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.

Abram S. Dorfman, Doctor of Science, Ph. D. was born in 1923 in Kiev. He was a leading scientist in fluid mechanic and heat transfer at the Institute of Thermophiysics of the Ukrainian Academy of Science and associate editor ofPromyshlennaya Teploteknika translated by Wiley asApplied Thermal Science. He earned his Ph.D. with a thesisInvestigation of Supersonic Flows in Nozzles and received a Doctor of Science degree with a thesis and a bookHeat Transfer in Flows around the Nonisothermal bodies. He emigrated to the United States in 1990 and continues his research as a visiting professor at the University of Michigan. During that time, he published several articles in leading American journals and two books.Dr. Dorfman has been an adviser to Ph. D. students and has published more than 140 papers and four books. More than 50 of his papers published in Russian have been translated into English.

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 Essential1

A Core of Conjugation3

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 Methods9

1.2.2 Using Universal Functions10

1.3 Integral Universal Function (Duhamels Integral) 10

1.3.1 Duhamels Integral Derivation10

1.3.2 Influence Function12

1.4 Differential Universal Function (Series of Derivatives) 13

1.5 General Forms of Universal Function 15

Exercises 1.11.32 16

1.6 Coefficientsgkand ExponentsC1 andC2 for Laminar Flow 19

1.6.1 Features of Coefficients gkof the Differential Universal Function19

1.6.2 Estimation of Exponents C1and C2for Integral Universal Function22

1.7 Universal Functions for Turbulent Flow 24

Exercises 1.331.47 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 Function36

1.13.2 Integral Inverse Universal Function37

1.14 Universal Function for Recovery Factor 38

Exercises 1.481.75 41

2 Application of Universal Functions 45

2.1 The Rate of Conjugate Heat Transfer Intensity 45

2.1.1 Effect of Temperature Head Distribution45

2.1.2 Effect of Turbulence50

2.1.3 Effect of Time-Variable Temperature Head58

2.1.4 Effects of Conditions and Parameters in the Inverse Problems60

2.1.5 Effect of Non-Newtonian Power-Law Rheology Fluid Behavior66

2.1.6 Effect of Mechanical Energy Dissipation67

2.1.7 Effect of Biot Number as a Measure of Problem Conjugation68

Exercises 2.12.33 70

2.2 The General Convective Boundary Conditions 73

2.2.1 Accuracy of Boundary Condition of the Third Kind73

2.2.2 Conjugate Problem as an Equivalent Conduction Problem76

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 Formulation87

2.6.2 Problem Formulation89

2.6.3 Problem Formulation92

Exercises 2.342.82 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 Plates102

Exercises 3.13.38 123

3.1.2 Conjugate Heat Transfer in Flows Past Bodies126

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 Effects166

4.1.3 Heat Exchanger Models170

4.1.4 Finned Surfaces175

4.2 Thermal Treatment and Cooling Systems 180

4.2.1 Treatment of Continuous Materials180

4.2.2 Cooling Systems185

4.3 Simulation of Industrial Processes 196

4.4 Technology Processes 202

4.4.1 Heat and Mass Transfer in Multiphase Processes202

4.4.2 Drying and Food Processing208

Summary of Part I 219

Effect of Conjugation219

Part II APPLICATIONS IN FLUID FLOW

5 Two Advanced Methods 225

5.1 Conjugate Models of Peristaltic Flow 225

5.1.1 Model Formulation225

5.1.2 The First Investigations228

5.1.3 Semi-Conjugate Solutions230

Exercises 5.15.19 236

5.1.4 Conjugate Solutions237

Exercises 5.205.31 243

5.2 Methods of Turbulence Simulation 244

5.2.1 Introduction244

5.2.2 Direct Numerical Simulation244

5.2.3 Large Eddy Simulation245

5.2.4 Detached Eddy Simulation247

5.2.5 Chaos Theory249

Exercises 5.325.44 249

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 Vessels251

6.1.2 Abnormal Flows in Disordered Human Organs261

6.1.3 Simulation of Biological Transport Processes267

6.2 Application of Fluid Flow Models in Engineering 273

6.2.1 Application of Peristaltic Flow Models273

6.2.2 Applications of Direct Simulation of Turbulence278

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 Processes295

7.1.2 Different Forms of Navier-Stokes, Energy, and Diffusion Equations297

7.2 Initial and Boundary Counditions 302

7.3 Exact Solutions of Navier-Stokes and Energy Equations 303

7.3.1 Two Stokes Problems303

7.3.2 Steady Flow in Channels and in a Circular Tube304

7.3.3 Stagnation Point Flow (Hiemenz Flow)304

7.3.4 Couette Flow in a Channel with Heated Walls306

7.3.5 Adiabatic Wall Temperature306

7.3.6 Temperature Distributions in Channels and in a Tube306

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 Sphere308

7.4.3 Oseens Approximation308

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 Plate315

7.5.2 Self-Similar Flows of Dynamic and Thermal Boundary Layers319

7.6 Approximate Karman-Pohlhausen Integral Method 320

7.6.1 Approximate Friction and Heat Transfer on a Flat Plate320

7.6.2 Flows with Pressure Gradients322

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 Aspects328

8.2.2 Reynolds Averaging329

8.2.3 Reynolds Equations and Reynolds Stresses330

8.3 Algebraic Models 331

8.3.1 Prandtls Mixing-Length Hypothesis331

8.3.2 Modern Structure of Velocity Profile in Turbulent Boundary Layer332

8.3.3 Mellor-Gibson Model334

8.3.4 Cebeci-Smith Model335

8.3.5 Baldwin-Lomax Model336

8.3.6 Application of the Algebraic Models337

8.3.7 The 1/2 Equation Model338

8.3.8 Applicability of the Algebraic Models339

8.4 One-Equation and Two-Equations Models 339

8.4.1 Turbulence Kinetic Energy Equation340

8.4.2 One-Equation Models340

8.4.3 Two-Equation Models341

8.4.4 Applicability of the One-Equation and Two-Equation Models343

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 Problems346

9.2.2 One-Dimensional Unsteady Problems347

9.2.3 Orthogonal Eigenfunctions348

9.2.4 Two-Dimensional Steady Problems351

9.3 Integral Transforms 353

9.3.1 Fourier Transform353

9.3.2 Laplace Transform356

9.4 Greens 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 Method368

9.7.2 Control-Volume Finite-Element Method371

10 Conclusion 373

References 376

Author Index 397

Subject Index 409