# Applications of Mathematical Heat Transfer and Fluid Flow Models in Engineering and Medicine

Abram S. Dorfman

Wiley-ASME Press Series

1

2017

en

9781119320739

**Applications of mathematical heat transfer and fluid flow models in engineering and medicine**

Abram S. Dorfman, University of Michigan, USA

*Engineering and medical applications of cutting-edge heat and flow models*

* *

This book presents innovative efficient methods in fluid flow and heat transfer developed and widely used over the last fifty years. The analysis is focused on mathematical models which are an essential part of any research effort as they demonstrate the validity of the results obtained.

The universality of mathematics allows consideration of engineering and biological problems from one point of view using similar models. In this book, the current situation of applications of modern mathematical models is outlined in three parts. Part I offers in depth coverage of the applications of contemporary conjugate heat transfer models in various industrial and technological processes, from aerospace and nuclear reactors to drying and food processing. In Part II the theory and application of two recently developed models in fluid flow are considered: the similar conjugate model for simulation of biological systems, including flows in human organs, and applications of the latest developments in turbulence simulation by direct solution of Navier-Stokes equations, including flows around aircraft. Part III proposes fundamentals of laminar and turbulent flows and applied mathematics methods. The discussion is complimented by 365 examples selected from a list of 448 cited papers, 239 exercises and 136 commentaries.

Key features:

- Peristaltic flows in normal and pathologic human organs.
- Modeling flows around aircraft at high Reynolds numbers.
- Special mathematical exercises allow the reader to complete expressions derivation following directions from the text.
- Procedure for preliminary choice between conjugate and common simple methods for particular problem solutions.
- Criterions of conjugation, definition of semi-conjugate solutions.

This book is an ideal reference for graduate and post-graduate students and engineers.

Series Preface xiii

Preface xv

Acknowledgments xxvii

About the Author xxix

Nomenclature xxxi

**Part I APPLICATIONS IN CONJUGATE HEAT TRANSFER**

Introduction 1

*When and why Conjugate Procedure is Essential *1

*A Core of Conjugation *3

**1 Universal Functions for Nonisothermal and Conjugate Heat Transfer 5**

1.1 Formulation of Conjugate Heat Transfer Problem 5

1.2 Methods of Conjugation 9

*1.2.1 Numerical Methods *9

*1.2.2 Using Universal Functions *10

1.3 Integral Universal Function (Duhamel’s Integral) 10

*1.3.1 Duhamel’s Integral Derivation *10

*1.3.2 Influence Function *12

1.4 Differential Universal Function (Series of Derivatives) 13

1.5 General Forms of Universal Function 15

Exercises 1.1–1.32 16

1.6 Coefficients *g**k *and Exponents *C*1 and *C*2 for Laminar Flow 19

*1.6.1 Features of Coefficients g**k **of the Differential Universal Function *19

*1.6.2 Estimation of Exponents C*1 *and C*2 *for Integral Universal Function *22

1.7 Universal Functions for Turbulent Flow 24

Exercises 1.33–1.47 27

1.8 Universal Functions for Compressible Low 28

1.9 Universal Functions for Power-Law Non-Newtonian Fluids 29

1.10 Universal Functions for Moving Continuous Sheet 32

1.11 Universal Functions for a Plate with Arbitrary Unsteady Temperature Distribution 34

1.12 Universal Functions for an Axisymmetric Body 35

1.13 Inverse Universal Function 36

*1.13.1 Differential Inverse Universal Function *36

*1.13.2 Integral Inverse Universal Function *37

1.14 Universal Function for Recovery Factor 38

Exercises 1.48–1.75 41

**2 Application of Universal Functions 45**

2.1 The Rate of Conjugate Heat Transfer Intensity 45

*2.1.1 Effect of Temperature Head Distribution *45

*2.1.2 Effect of Turbulence *50

*2.1.3 Effect of Time-Variable Temperature Head *58

*2.1.4 Effects of Conditions and Parameters in the Inverse Problems *60

*2.1.5 Effect of Non-Newtonian Power-Law Rheology Fluid Behavior *66

*2.1.6 Effect of Mechanical Energy Dissipation *67

*2.1.7 Effect of Biot Number as a Measure of Problem Conjugation *68

Exercises 2.1–2.33 70

2.2 The General Convective Boundary Conditions 73

*2.2.1 Accuracy of Boundary Condition of the Third Kind *73

*2.2.2 Conjugate Problem as an Equivalent Conduction Problem *76

2.3 The Gradient Analogy 78

2.4 Heat Flux Inversion 82

2.5 Zero Heat Transfer Surfaces 84

2.6 Optimization in Heat Transfer Problems 86

*2.6.1 Problem Formulation *87

*2.6.2 Problem Formulation *89

*2.6.3 Problem Formulation *92

Exercises 2.34–2.82 95

**3 Application of Conjugate Heat Transfer Models in External and Internal Flows 102**

3.1 External Flows 102

*3.1.1 Conjugate Heat Transfer in Flows Past Thin Plates *102

Exercises 3.1–3.38 123

*3.1.2 Conjugate Heat Transfer in Flows Past Bodies *126

3.2 Internal Flows-Conjugate Heat Transfer in Pipes and Channels Flows 141

**4 Specific Applications of Conjugate Heat Transfer Models 155**

4.1 Heat Exchangers and Finned Surfaces 155

*4.1.1 Heat Exchange Between Two Fluids Separated by a Wall (Overall Heat Transfer Coefficient) *155

*4.1.2 Applicability of One-Dimensional Models and Two-Dimensional Effects *166

*4.1.3 Heat Exchanger Models *170

*4.1.4 Finned Surfaces *175

4.2 Thermal Treatment and Cooling Systems 180

*4.2.1 Treatment of Continuous Materials *180

*4.2.2 Cooling Systems *185

4.3 Simulation of Industrial Processes 196

4.4 Technology Processes 202

*4.4.1 Heat and Mass Transfer in Multiphase Processes *202

*4.4.2 Drying and Food Processing *208

Summary of Part I 219

*Effect of Conjugation *219

**Part II APPLICATIONS IN FLUID FLOW**

**5 Two Advanced Methods 225**

5.1 Conjugate Models of Peristaltic Flow 225

*5.1.1 Model Formulation *225

*5.1.2 The First Investigations *228

*5.1.3 Semi-Conjugate Solutions *230

Exercises 5.1–5.19 236

*5.1.4 Conjugate Solutions *237

Exercises 5.20–5.31 243

5.2 Methods of Turbulence Simulation 244

*5.2.1 Introduction *244

*5.2.2 Direct Numerical Simulation *244

*5.2.3 Large Eddy Simulation *245

*5.2.4 Detached Eddy Simulation *247

*5.2.5 Chaos Theory *249

Exercises 5.32–5.44 249

**6 Applications of Fluid Flow Modern Models 251**

6.1 Applications of Fluid Flow Models in Biology and Medicine 251

*6.1.1 Blood Flow in Normal and Pathologic Vessels *251

*6.1.2 Abnormal Flows in Disordered Human Organs *261

*6.1.3 Simulation of Biological Transport Processes *267

6.2 Application of Fluid Flow Models in Engineering 273

*6.2.1 Application of Peristaltic Flow Models *273

*6.2.2 Applications of Direct Simulation of Turbulence *278

**Part III FOUNDATIONS OF FLUID FLOW AND HEAT TRANSFER**

**7 Laminar Fluid Flow and Heat Transfer 295**

7.1 Navier-Stokes, Energy, and Mass Transfer Equations 295

*7.1.1 Two Types of Transport Mechanism: Analogy Between Transfer Processes *295

*7.1.2 Different Forms of Navier-Stokes, Energy, and Diffusion Equations *297

7.2 Initial and Boundary Counditions 302

7.3 Exact Solutions of Navier-Stokes and Energy Equations 303

*7.3.1 Two Stokes Problems *303

*7.3.2 Steady Flow in Channels and in a Circular Tube *304

*7.3.3 Stagnation Point Flow (Hiemenz Flow) *304

*7.3.4 Couette Flow in a Channel with Heated Walls *306

*7.3.5 Adiabatic Wall Temperature *306

*7.3.6 Temperature Distributions in Channels and in a Tube *306

7.4 Cases of Small and Large Reynolds and Peclet Numbers 307

*7.4.1 Creeping Approximation (Small Reynolds and Peclet Numbers) *307

*7.4.2 Stokes Flow Past Sphere *308

*7.4.3 Oseen’s Approximation *308

*7.4.4 Boundary Layer Approximation (Large Reynolds and Peclet Numbers) *309

7.5 Exact Solutions of Boundary Layer Equations 315

*7.5.1 Flow and Heat Transfer on Isothermal Semi-infinite Flat Plate *315

*7.5.2 Self-Similar Flows of Dynamic and Thermal Boundary Layers *319

7.6 Approximate Karman-Pohlhausen Integral Method 320

*7.6.1 Approximate Friction and Heat Transfer on a Flat Plate *320

*7.6.2 Flows with Pressure Gradients *322

7.7 Limiting Cases of Prandtl Number 323

7.8 Natural Convection 324

**8 Turbulent Fluid Flow and Heat Transfer 327**

8.1 Transition from Laminar to Turbulent Flow 327

8.2 Reynolds Averaged Navier-Stokes Equation (RANS) 328

*8.2.1 Some Physical Aspects *328

*8.2.2 Reynolds Averaging *329

*8.2.3 Reynolds Equations and Reynolds Stresses *330

8.3 Algebraic Models 331

*8.3.1 Prandtl’s Mixing-Length Hypothesis *331

*8.3.2 Modern Structure of Velocity Profile in Turbulent Boundary Layer *332

*8.3.3 Mellor-Gibson Model *334

*8.3.4 Cebeci-Smith Model *335

*8.3.5 Baldwin-Lomax Model *336

*8.3.6 Application of the Algebraic Models *337

*8.3.7 The 1/2 Equation Model *338

*8.3.8 Applicability of the Algebraic Models *339

8.4 One-Equation and Two-Equations Models 339

*8.4.1 Turbulence Kinetic Energy Equation *340

*8.4.2 One-Equation Models *340

*8.4.3 Two-Equation Models *341

*8.4.4 Applicability of the One-Equation and Two-Equation Models *343

**9 Analytical and Numerical Methods in Fluid Flow and Heat Transfer 344**

Analytical Methods 344

9.1 Solutions Using Error Functions 344

9.2 Method of Separation Variables 345

*9.2.1 General Approach, Homogeneous, and Inhomogeneous Problems *346

*9.2.2 One-Dimensional Unsteady Problems *347

*9.2.3 Orthogonal Eigenfunctions *348

*9.2.4 Two-Dimensional Steady Problems *351

9.3 Integral Transforms 353

*9.3.1 Fourier Transform *353

*9.3.2 Laplace Transform *356

9.4 Green’s Function Method 358

Numerical Methods 361

9.5 What Method is Proper? 361

9.6 Approximate Methods for Solving Differential Equations 363

9.7 Computing Flow and Heat Transfer Characteristics 368

*9.7.1 Control-Volume Finite-Difference Method *368

*9.7.2 Control-Volume Finite-Element Method *371

**10 Conclusion 373**

References 376

Author Index 397

Subject Index 409

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