Although the solution of Partial Differential Equations by numerical methods is the standard practice in industries, analytical methods are still important for the critical assessment of results derived from advanced computer simulations and the improvement of the underlying numerical techniques. Literature devoted to analytical methods, however, often focuses on theoretical and mathematical aspects and is therefore useless to most engineers. Analytical Methods for Heat Transfer and Fluid Flow Problems addresses engineers and engineering students. It describes useful analytical methods by applying them to real-world problems rather than solving the usual over-simplified classroom problems. The book demonstrates the applicability of analytical methods even for complex problems and guides the reader to a more intuitive understanding of approaches and solutions.
Author(s): Bernhard Weigand
Publisher: Springer
Year: 2004
Language: English
Pages: 280
10.1007/978-3-540-68466-4......Page 1
Analytical Methods for Heat Transfer and Fluid Flow Problems......Page 4
Copyright Page - ISBN: 3540222472......Page 5
Preface......Page 8
Table of Contents......Page 12
List of Symbols......Page 15
List of Copyrighted Figures......Page 20
1 Introduction......Page 23
Problems......Page 31
2.1 Classification of Second-Order Partial Differential Equations......Page 33
2.2.1 Parabolic Second-Order Equations......Page 41
2.2.2 Elliptic Second-Order Equations......Page 43
2.2.3 Hyperbolic Second-Order Equations......Page 45
2.3.1 One-Dimensional Transient Heat Conduction in a Flat Plate......Page 46
2.3.2 Steady-State Heat Conduction in a Rectangular Plate......Page 53
2.3.3 Separation of Variables for the General Case of a Linear Second-Order Partial Differential Equation......Page 59
Problems......Page 61
3.1 Heat Transfer in Pipe and Channel Flows with Constant Wall Temperature......Page 65
3.1.1 Velocity Distribution of Hydrodynamically Fully Developed Pipe and Channel Flows......Page 66
3.1.2 Thermal Entrance Solutions for Constant Wall Temperature......Page 68
Properties of the Sturm-Liouville System......Page 72
1. The Problem is Self-Adjoint and Positive Definite......Page 73
2. Eigenvalues and Eigenfunctions......Page 75
4. Eigenfunction Expansions for an Arbitrary Function......Page 76
Laminar Flows......Page 80
Turbulent Flows......Page 82
Heat Transfer in Turbulent Pipe Flow......Page 85
3.2 Thermal Entrance Solutions for an Arbitrary Wall Temperature Distribution......Page 88
3.3 Flow and Heat Transfer in Axially Rotating Pipes with Constant Wall Heat Flux......Page 91
3.3.1 Velocity Distribution for the Hydrodynamically Fully Developed Flow in an Axial Rotating Pipe......Page 94
3.3.2 Thermal Entrance Solution for Constant Wall Heat Flux......Page 97
Temperature Distribution for the Fully Developed Flow ( Θ1 )......Page 99
The Temperature Distribution Θ2......Page 100
Problems......Page 105
4 Analytical Solutions for Sturm - Liouville Systems with Large Eigenvalues......Page 109
4.1 Heat Transfer in Turbulent Pipe Flow with Constant Wall Temperature......Page 125
4.2 Heat Transfer in an Axially Rotating Pipe with Constant Wall Temperature......Page 131
4.3 Asymptotic Expressions for other Thermal Boundary Conditions......Page 135
Problems......Page 137
5 Heat Transfer in Duct Flows for Small Peclet Numbers (Elliptic Problems)......Page 139
5.1 Heat Transfer for Constant Wall Temperatures for x≤ 0 and x > 0......Page 142
Laminar Pipe Flow......Page 153
Laminar Channel Flow......Page 157
5.1.2 Heat Transfer in Turbulent Pipe and Channel Flows for Small Peclet Numbers......Page 160
Turbulent Pipe Flow......Page 161
5.2 Heat Transfer for Constant Wall Heat Flux for x ≤ 0 and x > 0......Page 164
5.2.1 Heat Transfer in Laminar Pipe and Channel Flows for Small Peclet Numbers......Page 169
5.2.2 Heat Transfer in Turbulent Pipe and Channel Flows for Small Peclet Numbers......Page 172
5.3 Results for Heating Sections with a Finite Length......Page 175
5.3.1 Piecewise Constant Wall Temperature......Page 176
5.3.1 Piecewise Constant Wall Heat Flux......Page 179
5.4 Application of the Solution Method to Related Problems......Page 181
Problems......Page 183
6.1 The Method of Separation of Variables......Page 187
6.2 Transformations Resulting in Linear Partial Differential Equations......Page 194
6.3.1 Incompressible Flow over a Heated Flat Plate......Page 196
6.3.2 Compressible Flow over a Flat, Heated Plate......Page 199
6.4.1 Similarity Solutions for a Transient Heat Conduction Problem......Page 201
6.4.2 Similarity Solutions of the Boundary Layer Equations for Laminar Free Convection Flow on a Vertical Flat Plate......Page 208
6.4.3 Similarity Solutions of the Compressible Boundary Layer Equations......Page 214
Problems......Page 220
Appendix A: The Fully Developed Velocity Profile for Turbulent Duct Flows......Page 225
Appendix B: The Fully Developed Velocity Profile in an Axially Rotating Pipe......Page 237
Appendix C: A Numerical Solution Method for Eigenvalue Problems......Page 249
C.1 Numerical Tools......Page 251
D.1 Symmetry of the Matrix Operator L......Page 257
D.2 The Eigenfunctions Constitute a Set of Orthogonal Functions......Page 258
D.3 A detailed Derivation of Eq. (5.31) and Eq. (5.61)......Page 259
D.4 Simplification of the Expression for the Temperature Distribution (for Constant Wall Temperature)......Page 261
D.5 Simplification of the Expression for the Temperature Distribution (for Constant Wall Heat Flux)......Page 262
D.6 The Vector Norm......Page 265
References......Page 269
Index......Page 279
