Inverse Heat Transfer. Fundamentals and Applications

299 Pages·2021·14.323 MB·English

by  M. Necati Özisik, Helcio R. B. Orlande

Inverse Heat Transfer
Heat Transfer
A Series of Reference Books and Textbooks
Afshin J. Ghajar
Regents Professor, School of Mechanical and Aerospace Engineering,
Oklahoma State University
Engineering Heat Transfer, Third Edition
William S. Janna
Conjugate Problems in Convective Heat Transfer
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Thermal Measurements and Inverse Techniques
Helcio R.B. Orlande, Olivier Fudym, Denis Maillet, Renato M. Cotta
Introduction to Thermal and Fluid Engineering
Allan D. Kraus, James R. Welty, Abdul Aziz
Advances in Industrial Heat Transfer
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Introduction to Compressible Fluid Flow, Second Edition
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District Cooling: Theory and Practice
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Advances in New Heat Transfer Fluids: From Numerical to
Experimental  Techniques
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Finite Difference Methods in Heat Transfer
M. Necati Özişik, Helcio R.B. Orlande, Marcelo J. Colaço, Renato M. Cotta
Convective Heat and Mass Transfer, Second Edition
S. Mostafa Ghiaasiaan
The Art of Measuring in Thermal Sciences
Josua Meyer and Michel De Paepe
Inverse Heat Transfer
Fundamentals and Applications
Second Edition
M. Necati Özisik
Helcio R. B. Orlande
Second edition published 2021
by CRC Press
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© 2021 Helcio R. B. Orlande
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Contents
Preface............................................................................................................................................xiii
Preface of the First Edition ............................................................................................................xvii
Authors ............................................................................................................................................xix
PART I Introduction and Parameter Estimation
Chapter 1 Basic Concepts .............................................................................................................3
1.1 Inverse Heat Transfer Problem Concept ............................................................4
1.2 Classification of IHTPs .....................................................................................6
1.3 Difficulties in the Solution of Inverse Heat Transfer Problems .........................8
1.4 An Overview of Solution Techniques for Inverse Heat Transfer Problems ......13
1.5 Basic Steps for the Solution of Inverse Heat Transfer Problems .....................18
Problems .....................................................................................................................19
Note 1: Statistical Concepts .......................................................................................21
Random Variable .............................................................................................21
Probability Distribution ...................................................................................21
Expected Value of X .........................................................................................21
Expected Value of a Function g(X) ..................................................................22
Variance of a Random Variable X ....................................................................22
Covariance of Two Random Variables X and Y ...............................................22
Gaussian Distribution ......................................................................................22
Uniform Distribution .......................................................................................23
Rayleigh Distribution ......................................................................................23
Gamma Distribution ........................................................................................24
Beta Distribution ..............................................................................................25
Chi-Square Distribution ...................................................................................25
Covariance Matrix ...........................................................................................26
Multivariate Gaussian Distribution ..................................................................27
Chapter 2 Parameter Estimation: Minimization of an Objective Function without Prior
Information about the Unknown Parameters .............................................................29
2.1 Objective Function ...........................................................................................30
2.2 Technique I: The Levenberg-Marquardt Method ............................................31
The Direct Problem .........................................................................................32
The Inverse Problem ........................................................................................32
The Iterative Procedure for Technique I ..........................................................33
The Stopping Criteria for Technique I .............................................................35
The Computational Algorithm for Technique I ...............................................36
2.3 Technique II: The Conjugate Gradient Method for Parameter Estimation .....37
The Direct Problem .........................................................................................37
The Inverse Problem ........................................................................................37
The Iterative Procedure for Technique II .........................................................38
The Stopping Criterion for Technique II .........................................................40
The Computational Algorithm for Technique II ..............................................40
vii
viii Contents
2.4 Sensitivity Coeffcients .................................................................................... 41
Methods of Determining the Sensitivity Coeffcients ..................................... 41
Direct Analytic Solution for Determining Sensitivity Coeffcients .......41
The Boundary Value Problem Approach for Determining the
Sensitivity Coeffcients ....................................................................... 47
Finite Difference Approximation for Determining Sensitivity
Coeffcients .........................................................................................49
2.5 Design of Optimum Experiments ....................................................................49
2.6 T he Use of Multiple Sensors ...........................................................................50
2.7 S tatistical Analysis .......................................................................................... 52
2.8 Estimation of Thermal Conductivity Components of an Orthotropic
Heat Conducting Medium ...............................................................................54
The Direct Problem ......................................................................................... 55
The Inverse Problem ........................................................................................57
Analysis of the Sensitivity Coeffcients and Design of Optimum
Experiments..................................................................................................... 58
Parameter Estimation and Statistical Analysis ................................................ 61
2.9 Technique III: The Conjugate Gradient Method with Adjoint Problem
for Parameter Estimation.................................................................................65
The Direct Problem .........................................................................................66
The Inverse Problem ........................................................................................67
The Sensitivity Problem .................................................................................. 67
The Adjoint Problem .......................................................................................68
The Gradient Equation .................................................................................... 70
The Iterative Procedure for Technique III ....................................................... 70
The Stopping Criterion for Technique III ........................................................71
The Computational Algorithm for Technique III............................................. 71
The Use of Multiple Sensors ...........................................................................72
2.10 E stimation of a Heat Source Term in a Heat Conduction Problem .................73
Problems ..................................................................................................................... 75
Note 1: Search Step-Size for Technique II .................................................................77
Note 2: Search Step-Size for Technique III ................................................................ 78
Chapter 3 Parameter Estimation: Minimization of an Objective Function with Prior
Information about the Unknown Parameters .............................................................81
3.1 O bjective Function. .......................................................................................... 81
Maximum a Posteriori Objective Function with a Uniform Prior ...................83
Maximum a Posteriori Objective Function with a Gaussian Prior ..................84
Maximum a Posteriori Objective Function with a Truncated
Gaussian  Prior .............................................................................................86
3.2 M inimization of the Objective Function .........................................................87
3.3 I dentifcation of the Thermophysical Properties of
Semi-Transparent Materials ............................................................................ 91
The Direct Problem ......................................................................................... 91
The Inverse Problem ........................................................................................94
Analysis of the Sensitivity Coeffcients and Design of Optimum
Experiments .....................................................................................................95
Parameter Estimation and Statistical Analysis ................................................96
Problems .....................................................................................................................98
Contents ix
Chapter 4 Parameter Estimation: Stochastic Simulation with Prior Information about the
Unknown Parameters ............................................................................................... 101
4.1 Markov Chains .............................................................................................. 102
4.2 Technique IV: Markov Chain Monte Carlo (MCMC) Method ..................... 103
Proposal Distribution .....................................................................................106
Random Walk ...................................................................................106
Independent Move ............................................................................107
4.3 MCMC Estimation of Thermal Conductivity Components of an
Orthotropic Heat Conducting Medium ......................................................... 108
The Direct Problem ....................................................................................... 108
The Inverse Problem ......................................................................................108
Stochastic Simulation .................................................................................... 108
4.4 MCMC Estimation of Thermal Conductivity and Volumetric Heat
Capacity of Viscous Liquids with the Line Heat Source Probe....................109
The Direct Problem ....................................................................................... 110
The Inverse Problem ...................................................................................... 112
Analysis of the Sensitivity Coeffcients and Design of Optimum
Experiments................................................................................................... 113
Stochastic Simulation .................................................................................... 113
4.5 MCMC Estimation of Thermophysical Parameters of Thin Metal Films
Heated by Fast Laser Pulses.......................................................................... 116
The Direct Problem ....................................................................................... 117
The Inverse Problem ...................................................................................... 118
Analysis of the Sensitivity Coeffcients and Design of Optimum
Experiments ................................................................................................... 119
Stochastic Simulation .................................................................................... 121
4.6 Analysis of Markov Chains ...........................................................................122
Statistics ......................................................................................................... 122
Convergence of the Markov Chain ................................................................125
Proposal Distribution ..................................................................................... 131
4.7 Reduction of the Computational Time for Solving Inverse Problems
with Technique IV.........................................................................................132
Delayed Acceptance Metropolis-Hastings (DAMH) Algorithm ................... 133
Approximation Error Model (AEM) Approach ............................................. 133
4.8 Approximation Error Model to Account for Convective Effects in the
Line Heat Source Probe Method................................................................... 135
Problems ................................................................................................................... 137
Note 1: M etropolis-Hastings Algorithm with Sampling by Blocks of Parameters .. ..138
PART II Function Estimation
Chapter 5 Function Estimation: Minimization of an Objective Functional without Prior
Information about the Unknown Functions ............................................................. 141
5.1 Technique V: The Conjugate Gradient Method with Adjoint Problem for
Function Estimation ...................................................................................... 141
The Direct Problem ....................................................................................... 142
The Inverse Problem ...................................................................................... 143
The Sensitivity Problem ................................................................................ 143