CHAPTE~ 1
D ESCRIPTION O F T HE I NVERSE
H EAT C ONDUCTION P ROBLEM
1.1
I NTRODUCTION
I f the heat flux o r t emperature histories at the surface o f a solid a re k nown as
functions o f time, then the temperature distribution can be found . This is
termed a direct problem. In many dynamic heat transfer situations, the surface
heat flux and temperature histories o fa solid must be determined from transient
temperature measurements a t o ne o r m ore interior locations; this is a n inverse
problem. In particular, during the past two decades the special case o f e stimating
a surface condition from interior measurements has come to be known as the
inverse heat conduction problem. There a re n umerous o ther inverse problems
in transient conduction a nd diffusion. but this particular problem has been so
named and is the main subject o f this book.
The inverse heat conduction problem is m uch more difficult t o solve analytically than the direct problem. But in the direct problem many experimental
impediments may arise in m easuring o r p roducing given b oundary conditions.
The physical situation a t the surface may be unsuitable for attaching a sensor,
o r the accuracy o f a surface measurement may be seriously impaired by the
presence o f the sensor. Although i t is often dilficult t o measure the temperature
history o f t he heated surface o f a solid, it is easier to measure accurately the
temperature history a t an interior location o r at an insulated surface o f the
body. T hus there is a choice between relatively inaccurate measurements o r a
difficult analytical problem. An accurate and tractable inverse problem solution
would thus minimize both disadvantages a t once.
The problems o f d etermining the surface temperature and the surface heat
flux histories are equivalent in t he sense that if one is k nown the other can be
found in a straightforward fashion. They cannot be independently found since
in direct heat conduction problems only one boundary condition can be imposed a t a given time and boundary. Even though this is true, the following
2
CHAP.1
SEC. 1 .2
D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM
seemingly contradictory statement can be made: the heat flux is m ore difficult
to c alculate accurately than the surface temperature. F or this reason the
emphasis in this book is o n the calculation of the surface heat flux history.
(The surface temperature is a b yproduct o f the heat flux calculations for difference procedures.)
F or t he purposes o f this book the inverse heat conduction problem ( IHCP)
is defined as follows: T he ! HCP is the estimation o f the surface heat flux history
given one o r m ore measured temperature histories inside a heatconducting
body. T he w ord " estimation" is used because in m easuring the internal temperatures, errors are always present t o s ome extent a nd they affect the accuracy
o f the heat flux calculation. F urthermore even if discrete d ata a ccurate to a
large but finite n umber o f significant figures are used, the heat flux c annot be
exactly determined.
O ne o f the earliest papers on the ! HCP was published by Stolz l in 1960;
it a ddressed calculation of heat transfer rates during quenching o f bodies of
simple finite shapes. Stolz I claimed use of his method as early as J une 1957.
F or semiinfinite geometries Mirsepassi2 maintained t hat he h ad used the same
technique b oth numerically2.3 a nd graphically3 for several years prior to 1960.
A Russian p aper by Shumakov 4 on the I HCP was translated in 1957. T he s pace
program, starting a bout 1956, gave considerable impetus to the study o f t he
inverse heat conduction problem. T he a pplications therein were related t o nose
cones o f missiles a nd probes, t o rocket nozzles, and o ther devices. Beck also
initiated his work on the ! HCP a bout t hat time a nd developed the basic concepts 5 II t hat p ermitted much smaller time steps than the Stolz method. I
O thers whose work h ad a pplication to the space program included Blackwell,12.13 Imber,1623 Mulholland,2427 a nd Williams a nd C urry.31 A nother
research area t hat extensively required solutions of the I HCP was the testing
of nuclear reactor components.J2  38 M any of the c omputer p rograms in
c urrent use in the United S tates 35  38 a ppear t o be based on the method described in a 1970 p aper. 9 O ther a pplications reported for the I HCP included
(1) p eriodic heating in combustion chambers o f i nternal combustion engines,39
(2) solidification o f glass,40 (3) indirect calorimetry for laboratory use,41 a nd
(4) t ransient boiling curve studies. 42 O ver 300 p apers have been 'w ritten to
date on the I HCP o r closely related problems.
There have been extremely varied approaches to the inverse heat conduction
problem. Thes'e have included the use o f D uhamel's theorem (or convolution
integral) which is restricted t o l inear problems. I  3.5 7.1 0.28.58 N umerical
procedures such as finite differences 8.9. 11  13.29  31.35  38 a nd finite elements 35 . 36 h ave also been employed due to their inherent ability to treat
nonlinear problems. Exact solution techniques were proposed by Burggraf,43
Imber a nd Khan,23 L angford,t4 a nd o thers; such techniques have limited use
for real,stic p roblems (as discussed in C hapter 2) b ut they can give considerable
insight into the I HCP. S ome techniques used Laplace transforms and were
also limited t o l inear cases. 41 .44
T he I HCP is o ne o f m any mathematically " illposed" p roblems. Such
E XAM PLES OF INVERSE P ROBLEMS
3
p roblems are typically inverse problems a nd a re extremely sensitive to measurement errors. (See Section 4.2 for further discussion o f illposed problems.) There
are a n um?er o f p rocedures that have been advanced for the solution of illposed
problems m general. O ne o f these was developed by T ikhonov a nd Arsenin in
45
1963. T ikhonov i ntroduced what he called the regularization method t o
reduce the sensitivity o f illposed problems t o m easurement errors. A modification o f this method is presented herein for m ore efficient solution o f the ! HCP
(see Section 4.5). N umerous o ther general procedures for illposed problems
have been proposed including a technique, well known t o geophysicists called
the Bac~usGil?:rt technique.28.46.47 T he m athematical techniques for ~olving
sets ~f IllconditIOned a lgebraic equations called singlevalue decomposition
techmques can also be used for the I HCP. 48 . 57
T he p urposes o f this c hapter a re t o i ntroduce the inverse heat conduction
a nd related problems and to provide a general description o f v arious aspects
of t~e I HCP. T he c ontents o f the remainder o f this c hapter a re as follows:
Secllon 1.2 p.rovides some explicit examples o f t he I HCP a nd related problems.
T he.IHCP IS related to function a nd p arameter e stimation in Section 1.3.
Sect~on 1.4 gives a description o f t he n ature o f t he measurement errors. In
.
SectIon. 1. ~ a n answe.r is gi:,en .why the ! HCP is difficult. T he i mportant subject
of sensltlVlty coeffiCIents IS dIscussed in Section 1.6. A b rief classification of
sol~tion m ethods o f t he I HCP is given in Section 1.7. C riteria for evaluating
varIOUS I HCP m ethods are suggested in Section 1.8. T he final section, 1.9, gives
the scope o f the book.
1 .2
1.2.1
;.
E XAMPLES O F I NVERSE P ROBLEMS
I nverse H eat C onduction P roblem E xamples
O ne e xample of t~e . IHCP is t.he e stimation o f the heating history experienced
by a s huttle o r mISSIle reenterIng the earth's atmosphere from space. T he h eat
flux a t the he~ted s~rfac~ is needed. Figure 1.1 depicts a reentering body a nd a n
enlar~ed sectIOn o f ItS s km. T hough the heat flux, d enoted q, may be in general a
functIon. o f b oth position y a nd time I, it is a ssumed at present that lateral
conduction can be neglected compared to the heat flow n ormal to the surface.
Thus the net surface heat flux as a function o f time is e stimated from measurements obtained from an interior temperature sensor at position x as shown in
F'
I
Igure 1.1b. T he measurements are made at discrete times, I I , 12 , • .. o r in
gener~,1 ~t tim~, I j a t which the temperature measurement is d enoted Y; . (The
word discrete means at several particular times, such as I second, 2 seconds,
3 seconds, etc., but not continuously.) Figure 1.2 is a n illustration of postulated
values.
An estimated surface heat flux, d enoted qj, is associated with the time /.
a t which the corresponding temperature measurement, y;, is made. T he Iru~
value o f t he surface heat flux is simply denoted qj. T he surface heat flux history,
4
CHAP.1
DESCRIPTION OF THE INVERSE HEAT C ONDUCTION P ROBLEM
SEC. 1.2
sources include hightemperature fluids that flow in reactor heat exchangers,
over reentry vehicle surfaces, o r across turbine blades. The heating can also be
by r adiation from any source o r by conduction from an adjacent solid t hat is in
thermal contact with the b oundary in question.
T o e stimate the surface heat flux history it is necessary to have a m athematical model o f t he heat transfer process. F or example, in the reentry vehicle
case shown in Figure 1.1 b, it is assumed that the section of the skin is o f a single
material, homogeneous and isotropic, a nd t hat it closely approximates a flat
plate. ( A radial segment o f a cylinder o r o ther one dimensional coordinate can
be treated in a similar manner.) Then a possible mathematical model for the
temperature T in the plate is:
Section A
(4)
(b)
!..(k
FIGURE 1.1 Example o f reentering vehicle for which the surface heat flux is needed. (a), Reentering vehicle s chematic; (b), section A.
ax
OT)=PC a T
ax
at
T(x, 0) = To(x)
a T = 0 a tx=L
ax
.......
~.
.
:3
... .. \ .
•
5
EXAMPLES OF INVERSE P ROBLEMS
Measured temperature.
(1.2.1)
(1.2.2)
(1.2.3a)
(1.2.3b)
Y i.
T he objective is t o estimate the surface heat flux a t discrete times, t;, from
at time I i
~
q(t i ) =  k iJT(x, t;)1
ax
% =0
Time,t;
FIGURE 1 .2 M easured temperatures a t discrete times.
•I
o•
r
This problem is q uite different from the direct problem in t hat t he boundary
condition is n ot specified a t x =O b ut instead a measured temperature history
is given a t o ne o r m ore internal locations. F urther c omplications are that the
measured temperatures a re o btained only a t discrete times a nd they inherently
have errors in them. Clearly interior measurements contain much less information than given for classical direct problems where the surface conditions are
continuous, errorless relations.
T he t hermal conductivity, k, density, p, a nd specific heat, c, a re postulated
to be known functions of temperature. If a nyone o f these thermal properties
varies with temperature, the I HCP becomes nonlinear. T he initial temperature
distribution, To(x), is a lso taken as known. The location, X I, o f t he sensor is
assumed to be measured a nd t o have negligible error. T he thickness of the plate,
L, is also known and considered errorless.
The known boundary condition o f perfect insulation given by Eq. (1.2.3a) is
only one o f m any t hat c an be prescribed a t x = L. T here can be a convective
a nd/or a r adiation condition a t x = L. F or a n I HCP with a single unknown heat
flux, it is only necessary t hat t he boundary condition a t x = L be known. F or a
t emperaturedependent heat transfer coefficient h a nd also for a radiation
condition, the inverse heat conduction problem again becomes nonlinear.
F or t he case of a single interior temperature history, the problem can be
Calculate.d surface heat
flux, q i. at time t;
.. ..
Time,t;
FIGURE 1 .3

(1.2.4)
Representations o f calculated surface heat fluxes.
q(t), can be a n a rbitrary singlevalued time function; see Figure 1.3, which is a
representation o f estimated values o f q(t) a t times ti • In general the heat flux
can rise and fall abruptly a nd c an be both positive and negative where negative
values indicate heat losses from the surface.
The source o f h eating is immaterial to the I HCP procedures. Convective
L
CHAP.1
6
DESCRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM
subdivided into two separate problems, one o f which is a direct problem as
shown in F igure 1.4. T he portion of the body from X=X I t o L, body 2, c an be
analyzed as a direct problem because there a re known b oundary c onditions a t
b oth boundaries [ T(t)=Y(t) a t X=X I , a T/ax=O a t x =L]. F rom this direct
problem the heat flux a t X I can be found from the solution for the temperature
distribution in X I ~x~L by using
qx,(t) =  k aT\
a
(1.2.5)
A
x
X =Xl
T his same heat flux must leave b ody 1 (O~x<xd. Consequently, two c onditions
are specified a t X=XI in body 1 a nd none a t x=O. Such a s et o f b oundary
conditions for the transient heat conduction equation, Eq. (1.2.1), is related to
the mathematical problem being illposed.
T he I HCP for a single unknown surface heat flux c an be complicated in
many ways, some o f which are illustrated in Figure 1.5. T here a re four temperature sensors shown which preclude the simple subdivision shown in Figure 1.4
because the heat flux calculated to leave one subdivision would n ot in general
equal the calculated heat flux entering the next subdivision. Another complication is t he composite body o f t hree different materials which may be joined
together with either perfect o r imperfect contacts. In addition the plates might
q(t)
=?
xl1
I
L
(0
=1
, Yet)
q(t)
Inverse
·1
yet)
CD
Direct
f XIl
FIGURE 1 .4 Subdivision o f a single interior sensor I HCP i nto inverse a nd direct problems.
SEC. 1.2
E XAMPLES OF INVERSE P ROBLEMS
7
n ot be flat b ut r ather be parts o f a cylindrical wall. A satisfactory solution o f t he
inverse heat conduction problem should permit treatment o f each o f these
complicating factors.
1 .2.2
O ther I nverse F unction E stimation P roblems
T here are se~eral o ther ~robl.ems related to the inverse heat conduction problem.
The IHC~ mv?lves estImatIOn o f t he surface heatflux timefunction utilizing
~easured m tenor t emperature histories. I t is a linear problem (see Section 1.6.1)
If th~ ~hermal p roperties a re i ndependent o f t emperature a nd t he boundary
c ondltlon a~ t he " known" b oundary is linear. A closely related problem involves
the convective b oundary c ondition,
aTI
k a
= h[T..,(t) T(O, t)]
X x =o
(1.2.6)
I f the heat transfer coefficient, h, is k nown either as a constant o r as a function
o f time, the estimation o f t he ambient temperature T..,(t) from given internal
temperature measurements is a linear, inverse function estimation problem.
If h is a k nown function o f T, then the inverse problem becomes nonlinear.49
An~ther import~nt .function estimation problem in connection with Eq.
(1.2.6~ IS t he ~etermmatlOn o f h as a function o f time. This is a nonlinear p roblem
even If t he dIfferential equation is linear. See Section 1.6 for further discussion
o f nonlinearity. T he d etermination o f t he transient heat transfer coefficient is
an important technique, for example, for investigating the complete boiling
42
curve. T he estimation o f h(t) is discussed in C hapter 8.
An interface contact conductance, he(t), is often used t o model imperfect
contact. F or t he interface in Figure 1.5 t he heat flux is related to he by
k

UTI
UX
x =(I. , + L,)
=hc[TI.,=(L, +L,)  TI.,=(L, + L,).]
h
T• . ambient
temperature
~X31~
~~~
FIGURE 1 .5 C omposite plate with mUltiple temperature sensors.
=_kiJTI
a
x x=(L, + L,)'
(1.2 .7)
where the ~ign +: m eans the material 3 side o f the interface and the sign  means
the m atena. 2 side. The problem o f e stimating hc(t) is very similar to that for
1
the convective heat transfer coefficient.
Endothermic o r e xothermic chemical reactions can occur inside materials.
These ca.n be o~ u nknown magnitudes. Also there can be an energy source due
to electnc heatmg, a nuclear sotirce, o r frictional heating. In these cases a n
a ppropriate describing equation for onedimensional plane geometries is
U( aT)
a x k a x + g(x, t )=pc
at
aT
(1.2.8)
8
C HAP .1
D ESCRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM
where g(x, t) is a volume energy source term. If g is a function of time only,
then estimation of g(t) from transient interior temperature measurements is
quite similar t o t he onedimensional I HCP. I fEq . (1.2.8) is linear and the boundary conditions are linear, estimation of g(t) is a linear problem; however, if
k = k(T)" t he problem of estimating g(t) becomes nonlinear. When g is a function
of both x a nd t, the estimation o f g(x, t) is similar to that of a twodimensional
I HCP.
T wo inverse function estimation problems that have received a great deal
o f a ttention from mathematicians are called the (improperly posed) Cauchy
problem for the twodimensional Laplace's e quation4s.Hss a nd the initialboundary value problem for the backward heat equation. S 3.S6
O ne form of the Cauchy problem for the equation,
a2 T a2 T
ax 2 + ay2 = 0
(1.2.9)
is for incomplete specification o f the boundary conditions but some interior
measurements o f temperature are given. The objective is to obtain an estimate
of T(x, y) for the complete domain including the boundaries.
O ne example of a backward heat equation problem is the determination
of the initial temperature distribution, To(x), in a finite body given the boundary
conditions and some internal transient measurements of temperature. (See
Problem 1.26.)
T he inverse problems mentioned become more complex as more functions
are determined simultaneously. F or example, one might attempt to simultaneously estimate for Figure 1.S t he heat flux q(t) o n the left boundary of the
body and TCXl(t) o n the right. This would involve simultaneous estimation of
two time functions.
If the surface heat flux is a function of position across the surface as shown in
Figure 1.6, a n umber of heat flux components would be simultaneously estimated; this is the twodimensional I HCP a nd is discussed further in C hapter 7.
S EC.1.4
F UNCTION E STIMATION V ERSUS P ARAMETER
E STIMATION
T he words "function estimation" were used in the previous section in connection
with the I HCP. In the I HCP, the heat flux is found as an arbitrary, singlevalued
function of time. The heat flux can be positive o r negative, constant o r a bruptly
changing, periodic o r non periodic, and so on. I t may be influenced by h uman
decisions. F or example, the pilot o f a shuttle can change the reentry trajectory.
In the I HCP problem the surface heat flux is a function of time and may require
hundreds of individually estimated heat flux components, qj, t o define it adequately.
Related estimation problems are those called " parameter estimation"
problems which are also inverse problems but with the emphasis on the estimation o f certain " parameters" o r c onstants o r physical properties. In the context
of heat conduction one might be interested in determining the thermal conductivity of a solid given some internal temperature histories and t he surface
heat flux a nd other boundary conditions. 49 T he thermal conductivity of
A RMCO iron near room temperature, for example, could be a parameter; it is
not a function and does not require hundreds o f values of k/ to describe it. The
parameter estimation and function estimation problems start to merge if
estimates are made o f the thermal conductivity, k, as a function of temperature,
T. However, the k(T) function is not arbitrary and is not adjustable by humans.
P arameter estimation is a c ompanion subject; a book by Beck a nd Arnold
has been written on the subject. 49 A background in parameter estimation is not
required to understand this book. The subject o f p arameter estimation has been
built on a statistical base but that is not as true for function estimation problems.
This book stresses the numerical and mathematical aspects of function estimation rather than the statistical aspects. Certain statistical aspects are included,
however, as in Section 1.4 in which measurement errors are discussed.
1.4.1
~~
•
Temperature·sensors
FIGURE 1 .6 Surface heat flux as a function of position for a flat plate.
9
1 .3
1 .4
r
M EASUREMENTS
M EASUREMENTS
D escription o f M easurement E rrors
In the inverse heat conduction problem there are a number of measured quantities in addition to temperature; such as time, sensor location, and specimen
thickness. Each is assumed to be accurately known except the temperature.
I f this is n ot true, then it may be necessary, for example, to simultaneously
estimate sensor location and the surface heat flux. T he latter problem would
involve both the inverse heat conduction and parameter estimation problems
and is beyond the scope of this book. If the thermal properties are not accurately
known, they should be determined as accurately as possible using parameter
estimation techniques.
10
CHAP.1
DESCRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM
T he temperature measurements are assumed to contain the major sources
o f e rror o r uncertainty. Any known systematic effects due to calibration errors,
presence o f the sensor, conduction and convection losses o r whatever are
assumed to be removed to the extent that the remaining errors may be considered to be random. These random errors can then be statistically described.
The information provided by the sensors inside the heatconducting body
is incomplete in several respects. First, these measurements are a t discrete
locations. There is only a finite number o f sensors, sometimes only one. Hence
the spatial variation of temperature is quite incompletely known. Moreover,
the measurements obtained from any sensor are available only at discrete times,
rather than continuously. Due to the nature o f the measurement errors, a
continuous t emperature record might contribute little more information than
the discrete values, however.
1 .4.2
S tatistical D escription o f E rrors
A set o f eight standard statistical assumptions regarding the temperature
measurements is given in this section. These are standard a ssumptions and may
not be valid for a particular case. These eight assumptions49 d o provide a
yardstick with which to compare the actual conditions. T he r andom errors in
the temperature measurements cause random errors in the surface heat flux
values. The standard assumptions permit simplifications in the analysis o f
r andom errors. T he eight standard assumptions discussed in Beck and Arnold 49 are:
1. T he first standard assumption is t hat the errors are additive o r
11 = 7; + ej
(additive errors)
(1.4.1)
where 11 is the temperature measurement at time tj, 7; is the " true" t emperature
at time tj, and ej is the random error at time t j •
2. T he second standard assumption is t hat the temperature errors, ej,
have a zero mean (a theoretical quantity),
E(ej)=O (zero mean errors)
(1 .4.2)
where E (') is the "expected value operator."49 A r andom error is o ne that varies
as the measurement is repeated b ut the theoretical mean does n ot have to be
equal to zero. There can be a bias; t hat is, the error might tend to be positive.
I t is frequently possible to calculate and remove the bias.
A sample mean is the average t hat is based on actual measurements. T he
t rue average Ofej c annot be determined becauseej is u nknown; a typical equation
for finding the sample mean o f a r andom variable such as 11 is
_
1
1 ';=J
J
L
j= 1
ljj
(1.4.3)
SEC.1 .4
M EASUREMENTS
11
where ljj is the j th measurement at time tj a nd there are J measurements at
time tj.1f Eq. (1.4.2) is true, the expected value o f ~ is 7;. T he expression given by
Eq. (1.4.3) is called the sample mean a nd c an sometimes be used to check the
assumption given by Eq. (1.4.2).
3. T he third standard assumption is t hat o f a c onstant variance,
V( 11) =
0 '2
(constant variance error)
(1.4.4)
where V (·) is the "variance operator" a nd is related to the expected value
operator by
(1.4.5)
The symbol 0 '2 does not contain an i subscript, thus Eq. (1.4.4) means that the
variance o f 11 is i ndependent o f time tj a nd is a constant. I f the constant variance
ass.umption embodied in Eq. (1.4.4) is valid and there is only a single sensor, an
estimate o f the "variance o f 1';", d enoted S 2, is
(1.4.6)
for n measurement times; p is the number o f p arameters being used to estimate
7;, the estimate o f which is denoted 9;, a nd e j is the residual defined by
(1.4.7)
Ex~r~ssions o f the type giv~n by Eq. (1.4.6) c an be employed to investigate the
validity o f the constant variance a ssumption given by Eq. (1.4.4).
4. T he fourth standard assumption relates to the correlations among
measurements. F or two measurement errors ej a nd e j where i of j , the two errors
are uncorrelated if the covariance o f ej a nd e j is zero o r
(1.4.8)
The different errors I l j a nd I lj a re uncorrelated if each has no effect on o r relationship to the other. An example o f correlated errors is ej = pe j _ 1 + U j where U j is
uncorrelated to the e/S a nd p is a constant. As the sampling rate o f a n automatic
data acquisition system increases, the errors tend to become more correlated.
High correlation between succeeding temperature measurements indicates that
each new measurement is c ontributing much less information than if the
correlation were zero. Very high sampling rates (which approach continuous
measurements) may contribute little more information than considerably lower
rates; that is, larger time steps, 61, between the measurements.
A measure o f the correlation between the two succeeding d ata points 1'; a nd
12
C HAP .1
DESCRIPTION OF T HE INVERSE H EAT C ONDUCTION P ROBLEM
S EC.1.5
~
Yi+1 is the sample correlation coefficient,
p, defined as :
I
ejej+1
.:,.j=...;.I:""I
13
t hat is, those that vary significantly between successive values, are not permitted,
however.
• 1
p
W HY IS T HE I HCP D IFFICULT?
.
(1.4.9)
I 1 er
1 .5
W HY IS T HE I HCP D IFFICULT?
i=
(This is a n appropriate estimator for p if Ilj = p llj _ I + Uj which was mentioned
previously.) A low correlation is n ear zero and a high correlation is n ear ± 1.
( For further discussion of correlated errors, see Reference 4 9, p p. 301  326.)
5. T he fifth standard assumption is t hat the temperature measurement
errors have a normal (that is, gaussian) distribution,
Ilj has a normal distribution
(1.4.10)
I f the second, third, a nd fourth standard assumptions are valid, the probability
density o f Ilj is given by
1
(Il~)
f (eJ = a .J2rt exp 2a;
(1.4.11)
T he assumption o f n ormality is frequently valid even if standard assumptions
2 ,3, a nd 4 are n ot; in t hat case a joint p robability density for the errors is needed
(Reference 49, p. 230).
.
6. T he sixth standard assumption is t hat the statistical parameters such as
( 12 a nd p a re known,
Known statistical parameters
(1.4.12)
7.
T he seventh standard assumption is t hat t he times t l , t 2 , • • • , t ., positions
X J> specimen dimensions, a nd t hermal properties are accurately
known.
X I, X 2,""
Errorless time, dimensions, and configuration o f object
in question and thermal property values
(1.4.13)
In other words, the only source o f e rror is in the measured temperatures. I n
statistical terms, the variances o f time, a nd s o on, are zero.
8. T he last standard assumption is t hat there is no prior information
regarding the shape of the surface heat flux,
N o p rior information regarding the surface heat flux
1.5.1
S ensitivity t o E rrors
T he inverse heat conduction problem is difficult because it is extremely sensitive
to measurement errors. The difficulties are particularly pronounced as one tries
to obtain the maximum a mount o f i nformation from the data. F or t he onedimensional I H C P when discrete values o f t he q c urve a re estimated, maximizing
the a mount o f i nformation implies small time steps between qj values (see
Figure 1.3). However, the use o f small time steps frequently introduces instabilities in the solution o f the I HCP unless restrictions are employed t hat will be
discussed in later chapters. Notice the condition o f small time steps has the
opposite effect in the I HCP c ompared to t hat in the numerical solution o f t he
heat conduction equation. In the latter, stability problems often can be corrected
by reducing the size o f t he time steps.
1 .5.2
E xamples o f D amping a nd L agging; E xact S olutions
T he transient temperature response o f a n internal point in an opaque, hcatconducting body is q uite different from that o f a p oint a t t he surface. The internal
temperature excursions a re much diminished internally compared to the surface
temperature changes. This is a d amping effect. A large time delay o r lag in the
internal response can also be noted. These damping a nd lagging effects for the
direct problem a re i mportant t o study because they provide engineering insight
into the difficulties encountered in the inverse problem.
O ne interesting case is t hat o f a semiinfinite body .heated by a sinusoidal
surface heat flux o f frequency w,
q =qo cos(wt)
(1.4.14)
" Prior i nformation" m eans information known before any temperature
measurements a re m ade for a particular case. I f p rior information exists, then
it can be utilized t o o btain better estimates. If, for example, from experience
with previous similar tests the heat flux is c onstant over some t.ime perio.d o r .is
periodic, this information can be used to improve upon the estimators given I n
this book. I t is a ssumed herein t hat little is known a bout t he surface heat flux
except t hat it can vary abruptly with time. Highfrequency fluctuations o f qj,
(1.5.1)
where qo is t he maximum value o f t he surface heat flux. After a sufficiently
long time, the temperature solution also becomes periodic and is given by
T =To+
:0 (~r2
exp [  x
(~r2}os [ wtx (~r2 
iJ
(1.5.2)
where a. is the thermal diffusivity, k is t hermal conductivity, and To, a c onstant,
is the initial temperature distribution. T he envelope o f Eq. (1.5.2) is
( TTo).nv=qok
I
(~r2 ex p [ x(~r2J
(1.5.3)
As t he frequency w increases, the envelope decreases. The maximum temperature
14
C HAP.l
DESCRIPTION OF THE INVERSE HEAT C ONDUCTION PROBLEM
SEC . l.5
rise occurs at x =O a nd is p roportional to w  II2. F or an interior location
( TTo).nv
(TTo).nv . x=O
= exp [ (W)II2]
x 20:
W HY IS THE I HCP DIFFICULT?
1.0 r~:~_..
x +;;::+
0 .9
~
(1.5 .4)
0 .8
which shows that the envelope of interior temperatures sharply decreases for
increased x values. The exponential in Eq. (1.5.4) also indicates a large effect
as w is increased. T o o btain some insight from this equation, the case is considered wherein the righthand side is less than om o r
x(~r2 > 4.6
15
.
0.7
qc~
f  L  l
·I ~ 0.6
' i.
. .... 0.5
II
t.
0.4
0.3
F or steel with ex = 1 0 5 m 2 /s and w = 27t r ad/s= 1 Hz, there is negligible response
for x~0.82 cm ( =0.32 in.). This is large damping but ifw were further increased,
say by a factor of 100, there would be negligible response for x > 0.08 cm
( = 0.03 in .).
T he lagging effect can also be investigated through a n examination of Eq.
(1.5.2) . The surface temperature lags 7t/4 radians o r 45 behind the surface
heat flux, a nd any interior location lags even more. F or example, for the values
of ex = 10  5 m 2 /S, w = 27t rad/s, and x = 0.82 cm, there is a lag of 4.6 rad o r 264
which corresponds to a 0.73s lag of the internal temperature compared t o the
surface T.
Returning now to the IHCP, consider a transient interior temperature with
small fluctuations imposed o n its changing value. These fluctuations can be
the result of highfrequency sinusoidal surface heat flux components o r r andom
measurement errors. F or a given sensitivity in the temperature sensor, it is
possible to specify many different heat flux curves (each having highfrequency
components) that will p roduce interior temperatures indistinguishable from
one another. This implies that the inverse problem does not have a unique
solution. However, the heat flux history t hat caused a thermocouple response
can usually be determined to acceptable accuracy for properly designed experiments using the methods to be given.
Similar effects to those just described can be found through an examination
of the problem of a flat plate exposed to a constant heat flux qc a t x = 0 and
insulated at x =L. See Figure 1.7 a nd Table 1.1. T he solution for the temperature
distribution for this problem is given by
0
0
(1.5.5)
where
+ _TTo
T = qc L /k'
+ _ ext
t = L2 '
(1.5 :6a,b,c)
T he dimensionless time defined by Eq. (1.5 .6b) is sometimes called the Fourier
number. F or x+ = 1, the insulated surface, the time t+ = 0 .05 can be considered
as small and above 0.5 as large since little temperature response occurs before
0.2
0.1
°o~~~~~~~~~~~J
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t += ;~
FIGURE 1 .7 T emperalure s inside a plate with a constant h eatllux a t x = 0 a nd insulated at x = L.
{+ = 0.05 a nd a "fullydeveloped"linearwithtime response occurs after t + = 0.5 .
F or small times the response at x+ = 0 is quite rapid; in fact, the dimensionless
te~perature is expressed by
t+)II2
T+(O, { +)=2 ( 1t
for t+ < 0 .3
(1.5.7)
As t+ +0 the time derivative of Eq. (1.5.7) goes to infinity, indicating an instantaneous change in the surface temperature when the surface heat flux is
applied. F or an interior point the response is slow, being both lagged and
damped. As an example, for x + = 1 a nd for small times the T + expression is
(see Reference 51, p. 484)
(1.5 .8)
These expressions yield very small temperatures at early times and the time
derivative is zero as t+ +0. See also Figure 1.7 for x+ = 1 a nd small t + values.
Some numerical values for T + a re provided by Table 1.1. F or t+ = 0.05, for
example, T +(O, ( +)=0.2523, whereas T+(1, t+)=0.000269, a factor of almost
1000 smaller. This factor increases as t+ becomes smaller. O n the other hand for
sufficiently large times, the factor approaches unity. This can be demonstrated
as follows : The summation in Eq . (1.5 .5) can be dropped for large times to get
(1.5.9)
(.
SEC. 1 .6
T ABLE 1.1 D imensionless T emperature V alues. T+ ( x+. t +).
f or V arious D imensionless T ime a nd D istances f or a
F inite P late H eated a t x=O a nd I nsulated a t x =L.
T +(x+. t +) = [T(x. t )To)/q,L/k): x + = x/L: t+ = a.tIP
1+
om
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0 .60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
...........
' .J
16
x+ = 0 .0
0.112838
0.159577
0. 195441
0.225676
0.252313
0.276395
0.298541
0.319154
0.338514
0.356826
0.374245
0.390892
0.406863
0.422240
0.437089
0.451466
0.465422
0.479000
0.492236
0.505165
0.566146
0.622842
0.676928
0.729423
0.780946
0.831876
0.882444
0.932790
0.983002
1.033131
1.083210
1.133258
1.183287
1.233305
1.283316
1.333323
x+ = 0 .25
0.004377
0.020235
0.039238
0.058510
0.077297
0.095405
0.112807
0.129537
0.145644
0.161180
0.176198
0.190745
0.204865
0.218598
0.231980
0.245044
0.257820
0.270335
0.282614
0.294679
0.352432
0.407165
0.460054
0.511818
0.562895
0.613553
0.663954
0 .714199
0.764349
0.814440
0.864496
0.914530
0.964551
1.014563
1.064571
1.114576
x+ =0.50
0.000014
0.000802
0.003722
0.008754
0.015366
0.023074
0.031528
0.040486
0.049784
0.059311
0.068992
0.078777
0.088632
0.098535
0.108469
0.118425
0.128395
0.138375
0.148361
0.158352
0.208336
0.258334
0.308333
0.358333
0.408333
0.458333
0.508333
0.558333
0.608333
0.658333
0 .708333
0:758333
0.808333
0.858333
0.908333
0.958333
x+ = 0 .75
0.000000
0.000008
0.000150
0.000702
0.001879
0.003764
0,006360
0.009630
0.013523
0.017986
0.022969
0.028422
0.034302
0.040569
0 .047187
0.054123
0.061347
0.068831
0.076553
0.084488
0. 126735
0.172002
0.219112
0.267348
0.316271
0.365614
0.415212
0.464967
0.514818
0.564726
0.614671
0.664637
0.714616
0.764603
0.814595
0.864591
W HY IS T HE I HCP DIFFICULT?
so that
T+(O, t +) ~t+
o.()()()()()()
+t
(1.5.10a)
T +(l, t +) ~t+
x+ = 1.0
0.000000
0.()()()()()5
0.000057
0.000269
0.000786
0.001735
0.003207
0.005251
0.007885
0.011104
0 .014887
0.019205
0.024024
0.029306
0.035017
0.041121
0.047584
0.054375
0.061464
0.100516
0.143824
0.189738
0.237244
0.285721
0.334791
0.384223
0.433877
0.483665
0.533536
0.583457
0.633409
0.683379
0 .733361
0.783351
0.833344
17
i
(1.5.10b)
Hence, the temperature ratio is
T +(O,I+)
T +(l, t +)
/ ++t
1
~ / +  i ~ 1+ 2t+ for t + ~ 1
(1.5.11)
This result is a ppropriate for a " thin" plate which is defined as one with a
negligible temperature difference across it compared to the temperature rise.
The detailed values for Eq. (1 .5.5) given in T able 1.1 are provided for examples
and problems in subsequent chapters.
A geometry related to the finite plate is the semiinfinite body. This case
with a constant heat flux at x = 0 is discussed further in Section 1.6.2.2 a nd
numerical values are given in Table 1.2. T wo other cases o f interest are the solid
cylinder and solid sphere, both subjected to a constant heat flux. The center
location for each has the greatest lagging and damping; for that reason only
T ABLE 1 .2 D imensionless T emperature V alues.
f or T+ ( t;). V arious D imensionless T imes f or a
S emiInfinite B ody. T +(t:J = [ T(x. t )To)/(qcxlk):
t ; =a.tlx 2
.
r+(t;l
t;
T +(r;l
I.:
T +(r;l
0.05
0.06
0.10
0.12
0. 15
0.18
0.20
0.24
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
0.000135
0.000393
0.003943
0.007444
0.014653
0.023792
0.030732
0.046147
0.050254
0.071893
0.094800
0. 118437
0.142456
0.166631
0.190810
0.214891
0.238808
0.262515
0.285982
0.309190
0.332128
0.354791
0.377175
0.399282
1.25
1.50
1.75
2.00
2.50
3.00
3.50
4.00
5.00
6.00
7.00
8.00
9.00
10.00
12.00
15.00
20.00
25.00
30.00
35.00
40.00
50.00
60.00
70.00
0.50579
0.60612
0.70101 .
0.79119
0.95962
1.11505
1.26002
1.39635
1.64825
1.87832
2.09140
2.29076
2.47874
2.65708
2.98997
3.44283
4. 10921
4.69822
5.23182
5.72321
6. 18105
7.01871
7.77678
8.47439
80 .0
90.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
1000.0
9.1241
9.7345
10.3120
14.9776
18 .5604
21.5817
24.2439
26.6510
28 .8648
30.9254
32.8608
34.6914
t+
18
CHAP.1
DESCRIPTION OF THE INVERSE HEAT CONDUCTION PROBLEM
SEC . 1.6
T ABLE 1 .4 D imensionless
T emperatures a t t he C enter o f a
S olid S phere
the center temperatures are tabulated in Table 1.3 for the solid cylinder a nd in
Table 1.4 for the solid sphere. T he e quation for the temperature distribution
in the cylinder is (see Reference 51, p. 203)
+
+
+_
+
!
T .(r " .)2,. + 2(r
where
P., n =
+ )2_ !
4
 2 ~ e  P:'; J o(r+P.)
.~I
fI:Jo(P.)
,+
.
(1.5.13)
a nd where a is the cylinder radius a nd
+(+ + )
T lJ r , to
[ T(r,I)To]k
qc a
+
'
al
1. == ' 2'
a
(1.5.14)
The equation for the temperature distribution in the sphere is (see Reference 51,
p.242)
2 ~ sin(r+ /3.) _ p',+
+ + +_ + I +2 3
T .(r , 1.)=31. + 2 (r)  10 + L... p2 • P e • •
r n = 1 n sin "
(1.5.15)
where /3., n = I, 2, . .. , a re the positive roots o f
tan
fl. = fin
(1.5.16)
T ABLE 1 .3 D imensionless
T emperatures a t t he C enter
o f a S olid C ylinder.
P (O. t n E [T(O. t )ToJ/(qca/k).
t : E flt/a 2
,+
.
...........
~
0.010
0.020
0.030
0 .040
0 .050
0.060
0.070
0.080
0.090
0. 100
0. 110
0. 120
0 .130
0. 140
0. 150
0 .160
0. 170
0 .180
0.190
0.200
T +(O,,:)
,+
.
r IO,':)
0.00000
0 .00000
0.00003
0.00028
0.00120
0.00325
0.00676
0.01187
0.01862
0.02692
0.03667
0.04771
0 .05993
0.07317
0.08731
0.10223
0.11784
0.13405
0. 15077
0.16794
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.550
0.600
0.650
0.700
0.750
0.800
0.850
0.900
0.950
1.000
0.02692
0.08731
0. 16794
0.25861
0.35413
0.45198
0.55095
0 .65046
0.75022
0.85011
0.95005
1.05002
1.15001
1.25001
1.35000
1.45000
1.55000
1.65000
1.75000
T+(O,,:1
':
T+(O,':1
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.100
0.110
0.120
0.130
0.140
0.150
0.160
0.170
0.180
0. 190
0.200
(1.5.12)
I , 2, . .. , a re the positive roots o f the Bessel function,
J dP.)=O
19
SENSITIVITY COEFFICIENTS
0.00000
0.00000
0.00009
0.00088
0.00342
0.00865
0.01699
0.02848
0.04288
0.05988
0.07911
0.10023
0.12293
0.14694
0.1 7203
0.19801
0.22472
0.25203
0.27983
0.30804
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.550
0.600
0.650
0.700
0.750
0.800
0.850
0.900
0.950
1.000
0.05988
0.17203
0.30804
0.45293
0.60107
0.75039
0 .90014
1.05005
1.20002
1.35001
1.50000
1.65000
1.80000
1.95000
2.10000
2.25000
2.40000
2.55000
2.70000
the first few of which are PI =4.4934, P2 = 7.7253 a nd P3 = 10.9041. Both Tables
1.3 and 1.4 clearly exhibit the same lagging behavior as the x + = 1 location in
Table 1.1.
1.6
S ENSITIVITY COEFFICIENTS
1.6.1
D efinitions o f S ensitivity C oefficients a nd L inearity
In function estimation as in parameter estimation a detailed examination of
the sensitivity coefficients can provide considerable insight into the estimation
problem. These coefficients can show possible areas o f difficulty and also lead
to improved experimental design. The sensitivity coetlicient is defined as the
first derivative o f a d ependent variable, such as temperature, with respect to an
unknown parameter, such as a heat flux component. I f t he sensitivity coeffici~nts
are either small o r c orrelated with one another, the estimation problem is
difficult and very sensitive to measurement errors.
For the inverse heat conduction problem, the sensitivity coefficients o f
interest are those o f the first derivatives o f t emperature T a t location X j a nd
time with respect to a heat flux c omponent, qM, a nd are defined by
'i
(1.6.la)
20
CHAP.1
DESCRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM
for j =I, . .. , J, i =I,2, . .. , n, a nd M =I,2, . .. , n. N ote t hat the number o f
times t j equals the number o f h eat flux components. The heat flux component,
qM, is s hown in Figures 1.8 a nd 1.9; qM is t he constant heat flux between times
t M 1 a nd t M' I f t here is only one interior location, t hat is, J = I, t he sensitivity
coefficient is simply given by
aT,
X M(t j )='
(1.6.1 b)
aqM
F or t he transient problems considered in the I HCP, t he sensitivity coefficients
a re z ero for M > i. In other words, the temperature a t time t j is i ndependent of a
yettooccur future heat flux component of qM, M > i.
O ne o f the important characteristics of the I HCP is t hat it is a linear problem
if the heat conduction equation is linear and the boundary conditions are lilJear.
T he thermal properties (k, p, a nd c), c an be functions o f p osition a nd n ot affect
the linearity. They cannot, however, be functions o f t emperature without causing
the I HCP to be nonlinear. Linearity, if it exists, is a n important property because
it allows superposition in various ways and it generally eliminates the need
for iteration in the solution. I f t he linear I HCP is t reated as if it were nonlinear,
excessive computer time would be used due to unnecessary iterations.
q (/)
SEC. 1 .6
S ENSITIVITY COEFFICIENTS
21
O ne way to determine the linearity o f a n estimation problem is t o inspect the
sensitivity coefficients. I f t he sensitivity coefficients are not functions o f t he
parameters, then the estimation problem is linear. I f they are, then the problem
is nonlinear. This can be illustrated using Eq. (1.5.7) and differentiating T with
respect to qc. F rom Eq. (1.5.7):
aT(O, t) =~ 2 (~)1/2
aqc
k
nL2
'
+
03
(1.6.2)
t<.
which is i ndependent o f qt. M ore generally, Eq. (1.5 .5) c an be used to arrive
at the same conclusion.
An example o f a n onlinear estimation problem is t hat o f e stimating ex.
T aking the derivative of T in Eq. (1.5.7) with respect to ex yields
(_t_)1/2 '
aT(O, t) = qc L
aex
k 7texL2
+ 03
t<.
(1.6.3)
The right side o f Eq. (1.6.3) is a function o f ex; t hus the estimation o f ex from
transient temperature measurements is a nonlinear problem. ( If t he m ore lengthy
Eq. (1.5.5) is used, the same conclusion is reached.)
This principle o f the sensitivity coefficients being independent o f t he parameter to be estimated can be employed in cases when the solution is n ot explicitly
known. The equations for a flat plate with temperatureindependent properties
are given as a n example:
aa
a [k(x)T ] = pc(x)  T
ax
qM
I
FIGURE 1 .8 Hea~ flux history with a conSlant
heat flux q. . and arbitrary elsewhere.
I
0
I MI
a TI
k
ax
(1.6.4)
at
 k a TI
= {qM=constant, t M
a x x =O
q(t), t >tM
I
I
I
ax
= qloss
 1 < t<t M
(1.6.5)
(1.6.6)
x =L
1M
(1.6.7)
where TM  I (x) d enotes the temperature distribution a t time tM _ 1 a nd qto,. is a
heat flux history due to losses that are independent o f qM' T he heat flux, q(t),
for t > tM is a n arbitrary function o f time. The thermal properties can be functions of x. And the parameter o f interest is the heat flux, qM, which is a c onstant
between times tM  I a nd tM' See Figure 1.9. T he temperature distribution at
time t M  1 is k nown and is given by Eq. (1.6.7).
I t is desired to find the differential equation and boundary conditions for
the qAt sensitivity coefficient defined by
X M(x, t )=aT(x, t)
DqAt
FIGURE 1 .9 H eat flux components.
(1.6.8)
F or t < tM  I , t he solution is X M(X, t) = 0; t hat is, the body has not yet been
22
C HAP. 1
DESCRIPTION OF T HE INVERSE H EAT C ONDUCTION P ROBLEM
exposed t o qM' F or times greater t han tM  I , Eqs. (1.6.4)(1.6.7) are differentiated
with respect to qM t o o btain,
aXM] = p c(x)aXM
a [ k(x) ax
ax
at
_{I,
 k aXMI
ax
x =O 
0,
S EC.1.6
23
S ENSITIVITY COEFFICIENTS
and thus XM(t)=O, for
then
t~IMI'
T he initial condition for XM(t) for l >t M_ 1 is
(1.6.15)
(1.6.9)
The complete solution for
X ,\ /
is
XM(t)=O,
(1.6. 10)
t~IMI
(1.6 .16a)
(1.6.l6b)
a XMI
0
ax x =L
(1.6. 11)
(1.6.16c)
(1.6.12)
Equations (1.6.9) (1.6.12) describe the mathematical problem for the sensitivity
coefficient X M which can b e explicitly found if the functions k(x) a nd pc(x) a re
known. Notice t hat it is n ot necessary t o k now qM, q(t), o r even TM_1(x) to
obtain a solution for the sensitivity coefficient X M(X, t) b ecause Eqs. (1.6.9)(1.6.12) are n ot functions o f qM, q(t), o r TM_1(x).
An i mportant c onclusion t hat c an be drawn from Eqs. (1.6.9) (1.6.12) is t hat
the estimation problem for qM is linear, a consequence o f X M(X, t) n ot being a
function o f qM' T his means t hat t he (unknown) value of qM is n ot needed to
find its sensitivity coefficient. I t is a lso significant t hat t he same differential
equation is given for X M(X, t), Eq. (1.6.9), as for T(x, t), Eq. (1.6.4). Also the
boundary conditions are o f t he same type; t hat is, the gradient conditions given
by Eqs. (1.6.5) a nd (1.6.6) for T(x, t) a re still gradient conditions, Eqs. (1.6.10)
a nd (1.6.11) for X M(x,t). T he m ain differences are t hat t he XM(x, t) b oundary
c onditions a re s impler a nd t he initial condition is zero. D ue t o t his similarity
between the T(x, t) a nd XM(x, t) problems, the same solution procedure o r
c omputer p rogram c an be used for the XM(x, t) s olution as for the T(x, t),
which can result in considerable programming a nd c omputational efficiency.
which is s hown in Figure 1.10a. T he sensitivity coefficients X I , . .. , X 4 a re
shown in F igure 1.10b .
In general, welldesigned experiments for estimating the surface heat flux
component qM w ould have large values of the sensitivity coefficient X M(I).
Hence, Eq. (1.6.16b) indicates t hat the r atio o f t he surface area A to the thermal
capacity (pc V) s hould be as large as possible. T hat is, a t hin foil is deSIred.
Several observations can be made regard ing the I HCP t hrough the examination of Figure 1.10. First, for the heat flux c omponent qM, m easurements before
time 1M  I yield n o i nformation regarding qM because Eq. (1.6.l6a) is t rue.
Consequently measurements after time 1M  I a re needed t o e stimate qM' Because
X M r emains greater than zero for t > 1M _ I ' t here is a n "infinite memory" o f
qM' In o ther words, the temperature a t a ny time subsequent t o 1M _ I is affected
by qM'
A second observation concerns the case of estimating m ore t han o ne component o f qM, such as q l' q2, a nd q3' T o e stimate several parameters simultaneously, it is necessary t hat t he sensitivity coefficients be linearly independent. 49 T his means t hat a t least one C;,/=O a nd t hat c onstants C 1 , C 2 , a nd C 3
1 .6.2 O neDimensional S ensitivity C oefficient E xamples
1 .6.2.1 l umped B ody C ase T he simplest transient heat conduction sensitivity coefficient is for a lumped body described by the differential equation
a nd initial condition,
q (t)A=pcV
dT
d r'
T(O) = To
( 1.6.l3a,b)
where pc, A, a nd V a re all constants. T he q(t) function can be such as those
given in Figures 1.8 o r 1.9. T he sensitivity coefficient for qM is o btained by
differentiating Eq. (1.6.13a) with respect t o qM t o get ( X M == a T/aqM),
C~dXM={l, t M1<t<t M
pA
dt
0,
o therwise
(1.6.14)
°o~~~L~
' 101 + 1
(a)
F IGURE 1.10
(b)
Sensili vil Y coefficients for the l umped body case .
24
CHAP. 1
DESCRIPTION OF T HE INVERSE H EAT C ONDUCTION P ROBLEM
c annot be found such that
(1.6.17a)
over the domain o f t he measurements. Referring to Figure 1.10b, if measurements o f T a re m ade a t t l , t 2, a nd t 3, t here would be no set o f c onstants that
would m ake Eq. (1.6.17a) true. Hence, q l' q2, a nd q3 can be estimated if Y1, Y2,
a nd Y3 a re known along with the temperature a t time zero.
Another way to look a t t he linear dependence of the sensitivity coefficients
X I' X 2, a nd X 3 is t o write Eq. (1.6.17a) in matrix form for the three different
times, t i> tj ' a nd t b
...
SEC. 1 .6
25
S ENSITIVITY COEFFICIENTS
ing q /s as revealed by the proportionality o f t he sensitivity coefficients to each
other over large time periods as shown in Figure 1.10b.
1 .6.2.2
S emiInfinite B ody. T he s olution for the temperature in a semi
infinite p lanar body (defined by
q " is
x :;': 0)
subjected to a constant surface heat flux,
i
T(x, 1 )= To+2 q (exl)112 ierfc [ (4cct)1/2]
7
(1.6.18a)
which can be written in dimensionless form as
T+(t;)=2(t;)112 i erfc[(4t;)1/2]
X I(ti)
X1(t j )
[ X I (tt)
(1.6.17b)
T+
F or linear dependence this equation must be true for arbitrary C i values, but
not all Ci c an equal zero. Equation (1.6.17b) is t rue if and only if the determinant
o f the square matrix o n t he left is e qual to zero. O ne such example is for the
matrix associated with t2, t2,5, a nd t 3,
[~
.1
~.5]
k [T(x, t ) To]
qc x
(1.6.18c, d)
This dimensionless temperature is t abulated for some values o f
1.2. The function, ierfc(z), is
ierfc(z) = 1t 1/ 2 exp( 
Z2)_ Z
(1.6.18b)
erfc(z)
t;
in Table
(1.6.19)
At t he heated surface the temperature is
(1.6. 17c)
T(O, t )=To+2
1
which has a determinant o f zero. Note that the first two columns o f Eq. (1.6.17c)
are equal and thus linear dependence is seen by simple inspection. Equation
(1.6.17a) is satisfied by setting C 1 =  C 2 a nd C 3 = 0. T he heat flux components
ql ' q2, a nd q3 c annot be simultaneously a nd uniquely estimated by using
Y;, l j, a nd Y,. if Eq. (1.6.17b) is valid.
The final observation regarding the sensitivities shown in Figure 1.10 is
that, though there is a n infinite memory o f q M, t here is a perfect correlation
between qM a nd qM + 1 for times greater than tM + I ' T his is because the sensitivity
coefficients for X M a nd X M + 1 a re equal for t >tM+1 as shown in Figure 1.10b .
Expressed another way, the measurement Y c ontains information regarding
100
ql b ut it may not be efficient t o use this information; Y also contains informa100
tion a bout q2," " qloo a nd is affected in exactly t he same m anner by changes
in each o f these q c omponents. O n t he o ther h and Y1 is n ot affected by q2, . .. ,
q 1 00 a nd neither is Y by q 1 0 1 , • • •• Consequently an estimation scheme that
100
simultaneously estimates q l,"" qloo by using Y1, . .. , Y may n ot be much
100
better than one that estimates the qi values in a sequential m anner t hat depends
mainly o n previous times. (By sequential, it is m eant that q M1 is estimated
before a nd is completely independent of q M') Even though a temperature a t a
large time contains information a bout a h eat flux component a t a n early time,
it may not be advisable to use t hat i nformation to estimate the early q component. This conclusion is a result o f t he high correlation between the interven
(~)(~r2
(1.6.20)
which is similar to Eq. (1.5.7) for early times in a finite plate. Notice that for
Eqs. (1.6.19) a nd (1.6.20) t he qc sensitivity coefficient,
X (x, t )= aT(x, t)
aqc
(1.6 .21)
is e qual t o t he temperature rise for qc= I . I t is c onvenient to examine the case
o f x =O a nd x fO separately. F or x=O,
1
(t
2X(O, t )=  (ext)1 2 = 2 )112
k 1t
1tkpc
(1.6.22a)
a nd for x fO
(1.6.22b)
T he dimensionless sensitivity coefficient,
X+=~aT=T+
x aqc
(1.6.23)
is t abulated in Table 1.2 a nd plotted in Figure 1.11. T he p lot for x = 0 is included
for comparison ; for the x =O case, the x in Eq. (1.6.18d) and in Eq. (1.6.23)
should be interpreted as the x o f t he interior location. [This is confusing b ut it
permits the x =O a nd x fO cases to be plotted in the same graph. N ote t hat
26
, CHAP,1
DESCRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM
100~~~~~~,
SEC. 1 .6
S ENSITIVITY COEFFICIENTS
27
In order to demonstrate this difference, the x = 0 a nd x + 0 cases are treated
separately. The x =O case is considered first. F or t <t M 1, the sensitivity coefficient at x = 0,
x M(O, t):; BT(O, t)
10
=~ {[CX(/~M  t)r2 _[CX(t~tM>J/}
0 .1
Interior location
( x" 0)
0.01
0.001
=
Note: For the surface, the
x in X+ a nd t+ is arbitrary
a nd may b e s et equal to
unity. (See Equations (1 .6.1&, d)
and (1.6.220.).)
S ensitivity coefficients for semiinfinite b ody h eated w ith a c onstant h eat flux.
Eq. (1.6.22a) can be normalized by multiplying by k /x only if x + 0.] F or a single
constant heat flux, the sensitivity coefficient behaves in the same manner as
the temperature. The temperature rise for a semiinfinite body is similar to
that o f a finite body at early times. T he surface temperatures (and sensitivity
coefficients) are numerically equal for the finite and semiinfinite bodies at
small times and differ by only 1%a t t + = I; = 0.3 (see Problem 1.20). Because o f
similarities between the temperature rise a nd the sensitivity coefficient and also
between the finite plate and semiinfinite body, many o f the same comments
made in Section 1.5 regarding the lagging and damping effects can be applied
to a semiinfinite body as well as to a finite plate.
In the I HCP t he complete heat flux history is sought. Hence, it is necessary
to 'examine the sensitivity coefficients for a general heat flux component, qM,
which is depicted in Figure 1.9. As c an be seen from Figure 1.11, the behavior
o f the surface temperature is quite different from that o f an interior temperature.
t >I M
(1.6.25)
The. ~econd form of Eq. (1.6.25) is constructed using the principle o f superpOSItIOn . See Problems 1.21 a nd 1.22. These expressions given by Eq. (1.6 .25)
can be plotted on a single curve if one lets I lt=t Mt M_ t and rewrites Eq.
(1.6.25) as
1l/)t/2
2 ( tIM_1)1/2
kXM(O , t) ( ;;= 1t 1/2  IlI,
0 .0001 L ______.....L.._ ___ L._ _I ..______....L..._ _____ J_ _____  '
0 .001
0.01
0.1
10
100
FIGURE 1.11
(1.6.24)
BqM
is zero. F or I> 1M  1 there are two nonzero expressions,
2 [CX(/t M_ d]1 /2
XM(O, t)=1<
1t
'
t M 1< t<tM
x + = 2(t+ /".)1/2
(See Equation (1.6.22a).)
IM 1< t<tM
1t~/ 2 [ C;;J/2 C~~1 1Y'],
t >I M
(1.6.26)
Th ~ s expression i.s plotted in Figure 1. 12 for M = 1, 2, 3, and 4. T he graph is
vahd for any chOice o f M provided I II is a positive constant.
T he sensitivity coefficients plotted in Figure 1.12 are for the surface o f a
semiinfinite body and are similar in some respects to those shown in Figure 1.10
which are for a lumped body. The similarities include the instantaneous response
to changes in the surface heat flux . A difference is t hat the semi i nfinite body
responds more rapidly and the effect o f heating over a time interval gradually
Semi·infinite body; x =0
1.2
1.0
~
;;1·
0.8
~ 0 .6
""
0 .2
°0~~~2~3L4~~
t /dt
FIGURE1 ,12
S ensitivily coefficients for hea t flux c omponents for surface o f a sem i infinite b ody .
.
_.__.__ .. _ _.       
28
C HAP. 1
D ESCRIPTION O F T HE I NVERSE HEAT C ONDUCTION P ROBLEM
S EC. 1 .6
diminishes. There is an "infinite memory" in the sense that the sensitivity
coefficient given for t > tM by Eq. (1.6.26) only approaches zero for t  tw+ 0 0.
T he decision regarding the times at which to make measurements to estimate
certain parameters is called experimental design. As noted previously, because
XM(O, t)=O for t <t M 1 t emperature measurements prior to t M 1 c annot be
used to estimate qM' O n the other hand, if only measurement times much
larger than tM are used to estimate qM, difficulties are encountered for two
reasons. First, the X M value is relatively small, indicating little sensitivity
regarding qM a nd hence little information. Second, the X M , X M + I , . •• functions
tend to become correlated, t hat is, have nearly the same shape and thus approach
linear dependence. Consequently the best choice of measurement times for
estimating qM from surface temperature measurements of a semiinfinite body
must include tM, t M+ I , a nd slightly larger t values.
The case o f a n interior temperature measurement history has substantially
different characteristics than that for x = 0 j ust discussed. The dimensionless
sensitivities for q l' q2, a nd q3 are shown in Figure 1.13. Unlike the x = 0 curve,
the results for x =1=0 c annot be combined into a single curve. The dimensionless
sensitivity curves for q l' q2, a nd q3 are denoted X i, X t, and X j ; the curves
have exactly the same shape but they are shifted to the right
for
I'lt; from X ;' t o X ;'+I'
T here are several important observations t hat can be made in regard to
Figure 1.13. First, the X;"s display a lag which is most clearly shown in Figure
1.13a by X i. Even though ql starts at t=O as shown by Figure 1.9, X i in Figure
>0.05. This is quite unlike the x =O
1.13a does n ot appreciably rise until
behavior for a semiinfinite body for which Figure 1.12 shows the greatest
gradient of X i as t+O.
T he second observation is t hat the magnitudes of the
increase with
I'lt;. Small values o f I'lt; give small values of
causing the estimation o f the
surface heat flux t o become inaccurate.
x~
2.5
t;
(b)
xt, xt, ...
0.4
0.3
X~
t;
0.2
xt's
xt
0.03 r r,.,r;r.rr,.,,
29
S ENSITIVITY C OEFFICIENTS
t .t;= 1
0.1
~
0
!
0
2
3
4
5
6
7
8
9
10
t;
(e)
ff
F IGURE 1 .13 Dimensionless sensitivity coefficients for q" q2 and qJ for an interior point in a
semiinfinite body. I 'll: =0.05, 0.25, a nd I .
xt
t .t; = 0.05
0.7
t;
( a)
0.8 0.9
l .0
Third, each
curve has a maximum which occurs farther from the time o f
the end o f the heating pulse as I'lt; becomes small. F or example, for I'lt; = 1,
X i has a maximum at t ; = 1.3. Also the value o f X i a t t ; equal to one I'lt; is
a bout 87% o f the maximum X i for I'lt; = 1. This is in contrast to the I'lt; =0.05
case where the maximum X i is a t t ; = 0.45 which is 8 I'lt; values after the
heating ends. Moreover, X i a t t ; =L'lt; is only 0.7% o f the maximum X i.
T he fourth and final observation is t hat all the X ;"s tend to become correlated
increases.
as
t;
30
C HAP. 1
DESCRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM
These characteristics of sensitivity coefficients provide some insight into the
design of an estimator for the I HCP utilizing an interior sensor. O ne o f the
most obvious points is t hat temperature measurements at times greater than
tM are needed t o estimate qM, particularly as IJ.t; becomes small. Because
X i for IJ.t; =O.OS is nearly zero a t time t; = O.OS {the end of nonzero q d, Y
1
would yield very little information regarding q I . T he Y2 measurement, however,
is significantly affected by q l, b ut insignificantly by q2. Hence, a more effective
strategy for estimating qM for the case of IJ.t; =O.OS would use "f. ~ 1 YM and
YM + 1 rather than omitting YM + I . Even a better procedure might be to utilize
more future (relative to tM) temperature measurements than YM+I . If, however,
a large number of future temperature measurements is used, for example,
Y2 , • •• , Y to estimate q l then the procedure might not be efficient due to
100
the high correlation among the sensitivity coefficients.
When r future temperatures such as YM, YM+I , • .. , YM+r  1 are used to
estimate qM, there is extra information for estimating qM which manifests itself
as extra algebraic equations involving qM . This information is needed, but due
to measurement errors, it is not completely consistent. O ne way to use all this
information is to employ the method of least squares which is discussed in
C hapter 4. Also see Problems 1.9  1.14.
1 . 6.2.3 P late I nsulated o n O ne S ide T he heat flux sensitivity coefficients for a flat plate heated on one side and insulated on the other are discussed
in this section. This is the same geometry as shown in t he inset of Figure 1.7. F or a
heat flux o f infinite duration the temperature distribution is given in equation
form by Eq. (1.S.5), in t abular form by Table 1.2, a nd in graphical form by
Figure 1.7. These results can also be interpreted as being equal to the dimensionless heat flux sensitivity coefficient o f
SEC. 1 .6
S ENSITIVITY COEFFICIENTS
31
and IJ.t is the duration of heating. This result is even more complicated than
that for a semiinfinite body because X i depends on t+ a nd 1J.t+ a nd also x+.
F or the large dimensionless times o f
t+
lJ.t+ > O.S
(1.6.31)
the second expression of Eq. (l.6.29) goes to the time and spaceindependent
value of
(l.6.32)
This expression shows the dependence of X; on the duration of heating;
hence, as more components of q are estimated over a fixed time period, the
sensitivity coefficients become smaller and t hus· the q{t) curve becomes more
difficult to estimate. Notice that as t + increases, the temperatures increase as
shown by Figure 1.7 b ut the sensitivity coefficients d o not.
The sensitivity coefficient X i normalized with respect to xtmax is plotted
in Figure 1.14. I n order to present results compactly, the time has been made
dimensionless by normalization with respect to the heating duration, 1J.t.
1 .6.3
T woDimensional S ensitivity C oefficient E xample
In this section the estimation of twodimensional heat flux histories is investigated. The heat flux is a function of time as previously discussed but also is a
function of position over the surface. Further, the heat flux is subdivided into a
(1.6.27)
0.8
% +=xIL=l
T he sensitivity coefficient for q l where
q= {
ql'
0,
O <t<tl=lJ.t
t >t l
( l.6.28)
can be readily found using Eq. (l.5.S). Because the other coefficients for q2, . . .
have exactly the same shape but are displaced IJ.t a part, it is only necessary to
examine X i which is given by
k aT
X i==T+{x+ t+)
L aq<
"
~
>< 0.4
0.2
O<t+<lJ.t+
(1.6.29)
where
I loll
A
ut
~.
+ _a.lJ.t
= 2
L
(1.6.30)
FIGURE 1 .14
H eal Hux sensilivilY coefficienls al insulaled surface o f Hal pl ale.
32
C HAP. 1
DESCRIPTION OF THE INVERSE HEAT C ONDUCTION PROBLEM
n umber o f s imple building blocks of s hort d uration o ver small regions. As
before, each h eat flux c omponent is a ssumed c onstant o ver position as well as
over time. See F igure 1.6 for the spatial variation a nd F igure 1.9 for the time
variation. O ther functional variations such as linear, parabolic, sinusoidal,
a nd so on, a re p ossible b ut t he basic ideas can be m ore easily presented for the
c onstant a pproximation.
T he t wodimensional a nd t imedependent h eat flux, q(y, t) is a pproximated
by (see P roblem 1.23)
q(y, t) =
L L jj;(J, t)
i
(1.6.33)
j
a nd
YjI/2~y<Yj+I / 2
(6)
1. .34
for constant h eat flux " building blocks." T he sensitivity coefficient for qji is t hen
_ aT(x, y, t)
X ji (x, y, t ) =
a
qji
m athematically by
(1.6.36)
,
,
(1.6.35)
~T =qo,
k
i
ir
x =O
At
y <o
(X
aT
(1.6.37)
{ ax = 0,
y>O
I
aT
t
JI
33
S ENSITIVITY COEFFICIENTS
f;
w here i refers t o t ime a nd j t o position. T he function j ji(y, t) is given by
t;_I~t<t;
J:.. ( y, t )={qjj, otherwise
0,
SEC. 1 .6
ay >0
I
!
An example o f a t wodimensional body is a semiinfinite b ody h eated with a
spacevariable flux as shown in Figure 1.15. T he c oordinate x goes into the
body s tarting a t t he surface; y is p arallel t o t he surface b ut t here is n o n atural
s tarting point. T he h eat flux q(y, t) is usually a c ontinuous f unction in b oth t ime
a nd p osition b ut it is a pproximated b y a series o f u niform pulses over s hort
d istances a nd t ime periods, as discussed previously. T he sensitivity coefficients
for this problem can be developed from the T(x, y, t) s olution for a semiinfinite
body heated continuously over negative y values; this problem can be described
as y> ± 0 0

T  > To
(1.6.38a)
for x> 0 0
(1.6.38b)
T(x, y, 0) = To
(1.6.38c)
T he s olution for all x's is given in Reference 50, b ut t o simplify the t reatment
only the solution for x =o is used, which is given in C arslaw a nd J aeger [ Reference 51, p. 264].
T he s olutio.n 51 for Egs. (1.6.36) (1.6.38) for x =o a nd y=/=O is
T+( +)
,y
T(O , y, t ) To
qo(at/n)1/2/k
~ ( y+)
er c
T
y+
2nl/2 E I
[ (y+)2]
 4 ,
T +( Y+ )=2erfc(!iJ)+ ly+1 E [ (y+)2]
Y
2
21[1/2 1
4
, Y<
°
y >o
(1.6.39a)
(1.6.39b)
where y+ a nd E1(z) (see Reference 52, p. 227) a re defined by
+_
y
=
Y
(at)1/2'
_ra) u
J
Iu
E 1(z)=
(1.6.40a,b)
e du
T he e xpression given by Eg. (1.6.39) can be plotted as a function o f y+ with
as a
attention to whether y is positive o r negative. Figure 1.16 gives a plot o f
function o f Iy+ 1/2 for positive a nd n egative y values. F or t he special locations
corresponding to y>  00 , y=O, a nd y> 0 0, t he surface temperatures are :
T;
t
T = To+2qo (  kn pc
"
Semi·infinite body
)1/2
,y> 
00
(1.6.41a)
(1.6.41 b)
(1.6.41c)
T he sensitivity coefficients for the various terms qji c an be constructed using
the principle o f s uperposition with Eg. (1.6.39). T o b e morl" specific, the surface
FIGURE 1 .15 Semiinfinite body with a space and timevariable heat flux t hat is a pproximated
by constant elements.
L
34
C HAP. 1
D EStRIPTION OF THE INVERSE H EAT C ONDUCTION P ROBLEM
S EC.1.6
S ENSITIVITY COEFFICIENTS
35
T:
The notation
[(2s  I)/(t: )1/2] means t hat y+ in Eq. (1.6.39) is t o be replaced
by ( 2s1)/(£:)1/2 . F or positive values of 2 s1 (or 2 s+ I). Eq. (1.6 .39a) is used.
whereas Eq. (1.6.39b) is used for negative values. F or the time o f n 6 t a nd
n = 2 .3 • .. . • the sensitivity coefficient for q OI is
+_
T+
y
T'1 
Xcii , sn = 1(n;)"2 { T:
T(O , y, t )  T.
qo(atl,,)"' 11t
y +a
(n_I)I+)1/2{
~ 'l (
(~i2
[(I~;S:~1~2]_ T:L~::~1~2]}
7t.
T:
[
2 s1
]
[2S+I]}
((11_1)1:)1/ 2  T: ( (nl)t:)112
(1.6.44)
0.3
00
0.1
0.2
0.3
0.4 0.5
0.6
0.7
0.8
0.9
0.9
1.0
IY+112
•
n
JJ
...
...:~
FIGURE 1 .16 Dimensionless surface temperature for a semiinfinite body heated uniformly
over onehalf the surface.
I\
0.6
I
shown in Figure 1.15 which is exposed to a number o f equally spaced heat flux
c omponents for time t j is considered. Let the heated regions be equal to 6 y= 2a.
A p rototype sensitivity coefficient is the one for q OI which is for heating over
the region  a< y <a a nd for a heating duration of 0 < t<6t. T he dimensionless
sensitivity coefficient for the location. x =O. y =2sa. (s=O. I. ...) a nd time
t =n6t(n=I.2• ... ) is needed. The notation is
0.5
: ,.
n
 k aT(O. y. . 6 t) . ( t:)1/2 { + [ 2SI]
+ [ 2S+ I ] }
Ol,$l=~
aqol
I,
Ty (t:)1/2  Ty (£:)1 1
2
(1.6.42)
Semi ·infinite
•
+0
>.;
.I
02
t,;
+ +
= 0.05
Use right axis
I
I
I
0 .1
0 .1
/
/
.  e
\
' ..
t: = 1
Use left axis
· ;. ~'.·IX~1'2.
1
................. ~.""'""
::::::.
_ee
4
ee __ :
5
Time index,
(1.6.43)
J2
body
3
where
0 .2 ::~
+0
0.3
X+
II
I
I
J2
>.;
where the first two subscripts refer to the location of the application of the heat
flux c omponent (i.e .• j = 0) a nd to the time o f the applied heat flux component
(i = I). and the last two subscripts are for the location and time o f evaluation.
The first two subscripts are for the cause (i.e .• the impulse) and the last two are
for the effect (i.e .• response). The sensitivity coefficient a t time t = 6 t. (n = I). a t
location y .=2sa is o btained by superimposing two solutions taken from Eq.
(1.6.39).
~
o
6
7
8
9
10
n
FIGURE 1 .17 Sensitivity coellicients for heat Hux components along surface of a semiinfinite
body.
36
C HAP. 1
D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM
T he general sensitivity coefficient for
q OI by
+
X ji • s
.= {O,
q ji
a t Ys a nd In is related t o the one for
n <i
X + I.lsjl . n O
i+l,
(1.6.45)
>.
n,....1
Hence if the Xcii
sensitivity coefficients are known for s = 0, 1,2, . .. and
1,
a t the ';~rface of a semiinfinite body, the sensitivity coefficients
a t the surface can be found for all other
F igure 1.17 gives plots o f XciI. O n' Xcii. I n, a nd Xci1.2n for = I . These curves
are those with dots and dashed lines. Use the left axis for these curves. T he
Xcil ,on curve is the largest since it is for the response directly below the heat
pulse; it is a maximum at n = 1, the end o f the heat pulse, a nd decays thereafter.
The Xcil.2n curve is much smaller in magnitude than the Xcil.o. curve and its
maximum is displaced in time. Also shown in Figure 1.17 are two curves for
= 0 .05, which is a relatively small value. These are the continuous curves
that have crosses for which the right axis is used. In this case the Xcii . o. curve
is similar to that for = 1 b ut the Xcil . l. curve is very small in magnitude and
there is a very large lag in response. This means that the heat impulse for the
small time steps o f = 0.05 affects mainly the temperature response at the
location o f the pulse. There is little correlation (or crosscoupling) of the individual heat pulses.
T he uncorrelated nature o f the spatial variation o f the sensitivity coefficients
for small time steps and with surface temperature measurements has experimental design implications. If, for example, the heat transfer coefficients (or
equivalently heat fluxes if the ambient temperatures are known) are needed
a round a j et engine turbine blade, a transient experiment with surface thermocouples could be used. T he calculational time steps should be small t.o reduce
spatial correlation between the components o f the heat transfer coeffiCient.
1 1=
i, ...,
X;;.•• .
t:
I:
t:
t:
1 .7
C LASSIFICATION OF M ETHODS
T he methods for solving the inverse heat conduction problem can be classified
in several ways, some o f which are discussed in this section.
.
O ne classification relates to the ability o f a m ethod to treat nonhnear as
well as linear IHCP's. This book emphasizes basic algorithms t hat c an be
employed for both linear a nd n onlinear problems. The. tW? basic procedures
given herein are the function specification and r:gulanzatlOn .methods. Both
o f these can be used for nonlinear problems prOVided the nonhnear heat conduction equation is solved. Some methods o f solution of the I HCP are inherently
linear such as t hose based o n the Laplace transform; such methods are not
considered because the nonlinear case is m ore important for industrial applications.
S EC.1 .7
C LASSIFICATION OF M ETHODS
37
T he m ethod o f solution o f the heat conduction equation is a nother way to
classify the I HCP . Methods o f solution include the use of Duhamel's theorem,
finite differences, finite elements and finite control volumes. The use o f
D uhamel's theorem restricts the I HCP algorithm to the linear case, whereas the
other procedures can treat the nonlinear problem. Duhamel's theorem is used
frequently in this book because the basic I HCP algorithms are easier to use
and program for simple calculations than the finite difference and other methods.
Moreover for linear problems, the answers for the surface heat flux are nearly
identical for all of the methods mentioned provided the heat conduction
equation is solved accurately. Consequently, experience acquired using
Duhamel's theorem incorporated in a basic I HCP a lgorithm is also relevant
to the other methods when used for linear I HCP 's.
The time domain utilized in the I HCP can also be used to classify the method
of solution. Three time domains have been proposed: (1) only to the present
time, (2) t o the present time plus a few time stops, a nd (3) the complete time
domain. The use o f measurements only t o the present time with a single temperature sensor allows the calculated temperatures to match the corresponding
measured temperatures in an exact manner; that is, the calculated temperatures
equal the measured values. This is called the Stolz method. I Such exact matching is intuitively appealing but the algorithms based on it frequently are extremely sensitive to measurcmenl errors. In the second method, a few future
temperatures (associated with future times) are used ; the associated algorithms
are called "sequential." Greatly improved algorithms are obtained compared
with exact matching. T he improvements are noted in the considerably reduced
sensitivity to measurement errors and in the much smaller time steps that are
possible. Small time steps permit more detailed information regarding the time
variation o f the surface heat flux t o be found. T he whole domain estimation
procedure is also very powerful because very small time steps can be taken but
it is n ot as computationally efficient as the use o f only a few future temperatures.
Both the function specification and regularization methods can be employed
in sequential a nd whole domain estimation forms.
The last classification to be mentioned is relative to the dimensionality of
the IHCP. If a single heat flux history is t o be determined, the I HCP c an be
considered as onedimensional. In the use o f D uhamel's theorem, the physical
dimensions o f the problem are not o f concern ; that is , the same procedure is
used for physically one, two, o r threedimension bodies provided a single
heat flux history is to be estimated. I f two o r more heat flux histories are estimated and Duhamel's theorem is used, the problem is multidimensional. When
the finite difference o r the other methods for nonlinear problems are employed,
the dimensionality o f the problem depends on the number o f space coordinates
needed to describe the heatconducting body; one coordinate would give a
onedimensional problem, twocoordinates a twodimensional problem, and
so on.
38
CHAP. 1
1 .8
DESCRIPTION OF THE INVERSE HEAT CONDUCTION PROBLEM
CRITERIA FOR E VALUATION OF IHCP M ETHODS
In o rder t o evaluate the several I HCP procedures, various criteria are needed.
The criteria proposed in Reference 10 a re given in this section and are as follows:
SEC.1.9
5 . T he m ethod should n ot require continuous firsttime derivatives o f the
surface ~eat flux. Furthermore, step changes o r even m ore a brupt
changes In the surface heat fluxes should be permitted.
6 . Knowledge o f t he precise starting time o f the application o f the surface
heat flux should n ot be required. The start o f h eating is frequently n ot
synchronized with the discrete times t hat t emperatures are measured.
R~so~s for this might be that the starting time is n ot accurately known
o r IS dIfficult t o measure. Precise times a t which a brupt changes in the
heat flux OCCur may also be unknown.
7 . T he method should n ot be restricted to any fixed number o f observations.
8 . C omposite solids should be permitted.
9.
1 0.
11 .
1 2.
T emperaturevariable properties should be permitted.
C ontact conductances should n ot be excluded.
T he m ethod should be easy t o p rogram.
T he c omputer cost should be moderate.
1 3. T he user should n ot have to be highly skilled in mathematics in o rder
t o use the method o r t o a dapt it to o ther geometries.
1 4. T he method should be capable o f t reating various onedimensional
coordinate systems.
1 5. T he m ethod should permit extension to more than one heating surface.
1 6. T he method should have a statistical basis a nd p ermit various statistical
assumptions for the measurement errors.
. T he functi~n s~cific~tion a nd regularization methods are capable o f satisfyIng all th~se c ntena prOVIded t he nonlinear heat conduction equation is a pproximated USIng m ethods such as the finite difference, finite element, o r finite control
volume methods.
If t he heat conduction equation is solved by Duhamel's theorem a nd the
39
function specification o r regularization method is used, all the criteria can be
satisfied except t hat o f t reating the nonlinear problem (criterion number 9).
1.9
1 . T he predicted temperatures and heat fluxes should be accurate if the
measured d ata a re o f high accuracy.
2 . T he m ethod should be insensitive t o measurement errors.
3 . T he method should be stable for small time steps o r intervals. This
permits the extraction o f m ore information regarding the time variation
o f surface conditions than is p ermitted by large time steps.
4 . T emperature m easurements from one o r m ore sensors should be
permitted.
SCOPE OF BOOK
SCOPE OF BOOK
The scope o f t he book is mainly limited t o the inverse heat conduction problem.
I t is o ne o f a class o f illposed problems. However, many o f t he techniques given
herein apply t o a wide variety o f illposed problems. A number o f s olution
methods are presented a nd t he emphasis is o n general methods t hat c an meet
the criteria in Section 1.8.
C hapter 1 h as given a n i ntroduction t o the subject.
Chapter 2 presents some exact solutions, one o f which is due to Burggraf. 43
T hough this exact solution is restricted in its application, the insight gained
from it is very important.
C hapter 3 presents two different basic procedures for solving direct transient
heat conduction problems. T he first is based o n a numerical form o f D uhamel's
integral. T he second method approximates the partial differential equation
for transient heat conduction by a set o f algebraic equations, t hat is, difference
equations. T he s econd method is more powerful in t hat it can treat the nonlinear problems.
C hapter 4 o pens with a discussion o f illposed problems a nd t hen presents a
number o f m ethods for the I HCP. M any researchers have contributed t o these
methods. T he t wo basic classes are the function specification a nd regularization
methods. A procedure called the trial function method unifies these approaches.
Two ways t hat the function specification a nd regularization methods can be
used are called sequential a nd whole domain. Various modifications o f t hem a re
discussed. Each o f these methods can utilize the Duhamel a nd difference
equation procedures a nd t hus is applicable for both linear a nd n onlinear
problems. An important p art o f this chapter is a discussion o f a digital filter
form o fthe I HCP. Such a form can be used for the very efficient implementation
o f the linear I HCP algorithms. T he final section o f C hapter 4 discusses some
criteria for comparing estimation procedures.
C hapter 5 p resents a n umber o f test cases. Utilizing a numerical approximation o f D uhamel's integral for the solution o f t he heat conduction equation,
these test cases are used t o investigate a number o f t he I HCP algorithms. T he
s tudy o f I HCP a lgorithms is facilitated by using a numerical form for Duhamel's
integral because only a single equation is needed a t each time rather t han a set
o f difference equations. C hapter 5 e nds with a discussion o f o ptimal choices o f
p arameters in the function specification a nd regularization methods.
C hapter 6 uses the finite control volume method t o a pproximate the heat
conduction equation for the onedimensional inverse heat conduction problem.
T he sequential function specification a nd regularization methods developed in
C hapter 4 a re used. In addition, some space marching techniques are discussed;
these methods are unique t o t he difference equation approach since analogous
40
CHAP.1
DESCRIPTION OF THE INVERSE HEAT C ONDUCTION PROBLEM
REFERENCES
e~uatio~s b ased o n D uhamel's integral a re n ot available. T he c hapter concludes
With a list o f c omputer p rograms available for the I HCP.
C hapter 7 is for multiple heat flux estimation. Two o r m ore h eat flux histories
c an be estimated a t t he same time. This case is commonly encountered in twoor t hreedimensional inverse heat conduction problems.
C hapter 8, t he last chapter, discusses methods for estimating the heat transfer
coefficient. O ne way is t o use the I HCP m ethods to calculate the surface heat
fl~x history. An alternate procedure calculates the heat transfer coefficient
directly.
16. Imber, M., A Temperature Extrapolation Method for Hollow Cylinders, , 1/,1,1 J. I I, 117  118
(1973).
17. Imber, M., Comments on " On Transient Cylindrical Surface Heat Flux Predicted from
Interior Temperature Responses," , 1/,1,1 J. 14,542  543 (1975).
18. Imber, M., Inverse Problem for a Solid Cylinder, , 1/,1,1 J. 17,91 94 (1979).
19. Imber, M., Nonlinear Heat Transfer in P lanar Solids: Direct and Inverse Applications,
A lAA J. 17,204 212 (1979).
20. Imber, M., A Temperature Extrapolation Mechanism for TwoDimensional Heat Flow,
A lA A J. 12, 1087  1093 (1974).
21. Imber, M., The Two Dimensional Inverse Problem in Heat Conduction, Fifth International
Heat Transfer Conference, Tokyo, Japan (1974).
22. Imber, M., TwoDimensional Inverse Conduction Problem Further Observations," A IAA J.
13,114 115 (1975).
23. Imber, M. and Khan, J., Prediction of Transient Temperature Distributions with Embedded
Thermocouples, A IAA J. 10,784 789 (1972).
24. Mulholland, G. P., Gupta, B. P., and San Martin, R. L., Inverse Problem o f Heat Conduction
in Composite Media, ASME Paper, 75WA/HT83 (1975).
25. Mulholland, G. P. and San Martin, R. L., Inverse Problem of Heat Conduction in Composite
Media, Third Canadian Congress of Applied Mechanics, Calgary, Alberta, Canada (May 1971).
26. Mulholland, G. P. and Cobble, M. H., Diffusion Through Composite Media, Int. J. Heal
Mass Transfer 15,147  152 (1972).
27. Mulholland, G. P. and San Martin, R. L., Indirect Thermal Sensing In Composite Media,
Int. J. Heal Mass Transfer 16, 1056  1060 (1973) .
28. Hills, R. G. and Mulholland, G. P. ; Accuracy and Resolving Power of OneDimensional
Transient Inverse Heat Conduction Theory as Applied to Discrete and Inaccurate Measurements, Int. J. Heal Mass Transfer 22,1221  1229 (1979).
29. Randall, J. D., Embedding Multidimensional Ablation Problems in Inverse Heat Conduction
Problems Using Finite Differences, 6th Int. Heat Transfer Conf., Toronto, Ont., Aug. 7  11,
1978. Publ. by Nail. Res. Council of Can., Toronto, Ont., 1978. Available from Hemisphere
Publ. Corp, Washington, D.C., Vol. 3, 129  134.
30. Randall, J. D., Finite Difference Solution of:the Inverse Heat Conduction Problem and
Ablation, Technical Report, Johns Hopkins University, Laurel, Maryland (1976), Proceedings
o f the 25th Heat Transfer and Fluid Mechanics Institute, Univ. o f California, Davis (1976).
31. Williams, S. D. and Curry, D. M., An Analytical Experimental Study for Surface Heat Flux
Determination, J. Spacecrafl 14,632 637 (1977).
32. France, D. M., Chiang, T., Carlson, R. D., and Minkowycz, W. J., Measurements and Correlation of Critical Heat Flux in a Sodium Heated Steam G enerator Tube, Technical Memorandum, ANLCT7815, Argonne National Laboratory, Argonne, IL (1978).
33. France, D. M., Carlson, R. D., Chiang, T., and Priemer, R., CHFInduced Thermal Oscillations
Measured in a n LMFBR Steam Generated Tube Wall, Technical Report, ANLCT78I,
Argonne National Laboratory Argonne, IL (1977).
34. France, D. M. and Chiang, T., Analytical Solution to Inverse Heat Conduction Problems with
Periodicity, J. Heat Transfer 102, 579  581 (1980).
35. Bass, B. R., Applications of the Finite Element to the Inverse Heat Conduction Problem
Using Beck's Second Method, J. Eng. Ind. 102, 168  176 (1980).
36. Bass, B. R .,lncap: A Finite Element Program for OneDimensional Nonlinear Inverse Heat
Conduction Analysis, Technical Report NRCjNUREG/CSD/TM8, O ak Ridge National
Laboratory (1979).
37. Muzzy, R. J., Avila, J. H. and Root, R. E., Topical Report: Determination of Transient Heat
Transfer Coefficients and the Resultant Surface Heat Flux from Internal Temperature
Measurements, General Electric, GEAP20731 (1975).
REFERENCES
I. Stolz, G., Jr., Numerical Solutions to an Inverse Problem of Heat Conduction for Simple
Shapes, J. Heat Transfer 8 2,2026 (1960).
2.
Mirsepassi, T. J., HeatTransfer Charts for TimeVariable Boundary Conditions, Br. Chem.
Eng. 4, 130  136 (1959).
3.
Mirsep~si, T. J., Graphical Evaluation of a Convolution Integral, Mathematical Tables and
Olher Aides to Computallon 13, 202  212 (1959).
4. Shumakov, N. V., A Method for the Experimental Study o f the Process o f Heating a Solid
Body, Soviet Physics Technical PhYSics (Translated by American Institute o f Physics) 2 771
. (1957).
'
5.
~
' \:)
Beck, J. V., Correction of Transient Thermocouple Temperature Measurements in HeatConducti?g Solids, Part II, The Calculation of Transient Heat Fluxes Using The Inverse
Convolution, AVCO Corp., Res. a nd Adv. Dev. Div., Wilmington, MA., Tech. Report RADTR76O38 ( Part II), March 30,1961.
6.
!J1
Beck, J. V., Calculation of Surface Heat Flux From an Internal Temperature History ASME
Paper 62HT46 (1962).
'
7. Beck, J. V., "Surface Heat Flux Determination Using an Integral M ethod" Nuc/. Eng Des 7
170  178 (1968).
'
.. ,
8. Beck, J. V. a nd Wolf, H., The Nonlinear Inverse Heat Conduction Problem ASME Paper
65HT40 (1965).
'
,
9. Beck, J. V., Nonlinear Estimation Applied to the Nonlinear Heat Conduction Problem
Int. J. H eat Mass Transfer 13, 703 716 (1970).
'
10. Beck, J. V., Criteria for Comparison o f Methods of Solution o f the Inverse Heat Conduction
Problem, Nuc/. Eng. Des. 5 3,1122 (1979).
I I. Beck, J. V., Litkouhi, B., a nd St. Clair, C. R., Jr., Efficient Sequential Solution o f the Nonlinear
Inverse Heat Conduction Problem, Numer. Heat Transfer 5 ,275 286 (1982).
12. ~Iackwell, B. F., A New iterative Technique for Solving the Implicit FiniteDifference Equalions for t.he Inverse Problem of Heat Conduction, unpublished technical report, Sandia
Laboratones, Albuquerque, NM (1968).
13. Blackwell, B. F., An Efficient Technique for the Numerical Solution o f the OneDimensional
Inverse Problem of Heat Cond\lction, Numer. Heat Transfer 4 ,229239 (1981).
14. Langford, D., New Analytic Solutions of the OneDimensional Heat Equation for Temperature
and Heat Flow Rate Both Prescribed at the Same Fixed Boundary (with Applications to the
Phase Change Problem), Q. Appl. Math. 2 4,315322 (1967).
15. Woo, K. C. a nd Chow, L. C .,lnverse Heat Conduction by Direct Inverse Laplace Transform
Numer. Heal Transfer 4, 499504 (1981).
'
2
2
42
CHAP.1
DESCRIPTION OF THE INVERSE HEAT CONDUCTION PROBLEM
38. Snider, D. M., INVERT 1.0 A Program for Solving the Nonlinear Inverse Heat Conduction
Problem for OneDimensional Solids. E G&G Idaho, Inc., Idaho Falls, Idaho, EGG2068
(1981).
39.
40 .
41 .
42
43 .
44.
45 .
46.
47 .
48 .
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
Alkidas, A. L., H eat Transfer Characteristics o f a SparkIgnition Engine," J . Heat Transfer
102, 189 193 (1980).
Howse, T. K. J., Kent, R., a nd Rawson, H., The Determination o f G lassMould Heat Fluxes
from Mould Temperature Measurements, Glass Technol. 12,91  93 (1971).
Sparrow, E. M~ HajiSheikh, A., a nd Lundgren, T. S., T he Inverse Problem in Transient Heat
Conduction, J . Appl. Mech., Trans. A SME, Series E, 86, 369  375 (1964).
Lin, D. Y. T~ a nd Westwater, J. W., Effect o f Metal Thermal Properties o n Boiling Curves
O btained by the Quenching Method, Heat Transfer 1982 Munchen Conference Proceedings,
Hemisphere Publ. Corp., New York, 1982, p p. 155  160, Vol. 4.
Burggraf, O. R., An Exact Solution o f the Inverse Problem in Heat Conduction Theory and
Applications, J . Heat Transfer 116C, 373  382 (1964).
Grysa, K., Cialkowski, M. J. a nd Kaminski, H ~ An Inverse Temperature Field Problem o f
the Theory o f T hermal Stresses, Nucl. Eng. Des. 64, 169 184 (1981).
Tikhonov, A. N. a nd Arsenin, V. Y., Solutions o f IIIPosed Problems, V. H . Winston & Sons,
Washington, D.C., 1977.
Backus, G. a nd Gilbert, F., Uniqueness in the Inversion o f I naccurate Gross Earth D ata
Phil. Trans. R . Soc. London Ser. A 266, 123  192 (1970).
'
Nolet, G., Simultaneous Inversion o f Seismic Data, Geophys. J . R. Astr. Soc. 55, 679  691
(1978).
Mandrel, J., Use o f t he Singular Value Decomposition in Regression Analysis, Am. Stat . 36,
15  24 (1982).
Beck, J . V. a nd Arnold, K. J ., Parameter Estimation in Engineering and Science, Wiley, New
York,1977.
Litkouhi, B. a nd Beck, J. V., T emperatures in SemiInfinite Body Heated by Constant Heat
Flux O ver H alf Space, Heat Transfer 1982 Munchen Conference Proceedings, Hemisphere
Publ. Corp., New York, 1982, pp. 21  27, Vol. 2.
Carslaw, H. S. a nd Jaeger, J. c., Conduction o f Heat in Solids, 2nd ed., Oxford Univ. Press,
L ondon, 1959.
Abramowitz, M. a nd Stegun, I. A., Handbook o f Mathematical Functions with Formulas,
Graphs and Mathematical Tables, N ational Bureau o f S tandards, Applied Mathematics Series,
Vol. 55, 1964.
Payne, L. E., Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia,
PA., 1975 .
H adamard; J., Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale
University Press, New Haven, CT, 1923.
Cannon, J. R. a nd Douglas, J., The Cauchy Problem for the Heat Equation, S IAM J . Numer.
Anal. 3, 317 336 (1967).
John, F., Numerical Solution o f t he Heat Equation for Preceeding Times, Ann. Mat . Para.
Appl.40, 129 142 (1955).
Lawson, C. L. a nd Hanson, R. J ., Solving Least Squares Problems, PrenticeHall, Englewood
Cliffs, NJ, 1974.
Murio, D. A., T he Mollification Method a nd the Numerical Solution o f a n Inverse Heat
Conduction Problem, S IAM J . Sci. Stat . Comput. 2, 17  34(1981).
r
,
43
PROBLEMS
PROBLEMS
1.1.
Give a mathematical description of the I Hep for a solid, homogeneous
sphere when the temperature is measured at the center point. The sphere
radius is denoted a. Make a sketch of a sphere showing the various
quantities. Form a graph that shows typical discrete temperatures
measured at time steps of fl.1 with an initial temperature of To = constant.
Also show an associated surface heat flux history.
1 2. A certain automobile brake is composed of a cast iron drum and two
brake shoes. The rotating drum has an inner radius of 10 cm and an
outer radius 10.7 cm. The heat transfer in the drum is assumed to be
only in the radial direction and there is a convective boundary condition
at the outer drum radius. The brake shoes are 0.5 cm thick, are inside
the drum, and are stationary. They cover only 75% of the angular area
of the drum. The inner surface of the brake shoes (the surface not in
contact with the drum) can also be considered to have a convective
boundary condition. Describe the inverse heat conduction problem(s)
for determining the surface heat flux based on the drum surface area.
Describe the problem mathematically and through the use of any needed
sketches.
1.3. The temperature distribution in a semiinfinite body (x ~O) is given by
T
To=~T,~ To) erfc [(4<%:)1 /2]
where T, is the surface temperature and To is the initial temperature.
a. Derive an expression for the heat flux for any x. Also give the expression for x=O.
Answer: k (T, To)(1tCXl)1/2 for x = 0.
b. Plot the heat flux versus CXI/E2 for x = E and also show on the same
plot the x = 0 curve.
1 .4.
F or a semiinfinite body with a surface temperature given by
T(O,t)=O,
1 <0
T(0,1)=Ct,/2,
n = 1, 2, 3, . .. ,
I~O
the temperature distribution is given by (Reference 51, p. 63)
T = cr
(n;
2 ) (4/)"/2i" erfc [ (:r)1/2]
where (Reference 51, p. 483)
~
i" erfc(z)= 1"0 i ,I erfc(u)du,
L
n = 1,2, . .. , jO erfc(z)=erfc(z)
44
CHAP.1
D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM
r
Time(s)
Derive the expression for the surface heat flux,
3
4
5
6
7
Find a n expression for T(O , t)/q(O, t) a nd c omment o n t he result. Does
this provide a relation t hat c an be used for the I HCP? G ive reasons.
1824
1920
2016
2112
2208
Time(s)
lj(O
F)
136.04
132.94
130.Q7
127.39
124.88
F ind t he average o f t he temperatures given in Problem 1.6. Calculate
a nd p lot the residuals defined by
for the ext/x values o f om, 0.1, 1, to, a nd 100. C ompare with the exact
result o f P roblem 1.3a a nd give some conclusions.
1 .6. A solid copper billet 1.82 in. long a nd 1.00 in. in diameter is heated in a
furnace a nd t hen removed. T he densityspecific heat p roduct o f t he copper
is 51 Btu/ft3F. Some o f t he temperature measurements a re given below.
T he h eat transfer model is
dT
e j= l j Y;
p cV Y t =qA
where Y; is t he average temperature. W hat is t he main reason for the
lack o f r andomness in the residuals?
where V is t he volume a nd A is the heated surface area (the cylindrical
sides a nd t he flat ends).
a. Using the forward difference approximation,
b.
c.
i =l
with respect t o t he parameters. F or t he model
where

1
2
lj(O
F)
1632
1728
142.93
139.34
Time(s)
8
9
1•
t = L tj
n j=1
l j+l lj  I
2 ~t
derive the estimators for /11 a nd /12 which a re respectively denoted
a nd P2'
Which approximation (forward, backward, o r central) gives the best
results?
Time(s)
W
'
I,
T he m ethod o f o rdinary least squares involves minimizing the s um o f
s quares function S,
~t
c alculate the heat flux for M = 9 6 s using all the given data. P lot
t he results versus time.
P lot results o n t he same figure for the backward difference approximation.
R epeat p art (a) using the central difference approximation,
dt
d.
1 .9.
l j+llj
I,
dTI
118.04
115 .97
114.13
112.35
1.8.
1'. T(x, t)
x
de
2496
2592
2688
2784
to
11
12
13
Use t he forward difference method t o a pproximate the first derivative o f
e  I a t t = 2. Use ~t = 0.001, om, 0.1, a nd 1. Use t he exact values o f e  I
first a nd t hen repeat the solution by adding the r andom e rrors o f 0.000464,
0.000137,0.002455,  0.000323 a nd  0.000068 t o e  I for t =2, 2.001,
2.01,2.1, a nd 3, respectively. F or example, Y = e 2 + 0 .000464=0.135799.
I
(These r andom values have a s tandard d eviation o f 0 .001 a nd a re t aken
from the first row o f T able 5.2.) W hat c onclusions c an be d rawn from the
approximations as ~t+O'fol' t he exact values a nd from those with the
r andom e rrors?
2
dTI
lj(O
F)
1.7.
1 .5. Use t he relation in Problem 1.3 for T a t x n ot equal to zero t o e valuate
q(O, t) ~k
45
P ROBLEMS
2304
2400
AI •
lj(O
F)
122.46
120.18
f 31=Y=  L l j
n j=1
P2
L
L (ejt)lj
L (t j t)2
PI
46
CHAP.1
DESCRIPTION OF THE INVERSE HEAT CONDUCTION PROBLEM
1 .10. Using the linear model in Problem 1.9, estimate PI and
P2 for the data
o f P roblem 1.6. Also calculate and plot the residuals. Comment on the
time variation o f the residuals.
r
47
PROBLEMS
1.11 for the d ata o f Problem 1.6. Calculate the average estimated standard
deviation a nd sample correlation coefficient. Discuss your results.
1.14. Derive estimators for PI a nd P2 in the model
1 .11. F or measurements equally spaced in time, orthogonal polynomials can
be used in least squares estimation. (See Reference 49, p. 248.) A polynomial model is
T j=Po+P l t j+P2t f+ . .. +Pr1i
T j=PIXjl +/12 X j2
using the ordinary least square method which requires that
s = L (1';~ Tj)2
(a)
where t j+ I = tj+i\t. This model can be rewritten as
Tj=cxoPo(tj)+CXIPI(tj)+CX2P2(tj)+ . .. +cxrPr(t;)
(b)
be minimized with respect to PI a nd P2'
Answer:
where Po(t;), PI(t j), a nd P2(t j) a re
Po(t j) = 1
t ·t
n+ 1
PI(t j)= 'i\t = i 2'
(
ft
ft
tj_~2
P2(t·)= ,
i \t
_1
t=
c.,= L Xj.X
ft
L
n j =1
i =1
tj
n 2_1
12
d .=
il ,
(c)
L
; =1
1';X jl
1.15. With ordinary least squares, show that the estimator, p, o f Pin the model
T =/1is
Verify that these Pj(t j ) functions for r =2 a re orthogonal; that is, show that
I•
P= L 1';
n
ft
L
Pj(tj)P.(t;)=O
for j fk
fO
for j =k
j =1
j =1
a nd for the standard statistical assumptions being valid, show that the
variance o f Pis
a ndj, k=O, 1 ,2.
Derive
(d)
for m=O, I , . .. , r.
1.12. F or the polynomial and orthogonal polynomial models of Problem 1.11,
prove for r = I t hat
1 .16. Show that for the random variables 1'; a nd 1),
a. V (1)=E(YJ>E 2(1)
b. cov( Y;, 1) = E( 1'; 1)  E( 1';)E( 1)
1.17. W hat is the expected value for the random variable I l t hat has the probability density o f
I
1.( y) = { 0
and for r = 2 t hat
1 [(1)2  1_1]
ni \t
2
a~
PO=CXOCXI  + CX 2
i\t
A
P (A I A 2t\
2
2
A
I
1 = CX CX i \c) i \t'
A
a2
/12 = (i\t)2
1.13. Using the r =2 o rthogonal polynomial model given in Problem 1.11,
estimate the parameters using ordinary least squares in Eq. (b) o f Problem
f orO<y<1
otherwise
What is t he variance o f e?
1 .18. W hat are the mean and the variance of a random variable I l t hat has the
probability density
J .(y)=aexp[  (yWJ,
 oo<y<oo
W hat is the value o f a ?
1 .19. Using the first row o f the random numbers o f Table 5.2, generate a set
r
48
CHAP.1
D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM
r
of 10 correlated random numbers using
I:j+I=0.5I:j+Uj+l,
49
P ROBLEMS
problems 1 and 2:
i =I,2, . .. , 9
ar;1
 k  '= qj(t)
a x x=o
where the Uj are the entries in Table 5.2, with 1:1 =U I the first entry.
Calculate the sample average and the associated residuals. Next calculate
the sample correlation coefficient. Repeat for the next two rows of Table
5.2. Give some conclusions.
a1x'; Ix =L0
a
1';(x, 0 )=0
1 .20. a. Plot the ratio of the temperature rise at x = 0 o f a finite plate with a
constant heat flux a t x = 0 and insulated at x = L t o the temperature
rise at x = 0 of a semiinfinite body heated at x = 0 by the same value
of constant heat flux. Evaluate the ratio (at least) for I1.t/ L2 = 0.05,
0.1,0.15,0.5,1, and 5. P lot on a semilogarithmic scale.
b. Repeat part (a) for the temperature rise at x = L in both bodies.
c. Repeat part (a) for the temperature rise a t x = L /2 in both bodies.
d. Compare the results for the different cases particularly for small
I1.t/L2 values.
e. F or the same additive measurement errors in both geometries,
over what regions of x a nd t would an I HCP procedure give the
same results? Different results?
for i = 1 for problem 1 and i = 2 for problem 2.
1 .22. Use the summation relation in Problem 1.21 t o verify Eq. (1.6.25). What
are q l(t) and q2(t) for Eq. (1.6.25)?
1 .23. Make a threedimensional plot of q (y, t) versus y (horizontal axis) and
t (drawn as an axis 30° t o the yaxis) for
q ll = 3, q12 = 5, ql3 = 7
q21 = 2,
11.
q 32=2,
q 33=3
1 .24. Derive Eq. (1.6.44).
1 .25. Derive Eq. (1.6.45).
11. =
constant
 k aaTI
= q l(t)+q2(t),
x x=o
1 .26. The solution for the temperature in an infinite body is
a x
T ITo [
J
a +x]
T =To +  2
ert.: 2(l1.t)I /2+ erh 2(l1.t)I/2
k =constant
where  00 < x < 0 0 a nd the initial temperature distribution is (Reference
51, pp. 54, 55)
a TI
0
a x x =L 
T =To
show that the solution T (x, t) is equal to the solution ofthe three problems,
where the three problems are described by
T =TI
2
a x2'=Tr'
Ixl<a
Using the principle of superposition give the temperature distribution
in a A . infinite body for the initial temperature distribution of
T (x, t )= To(x, t )+ TI(x, t )+ T2(x, t)
a r; ar;
Ixl>a
T =TI
T(x,O)=F(x)
11.
q 23=4
q31=1,
1 .21. F or the problem
a 2T a T
a x 2 = a t'
q 22=3,
Ixl<a
T =T2 a <x<3a
i=O, 1 ,2
T = To
problem 0:
otherwise
Plot the sensitivity coefficients,
aTo
 =0
ax
a t x =O a nd
X = aT(O, t)
loTI
L
and
X
2=
oTtO, t)
ar;
versus t for I1.t/a 2 =0 to 5. (See Reference 51, p. 55.) The backward heat
T o(x,O)=F(x)
I
f",
50
C HAP.1
D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION PROBLEM
c.onduction problem is the estimation o f the initial temperature distributIOn from temperatures measured at later times. Comment on the relative
difficulty o f recovering I i and T2 from measurements o f T a t x = 0 and
2a for the two cases of measurements of (1.t/a 2 equal to (a) 0.2, 0.4, 0.6,
0.8, and I ; and (b) 2, 4, 6, 8, a nd 10.
r
I
I
,
I
I
i
1 .27. a.
F or Eqs. (1.6.4), (1.6.5), and (1.6.7) a nd the convective boundary
condition at x = L o f
k
Ox L = h[T(L, t) oT/
T",,(t)]
wher~ .h is constant, derive the differential equation and boundary
b. condltl?ns for the sensitivity coefficient defined by Eq. (1.6.8).
~elate m words the results of part (a) to the problem of a unit step
IDcrease o f surface heat flux.
c. F or the case of x = L , t M _ I = 0, t < t M , k (x) = constant, c (x) = constant
and hL/k = I, find numerical values for k X M/L for t + = 0.25, 0.5, a nd
1. C ompare the values with those for x /L in Table 1.1.
CHAPTER
2
EXACT S OLUTIONS OF T HE
I NVERSE H EAT C ONDUCTION
P ROBLEM
2.1
I NTRODUCTION
Exact solutions of the inverse heat conduction problem are very important
because ( I) they provide closed form expressions for the heat flux in terms of
temperature measurements, (2) they give considerable insight into the characteristics of inverse problems, and (3) they provide standards of comparison for
approximate methods. Inverse heat conduction problems can be divided into
steadystate and transient problems. The steadystate inverse problem is
simpler in that the only necessary thermal property is the thermal conductivity
k, and a temperature history is not required. The transient inverse heat conduction problem can be divided into two categories: lumped thermal capacitance and distributed thermal capacitance. The transient case requires many
discrete temperature measurements. F or lumped thermal capacitance, the
important thermal property is the volumetric heat capacity p c. If the thermal
capacitance is distributed, then the thermal conductivity k must be known in
addition to the volumetric heat capacity. Throughout Chapter 2, the thermal
properties are assumed to be independent of temperature ; this assumption is
one of the weaknesses of exact solutions.
Section 2.2 considers onedimensional steadystate problems in which the
temperature is known at two or more locations. Section 2.3 examines the
lumped thermal capacitance case and some numerical approximations to the
exact solution. Section 2.4 considers a planar semiinfinite body for which the
surface temperature history is known; an approximate technique for numerically
evaluating the resulting integral and an example problem are presented. Section
2.5 presents the development of an exact solution for a onedimensional planar
body with a temperature sensor at an arbitrary depth E below the heated
51
