- Preface
- Chapter 1: Description of the Inverse Heat Conduction Problem
- Chapter 2: Exact Solutions of the Inverse Heat Conduction Problem
- Chapter 3: Approximate Methods For Direct Heat Conduction Problems
- Chapter 4: Inverse Heat Conduction Estimation Procedures
- Chapter 5: Inverse Convolution Procedures For a Single Surface Heat Flux
- Chapter 6: Difference Methods For the Solution of the One-Dimensional Inverse Heat Conduction Problem
- Chapter 7: Multiple Heat Flux Estimation
- Chapter 8: Heat Transfer Coefficient Estimation
- Index

I NVERSE H EAT
C ONDUCTION
I II-posed P roblems
J AMES V . B ECK
D"pClfl",etrl o f Medlatrkal
M it'IIiga" Slclle Utrivc·r,o;iIJ'
£"yi"c'('r;tr~1
BEN B LACKWELL
Sanaia NUlicmal Laboratories
Alhuqu('rqllt'. N"It' M exim
C HARLES R. ST. CLAIR. J R.
or
Depurlme", Medlatrit'al £trg;""('r;"y
Michigan Slale Ullivl'rsil),
A W ilev-Interscience P ublication
N ew Y ork • C hichester • B risbane • T oronto • S ingapore
To my wife, Barbara; children, Sharon and Douglas; and father
and mother, Peter and Louise Beck
J.Y.B
To my wife, Betty; and children, Jeffrey and Gregory Blackwell
B.E
Copyril:hl
© 1985 by John Wiley & Sons. Inc.
All righls reserved. Published simullaneously in Canada.
Reproduclion or lranslalion o f any pari o f lhis work
beyond lhal pcrmilled by Seclion 107 o r 108 o f lhe . .
1976 Uniled Slales Copyrighl ACI w ilhoullhe pcrmls."on
o f lhe copyrighl owner is unlawful. Requesls for
permission o r furlher informalion sh~uld be addressed 10
lhe Permissions Deparl~enl, J ohn Wiley & Sons, Inc.
Librllry o f C ongrus CIIIIIIOf/ing in Publicli/ion DIIIII:
Beck, J . V. (James Vere), 1930Inverse heal conduclion .
" A Wiley·lnlerscience publicalion."
Includes bibliographies and index.
\. Heal .-Conduclion .
2. Numerical analysis -·
Improperly posed problems.
I. Blackwell, Ben.
II . Sl . Clair, Charles R .
I ll. Tille.
QC320.B4
1985
536' ,23
85-5391
ISBN 0-471-08319-4
Prinled in lhe Uniled Slales o f America
10 9 8 7 6 5 4 3 2 I
T o the greatest engineers in the family : Charles R. St. Clair, Sr. and
Deborah S. Short (my daughter). And to those for whom the
engineers live and labor: my mother Erla, my wife Jeanette, and our
children and grandchildren from oldest to youngest with the greatest
in doubt: Charles III, Scott, Judy, Timothy B. Short, Gregory, and
Kevin. And to those who are as family: John and Ann Polomsky
C.R.S
P REFACE
This book presents a study o f the Inverse Heat Conduction Problem ( IHCP)
which is the estimation o f t he surface heat flux history o f a heat conductinr
body. Transient temperature measurements inside t he body are utilized in the
calculational procedure. The presence o f e rrors in the measurements as well
as the ill-posed nature o f the problem lead to " estimates" r ather than the "true"
surface heat flux a nd/or temperature.
This book was written because o f the importance and practical nature o f the
I HCP; furthermore, at the time o f writing there is no available book on the
subject written in English. T he specific problem treated is only one o f man}
ill-posed problems but the techniques discussed herein c an be applied to many
others. The basic objective is to estimate a function given measurements that
are " remote" in s ome sense. O ther applications include remote sensing, oil
exploration, nondestructive evaluation o f materials, and determination o f the
Earth's interior structure.
The authors became interested in the I HCP over two decades ago while
employed in the aerospace industry. O ne o f the applications was the determination o f the surface heat flux histories o f reentering heat shields.
This book is written as a textbook in engineering with numerical examples
and exercises for students. These examples will be useful to practicing engineers
who use the book to become acquainted with the problem and methods o f
solution. A companion book, Parameter Estimation in Engineering (lnd Science
by J . V. Beck and K . J . A rnold (Wiley, 1977), discusses estimation o f certain
constants o r p arameters rather than functions as in t he I HCP. T hough many
o f the ideas relating to least squarcs and sensitivity coefficients are present in
both books, the present book docs not re,!uire a mastery o f p aramcter
estimation.
The book is written at the advanced B.S. o r the M.S. level. A course in
heat conduction a t the M .S. level o r courses in partial differential equations
and numerical methods are recommended as prerequisite materials.
vii
.'!"
v iii
PREFACE
O ur philosophy in writing this book was to emphasize general techniques
rather than specialized procedures unique to the IHCP. F or example, basic
techniques developed in Chapter 4 can be applied either to integral equation
representations o f the heat diffusion phenomena o r t o finite difference (or
element) approximations o f the heat conduction equation. The basic procedures
in C hapter 4 can treat nonlinear cases, mUltiple sensors, nonhomogeneous
media, multidimensional bodies, and many equations, in addition to the
transient heat conduction equation .
The two general procedures that are used are called (a) function specification
and (b) regularization. A method o f combining these (the trial function method)
is also suggested. One o f the important contributions o f this book is the demonstration t hat all o f these methods can be implemented in a sequential manner.
The sequential method in some case gives nearly the same result as whole
domain estimation and yet is much more computationally efficient.
One o f o ur goals was to provide the reader with an insight into the basic
procedures that provide analytical tools to compare various procedures. We
do this by using the concepts o f sensitivity coefficients, basic test cases, and the
mean squared error. The reader is also shown that optimal estimation involves
the compromise between minimum sensitivity to random measurement errors
and the minimum bias.
Preliminary notes have been used for an ASME short course and for a
graduate course a t Michigan State University.
There are many people who have helped in the preparation o f this text and
to whom we express o ur appreciation. These include D. Murio, M. Raynaud,
and other colleagues and students who have read and commented on the notes.
Thanks are also due to Judy Duncan, Phyllis Murph, Terese Stuckman, Alice
Montoya, a nd J eana Pineau, who have aided in typing the manuscript.
James V. Beck wishes to express appreciation for the contributions to his
education made by Kenneth Astill o f T ufts University, Warren Rohsenow o f
Massachusetts Institute o f Technology, and A. M. Dhanak o f Michigan State
University.
Ben Blackwell would like to acknowledge the contributions that several
people made to his heat transfer education : H. Wolf o f the University o f
Arkansas, M. W. Wildin o f the University o f New Mexico, and W. M. Kays o f
S tanford University.
A special a nd deep appreciation is extended to George A. Hawkins for the
education a nd philosophy that he imparted to Charles R. St. Clair, Jr. as his '
graduate student.
JAMES V. B ECK,
BEN BLACKWELL
CHARLES R. S T. C LAIR, J R.
Eos/ Lallsing. Michigan
Alhuquerque. Nitl\' M exico
East Lal/sillg. Michigan
August 1985
C ONTENTS
N omenclature
1.
xv
D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION
P ROBLEM
1
1 .1
1 .2
Introduction. 1
Examples o f Inverse P roblems. 3
1.2.1 Inverse H eat C onduction P roblem Examples. 3
1 .2.2 . O ther Inverse F unction Estimation Problems 7
1 .3
Function Estimation Versus Parameter Estimation 9'
1 .4
Measurements. 9
.
1.4.1 Description of M easurement Errors 9
1 .4 .2. S tatistical Description of Errors. 1 0'
1.5
Why IS t he IHCP Difficult? 1 3
1 .5.1 Sensitivity to Errors. 1 3
1 .5.2 Exam?'es of Damping a nd L agging; Exact
Solutions. 1 3
1.6
S ensitivity Coefficients. 1 9
1 .6.1 Definition of Sensitivity C oefficients a nd
Linearity. 19
1.6.2 O ne- Dimensional Sensitivity C oefficient
Examples. 22
1.6.2.1 Lumped Body Case. 22
1.6.2 .2 Semi-Infinite Body. 25
1.6.2 .3 Plate I nsulated o n O ne S ide 3 0
1 .6.3 T wo-Dimensional S ensitivity Coeffici~nt
Example. 31
1 .7
Classification of Methods. 3 6
1 .8
Criteria for Evaluation of IHCP M ethods 3 8
1 .9
S cope o f B ook. 39
.
References. 4 0
P roblems. 4 3
ix
CONTENTS
CONTENTS
x
2.
EXACT SOLUTIONS OF T HE I NVERSE H EAT
C ONDUCTION P ROBLEM
4.
51
I ntroduction. 51
Steady-State Solution. 52
Transient Analysis of Bodies w ith Small Internal
Thermal Resistance. 54
2.3.1 Exact Solution. 54
2.3.2 Approximate Solutions. 54
2.3.3 Temperature Errors and Approximate Solutions. 55
2.4
Heat Flux From Measured Surface Temperature History. 59
2.4.1 Exact Results for C ontinuous Surface
Temperature History. 59
2.4.2 Approximate Results f or S emi-Infinite B ody
w ith Surface Temperature Measured at
Discrete Times. 61
2.4 .3 Temperature Error Propagation in Eq . (2.4 .8). 63
2.5
Exact Solutions o f Inverse Heat C onduction Problems. 67
2.5.1 Literature Review. 67
2.5.2 Derivation of Exact Solution f or Planar Geometry. 67
2.5.3 Expressions for Cylinders and Spheres. 71
2 .5.4 Example Results for Planar Geometry. 7 2
References. 75
Problems. 76
3.
A PPROXIMATE M ETHODS FOR DIRECT H EAT
C ONDUCTION P ROBLEMS
I NVERSE H EAT C ONDUCTION E STIMATION
PROCEDURES
4 .1
4.2
2.1
2 .2
2.3
4.3
4.4
4.5
78
Introduction. 78
3.1.1 Various Numerical Approaches. 78
3.1.2 Scope o f Chapter. 79
3.2
Duhamel's Theorem. 80
3 .2.1
D erivation of D uhamel's Theorem. 8 0 .
3.2.2 Numerical Approximation o f D uhamel's Theorem. 82
3.2.3 M atrix Form o f D uhamel's Theorem . 8 3
3.3
Difference Methods. 87
3.3 .1 Finite C ontrol V olume Procedure for C onstant
Property Planar Geometries. 87
3.3.2 Other B oundary C onditions and Material
Interfaces. 93
3.3 .3 Numerical Techniques for Solving Systems of
First -Order Ordinary Differential Equations. 9 4
3.3.4 General Form of Difference Equations for Heat
C onduction i n Planar Body. 96
3.3.5 Standard Form for Temperature Equation
for I HCP. 99
References. 1 02
Problems. 1 03
3 .1
4.6
4.7
4.8
i
xi
108
Introduction. 1 08
111- Posed Problems. 1 10
4.2.1
Partial Differential Equation Perspective. 1 10
4.2.2 I ntegral Equation Perspective. 112
4.2.3 Difference Equati on Perspective. 1 13
S ingle Future Time Step M ethod. 115
4.3.1 I ntroduction. 115
4.3.2 Exact M atching o f Measured Temperatures
(Single Sensor) . 1 15
4 .3.3 M ultiple Temperature Sensors. 1 18
F unction Specification M ethod . 119
4.4 .1 Introduction. 1 19
4.4.2 Whole Domain Estimation. 1 19
4.4.2.1 Smoothly C hanging H eat Flux. 1 20
4.4.2 .2 A bruptly C hanging Heat Flux Histories. 122
4.4 .3 Sequential Estimation. 125
4.4 .3.1 C onstant Heat Flux Functional Form. 125
4.4 .3.2 Linear Heat Flux Functional Form. 131
4.4 .3.3 Alternative Interpretation. 1 33
Regularization M ethod . 1 34
4 .5.1 Introduction. 1 34
4 .5.2 Physical Significance of Regularization Terms. 135
4.5.3 Whole Domain Regularization Method. 1 37
4 .5.3.1 Algebra ic Formulation. 1 37
4 .5.3.2 M atrix F ormulation . 1 38
4.5.3 .3 Selection o f Regularization Parameter. 1 40
4 .5.4 Sequential Regularization Method. 141
Trial Function M ethod . 1 45
4 .6.1
I ntroduction. 1 45
4.6 .2 Matrix Analysis. 1 45
4 .6.3 Zeroth-Order Regularization M ethod . 1 47
4 ,6.4 Generalized Sequential Function Specification
M ethod . 1 47
Filter Form o f Linear I H CPo 1 48
4 .7.1 I ntroduction. 1 48
4 .7.2 Sequential Filter Algorithm. 1 48
4 .7.3 Prefiltering Temperature Measurements. 1 53
T wo C onflicting Objectives. 1 53
4 .8.1
M in imum Deterministic Bias. 1 53
4 .8.2 M inimum S ensitivity to Random Errors. 1 54
4.8.3 Mean Squared Error. 1 54
4 .8.4 Variance o f Estimated Heat Flux Component. 1 56
4 .8.5 Estimate o f Determin istic Error in Surface
Heat Flux. 1 57
C ONTENTS
xii
C ONTENTS
5.5
Digital Filter Algorithm. 196
5.5.1
Introduction. 1 96
5.5.2 Function Specification- Based Filter. 197
5.5.2.1
Finite Plate Case. 1 97
5.5.2.2 S emi-Infinite Body. 200
5.5.3 W hole Domain Regularization Filter. 201
5.6 Optimal Considerations. 203
5.6.1
Optimal Function Specificatio'n Algorithm. 204
5.6.2 Optimal Whole Domain Regularization Method. 210
References. 212
Problems. 213
References. 159
Problems. 161
5.
I NVERSE C ONVOLUTION P ROCEDURES FOR A
S INGLE S URFACE H EAT F LUX
5.1
5.2
5.3
5.4
xiii
1 65
I ntroduction. 165
Test Cases. 1 67
5.2.1
I ntroduction. 1 67
5.2.2 Step Change in Surface Heat Flux. 1 68
5.2.3 Triangular Heat Flux. 169
5.2.4 Random Errors. 1 70
5.2.5 Heat Flux Impulse Test Case (C5qM/Dqf)' 1 73
5.2.6 Temperature Impulse Test Case (i>qM/CW f ). 1 74
5.2.7 Test Cases w ith Units. 1 74
Function Specification Algorithms. 176
5.3.1
I ntroduction. 1 76
5.3.2 Single Future Temperature Algorithm
(Stolz Method). 176
5.3.2.1
Step Heat Flux Test Case. 1 77
5.3.2.2 Triangular Heat Flux Test Case. 1 77
5.3.2.3 Heat Flux Impulse Test Case (OqM/Oqf)' 1 78
5.3.2.4 Temperature Impulse Test Case
(OC,M/cW f ).179
5.3.3 M ultiple Future Temperatures Algorithm. 181
5.3.3.1
Step Heat Flux Test Case. 181
5.3.3.2 Triangular Heat Flux Test Case. 182
5.3.3.3 Heat Flux Impulse Test Case (oq M/Oq f )' 184
5.3.3.4 Temperature Impulse Test Case
(fJqM/fJY f ). 1 84
Regularization Algorithms. 186
5.4.1
I ntroduction. 1 86
5.4.2 W hole Domain Regularization Method. 187
5.4.2.1 Triangular Heat Flux Test Case. 189
5.4.2.2 Heat Flux Impulse Test Case. 1 90
5.4.2.3 Temperature Impulse Test Case
( oqM/oY f ).191
5.4.3 Sequential Regularization Method. 191
5.4.3.1 Triangular Heat Flux Test Case. 1 93
5.4.3.2 Heat Flux Impulse Test Case. 194
5.4.3.3 Temperature Impulse Test Case. 1 94
5.4.3.4 Comparison of Whole Domain and
Sequential Regularization Methods. 196
5.4.3.5 Comparison of Sequential
Regularization and Function
Specification Methods. 196
6.
D IFFERENCE M ETHODS FOR T HE S OLUTION O F T HE
O NE-DIMENSIONAL I NVERSE H EAT C ONDUCTION
P ROBLEM
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
I ntroduction. 218
Sensitivity Coefficients and Their Calculation by
Difference Methods. 219
Single Temperature Sensor. Function Specification
( q=C). Single Future Time Step (Exact Matching o f
Data). 222
6.3.1
M odification o f Difference Equations o f the
Direct Heat C onduction Problem for the
Solution o f the IHCP. 222
6.3.2 Sensitivity Coefficient Approach for Exactly
Matching Data from a Single Sensor. 223
M ultiple Temperature Sensors. Function Specification
( q=C). Single Future Time Step. 230
W hole Domain Estimation With Difference Methods. 233
Single Temperature Sensor. Function Specification
( q=C). r Future Time Steps. 237
M ultiple Temperature Sensors; Function Specification
( q=C). Arbitrary Future Time Steps. 241
Single Temperature Sensor. Function Specification.
Linear Heat Flux (Connected Segments). 242
Second Order Sequential Regularization Methods. 243
Space Marching Techniques for One-Dimensional
Problems. 247
6.10.1 Analytical Solution. 248
6.10.2 M ethod o f D·Souza. 249
6.10.3 M ethod o f Weber. 252
6.10.4 M ethod of Raynaud and Bransier. 253
6.10.5 M ethod o f Hills and Hensel. 254
6.10.6 Comparison w ith Prior Methods. 256
Numerical Calculations. 256
Computer Programs. 262
218
C ONTENTS
x iv
/
'J
~
References. 264
Problems. 265
7.
M ULTIPLE H EAT F LUX E STIMATION
267
7.1
7.2
I ntroduction. 267
T wo Independent Heat Fluxes Case. 268
7.2.1 Sequential Function Specification M ethod . 271
7.2.2 Sequential Regularization Method. 273
7.3
Multiple Heat Flux Case . 275
7.3.1
Sequential Function Specification Method for
Multiple Heat Flux Components. 276
7.3.2 Sequential Regularization Method for
Multiple Heat Fluxes. 277
References. 279
Problems. 279
8.
N OMENCLATURE
H EAT T RANSFER C OEFFICIENT E STIMATION
281
8.1
I ntroduction. 281
8 .2
Sensitivity Coefficients. 283
8.2.1
Lumped Body Case . 283
8.2.2 Semi - Infinite Body. 287
8.3
Lumped Body Analyses . 290
8.3.1
Exact Matching of the Measured Temperatures. 290
8.3.2 Regression Method. 292
8.3.3 Function. Specification Procedure w ith
q = Constant. 293
8.3.4 Function Specification Procedure w ith
h =Constant. 294
8.4
Bodies With Internal Temperature Gradients. 297
8.4.1 Analysis for r Future Temperatures -Using
q =C Function Specification Method. 297
8.4.2 Examples. 299
8.5
Estimation of Contact Conductance. 301
References. 301
Problems. 301
A uthor I ndex
3 05
"
c
cov (., .)
ej
e rf
erfe
E
E (· )
jj
If
G
h
he
H o• H b
ierfc
3 03
S ubject I ndex
a, b, c, d, e,1 Coefficients in tridiagonal matrix algorithm; see (6.3.9-10)
a
J
...
Radius o f cylinder o r sphere
Heat capacity
Covariance o perator; see (104.8)
Residual temperature e rror; see (104.7)
E rror function
Complementary e rror function
Sensor depth below heated surface
Expected value operator
Filter coefficient; see (4.7.2)
Dimensionless filter coefficient
Green's function
Convective heat transfer coefficient
Contact conductance
Regularization matrices; see (4.5.16)
Integral error function
A vector o f ones
Identity matrix
Number o f t emperature sensors
Bessel functions o f the first kind, order 0, I, . . _
Thermal conductivity
Gain coefficient at time I j ; see (4.4.25)
= (q,L/k)K j , q ,= I in consistent units, dimensionless gain
coefficien t
xv
NOMENCLATURE
x vi
G ain coefficient for sensor j a t time I j
Slab thickness
General time index
a il tl!:,.x2, grid scale Fourier number
Heat flux
C onstant value o f heat flux
H eat flux a t time Ij
Estimated value o f qj
H eat flux a t time t j t hat exactly matches the temperature d ata
Y j ; see (4.4.3.3)
Estimated heat flux for interval 1 M-I t o 1M
H eat flux vector; see (4.6.4)
Heat flux vector; see (4.6.4)
Trial value o f q
N umber o f future time steps
Radial coordinate
qM
q
ql
q*
r
r
r;
rIa
S
Least square function
Time
Dimensionless time, atl L 2
atla 2
T emperature
Initial temperature
Ambient temperature a t which convection o r r adiation is taking
place
( T - To)/(qLlk)
Estimated value o f T
Vector o f estimated temperatures for q = 0
T emperature corresponding t o q = q*; see (6.3.8)
T emperature response function for a body at zero initial
temperature and subjected to a unit step in surface temperature;
see (2.4.1)
Variance operator
Heat flux weighting factor; see (4.4.40c)
Regularization constants; see (4 .5.1)
Spatial coordinate
x lL
Sensitivity coefficient for heat flux pulse
= (klx)oTloq<
T
To
T..,
T+
T
t\q=O
t
u(x, I)
v (·)
Wj
W o, W h
X
x+
X
X+
·••
NOMENCLATURE
X
Yj
Z
Matrix o f sensitivity coefficients
Measured value o f t emperature at time I j
Sensitivity coefficient for heat flux step
GREEK S YMBOLS
PI,P2,· "
y
fiq,
c5Y,
c5qM
f iYj
I lt
illM
!:,.x
i l¢(r, In)
ilT1
;
o
}.
p
a
¢ (x, I )
w
T hermal diffusivity, also regularization parameter
See (5.6.14)
Parameters; see (4.4.1-5)
Thermal wave speed
Heat flux impulse
Temperature impulse o r e rror
Heat flux e rror for a uni"t e rror in t emperature
Time step
= IM+I-I M
Spatial grid size for difference methods
= ¢(r, I n + , )-¢(r, t n )
a illlE 2
T emperature e rror; see (1.4.1)
Time difference weighting parameter
Dummy time variable
Density
Standard deviation
Temperature response to a unit step in heat flux
Tridiagonal matrix coefficient; see (6.3.10)
x vii