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

52
C HAP.2
S OLUTIONS OF THE INVERSE H EAT C ONDUCTION PROBLEM
S EC.2.2
surface; this solution requires the existence o f all o rder derivatives o f b oth the
experimental temperature d ata Yet) a nd t he heat flux q f: passing through depth
E. Results for solid spheres a nd cylinders a re a lso presented.
STEADYSTATE SOLUTION
by t he weighting factors
Wi '
I
A weighted least squares criterion is defined as
J
L wf()j _ 1)2
S=
I
,
II
(2.2.3)
j= t
2.2
F ourier's law indicates the temperature profile must be linear,
STEADYSTATE SOLUTION
X
(2.2.4)
T (x)=ax+b=  q"k+ To
O ne o f t he simpler inverse solutions is for steadystate heat flow t hrough a
p lanar o nedimensional body with constant thermal properties. F or this
situation, Fourier's Law,
dT
q =kdx
There are two parameters (q, To) in Eq. (2.2.4) t hat a re d etermined such t hat t he
weighted least squares e rror is a minimum. (To is the temperature a t x~O.)
Differentiating Eq. (2.2.3) with respect t o t he two unknown parameters gives
(2.2.1)
as
a
 =  2 ~ Wj(lj1)  1) = 0
L 2
aTo
A
q=
k ( YI

Y2 )
X 2 X I
i=1
a
 s =2
is a differential equation and can be integrated directly. T he e ntering heat flux,
q, is the same as a t a ny location x. S uppose t he steadystate temperature is
k nown a t two depths ( XI' x 2 ) below the surface, as shown in Figure 2.1. Integrating Eq. (2.2.1) between locations X I a nd X2 a nd solving for q gives,
L
aq
a1) 0
2
..
IwJ(yJT.)=
J aq
(2.2.6)
Equations (2.2.5) a nd (2.2.6) involve two sensitivity coefficients which c an be
evaluated from Eq. (2 .2.4),
(2.2 .2)
_ J=I,
aT,
aT,
x·
_ J=_...J.
aTo
where } j represents the experimental value o f t emperature a t d epth X i' E quation
(2.2.2) demonstrates t hat a m inimum o f two experimental temperature m easurements a nd t heir corresponding locations along with the thermal conductivity
must be known t o d etermine the heat flux. N ote t hat t he heat flux is l inear in the
experimental temperature measurements 1'1; this linearity occurs repeatedly for
constantproperty I HCP's.
As i ndicated previously, a minimum o f two temperature measurements is
necessary for the steadystate determination o f h eat flux. S uppose t here a re J
t emperature sensors located in a p lanar b ody for which the heat flow is o nedimensional. While there a re J sensors, the n umber o f d istinct sensor locations
may be less t han J d ue t o m ultiple sensors a t t he same depth. T he s teadystate
heat flux c an be calculated by minimizing the least squares e rror between the
computed a nd e xperimental temperatures. I n o rder t o generalize the analysis,
assume t hat s ome o f t he sensors a re m ore accurate t han others, as indicated
aq
(2.2.7)
k
S ubstituting Eqs. (2.2.4) a nd (2.2.7) i nto Eqs. (2.2.5) a nd (2.2.6), replacing q by
its estimate q, a nd simplifying, the following two normal equations a re o btained :
1'1
I,
I
,
I
(2.2.8)
q
To Lt W2jXj+ "2 L W2jXj2 = k1 L Wj2 ljXj
k
k
J
J
J
j= I
j=
j= I
Solving this system o f e quations for the unknown heat flux
q=  k
JI ( JI)(
J
J
L wf J.=I w fxf
L
)JI(

J
(2.2.9)
qgives,
(.t wf)(.t WfXj lj) (.t wfXj)(.t wflj)
j =1
J)1 2
.
il
(2.2.10)
.L
JIWfXj
I t c an be demonstrated t hat Eq. (2.2.10) reduces to Eq. (2.2.2) for J = 2 a nd
W I' W2'" 1. Again, note t hat t he unknown heat flux is linear in the temperature
measurements.
Equation (2.2.10) could also be developed by (1) d etermining the two c onstants a a nd b in Eq. (2.2.4) by fitting a weighted least squares curve t o t he experimental temperature d ata a nd (2) differentiating the curve t o d etermine the
heat flux (see Problem 2.1). However, the a pproach o f using sensitivity coefficients (aTjaq) is m ore c onsistent with the remainder o f t he text. Results
~~~~
FIGURE 2.1
J
j=
( 225)
..
aTo
Steadystate temperature measurements at J locations.
b
4
64
C HAP.2
SEC. 2.3
S OLUTIONS OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM
2.3 T RANSIENT A NALYSIS O F BODIES W ITH S MALL
I NTERNAL T HERMAL R ESISTANCE
E xact S olution
F or bodies in which the thermal conductivity is very large a nd/or t he c haracteristic length scale L ( = volume/surface area) is small, it is possible t o ignore the
internal thermal resistance. This analysis is referred t o as the lumped (thermal)
capacitance analysis. T he energy balance on a body o f a rbitrary shape a nd o f
surface area A, volume V, a nd uniform temperature T is
V dT
q (t)=pc A d t
d TI
(2.3.6)
dt ' "
Since the truncation e rror becomes smaller with increasing polynomial order,
one might be inclined t o t hink t hat t he fivepoint equation is superior t o t he
threepoint (CD) equation. This would be true if the d ata were errorless; however, it is demonstrated t hat e rrors in the temperature d ata p laya crucial role
in determining the accuracy o f t he computed heat flux. Instead o f r equiring a
fourthorder polynomial t o pass exactly through five d ata p oint, an alternative
approach is t o leastsquares fit a straight line through five p oints a nd e valuate
the slope a t the center point. This approximation can be determined from
Eq. (2.2.10) by choosing all the weighting factors equal to unity a nd considering
the five equally spaced points o f :q... CIA', aI, 9, m;z~t; t he result is
(2.3.1)
N ote t hat q(t) d epends o n t he rate o f c hange o f t emperature a t time t a nd n ot
o n temperature d ata for all times past a nd future. I f t he heat flux is known, the
temperature response o f t he body can be calculated by integrating Eq. (2.3.1)
T(t) = To +
i'
o
A
q(l) dl
p cV
(2.3.2)
i ;<12 > cf'll
d TI
d t '"
N ote t hat the temperature a t any time depends only o n t he p ast history a nd n ot
o n the future values o f q(t). H eat flux measurement devices t hat satisfy the
assumption o f negligible internal resistance are often referred t o as slug calorimeters.
2.3.2
55
T he symbol O( ) d enotes o rder o f m agnitude o f the truncation error, as
determined by a Taylor series expansion. N ote t hat the C D a pproximation is a
higherorder approximation a nd thus for small At h as a smaller truncation
error than either B D o r F D. T he B O a nd F D results can be derived by passing a
straight line through two points, a nd t he C D results can be developed by
passing a parabola through three points. Higherorder results can be obtained
by increasing the number o f p oints through which a polynomial is required t o
pass. F or example, a fourthorder polynomial passing through five equally
spaced points yields
similar t o Eq. (2.2.10) c an also be developed for cylindrical a nd spherical
geometries (see Problems 2.2 a nd 2.3).
2.3.1
T RANSIENT A NALYSIS O F BODIES
 2YM 
1
(.)t,..,t" M f
YM 1 + YM+1 + 2YM + 1
,
~ t~+ l ..
10At
(2.3.7)
N ote t hat when using any o f t he foregoing derivative approximations in c onjunction with the exact solution Eq. (2.3.1), t he heat flux is linear in the temperature measurements provided the volumetric heat capacity pc is constant. All
o f these results have the appearance o f a digital filter.
A pproximate S olutions
2 .3.3
Although Eq. (2.3.1) is a n exact solution for the heat flux, practical application
o f this equation is generally with d ata available only a t discrete times. C onsequently, some difference approximations for d T/dt a re needed. I n t he discussion t hat follows, the temperature d ata (li) a re taken at equally spaced time
increments At. T hree common twopoint difference approximations t o d T/dt
a re a s follows:
d TI
dt ' /II
d TI
dt '/11
dTI
dt '/11
YM y;M 1
A t  +O(At) B ackward Difference (BD)
YM  YM
+~t
+ O(At)
F orward Difference (FD)
YM  Y;
+ 1 M  I + O(At 1 )
2At
C entral Difference (CD)
T emperature E rrors a nd A pproximate S olutions
Any measurement o f t emperature contains errors, a nd these errors have a n
i mpact o n t he computed heat flux . A convenient assumption to be used t hroughout this book is t hat t he temperature errors are additive (see Section 1.4.2).
T his allows the measured temperature li t o be written as the sum o f t he true
temperature T; a nd a n associated e rror 0 li
(2.3.3)
(2.3.8)
Let us determine the e rror in the derivative approximation caused by temperature errors. F or the BD method,
(2.3.4)
d TI
T MTM 1 OYMOYM 1
~
+=~~
(2.3.9)
dt ' "
At
At
T he effect o f t he temperature errors is c ontained in the term (0 YM  0 YM _ d/At.
(2.3.5)
L
56
C HAP. 2
SOLUTIONS OF THE INVERSE H EAT C ONDUCTION PROBLEM
r
Difference
Approximation
Backward
Forward
Central
Fivepoint, fourthorder
Leastsquares,
straightline, fivepoint
(2 .3.10)
Equation
Number
Standard Deviation
Derivative Approximation
(2 .3.3)
(2 .3.4)
(2.3.5)
(2 .3.6)
(2 .3.7)
.j~0'/61 = 1.4140'/61
. j20'/6t = 1.414O'/6t
.j20'/(2/lt) = 0.7070'/61
0.950'/61
0'/ (.j1061) = 0.316a/61
TABLE 2 .2 T emperature D ata
f or E xample 2 .1; f rom B eck a nd
A rnold'
(2 .3.11)
(2.3.12)
Observation
Number
d
t '"
;::;
·
L
a jY",_j+j
Time
(s)
Yj
Temperature
o f Dillet (O
F)
9
10
11
12
13
14
15
16
17
The single error bY", affects only the calculations of q", and q", + 1 for the BD
method, with all other q calculations being unaffected. For the F D method,
only q", _ I a nd q", are affected by the single error bY", . At first glance, it might
seem inappropriate to study only a single error because all temperature measurements are likely to have errors. However, the effects of all temperature errors
can be superposed for linear problems (constant properties). I f all errors b Yj
are the same, there will n ot be any heat flux error because they cancel out at
each time step. This can be understood by realizing that the slope of the temperature response curve does not change if all temperature measurements are
shifted by the same amount. In general, all difference approximations can be
written as
dTI
67
T RANSIENT A NAL VSIS OF BODIES
T ABLE 2.1 S tandard D eviation o f E rror i n D erivative
A pproximation f or T emperature E rrors t hat a re
I ndependent, A dditive, a nd o f C onstant V ariance ( 0'2)
N ote that the form of the error term is the same as the difference approximation
itself. This will always occur in linear problems with additive errors. Similar
results can be written down by inspection for the other difference approximations considered.
I t is i mportant to understand how a single temperature error affects the heat
flux computed a t the same time the error was made and how the error affects
any subsequent calculations o f heat flux . Suppose the backward difference
approximation is used a nd all temperature measurement errors are identically
zero except bY",. F rom Eqs. (2.3.1) a nd (2.3.9),
V
bY",)
q (t",)=q",=pc  ( T",T"'_I +A
L\t
L\t
SEC. 2.3
768
864
960
1056
1152
1248
1344
1440
1536
191.65
184.44
177.64
171.41
165.04
159 .89
155.19
150.78
146.68
EXAMPLE 2. I .
A solid copper billet 0.0462 m (1.82 in.) long and 0.0254 m (1 in.) in
diameter is heated in a furnace and then removed. Two thermocouples are attached to
the billet. Some temperatures, Y" given by one of the thermocouples are listed in Table
2.2 as functions of time. See also the plot of Yj versus time in Figure 2.2. F or this test,
pcV/(A61) = 200 W/m2K (31 Dtu/hrft 2 F). Compare the five methods presented i~
this section for calculating the derivative d T/dl. These data are from an actual expenment presented on p. 243 o f Deck a nd Arnold I .
(2.3.13)
j =1
where n is the number of points used in the difference approximation; Eq.
(2.3.13) is a compact way o f writing Eqs. (2.3.3)  (2.3.7). F or all of the difference
approximations considered in this section, L~ . I a j=O ; it is this characteristic
that allows a uniform shift in all Yj to have no effect on the slope calculation.
If certain statistical assumptions are made about the temperature errors,
then it is possible to calculate the standard deviation of the error in the difference
approximation. For example, if the temperature errors are independent, additive, and of constant variance (0'2), the methods of Beck and ArnoldI can be
used to calculate the results in Table 2.1 (see also Section 1.4). These theoretical
results favor the least squares approach because the standard deviation of the
heat flux estimate is smallest.
Solution. T he results of Example 2.1 a re summarized in Table 2.3 and Figure 2.3.
Several conclusions can b e drawn from this example.
1 . Doth F D and DD give the same numerical results but they are displaced in time
by 61 .
2 . All methods except DD use temperature measurements at times greater than the
calculation time. This will be referred to as using "future (temperature) information."
I
b
68
CHAP. 2
SOLUTIONS OF THE INVERSE HEAT C ONDUCTION PROBLEM
I
,
S EC.2.4
!
410
HEAT FLUX F ROM MEASURED SURFACE TEMPERATURE HISTORY 59
8 .0 ~"'"'"T"r...".rrr'"
400
7.0
390
'!"
~
380
!
E 370
~
..
..
!
~
.
220
.... 6.0
!
E
~
360
5.0
180
0
0
350
0
160
BO
FO
CO
t::..
4 .0
2
4 6 8 10 12 14 16
Number of time steps, i
0
0
Eq . (2.3.6)
V
3 40
Eq . (2.3.7)
3
2
4
5
6
7
8
9
10
Time index, M
FIGURE 2.2 Temperatures for cooling billet example (Example 2.1).
FIGURE 2 .3 Average temperature difference (AT) for Example 2.1.
T ABLE 2.3
A verage T emperature D ifference ( AT) f or E xample 2.1.
pcV   11M =A At AT. A T i n ' F
M
YJI  Y  1
JI
9
10
 7.21
 6.80
6.23
 6.37
 5.05
 4.70
 4.41
 4.10
11
12
13
14
15
16
17
3.
YJI + 1  YJI
 7.21
 6.80
 6.23
6.37
 5 .05
 4.70
 4.41
 4.10
! (YM + 1  YM 
 7.01
 6.52
 6.30
 5.76
 4.93
 4.56
 4.26
1)
Fivepoint,
f ourth o rder
Fivepoint linear
least squares
6 . T he l inear leastsquares result is appealing because t he s lope continuously
D
decays with time in a s mooth m anner.
2.4 H EAT F LUX F ROM M EASURED S URFACE
TEMPERATURE H ISTORY
2.4.1 E xact R esults f or C ontinuous S urface
T emperature H istory
 6.47
 6.35
 5.81
 4.85
4.54
 6.63
 6.17
 5 .64
 5.11
 4.58
All m ethods except F D a nd B D show a continuous decay o f s lope with time.
T he physics o f t he p roblem dictates t hat t he magnitude o f t he average temperature difference s hould decrease continuously with time.
4 . By using m ore i nformation (additional temperature measurement) t o c alculate
h eat flux a t a given time, some values c annot be c alculated a t b oth early a nd late
times.
I f the surface temperature, Y(t), o f an object is known continuously as a function
of time, some relatively simple exact solutions exist for the heat flux variation
with time. This surface temperature specification yields a simpler inverse
problem because the known surface temperature can be treated as a boundary
condition in the traditional sense. O ne approach to this problem is to determine
the temperature distribution within the body and then the temperature gradient
at the surface is used to determine the heat flux. I f the thermal properties are
treated as constant, then Duhamel's theorem provides a convenient means of
calculating the temperature field. The analysis starts with the temperature form
of Duhamel's theorem 2 ,3 ,4
,
d Y(,i)
N I
T (x, t) = To + ' 0 u(x, t  ,i) ~ d,i + i~O u(x, t  ,ii)~ l i
f.
(2.4.1)
where u(x, t) is the temperature response function for a body a t zero initial
60
CHAP. 2
S OLUTIONS OF T HE INVERSE H EAT C ONDUCTION PROBLEM
r
I
,
S EC.2.4
HEAT FLUX F ROM M EASURED SURFACE TEMPERATURE HISTORY 61
I
t emperature a nd subjected to a unit step in surface temperature, Y(t) is the
surface temperature variation with time, and To is the uniform initial (for t ~ to)
temperature. T he integral in Eq. (2.4.1) allows a continuous surface t emperature
in time, and the summation term allows N discrete steps in surface temperature
occurring a t A.; = illt. Some understanding o f D uhamel's theorem can be gained
by considering the discrete version o f Eq. (2.4.1). I f a series o f steps in the surface
temperature occurs as shown in Figure 2.4, t he temperature a t p osition x a nd
a t time t for 21lt < t < 31lt is given by
I
I
I
,
T (x, t) = To + u(x, t)1l Yo + u(x, t  Ilt)1l Y + u(x, t  21lt)1l Y
1
2
i
(2.4.2)
I
L
~~Ix=o =  k au(xa~A.)lx=o Y'(A.)dA.
i
x)
(2",l1.t
C'
a
 ul
a
I
= c=
x x=O ",Xl1.t
=
~
f
e  o y.'I L 2
(2.4.5)
L .=0
a x x=O
x
1'.=(2n+ 1) 2' n =O, 1,2, . ..
O ther step function solutions for various geometries a nd/or "inactive surface"
b oundary conditions are available in the literature. The analysis t hat follows
is restricted t o t he semiinfinite planar solid; for this case, Eq. (2.4.3) becomes
q(t) =
fkPC
vn J[r.' ~dA.
fkPC
vn {Jt
(2.4.6)
" ,tA.
=
1
1 [ ' Y(t)  Y(A.) }
[ Y(t) Y(t o)] +2
(t_A.)312 dA.
J,.
(2.4.7)
Equation (2.4.7) comes from integration by parts o f Eq. (2.4.6). Note that b oth
o f these forms have singularities a t A.=t. I f a purely numerical procedure is
used, these singularities can cause some difficulty. Therefore, a n analytical
integration procedure is considered. T he r eader is reminded t hat Eqs. (2.4.6)
and (2.4.7) presume that the body is a t a uniform temperature Y(t o) for t~ to.
l t may be possible to use a gaussian q uadrature S t o t reat this singularity;
however, such an approach would not allow the experimenter freedom in
choosing times a t which the surface temperature is sampled. I n the next section,
a combined analytical numerical procedure will be explored.
(2.4.3)
where Y'(A.) = d Y /dA.. E quation (2.4.3) is exact within the restrictions o f
D uhamel's theorem. I n o rder t o apply Eq. (2.4.3), the derivative o f the unit
step response must be known. The simplest case o f a unit step function is for a
semiinfinite planar solid.
u(x, t )= I erf
au I
I
The magnitude o f t he actual step in surface temperature is multiplied by the
response due to a unit step in surface temperature, u(x, t). T he unit step response,
u, must be shifted in time to correspond t o the time when the temperature step
actually occurs. Additional information on Duhamel's theorem c an be found in
Carslaw a nd Jaeger 2 , Schneider 3, Myers,4 and C hapter 3.
I f only the heat flux is o f interest, it is n ot necessary to calculate the entire
temperature field. The heat flux a t the active surface can be determined from
Fourier's law by differentiating Eq. (2.4.1) t o get [for Y(t) c ontinuous in time t],
q(t) =  k
"
.
u(x, t ) = I  2 ~ I e _o y. I L' sm 1'.x
~
.=01'.
L
I
(2.4.4)
2,4.2 A pproximate R esults f or S emiInfinite B ody w ith
S urface T emperature M easured a t D iscrete T imes
Another simple case is t he infinite plate o f thickness L where the step change in
surface temperature occurs a t x=o a nd the "inactive surface" is perfectly insulated.
Assume t hat t he surface temperature l j is measured only a t discrete times t j •
Between successive times, the surface temperature is a ssumed to vary linearly
with time. F or these assumptions, Eq. (2.4.6) can be integrated analytically t o
give
(2.4.8a)
=2
fkPC
vn f
; =1
(2.4.8b)
where the symbol denotes a n e stimated value.
The important material property group is (kpc)tI2; values o f this parameter
are presented in Table 2.4. F or a given heat flux into a semiinfinite body, a
A
FIGURE 2 .4 Illustration o f temperature form o f Duhamel's theorem.

11111
J tMt;+JtMt;t
b
62
C HAP. 2
S OLUTIONS O F T HE INVERSE HEAT C ONDUCTION PROBLEM
T ABLE 2 .4
T hermal P roperty
S EC.2.4
JiPC
HEAT FLUX F ROM M EASURED S URFACE T EMPERATURE HISTORY 6 3
2 000r.,.,.,.r.1 .5
Material
Copper, pure
Silver, pure
Aluminum, pure
Steel, low carbon
Steel, 20% C r
Steel, 40% Ni
Glass, plate
106.8
1150
1000
703
( q"",. = 12,000 Btu/ft2  s)
92.9
65 .3
41.0
26.2
18.4
3.9
442
282
198
42
k pc=3.2855 (ft2~:R)
1500
2 S
1.0
.
I t:
1 '
1000
body with a small value o f kpc has a larger surface temperature rise than bodies
with large kpc. Physically, this occurs because a largek body is able to remove
the heat from the surface more rapidly. Note that the heat flux is linear in the
temperature measurements, provided (kpC)I/2 is independent of temperature.
E XAMPLE 2.2. I n o rder t o d emonstrate the method given by Eq. (2 .4.8), a n example
problem is selected for which a n exact solution is available. Consider a heat ftux with a
parabolic variation with time
q
 = 4 t ( 1 t)

q....
I....
(2.4.9)
t.....
Using t he analytical solution o f Carslaw and Jaegerl for q t" a nd superposition, the
surface temperature variation is given by (see also Problem 1.4).
7 ;(t) To=
4qm•• [ r (2) ( t
~ r(5/2) tma•
)3/1

r (3) ( t
rm tmax
)5/1J
(2.4.10)
.Jr=
where r (x) is the G amma function. [ r(2) = 1, r (3)=2, r (5/2)=37t lll/4 a nd r (7/2)=
157t l/l/8.] T he results o f t he foregoing analytical solution are given in Figure 2.5 for
the following parameters:
q. ... = 12,000 Btu/ftlsec,
Btu
k pc=3.2855 (  1I t sR
)1
To = 5400 R
s (Copper a t elevated temperature)
These conditions are intended t o b e representative o f those encountered in traversing a
copper calorimeter across a high heat ftux arc jet. T he exact temperature d ata calculated
from Eq. (2.4.10) for a time step o f AI =0.001 s applied t o t he inverse solution given by
Eq. (2.4.8), yield t he results shown in Figure 2.6. At early times, the percent e rror in the
computed heat ftux is in excess o f  15%; after a time o f 0.01 s t he heat ftux error remains
nearly constant a t approximately  2% until approximately 0.065 s. As the final time
o f 0.07 s is approached, the errors become large a nd positive. This is because the heat
ftux is a pproaching zero a l 0.07 s.
0
0.01
0.02
0 .03
0.04
Time,s
FIGURE 2 .6 Parabolic heat flux history and corresponding surface temperature variation.
Although the results for the preceding example are good, the use of exact
temperature data is not a very severe test of any inverse method. In order to
simulate the effect of temperature errors on the computed heat flux history, a
random error term was added to the results computed from Eq. (2.4.10)
Y. = 'T.(t.. ) + (J.l1 T
.
where (J is a random variable o f uniform distribution with values in the range
[  1,1] and l1 T is the maximum magnitude of the temperature error. F or the
calculations that follow, it is assumed that l 1T=SR; the calculated errors are
shown in Figure 2.6. In general, the results using simulated temperature errors
scatter about the results using errorless temperature data. However, the realistic
temperature d ata produce a much wider variation in the heat flux error.
2 .4.3
L""';O
T emperature E rror P ropagation i n Eq. ( 2.4.8)
I t was demonstrated in Section 2.3.3 t hat if an error {, Yj is made in a single
temperature measurement Yj, the corresponding heat flux e rror becomes
identically zero after n time steps where n is the number of measurement points
a pp,.,imal;n, Iho dod. .I ;" d Tld,. F o, a "m;;ofiruto b od, Ihal . ...
C HAP.. 2
64
S OLUTIONS OF T HE INVERSE H EAT C ONDUCTION PROBLEM
5
I
I
a
a
•
; ;;0
G0
.
..
g
°
a
a
e
a
••
e
a
•
a
ee
e
0
Q
e
e
e
e
e
e
a
.
e
a
e
•
..a
.
e
,
a
~
1t~1
0
represents the dimensionless heat flux e rror and shows how it decays for subsequent times. The normalization is conveniently chosen so that the first value
is unity. Table 2.5 presents results showing how the dimensionless heat flux
error due to an initial temperature error, b Yo, decays with time. The results
of Table 2.5 indicate that a positive temperature error b Yo causes a negative
error in heat flux and this error damps very slowly with time. Even after 100
time steps, the dimensionless heat flux e rror is still 5% o f its initial value. I f
the initial heat flux e rror was large, then this heat flux may be very inaccurate.
For a given value of temperature error b Yo , the heat flux e rror is proportional
to ( kpLf/l and inversely proportional to I/(~r)l/l; small time steps and large
values o f kpc b oth produce large heat flux errors.
Equation (2.4.11) reveals that an error in Y1 may be quite different from an
error in Yo because Y appears in two places in the heat flux equation. I f a single
1
temperature error ( i YM occurs, the resulting heat flux e rror and its decay with

~  10
co
~
Exact thermocouple data
e 5R" Random thermocouple error
e
1
0.02
J
J
J
1
0.03
0.04
0.05
0.06
0.07
Time,s
FIGURE 2.6 Relative heat ftux error (%) using exact and inexact thermocouple data.
finite internal thermal resistance, the heat flux e rror corresponding to a single
temperature error takes an infinite number o f time steps to decay to zero.
This can be demonstrated by considering the first few terms o f Eq. (2.4.8),
Jfffr
TABLE 2 .5 H eat F lux
E rror f or 6 Y 0 a t I nitial
T ime
[ (Y1  YoX.JM  .JM  1)+(Y1  Y1X.JM  I.JM  2)+ . . 'J,
M = 1 ,2,...
M
(time index)
(2.4.11)
where the data were taken with equal increments of ~t. Note that the initial
temperature measurement Yo appears in only one place in Eq. (2.4.11). Suppose
there is a n error b Yo in the temperature Yo( = To + b Yo) b ut all other temperature
measurements are exact. The corresponding error in heat flux can be determined
from Eq. (2.4.11),
q u=2
fkPC
.Jruft [(Y1
2
(2.4.13)
a
~
q u=2
M = I , 2, . ..
Jfffr
0
'l;j
°
201
0.0
0.01
H EAT FLUX F ROM M EASURED S URFACE T EMPERATURE H ISTORY 6 5
 I (iq'iP
: =.JM  .JM  I,
kpc ( i Yo
2 •
.
0
><
::J
: ;:
S EC.2.4
the term in Eq. (2.4.12) that contains the temperature error ( i Yo represents the
corresponding heat flux error. Let (iqC;:' be the error in q u corresponding to
the temperature error ( i Yo; the superscript on (iqM is important because it
indicates the time at which the temperature error occurred. Then
I
I
e
0
#.
I
I
r
ToX.JM  .JM  1)+(Y1  Y1X.JM  I.JM  2)+ . ..J
.IfkPC bYo(.JM.JMl)
n&i
(2.4.12)
The term inside the brackets in Eq. (2.4.12) is the heat flux for errorless data;
I
L
I
2
3
4
5
6
7
II
9
10
100

I b,l;:'
  
2
·PC bYo
~

7tt'11
1.000
0.414
0.31!!
0.26!!
0.236
0.213
0.196
0.1!!3
0.172
0.162
0.050
66
C HAP.2
S OLUTIONS OF THE INVERSE H EAT C ONDUCTION PROBLEM
subsequent time steps are as follows:
lJqW'
=10
Pc lJY",
.
21tdt
1
(2.4.14a)
~
1
2
~
kpc o ",
i = I, 2, . . .
M =I,2, . . .
(2.4.14b)
1 tdt
The subscript on q represents the computation time t ",=Mdt, a nd the superscript indicates the time a t which the temperature error occurred. Note that
the dimensionless heat flux error in Eq. (2.4.14) is independent of the time
( M) a t which it occurred, provided M + 0. T he results of Eq. (2.4.14) are given
in Table 2.6.
T he heat flux e rror corresponding to lJY",(M + 0) decays considerably faster
than the error corresponding to lJYo. This implies that greater care should be
taken in measuring Yo t han other values o f Y.
I t should be reiterated that all temperature measurements will contain
errors. A complete error analysis can be accomplished by superposing the heat
flux e rror calculations for a single temperature error because the problem
under consideration is linear.
T ABLE 2 .6 D ecay o f H eat
F lux E rror R esulting F rom
T emperature E rror CSY", a t
T ime t,w=MIlt
i
(time index)
I
2
3
4
5
6
7
8
9
10
100
1.00000
 0.58579
0.09638
 0.04989
 0.03188
0.02265
0.01716
0.01359
0.01110
0.00930
 0.00793
0.00025
EXACT S OLUTIONS OF INVERSE HEAT C ONDUCTION PROBLEMS
67
2.5 EXACT S OLUTIONS OF INVERSE H EAT
C ONDUCTION P ROBLEMS
2.5.1
lJqW! I
r :;t
r;
r:t
~y. = ",,+ 1 2", . +",.1
o
SEC. 2.6
L iterature R eview
Few exact solutions to the inverse problem of heat conduction for which the
temperature sensor is a t an arbitrary location are available in the literature.
This is in contrast to the direct problem of heat conduction for which a wide
range of solutions is available. Burggraf6 presented one of the earliest exact
solutions. He approached the problem by assuming that both the temperature
yet) and heat flux q dt) were known a t a sensor location. The temperature field
.was developed in terms of an infinite series of allorder derivatives of both yet)
and qE(t). I f the temperature sensor was located a t the center of a solid cylinder
or sphere, then qE(t) was identically zero (for onedimensional radial heat
conduction). Langford 7 independently developed results similar to the Burggraf
solution. Kover'yanov 9 deVeloped results for hollow cylinders and spheres.
The heat flux a t the exposed surface was determined by differentiating the
temperature field. Imber and KhanS obtained an exact solution for the temperature field using Laplace transforms when the temperature was known at two
distinct interior points. Their temperature solution can be extrapolated in b oth
directions toward the boundaries. The extrapolation distance is limited to the
distance between the two temperature sensors. N o computational results were
presented for the more difficult prob!em o f calculating heat flux a t the exposed
surface.
2.5.2
D erivation o f E xact S olution f or P lanar G eometry
The analysis that follows closely parallels that of Burggraf. 6 The body is divided
into (I) an inverse region and (2) a direct region as indicated in Figure 1.4.
The direct region has conventional boundary conditions: specified temperature
yet) a t the left face a nd arbitrary boundary conditions a t the "inactive surface,"
L. By some means, it is necessary to solve for the temperature field in region 2.
Next, the solution is differentiated a t the location of the temperature sensor in
order to calculate the heat flux qE a t X I = E. Once the heat flux a t the sensor
location is calculated, the inverse problem has two boundary conditions
specified a t the same boundary. The Burggraf solution requires that qE(t) and
all of its derivatives are known.
Starting with the constantproperty form of the energy equation,
(2.5.1)
and differentiating it with respect to time yields,
68
C HAP.2
S OLUTIONS OF THE INVERSE H EAT C ONDUCTION PROBLEM
a2 T a
aT
ai2=ar (exV 2 T)=cxV 2 a r =cxV2(cxV2T)
t;
2
aa
=ex 2V4 T
(2.5.2)
SEC. 2.5
EXACT S OLUTIONS OF INVERSE H EAT CONDUCTION P ROBLEMS
69
E quation (2.5.10) is t he general solution for the t emperature field in the inverse
region.
T he r emaining p roblem is t o d etermine t he functions I (r) a nd g(r). T hese
functions a re f ound by substituting Eq. (2.5.10) i nto t he differential equation,
Eq. (2.5.1),
G eneralizing for a n a rbitrary t ime derivative gives
anT =cx"V 20 T
at"
(2.5.3)
T he s ame t ype o f p rocedure is applied t o F o urier's law :
q =  kVT
(2.5.4)
(2.5.5)
F or t ime derivatives o f t he heat flux o f a rbitrary o rder, t he d erivative is
A solution is o btained b y requiring t hat e ach t erm i nside the brackets o f Eq.
(2.5.12) is identically zero,
V2J,,=  101 }
1
cx
" =1,2• ...
T he t emperature field is assumed t o b e a n infinite series involving t he t emperature g radients a t t he t emperature s ensor location r = E,
T(r, t )=
Jo
Ho(r)VOTL£
2I
7
(2.5. ~d o dd terms,
H 2i(r)V TI.=£ +
i~O H
2 i+ 1(r)V
2i
+I
T I.:£
(2.5.9)
1 aiTI
co
1 aiql
H 2i (r) i a i
+ L H 2i + l (r)( k i) a i
ex t r=f: i=O
 CX t r=E
ao
d iy
1 ao
diqE
d'
= L / i(r)i   L gi(r)  i
q l.
i=O
dt
k i=o
dt
/
~
Y
w here
H 2i } ;(r)=.(r),
ex'
gi(r)
H 2i + l (r)
exi
alTI
at i r=£
~
iY
i i1'
10(E) = I , go(E) = 0,
J,,(E)=go(E)=O.
" =1,2, . . .
(2 .5.14)
a TI
ao
d "Y
co
dOq .
q E=k=k L I~(E)  + L g~(E),t.
o
ar r=E
0=0
dt
0=0
dt
co
L
i=O
or
Also t he h eat flux is given by
S ubstituting Eqs. (2.5.3) a nd (2.5.5) i nto E q. (2.5.8) yields
T(r, t )=
The b oundary c onditions o n t he 1 a nd 9 f unctions a re d etermined from t he
r equirement t hat t he s olution exactly matches the t emperature d ata Y(t),
(2.5.8)
m = 2 s phencal
I t is convenient t o d ivide t he series in Eq.
ex
(2.5.7)
1 d ( d T) m =O p lanar
V2 T=  t "  m=lcylindrical
t " dr
dr
.
Jo
(2.5.13)
1
V go=goI
2
I n t he analysis t hat follows, the geometry is restricted t o o ne d imension such t hat
T(r. t) =
V 2 g 0 =0
V210 = 0
(2.5.6)
Jt C
~
aiql
a t i r= £ = at i
(2.5.10)
(2.5.11)
L J'~ ~
d~C
go(E) = 1,
lo(E) = 0,
1~(E)=g~(E)=O,
" =1.2, . . .
(2.5.15)
The s olution t o Eq. (2.5.13) subject t o t he b oundary c onditions given by Eqs.
(2.5.14) a nd (2.5.15) completely determines t he 1 a nd 9 functions. N ote t hat
these functions m ust be determined in a sequential m anner s tarting with 10
a nd go.
C HAP.2
70
S OLUTIONS OF T HE INVERSE HEAT C ONDUCTION PROBLEM
Observations
1.
F or an insulated surface a t r = E, the fseries alone determines the
temperature profile.
2 . F or an isothermal surface a t r =E, the gseries alone determines the
temperature profile.
3 . The functions fo a nd go represent steadystate solutions.
4 . Allorder derivatives of Y(t) a nd qE(t) must exist.
By direct substitution, it is shown that
1 ( E_x)2n
 1 ( E_x)2n+1
= (2n)! ]. (Xn
' ;n (2n + I)!
(Xn
(2.5.16)
Solution f or Planar Geometry ( r=x).
.r.
is a solution to Eqs. (2f."1~ (2.!14f. The temperature distribution is written as
co
1 ( E_x)2n d ny
T (x, t )= Y (t)+ n~1 (2n)!
(Xn
dtn
(E  x) [
co
1
(E  x)2n d nqE ]
+ k qdt) + J d2n + I)!
(Xn
dtn
(2.5.17)
The planar geometry solution is similar in appearance to a Taylor series expansion about the temperature sensor depth.
The heat flux a t the active surface is o f primary interest; it is determined by
using Fourier's law and Eq. (2.5.17) and evaluating at x = 0:
co E2n1 1 d ny co E 2n dnqE ,
(2.5.18)
q (t)=qE+k n~1 ( 2n1)!;'; de" + n~1 (2n)! dtn ;foI t is convenient to normalize the heat fluxes and introduce a dimensionless time
as follows:
qE
(Xt
qEE
Q =T' Q E=T' ' E= E2
f
f.+n=1
f
1
d ny
_ 1_ dnQE
( 2nl)! d'E +n=1 (2n)! d'E
EXACT S OLUTIONS OF INVERSE H EAT C ONDUCTION PROBLEMS
(2.5.20)
This is a very important result because it is exact for continuous temperature
measurements and has a simple form. The solution clearly shows the dependence
of the surface heat flux on all orders oftime derivatives of the measured temperature and the related heat flux, both at x = E. The temperature level itself is not
significant since only derivatives are needed. I t is surprising that the solution
given by Eq. (2.5.20) shows no explicit dependence on the initial temperature
distribution in the body. Burggraf6 has pointed out, however, that a polynomial
fit of finite order to the experimental temperature data, Y(t), implies that the
initial temperature profile must be nonuniform.
71
Since the inverse solution given by Eq. (2.5.20) is exact, one might ask if
there is any need for further investigation. The answer to this is a definite yes
because the exact solution has several practical limitations:
1.
2.
Highorder derivatives of discrete temperature data Y(t) and heat flux
QE must be evaluated numerically.
I t is awkward for composite bodies and not appropriate for temperature
dependent properties.
3 . I t may require a numerical procedure to solve the direct problem and
determine Q,;{t).
4 . I t is not applicable to the overspecified problem of more than one interior
temperature sensor.
S . The method does not lend itself readily to the case of multiple heat
flux determination such as the twodimensional case.
Although the above exact solution has limited applicability in a practical
sense, it is extremely important because o f the insights provided.
2.5.3
Expressions f or C ylinders a nd S pheres
Solid cylinders and spheres in which the heat flux is one dimensional radially
and the temperature sensor is located a t the center have simple exact solutions
for the IHCP. Consequently, the heat flux a t the exterior surface depends only
on the temperature response and its derivatives. From Burggraf6 and Langford,'
the temperature fields for solid cylinders and spheres are given by
co
(2.5.19)
Note that Q has the units of temperature. The normalized heat flux is written as
Q =Q.
SEC. 2.5
L
"=
( R_x)2n d ny
.
22n( ,)2 n d n (cylinder)
1
n. (X t
co ( R_x)2n d Ry
T (x, t )= Y (t)+ L (2
1 )'" d" (sphere)
n=1 n + .(X t
T (x, t) = Y (t)+
(2.5.21)
(2.5.22)
Note that the coordinate x is attached to the exterior surface and that R is the
external radius. The heat flux is evaluated from Fourier's law and Eqs. (2.5.21)
and (2.5.22); the results are
q=  k
a TI
co
nR 2n  1 d Ry
I( ,)2 n d "
n. (X t
 a x=O = k "L1 22"
x
=
a TI
co 2 nR 2n  1 d ny
q=  k =k
a x x=O
"= 1 (2n + 1)!(X" dt"
L

(cylinder)
(sphere)
(2.5.23)
(2.5.24)
Comments on hollow cylinders and spheres can be found in Burggraf,6 Langford,' and Kover'yanov. 9
72
C HAP.2
S OLUTIONS OF T HE INVERSE H EAT C ONDUCTION PROBLEM
r
I
2 .5.4
E xample R esults f or P lanar G eometry
..,
1
d "Y
.=d2nl)! dtE
<:i
0
II
......
<l
.,..
N
.q
...

<Xl
Q=I
(2.5.25)
0
II
......
T he first five derivatives are approximated as follows:
d lj
drE
~
<l
l j+llj_1
(2.5.26)
2ArE
N
>D
0
0
II
.
.....
dr~ ~
<l
r:
2Ar~
(2.5.28)
.,..
.0
;;
N
0
::J
.
0
I I)
II
II
.
.....
<l
....
N
0
u
(2.5.29)
IC
W
dr~ ~
..
..
II
 lj3+ 4 lj2 5 lj1 + 5lj+I4lj+2+ lj+3
2Ar~
(2.5.30)
II
......
CI
CI
<l
::J
Some insight is gained by looking a t the coefficients o f the various order differences; the appropriate term to consider is
10
r::
..
.,..
I /)
0
GI
<l
II
.
.....
r::
(2 .5.31)
( 2nl)!Ar1.
Equation (2.5.31) is tabulated in Table 2.7 for various values o f the dimensionless
time step. F or each value of ArE, the maximum coefficient is underlined. Note
that as ArE becomes smaller, higherorder derivatives appear to become more
important. If ArE is large ( > 10 2 ) then the first derivative is the dominant
term; if ArE is much smaller than unity, then several highorder terms appear
to be significant.
Equation (2.5.25) can be written in an al~ernative form by using the difference
approximations given by Eqs. (2.5.26)(2.5.30) and simplifying to obtain
~
V \)
Qj=D_ 3lj3 + D_ 2lj  2+ D_,lj_1 + Dolj+D1lj+ I + D 2lj+2 + D 3lj+3
(2.5.32)
. .......
..,
8:8~~~~~
_ ""';oci""";NN""':
r r v 0 '\ t V'l
0 '\0'\.::1"
:8:8 r o..,oo
' D'D o rO'\r1 1")\00\o~~ 0 "'00.,..
· 0 .... oON~vi
N rr....  N ::<lO\N
r MI,f"')VI,f')OO
>D .... r O\oo
'OM_V)M_
\ Of"I"'IMO\O("t)
0:8:::!8gg~;g
~~~:iNOO
r
~~8~~
q~  .ioooo
\ O\OMI.I')t'C")
'DN~>D§
~;;;;
8
oo~
.,..
I /)
dSlj
..,
00000
r X X r X X
X
\ 0 f"I"\ _
N 0\
1 oC("t)"Ct'v)V')V'I
.,..
(2.5.27)
d3lj  lj_2+ 2 lj_I2lj+l + lj+2
x  
0'"
In order to apply the exact solution, the time derivatives must be approximated
numerically. F or simplicity, suppose the temperature sensor is attached at a
perfectly insulated boundary and central differences are used. Only the first
five derivatives are considered; in order for the method t o be of practical utility,
the series must be truncated at a reasonable limit. Applying Eq . (2.5.20),
·u
I:r .... ..,\NO
8 .... rCXl_
o N ........
M
: 8 .... 0 N
q,,<~~~§~
. .,.NOOOOO
I\r r .,.. 00 N
0\01""'00
q~~§~~
N OOOOO
;:
....I
G
0
(J
.
.....
GI
u
<l
r::
...
GI
GI
.....
...
C~
..
.
...
, ...<l
~
N ::::..
w ....
.0
..
......
<l
q~a§~
0000
..,
I
'"
I
I
I
::~~8
xx
,..... M _x
x
c
I oOrt"iVv)
_ '0("1"')001.1')
~~"'1~r:
O OON
....I I
l Ot:
c (N
1"':'
D
_
I'>D .... 00 ....
r .... 0 \
'"
NMVI.I')\Or
73
75
REFERENCES
where the D's a re independent o f t he time index j a nd can be written as
1
D3=2.9!L\'t~
4
01
10
111'\
. ..
I I')
( "4
I
I
I
I
I
I
I
0
....
'"
S
01

00000000000
xxxxxxxxxx
O\~OOI,f')O\\OOO
\ 00000
6
~oci\CiV'i~~c_r'iNN.....;..n
fO\
C"4
_
_
1
I
I
I
0
0
0
_
,...
xxxxxxxxxxx
~~~~~~~~~~~
~~~~&1~C;;~~~q
D Z=2
I t'')1rD ...... N \ O  M r  N \ C \ o
I
It'I
'"
N
C"4
I
I
I
I
I
I
_
0
0
_
0
N
xxxxxxxxxx
~~~~N~~Y"l~$~
t"'lNO("f'lt"'lt'f")O~f"'lO
f"I"iviriociM"":..n . . r 'o\"":
IIIIIIIII
... CD
0 '0
. .. : I
I
CI
e(
:=~
CD 1 0
ON
O1·
0
~.
c~
.;;
.
.s;CT
...
' i ...
CI
~&&I
~~
I
....!
.
CD CD
... >
:I .
~
0'1
CD ...
I
c ,.
eO
..,
I >
CD CD
CI
..
ii:
00 . .
N ...
&&Iii:
. ...I>
a l_
e( c
1 0
74
I
I
10
10
111'\
I I')
,
I
I
I
I
1
N
I
4
5 +4+
3
·9!L\tt: 7!L\tt: 2·S!L\tE
Equation (2.5.32) has the appearance o f a sevenpoint moving average filter,
with the sum o f t he coefficients equal to zero. Again, note t hat q is linear in the
temperature measurements. T he coefficients a re n ot symmetrical; t hat is,
D  i'/=Dj except for certain values o f i. T able 2.8 presents the temperature coefficients defined in Eq. (2.5.33) as functions o f t he dimensionless time step L\tf:For large dimensionless time steps (> lOz), DI a nd D _I a re the dominant terms;
only D _I a nd DI c ontain IjL\tE terms. This suggests, in turn, t hat for large time
steps, a twopoint central difference might be adequate. T he o pposite is t rue
for small time steps; all o f t he D's a ppear t o be approximately the same order o f
magnitude. Thus, the sevenpoint formula considered in this example might
not be adequate for small time steps.
In summary, an exact solution has been presented for the inverse problem
of heat conduction, provided the properties are constant. The solution requires
that infiniteorder derivatives o f t he experimental d ata Y(t) must exist. If small
dimensionless time steps are used, highorder derivatives can dominate. F rom a
practical point o f view, the utility o f t he Burggrafsolution is t o provide beneficial
insight into inverse heat conduction problems.
X
t "'lOO\Q'\.:ttt')IrDrrt'lt ......
&&1'0
•
(2 .S.33)
2·9!L\~
01
00000000000
.u .C..
!D
2
S
4
1
1
1
D I =          +   + 2.9!L\t~
7 !L\tt
S!M~ 3 !L\tl 2L\'tf:
0'1
00000000000
: lU
c .E
1
2L\tE
_
I
cr
1
D o=     7 ! L\tt 3 !L\t~
I
o
1
1 .I")t"')\Ot"IO\O
_V'l("f'lt"'l\O_r~V')C"t')
CI
4
1
Dl=2.9!L\'t~ 7!L\'t:+S!L\'t~+3!M~ 
X
r_M~~'DNN~.,..'"
o o\Or
1
2.9!L\'t~+7!L\'tt  2·S!L\'t~
_
0
....
0

00000000000
xxxxxxxxxxX
~~~~~N~~OO~~
r_o~Or_~ _ _ : :!Nr_
REFERENCES
f""')N("f'lMf'f"I:;f"l:t""'~\oM
...;.:i"....;~....;~"....;~"....;~"....;
I.
B eck.J. V. a nd A rnold. K. J .• Parameter Estimation in Ellgineeringand Science. Wiley. New York •
1977.
Carslaw. H. S. a nd Jaeger. J. c.. Conduction 0/ Heat in Solids. 2nd ed .• O xford University Press•
L ondon. 1959.
L' ';du.
i
II'
2.
p, J .. C "",",, . . H~' " . ...' .
Add""W~", • . ...,;'. MA, " SS,
76
C HAP.2
S OLUTIONS OF THE INVERSE H EAT C ONDUCTION PROBLEM
4. Meyers, G. E., Analytical Methods in Conduction Heat Transfer, McGrawHili, New York,I971.
5. Carnahan, B., Luther, H. A., and Wilkes, J . 0 ., Applied Numerical Method;, Wiley, New York,
1969.
6. Burggraf, O. R., An Exact Solution o f the Inverse Problem in Heat Conduction Theory and
Applications, A SME J . Heal Transfer,86C, 373  382, August 1964.
7. Langford, D., New Analytical 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. 14 (4), 315  322 (1976).
8. Imber, M. and Khan, J., Prediction of Transient Temperature Distributions with Embedded
Thermocouples, A lA A J . 10 (6),784 789 (1972) .
9. Kover'yanov, V. A., Inverse Problem of NonsteadyState Thermal Conductivity, TeploJizika
Vysokikh Temperatur, 5 (1),141  143 (1967).
P ROBLEMS
2 .1. Develop Eq. (2.2.10) by (a) using a weighted leastsquares procedure
that fits a linear equation to the experimental temperature data and
(b) differentiating the curve fit to determine the average temperature
gradient, and (c) use of Fourier's law to obtain the heat flux.
2 .2. F or cylindrical geometries, the temperature profile is linear in the
variable lnr; T  7; = [  QIn(r/ri)]/(21tkL), Q= q(21trL) where Q is
constant. Develop a heat flux estimating equation analogous to Eq.
(2.2.10) t hat minimizes the weighted leastsquares error between the
computed and experimental temperatures. The heat flux is to be found
at r = rio Is this result the same as the one obtained for a temperature
profile linear in r ?
2 .3. F or spherical geometries, the temperature profile is linear in the variable
l /r; T  7; = (Q/41tk)(l/rl/rj), Q = q(41tr2) = constant. Develop a heat
flux equation analogous to Eq. (2.2.10) that minimizes the weighted
leastsquares error between the computed and experimental temperatures. Is this result the same as that obtained for a temperature profile
linear in r?
2 A.
Develop Eq. (2.3.7).lt is acceptable to start with Eq. (2.2.10) as suggested
in the text immediately above Eq. (2.3.7).
2.5. Demonstrate that if five temperaturetime points are used to determine
a leastsquares straight line [e.g., Eq. (2.3.7)], then a single temperature
error eM will cause errors in the calculation of qM  2, qM  I , q M, qM + I ,
and q M+2' W hat are the heat flux errors?
2 .6. Verify the numerical results in Table 2.3.
2 .7. Derive Eq. (2.4.7) from Eq. (2.4.6) by integration by parts.
Hint: rewrite the integrand as
. 77
PROBLEMS
Y '().) + Y '(t) Y'(t)
. jt).
and express in terms o f two integrals.
2 .8. Verify that Eq. (2.5.16) is a solution to Eqs. (2.5.13,14,15).
2 .9.
Prove that the sum of terms in Eq. (2.4.14a, b) adds to zero.
What is the physical significance of this result?
2.10. F or the spherical heat flux equation given by Eq . (2.5 .24), generate a
table analogous to Table 2.7. W hat conclusions can be made?
2.11. Starting with Eq. (2.4.7) a nd the assumption that the surface temperature
response varies piecewise linearly with time, show that the heat flux
can be expressed as
(k P
A
A
c)tI2 { YM  Yo
YM  YM  t
q (tM)=qM=  ;(CM  t )1I2+(CM  CM _ I)t /2
o
~ t[
+ L...
i =1
Y MYi
YM  Yi  I
2 (YiYi  I ) ] }
1/2 1/2 +
1/2
I2
( tMt i )
(CM  ti  tl
( CMt l ) +(CM  Ci  tl I
2 .12. The result given in Problem 2.11 has been given in numerous places in
the literature and is algebraically more complicated than Eq. (2.4.8).
Show t hat the above expression can be simplified to Eq. (2.4.8b). Hint:
Expand the first tw.o summations and show that most ofthe terms cancel.