99174 0.96444 0.94579 0.91991 0.90215 0.88929 0.87967 0.87209 0.86574 0.86008 0.84960 0.83927

50 100 150 200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 1400 1600

0.54854 0.28425 0.18502 0.13670 0.10799 0.086433 0.022177 0.024553 0.026802 0.028966 0.033117 0.037098 0.040946 0.044679 0.048306 0.051835 0.055269 0.061876 0.068161

0.64574 0.68279 0.68744 0.66697 0.62346 0.55506 0.069107 0.068716 0.071561 0.075923 0.087077 0.10000 0.11390 0.12834 0.14307 0.15797 0.17295 0.20297 0.23288

3.5331 1.7458 1.1512 0.91191 0.83023 0.88459 1.2874 1.1060 1.0289 0.98547 0.93470 0.90406 0.88328 0.86867 0.85798 0.84979 0.84314 0.83229 0.82276

50 100 150 200 250 300 350 400 450 500 600 700 800 900 1000 1100 1200 1400 1600

P = 5000 kPa (5 MPa) 0.70150 1.3034 1.8368 2.3251 2.7910 6.2110 6.4516 6.6483 6.8210 6.9781 7.2605 7.5136 7.7458 7.9618 8.1648 8.3566 8.5388 8.8784 9.1900

4.1702 4.2045 4.2926 4.4761 4.8562 3.1722 2.6608 2.4610 2.3717 2.3323 2.3216 2.3515 2.3986 2.4528 2.5091 2.5649 2.6186 2.7168 2.8016

P = 10,000 kPa (10 MPa) 217.94 426.62 638.11 855.80 1085.8 1343.3 2924.0 3097.4 3242.3 3375.1 3625.8 3870.0 4114.5 4362.0 4613.8 4870.3 5131.7 5668.7 6223.1

0.69920 1.2996 1.8313 2.3174 2.7792 3.2488 5.9459 6.2141 6.4219 6.5995 6.9045 7.1693 7.4085 7.6290 7.8349 8.0288 8.2126 8.5543 8.8671

4.1592 4.1935 4.2773 4.4491 4.7934 5.6807 4.0117 3.0953 2.7473 2.5830 2.4576 2.4370 2.4571 2.4952 2.5411 2.5898 2.6384 2.7301 2.8110

344

Table A.5 Thermophysical Properties of Single-Phase Water (SI Units)

Appendix A

345

Appendix A 472

Appendix B

Table A.6 B.3 APPENDIX Thermophysical Properties of atLiquids at (SI Saturation (SI Units) Thermophysical Properties of Liquids Saturation Units) T °C

Psat kPa

ρ kg/m3

h fg kJ/kg

cp kJ/ kg-K

5 10 15 20 25 30 35 40 45 50 60 70 80 90 100 150 200 250 300

0.87258 1.2282 1.7058 2.3393 3.1699 4.2470 5.6290 7.3849 9.5950 12.352 19.946 31.201 47.414 70.182 101.42 476.16 1554.9 3976.2 8587.9

999.92 999.65 999.06 998.16 997.00 995.61 993.99 992.18 990.17 988.00 983.16 977.73 971.77 965.30 958.35 917.01 864.66 798.89 712.14

2489.0 2477.2 2465.4 2453.5 2441.7 2429.8 2417.9 2406.0 2394.0 2381.9 2357.7 2333.0 2308.0 2282.5 2256.4 2113.7 1939.7 1715.2 1404.6

4.2055 4.1955 4.1888 4.1844 4.1816 4.1801 4.1795 4.1796 4.1804 4.1815 4.1851 4.1902 4.1969 4.2053 4.2157 4.3071 4.4958 4.8701 5.7504

–60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90 100 110 120

21.893 40.836 71.692 119.43 190.08 290.71 429.38 615.05 857.48 1167.2 1555.4 2034.0 2615.6 3313.5 4142.0 5116.7 6255.3 7578.3 9112.5

713.62 702.09 690.15 677.83 665.14 652.06 638.57 624.64 610.20 595.17 579.44 562.86 545.24 526.31 505.67 482.75 456.63 425.61 385.49

1441.8 1415.9 1388.6 1359.7 1329.1 1296.7 1262.2 1225.5 1186.4 1144.4 1099.3 1050.5 997.30 938.90 873.97 800.58 715.63 613.39 480.31

4.3031 4.3599 4.4137 4.4645 4.5138 4.5636 4.6165 4.6757 4.7448 4.8282 4.9318 5.0635 5.2351 5.4648 5.7837 6.2501 6.9912 8.3621 11.940

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

1.5183 1.3060 1.1376 1.0016 0.89004 0.79722 0.71912 0.65272 0.59575 0.54650 0.46602 0.40353 0.35404 0.31417 0.28158 0.18261 0.13458 0.10628 0.085855

1.5184 1.3064 1.1387 1.0035 0.89271 0.80074 0.72347 0.65786 0.60167 0.55314 0.47400 0.41272 0.36432 0.32546 0.29382 0.19914 0.15565 0.13304 0.12056

0.56772 0.57871 0.58874 0.59795 0.60646 0.61434 0.62165 0.62844 0.63474 0.64057 0.65096 0.65972 0.66697 0.67277 0.67721 0.68102 0.66001 0.61689 0.55265

0.13501 0.13798 0.14068 0.14317 0.14547 0.14762 0.14964 0.15154 0.15334 0.15505 0.15820 0.16103 0.16354 0.16573 0.16762 0.17243 0.16978 0.15855 0.13495

11.247 9.4682 8.0940 7.0092 6.1369 5.4245 4.8348 4.3411 3.9236 3.5674 2.9961 2.5630 2.2278 1.9638 1.7529 1.1549 0.91675 0.83908 0.89334

5 10 15 20 25 30 35 40 45 50 60 70 80 90 100 150 200 250 300

Ammonia 0.39129 0.32887 0.28124 0.24407 0.21441 0.19022 0.17009 0.15303 0.13832 0.12545 0.11404 0.10379 0.094483 0.085933 0.077979 0.070468 0.063231 0.056028 0.048340

0.54832 0.46841 0.40751 0.36008 0.32235 0.29172 0.26636 0.24499 0.22668 0.21078 0.19681 0.18440 0.17329 0.16328 0.15421 0.14597 0.13847 0.13164 0.12540

0.75700 0.72228 0.68811 0.65463 0.62196 0.59014 0.55920 0.52912 0.49986 0.47135 0.44354 0.41632 0.38959 0.36324 0.33710 0.31102 0.28477 0.25813 0.23124

0.24652 0.23596 0.22590 0.21632 0.20716 0.19832 0.18969 0.18117 0.17265 0.16403 0.15521 0.14607 0.13649 0.12629 0.11526 0.10308 0.089202 0.072528 0.050238

2.2242 1.9851 1.8040 1.6646 1.5560 1.4710 1.4042 1.3523 1.3130 1.2850 1.2680 1.2624 1.2696 1.2928 1.3379 1.4161 1.5523 1.8151 2.4961

–60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90 100 110 120

μ mPa-s Water

346

Appendix A 473

Appendix B

Table A.7 B.3 (CONTINUED) APPENDIX Thermophysical Properties of atLiquids at (SI Saturation (SI Units) Thermophysical Properties of Liquids Saturation Units) Psat kPa

ρ kg/m3

h fg kJ/kg

cp kJ/ kg-K

–100 –80 –60 –40 –30 –20 –10 –5 0 5 10 20 30 40 50 60 70 80 90

2.8994 13.049 42.693 111.12 167.83 244.52 345.28 406.04 474.46 551.12 636.60 836.46 1079.0 1369.4 1713.3 2116.8 2586.8 3131.9 3764.1

643.74 622.76 601.08 578.43 566.64 554.45 541.80 535.27 528.59 521.75 514.73 500.06 484.39 467.46 448.87 427.97 403.62 373.29 328.83

480.44 462.41 443.63 423.36 412.41 400.77 388.30 381.72 374.87 367.73 360.28 344.31 326.70 307.07 284.86 259.23 228.62 189.80 132.77

2.0538 2.1059 2.1720 2.2558 2.3054 2.3608 2.4230 2.4570 2.4932 2.5318 2.5733 2.6662 2.7767 2.9127 3.0893 3.3375 3.7350 4.5445 7.6233

–80 –40 –20 –10 0 10 20 30 40 50 60 70 80 90 100 120 140 160 180

0.10043 2.7187 9.0283 15.191 24.448 37.835 56.558 81.993 115.67 159.25 214.54 283.46 368.01 470.34 592.65 906.71 1330.5 1887.9 2609.9

717.29 681.58 663.44 654.22 644.87 635.37 625.70 615.82 605.70 595.30 584.59 573.51 562.01 550.01 537.42 510.00 478.31 439.49 385.55

440.72 412.64 398.87 391.89 384.81 377.56 370.11 362.40 354.38 346.01 337.22 327.94 318.11 307.63 296.40 271.14 240.75 202.15 146.99

1.9990 2.0792 2.1384 2.1727 2.2099 2.2500 2.2929 2.3385 2.3870 2.4382 2.4925 2.5501 2.6113 2.6770 2.7482 2.9137 3.1340 3.4903 4.4391

T °C

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

0.66127 0.50726 0.40526 0.33288 0.30411 0.27906 0.25706 0.24704 0.23759 0.22866 0.22021 0.20455 0.19032 0.17722 0.16501 0.15341 0.14212 0.13069 0.11805

0.16437 0.15206 0.13981 0.12795 0.12221 0.11662 0.11121 0.10857 0.10597 0.10343 0.10093 0.096073 0.091409 0.086923 0.082598 0.078398 0.074277 0.070213 0.067139

0.12432 0.11595 0.10709 0.098058 0.093552 0.089096 0.084712 0.082551 0.080412 0.078295 0.076198 0.072059 0.067961 0.063839 0.059565 0.054887 0.049271 0.041389 0.026783

5.3190 4.3750 3.7844 3.3948 3.2508 3.1322 3.0346 2.9926 2.9547 2.9205 2.8899 2.8387 2.8004 2.7760 2.7702 2.7950 2.8845 3.1577 4.4077

–100 –80 –60 –40 –30 –20 –10 –5 0 5 10 20 30 40 50 60 70 80 90

1.0947 0.61496 0.50136 0.45851 0.42210 0.39071 0.36331 0.33911 0.31752 0.29806 0.28038 0.26416 0.24919 0.23524 0.22216 0.19803 0.17577 0.15439 0.13239

0.15551 0.13791 0.12935 0.12520 0.12113 0.11716 0.11329 0.10951 0.10584 0.10227 0.098799 0.095426 0.092149 0.088964 0.085868 0.079923 0.074275 0.068887 0.063884

0.10846 0.097314 0.091172 0.088078 0.084997 0.081953 0.078964 0.076044 0.073207 0.070459 0.067805 0.065249 0.062789 0.060421 0.058139 0.053785 0.049548 0.044908 0.037326

10.093 6.3193 5.4990 5.2057 4.9660 4.7675 4.6010 4.4594 4.3373 4.2303 4.1350 4.0486 3.9687 3.8934 3.8212 3.6819 3.5474 3.4379 3.5468

–80 –40 –20 –10 0 10 20 30 40 50 60 70 80 90 100 120 140 160 180

Propane 0.42569 0.31590 0.24359 0.19255 0.17232 0.15473 0.13928 0.13223 0.12559 0.11930 0.11335 0.10229 0.092188 0.082844 0.074066 0.065654 0.057364 0.048787 0.038819 Pentane 0.78523 0.41915 0.33262 0.29997 0.27220 0.24825 0.22732 0.20883 0.19232 0.17744 0.16391 0.15150 0.14005 0.12939 0.11940 0.10100 0.084073 0.067854 0.051042

347

Appendix A 474

Appendix B

Table A.8 B.3 (CONTINUED) APPENDIX Thermophysical Properties of atLiquids at (SI Saturation (SI Units) Thermophysical Properties of Liquids Saturation Units) T °C

Psat kPa

ρ kg/m3

h fg kJ/kg

cp kJ/ kg-K

–50 –40 –30 –20 –10 0 10 20 30 40 60 80 100 120 140 160 180 200 220

0.20366 0.46089 0.96456 1.8855 3.4719 6.0652 10.114 16.187 24.973 37.292 76.424 142.54 246.29 399.76 616.32 910.70 1299.6 1803.4 2450.9

721.49 712.71 703.92 695.11 686.25 677.34 668.35 659.27 650.07 640.74 621.60 601.63 580.60 558.16 533.84 506.85 475.83 437.73 382.16

412.78 406.50 400.26 394.03 387.79 381.52 375.18 368.74 362.17 355.45 341.38 326.24 309.70 291.35 270.63 246.68 218.02 181.55 127.74

1.9903 2.0189 2.0493 2.0817 2.1161 2.1525 2.1909 2.2312 2.2733 2.3172 2.4100 2.5094 2.6161 2.7318 2.8612 3.0142 3.2169 3.5578 4.6726

–60 –40 –20 0 10 20 30 40 60 80 100 120 140 160 180 200 220 240 260

0.010397 0.077005 0.39420 1.5223 2.7467 4.7222 7.7770 12.326 28.039 57.090 106.23 183.60 298.61 461.74 684.66 980.41 1364.2 1855.1 2479.4

750.18 733.56 717.02 700.45 692.11 683.72 675.27 666.72 649.32 631.36 612.67 593.03 572.12 549.53 524.57 496.14 462.01 416.64 335.09

419.34 405.97 393.11 380.59 374.39 368.21 362.01 355.77 343.04 329.77 315.71 300.58 284.03 265.61 244.66 220.14 190.09 149.97 82.509

2.0026 2.0420 2.0922 2.1526 2.1863 2.2221 2.2597 2.2992 2.3827 2.4717 2.5658 2.6653 2.7714 2.8874 3.0201 3.1853 3.4269 3.9284 7.2271

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

1.1262 0.96000 0.83087 0.72839 0.64556 0.57753 0.52088 0.47313 0.43245 0.39746 0.34056 0.29647 0.26141 0.23289 0.20915 0.18887 0.17097 0.15426 0.13644

0.14730 0.14312 0.13912 0.13528 0.13160 0.12809 0.12472 0.12149 0.11839 0.11542 0.10984 0.10471 0.099987 0.095627 0.091587 0.087815 0.084245 0.080798 0.077711

0.10257 0.099466 0.096439 0.093491 0.090626 0.087857 0.085177 0.082595 0.080115 0.077741 0.073325 0.069357 0.065829 0.062714 0.059963 0.057480 0.055037 0.051882 0.043519

10.980 9.6516 8.6156 7.7911 7.1233 6.5735 6.1152 5.7283 5.3979 5.1126 4.6445 4.2745 3.9711 3.7136 3.4879 3.2859 3.1065 2.9733 3.1352

–50 –40 –30 –20 –10 0 10 20 30 40 60 80 100 120 140 160 180 200 220

2.0455 1.3601 0.98282 0.75255 0.66946 0.60090 0.54358 0.49507 0.41778 0.35924 0.31354 0.27693 0.24690 0.22171 0.20007 0.18094 0.16332 0.14588 0.12533

0.14821 0.14210 0.13591 0.12973 0.12666 0.12363 0.12063 0.11766 0.11186 0.10624 0.10082 0.095614 0.090607 0.085784 0.081105 0.076509 0.071899 0.067211 0.064864

0.098656 0.094868 0.090599 0.086037 0.083705 0.081374 0.079052 0.076757 0.072299 0.068077 0.064136 0.060493 0.057145 0.054064 0.051193 0.048413 0.045413 0.041064 0.026785

20.734 14.337 10.848 8.7468 7.9978 7.3844 6.8762 6.4498 5.7785 5.2770 4.8887 4.5778 4.3206 4.1009 3.9082 3.7375 3.5963 3.5525 4.6792

–60 –40 –20 0 10 20 30 40 60 80 100 120 140 160 180 200 220 240 260

Hexane 0.81257 0.68420 0.58487 0.50631 0.44302 0.39118 0.34813 0.31192 0.28112 0.25467 0.21169 0.17836 0.15178 0.12999 0.11165 0.095730 0.081352 0.067525 0.052142 Heptane 1.5345 0.99774 0.70470 0.52712 0.46334 0.41085 0.36706 0.33007 0.27127 0.22681 0.19210 0.16422 0.14126 0.12184 0.10495 0.089771 0.075454 0.060780 0.041997

348

Appendix A 475

Appendix B

Table A.9 B.3 (CONTINUED) APPENDIX Thermophysical Properties of atLiquids at (SI Saturation (SI Units) Thermophysical Properties of Liquids Saturation Units) T °C

Psat kPa

ρ kg/m3

h fg kJ/kg

cp kJ/ kg-K

–40 –20 10 0 10 20 40 60 80 100 120 140 160 180 200 220 240 260 280

0.012988 0.083067 0.75269 0.38526 0.75269 1.3916 4.1263 10.450 23.298 46.824 86.423 148.66 241.17 372.53 552.24 790.84 1100.3 1495.1 1995.0

750.20 734.17 710.22 718.21 710.22 702.20 686.04 669.61 652.79 635.45 617.43 598.53 578.48 556.91 533.27 506.68 475.51 436.27 378.16

404.85 391.98 373.52 379.58 373.52 367.53 355.65 343.76 331.67 319.17 306.03 292.02 276.85 260.16 241.43 219.89 194.21 161.58 113.81

2.0404 2.0866 2.1752 2.1433 2.1752 2.2090 2.2821 2.3610 2.4445 2.5318 2.6225 2.7169 2.8161 2.9223 3.0403 3.1805 3.3681 3.6869 4.6731

–40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320

0.0023016 0.018188 0.10056 0.42095 1.4136 3.9784 9.7038 21.057 41.500 75.519 128.59 207.09 318.26 470.16 671.82 933.60 1268.1 1692.7 2238.0

765.37 749.46 733.69 717.95 702.14 686.18 669.98 653.42 636.39 618.75 600.34 580.92 560.20 537.72 512.80 484.23 449.57 402.26 289.18

401.57 389.10 377.09 365.44 354.04 342.75 331.42 319.91 308.02 295.58 282.38 268.17 252.65 235.40 215.80 192.78 164.28 124.95 37.983

2.0099 2.0606 2.1212 2.1896 2.2639 2.3425 2.4245 2.5090 2.5957 2.6843 2.7753 2.8694 2.9685 3.0761 3.1993 3.3544 3.5893 4.1366 25.073

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

1.9747 1.3442 0.86853 0.98999 0.86853 0.77102 0.62543 0.52281 0.44686 0.38829 0.34151 0.30300 0.27045 0.24230 0.21745 0.19510 0.17463 0.15548 0.13710

0.14666 0.13930 0.12914 0.13242 0.12914 0.12595 0.11987 0.11410 0.10860 0.10335 0.098299 0.093434 0.088722 0.084137 0.079653 0.075248 0.070916 0.066742 0.063640

0.095817 0.090930 0.083592 0.086020 0.083592 0.081198 0.076565 0.072172 0.068058 0.064237 0.060708 0.057457 0.054463 0.051699 0.049129 0.046695 0.044279 0.041493 0.036012

20.609 14.783 10.390 11.509 10.390 9.4956 8.1687 7.2439 6.5659 6.0447 5.6255 5.2734 4.9657 4.6867 4.4262 4.1783 3.9438 3.7471 3.8071

–40 –20 10 0 10 20 40 60 80 100 120 140 160 180 200 220 240 260 280

3.0808 1.8399 1.2814 0.97158 0.77694 0.64376 0.54675 0.47260 0.41372 0.36550 0.32497 0.29015 0.25964 0.23240 0.20760 0.18448 0.16223 0.13940 0.10810

0.14782 0.14066 0.13410 0.12807 0.12249 0.11732 0.11250 0.10798 0.10373 0.099707 0.095872 0.092184 0.088598 0.085061 0.081502 0.077827 0.073892 0.069525 0.078846

0.096089 0.091083 0.086167 0.081466 0.077060 0.072986 0.069256 0.065864 0.062796 0.060031 0.057543 0.055303 0.053277 0.051424 0.049678 0.047915 0.045792 0.041783 0.010875

32.062 20.200 14.871 11.926 10.082 8.8203 7.8947 7.1754 6.5883 6.0884 5.6475 5.2467 4.8735 4.5193 4.1788 3.8503 3.5427 3.3364 9.9406

–40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320

Octane 1.4814 0.98686 0.61684 0.71102 0.61684 0.54141 0.42907 0.35007 0.29170 0.24674 0.21086 0.18135 0.15645 0.13494 0.11596 0.098855 0.083037 0.067831 0.051845 Nonane 2.3580 1.3789 0.94012 0.69754 0.54552 0.44174 0.36631 0.30880 0.26328 0.22615 0.19509 0.16856 0.14545 0.12497 0.10646 0.089332 0.072933 0.056076 0.031260

349

Appendix A 476

Appendix B

Table A.10B.3 (CONTINUED) APPENDIX Thermophysical Properties of atLiquids at (SI Saturation (SI Units) Thermophysical Properties of Liquids Saturation Units) T °C

Psat kPa

ρ kg/m3

h fg kJ/kg

cp kJ/ kg-K

–20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340

0.0039896 0.026319 0.12780 0.48671 1.5248 4.0763 9.5701 20.184 38.944 69.756 117.39 187.42 286.20 420.86 599.39 830.94 1126.4 1500.1 1974.8

761.59 745.90 730.33 714.78 699.16 683.38 667.35 650.97 634.13 616.67 598.44 579.21 558.65 536.33 511.52 483.02 448.49 402.23 317.26

386.87 375.18 363.83 352.74 341.82 330.95 320.02 308.87 297.37 285.32 272.55 258.83 243.88 227.30 208.51 186.56 159.62 123.34 57.874

2.0404 2.1040 2.1742 2.2494 2.3282 2.4096 2.4929 2.5775 2.6632 2.7499 2.8380 2.9284 3.0228 3.1245 3.2397 3.3828 3.5932 4.0398 8.5345

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

0.0017553 0.011684 0.057945 0.22694 0.73371 2.0269 4.9161 10.699 21.267 39.168 67.635 110.58 172.59 258.91 375.49 529.05 727.34 979.60 1297.7

764.28 749.36 734.56 719.77 704.91 689.88 674.60 658.97 642.88 626.21 608.82 590.53 571.10 550.25 527.53 502.35 473.69 439.82 396.77

376.34 364.36 352.90 341.84 331.08 320.52 310.04 299.50 288.77 277.70 266.11 253.82 240.61 226.23 210.32 192.41 171.72 146.96 115.18

2.1375 2.1964 2.2626 2.3340 2.4091 2.4870 2.5666 2.6475 2.7292 2.8118 2.8952 2.9800 3.0669 3.1575 3.2545 3.3630 3.4942 3.6770 4.0295

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

2.5196 1.7068 1.2493 0.96499 0.77543 0.64203 0.54401 0.46935 0.41074 0.36349 0.32449 0.29162 0.26335 0.23857 0.21641 0.19611 0.17697 0.15801 0.13622

0.14144 0.13604 0.13074 0.12557 0.12053 0.11563 0.11090 0.10633 0.10195 0.097764 0.093789 0.090036 0.086512 0.083224 0.080176 0.077367 0.074807 0.072646 0.075060

0.091021 0.086682 0.082337 0.078101 0.074047 0.070222 0.066658 0.063370 0.060367 0.057651 0.055223 0.053082 0.051229 0.049664 0.048381 0.047349 0.046421 0.044707 0.027721

27.682 19.690 15.173 12.356 10.472 9.1428 8.1612 7.4065 6.8041 6.3050 5.8761 5.4937 5.1407 4.8038 4.4730 4.1419 3.8123 3.5345 4.9138

–20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340

2.9652 1.9838 1.4425 1.1103 0.89033 0.73580 0.62193 0.53462 0.46533 0.40871 0.36122 0.32049 0.28484 0.25306 0.22425 0.19769 0.17280 0.14901 0.12558

0.14146 0.13647 0.13172 0.12715 0.12274 0.11845 0.11427 0.11017 0.10614 0.10217 0.098248 0.094367 0.090527 0.086729 0.082981 0.079303 0.075730 0.072340 0.069335

0.086590 0.082916 0.079254 0.075689 0.072276 0.069041 0.065998 0.063151 0.060495 0.058026 0.055738 0.053626 0.051685 0.049919 0.048334 0.046941 0.045753 0.044731 0.043367

34.245 23.926 18.201 14.670 12.319 10.657 9.4235 8.4658 7.6921 7.0435 6.4807 5.9765 5.5110 5.0694 4.6396 4.2115 3.7767 3.3313 2.8958

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

Decane 1.9189 1.2731 0.91238 0.68976 0.54215 0.43875 0.36304 0.30554 0.26046 0.22415 0.19419 0.16891 0.14712 0.12795 0.11070 0.094727 0.079368 0.063558 0.043216 Dodecane 2.2663 1.4866 1.0596 0.79919 0.62760 0.50761 0.41956 0.35230 0.29915 0.25594 0.21992 0.18926 0.16267 0.13925 0.11830 0.099309 0.081853 0.065539 0.049828

350

Appendix A 477

Appendix B

Table A.11B.3 (CONTINUED) APPENDIX Thermophysical Properties of atLiquids at (SI Saturation (SI Units) Thermophysical Properties of Liquids Saturation Units) T °C

Psat kPa

ρ kg/m3

h fg kJ/kg

cp kJ/ kg-K

–80 –60 -40 –20 10 0 10 20 40 60 80 100 120 140 160 180 200 220 240

0.0025173 0.028074 0.20096 1.0283 7.4384 4.0562 7.4384 13.032 35.518 84.713 181.11 353.73 640.81 1090.2 1759.6 2715.8 4027.0 5787.7 8182.6

887.13 867.24 847.70 828.52 800.28 809.65 800.28 790.93 772.10 752.79 732.58 710.95 687.29 660.83 630.42 594.32 549.21 484.87 310.87

1296.9 1276.4 1254.6 1231.0 1191.2 1205.1 1191.2 1176.6 1145.0 1109.6 1069.2 1022.1 966.15 898.89 817.72 723.80 621.13 467.93 73.259

2.2124 2.2267 2.2652 2.3224 2.4496 2.4011 2.4496 2.5047 2.6340 2.7880 2.9658 3.1689 3.4024 3.6783 4.0224 4.4939 5.2742 7.5369 390.28

–80 –60 –40 –20 10 0 10 20 40 60 80 100 120 140 160 180 200 220 240

0.00040787 0.0061203 0.055632 0.34589 3.1485 1.6017 3.1485 5.8759 17.880 46.734 107.81 224.17 427.30 756.55 1257.6 1980.7 2980.6 4320.5 6094.7

875.68 857.77 840.37 823.28 797.86 806.34 797.86 789.34 772.01 753.99 734.85 714.02 690.78 664.23 633.44 597.60 554.99 497.13 333.83

1014.2 997.73 981.23 964.14 936.24 945.94 936.24 926.01 903.52 877.53 847.02 810.92 768.09 717.34 657.70 588.18 505.72 397.54 122.67

1.9428 1.9811 2.0479 2.1414 2.3236 2.2571 2.3236 2.3961 2.5590 2.7438 2.9481 3.1739 3.4249 3.6958 3.9678 4.2485 4.6872 6.0630 63.991

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

5.8336 2.9632 1.7755 1.1615 0.68284 0.80548 0.68284 0.58498 0.44145 0.34372 0.27424 0.22275 0.18306 0.15129 0.12497 0.10251 0.082795 0.064314 0.038619

6.5758 3.4169 2.0945 1.4019 0.85325 0.99485 0.85325 0.73962 0.57176 0.45659 0.37435 0.31332 0.26635 0.22894 0.19823 0.17248 0.15075 0.13264 0.12423

0.25568 0.24074 0.22826 0.21794 0.20571 0.20941 0.20571 0.20233 0.19637 0.19119 0.18651 0.18212 0.17782 0.17346 0.16893 0.16421 0.15957 0.15743 0.26891

0.13027 0.12467 0.11887 0.11326 0.10493 0.10772 0.10493 0.10214 0.096555 0.091094 0.085844 0.080837 0.076041 0.071360 0.066617 0.061482 0.055089 0.043078 0.0022164

50.479 27.408 17.620 12.378 8.1314 9.2356 8.1314 7.2416 5.9216 5.0123 4.3608 3.8760 3.5027 3.2082 2.9757 2.8053 2.7365 3.0791 56.049

–80 –60 –40 –20 10 0 10 20 40 60 80 100 120 140 160 180 200 220 240

Ethanol 3.8662 2.2846 1.5075 2.9166 1.4638 1.8177 1.4638 1.1931 0.81899 0.58416 0.43008 0.32453 0.24908 0.19300 0.15017 0.11713 0.091299 0.069103 0.037405

4.4150 2.6634 1.7938 3.5427 1.8347 2.2542 1.8347 1.5115 1.0609 0.77476 0.58527 0.45451 0.36058 0.29057 0.23707 0.19600 0.16450 0.13900 0.11205

0.19801 0.18843 0.18060 0.17418 0.16657 0.16888 0.16657 0.16445 0.16065 0.15726 0.15408 0.15096 0.14777 0.14444 0.14105 0.13779 0.13490 0.13288 0.17769

0.11639 0.11088 0.10494 0.098800 0.089848 0.092789 0.089848 0.086949 0.081318 0.076015 0.071124 0.066616 0.062460 0.058839 0.056118 0.054273 0.051859 0.044087 0.0083181

37.933 24.020 17.094 35.857 20.420 24.294 20.420 17.384 13.046 10.192 8.2289 6.8229 5.7729 4.9383 4.2245 3.6113 3.1722 3.1529 13.470

–80 –60 –40 –20 10 0 10 20 40 60 80 100 120 140 160 180 200 220 240

μ mPa-s Methanol

351

Appendix A 478

Appendix B

Table A.12B.3 (CONTINUED) APPENDIX Thermophysical Properties of atLiquids at (SI Saturation (SI Units) Thermophysical Properties of Liquids Saturation Units) T °C

Psat kPa

ρ kg/m3

h fg kJ/kg

cp kJ/ kg-K

–80 –60 –40 –20 10 0 10 20 40 60 80 100 110 120 140 160 180 200 220

0.015650 0.13085 0.72172 2.9169 15.454 9.2991 15.454 24.662 56.582 115.67 215.48 372.30 477.84 604.83 934.13 1383.8 1980.9 2757.9 3757.4

897.33 876.00 854.82 833.58 801.21 812.10 801.21 790.19 767.66 744.28 719.79 693.84 680.19 666.00 635.64 601.79 562.85 515.54 449.93

634.81 615.66 596.79 577.95 549.13 558.87 549.13 539.22 518.73 497.07 473.88 448.71 435.21 420.97 389.81 353.96 311.35 257.85 182.00

1.9969 2.0099 2.0281 2.0540 2.1087 2.0883 2.1087 2.1311 2.1822 2.2417 2.3102 2.3895 2.4341 2.4828 2.5969 2.7457 2.9628 3.3565 4.6149

10 20 30 40 50 60 70 80 90 100 110 120 140 160 180 200 220 240 260

0.0060742 0.010030 0.015919 0.024388 0.036204 0.052249 0.073517 0.10111 0.13623 0.18016 0.23428 0.30002 0.47249 0.71033 1.0272 1.4378 1.9582 2.6067 3.4064

889.31 878.84 868.31 857.69 846.96 836.12 825.12 813.97 802.63 791.10 779.33 767.32 742.43 716.12 687.98 657.34 623.05 582.87 531.50

444.41 437.15 429.95 422.79 415.62 408.41 401.13 393.75 386.23 378.54 370.65 362.51 345.39 326.81 306.31 283.22 256.47 224.10 181.64

1.6954 1.7204 1.7476 1.7766 1.8071 1.8388 1.8717 1.9054 1.9401 1.9756 2.0120 2.0494 2.1277 2.2127 2.3087 2.4238 2.5756 2.8102 3.3008

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

1.8870 1.1455 0.80102 0.61003 0.44504 0.48978 0.44504 0.40739 0.34787 0.30346 0.26966 0.24363 0.23291 0.22347 0.20786 0.19594 0.18715 0.18139 0.17964

0.19740 0.18978 0.18183 0.17366 0.16130 0.16540 0.16130 0.15720 0.14907 0.14109 0.13332 0.12578 0.12210 0.11850 0.11148 0.10471 0.098144 0.091688 0.085422

0.11016 0.10779 0.10488 0.10143 0.095471 0.097527 0.095471 0.093349 0.088987 0.084564 0.080171 0.075864 0.073749 0.071660 0.067534 0.063371 0.058852 0.052986 0.041139

17.130 10.628 7.6375 6.0144 4.6615 5.0220 4.6615 4.3642 3.9093 3.5886 3.3636 3.2115 3.1581 3.1184 3.0779 3.0919 3.1800 3.4234 4.3665

–80 –60 –40 –20 10 0 10 20 40 60 80 100 110 120 140 160 180 200 220

0.86575 0.75141 0.65749 0.57964 0.51458 0.45982 0.41338 0.37376 0.33973 0.31034 0.28481 0.26250 0.22564 0.19667 0.17350 0.15471 0.13937 0.12690 0.11726

0.14637 0.14289 0.13952 0.13624 0.13305 0.12992 0.12683 0.12378 0.12076 0.11777 0.11480 0.11184 0.10597 0.10013 0.094325 0.088562 0.082853 0.077243 0.071954

0.097079 0.094507 0.091944 0.089410 0.086932 0.084500 0.082125 0.079809 0.077553 0.075356 0.073214 0.071124 0.067083 0.063189 0.059386 0.055586 0.051631 0.047156 0.041014

8.9180 7.9509 7.1510 6.4829 5.9194 5.4416 5.0336 4.6832 4.3806 4.1183 3.8900 3.6908 3.3636 3.1124 2.9216 2.7833 2.6993 2.6911 2.8589

10 20 30 40 50 60 70 80 90 100 110 120 140 160 180 200 220 240 260

Acetone 1.6933 1.0035 0.68472 0.50851 0.35657 0.39775 0.35657 0.32192 0.26705 0.22586 0.19410 0.16904 0.15842 0.14883 0.13213 0.11791 0.10534 0.093515 0.080823 Benzene 0.76992 0.66038 0.57090 0.49715 0.43583 0.38446 0.34109 0.30423 0.27268 0.24551 0.22196 0.20142 0.16752 0.14084 0.11937 0.10170 0.086833 0.073968 0.062322

352

Appendix A 479

Appendix B

Table A.13B.3 (CONTINUED) APPENDIX Thermophysical Properties of atLiquids at (SI Saturation (SI Units) Thermophysical Properties of Liquids Saturation Units) T °C

Psat kPa

ρ kg/m3

h fg kJ/kg

cp kJ/ kg-K

10 20 30 40 50 60 70 80 90 100 110 120 130 140 160 180 200 220 240

6.3407 10.343 16.240 24.642 36.267 51.936 72.566 99.166 132.83 174.73 226.10 288.24 362.51 450.30 672.31 966.43 1345.7 1824.4 2419.0

787.96 778.60 769.15 759.61 749.95 740.17 730.25 720.19 709.96 699.54 688.92 678.07 666.96 655.55 631.60 605.71 577.00 544.05 504.04

401.90 395.65 389.38 383.06 376.67 370.19 363.59 356.85 349.94 342.84 335.50 327.90 319.98 311.72 293.91 273.93 251.01 223.93 190.40

1.7893 1.8357 1.8826 1.9299 1.9777 2.0260 2.0748 2.1242 2.1745 2.2257 2.2782 2.3323 2.3881 2.4461 2.5703 2.7097 2.8735 3.0833 3.4051

–60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

0.0054177 0.042077 0.22511 0.90575 2.9189 7.8923 18.540 38.868 74.246 131.37 218.12 343.44 517.19 750.13 1054.0 1441.7 1928.2 2531.6 3275.7

941.08 922.37 903.83 885.35 866.82 848.12 829.15 809.79 789.93 769.40 748.04 725.61 701.83 676.28 648.38 617.21 581.17 537.13 476.60

466.17 452.76 440.01 427.75 415.81 404.01 392.16 380.07 367.57 354.43 340.46 325.40 308.96 290.75 270.24 246.61 218.52 183.35 134.38

1.4936 1.5274 1.5723 1.6257 1.6855 1.7501 1.8184 1.8896 1.9632 2.0390 2.1175 2.1993 2.2859 2.3803 2.4874 2.6175 2.7945 3.0901 3.8919

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

1.4486 1.2301 1.0586 0.92139 0.80997 0.71815 0.64149 0.57676 0.52151 0.47393 0.43261 0.39644 0.36458 0.33632 0.28849 0.24956 0.21709 0.18924 0.16440

0.11984 0.11772 0.11559 0.11345 0.11132 0.10920 0.10709 0.10500 0.10294 0.10090 0.098900 0.096936 0.095014 0.093137 0.089538 0.086170 0.083070 0.080297 0.078017

0.084994 0.082363 0.079827 0.077391 0.075056 0.072821 0.070682 0.068637 0.066679 0.064806 0.063012 0.061295 0.059653 0.058082 0.055153 0.052501 0.050102 0.047868 0.045457

17.044 14.935 13.261 11.906 10.791 9.8618 9.0757 8.4031 7.8212 7.3131 6.8654 6.4677 6.1117 5.7904 5.2307 4.7534 4.3330 3.9534 3.6166

10 20 30 40 50 60 70 80 90 100 110 120 130 140 160 180 200 220 240

2.9390 1.7329 1.1725 0.86611 0.67674 0.54945 0.45887 0.39166 0.34015 0.29953 0.26670 0.23957 0.21667 0.19700 0.17980 0.16454 0.15080 0.13823 0.12639

0.15229 0.14761 0.14253 0.13720 0.13171 0.12615 0.12065 0.11522 0.10995 0.10489 0.10010 0.095618 0.091490 0.087750 0.084424 0.081531 0.079082 0.077109 0.076019

0.10835 0.10477 0.10030 0.095323 0.090150 0.084992 0.080020 0.075300 0.070901 0.066860 0.063196 0.059918 0.057027 0.054512 0.052347 0.050466 0.048693 0.046457 0.040984

27.126 16.540 11.691 9.0860 7.5067 6.4648 5.7344 5.2014 4.7975 4.4800 4.2203 3.9983 3.7995 3.6138 3.4347 3.2603 3.0969 2.9755 3.0839

–60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Cyclohexane 1.1414 0.95778 0.81419 0.69989 0.60744 0.53155 0.46845 0.41537 0.37025 0.33154 0.29803 0.26882 0.24316 0.22047 0.18221 0.15116 0.12526 0.10296 0.082864 Toluene 2.7659 1.5984 1.0598 0.76681 0.58660 0.46600 0.38047 0.31717 0.26869 0.23046 0.19950 0.17383 0.15207 0.13323 0.11658 0.10155 0.087639 0.074250 0.060238

353

Appendix A 480

Appendix B

Table A.14B.3 (CONTINUED) APPENDIX Thermophysical Properties of atLiquids at (SI Saturation (SI Units) Thermophysical Properties of Liquids Saturation Units) Psat kPa

ρ kg/m3

h fg kJ/kg

cp kJ/ kg-K

μ mPa-s

–100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80

2.0102 4.8130 10.372 20.469 37.505 64.530 105.23 163.89 245.31 354.79 497.99 680.95 910.02 1191.9 1533.6 1942.7 2427.5 2997.4 3663.8

1571.3 1544.9 1518.2 1491.2 1463.7 1435.6 1406.8 1377.2 1346.5 1314.7 1281.5 1246.7 1209.9 1170.7 1128.5 1082.3 1030.4 969.74 893.74

268.26 262.53 256.84 251.12 245.33 239.39 233.24 226.81 220.02 212.79 205.05 196.69 187.60 177.64 166.60 154.19 139.94 123.00 101.57

1.0612 1.0612 1.0624 1.0655 1.0710 1.0793 1.0905 1.1049 1.1227 1.1439 1.1692 1.1993 1.2356 1.2807 1.3389 1.4191 1.5392 1.7434 2.1814

0.89602 0.73051 0.60936 0.51759 0.44603 0.38879 0.34197 0.30289 0.26971 0.24106 0.21598 0.19371 0.17370 0.15547 0.13867 0.12296 0.10799 0.093339 0.078270

–110 –100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70

1.4525 3.8130 8.8687 18.654 36.067 64.955 110.14 177.41 273.44 405.75 582.63 813.10 1106.9 1474.6 1927.5 2478.3 3141.2 3933.2 4876.8

1363.8 1339.0 1313.9 1288.4 1262.4 1235.7 1208.4 1180.2 1151.0 1120.6 1088.8 1055.3 1019.7 981.38 939.62 893.04 839.26 773.31 680.93

438.66 429.48 420.20 410.71 400.92 390.73 380.06 368.79 356.83 344.03 330.25 315.30 298.92 280.78 260.41 237.09 209.62 175.51 127.78

1.5647 1.5600 1.5586 1.5606 1.5663 1.5758 1.5895 1.6077 1.6311 1.6607 1.6980 1.7450 1.8056 1.8859 1.9973 2.1629 2.4385 3.0007 4.8653

0.66291 0.55456 0.47122 0.40512 0.35146 0.30715 0.27004 0.23861 0.21173 0.18852 0.16830 0.15049 0.13463 0.12033 0.10723 0.094988 0.083217 0.071369 0.058172

T °C

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

0.57023 0.47286 0.40137 0.34711 0.30474 0.27082 0.24308 0.21994 0.20030 0.18336 0.16853 0.15538 0.14356 0.13280 0.12288 0.11361 0.10481 0.096252 0.087576

0.14312 0.13778 0.13259 0.12752 0.12258 0.11774 0.11299 0.10836 0.10378 0.099250 0.094743 0.090247 0.085742 0.081205 0.076604 0.071900 0.067040 0.061948 0.056583

0.085832 0.084043 0.082203 0.080264 0.078197 0.075990 0.073646 0.071209 0.068654 0.065992 0.063230 0.060359 0.057353 0.054161 0.050696 0.046812 0.042270 0.036643 0.029024

6.6436 5.6264 4.8826 4.3245 3.8970 3.5639 3.3007 3.0886 2.9175 2.7785 2.6654 2.5743 2.5031 2.4520 2.4239 2.4269 2.4795 2.6268 3.0174

–100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80

0.48607 0.41415 0.35864 0.31444 0.27842 0.24856 0.22347 0.20219 0.18396 0.16824 0.15458 0.14261 0.13204 0.12262 0.11412 0.10637 0.099156 0.092290 0.085431

0.23063 0.22432 0.21740 0.20999 0.20222 0.19421 0.18604 0.17779 0.16954 0.16135 0.15324 0.14525 0.13741 0.12970 0.12210 0.11458 0.10703 0.099377 0.092046

0.10808 0.10739 0.10616 0.10444 0.10227 0.099731 0.096860 0.093707 0.090311 0.086700 0.082891 0.078879 0.074632 0.070075 0.065062 0.059318 0.052300 0.042826 0.027784

4.4975 3.8564 3.3782 3.0108 2.7223 2.4923 2.3072 2.1576 2.0370 1.9405 1.8648 1.8079 1.7692 1.7498 1.7541 1.7931 1.8959 2.1550 3.0748

–110 –100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70

R22

R32

354

Appendix A 481

Appendix B

Table A.15B.3 (CONTINUED) APPENDIX Thermophysical Properties of atLiquids at (SI Saturation (SI Units) Thermophysical Properties of Liquids Saturation Units) cp kJ/ kg-K

Psat kPa

ρ kg/m3

h fg kJ/kg

–100 –80 –60 –40 –30 –20 –10 0 10 20 30 40 60 80 100 120 140 160 180

0.011610 0.12548 0.80750 3.5752 6.7480 11.997 20.247 32.645 50.567 75.610 109.58 154.47 285.89 489.09 785.53 1199.0 1756.3 2490.1 3450.6

1754.5 1709.6 1665.1 1620.0 1597.0 1573.8 1550.1 1526.1 1501.6 1476.6 1451.0 1424.8 1370.0 1311.2 1246.9 1174.4 1088.3 975.68 765.91

220.49 212.07 204.21 196.63 192.87 189.11 185.30 181.44 177.49 173.44 169.27 164.94 155.73 145.54 134.01 120.53 104.02 81.886 40.596

0.92605 3.7133 0.92359 2.0932 0.93199 1.3834 0.94805 0.98641 0.95776 0.84796 0.96816 0.73539 0.97903 0.64240 0.99023 0.56459 1.0017 0.49878 1.0135 0.44260 1.0257 0.39427 1.0385 0.35240 1.0663 0.28386 1.0996 0.23054 1.1433 0.18809 1.2072 0.15336 1.3178 0.12382 1.5836 0.096802 4.5486 0.064292

–100 –90 –80 –70 –60 –50 –40 –30 –20 –10 –5 0 5 10 20 30 40 50 60

3.0883 7.2856 15.468 30.076 54.318 92.164 148.30 228.06 337.33 482.52 570.72 670.52 782.88 908.75 1205.2 1568.5 2008.5 2536.8 3170.3

1688.7 1656.2 1623.4 1589.9 1555.7 1520.5 1484.0 1446.1 1406.4 1364.5 1342.6 1319.8 1296.2 1271.5 1218.3 1158.4 1088.4 1001.1 872.09

189.96 185.18 180.37 175.47 170.41 165.14 159.59 153.71 147.41 140.60 136.97 133.16 129.14 124.90 115.57 104.81 92.024 75.924 52.112

1.0351 1.0450 1.0581 1.0736 1.0912 1.1107 1.1323 1.1565 1.1840 1.2161 1.2344 1.2547 1.2773 1.3029 1.3666 1.4575 1.6052 1.9102 3.1392

T °C

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

2.1164 1.2244 0.83082 0.60891 0.53097 0.46728 0.41441 0.36995 0.33216 0.29974 0.27172 0.24733 0.20720 0.17582 0.15084 0.13059 0.11378 0.099215 0.083942

0.11303 0.10738 0.10200 0.096071 0.092972 0.089848 0.086742 0.083686 0.080707 0.077821 0.075037 0.072359 0.067311 0.062621 0.058190 0.053884 0.049522 0.044842 0.039715

0.069566 0.068005 0.065730 0.062555 0.060783 0.058969 0.057156 0.055377 0.053654 0.051998 0.050416 0.048906 0.046078 0.043431 0.040818 0.038006 0.034532 0.029023 0.011400

30.423 18.004 12.640 9.7340 8.7354 7.9242 7.2506 6.6807 6.1909 5.7645 5.3896 5.0574 4.4967 4.0484 3.6955 3.4360 3.2949 3.4185 7.3635

–100 –80 –60 –40 –30 –20 –10 0 10 20 30 40 60 80 100 120 140 160 180

0.67133 0.53782 0.44389 0.37438 0.32097 0.27870 0.24446 0.21615 0.19233 0.17199 0.16287 0.15435 0.14636 0.13882 0.12494 0.11230 0.10052 0.089107 0.076885

0.11573 0.11103 0.10630 0.10157 0.096846 0.092188 0.087570 0.083016 0.078538 0.074145 0.071982 0.069840 0.067720 0.065619 0.061473 0.057382 0.053317 0.049261 0.045865

0.066213 0.064151 0.061885 0.059500 0.057052 0.054591 0.052114 0.049637 0.047162 0.044681 0.043432 0.042174 0.040902 0.039610 0.036923 0.033988 0.030518 0.025760 0.016753

10.139 8.3836 7.1729 6.2921 5.6259 5.1052 4.6908 4.3546 4.0782 3.8494 3.7501 3.6599 3.5782 3.5048 3.3837 3.3041 3.2937 3.4591 4.5892

–100 –90 –80 –70 –60 –50 –40 –30 –20 –10 –5 0 5 10 20 30 40 50 60

R123

R125 1.1336 0.89075 0.72062 0.59524 0.49932 0.42375 0.36278 0.31258 0.27051 0.23469 0.21867 0.20372 0.18970 0.17651 0.15221 0.13008 0.10940 0.089205 0.067051

355

Appendix A 482

Appendix B

Table A.16B.3 (CONTINUED) APPENDIX Thermophysical Properties of atLiquids at (SI Saturation (SI Units) Thermophysical Properties of Liquids Saturation Units) Psat kPa

ρ kg/m3

h fg kJ/kg

cp kJ/ kg-K

–100 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 100

0.55940 3.6719 7.9814 15.906 29.451 51.209 84.378 132.73 200.60 292.80 414.61 571.71 770.20 1016.6 1317.9 1681.8 2116.8 2633.2 3972.4

1582.4 1529.0 1501.9 1474.3 1446.3 1417.7 1388.4 1358.3 1327.1 1294.8 1261.0 1225.3 1187.5 1146.7 1102.3 1052.9 996.25 928.24 651.18

261.49 249.67 243.82 237.95 231.98 225.86 219.53 212.92 205.97 198.60 190.74 182.28 173.10 163.02 151.81 139.12 124.37 106.42 34.385

1.1842 1.1981 1.2096 1.2230 1.2381 1.2546 1.2729 1.2930 1.3156 1.3410 1.3704 1.4049 1.4465 1.4984 1.5661 1.6602 1.8039 2.0648 17.592

–100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 80 100

0.28879 0.82377 2.0645 4.6422 9.5257 18.087 32.140 53.960 86.272 132.23 195.36 279.57 389.08 528.42 702.45 916.39 1175.9 1858.3 2821.6

1803.4 1774.7 1745.7 1716.0 1685.8 1654.8 1623.1 1590.5 1556.9 1522.1 1486.0 1448.2 1408.4 1366.2 1320.9 1271.4 1216.5 1078.2 786.76

164.26 160.44 156.67 152.92 149.15 145.35 141.46 137.47 133.33 129.00 124.44 119.58 114.37 108.72 102.52 95.639 87.846 67.813 26.802

0.95223 1.8003 0.96521 1.4397 0.97846 1.1758 0.99209 0.97631 1.0062 0.82125 1.0211 0.69788 1.0369 0.59775 1.0539 0.51512 1.0722 0.44597 1.0921 0.38741 1.1141 0.33731 1.1386 0.29406 1.1663 0.25640 1.1983 0.22334 1.2364 0.19403 1.2837 0.16778 1.3463 0.14393 1.5960 0.10076 7.1411 0.051620

T °C

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

R134a 1.8824 1.0203 0.80910 0.66051 0.55089 0.46703 0.40095 0.34758 0.30355 0.26653 0.23487 0.20737 0.18313 0.16145 0.14177 0.12361 0.10651 0.089846 0.047429

1.1896 0.66728 0.53873 0.44800 0.38089 0.32943 0.28879 0.25590 0.22873 0.20585 0.18626 0.16923 0.15422 0.14079 0.12861 0.11741 0.10691 0.096791 0.072835

0.14323 0.076431 0.13154 0.071806 0.12603 0.069373 0.12071 0.066943 0.11557 0.064541 0.11059 0.062175 0.10576 0.059846 0.10107 0.057547 0.096491 0.055267 0.092013 0.052992 0.087618 0.050705 0.083284 0.048381 0.078992 0.045989 0.074716 0.043483 0.070427 0.040795 0.066091 0.037811 0.061672 0.034316 0.057147 0.029815 0.058884 0.0051403

15.564 9.2927 7.7656 6.6923 5.9016 5.2983 4.8255 4.4469 4.1386 3.8845 3.6734 3.4979 3.3533 3.2378 3.1527 3.1051 3.1153 3.2464 14.169

–100 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 100

0.99829 0.81121 0.67357 0.56894 0.48717 0.42172 0.36827 0.32387 0.28645 0.25452 0.22699 0.20305 0.18205 0.16347 0.14690 0.13196 0.11832 0.093454 0.065611

0.090519 0.052712 0.088723 0.051794 0.086772 0.050802 0.084685 0.049743 0.082480 0.048623 0.080159 0.047438 0.077740 0.046191 0.075236 0.044886 0.072659 0.043527 0.070021 0.042121 0.067335 0.040672 0.064613 0.039186 0.061868 0.037665 0.059114 0.036109 0.056364 0.034514 0.053634 0.032861 0.050941 0.031104 0.045838 0.026638 0.048251 0.0085882

18.939 15.662 13.259 11.438 10.019 8.8900 7.9729 7.2154 6.5808 6.0425 5.5810 5.1817 4.8334 4.5271 4.2563 4.0158 3.8039 3.5083 7.6397

–100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 80 100

R227ea

356

Appendix A

494

Appendix B

Table A.17 Thermophysical Properties of Ethylene APPENDIX B.4 Glycol–Water Solutions (SI Units) Thermophysical Properties of Ethylene Glycol–Water Solutions (SI Units) Conc mass %

Tfreeze °C

T °C

ρ kg/m3

cp kJ/kg-K

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

10

–3.357

20

–7.949

30

–14.58

40

–23.81

0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180

1013.6 1010.6 1004.3 995.23 983.55 969.62 953.75 936.25 917.44 897.63 1029.1 1024.1 1016.4 1006.2 993.76 979.32 963.11 945.40 926.42 906.43 1045.0 1038.0 1028.8 1017.5 1004.3 989.41 973.15 955.71 937.30 918.16 1060.4 1051.9 1041.4 1029.1 1015.4 1000.5 984.55 967.93 950.85 933.57

4.0374 4.0459 4.0619 4.0831 4.1071 4.1316 4.1543 4.1727 4.1847 4.1878 3.8608 3.8962 3.9328 3.9693 4.0048 4.0379 4.0675 4.0926 4.1118 4.1240 3.6581 3.7183 3.7754 3.8287 3.8777 3.9215 3.9596 3.9913 4.0158 4.0326 3.4342 3.5190 3.5978 3.6697 3.7338 3.7893 3.8353 3.8708 3.8952 3.9073

2.3449 1.2745 0.80135 0.56338 0.42808 0.33982 0.27241 0.21315 0.15735 0.10594 3.1776 1.6624 1.0133 0.69688 0.52381 0.41676 0.33995 0.27534 0.21448 0.15563 4.2976 2.1664 1.2856 0.86604 0.63884 0.49766 0.39488 0.30780 0.22732 0.15342 5.8107 2.8191 1.6351 1.0821 0.77997 0.58440 0.43445 0.30586 0.19464 0.10686

2.3134 1.2612 0.79789 0.56608 0.43524 0.35047 0.28562 0.22766 0.17151 0.11802 3.0878 1.6233 0.99691 0.69258 0.52710 0.42556 0.35297 0.29124 0.23152 0.17169 4.1126 2.0870 1.2496 0.85118 0.63613 0.50299 0.40577 0.32207 0.24253 0.16709 5.4798 2.6801 1.5702 1.0515 0.76814 0.58413 0.44126 0.31599 0.20470 0.11447

0.52328 0.55289 0.57833 0.59961 0.61678 0.62987 0.63890 0.64390 0.64490 0.64194 0.48414 0.50766 0.52908 0.54832 0.56533 0.58004 0.59241 0.60235 0.60982 0.61475 0.44592 0.46490 0.48303 0.50018 0.51624 0.53106 0.54452 0.55649 0.56684 0.57544 0.40990 0.42530 0.44050 0.45535 0.46966 0.48326 0.49598 0.50764 0.51806 0.52708

0.12787 0.13523 0.14176 0.14756 0.15269 0.15723 0.16125 0.16482 0.16798 0.17077 0.12186 0.12723 0.13236 0.13729 0.14205 0.14668 0.15122 0.15568 0.16009 0.16445 0.11665 0.12045 0.12436 0.12840 0.13257 0.13687 0.14131 0.14589 0.15059 0.15542 0.11256 0.11490 0.11757 0.12057 0.12388 0.12748 0.13135 0.13549 0.13988 0.14449

18.092 9.3265 5.6283 3.8363 2.8505 2.2290 1.7712 1.3813 1.0211 0.69113 25.340 12.759 7.5318 5.0448 3.7106 2.9012 2.3341 1.8708 1.4462 1.0440 35.255 17.327 10.048 6.6293 4.7986 3.6749 2.8715 2.2076 1.6105 1.0751 48.682 23.326 13.355 8.7209 6.2008 4.5823 3.3594 2.3322 1.4634 0.79220

0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180

357

Appendix A 496

Appendix B

Table A.18B.4 (CONTINUED) APPENDIX Thermophysical Properties of Propylene Glycol–Water Solutions (SI Units) Thermophysical Properties of Propylene Glycol–Water Solutions (SI Units) Conc mass %

T freeze °C

T °C

ρ kg/m3

cp kJ/kg-K

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

10

–2.867

20

–7.173

30

–12.79

40

–20.57

0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180

1009.3 1006.2 999.42 989.81 978.22 965.50 952.50 940.08 929.07 920.33 1020.1 1014.8 1006.4 995.59 983.18 969.86 956.38 943.47 931.86 922.28 1031.6 1023.8 1013.4 1001.2 987.60 973.42 959.27 945.79 933.65 923.48 1042.4 1032.3 1020.1 1006.3 991.63 976.62 961.88 948.00 935.59 925.25

4.0590 4.0763 4.1011 4.1303 4.1610 4.1901 4.2147 4.2317 4.2380 4.2308 3.9359 3.9768 4.0190 4.0612 4.1016 4.1389 4.1714 4.1976 4.2159 4.2248 3.8026 3.8570 3.9104 3.9623 4.0122 4.0595 4.1036 4.1441 4.1803 4.2118 3.6416 3.7067 3.7708 3.8339 3.8958 3.9567 4.0164 4.0749 4.1324 4.1886

2.7446 1.4323 0.88380 0.61366 0.45625 0.34567 0.25395 0.17216 0.10249 0.050990 4.3124 2.0301 1.1693 0.77776 0.56382 0.42044 0.30437 0.20190 0.11581 0.054220 7.1171 2.9650 1.5731 0.99458 0.70106 0.51549 0.36996 0.24246 0.13577 0.060780 11.879 4.3838 2.1408 1.2827 0.87425 0.62846 0.44177 0.28154 0.15082 0.062967

2.7192 1.4234 0.88431 0.61998 0.46641 0.35802 0.26661 0.18314 0.11032 0.055403 4.2275 2.0005 1.1619 0.78120 0.57347 0.43350 0.31826 0.21400 0.12428 0.058789 6.8994 2.8961 1.5523 0.99343 0.70987 0.52957 0.38567 0.25636 0.14542 0.065816 11.397 4.2467 2.0987 1.2746 0.88163 0.64351 0.45928 0.29698 0.16120 0.068054

0.51549 0.54381 0.56879 0.59018 0.60775 0.62128 0.63053 0.63528 0.63528 0.63031 0.47112 0.49221 0.51228 0.53100 0.54808 0.56320 0.57607 0.58636 0.59376 0.59798 0.42845 0.44443 0.46056 0.47657 0.49215 0.50703 0.52089 0.53346 0.54445 0.55356 0.38784 0.40026 0.41321 0.42651 0.43998 0.45345 0.46673 0.47964 0.49201 0.50367

0.12583 0.13258 0.13877 0.14436 0.14931 0.15357 0.15706 0.15969 0.16134 0.16188 0.11734 0.12197 0.12666 0.13133 0.13591 0.14030 0.14440 0.14806 0.15114 0.15347 0.10923 0.11255 0.11622 0.12014 0.12420 0.12831 0.13232 0.13611 0.13950 0.14232 0.10218 0.10461 0.10743 0.11055 0.11389 0.11735 0.12081 0.12416 0.12726 0.12996

21.611 10.736 6.3724 4.2947 3.1238 2.3313 1.6975 1.1468 0.68374 0.34226 36.028 16.402 9.1737 5.9484 4.2195 3.0897 2.2040 1.4453 0.82230 0.38307 63.166 25.732 13.357 8.2692 5.7153 4.1273 2.9145 1.8835 1.0425 0.46245 111.54 40.597 19.536 11.530 7.7411 5.4838 3.8016 2.3919 1.2667 0.52365

0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180

358

Appendix A

498

Appendix B

Table A.19B.4 (CONTINUED) APPENDIX Thermophysical Properties of Methanol–Water Solutions (SI Units) Thermophysical Properties of Methanol–Water Solutions (SI Units) Conc mass %

T freeze °C

T °C

ρ kg/m3

cp kJ/kg-K

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

10

–6.548

20

–15.09

30

–25.69

40

–38.70

–5 0 5 10 15 20 25 30 35 40 –15 –10 –5 0 5 10 15 20 30 40 –25 –20 –15 –10 –5 0 10 20 30 40 –35 –30 –25 –20 –10 0 10 20 30 40

984.45 984.39 984.05 983.46 982.62 981.54 980.23 978.70 976.96 975.02 974.91 974.30 973.51 972.52 971.35 970.00 968.48 966.79 962.90 958.37 968.53 967.11 965.56 963.90 962.12 960.22 956.08 951.48 946.42 940.90 962.74 960.50 958.19 955.83 950.91 945.73 940.29 934.57 928.56 922.27

4.2513 4.2408 4.2301 4.2192 4.2083 4.1975 4.1871 4.1771 4.1677 4.1590 4.0064 4.0296 4.0502 4.0681 4.0832 4.0953 4.1043 4.1101 4.1114 4.0980 3.6331 3.6871 3.7381 3.7859 3.8301 3.8705 3.9382 3.9865 4.0127 4.0144 3.2421 3.3122 3.3800 3.4451 3.5659 3.6712 3.7578 3.8224 3.8618 3.8728

3.1070 2.5348 2.1026 1.7709 1.5122 1.3073 1.1426 1.0083 0.89694 0.80327 6.4493 5.0298 3.9961 3.2301 2.6531 2.2115 1.8684 1.5979 1.2063 0.94179 12.405 9.3562 7.1951 5.6353 4.4897 3.6346 2.4870 1.7888 1.3399 1.0355 20.860 15.400 11.597 8.8989 5.5182 3.6395 2.5306 1.8385 1.3834 1.0684

3.1561 2.5750 2.1367 1.8007 1.5389 1.3319 1.1657 1.0302 0.91809 0.82385 6.6153 5.1625 4.1048 3.3214 2.7313 2.2798 1.9292 1.6528 1.2527 0.98270 12.808 9.6744 7.4518 5.8463 4.6665 3.7851 2.6013 1.8800 1.4157 1.1006 21.668 16.034 12.103 9.3101 5.8030 3.8484 2.6913 1.9673 1.4898 1.1585

0.50195 0.50955 0.51700 0.52430 0.53144 0.53840 0.54519 0.55179 0.55819 0.56439 0.44563 0.45117 0.45665 0.46209 0.46748 0.47283 0.47813 0.48339 0.49378 0.50402 0.39857 0.40231 0.40604 0.40975 0.41345 0.41715 0.42458 0.43210 0.43976 0.44762 0.35960 0.36196 0.36430 0.36663 0.37127 0.37595 0.38074 0.38571 0.39093 0.39646

0.11993 0.12206 0.12420 0.12636 0.12852 0.13068 0.13283 0.13497 0.13709 0.13918 0.11409 0.11492 0.11582 0.11680 0.11787 0.11903 0.12029 0.12165 0.12473 0.12833 0.11327 0.11283 0.11249 0.11228 0.11220 0.11224 0.11276 0.11392 0.11580 0.11851 0.11521 0.11378 0.11249 0.11134 0.10949 0.10828 0.10775 0.10797 0.10902 0.11100

26.315 21.097 17.204 14.251 11.974 10.192 8.7754 7.6326 6.6969 5.9193 57.983 44.923 35.442 28.437 23.173 19.154 16.038 13.586 10.044 7.6574 113.08 85.747 66.241 52.068 41.593 33.723 23.068 16.503 12.226 9.2868 188.07 140.92 107.60 83.622 53.001 35.541 24.976 18.220 13.666 10.437

–5 0 5 10 15 20 25 30 35 40 –15 –10 –5 0 5 10 15 20 30 40 –25 –20 –15 –10 –5 0 10 20 30 40 –35 –30 –25 –20 –10 0 10 20 30 40

359

Appendix A 500

Appendix B

Table A.20B.4 (CONTINUED) APPENDIX Thermophysical Properties of Ethanol–Water Solutions (SI Units) Thermophysical Properties of Ethanol–Water Solutions (SI Units) Conc mass %

T freeze °C

T °C

ρ kg/m3

cp kJ/kg-K

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

10

–4.388

0 5

984.99 984.67

4.4017 4.3764

3.3170 2.6376

3.3676 2.6787

0.50270 0.50927

0.11595 0.11818

29.044 22.667

0 5

10 15 20 25 30

984.02 983.07 981.84 980.37 978.67

4.3506 4.3257 4.3033 4.2850 4.2722

2.1531 1.7975 1.5287 1.3194 1.1513

2.1881 1.8284 1.5570 1.3459 1.1764

0.51577 0.52219 0.52852 0.53476 0.54089

0.12048 0.12280 0.12509 0.12730 0.12937

18.162 14.890 12.447 10.572 9.0939

10 15 20 25 30

35 40

976.77 974.70

4.2666 4.2696

1.0118 0.89217

1.0359 0.91532

0.54689 0.55276

0.13123 0.13282

7.8939 6.8914

35 40

45 –10 –5 0

972.49 977.60 976.83 975.77

4.2829 4.3827 4.3774 4.3694

0.78623 9.5644 6.9767 5.2493

0.80846 9.7836 7.1422 5.3796

0.55848 0.43871 0.44324 0.44782

0.13408 0.10240 0.10366 0.10504

6.0295 95.547 68.902 51.217

45 –10 –5 0

20

30

40

–11.13

–20.14

–29.53

5

974.42

4.3595

4.0621

4.1687

0.45243

0.10650

39.142

5

10 15 20 25 30

972.82 970.98 968.92 966.65 964.20

4.3488 4.3382 4.3287 4.3211 4.3165

3.2237 2.6161 2.1648 1.8212 1.5534

3.3137 2.6943 2.2342 1.8841 1.6111

0.45706 0.46169 0.46631 0.47091 0.47546

0.10803 0.10960 0.11118 0.11274 0.11424

30.673 24.582 20.095 16.712 14.102

10 15 20 25 30

40

958.81

4.3200

1.1640

1.2140

0.48440

0.11695

10.381

40

–20 –15 –10 –5

973.10 971.36 969.42 967.28

4.0715 4.0998 4.1242 4.1451

25.213 17.304 12.258 8.9429

25.910 17.814 12.644 9.2453

0.38499 0.38802 0.39113 0.39431

0.097171 0.097434 0.097830 0.098343

266.65 182.83 129.25 94.012

–20 –15 –10 –5

0 5

964.96 962.46

4.1632 4.1789

6.7046 5.1538

6.9480 5.3548

0.39754 0.40081

0.098956 0.099655

70.213 53.733

0 5

10 20 30

959.79 954.01 947.72

4.1926 4.2161 4.2376

4.0530 2.6599 1.8618

4.2227 2.7881 1.9645

0.40411 0.41071 0.41722

0.10042 0.10211 0.10389

42.049 27.305 18.910

10 20 30

40 –25 –20 –15 –10 –5 0

940.99 965.29 962.30 959.22 956.04 952.78 949.43

4.2610 3.6450 3.7020 3.7551 3.8045 3.8503 3.8926

1.3652 35.834 24.550 17.312 12.542 9.3164 7.0833

1.4508 37.122 25.512 18.048 13.118 9.7782 7.4605

0.42353 0.34154 0.34361 0.34574 0.34791 0.35012 0.35235

0.10563 0.097071 0.096455 0.095987 0.095653 0.095441 0.095339

13.735 382.43 264.50 188.03 137.14 102.45 78.252

40 –25 –20 –15 –10 –5 0

10 20 30 40

942.51 935.32 927.89 920.26

3.9672 4.0293 4.0799 4.1199

4.3572 2.8757 2.0062 1.4573

4.6229 3.0746 2.1621 1.5836

0.35681 0.36120 0.36541 0.36934

0.095427 0.095843 0.096525 0.097417

48.445 32.080 22.399 16.256

10 20 30 40

360

Appendix A

502

Appendix B

Table A.21B.4 (CONTINUED) APPENDIX Thermophysical Properties of Calcium Chloride–Water Solutions (SI Units) Thermophysical Properties of Calcium Chloride–Water Solutions (SI Units) Conc mass %

T freeze °C

T °C

ρ kg/m3

cp kJ/kg-K

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

5

–2.361

10

–5.840

15

–11.05

20

–18.26

0 5 10 15 20 25 30 35 40 45 –5 0 5 10 15 20 25 30 35 40 –10 –5 0 5 10 15 20 25 30 40 –15 –10 –5 0 5 10 15 20 30 40

1043.1 1041.7 1040.8 1040.2 1039.6 1038.8 1037.4 1035.2 1031.9 1027.2 1089.8 1088.3 1087.4 1086.8 1086.4 1085.9 1085.0 1083.6 1081.4 1078.1 1137.8 1136.3 1135.4 1134.8 1134.4 1133.9 1133.2 1132.1 1130.4 1124.6 1188.4 1186.9 1185.7 1184.8 1183.9 1183.0 1181.9 1180.6 1176.5 1169.8

3.8720 3.8734 3.8764 3.8809 3.8866 3.8932 3.9005 3.9083 3.9164 3.9244 3.5602 3.5688 3.5781 3.5879 3.5981 3.6086 3.6192 3.6298 3.6403 3.6505 3.2651 3.2792 3.2929 3.3063 3.3193 3.3319 3.3440 3.3556 3.3667 3.3868 3.0056 3.0232 3.0399 3.0558 3.0709 3.0851 3.0985 3.1110 3.1336 3.1527

1.9790 1.6917 1.4659 1.2855 1.1392 1.0185 0.91727 0.83087 0.75577 0.68928 2.6193 2.2273 1.9163 1.6668 1.4642 1.2981 1.1603 1.0447 0.94679 0.86283 3.5905 3.0425 2.6052 2.2529 1.9664 1.7315 1.5371 1.3749 1.2384 1.0238 5.1425 4.3363 3.6963 3.1830 2.7673 2.4273 2.1468 1.9131 1.5504 1.2852

1.8971 1.6240 1.4084 1.2358 1.0958 0.98048 0.88422 0.80265 0.73243 0.67102 2.4034 2.0466 1.7623 1.5336 1.3478 1.1954 1.0693 0.96412 0.87553 0.80030 3.1557 2.6774 2.2945 1.9852 1.7335 1.5270 1.3564 1.2145 1.0956 0.91041 4.3271 3.6534 3.1173 2.6866 2.3374 2.0519 1.8164 1.6205 1.3178 1.0986

0.55904 0.56753 0.57601 0.58444 0.59281 0.60108 0.60924 0.61726 0.62512 0.63280 0.54737 0.55523 0.56312 0.57102 0.57894 0.58687 0.59480 0.60273 0.61065 0.61857 0.53586 0.54333 0.55080 0.55829 0.56581 0.57335 0.58093 0.58856 0.59623 0.61175 0.52391 0.53117 0.53841 0.54564 0.55287 0.56011 0.56737 0.57466 0.58936 0.60431

0.13841 0.14066 0.14276 0.14477 0.14671 0.14863 0.15057 0.15257 0.15469 0.15698 0.14108 0.14296 0.14473 0.14644 0.14811 0.14977 0.15147 0.15324 0.15512 0.15716 0.14424 0.14581 0.14732 0.14880 0.15027 0.15176 0.15330 0.15493 0.15667 0.16062 0.14667 0.14803 0.14937 0.15071 0.15207 0.15347 0.15493 0.15647 0.15986 0.16385

13.707 11.546 9.8652 8.5364 7.4687 6.5967 5.8725 5.2608 4.7349 4.2747 17.036 14.316 12.176 10.473 9.1002 7.9817 7.0599 6.2916 5.6441 5.0921 21.878 18.362 15.575 13.342 11.536 10.062 8.8479 7.8389 6.9930 5.6681 29.502 24.680 20.870 17.826 15.371 13.370 11.724 10.357 8.2431 6.7047

0 5 10 15 20 25 30 35 40 45 –5 0 5 10 15 20 25 30 35 40 –10 –5 0 5 10 15 20 25 30 40 –15 –10 –5 0 5 10 15 20 30 40

361

Appendix A 504

Appendix B

Table A.22 APPENDIX B.4 (CONTINUED) Thermophysical Properties of Magnesium Chloride–Water Solutions (SI Thermophysical Properties of Magnesium Chloride–Water Solutions (SI Units) Units) Conc mass %

T freeze °C

T °C

ρ kg/m3

cp kJ/kg-K

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

5

–2.999

10

–8.283

15

–16.79

20

–28.71

0 5 10 15 20 25 30 35 40 45 –5 0 5 10 15 20 25 30 35 40 –15 –10 –5 0 5 10 15 20 30 40 –25 –20 –15 –10 –5 0 10 20 30 40

1043.8 1043.2 1042.3 1041.3 1040.0 1038.6 1037.0 1035.2 1033.5 1031.6 1089.9 1089.0 1087.9 1086.6 1085.1 1083.5 1081.7 1079.7 1077.6 1075.4 1137.4 1136.6 1135.7 1134.5 1133.2 1131.7 1130.0 1128.2 1124.1 1119.6 1187.5 1187.1 1186.3 1185.3 1184.0 1182.5 1178.9 1174.9 1170.6 1166.4

3.8744 3.8786 3.8833 3.8885 3.8939 3.8993 3.9047 3.9098 3.9145 3.9185 3.5633 3.5731 3.5826 3.5917 3.6004 3.6088 3.6169 3.6247 3.6323 3.6395 3.2601 3.2732 3.2858 3.2981 3.3100 3.3218 3.3333 3.3448 3.3677 3.3910 2.9861 2.9996 3.0133 3.0269 3.0407 3.0545 3.0822 3.1100 3.1376 3.1651

2.2022 1.8527 1.5858 1.3774 1.2107 1.0741 0.95932 0.86025 0.77246 0.69275 3.3696 2.7921 2.3499 2.0031 1.7246 1.4955 1.3025 1.1361 0.98975 0.85876 6.2748 5.1741 4.3202 3.6455 3.1027 2.6581 2.2877 1.9742 1.4700 1.0802 12.758 10.421 8.5696 7.0933 5.9089 4.9529 3.5428 2.5924 1.9379 1.4777

2.1099 1.7760 1.5214 1.3228 1.1641 1.0342 0.92513 0.83096 0.74746 0.67153 3.0916 2.5640 2.1601 1.8435 1.5893 1.3803 1.2041 1.0522 0.91844 0.79853 5.5168 4.5521 3.8042 3.2133 2.7380 2.3488 2.0245 1.7499 1.3077 0.96486 10.744 8.7787 7.2235 5.9844 4.9906 4.1885 3.0051 2.2065 1.6554 1.2669

0.55226 0.56160 0.57057 0.57921 0.58757 0.59570 0.60364 0.61145 0.61915 0.62682 0.53090 0.54025 0.54949 0.55860 0.56755 0.57633 0.58492 0.59330 0.60144 0.60933 0.50140 0.50986 0.51850 0.52726 0.53609 0.54495 0.55377 0.56251 0.57951 0.59555 0.47376 0.48133 0.48918 0.49727 0.50554 0.51395 0.53098 0.54796 0.56448 0.58015

0.13656 0.13880 0.14096 0.14305 0.14509 0.14710 0.14908 0.15106 0.15305 0.15506 0.13670 0.13884 0.14099 0.14313 0.14527 0.14740 0.14951 0.15159 0.15365 0.15568 0.13522 0.13704 0.13895 0.14091 0.14293 0.14497 0.14702 0.14907 0.15308 0.15686 0.13360 0.13517 0.13684 0.13860 0.14042 0.14229 0.14612 0.14996 0.15368 0.15714

15.450 12.795 10.793 9.2467 8.0231 7.0309 6.2054 5.5007 4.8837 4.3306 22.616 18.467 15.321 12.880 10.941 9.3643 8.0539 6.9410 5.9774 5.1294 40.799 33.217 27.378 22.803 19.157 16.203 13.771 11.739 8.5428 6.1509 80.416 64.945 52.787 43.178 35.541 29.436 20.565 14.713 10.771 8.0620

0 5 10 15 20 25 30 35 40 45 –5 0 5 10 15 20 25 30 35 40 –15 –10 –5 0 5 10 15 20 30 40 –25 –20 –15 –10 –5 0 10 20 30 40

362

Appendix A

506

Appendix B

Table A.23B.4 (CONTINUED) APPENDIX Thermophysical Properties of Chloride–Water Sodium Chloride–Water Solutions (SI Units) Thermophysical Properties of Sodium Solutions (SI Units) Conc mass %

T freeze °C

T °C

ρ kg/m3

cp kJ/kg-K

μ mPa-s

ν mm 2 /s

k W/m-K

α mm 2 /s

Pr

T °C

5

–3.054

10

–6.553

15

–10.90

20

–16.46

0 5 10 15 20 25 30 35 40 45 –5 0 5 10 15 20 25 30 35 40 –10 –5 0 5 10 15 20 25 30 40 –15 –10 –5 0 5 10 15 20 30 40

1038.1 1037.4 1036.4 1035.3 1034.0 1032.5 1030.7 1028.8 1026.7 1024.5 1078.0 1076.8 1075.4 1073.9 1072.3 1070.6 1068.7 1066.7 1064.6 1062.3 1119.6 1118.0 1116.2 1114.4 1112.5 1110.5 1108.5 1106.3 1104.0 1099.3 1162.7 1160.7 1158.7 1156.7 1154.5 1152.3 1150.1 1147.8 1142.9 1137.8

3.9121 3.9157 3.9196 3.9237 3.9281 3.9327 3.9376 3.9428 3.9483 3.9540 3.6780 3.6883 3.6979 3.7068 3.7150 3.7226 3.7294 3.7356 3.7411 3.7459 3.4952 3.5075 3.5188 3.5289 3.5380 3.5460 3.5529 3.5588 3.5635 3.5699 3.3533 3.3642 3.3742 3.3833 3.3915 3.3988 3.4052 3.4106 3.4189 3.4234

1.8918 1.6223 1.4046 1.2279 1.0837 0.96578 0.86897 0.78943 0.72409 0.67059 2.4367 2.0715 1.7784 1.5417 1.3498 1.1933 1.0654 0.96053 0.87452 0.80405 3.2795 2.7510 2.3318 1.9972 1.7286 1.5118 1.3360 1.1930 1.0766 0.90447 4.6494 3.8326 3.1960 2.6960 2.3006 1.9860 1.7342 1.5319 1.2375 1.0467

1.8223 1.5638 1.3552 1.1860 1.0481 0.93542 0.84306 0.76730 0.70523 0.65457 2.2604 1.9238 1.6537 1.4356 1.2587 1.1146 0.99689 0.90047 0.82148 0.75689 2.9292 2.4607 2.0890 1.7921 1.5537 1.3613 1.2053 1.0784 0.97510 0.82279 3.9989 3.3019 2.7582 2.3308 1.9927 1.7234 1.5079 1.3347 1.0827 0.91998

0.55832 0.56748 0.57641 0.58511 0.59357 0.60180 0.60980 0.61757 0.62511 0.63241 0.54719 0.55567 0.56406 0.57236 0.58058 0.58871 0.59675 0.60470 0.61256 0.62033 0.53646 0.54432 0.55219 0.56006 0.56793 0.57581 0.58370 0.59159 0.59948 0.61528 0.52535 0.53274 0.54019 0.54768 0.55521 0.56280 0.57043 0.57810 0.59359 0.60927

0.13747 0.13970 0.14189 0.14404 0.14615 0.14822 0.15025 0.15224 0.15420 0.15612 0.13801 0.13992 0.14184 0.14378 0.14574 0.14772 0.14972 0.15175 0.15381 0.15589 0.13709 0.13881 0.14059 0.14241 0.14429 0.14622 0.14821 0.15026 0.15237 0.15679 0.13475 0.13643 0.13816 0.13995 0.14179 0.14370 0.14566 0.14768 0.15191 0.15642

13.256 11.194 9.5512 8.2339 7.1718 6.3112 5.6111 5.0400 4.5735 4.1926 16.378 13.750 11.659 9.9847 8.6369 7.5457 6.6583 5.9339 5.3410 4.8553 21.367 17.727 14.859 12.584 10.768 9.3097 8.1320 7.1769 6.3995 5.2478 29.677 24.203 19.963 16.655 14.053 11.993 10.352 9.0379 7.1272 5.8816

0 5 10 15 20 25 30 35 40 45 –5 0 5 10 15 20 25 30 35 40 –10 –5 0 5 10 15 20 25 30 40 –15 –10 –5 0 5 10 15 20 30 40

363

Appendix A 508

Appendix B

Table A.24B.5 APPENDIX Thermophysical Properties of (SI Dry Air (SI Units) Thermophysical Properties of Dry Air Units) T °C

ρ kg/m3

v m3/kg

u kJ/kg

h kJ/kg

0 10 20 30 40 50 60 70 80 90 100 150 200 250 300 350 400 600 800

0.012751 0.012301 0.011881 0.011489 0.011122 0.010778 0.010454 0.010150 0.0098624 0.0095908 0.0093338 0.0082309 0.0073611 0.0066576 0.0060768 0.0055892 0.0051740 0.0039889 0.0032455

78.425 81.294 84.168 87.040 89.912 92.782 95.657 98.522 101.40 104.27 107.14 121.49 135.85 150.20 164.56 178.92 193.27 250.70 308.12

195.14 202.31 209.49 216.66 223.85 231.04 238.23 245.43 252.64 259.86 267.09 303.41 340.08 377.19 414.81 452.99 491.76 652.80 822.50

273.57 283.61 293.65 303.70 313.76 323.82 333.88 343.96 354.04 364.13 374.23 424.90 475.93 527.39 579.37 631.91 685.03 903.49 1130.6

0 10 20 30 40 50 60 70 80 90 100 150 200 250 300 350 400 600 800

1.2927 1.2469 1.2043 1.1644 1.1272 1.0922 1.0594 1.0284 0.99926 0.97170 0.94563 0.83378 0.74562 0.67433 0.61549 0.56611 0.52406 0.40403 0.32875

0.77357 0.80199 0.83036 0.85881 0.88715 0.91558 0.94393 0.97238 1.0007 1.0291 1.0575 1.1994 1.3412 1.4830 1.6247 1.7664 1.9082 2.4751 3.0418

194.91 202.09 209.28 216.46 223.66 230.85 238.06 245.27 252.48 259.71 266.94 303.28 339.97 377.09 414.73 452.92 491.70 652.75 822.46

273.29 283.35 293.41 303.48 313.55 323.62 333.70 343.79 353.88 363.98 374.09 424.81 475.86 527.35 579.35 631.90 685.04 903.54 1130.7

s cp kJ/kg-K kJ/kg-K

μ mPa-s

k W/m-K

Pr

T °C

0.017241 0.017740 0.018232 0.018717 0.019195 0.019667 0.020132 0.020592 0.021046 0.021494 0.021937 0.024076 0.026103 0.028034 0.029883 0.031658 0.033370 0.039708 0.045453

0.023970 0.024723 0.025467 0.026202 0.026929 0.027649 0.028361 0.029066 0.029763 0.030454 0.031139 0.034469 0.037664 0.040743 0.043721 0.046612 0.049425 0.060071 0.070012

0.72210 0.72064 0.71928 0.71803 0.71687 0.71581 0.71485 0.71397 0.71319 0.71250 0.71190 0.71010 0.71013 0.71163 0.71424 0.71760 0.72142 0.73722 0.74948

0 10 20 30 40 50 60 70 80 90 100 150 200 250 300 350 400 600 800

0.017258 0.017756 0.018247 0.018731 0.019209 0.019680 0.020145 0.020604 0.021058 0.021505 0.021948 0.024085 0.026111 0.028042 0.029889 0.031664 0.033375 0.039712 0.045456

0.024009 0.024760 0.025503 0.026237 0.026963 0.027681 0.028392 0.029096 0.029793 0.030483 0.031167 0.034494 0.037686 0.040763 0.043740 0.046629 0.049441 0.060083 0.070021

0.72307 0.72151 0.72007 0.71875 0.71752 0.71641 0.71539 0.71448 0.71366 0.71293 0.71229 0.71038 0.71033 0.71177 0.71434 0.71767 0.72147 0.73723 0.74948

0 10 20 30 40 50 60 70 80 90 100 150 200 250 300 350 400 600 800

P = 1 kPa (0.001 MPa) 8.0991 8.1352 8.1700 8.2037 8.2364 8.2680 8.2987 8.3285 8.3574 8.3856 8.4130 8.5405 8.6544 8.7578 8.8527 8.9406 9.0226 9.3065 9.5405

1.0039 1.0043 1.0047 1.0052 1.0057 1.0063 1.0070 1.0078 1.0086 1.0095 1.0105 1.0167 1.0247 1.0342 1.0450 1.0566 1.0685 1.1153 1.1544

P = 101.325 kPa (1 atm) 6.7722 6.8084 6.8433 6.8771 6.9098 6.9414 6.9721 7.0020 7.0310 7.0592 7.0866 7.2142 7.3282 7.4317 7.5266 7.6145 7.6965 7.9804 8.2145

1.0059 1.0061 1.0064 1.0067 1.0072 1.0077 1.0083 1.0089 1.0097 1.0105 1.0115 1.0174 1.0252 1.0347 1.0454 1.0568 1.0688 1.1154 1.1545

364

Appendix A 509

Appendix B

Table A.25 APPENDIX B.5 (CONTINUED) Thermophysical Properties of (SI Dry Air (SI Units) Thermophysical Properties of Dry Air Units) T °C

ρ kg/m3

v m3/kg

u kJ/kg

h kJ/kg

0 10 20 30 40 50 60 70 80 90 100 150 200 250 300 350 400 600 800

6.3933 6.1638 5.9505 5.7516 5.5658 5.3917 5.2283 5.0746 4.9298 4.7931 4.6638 4.1101 3.6745 3.3228 3.0326 2.7892 2.5821 1.9910 1.6202

0.15641 0.16224 0.16805 0.17386 0.17967 0.18547 0.19127 0.19706 0.20285 0.20863 0.21442 0.24330 0.27215 0.30095 0.32975 0.35853 0.38728 0.50226 0.61721

193.99 201.22 208.44 215.67 222.90 230.12 237.36 244.59 251.84 259.09 266.35 302.78 339.54 376.72 414.40 452.63 491.44 652.58 822.34

272.20 282.34 292.47 302.60 312.73 322.86 332.99 343.12 353.26 363.41 373.55 424.43 475.61 527.20 579.27 631.89 685.08 903.72 1130.9

0 10 20 30 40 50 60 70 80 90 100 150 200 250 300 350 400 600 800

12.821 12.353 11.920 11.516 11.140 10.788 10.458 10.148 9.8561 9.5808 9.3208 8.2092 7.3368 6.6333 6.0537 5.5677 5.1542 3.9749 3.2354

0.077997 0.080952 0.083893 0.086836 0.089767 0.092696 0.095621 0.098542 0.10146 0.10438 0.10729 0.12181 0.13630 0.15075 0.16519 0.17961 0.19402 0.25158 0.30908

192.84 200.12 207.40 214.67 221.94 229.21 236.48 243.76 251.03 258.31 265.60 302.15 339.00 376.25 413.99 452.26 491.11 652.36 822.19

270.84 281.07 291.29 301.50 311.71 321.91 332.10 342.30 352.49 362.69 372.88 423.97 475.30 527.01 579.18 631.87 685.13 903.94 1131.3

s kJ/ kg-K

cp kJ/ kg-K

μ mPa-s

k W/m-K

Pr

T °C

0.017324 0.017820 0.018308 0.018790 0.019265 0.019734 0.020198 0.020655 0.021106 0.021553 0.021994 0.024124 0.026146 0.028072 0.029916 0.031688 0.033397 0.039728 0.045468

0.024169 0.024915 0.025651 0.026380 0.027102 0.027816 0.028522 0.029222 0.029915 0.030602 0.031282 0.034595 0.037777 0.040844 0.043814 0.046697 0.049504 0.060131 0.070060

0.72679 0.72487 0.72312 0.72151 0.72004 0.71871 0.71750 0.71641 0.71543 0.71457 0.71380 0.71141 0.71106 0.71230 0.71472 0.71796 0.72168 0.73728 0.74947

0 10 20 30 40 50 60 70 80 90 100 150 200 250 300 350 400 600 800

0.017412 0.017904 0.018388 0.018867 0.019339 0.019806 0.020266 0.020721 0.021170 0.021614 0.022053 0.024175 0.026190 0.028111 0.029951 0.031720 0.033426 0.039749 0.045484

0.024386 0.025122 0.025851 0.026572 0.027287 0.027994 0.028695 0.029389 0.030077 0.030759 0.031434 0.034728 0.037894 0.040950 0.043910 0.046784 0.049585 0.060192 0.070110

0.73125 0.72888 0.72674 0.72479 0.72303 0.72143 0.71999 0.71870 0.71754 0.71650 0.71559 0.71264 0.71192 0.71292 0.71517 0.71828 0.72192 0.73733 0.74945

0 10 20 30 40 50 60 70 80 90 100 150 200 250 300 350 400 600 800

P = 500 kPa (0.05 MPa) 6.3106 6.3470 6.3822 6.4162 6.4490 6.4809 6.5117 6.5417 6.5708 6.5992 6.6267 6.7547 6.8690 6.9726 7.0677 7.1557 7.2378 7.5219 7.7561

1.0140 1.0135 1.0132 1.0130 1.0129 1.0130 1.0132 1.0136 1.0140 1.0146 1.0153 1.0202 1.0274 1.0364 1.0467 1.0580 1.0697 1.1159 1.1548

P = l000 kPa (1 MPa) 6.1073 6.1441 6.1796 6.2139 6.2470 6.2790 6.3101 6.3403 6.3696 6.3980 6.4257 6.5542 6.6688 6.7727 6.8680 6.9561 7.0383 7.3227 7.5569

1.0241 1.0228 1.0217 1.0208 1.0202 1.0197 1.0195 1.0194 1.0194 1.0196 1.0200 1.0237 1.0301 1.0385 1.0485 1.0594 1.0709 1.1166 1.1552

365

Appendix A 510

Appendix B

Table A.26B.5 (CONTINUED) APPENDIX Thermophysical Properties of (SI Dry Air (SI Units) Thermophysical Properties of Dry Air Units) T °C

ρ kg/m3

v m /kg

u kJ/kg

h kJ/kg

0 10 20 30 40 50 60 70 80 90 100 150 200 250 300 350 400 600 800

65.195 62.539 60.116 57.893 55.844 53.949 52.189 50.550 49.018 47.583 46.235 40.552 36.165 32.663 29.796 27.403 25.372 19.597 15.978

0.015339 0.015990 0.016635 0.017273 0.017907 0.018536 0.019161 0.019782 0.020401 0.021016 0.021629 0.024660 0.027651 0.030616 0.033562 0.036492 0.039414 0.051028 0.062586

183.56 191.34 199.06 206.73 214.37 221.98 229.56 237.12 244.66 252.19 259.71 297.22 334.78 372.59 410.78 449.42 488.57 650.67 821.01

260.26 271.29 282.23 293.10 303.91 314.66 325.37 336.03 346.67 357.27 367.85 420.51 473.04 525.67 578.58 631.88 685.64 905.81 1133.9

0 10 20 30 40 50 60 70 80 90 100 150 200 250 300 350 400 600 800

131.34 125.47 120.19 115.40 111.04 107.04 103.35 99.949 96.788 93.844 91.093 79.629 70.903 64.000 58.380 53.704 49.746 38.510 31.469

0.0076138 0.0079700 0.0083202 0.0086655 0.0090058 0.0093423 0.0096759 0.010005 0.010332 0.010656 0.010978 0.012558 0.014104 0.015625 0.017129 0.018621 0.020102 0.025967 0.031777

172.15 180.58 188.89 197.09 205.20 213.23 221.21 229.12 237.00 244.83 252.63 291.30 329.73 368.21 406.93 446.00 485.52 648.63 819.57

248.29 260.28 272.09 283.74 295.26 306.66 317.96 329.17 340.31 351.39 362.41 416.88 470.77 524.46 578.22 632.21 686.55 908.30 1137.3

3

s kJ/ kg-K

cp kJ/ kg-K

μ mPa-s

k W/m-K

Pr

T °C

0.018284 0.018731 0.019175 0.019616 0.020054 0.020489 0.020921 0.021349 0.021773 0.022194 0.022612 0.024645 0.026594 0.028464 0.030263 0.031999 0.033678 0.039929 0.045621

0.026560 0.027191 0.027824 0.028458 0.029093 0.029727 0.030361 0.030993 0.031623 0.032251 0.032877 0.035963 0.038974 0.041910 0.044772 0.047568 0.050302 0.060729 0.070539

0.76270 0.75670 0.75150 0.74695 0.74295 0.73942 0.73629 0.73351 0.73104 0.72885 0.72691 0.72016 0.71708 0.71652 0.71771 0.72006 0.72314 0.73743 0.74915

0 10 20 30 40 50 60 70 80 90 100 150 200 250 300 350 400 600 800

0.019810 0.020156 0.020512 0.020876 0.021245 0.021618 0.021994 0.022372 0.022750 0.023129 0.023507 0.025385 0.027221 0.029006 0.030739 0.032422 0.034058 0.040193 0.045819

0.029882 0.030327 0.030797 0.031287 0.031791 0.032307 0.032833 0.033367 0.033906 0.034451 0.034999 0.037767 0.040543 0.043296 0.046013 0.048690 0.051326 0.061484 0.071133

0.80213 0.79062 0.78091 0.77264 0.76552 0.75937 0.75400 0.74931 0.74519 0.74156 0.73835 0.72719 0.72156 0.71940 0.71953 0.72116 0.72373 0.73704 0.74847

0 10 20 30 40 50 60 70 80 90 100 150 200 250 300 350 400 600 800

P = 5000 kPa (5 MPa) 5.6122 5.6519 5.6899 5.7263 5.7614 5.7952 5.8278 5.8594 5.8899 5.9195 5.9483 6.0807 6.1980 6.3038 6.4004 6.4895 6.5725 6.8586 7.0937

1.1079 1.0985 1.0905 1.0836 1.0778 1.0728 1.0685 1.0649 1.0618 1.0591 1.0569 1.0509 1.0509 1.0550 1.0618 1.0704 1.0801 1.1216 1.1583

P = 10,000 kPa (10 MPa) 5.3747 5.4178 5.4588 5.4979 5.5352 5.5711 5.6055 5.6387 5.6707 5.7016 5.7315 5.8686 5.9889 6.0968 6.1949 6.2853 6.3691 6.6574 6.8934

1.2099 1.1896 1.1725 1.1579 1.1455 1.1348 1.1256 1.1176 1.1106 1.1046 1.0993 1.0819 1.0747 1.0738 1.0771 1.0830 1.0907 1.1275 1.1620

366

Appendix A

514

Appendix B

Table A.27B.6 APPENDIX Thermophysical Properties of Saturated R134a (SI Units) Thermophysical Properties of Saturated R134a (SI Units) T °C –103.3 –95 –90 –85 –80 –75 –70 –65 –60 –55 –50 –45 –40 –35 –30 –25 –20 –15 –10 –5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 101.06

P kPa

ρf kg/m3

vg m3/kg

uf kJ/kg

ug kJ/kg

hf kJ/kg

hg kJ/kg

sf kJ/kg-K

sg kJ/kg-K

T °C

0.38956 0.93899 1.5241 2.3990 3.6719 5.4777 7.9814 11.380 15.906 21.828 29.451 39.117 51.209 66.144 84.378 106.40 132.73 163.94 200.60 243.34 292.80 349.66 414.61 488.37 571.71 665.38 770.20 886.98 1016.6 1159.9 1317.9 1491.5 1681.8 1889.8 2116.8 2364.1 2633.2 2925.8 3244.2 4059.3

1591.1 1569.1 1555.8 1542.5 1529.0 1515.5 1501.9 1488.2 1474.3 1460.4 1446.3 1432.1 1417.7 1403.1 1388.4 1373.4 1358.3 1342.8 1327.1 1311.1 1294.8 1278.1 1261.0 1243.4 1225.3 1206.7 1187.5 1167.5 1146.7 1125.1 1102.3 1078.3 1052.9 1025.6 996.25 964.09 928.24 887.16 837.83 511.90

35.496 15.435 9.7698 6.3707 4.2682 2.9312 2.0590 1.4765 1.0790 0.80236 0.60620 0.46473 0.36108 0.28402 0.22594 0.18162 0.14739 0.12067 0.099590 0.082801 0.069309 0.058374 0.049442 0.042090 0.035997 0.030912 0.026642 0.023033 0.019966 0.017344 0.015089 0.013140 0.011444 0.0099604 0.0086527 0.0074910 0.0064483 0.0054990 0.0046134 0.0019535

71.455 81.287 87.225 93.180 99.158 105.16 111.19 117.26 123.35 129.48 135.65 141.86 148.11 154.40 160.73 167.11 173.54 180.02 186.55 193.13 199.77 206.48 213.25 220.09 227.00 233.99 241.07 248.25 255.52 262.91 270.43 278.09 285.91 293.92 302.16 310.68 319.55 328.93 339.06 381.71

321.11 325.29 327.87 330.49 333.15 335.85 338.59 341.35 344.15 346.96 349.80 352.65 355.51 358.38 361.25 364.12 366.99 369.85 372.69 375.51 378.31 381.08 383.82 386.52 389.17 391.77 394.30 396.76 399.13 401.40 403.55 405.55 407.38 408.99 410.33 411.32 411.83 411.67 410.45 381.71

71.455 81.288 87.226 93.182 99.161 105.17 111.20 117.26 123.36 129.50 135.67 141.89 148.14 154.44 160.79 167.19 173.64 180.14 186.70 193.32 200.00 206.75 213.58 220.48 227.47 234.55 241.72 249.01 256.41 263.94 271.62 279.47 287.50 295.76 304.28 313.13 322.39 332.22 342.93 389.64

334.94 339.78 342.76 345.77 348.83 351.91 355.02 358.16 361.31 364.48 367.65 370.83 374.00 377.17 380.32 383.45 386.55 389.63 392.66 395.66 398.60 401.49 404.32 407.07 409.75 412.33 414.82 417.19 419.43 421.52 423.44 425.15 426.63 427.82 428.65 429.03 428.81 427.76 425.42 389.64

0.41262 0.46913 0.50201 0.53409 0.56544 0.59613 0.62619 0.65568 0.68462 0.71305 0.74101 0.76852 0.79561 0.82230 0.84863 0.87460 0.90025 0.92559 0.95065 0.97544 1.0000 1.0243 1.0485 1.0724 1.0962 1.1199 1.1435 1.1670 1.1905 1.2139 1.2375 1.2611 1.2848 1.3088 1.3332 1.3580 1.3836 1.4104 1.4390 1.5621

1.9639 1.9201 1.8972 1.8766 1.8580 1.8414 1.8264 1.8130 1.8010 1.7902 1.7806 1.7720 1.7643 1.7575 1.7515 1.7461 1.7413 1.7371 1.7334 1.7300 1.7271 1.7245 1.7221 1.7200 1.7180 1.7162 1.7145 1.7128 1.7111 1.7092 1.7072 1.7050 1.7024 1.6993 1.6956 1.6909 1.6850 1.6771 1.6662 1.5621

–103.3 –95 –90 –85 –80 –75 –70 –65 –60 –55 –50 –45 –40 –35 –30 –25 –20 –15 –10 –5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 101.06

Appendix A

367

Appendix B

515

Table A.28B.6 (CONTINUED) APPENDIX Thermophysical Properties of Saturated R134a (SI Units) Thermophysical Properties of Saturated R134a (SI Units) T °C –103.3 –95 –90 –85 –80 –75 –70 –65 –60 –55 –50 –45 –40 –35 –30 –25 –20 –15 –10 –5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 101.06

P kPa

c pf kJ/kg-K

c pg kJ/kg-K

μf mPa-s

μg mPa-s

kf W/m-K

kg W/m-K

Pr f

0.38956 0.93899 1.5241 2.3990 3.6719 5.4777 7.9814 11.380 15.906 21.828 101.325 39.117 51.209 66.144 84.378 106.40 132.73 163.94 200.60 243.34 292.80 349.66 414.61 488.37 571.71 665.38 770.20 886.98 1016.6 1159.9 1317.9 1491.5 1681.8 1889.8 2116.8 2364.1 2633.2 2925.8 3244.2 4059.3

1.1838 1.1861 1.1892 1.1933 1.1981 1.2036 1.2096 1.2161 1.2230 1.2304 1.2381 1.2462 1.2546 1.2635 1.2729 1.2827 1.2930 1.3040 1.3156 1.3279 1.3410 1.3552 1.3704 1.3869 1.4049 1.4246 1.4465 1.4709 1.4984 1.5298 1.5661 1.6089 1.6602 1.7234 1.8039 1.9115 2.0648 2.3064 2.7559 –

0.58530 0.60524 0.61730 0.62943 0.64165 0.65401 0.66654 0.67932 0.69239 0.70582 0.71969 0.73406 0.74900 0.76458 0.78087 0.79792 0.81580 0.83458 0.85435 0.87520 0.89723 0.92059 0.94546 0.97206 1.0007 1.0316 1.0655 1.1028 1.1445 1.1917 1.2461 1.3099 1.3868 1.4822 1.6051 1.7714 2.0122 2.3971 3.1207 –

2.1536 1.5720 1.3410 1.1624 1.0203 0.90473 0.80910 0.72879 0.66051 0.60182 0.55089 0.50633 0.46703 0.43213 0.40095 0.37292 0.34758 0.32456 0.30355 0.28428 0.26653 0.25011 0.23487 0.22066 0.20737 0.19489 0.18313 0.17200 0.16145 0.15139 0.14177 0.13253 0.12361 0.11496 0.10651 0.098169 0.089846 0.081372 0.072450 –

0.0068294 0.0071588 0.0073562 0.0075528 0.0077484 0.0079430 0.0081364 0.0083286 0.0085194 0.0087089 0.0088970 0.0090837 0.0092690 0.0094531 0.0096361 0.0098181 0.0099995 0.010181 0.010362 0.010543 0.010726 0.010911 0.011099 0.011291 0.011488 0.011693 0.011907 0.012132 0.012373 0.012633 0.012917 0.013232 0.013587 0.013996 0.014475 0.015053 0.015773 0.016711 0.018023 –

0.14524 0.14022 0.13727 0.13438 0.13154 0.12876 0.12603 0.12335 0.12071 0.11812 0.11557 0.11306 0.11059 0.10816 0.10576 0.10340 0.10107 0.098767 0.096491 0.094241 0.092013 0.089806 0.087618 0.085444 0.083284 0.081134 0.078992 0.076853 0.074716 0.072575 0.070427 0.068267 0.066091 0.063894 0.061672 0.059421 0.057147 0.054880 0.052755 –

0.0030801 0.0037452 0.0041459 0.0045469 0.0049479 0.0053493 0.0057509 0.0061529 0.0065554 0.0069586 0.0073625 0.0077675 0.0081736 0.0085812 0.0089906 0.0094022 0.0098164 0.010234 0.010655 0.011082 0.011514 0.011954 0.012402 0.012862 0.013335 0.013825 0.014336 0.014874 0.015446 0.016062 0.016734 0.017481 0.018326 0.019305 0.020471 0.021903 0.023735 0.026205 0.029819 –

17.553 13.298 11.618 10.322 9.2927 8.4569 7.7656 7.1854 6.6923 6.2688 5.9016 5.5807 5.2983 5.0482 4.8255 4.6261 4.4469 4.2851 4.1386 4.0056 3.8845 3.7741 3.6734 3.5815 3.4979 3.4219 3.3533 3.2920 3.2378 3.1912 3.1527 3.1234 3.1051 3.1007 3.1153 3.1579 3.2464 3.4198 3.7848 –

PRg

T °C

1.2978 –103.3 1.1569 –95 1.0953 –90 1.0456 –85 1.0048 –80 0.97112 –75 0.94303 –70 0.91952 –65 0.89982 –60 0.88336 –55 0.86968 –50.000 0.85845 –45 0.84939 –40 0.84228 –35 0.83693 –30 0.83322 –25 0.83101 –20 0.83022 –15 0.83079 –10 0.83267 –5 0.83584 0 0.84031 5 0.84612 10 0.85335 15 0.86211 20 0.87256 25 0.88493 30 0.89953 35 0.91679 40 0.93728 45 0.96180 50 0.99154 55 1.0282 60 1.0745 65 1.1350 70 1.2174 75 1.3372 80 1.5287 85 1.8862 90 – 101.06

368

Appendix A

518

Appendix B

Table A.29B.7 APPENDIX Thermophysical Properties Single-Phase R134a (SI Units) Thermophysical Properties of SingleofPhase R134a (SI Units) T °C

ρ kg/m3

v m3/kg

u kJ/kg

h kJ/kg

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260

1582.4 1529.0 0.58001 0.52903 0.48655 0.45051 0.41950 0.39253 0.36884 0.34786 0.32916 0.31236 0.29721 0.28346 0.27092 0.25945 0.24892 0.23921 0.23022

0.00063195 0.00065402 1.7241 1.8903 2.0553 2.2197 2.3838 2.5476 2.7112 2.8747 3.0380 3.2014 3.3646 3.5278 3.6911 3.8543 4.0174 4.1804 4.3437

75.360 99.157 344.41 356.77 369.85 383.66 398.19 413.42 429.36 445.97 463.25 481.18 499.75 518.95 538.77 559.20 580.23 601.85 624.05

75.366 99.163 361.65 375.67 390.40 405.85 422.02 438.90 456.47 474.72 493.63 513.19 533.39 554.23 575.68 597.74 620.40 643.65 667.48

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260

1582.5 1529.1 1474.4 2.7023 2.4699 2.2780 2.1157 1.9759 1.8541 1.7469 1.6516 1.5664 1.4897 1.4202 1.3569 1.2992 1.2461 1.1973 1.1521

0.00063191 0.00065398 0.00067824 0.37006 0.40487 0.43898 0.47266 0.50610 0.53935 0.57244 0.60547 0.63841 0.67128 0.70413 0.73697 0.76970 0.80250 0.83521 0.86798

75.353 99.148 123.34 355.55 368.97 382.98 397.64 412.97 428.98 445.64 462.96 480.93 499.53 518.75 538.59 559.03 580.08 601.71 623.92

75.384 99.181 123.38 374.05 389.21 404.92 421.27 438.28 455.94 474.26 493.24 512.85 533.09 553.96 575.44 597.52 620.20 643.47 667.32

s cp kJ/kg-K kJ/kg-K

μ mPa-s

k W/m-K

Pr

T °C

1.8827 1.0204 0.0085303 0.0093211 0.010105 0.010883 0.011655 0.012422 0.013185 0.013942 0.014696 0.015445 0.016190 0.016931 0.017669 0.018403 0.019133 0.019860 0.020583

0.14323 0.13155 0.0065526 0.0081541 0.0097557 0.011357 0.012959 0.014561 0.016163 0.017764 0.019366 0.020968 0.022570 0.024172 0.025774 0.027376 0.028977 0.030579 0.032181

15.566 9.2932 0.89115 0.82130 0.78156 0.75764 0.74314 0.73474 0.73048 0.72916 0.72998 0.73239 0.73600 0.74054 0.74579 0.75161 0.75787 0.76448 0.77137

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260

1.8838 1.0208 0.66074 0.0092706 0.010071 0.010860 0.011641 0.012414 0.013181 0.013943 0.014699 0.015451 0.016198 0.016941 0.017680 0.018415 0.019146 0.019874 0.020598

0.14324 0.13156 0.12072 0.0081729 0.0097718 0.011372 0.012972 0.014573 0.016174 0.017775 0.019376 0.020977 0.022579 0.024180 0.025782 0.027383 0.028985 0.030586 0.032188

15.574 9.2960 6.6936 0.84835 0.79410 0.76508 0.74815 0.73837 0.73326 0.73138 0.73181 0.73394 0.73736 0.74175 0.74689 0.75263 0.75882 0.76538 0.77223

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260

P = l0 kPa (0.01 MPa) 0.43539 0.56544 1.8400 1.9029 1.9635 2.0222 2.0793 2.1350 2.1894 2.2425 2.2946 2.3457 2.3958 2.4450 2.4934 2.5411 2.5880 2.6342 2.6798

1.1842 1.1981 0.68454 0.71847 0.75455 0.79068 0.82629 0.86123 0.89548 0.92905 0.96198 0.99429 1.0260 1.0572 1.0879 1.1181 1.1478 1.1771 1.2060

P = 50 kPa (0.05 MPa) 0.43535 0.56539 0.68458 1.7665 1.8288 1.8885 1.9463 2.0024 2.0571 2.1105 2.1627 2.2139 2.2641 2.3134 2.3619 2.4096 2.4565 2.5028 2.5484

1.1842 1.1981 1.2230 0.74790 0.77053 0.80114 0.83373 0.86677 0.89974 0.93240 0.96466 0.99647 1.0278 1.0587 1.0892 1.1192 1.1488 1.1779 1.2067

Appendix A

369

Appendix B

519

Table A.30B.7 (CONTINUED) APPENDIX Thermophysical Properties Single-Phase R134a (SI Units) Thermophysical Properties of SingleofPhase R134a (SI Units) T °C

P kg/m3

v m /kg

u kj/kg

h kj/kg

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260

1582.5 1529.2 1474.5 1417.8 5.0401 4.6232 4.2784 3.9860 3.7336 3.5130 3.3181 3.1444 2.9885 2.8476 2.7198 2.6031 2.4961 2.3977 2.3069

0.00063191 0.00065394 0.00067820 0.00070532 0.19841 0.21630 0.23373 0.25088 0.26784 0.28466 0.30138 0.31803 0.33462 0.35117 0.36767 0.38416 0.40062 0.41707 0.43348

75.344 99.137 123.33 148.09 367.81 382.10 396.94 412.40 428.49 445.23 462.61 480.61 499.25 518.50 538.36 558.83 579.89 601.53 623.76

75.407 99.202 123.40 148.16 387.65 403.73 420.31 437.49 455.28 473.70 492.74 512.42 532.71 553.62 575.13 597.25 619.95 643.24 667.11

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260

1583.0 1529.8 1475.2 1418.7 1359.4 1295.6 23.744 21.526 19.808 18.406 17.225 16.211 15.324 14.540 13.840 13.209 12.637 12.116 11.638

0.00063171 0.00065368 0.00067787 0.00070487 0.00073562 0.00077184 0.042116 0.046455 0.050485 0.054330 0.058055 0.061687 0.065257 0.068776 0.072254 0.075706 0.079133 0.082535 0.085925

75.271 99.048 123.22 147.96 173.38 199.66 390.55 407.40 424.40 441.78 459.65 478.04 496.98 516.47 536.54 557.17 578.38 600.14 622.47

75.587 99.375 123.56 148.31 173.75 200.05 411.61 430.63 449.64 468.95 488.68 508.88 529.60 550.86 572.67 595.03 617.94 641.41 665.43

3

s kJ/ kg-K

cp kJ/ kg-K

μ mPa-s

k W/m-K

Pr

T °C

1.8852 1.0214 0.66109 0.46727 0.010028 0.010832 0.011623 0.012404 0.013177 0.013944 0.014704 0.015458 0.016208 0.016953 0.017694 0.018430 0.019163 0.019892 0.020617

0.14326 0.13158 0.12074 0.11061 0.0097966 0.011392 0.012990 0.014589 0.016189 0.017789 0.019389 0.020990 0.022590 0.024191 0.025792 0.027393 0.028994 0.030595 0.032197

15.583 9.2995 6.6956 5.2996 0.81382 0.77529 0.75474 0.74307 0.73683 0.73421 0.73414 0.73592 0.73907 0.74328 0.74828 0.75391 0.76002 0.76651 0.77331

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260

1.8967 1.0260 0.66384 0.46928 0.34910 0.26730 0.011504 0.012349 0.013169 0.013970 0.014758 0.015535 0.016303 0.017062 0.017816 0.018563 0.019306 0.020043 0.020776

0.14338 0.13172 0.12090 0.11080 0.10126 0.092142 0.013250 0.014788 0.016353 0.017931 0.019516 0.021105 0.022697 0.024290 0.025885 0.027480 0.029077 0.030674 0.032271

15.660 9.3278 6.7113 5.3099 4.4535 3.8869 0.83654 0.79066 0.77026 0.75974 0.75464 0.75301 0.75376 0.75622 0.75992 0.76456 0.76992 0.77583 0.78217

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260

P = l00 kPa (0.1 MPa) 0.43529 0.56533 0.68451 0.79554 1.7677 1.8288 1.8874 1.9441 1.9991 2.0528 2.1053 2.1566 2.2070 2.2564 2.3049 2.3527 2.3997 2.4460 2.4916

1.1842 1.1980 1.2229 1.2545 0.79505 0.81542 0.84352 0.87396 0.90521 0.93667 0.96806 0.99923 1.0301 1.0606 1.0908 1.1205 1.1499 1.1790 1.2076

P = 500 kPa (0.5 MPa) 0.43487 0.56487 0.68400 0.79496 0.89964 0.99959 1.7339 1.7967 1.8555 1.9118 1.9661 2.0189 2.0703 2.1205 2.1697 2.2180 2.2654 2.3121 2.3580

1.1839 1.1976 1.2223 1.2536 1.2918 1.3399 0.96352 0.94679 0.95652 0.97514 0.99794 1.0230 1.0494 1.0765 1.1041 1.1318 1.1596 1.1873 1.2149

370

Appendix A

520

Appendix B

Table A.31B.7 (CONTINUED) APPENDIX Thermophysical Properties Single-Phase R134a (SI Units) Thermophysical Properties of SingleofPhase R134a (SI Units) T °C

ρ kg/m3

v m /kg

u kJ/kg

h kJ/kg

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260

1583.6 1530.5 1476.1 1419.9 1360.8 1297.5 1227.7 49.004 43.350 39.372 36.295 33.792 31.692 29.889 28.314 26.922 25.678 24.557 23.539

0.00063147 0.00065338 0.00067746 0.00070427 0.00073486 0.00077071 0.00081453 0.020406 0.023068 0.025399 0.027552 0.029593 0.031554 0.033457 0.035318 0.037144 0.038944 0.040722 0.042483

75.180 98.937 123.09 147.79 173.18 199.39 226.69 399.45 418.46 437.00 455.65 474.62 494.00 513.84 534.19 555.05 576.45 598.37 620.84

75.811 99.591 123.76 148.50 173.91 200.16 227.50 419.86 441.53 462.40 483.21 504.21 525.55 547.30 569.51 592.20 615.39 639.10 663.32

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260

1587.2 1534.9 1481.5 1426.6 1369.3 1308.6 1242.9 1169.4 1082.4 967.24 677.84 200.28 165.99 146.16 132.37 121.91 113.54 106.60 100.71

0.00063004 0.00065151 0.00067499 0.00070097 0.00073030 0.00076418 0.00080457 0.00085514 0.00092387 0.0010339 0.0014753 0.0049930 0.0060245 0.0068418 0.0075546 0.0082028 0.0088075 0.0093809 0.0099295

74.643 98.286 122.29 146.82 171.97 197.86 224.65 252.65 282.39 315.32 364.80 445.13 471.28 495.07 518.11 540.97 563.91 587.07 610.55

77.163 100.89 124.99 149.62 174.89 200.91 227.87 256.07 286.09 319.46 370.70 465.10 495.38 522.43 548.33 573.78 599.14 624.60 650.27

3

s kJ/ kg-K

cp kJ/ kg-K

μ mPa-s

k W/m-K

Pr

T °C

1.9111 1.0317 0.66729 0.47179 0.35116 0.26915 0.20893 0.012370 0.013232 0.014065 0.014878 0.015674 0.016458 0.017231 0.017996 0.018753 0.019504 0.020250 0.020990

0.14354 0.13190 0.12110 0.11102 0.10152 0.092451 0.083611 0.015410 0.016749 0.018226 0.019755 0.021307 0.022873 0.024448 0.026028 0.027612 0.029199 0.030788 0.032378

15.757 9.3634 6.7310 5.3228 4.4625 3.8929 3.4999 0.91026 0.83229 0.80156 0.78613 0.77820 0.77477 0.77430 0.77590 0.77897 0.78315 0.78815 0.79379

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260

2.0024 1.0671 0.68828 0.48695 0.36351 0.28014 0.21952 0.17257 0.13383 0.098907 0.050372 0.019036 0.019013 0.019427 0.019992 0.020626 0.021295 0.021986 0.022690

0.14449 0.13295 0.12228 0.11235 0.10304 0.094232 0.085770 0.077471 0.069049 0.059942 0.054130 0.026830 0.026158 0.026800 0.027871 0.029134 0.030500 0.031925 0.033386

16.371 9.5843 6.8525 5.4027 4.5191 3.9322 3.5207 3.2266 3.0317 2.9940 8.7507 1.1949 1.0210 0.95331 0.91801 0.89781 0.88616 0.87988 0.87726

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200 220 240 260

P = l000 kPa (1 MPa) 0.43435 0.56430 0.68337 0.79425 0.89881 0.99860 1.0952 1.7135 1.7806 1.8414 1.8988 1.9536 2.0065 2.0579 2.1080 2.1570 2.2050 2.2522 2.2985

1.1835 1.1971 1.2216 1.2526 1.2901 1.3372 1.4006 1.1340 1.0535 1.0387 1.0438 1.0579 1.0768 1.0986 1.1222 1.1470 1.1724 1.1983 1.2244

P = 4000 kPa (4 MPa) 0.43123 0.56090 0.67962 0.79005 0.89400 0.99291 1.0881 1.1812 1.2740 1.3713 1.5117 1.7616 1.8368 1.9008 1.9592 2.0142 2.0667 2.1173 2.1664

1.1813 1.1941 1.2174 1.2465 1.2810 1.3227 1.3756 1.4485 1.5642 1.8145 9.4035 1.6841 1.4047 1.3151 1.2798 1.2682 1.2692 1.2776 1.2908

371

Appendix A 524

Appendix B

APPENDIX Table A.32B.8 Thermophysical Properties of Metals (SI Units) Thermophysical Properties of Metals (SI Units) T °C

ρ kg/m3

30 90 150 210 270 330

2701 2690 2678 2665 2652 2640

30 90 150 210 270 330

7160 7151 7141 7129 7118 7106

30 90 150 210 270 330

7867 7815 7782 7782 7781 7779

c kJ/kg-K

k W/m-K

ρ kg/m3

c kJ/kg-K

236.1 238.5 239.1 236.7 234.3 231.8

8530 8530 8544 8581 8618 8655

0.3781 0.3886 0.3985 0.4075 0.4165 0.4255

93.61 91.93 89.72 86.66 83.60 80.55

8932 8904 8876 8847 8818 8789

72.48 68.28 64.24 60.46 56.68 52.93

7831 7814 7797 7777 7757 7737

14.95 15.97 16.97 17.93 18.89 19.84

8237 8214 8190 8164 8139 8113

35.26 34.48 33.70 32.92 32.14 31.36

8899 8876 8853 8829 8805 8781

Aluminum 0.9044 0.9320 0.9587 0.9839 1.0090 1.0350

0.4213 0.4453 0.4698 0.4953 0.5208 0.5463

0.3854 0.3926 0.3993 0.4053 0.4113 0.4173

400.7 395.9 391.4 387.2 383.0 378.8

7869 7851 7833 7813 7793 7772

0.4484 0.4742 0.4966 0.5137 0.5308 0.5491

0.4354 0.4618 0.4874 0.5117 0.5360 0.5610

7899 7877 7854 7829 7805 7780

30 90 150 210 270 330

11337 11277 11214 11147 11081 11014

30 90 150 210 270

10498 10462 10424 10386 10348

0.2351 0.2375 0.2403 0.2436 0.2469

248.9 426.5 423.6 419.6 415.7

4500 4492 4485 4477 4469

330

10309

0.2502

411.7

4461

Lead

0.4691 0.4907 0.5093 0.5231 0.5369 0.5504

k W/m-K 52.00 52.00 52.81 54.91 57.01 59.11

Iron

63.74 60.62 57.55 54.58 51.61 48.65

79.86 73.44 67.79 63.35 58.91 54.52

AISI 302 Stainless Steel 8052 8031 8008 7983 7958 7932

0.4825 0.5008 0.5174 0.5315 0.5456 0.5594

15.27 16.53 17.61 18.42 19.23 20.04

AISI 347 Stainless Steel

13.46 14.54 15.56 16.49 17.42 18.35

7975 7954 7931 7907 7882 7857

90.37 84.07 78.51 74.13 69.75 65.63

8509 8490 8470 8448 8426 8405

0.5229 0.5403 0.5556 0.5676 0.5796

21.85 20.95 20.28 19.98 19.68

6570 6563 6556 6548 6540

0.2787 0.2919 0.3025 0.3091 0.3157

22.67 22.01 21.50 21.23 20.96

0.5917

19.40

6533

0.3223

20.71

Nickel

Silver

c kJ/kg-K Bronze

AISI 316 Stainless Steel

30 90 150 210 270 330

0.1291 0.1309 0.1332 0.1362 0.1392 0.1422

8799 8771 8742 8712 8682 8652

AISI 1010 Carbon Steel

AISI 304 Stainless Steel 0.4782 0.5010 0.5199 0.5325 0.5451 0.5574

116.7 129.3 138.4 142.0 145.6 149.2

Copper

Iron-Armco 0.4484 0.4742 0.4966 0.5137 0.5308 0.5491

ρ kg/m3

Brass

Chromium 0.4501 0.4711 0.4907 0.5081 0.5255 0.5426

k W/m-K

0.4453 0.4699 0.4974 0.5295 0.5616 0.5910

0.4826 0.5014 0.5183 0.5321 0.5459 0.5594

14.32 15.24 16.16 17.09 18.02 18.95

Inconel X-750

Titanium

0.4401 0.4605 0.4773 0.4884 0.4995 0.5106

11.76 12.84 13.91 14.96 16.01 17.06

Zirconium

Properties B Thermophysical (Exergy Calculation) The thermophysical properties for exergy calculations have been adapted from the following reference:

A. Bejan, G. Tsatsaronis, and M. Moran. Thermal Design and Optimization. John Wiley & Sons, 1996.

DOI: 10.1201/9781003049272-B

373

374

Appendix B

Table B.1 Property Data Table as a Function of Temperature for Various Substances at 1 Bar

Appendix B

Figure B.1 Constant values for equations.

375

376

Table B.2 Standard Chemical Exergy for Different Substances

Appendix B

Properties C Thermophysical (Emissivity) The radiation properties provided in Table C.1 have been adapted from the following reference:

Y. Jaluria Design and Optimization of Thermal Systems. CRC Press, 2007.

DOI: 10.1201/9781003049272-C

377

378

Appendix C Appendix B

677

Table C.1 TABLE B.7 Emissivities Emissivities εn of theEnRadiation in the Normal of the Radiation in theDirection Direction of of the the Normal to the to the SurSurface E of the Total Hemispherical Radiation for Materials face and ε of the and Total Hemispherical Radiation forVarious Various Materials for TemperatureforTthe Temperature T†‡ Surface

T, nC

En

E

Gold, polished

130 0.018 400 0.022 Silver 20 0.020 Copper, polished 20 0.030 Lightly oxidized 20 0.037 Scraped 20 0.070 Black oxidized 20 0.78 Oxidized 131 0.76 0.725 Aluminum, bright rolled 170 0.039 0.049 500 0.050 Aluminum paint 100 0.20–0.40 Silumin, cast polished 150 0.186 Nickel, bright matte 100 0.041 0.046 Polished 100 0.045 0.053 Manganin, bright rolled 118 0.048 0.057 Chrome, polished 150 0.058 0.071 Iron, bright etched 150 0.128 0.158 Bright abrased 20 0.24 Red rusted 20 0.61 Hot rolled 20 0.77 130 0.60 Hot cast 100 0.80 Heavily rusted 20 0.85 Heat-resistant oxidized 80 0.613 200 0.639 Zinc, gray oxidized 20 0.23–0.28 Lead, gray oxidized 20 0.28 Bismuth, bright 80 0.340 0.366 Corundum, emery rough 80 0.855 0.84 678 Design and Optimization of Thermal Systems Clay, fired 70 0.91 0.86 Lacquer, white 100 0.925 Red lead 100 0.93 Enamel, 20 0.85–0.95 TABLElacquer B.7 (CONTINUED) Lacquer, black matte Emissivities En of the Radiation in 80 the Direction0.970 of the Normal to the Bakelite lacquer 80 0.935 Surface and E of the Total Hemispherical Radiation for Various Materials Brick, mortar, plaster 20 0.93 for the Temperature T†‡ Porcelain 20 0.92–0.94 Glass 90nC 0.940E 0.876 Surface T, E n

Ice, smooth, water

0

0.966

Rough crystals

0

0.985

0.918 (Continued)

Waterglass

20

0.96

Paper

95

0.92

0.89

Wood, beech

70

0.935

0.91

Tarpaper

20

0.93

†: From measurements by E. Schmidt and E. Eckert. ‡: For metals, the emissivities rise with rising temperature, but for nonmetallic substances (metal oxides, organic substances) this rule is sometimes not correct. Where the exact measurements are not given, take for bright metal surfaces an average ratio E/En = 1.2 and for other substances with smooth surfaces E/En = 0.95; for rough surfaces use E/En = 0.98. Source: Eckert, E.R.G., and Drake, R.M., (1972) Analysis of Heat and Mass Transfer, McGraw-Hill, New York.

D Standard Pipe Dimension The thermophysical properties presented here have been adapted from the following reference:

Steven G. Penocello Thermal Energy Systems Design and Analysis. CRC Press, 2019.

DOI: 10.1201/9781003049272-D

379

Appendix C: Standard Pipe Dimensions This appendix shows standard pipe dimensions for steel, iron, and stainless steel (SS) pipes. The data shown here are a subset of a much more comprehensive table published by The Crane Company (2017).

380

Appendix D

Reference The Crane Table D.1Company. Flow of Fluids Through Valves, Fittings and Pipe. Stamford, Connecticut: The Crane Company, 2017. Standard Pipe Dimension Nominal Diameter 1/8

1/4

Outside Diameter

Schedule

in

ft

cm

0.405

0.033750

1.0287

0.540

0.045000

Steel (Iron)

SS

in

ft

cm

40 (std) 80 (xs)

10S 40S 80S

0.307 0.269 0.215

0.025583 0.022417 0.017917

0.77978 0.68326 0.54610

40 80

10S 40S 80S

0.410 0.364 0.302

0.034167 0.030333 0.025167

1.0414 0.92456 0.76708

10 40 (std) 80 (xs)

10S 40S 80S

0.545 0.493 0.423

0.045417 0.041083 0.035250

1.3843 1.2522 1.0744

5S 10S 40S 80S

0.710 0.674 0.622 0.546 0.466 0.252

0.059167 0.056167 0.051833 0.045500 0.038833 0.021000

1.8034 1.7120 1.5799 1.3868 1.1836 0.64008

5S 10S 40S 80S

0.920 0.884 0.824 0.742 0.612 0.434

0.076667 0.073667 0.068667 0.061833 0.051000 0.036167

2.3368 2.2454 2.0930 1.8847 1.5545 1.1024

5S 10S 40S 80S

1.185 1.097 1.049 0.957 0.815 0.599

0.098750 0.091417 0.087417 0.079750 0.067917 0.049917

3.0099 2.7864 2.6645 2.4308 2.0701 1.5215

1.3716

3/8

0.675

0.056250

1.7145

1/2

0.840

0.070000

2.1336 40 (std) 80 (xs) 160 (xxs)

3/4

1.050

0.087500

2.6670 40 (std) 80 (xs) 160 (xxs)

1

1.315

0.10958

Inside Diameter

3.3401 40 (std) 80 (xs) 160 (xxs)

(Continued)

527

528

Appendix C

381

Appendix D

Nominal Diameter 1 1/4

Outside Diameter

Schedule

in

ft

cm

1.660

0.13833

4.2164

Steel (Iron)

40 (std) 80 (xs) 160 (xxs) 1 1/2

1.900

0.15833

4.8260 40 (std) 80 (xs) 160 (xxs)

2

2.375

0.19792

6.0325 40 (std) 80 (xs) 160 (xxs)

2 1/2

2.875

0.23958

7.3025 40 (std) 80 (xs) 160 (xxs)

3

3.500

0.29167

8.8900 40 (std) 80 (xs) 160 (xxs)

3 1/2

4.000

0.33333

10.160 40 (std) 80 (xs)

4

4.500

0.37500

11.430 40 (std) 80 (xs) 120 160 (xxs)

Inside Diameter SS

in

ft

cm

5S 10S 40S 80S

1.530 1.442 1.380 1.278 1.160 0.896

0.12750 0.12017 0.11500 0.10650 0.096667 0.074667

3.8862 3.6627 3.5052 3.2461 2.9464 2.2758

5S 10S 40S 80S

1.770 1.682 1.610 1.500 1.338 1.100

0.14750 0.14017 0.13417 0.12500 0.11150 0.091667

4.4958 4.2723 4.0894 3.8100 3.3985 2.7940

5S 10S 40S 80S

2.245 2.157 2.067 1.939 1.687 1.503

0.18708 0.17975 0.17225 0.16158 0.14058 0.12525

5.7023 5.4788 5.2502 4.9251 4.2850 3.8176

5S 10S 40S 80S

2.709 2.635 2.469 2.323 2.125 1.771

0.22575 0.21958 0.20575 0.19358 0.17708 0.14758

6.8809 6.6929 6.2713 5.9004 5.3975 4.4983

5S 10S 40S 80S

3.334 3.260 3.068 2.900 2.624 2.300

0.27783 0.27167 0.25567 0.24167 0.21867 0.19167

8.4684 8.2804 7.7927 7.3660 6.6650 5.8420

5S 10S 40S 80S

3.834 3.760 3.548 3.364

0.31950 0.31333 0.29567 0.28033

9.7384 9.5504 9.0119 8.5446

5S 10S 40S 80S

4.334 4.260 4.026 3.826 3.624 3.438 3.152

0.36117 0.35500 0.33550 0.31883 0.30200 0.28650 0.26267

11.008 10.820 10.226 9.7180 9.2050 8.7325 8.0061 (Continued)

529

Appendix C

382

Nominal Diameter 5

Appendix D Outside Diameter

Schedule

in

ft

cm

5.563

0.46358

14.130

Steel (Iron)

40 (std) 80 (xs) 120 160 (xxs) 6

6.625

0.55208

16.828

8

8.625

0.71875

21.908

5 10 40 (std) 80 (xs) 120 160 xxs

20 30 40 (std) 60 80 (xs) 100 120 140 (xxs) 160 10

10.750

0.89583

27.305

Inside Diameter SS

in

ft

cm

5S 10S 40S 80S

5.345 5.295 5.047 4.813 4.563 4.313 4.063

0.44542 0.44125 0.42058 0.40108 0.38025 0.35942 0.33858

13.576 13.449 12.819 12.225 11.590 10.955 10.320

5S 10S 40S 80S

6.407 6.357 6.065 5.761 5.501 5.187 4.897

0.53392 0.52975 0.50542 0.48008 0.45842 0.43225 0.40808

16.274 16.147 15.405 14.633 13.973 13.175 12.438

5S 10S

8.407 8.329 8.125 8.071 7.981 7.813 7.625 7.437 7.187 7.001 6.875 6.813

0.70058 0.69408 0.67708 0.67258 0.66508 0.65108 0.63542 0.61975 0.59892 0.58342 0.57292 0.56775

21.354 21.156 20.638 20.500 20.272 19.845 19.368 18.890 18.255 17.783 17.463 17.305

10.482 10.420 10.250 10.136 10.020 9.750 9.562 9.312 9.062 8.750 8.500

0.87350 0.86833 0.85417 0.84467 0.83500 0.81250 0.79683 0.77600 0.75517 0.72917 0.70833

26.624 26.467 26.035 25.745 25.451 24.765 24.287 23.652 23.017 22.225 21.590

40S 80S

5S 10S 20 30 40 (std) 60 (std) 80 100 120 140 (xxs) 160

40S 80S

(Continued)

530

Appendix C

383

Appendix D

Nominal Diameter 12

Outside Diameter

Schedule

in

ft

cm

12.75

1.0625

32.385

Steel (Iron)

20 30 (std) 40 (xs) 60 80 100 120 (xxs) 140 160 14

14

1.1667

35.560

Inside Diameter SS

in

ft

cm

5S 10S

12.438 12.390 12.250 12.090 12.000 11.983 11.750 11.626 11.374 11.062 10.750 10.500 10.126

1.0365 1.0325 1.0208 1.0075 1.0000 0.99858 0.97917 0.96883 0.94783 0.92183 0.89583 0.87500 0.84383

31.593 31.471 31.115 30.709 30.480 30.437 29.845 29.530 28.890 28.097 27.305 26.670 25.720

5S 10S

13.688 13.624 13.500 13.376 13.250 13.124 13.000 12.812 12.500 12.124 11.812 11.500 11.188

1.1407 1.1353 1.1250 1.1147 1.1042 1.0937 1.0833 1.0677 1.0417 1.0103 0.98433 0.95833 0.93233

34.768 34.605 34.290 33.975 33.655 33.335 33.020 32.542 31.750 30.795 30.002 29.210 28.418

5S 10S

15.670 15.624 15.500 15.376 15.250 15.000 14.688 14.312 13.938 13.562 13.124 12.182

1.3058 1.3020 1.2917 1.2813 1.2708 1.2500 1.2240 1.1927 1.1615 1.1302 1.0937 1.0152

39.802 39.685 39.370 39.055 38.735 38.100 37.308 36.352 35.403 34.447 33.335 30.942

40S 80S

10 20 30 (std) 40 (xs) 60 80 100 120 140 160 16

16

1.3333

40.640 10 20 30 (std) 40 (xs) 60 80 100 120 140 160

(Continued)

531

Appendix C

384

Appendix D Outside Diameter

Schedule

Nominal Diameter

in

ft

cm

18

18

1.5000

45.720

Steel (Iron)

Inside Diameter SS

in

ft

cm

5S 10S

17.670 17.624 17.500 17.376 17.250 17.124 17.000 16.876 16.500 16.124 15.688 15.250 14.876 14.438

1.4725 1.4687 1.4583 1.4480 1.4375 1.4270 1.4167 1.4063 1.3750 1.3437 1.3073 1.2708 1.2397 1.2032

44.882 44.765 44.450 44.135 43.815 43.495 43.180 42.865 41.910 40.955 39.848 38.735 37.785 36.673

5S 10S

19.624 19.564 19.500 19.250 19.000 18.812 18.376 17.938 17.438 17.000 16.500 16.062

1.6353 1.6303 1.6250 1.6042 1.5833 1.5677 1.5313 1.4948 1.4532 1.4167 1.3750 1.3385

49.845 49.693 49.530 48.895 48.260 47.782 46.675 45.563 44.293 43.180 41.910 40.797

5S 10S

21.624 21.564 21.500 21.250 21.000 20.250 19.750 19.250 18.750 18.250 17.750

1.8020 1.7970 1.7917 1.7708 1.7500 1.6875 1.6458 1.6042 1.5625 1.5208 1.4792

54.925 54.773 54.610 53.975 53.340 51.435 50.165 48.895 47.625 46.355 45.085

10 20 (std) 30 (xs) 40 60 80 100 120 140 160 20

20

1.6667

50.800 10 20 (std) 30 (xs) 40 60 80 100 120 140 160

22

22

1.8333

55.880 10 20 (std) 30 (xs) 60 80 100 120 140 160

(Continued)

532

Appendix C

385

Appendix D Outside Diameter

Schedule

Nominal Diameter

in

ft

cm

24

24

2.0000

60.960

Steel (Iron) 10 20 (std) (xs) 30 40 60 80 100 120 140 160

Inside Diameter SS

in

ft

cm

5S 10S

23.564 23.500 23.250 23.000 22.876 22.624 22.062 21.562 20.938 20.376 19.876 19.312

1.9637 1.9583 1.9375 1.9167 1.9063 1.8853 1.8385 1.7968 1.7448 1.6980 1.6563 1.6093

59.853 59.690 59.055 58.420 58.105 57.465 56.037 54.767 53.183 51.755 50.485 49.052

26

26

2.1667

66.040

10 (std) 20 (xs)

25.376 25.250 25.000

2.1147 2.1042 2.0833

64.455 64.135 63.500

28

28

2.3333

71.120

10 (std) 20 (xs) 30

27.376 27.250 27.000 26.750

2.2813 2.2708 2.2500 2.2292

69.535 69.215 68.580 67.945

30

30

2.5000

76.200

29.500 29.376 29.250 29.000 28.750

2.4583 2.4480 2.4375 2.4167 2.3958

74.930 74.615 74.295 73.660 73.025

10 (std) 20 (xs) 30

5S 10S

32

32

2.6667

81.280

10 (std) 20 (xs) 30 40

31.376 31.250 31.000 30.750 30.624

2.6147 2.6042 2.5833 2.5625 2.5520

79.695 79.375 78.740 78.105 77.785

34

34

2.8333

86.360

10 (std) 20 (xs) 30 40

33.376 33.250 33.000 32.750 32.624

2.7813 2.7708 2.7500 2.7292 2.7187

84.775 84.455 83.820 83.185 82.865

36

36

3.0000

91.440

10 (std) 20 (xs) 30 40

35.376 35.250 35.000 34.750 34.500

2.9480 2.9375 2.9167 2.8958 2.8750

89.855 89.535 88.900 88.265 87.630

E Pump Performance Curve (SOURCE: GOULD’S MANUAL)

DOI: 10.1201/9781003049272-E

387

388

Appendix E

Appendix E

50 Hz Performance Curves RPM 2850

CENTRIFUGAL PUMP CHARACTERISTICS

0.9m

65

40

203mm

60

1.5m 50 1.8m 55 2.1m 58 60 62 2.7m 63 63 3.4m

55 190mm 50 178mm 45

200 175

60 58

35 152mm

150

30 140mm

125

10kW

55

20

100

0

10

0

40

20 80

40

120

160

50

60

200

70

240

280

80

320

360

2850 RPM RPM 2850

CENTRIFUGAL PUMP CHARACTERISTICS

55

183mm

40

67 69

2.6m 69

3m 67

140mm

175

MTX 3 x 4-7 A70

10

0

100

CAPACITY

200

300

100

120

400

140

500

160

600

700

2850 RPM RPM 2900

CENTRIFUGAL PUMP CHARACTERISTICS

1.4m 1.7m 45 50 2m 55 2.4m 60 63 65

65

203mm

60

3m

10

0

40

20 80

120

50

200

60 240

70 280

80

320

360

RPM 2900

CENTRIFUGAL PUMP CHARACTERISTICS

70 213mm

60

203mm

55

m3 /h

60 65

70

74

78

3.7m 83

81

4.3m

4.9m

191mm

83

50

5.5m

178mm

40 35

74

152mm

30

70

25 65

20 15 0 0

50

0

75 200

100

15kW

300

2900 RPM

CHEM-1A

CAPACITY

125 400

150

500

175 600

700

200

200 800

225 900

m3 /h GPM

GPM

CDS 2701-3

25 20

55 1.5kW

15

1.1kW

10 5

0 0

10

0

40

20

CAPACITY

30

80

40

120

160

50

60

200

240

70 280

320

80 360

m3 /h

RPM 1450

GPM

CDS:3085-3

Model: 3196 Size: 2X3-8 Imp. Dwg. 100-161 Pattern 53811 Eye Area: 35 cm2 FOR ALLOY CONSTRUCTION

24 22 20 18

0.3m 35

213mm

45 50

56

59

0.9m

50

1.5m 59 56

165mm

40 2.2kW

152mm

50 45

30

1.5kW

20

1.1kW

.75kW

10

2 5

0

20

10

CAPACITY

15

40

60

20 80

25

30

100

120

35 140

160

40 180

m3 /h

RPM 1450

GPM

CDS 1588-4

Model: 3196 Size: 3X4-8 Imp. Dwg. 100-164 / 100-163 Pattern 53814 / 53813 Eye Area 12.55 cm2

24 22 20

16

203mm

14 190mm

50 76

12 178mm

1.5m 75 73

40

10 165mm 70

8 152mm

4

50

2

0

0 0

20 100

1450 RPM

30

20 10

1.5kW

0

CDS 1780-4

30 3.7kW

65 60 2.2kW 55

6

40 200

CAPACITY

60

80 300

100

400

70 60

55 60 650.9m 70 73 7576 1.2m

213mm

0

CDS 3085-3

1450 RPM

18

70 60

0.6m 61

61

0

0

CDS 2701-3

4

MTX 3 x 4-8 A70

35 30

60

50

0

CDS 5015-1

1.5m

63

CENTRIFUGAL PUMP CHARACTERISTICS

225

m3 /h

1.2m 65

6

75

11kW

1m

65

8 140mm

100

18.5kW

63

32

Model: 3196 Size: 3x4-7 Imp. Dwg. C00143A01 / A02 Pattern 57720/ 57725 Eye Area581 cm 2

127mm

10

125

22kW

RPM 1450

12 178mm

150

30kW

60

28

90 100 110 120 130 140

14 190mm

175

78

165mm

80

9 165mm

0

CDS 1780-4

81

45

50 55

10

GPM

Model: 3196 Size: 3x4-8 Imp. Dwg. 100-164 / 100-163 Pattern 53814 / 53813 Eye Area281 cm 2

3m

70

0.8m 40

16 203mm

CDS 3086-3

2900 RPM

65

200

50

40 160

225

5.5kW

CAPACITY

30

60

24

CENTRIFUGAL PUMP CHARACTERISTICS

MTX 2 x 3-8 A60

100

3.7kW

50

20

1450 RPM

75

55

20

0

20

1

125

7.5kW

25

15

30

2

150

11kW

30 140mm

40

16

3

0

175

60

35 152mm

30

CDS:3086-3

15kW

165mm

40

20

11 183mm

GPM

178mm

45

40

10 CAPACITY

12

4

CDS 2700-3

65 3.5m 63

55 191mm 50

m3 /h

Model: 3196 Size: 2X3-8 Imp. Dwg. 100-161 Pattern 53811 Eye Area: 35 cm2 FOR ALLOY CONSTRUCTION

1.1m

70 213mm

8

5

50

7.5kW 5.5kW

80

10

12

6

25 60

1.5m

.75kW

7 140mm

100

5 40

4

8 152mm

11kW 55

20

55

1.1kW

45 40

0

75

127mm

0 0

127mm

1450 RPM

125

62

15

58

CENTRIFUGAL PUMP CHARACTERISTICS

2.3m

30 152mm

20

60

50

CDS 2700-3

150

35 165mm

25

61

0.9m

165mm

0

GPM

2m

62

50

0.6m

178mm

4

CDS 5014-1

50 50 55

55 58 60 61

14 190mm

0

m3 /h

Model: 3196 Size: 3x4-7 Imp. Dwg. C00143A01 / A02 Pattern 57720/ 57725 Eye Area 81 cm2

60

45

60 40 45 50

16 203mm

2 CAPACITY

30

70

18

6

3kW

10

20

8 140mm

50

15

22

10 152mm

75

5kW

50

STX 1 ⁄2 x 3-8 AB 1

CDS:5015-1

Model: 3196 Size: 1.5X3-8 Imp. Dwg. 76794 Pattern 56209 Eye Area: 28.57 cm2 ALLOY CONSTRUCTION

12

7.5kW

25 127mm

RPM 1450

CENTRIFUGAL PUMP CHARACTERISTICS

24

62

165mm

40

Model 3196

CDS:5014-1

Model: 3196 Size: 1.5X3-8 Imp. Dwg. 76794 Pattern 56209 Eye Area: 28.6 cm2 ALLOY CONSTRUCTION

1.2m

389

120 500

140 600

160 700

m3 /h

0

GPM

CDS 1588-4

390

50 Hz Performance Curves RPM 2900

CENTRIFUGAL PUMP CHARACTERISTICS

65

1.8m 2.4m 45 55 60 64 68

213mm

60

3m 70

3.7m

72

203mm

55

4.9m

72

50 190mm

70

45 178mm

68 64

30 152mm

11kW

0 0

20

0

40

100

CAPACITY

60

80

200

2900 RPM

100

300

140

500

600

160

m3 /h

700

GPM

200 180 160

254mm

100

1.2m

40

1.5m 47 48

45

600

48 47

178mm 40

127mm

20

165mm

20

CAPACITY

30

80

40

0

4 10

20

8

CAPACITY

12

30

40

16

50

60

70

80

20

24

28

32

m3 /h

90 100 110 120 130 140

120

160

60

200

240

27

32

37

RPM 2900

CENTRIFUGAL PUMP CHARACTERISTICS

110 100 40

90 254mm 80

1.2m 50

59

2.4m 2.7m 62 61 59

229mm

70

60 203mm 50 178mm 40

37

152mm

30

0

40

20

CAPACITY

30

80

120

40 160

50

60

200

240

70 280

320

2900 RPM RPM 2900

CENTRIFUGAL PUMP CHARACTERISTICS

110 100 90

2.4m 55

254mm

3m 3.6m 60 62 4.6m 64

62

70

360

152mm

30

55

20

11kW

0 0 0

350

MTX 2 x 3-10 A60

20 100

2900 RPM

40 200

CAPACITY

60

80 300

100

400

500

140 600

160

m3 /h

700

GPM

45

50

55

60

65

70

GPM

RPM 1450

CDS:5023-1

50

55

58

60 61

55

10

20

30

40

60 58

2.2kW

CAPACITY

50

16 60

70

20

80

24

28

1m

30

62

1.6m 1.9m 2.2m

62

20 229mm

12

m3 /h

RPM 1450

CDS 5027-1

60 55

50

178mm

40

3.7kW

30

8

20 1.5kW

4

10

1.1kW

2 0 0

10

0

40

1450 RPM

31

2.2kW

50

6

CDS 5026-1

70

2.5m

60

10 152mm

0

0

GPM

Model: 3196MT Size: 2X3-10 Imp. Dwg. 100-596 / 100-595 Pattern 54020/ 54019 Eye Area650cm 2

1.3m

60

22

200

40

20 32

90 100 110 120 130 140

CENTRIFUGAL PUMP CHARACTERISTICS

16 203mm

50

CDS 5023-1

50 55

24 254mm

14

60

1.5kW

45

12

80 70

0.7m 0.8m

8

90

0.6m

61

4

250

50 120

45

0

CDS 5019-1

18

100

7.5kW

10

40

1.1kW

0

150

18.5kW

35

m3 /h

1450 RPM

300

15kW

30

16

50

0

GPM

22kW

25

14

6

30kW

50 178mm

20

12

152mm

8

0

CDS 5026-1

20

.5kW

27

10

10

CDS 5022-1

60

60 203mm

40

m3 /h

6m

64

80 229mm

80

Model: 3196MT Size: 2X3-10 Imp. Dwg. 100-596 / 100-595 Pattern 54020/ 54019 Eye Area 50 cm 2

120

15

8

12 178mm

50 10

10

CAPACITY

6

203mm

14

100

10

30

229mm

16

150

11kW

0

4

254mm

18

7.5kW

20

2

26

20

200

40

.75kW

Model: 3196 Size: 1.5X3-10 Imp. Dwg. 100-593 Pattern 54017 Eye Area: 42 cm2 FOR ALLOY CONSTRUCTION

22

60 50

1.1kW

28

24

90

70

32

5

100

80 45

1.5kW

42

1450 RPM

250

15kW

55

0.6m

45 46

CENTRIFUGAL PUMP CHARACTERISTICS

300

1.8m 2.1m 61 62

CDS:5019-1

Model: 3196 Size: 1X2-10 Imp. Dwg. B10009 Pattern 56381 Eye Area: 31.6 cm2 FOR ALLOY CONSTRUCTION

152mm

0

MTX 11⁄2 x 3-10 A50

GPM

203mm

10

0

CDS 2321-4

2.2kW

CDS:5022-1

1.5m

55

360

229mm

0

350

m3 /h

46

GPM

Model: 3196 Size: 1.5X3-10 Imp. Dwg. 100-593 Pattern 54017 Eye Area: 42 cm2 FOR ALLOY CONSTRUCTION

120

320

178mm

0

80

RPM 1450

42

254mm

70 280

30

15

CDS 5018-1

2900 RPM

10

1.1kW

50

5 0

20

2.2kW

40

40

20

100

5kW

30

1.5kW

0

25

40 63 58

50

10

200

10kW

45

40 152mm

MTX 1 x 2-10 A05

300

15kW

203mm

60

1.5m 70 68

72

178mm

1450 RPM

400

1.8m

229mm

80

190mm

CENTRIFUGAL PUMP CHARACTERISTICS

700

50

1.2m

0 0

70 60

0.9m

4

0

500

140 0.9m 35

50

203mm

14

CDS:5018-1

Model: 3196 Size: 1X2-10 Imp. Dwg. B10009 Pattern 56381 Eye Area: 31.6 cm2 FOR ALLOY CONSTRUCTION

30

40

213mm

16

0.6m 58 63 68 70 72

6 140mm

CDS 1591-3

RPM 2900

CENTRIFUGAL PUMP CHARACTERISTICS

25

18

2

220

120

20

8 152mm

50

120

400

22

10

45 7.5kW 5.5kW

10

CDS 2321-4

Model: 3196 Size: 3X4-8G Imp. Dwg. 100-166 / 100-165 Pattern 53816 / 53815 Eye Area 11 cm2

12

75

15kW

15

MTX 3 x 4-8G A70

100

22kW 18.5kW

55

140mm

20

125

60

35 165mm

25

200

150

6.1m

RPM 1450

CENTRIFUGAL PUMP CHARACTERISTICS

24

175

40

Model 3196

CDS 1591-3

Model: 3196 Size: 3X4-8G Imp. Dwg. 100-166 / 100-165 Pattern 53816 / 53815 Eye Area211 cm 2

Appendix E

20 80

CAPACITY

30 120

40 160

200

50

60 240

70 280

320

80 360

m3 /h

0

GPM

CDS 5027-1

CHEM-1A

Appendix E

50 Hz Performance Curves RPM 2900

CENTRIFUGAL PUMP CHARACTERISTICS

2.1m

110 100

50

254mm

2.4m

60

65

2.7m

72

70 60 203mm 50

69

65

30 152mm 60

20

300

71

250

37kW

200

30kW

178mm

40

350

4.3m

80 229mm

0 0

20

0

40

22kW 18.5kW 15kW

100

200

2900 RPM

80

100

300

400

500

600

0.9m 50

60

1.2m

65

70

74 73

203mm

50

20

0

40

100

200

80

100

300

400

120

140

500

600

1450 RPM RPM 2965

CENTRIFUGAL PUMP CHARACTERISTICS

110 100

40

254mm

90

4.5m 60 65

50

5.8m 70 72 7.3m 7475

241mm

80

229mm

70

160

m3 /h

700

GPM

40

60

30

50

0

200

100

150

400

600

200 800

250

1000

350

1400

1600

400

24

20 18 16 14 12

m3 /h

1800

RPM 1450

70

75 77

80

82

241mm

3m

82

229mm

MTX 4 x 6-10G A80

75

10 70

8

0

40 200

80 400

1450 RPM

CHEM-1A

CAPACITY

120

160 600

200

800

30

20 15

1.5kW

10

1.1kW

.75kW

100

40

5 CAPACITY

60

200

80

100

300

400

120

140

500

160

600

700

22 20

m3 /h

50 254mm

60

RPM 1475

GPM

69

72

CDS 5382

Model: 3196 Size: 4X6-10G Imp. Dwg. C06088A Pattern 1E645 Eye Area 167.5 cm2

1.5m

65

74

80

1.8m

70

2m

241mm

74

60

2.2m72 69

216mm

50

65

203mm

40 60

10 8

7.5kW

20

3.7kW

4 0 0 0

20 100

40

CAPACITY

60

200

80 300

100

400

120 500

140 600

160

m3 /h

700

GPM

RPM 1450

CENTRIFUGAL PUMP CHARACTERISTICS

24 70 74 76

20

78

CDS:4029-4

Model: 3196 Size: 4x6-10H Imp. Dwg. C02478A Pattern 63702 Eye Area: 174.8 cm2 70 3m DUCTILE IRON CONSTRUCTION

2m 60

22 260mm

80

18 241mm

80

16 229mm

12

216mm

5m

60

78 76 74

50

203mm

40

11kW

10

30

7.5kW 5.5kW

6

20

4

10

2 240 1000

280 1200

320

m3 /h

1400

GPM

0

0 0

CDS 4034-3

40 200

1450 RPM

32

0

CDS 5382

1475 RPM

MTX 4 X 6-10H A80

30

5.5kW

50

6

0

0

CDS 2991-4

12

10

2 0

24

14

20

4

35

25

8

6

CDS 2991-4

2.2kW

26

16

30

7.5kW

GPM

70

20

0

CDS 2241-6

1.2m

18 229mm

40

11kW

m3 /h

1

14

203mm

360

950 RPM

50

216mm

320

RPM 950

CENTRIFUGAL PUMP CHARACTERISTICS

60

80

0.9m

60

0

70

77

280

0.6m

73

CDS:4034-3

5m 80

240

10

70

Model: 3196MTX Size: 3X4-10H Imp. Dwg. C00262A Pattern 58009 / 58434 Eye Area 17.5 cm2

73

2

GPM

Model: 3196 Size: 4X6-10H Imp. Dwg. C02521A Pattern 63773 Eye Area: 174.8 cm2 ALLOY CONSTRUCTION

2m

60

260mm

60

200

178mm

0 0

0

CDS 5380 CENTRIFUGAL PUMP CHARACTERISTICS

22

.8kW

50

300 1200

70

3

100

2965 RPM

160

50

20

65

4

55kW

CAPACITY

40

120

60 65

254mm

9

5

150

10 0 0

80

11

6 203mm

250

45kW 37kW 30kW

20

30

12

50

300

75kW

CAPACITY

30

8 229mm

200

65

40

7

350

8.8m 9.7m 75 10.4m 74 72 70

203mm

50

0

20

CDS 5380

216mm

60

10

60

CDS 2825-5

Model: 3196 Size: 4X6-10G Imp. Dwg. C06088A Pattern 1E645 Eye Area 167.5 cm2

120

0 0

10

40

3.7kW

2.2kW 55 50 1.5kW 1.1kW

1450 RPM

10

CAPACITY

60

60

60

2.5

20

2.2kW

2.5 0 0

3.7kW

70

69 68

50 65

5

MTX 3 x 4-10H A40

90 80

1.2m

178mm

10

30

5.5kW 60

0.9m 70

17.5

40

7.5kW

152mm

5

70

65

6869

CENTRIFUGAL PUMP CHARACTERISTICS

90

70

65

70

CDS 2825-5

2.1m

60

7.5 152mm

CDS 1695-8

1.8m 75

10 178mm 7.5

GPM

0.6m

50 55

254mm

20 229mm

0

80

20

12.5

700

1.5m

7374 75

17.5 229mm 15

m3 /h

Model: 3196 Size: 3X4-10H Imp. Dwg. C00262A Pattern 58006 / 58434 Eye Area 112 cm 2

27.5 25

160

RPM 1450

CENTRIFUGAL PUMP CHARACTERISTICS

30

22.5 254mm

140

25 22.5

CDS 2241-6

Model: 3196 Size: 3x4-10 Imp. Dwg. 100-598 / 100-597 Pattern 54022 / 54021 Eye Area381 cm 2

15 203mm

50

120

27.5

12.5

100

11kW

CAPACITY

60

MTX 3 x 4-10 A70

150

7.5kW

10

RPM 1450

CENTRIFUGAL PUMP CHARACTERISTICS

30

3.4m

69 71 72

90

Model 3196

CDS 1695-8

Model: 3196 Size: 3X4-10 Imp. Dwg. 100-598/100-597 Pattern 54022/54021 Eye Area 12.57 cm2

120

391

80 400

CAPACITY

120

160 600

200

800

240 1000

280 1200

320 1400

m3 /h

0

GPM

CDS 4029-4

392

50 Hz Performance Curves RPM 2900

CENTRIFUGAL PUMP CHARACTERISTICS

150

1.5m

35

140 330mm

45

2m

50

130

53

56

120 305mm

3m 58

59

5m

110

60

90

45kW

60

20

0

40

100

200

2900 RPM

CAPACITY

60

80

100

300

120

400

600

2m

150

45

55

330mm

140

3m

4m

60

700

GPM

229mm

60

MTX/LTX 2 x 3-13 A30

40

0

60

80

200

22kW 120

100 300

400

0

10

0

40

140

160

180

600

110 100 90 80

60

65

70

279mm

6m

73

254mm

300

55kW

0

40 200

80

160 600

200

800

240 1000

280 1200

320

m3 /h

1400

GPM

55

25 20

2m

77

279mm

229mm 60

75

73

65

70

0

40 200

1470 RPM

80 400

CAPACITY

120

160 600

200

800

90 80 70 60 50 40

3.7kW

25

35

CAPACITY

45 120

55 160

65

75

200

240

85 280

95

320

m3 /h GPM

360

RPM 1450

55

175

50 0.9m

45

55

40 330mm

60

1.2m

150

1.5m 1.8m 70 72

65

2.4m

3m

73

125

73

72 18.5kW

15kW

70 65

75 50

5.5kW

240

280

200

320

m3 /h

1400

GPM

80 300

100

25

400

120

140

500

160

600

700

RPM 950

0

GPM

CDS 2323-4

Model: 3196 Size: 4X6-13 MTX Imp. Dwg. 101-499/101-500 Pattern 54611/54612 Eye Area 181 cm2

25 22.5 20

0.6m 60 65 70

17.5 330mm

74

15 305mm

7.5

m3 /h

CDS 2337-3

80 70 60

0.8m 76

1m 76

50 74

70

10 254mm

40

1.25m

30

7.5kW

65

229mm

5.5kW

5

60

20

3.7kW

2.5

10

0

25

1200

100

CAPACITY

60

12.5 279mm

50

11kW

1000

40

100

11kW

7.5kW

1450 RPM

MTX 4 x 6-13 A80

440

CDS 2337-3

Model: 3196 Size: 3x4-13 Imp. Dwg. B10541 / B10542 Pattern 56693 / 56849 Eye Area592 cm 2

20

0

CDS 3833-3

60

0 0

75

22kW 18.5kW 15kW

7.5kW

0 0

100

5.5kW

10

100

2.5m

254mm

5

110

3m

7.5kW

CENTRIFUGAL PUMP CHARACTERISTICS

125

2.2m

305mm

10

120

61

229mm

0

150

1.4m

15

GPM

CDS 3833-3

63

279mm

25

0

CDS 4997-1

2.5m

64 64

15

0

175

50

30

2m

305mm

30

CDS 1785-5

Model: 3196 Size: 4X6-13 MTX Imp. Dwg. 101-499/101-500 Pattern 54611/54612 Eye Area 181 cm 2

1.6m 60 65 1.8m 70 73 75 77

63

20 229mm

CDS 2338-3

RPM 1470

CENTRIFUGAL PUMP CHARACTERISTICS

40 330mm

RPM 1475

25 254mm

100

60

45

61

360

Model: 3196 Size: 2X3-13MTX Imp. Dwg. 100-539/100-540 Pattern 68643 / 68642 Eye Area 49.4 cm2

1.5m

57

320

m3 /h

5

CAPACITY

120

400

280

80

30 279mm

22kW

0

240

35 305mm

150

45kW 37kW

2900 RPM

35

MTX/LTX 3 x 4-13 A40

200

30kW

20

200

70

1475 RPM

250 75kW

30

160

60

CENTRIFUGAL PUMP CHARACTERISTICS

350

229mm

65

30

50

1m

50

0

400

70

60

40

120

330mm

35

0 0

73

178mm

80

CDS 2338-3

7.6m 9.4m

60 203mm

40

0

GPM

Model: 3196 Size: 3x4-13 Imp. Dwg. B10591 / B10592 Pattern 56693 / 56849 Eye Area 92 cm2

70

50

m3 /h

700

RPM 2900

CENTRIFUGAL PUMP CHARACTERISTICS

120

CAPACITY

30

10

500

4.6m

20

15

CDS 3831-3

3m

40

200

2965 RPM

130

60

2.2kW

20

250

45kW

CAPACITY

70

50

1450 RPM

30kW

0 0

80

5.5kW

10

300

37kW

50

90 1.5m

254mm

55kW

70

100

3.7kW

350

75kW

110

59 58

254mm

254mm

80

0

400

63

66

90

60

CENTRIFUGAL PUMP CHARACTERISTICS

500

65

279mm

100

0.8m

60

0.6m

9m

66

59

1.2m

CDS 3831-3

11m

110

58

25 279mm

0

450

305mm

120

m3 /h

7m

65

130

160

Model: 3196 Size: 2x3-13LTX Imp. Dwg. B10588/B10589 Pattern 68643 / 68642 Eye Area 49.4 cm2

5m

63

56

15

CDS 4995-1

RPM 2965

CENTRIFUGAL PUMP CHARACTERISTICS

160

140

500

0.6m

53

229mm

150

22kW 18.5kW

40

50

CDS:4997-1

Model: 3196 Size: 1.5x3-13 Imp. Dwg. 100-537 Pattern 68720 Eye Area: 31.6 cm2 FOR ALLOY CONSTRUCTION

200

37kW 30kW

50

45

30 305mm

20

250

70 229mm

0

MTX/LTX 11⁄2 x 3-13 A20

300

59 58

254mm

80

450

350

6m

100 279mm

RPM 1450

CENTRIFUGAL PUMP CHARACTERISTICS

35

35 330mm

400

4m

60

Model 3196

CDS:4995-1

Model: 3196 Size: 1.5x3-13 Imp. Dwg. 100-537 Pattern 68720 Eye Area: 31.6 cm2 FOR ALLOY CONSTRUCTION

Appendix E

-2.5

0

0

CDS 1785-5

0

25

0

100

950 RPM

33

50 200

75 300

CAPACITY

100 400

125

500

150 600

700

175 800

200 900

m3 /h

0

GPM

CDS 2323-4

CHEM-1A

Appendix E

50 Hz Performance Curves RPM 1470

CENTRIFUGAL PUMP CHARACTERISTICS

55

175 150

35

2m

50 60

40 330mm

70

75

78

305mm

30

2.5m 80

125 3m 80

279mm

25

254mm

20

100

3.5m 78 75

0 0 0

100

200

500

1000

1470 RPM

CAPACITY

300

400

1500

500

2000

600 2500

700 3000

800

m3 /h

3500

GPM

59

69

3m 74 77 80

305mm 292mm 25 279mm 267mm

30

84 85

125

4m 5m

74

15

37kW 30kW

10

0 0 0

100

200

500

CAPACITY

300

1000

400

1500

500

2000

600 2500

700 3000

RPM 1470

CENTRIFUGAL PUMP CHARACTERISTICS

60

2m 50

381mm

50

2.2m 60 65 70

45 356mm 40

2.4m 75 77 7879 2.7m

35

XLT-X 8 x 10-13 A100

m3 /h

3500

GPM

60

10

100

0 0

100

0

500

200

45kW

1000

400

1500

2000

600 2500

700 3000

RPM 1470

CENTRIFUGAL PUMP CHARACTERISTICS

55 50 381mm 45

65 70 4.3m 75

368mm 356mm

79 815.5m 8283

40 343mm 35

800

m3 /h

3500

GPM

XLT-X 6 x 8-15 A110

83 7.9m 82 81 79 75

330mm 317mm

110kW

15

0 0 0

200 1000

400 2000

1470 RPM

CHEM-1A

600 3000

800

1000

4000

1200 5000

1400 6000

1600

m3 /h

7000

GPM

1.8m

75 79 81

84

2m

70

50

2.2m 84 81

40

3m 79

30

15kW

6

75

4

7.5kW

20

11kW

80 250

160 500

CAPACITY

240

750

320

400

10 480

560

640

m3 /h

1000 1250 1500 1750 2000 2250 2500 2750

1.2m 50

22 381mm

RPM 960

GPM

60

65

70 72

74

76 77

CDS 2326-1

Model: 3196 Size: 6x8-15 Imp. Dwg. 256-115 / 256-116 Pattern 55436 / 55437 Eye Area 323 cm2

1.4m

356mm

18

78 1.6m

70

1.8m

60

2.1m

78

77

16 330mm

76

74

72

40 70

30

11kW 65

20

7.5kW

10

2 0 0

40 200

80

CAPACITY

120

160

400

600

200

800

240 1000

280

320

1200

1400

m3 /h

RPM 960

24

GPM

22 381mm

65

20 368mm

70

75

18 356mm

1.8m 79 81 8283 2.1m

CDS 1886-2

Model: 3196 Size: 8X10-15XLT Imp. Dwg. 256-118 / 256-119 Pattern 55464 / 55465 Eye Area 481.3 cm2

16 330mm

50 40

12 10

65

8

30

30kW

22kW 18.5kW

6

20

4

10

2 0 0 0

100 500

960 RPM

34

70 60

2.4m 8382 81 79 2.7m 75 70

343mm

14 317mm

CDS 2457-2

0

CDS 2326-1

960 RPM

0

50

15kW

4

XLT-X 8 x 10-15 A120

0

CDS 3614-1

6

25

5

69

8

50

10

CAPACITY

59 64

317mm 305mm

14

10

75

90kW 75kW

20

GPM

CDS 2324-1

CDS 3614-1

12 279mm

100

30 25

48

337mm

16

20

0

60

18

CENTRIFUGAL PUMP CHARACTERISTICS

125

1400

20

0

175

m3 /h

14 305mm

0

150

6.7m

1200

RPM 960

22

CDS 2457-2

Model: 3196 Size: 8X10-15XLT Imp. Dwg. 256-118 / 256-119 Pattern 55464 / 55465 Eye Area 481.3 cm2

60

1000

320

960 RPM

CDS 1890-4

1470 RPM

800

280

Model: 3196 Size: 8X10-13 Imp. Dwg. D02490A01 / A02 Pattern 63334 / 63335 Eye Area 389 cm2

0 0

25

500

20

2

50

37kW 30kW

CAPACITY

300

600

240

279mm

75

55kW

22kW

5

200

CENTRIFUGAL PUMP CHARACTERISTICS

70

65

15

160

400

24

125

20

200

CAPACITY

120

8

279mm

25

80

CDS 1890-4

79 3.4m 78 77 4m 75

30

7.5kW

10 40

0

150

11kW

5.5kW

10 267mm

0

175

40

78 75

12 292mm

25 800

50

1.5m

70

0

50

3m

305mm

30

55kW

45kW

Model: 3196 Size: 6x8-15 Imp. Dwg. 256-115 / 256-116 Pattern 55436 / 55437 Eye Area2323 cm 2

330mm

80

80

950 RPM

CDS 3616-1

1470 RPM

78

254mm

0 0

75

22kW

5

55

80

77

1.2m

2

100

85 84

20

75

CENTRIFUGAL PUMP CHARACTERISTICS

150 49

70

4

0

175

45

35 318mm

60

60

14 305mm

24

50

40 337mm

50

330mm

16

CDS 3616-1

Model: 3196 Size: 8x10-13 Imp. Dwg. D02490A Pattern 63334 / 63335 Eye Area 389 cm2

55

1m

18

6

CDS 1894-1

RPM 1470

CENTRIFUGAL PUMP CHARACTERISTICS

60

70

20

8

25

5

22

10

50

30kW

CDS 2324-1

Model: 3196 Size: 6x8-13 Imp. Dwg. 103-608/103-609 Pattern 55329 / 55330 Eye Area5293.5 cm 2

12 279mm

75

22kW 18.5kW

10

XLT-X 6 x 8-13 A90

37kW

70

15

RPM 950

CENTRIFUGAL PUMP CHARACTERISTICS

24

50 45

Model 3196

CDS 1894-1

Model: 3196 Size: 6x8-13 Imp. Dwg. 103-608/103-609 Pattern 55329 / 55330 Eye Area 293.5 cm2

60

393

200 1000

300 1500

CAPACITY

400

500

2000

600 2500

700 3000

800 3500

m3 /h

0

GPM

CDS 1886-2

394

50 Hz Performance Curves RPM 1480

CENTRIFUGAL PUMP CHARACTERISTICS

55 50

381mm

45

73

76

330mm

30

305mm

150 4.1m 78

125

4.6m

76

75kW

20

50

45kW

5 0 0

250

0

350

550

750

2000

850

2500

3000

950

m3 /h

3500

GPM

RPM 1485

CENTRIFUGAL PUMP CHARACTERISTICS

50

60

70

4.9m 75

80 82

45 381mm 40

5.5m

150

7.3m 84

82 80

8.5m

125

75

100

30 330mm 25 305mm

150kW

70

20 50

10 5

90kW

0

1000

CAPACITY

600

2000

800

3000

1000

4000

1200

1400

5000

1600

6000

m3 /h

7000

RPM 1470

CENTRIFUGAL PUMP CHARACTERISTICS

1.8m

59

425mm

64

69

60 406mm

2.1m 72

2.4m

74

2.7m

76

50 381mm

75kW

55kW

30

200

400

CAPACITY

600

800

250

1000

300 1200

350

1400

400

1600

CENTRIFUGAL PUMP CHARACTERISTICS

65

2.1m 60 65 70 74

425mm

60 406mm 55 50

RPM

1470

2.7m 76 78 3.4m 79 4m

381mm

4.6m

200 175

76 74

40 35

330mm

20 0 0

100 500

1470 RPM

200 1000

300 1500

CAPACITY

400

500

2000

55kW 45kW 600 2500

3000

m3 /h

3500

GPM

81

70

78 5.5m 75

60 50

55kW

40 30 20

200 1000

400

CAPACITY

600

2000

800

1000

3000

1200

4000

1400

5000

6000

1600

m3 /h

7000

GPM

58

425mm

RPM 960

27.5 25

63

68

0.9m 71

406mm

22.5

1.2m

73

75

CDS 4359-1

90 80

1.5m 75

381mm

70

73 71

60

356mm 68

50

18.5kW

330mm 63

12.5

40

15kW

58

10

30

7.5

11kW

20

5

10

7.5kW

2.5 0 0

40 200

80

CAPACITY

120

160

400

600

200

800

240 1000

280 1200

320 1400

CENTRIFUGAL PUMP CHARACTERISTICS

32.5

27.5 25

425mm 406mm

22.5 381mm 20 17.5

356mm

CDS 4363-1

0

500

960 RPM

100

80 70

37kW

50 40

22kW 18.5kW 15kW 100

110

90

30kW

65

0 0

35

960

60

330mm

10

CDS 4361-1

RPM

0.9m 6065 1.2m 70 74 1.5m 76 7778 1.8m 2.1m 78 77 76 74 70

12.5

0

0

GPM

Model: 3196XLT Size: 6X8-17 Imp. Dwg. D04564A01 / A02 Pattern 68004 / 68016 Eye Area 345.2 cm 2

35

30

m3 /h

CDS 4359-1

960 RPM

X-17 6 x 8-17

0

CDS 5420

Model: 3196XLT Size: 4X6-17 Imp. Dwg. D04563A Pattern 68003 / 68015 Eye Area 248.4 cm 2

30

7.5 800

80

4.9m

22kW

15

75 700

CDS 5420

30kW

17.5

100

75kW

70

25

83

0

CDS 2327-2

37kW

15

90kW

4.3m

83

20

125

110kW

30

0

X-17 4 x 6-17 A120

150

45 356mm

GPM

70

0

79 78

m3 /h

RPM 985

330mm

0

0

225

800 3500

985 RPM

125

CDS 4361-1

700 3000

Model: 3196 Size: 8X10-16H Imp. Dwg. C06161A02 Pattern 1E684 Eye Area: 532.3 cm2

3.7m 81

381mm

0 0

GPM

Model: 3196XLT Size: 6X8-17 Imp. Dwg. D04564A01 / A02 Pattern 68004 / 68016 Eye Area 345 cm 2

75 70

1800

m3 /h

600 2500

CENTRIFUGAL PUMP CHARACTERISTICS

22 406mm 20

500

2000

3m 70 75 78

24

400

1500

4

CDS 4357-1

1470 RPM

50 60

26 425mm

CAPACITY

300

1000

6

75

30kW

20

200

500

8

100

45kW 37kW

64

25

0

20 100

10

150

35 330mm

200

30

CENTRIFUGAL PUMP CHARACTERISTICS

225

72

69

150

40 22kW 18.5kW

12

175

356mm

100

50

15kW

14

3.4m 74

50

60

76 2m

CDS 4357-1

76

0

70 1.6m

16 356mm

0

200

3m

55

0

78

1.4m

73

18

GPM

Model: 3196XLT Size: 4X6-17 Imp. Dwg. D04563A Pattern 68003 / 68015 Eye Area4248.4 cm 2

75

40

X-17 8 x 10-16H A120

CDS 5418

1485 RPM

70

76

78 1.8m

960 RPM

25

55kW

400

80

1.2m

330mm

0

50

75kW

200

18

0 0

75

110kW

60

15

0 0

0

175

6.4m

84

356mm

35

73

20 356mm

CDS 5418

Model: 3196 Size: 8x10-16H Imp. Dwg. C06161A02 Pattern 1E684 Eye Area: 532.3 cm 2

60 50 406mm

70

8

CDS 2035-6

1480 RPM

55

381mm

22

90

6

650

1500

60 65

10

25 CAPACITY

450

1000

24

12

MINIMUM CONTINUOUS OPERATING POINT

10

26

14 305mm

75

CDS 2327-2

Model: 3196 Size: 8x10-15G Imp. Dwg. 256-121/256-122 Pattern 55758 / 55759 Eye Area9406.5 cm 2

16

55kW

15

45

XLT-X 8 x 10-15G A120

100

25

65

175

3.2m 78 79 3.6m 79

35

RPM 960

CENTRIFUGAL PUMP CHARACTERISTICS

28

2.8m

356mm

40

Model 3196

CDS 2035-6

Model: 3196 Size: 8x10-15G Imp. Dwg. 256-121/256-122 Pattern 55758 / 55759 Eye Area 406.5 cm 2

60

Appendix E

200 1000

300 1500

CAPACITY 400 500

2000

30

600 2500

700 3000

800 3500

m3 /h

0

GPM

CDS 4363-1

CHEM-1A

Appendix E

50 Hz Performance Curves RPM 1470

CENTRIFUGAL PUMP CHARACTERISTICS

65 60 55 50 45 40 35 30

416mm

59

69 74

3m 79

3.7m 4.6m 6.1m 82

357mm

20

0 0

200 1000

400 2000

1470 RPM

CHEM-1A

600

150

79

3000

1000

4000

27.5 25 22.5

17.5

58 416mm

68

73

78

1.5m

2.1m

82

383mm

100

357mm

5000

6000

1600 7000

GPM

70 60

73

50 37kW

40

30kW

30

22kW

7.5

20

15kW

5

1400

80

78

10

75

1200

90

3m

82

12.5

m3 /h

CDS 4367-2

Model: 3196XLT Size: 8X10-17 Imp. Dwg. D04568A Pattern 68028 / 68031 Eye Area5457.4 cm 2

1.2m

15 330mm

125

50

CAPACITY

800

30

20

55kW

15

X-17 8 x 10-17

175

110kW 90kW 75kW

25

0

200

150kW

330mm

RPM 960

CENTRIFUGAL PUMP CHARACTERISTICS

225

82

383mm

Model 3196

CDS 4365-2

Model: 3196XLT Size: 8X10-17 Imp. Dwg. D04568A Pattern 68028 / 68031 Eye Area 457.4 cm 2

70

395

10

2.5

0

0 0 0

CDS 4365-2

100 500

960 RPM

36

200 1000

300 1500

CAPACITY

400

500

2000

600 2500

700 3000

800 3500

m3 /h GPM

0

CDS 4367-2

F Minor Loss Coefficient The table on the minor loss coefficient presented here has been adapted from the following reference:

Ronald Darby, Ron Darby, and Ravi P. Chhabra. Chemical Engineering Fluid Mechanics. CRC Press, 2017.

DOI: 10.1201/9781003049272-F

397

398

Table F.1 Minor Loss Coefficient

Appendix F

G Sample Project Topics Examples for possible thermal system design project topics are presented here. For more details, readers can refer to the Further Reading provided at the end. The sample project topics can also be expanded for different applications. 1. Cooling of battery banks of an electric car using combined radiator and evaporator system 2. Cooling of battery banks of an electric car using phase change material 3. A micro gas turbine power plant for campus use 4. Centralized air conditioning plant for student activity center 5. Supercomputer/Data center cooling system 6. Compact heat exchanger for electronic cooling 7. High power laser cooling system 8. Piping network for residential units 9. Cold storage system design 10. Heat exchanger system for waster heat recovery 11. Solar energy based heating system 12. Solar power generation system using low boiling point fluid 13. Design of sports car structures 14. Design of Stirling refrigerator for space shuttle 15. Solar thermoelectric generator 16. Thermoelectric cooling for mobile devices/laser 17. Thermoelectric cooling helmet 18. Helmet cooling using phase change materials 19. Thermoelectric power generation from exhaust 20. Automotive thermoelectric air conditioner from exhaust 21. Radioisotope thermoelectric generator 22. Car seat climate control 23. Thermoelectric air heating/cooling system 24. Wearable thermoelectric cooler 25. Thermoelectric power generation from body heat

FURTHER READING L. Aichmayer, J. Spelling, B. Laumert, and T. Fransson. Micro gas-turbine design for small-scale hybrid solar power plants. Journal of Engineering for Gas Turbines and Power, 135(11), 2013. doi: 10.1115/1.4025077. L. Cao, J. Han, L. Duan, and C. Huo. Design and experiment study of a new thermoelectric cooling helmet. Procedia Engineering, 205:1426–1432, 2017. doi: 10.1016/j.proeng.2017.10.339.

DOI: 10.1201/9781003049272-G

399

400

Appendix G

A. Capozzoli and G. Primiceri. Cooling systems in data centers: State of art and emerging technologies. Energy Procedia, 83:484–493, 2015. doi: 10.1016/j. egypro.2015.12.168. M. Cosnier, G. Fraisse, and L. Luo. An experimental and numerical study of a thermoelectric air-cooling and air-heating system. International Journal of Refrigeration, 31(6):1051–1062, 2008. doi: 10.1016/j.ijrefrig.2007.12.009. Y. Du, K. Cai, S. Chen, H. Wang, S. Z. Shen, R. Donelson, and T. Lin. Thermoelectric fabrics: Toward power generating clothing. Scientific Reports, 5(1), 2015. doi: 10.1038/srep06411. A. Elarusi, A. Attar, and H. Lee. Optimal design of a thermoelectric cooling/heating system for car seat climate control (CSCC). Journal of Electronic Materials, 46(4):1984–1995, 2016. doi: 10.1007/s11664-016-5043-y. M. N. H. M. Hilmin, M. F. Remeli, B. Singh, and N. D. N. Affandi. Thermoelectric power generations from vehicle exhaust gas with TiO2 nanofluid cooling. Thermal Science and Engineering Progress, 18:100558, 2020. doi: 10.1016/ j.tsep.2020.100558. T. C. Holgate, R. Bennett, T. Hammel, T. Caillat, S. Keyser, and B. Sievers. Increasing the efficiency of the multi-mission radioisotope thermoelectric generator. Journal of Electronic Materials, 44(6):1814–1821, 2014. doi: 10.1007/s11664-014-3564-9. Z. Z. Jiling Li. Battery thermal management systems of electric vehicles. Master’s thesis, Chalmers University of Technology, 2014. R. Jilte and R. Kumar. Numerical investigation on cooling performance of Li-ion battery thermal management system at high galvanostatic discharge. Engineering Science and Technology, an International Journal, 21(5):957–969, 2018. doi: 10.1016/j.jestch.2018.07.015. H. Jouhara, N. Khordehgah, S. Almahmoud, B. Delpech, A. Chauhan, and S. A. Tassou. Waste heat recovery technologies and applications. Thermal Science and Engineering Progress, 6:268–289, 2018. doi: 10.1016/j.tsep.2018.04. 017. S. S. Katoch and M. Eswaramoorthy. A detailed review on electric vehicles battery thermal management system. IOP Conference Series: Materials Science and Engineering, 912:042005, 2020. doi: 10.1088/1757-899x/912/4/042005. J. Kim, J. Oh, and H. Lee. Review on battery thermal management system for electric vehicles. Applied Thermal Engineering, 149:192–212, 2019. doi: 10.1016/j.applthermaleng.2018.12.020. R. A. Kishore, A. Nozariasbmarz, B. Poudel, M. Sanghadasa, and S. Priya. Ultrahigh performance wearable thermoelectric coolers with less materials. Nature Communications, 10(1), 2019. doi: 10.1038/s41467-019-09707-8. D. Laing, C. Bahl, T. Bauer, D. Lehmann, and W.-D. Steinmann. Thermal energy storage for direct steam generation. Solar Energy, 85(4):627–633, 2011. doi: 10.1016/j.solener.2010.08.015.

Appendix G

401

Y. Lee, E. Kim, and K. G. Shin. Efficient thermoelectric cooling for mobile devices. In 2017 IEEE/ACM International Symposium on Low Power Electronics and Design (ISLPED). IEEE, 2017. doi: 10.1109/islped.2017.8009199. H. Manchanda and M. Kumar. Study of water desalination techniques and a review on active solar distillation methods. Environmental Progress & Sustainable Energy, 37(1):444–464, 2017. doi: 10.1002/ep.12657. J. B. Marcinichen, J. A. Olivier, and J. R. Thome. On-chip two-phase cooling of datacenters: Cooling system and energy recovery evaluation. Applied Thermal Engineering, 41:36–51, 2012. doi: 10.1016/j.applthermaleng.2011.12. 008. G. J. Marshall, C. P. Mahony, M. J. Rhodes, S. R. Daniewicz, N. Tsolas, and S. M. Thompson. Thermal management of vehicle cabins, external surfaces, and onboard electronics: An overview. Engineering, 5(5):954–969, 2019. doi: 10.1016/j.eng.2019.02.009. A. R. Mukaffi, R. S. Arief, W. Hendradjit, and R. Romadhon. Optimization of cooling system for data center case study: PAU ITB data center. Procedia Engineering, 170:552–557, 2017. doi: 10.1016/j.proeng.2017.03.088. M. Olsen, E. Warren, P. Parilla, E. Toberer, C. Kennedy, G. Snyder, S. Firdosy, B. Nesmith, A. Zakutayev, A. Goodrich, C. Turchi, J. Netter, M. Gray, P. Ndione, R. Tirawat, L. Baranowski, A. Gray, and D. Ginley. A hightemperature, high-efficiency solar thermoelectric generator prototype. Energy Procedia, 49:1460–1469, 2014. doi: 10.1016/j.egypro.2014.03.155. F. Tan and S. Fok. Cooling of helmet with phase change material. Applied Thermal Engineering, 26(17–18):2067–2072, 2006. doi: 10.1016/j.applthermaleng. 2006.04.022. X. Wang, W. Dai, J. Zhu, S. Chen, H. Li, and E. Luo. Design of a two-stage high-capacity stirling cryocooler operating below 30 K. Physics Procedia, 67:518–523, 2015. doi: 10.1016/j.phpro.2015.06.069.

Index A concurrent, 8 Actuator, 242, 244 detail, 24, 42, 234 ADI, 310 evaluation, 10, 14, 17, 43, 47, 57, AI, 35, 41, 149, 170, 200, 214, 216, 223, 281, 331 244–245, 258, 264–266, DFX, 9 269–270 Differential equations ANN, 223, 228–234, 236–245, 265–266 elliptic problem, 291–293 Approximation, 2, 15, 28, 31, 37, 231, equilibrium problem, 290, 293 273, 284–287, 322 hyperbolic problem, 291 Artificial intelligence, 130, 223, 225, marching problem, 290, 308 227, 229, 231, 233, 235, 237, ordinary, 15, 19, 28, 35, 41–42, 239, 241, 243, 245–247 273, 275, 277, 279, 289 Artificial neural network, 228, 246–247 parabolic problem, 291, 293 Automation, 14, 18 partial, 15, 28, 42, 66, 113, 207, 240, 266, 289, 291, 293, 295, B 297, 299, 301, 303, 305, 307, Battery, 323–324 309, 311 Bell–Delaware, 161, 183–184 Differentiation, 19, 273, 281, 283, 285, Bisection, 42, 249–253, 278 287, 293, 305–306 Blasius, 277–278, 322 Diffusion, 290, 292–299, 302, 304–308, Boundary conditions, 28, 32, 34–35, 38, 311, 314, 318 58, 241, 277–278, 291, 320, Dimensional analysis, 15, 35, 195, 204 331 Dimensionless, 15, 27, 32, 35–38, 85–87, 94, 113, 173, 190, 196, C 209, 314, 321, 331 CFD, 35, 149, 227, 241, 313 Discrete, 38, 44, 258–260, 262, 281, Code, 24, 42, 227 284, 338 Composite, 23, 53–54, 56, 118 Consistency, 43, 210, 296–299 Convergence, 47–48, 217, 250–252, 255–256, 268–269, 271, 338 Copyright, 18 Crank–Nicolson, 293, 295–299, 302, 306–307, 311, 318 Curve fitting, 19, 24, 30, 38–39, 41, 51–52, 102, 195, 199, 214, 233, 258, 261–262 D Decomposition, 41 Design CAD, 19–20, 71

E Eigenvalue, 253–254, 256–257, 290 Eigenvector, 253–257 Electric, 25, 30, 32–33, 46, 53, 113 Electrochemical, 110, 323–325, 327–329 Electrolyzer, 104–106, 108–109, 111–112 Electronic, 1, 10–11, 13, 21, 25–26, 36, 121, 133

403

404

Energy solar, 1, 11, 21, 26, 30, 32, 94–96, 99–100, 104–106, 112, 114–115, 121, 126–128, 130–131, 191 source, 6, 12, 27, 37, 81–82, 117–118, 129, 141–143, 148, 151–155, 162, 172, 225, 237, 239, 244, 290, 296, 308, 310, 313–314 storage, 4, 11, 19, 21, 23, 30–31, 56, 91–92, 108, 121–123, 125–126, 130–131, 265, 270 Engine, 1, 21, 142–143, 180, 187, 225 Error estimation, 261 Euler method, 273–274 Euler number, 158 Experimental, 3, 15–16, 24, 43, 152–153, 206, 214, 228, 233–234, 236, 238, 241, 258, 331–332, 336–337 Expert system, 223–225, 246 Explicit scheme, 293–294, 298–299, 302, 304–305, 310 F Fabrication, 7, 11, 18 Finite difference, 15, 19, 24, 42, 273, 278, 280–281, 296, 314 Formulation, 10, 21, 47, 178, 314–315, 320 Fuel, 6–7, 51–52, 67–69, 73–79, 117, 130, 142, 323–329 Fuel cell, 323–329 Furnace, 12–13, 21, 30, 32–33, 35, 43, 46, 48, 57 G GA, 228, 244–245 Gauss–Jordan elimination, 41 Gauss–Seidel, 266, 269–270, 308, 310, 320 Gaussian elimination, 41, 263–266, 270 Goodness of fit, 262

Index

Grid, 24, 43, 273, 281–284, 288, 293, 305–306, 309, 318 H Hardware, 11, 16, 224 Hardy Cross, 208, 211, 214–217, 220 Hazen–Williams, 208, 215–216, 220 Heat conduction, 2, 23, 27, 34, 53, 331–332, 337–338 convection, 28, 53–54, 57, 96–98, 114, 134, 150–152, 154–155, 210, 241, 314, 318 exchanger, 6–7, 16–17, 45–46, 58, 70, 74, 77–78, 81–83, 85, 87–89, 91, 102, 106, 114, 121, 127, 133–141, 143–151, 153, 155–157, 159–161, 163–183, 185–188, 214–216, 234, 236–237, 242–244, 247 fin, 53–58, 133–136, 139–141, 150, 172–173, 175–178, 234–236, 242, 246 fouling resistance, 53, 138, 140, 142–143, 174, 182, 187 lumped, 25, 28, 31, 33, 35, 57 radiation, 29, 33, 95, 98–99, 106, 112, 115 radiative, 29, 95, 98 radiator, 21, 135, 180 rejection, 21, 180 removal, 4, 10, 13, 21, 106 transfer coefficient, 24–25, 28–29, 33, 36, 53, 57–58, 92, 95–98, 100, 103, 136, 138, 140, 147, 150, 156, 160–161, 164, 168–170, 174, 180, 183, 185–188, 235 treatment, 4, 21, 30, 32–33, 35, 44, 46, 48, 56–57 HHV, 69 Hydraulic diameter, 155–156, 174, 176–178

Index

Hydrogen, 104–105, 108–112, 114–115, 323 I Idealization, 15 Implicit scheme, 294 Initial condition, 32, 34, 241, 273, 276–278, 290, 292–294, 305, 308, 310–311 Insulation, 32–35, 43, 57, 95–96, 121, 127, 130 Integration, 19, 143, 281, 283–285, 287 Inverse problem adjoint, 334–335, 337 conjugate gradient, 231, 332 descent, 333, 335, 337 functional, 39, 119–121, 123, 333–337 regularization, 332, 336–338 sensitivity, 44, 206–207, 239, 333, 337 step size, 243, 333, 335, 337 Inversion, 19, 41, 279, 310 Iteration, 8, 49, 58, 178, 210, 213, 217, 220, 233, 250–257, 266–272, 333, 335, 337 J Jacobi iteration, 266–270 L Least square, 39, 258 LMTD, 140, 143–146, 160, 170, 186 M Manufacturing, 1, 3, 8–9, 17, 21 Materials, 1–3, 5, 7, 10–11, 18, 44, 117–119, 121–124, 126–127, 130–131, 216, 324 Matrix, 19, 41, 164, 172–174, 176–178, 186, 188, 211, 253–255, 257–258, 263, 265–270, 279, 308–310

405

Mechanics, 1, 35, 289, 313–314, 319, 321 Minor loss, 49–50, 189–190, 207, 214–215, 219 Model analog, 23–24, 228 descriptive, 23, 224 empirical, 24, 35, 38, 50, 96, 190, 237 mathematical, 20, 23–25, 29–33, 35, 41–43, 46, 57–58, 225, 238, 258, 308, 331 physical, 4, 14–16, 18, 23–24, 30, 35, 38, 43, 61–62, 65–66, 108, 113, 117–118, 128, 147, 150, 153, 157, 228, 233, 236, 244, 273, 289–290, 308, 331, 337 predictive, 23 N Navier–Stokes, 289, 308, 313 Newton’s method, 42, 47 Newton–Cotes, 284 Newton–Raphson, 42, 47, 52, 58, 249, 251–253 Nonlinear, 15, 20, 41–42, 47, 228, 230, 258, 263, 270–271, 289 NPSH, 193, 202–204, 226–227 NTU, 84, 87–88, 91–94, 147–148, 180–181, 187 Numerical, 2–3, 15–16, 18–20, 23–24, 38, 41–44, 149, 152, 223, 225, 231, 240, 249, 262, 265, 273, 276, 278–279, 281–285, 287, 293, 297–298, 302, 305–306, 310, 313, 338 Nusselt number, 97, 114, 150–151, 156–157 O Optimal, 6–7, 11, 14, 17, 19, 47, 87, 91, 123

406

Optimization constraints, 4, 10–11, 13, 16–17, 21, 44, 47, 85, 119, 121–122, 124, 177–178, 226 objective function, 1, 17, 120, 123, 177–178, 258–260, 262 OTEC, 101, 103–105, 107, 110–111, 113, 115 P Patent, 6 PEM, 105–106, 108, 110–112, 114 PEMFC, 327–328 Physics informed, 240, 244 Pilot, 8–9 PINN, 240–241 Pollution, 4–5 Polynomial, 39–40, 214, 233, 253, 258, 260–262, 284, 286, 338 Power iteration, 254–257 Prandtl number, 24, 150, 157, 173 Predictor–corrector, 42, 274 Pseudo-transient, 310–311 Pump performance curve, 40, 50, 192, 194–195, 198–199, 201–202, 214 piping network, 49–50, 189, 191, 206–207, 209, 212, 220 selection, 4, 7, 13, 43, 47–48, 117–123, 125–127, 129–131, 150, 188, 192, 195, 198, 201, 204–205, 207, 226–227, 236, 245–246, 257 specific speed, 195–197, 203, 226 R Rankine, 14, 101, 107, 111, 115 Reynolds number, 50, 85–86, 114, 147, 150, 156–158, 164–165, 173, 181, 190, 235–236, 314 Round-off, 265, 297 Runge–Kutta, 42, 274

Index

S Safety, 1–2, 4–5, 10, 12, 14, 17–18, 21, 44, 205, 246 Sensor, 241 Shooting method, 277–279 Simulation, 5, 14–16, 18, 20, 23–25, 27, 29, 31, 33, 35, 37, 39, 41, 43–49, 51, 53, 55, 57, 59, 111–112, 118, 130, 227, 320 SOFC, 327–328 Software, 19–20, 42, 117–118, 208, 225 Specification, 8, 10, 45, 63, 117, 121, 169, 201, 205, 226–227 Spectral, 42 Stability, 127–129, 291, 296–297, 302, 304–305, 309, 311, 332 Standard, 4, 42, 63, 66–67, 69, 118, 187, 204, 237, 278–279, 311, 324, 336–337 Stochastic, 45, 245 Stream function, 314–315, 320–321 Streamline, 315, 320 T Testing, 9, 18, 220, 233, 243 Thermal conductivity, 27, 30, 37–38, 53, 56, 96–97, 117, 119, 121–122, 124–127, 139, 150, 154, 182, 187, 237, 292, 331–334, 338 Thermodynamics air-conditioning, 12, 15–16, 172 boiler, 12, 14, 75–76, 143 Carnot, 64 closed system, 62–64, 70–71 compressor, 10, 13, 15, 23, 45–46, 48, 51–52, 72, 74–75, 78, 80–82, 89–90, 103, 172, 207, 225–226 entropy, 62–63, 65, 68–69, 79, 83, 85–86, 88, 93–94, 113–114, 325

407

Index

evaporator, 13, 15, 89–90, 102–104, 107–108, 110, 140, 147, 149 exergy, 61–83, 85, 87–89, 91–95, 97, 99–101, 103–105, 107, 109, 111–115 expander, 74 gas mixture, 67, 113 open system, 64 refrigeration, 10, 44, 57, 76, 88–89, 91, 121, 133, 159, 172 response time, 12–13, 25 reversible, 64–65, 69, 90, 109, 125, 325–327, 329 specific heat, 25, 33, 82, 93, 96, 102, 106, 113, 117, 125, 180 thermo-economics, 20

throttle, 205 vapor compression, 46–47, 89, 114 Thermophysical, 117, 123, 126, 131, 236 Transient, 25–26, 28, 92, 117, 292–293, 296, 298–299, 302, 304–305, 310–311 Trapezoidal, 284–288 Truncation error, 274, 282, 297–298 Turbine, 1, 14, 23, 49, 51–52, 61, 72–75, 78, 80–81, 102–104, 107, 111–112, 127, 130, 189, 207, 214, 225–226 V Validation, 43, 223, 237 Vorticity, 314–318, 320