212
C HAP. 6
INVERSE CONVOLUTION FOR A SINGLE SURFACE H EAT FLUX
P ROBLEMS
2 13
EXAMPLE 5 .6. Repeat Example 5.4 for the whole d omain regularization method. Find
a n o ptimal value o f a +.
10. Beck, J. V., C riteria for Comparison o f M ethods o f S olution o f the Inverse H eat C onduction
P roblem, Nuc/. Eng. Des. 53, 11  22 (1979).
Using t he R value o f 0 .0283 in Eq. (5 .6.15) gives a+ = 0 .0008 which is quite
close t o t he a+ =0.001 values used in the examples for the triangular heat flux test
P ROBLEMS
Solurion.
0
~
5.1. a. C alculate the 4>(r, t) function for a solid cylinder for r = 0. Give the
EXAMPLE 5.7. Repeat Example 5.5 for the whole domain regularization method.
values for a steel cylinder 0.664 cm in diameter. The dimensionless
.
Solution.
T he A t+ value is 0.028 a nd thus Eq. (5 .6.15) gives
a +=R 1
Since the standard statistical assumptions are n ot valid, a value o f R =O.OI is used
a nd then Eqs. (S.6.14) a nd (S.6.1S) give
U
a = 0.0000 k l
=0.0001
+
2
.
GIve 4> values I n SI units for 6 time steps a nd give the times in
seconds. Use values given in Table 1.3 a nd the properties given in
Table 5.4.
b . Repeat the problem for a cylinder 4 cm in diameter.
c. Repeat p art (a) for brick.
5 2. a. A solid steel sph.ere is 4 c m in diameter. Calculate the center temper
e·~!S4y = U S x 10 12 m4_KI/Wl
N ote t hat t he temperatures are in K and t hat t he heat flux has units o f W1m2.
.
tl~e s tep IS A t•. = aAt/a = 0.05 where a = radIUS o f the cylinder.
atures for ten tIme steps o f A t = 1.818 s for the heat flux given by
t<O, q =O
O <t<4At,
0
T he time step did enter the above selection o f a. S uppose that t he R =0.01
value is appropriate for A t=0.9 s. I t is interesting t o see t he 'effect o f choosing
the time step o f 0.3 s. W ith the time step decreased by a factor o f 3, the ()qr value
is also decreased by this factor. T hen Eq. (5.6.16) shows t hat ex is increased by a
factor o f9.
q=I00,OOOW/m 2
O <t<8At, q=200,OOO W /m 2
t >8At, q =OW/m 2
T he initial temperature is 50°C.
b. Add to the temperatures some random errors that have zero mean
have a s tandard deviation o f2 C, are uncorrelated, and are normal.'
5.3. Show that the expressions
M
REFERENCES
~
0)
1. Keltner, N. R. a nd Beck, J. V., U nsteady Surface Element Method, J . Heat Transfer 103,
759 764 (1981).
2. Beck, J. V. a nd Keltner, N. R., T ransient Thermal Contact o f T wo SemiInfinite Bodies O ver a
C ircular Area, Spacecraft Radiative Transfer and Temperature Control, T . E. Horton, Ed. Vol.
83 o f P rogress in Astronautics a nd Aeronautics, AIAA, New York, 1982, p p. 61  82.
3. Litkouhi, B. a nd Beck, J . V~ Intrinsic Thermocouple Analysis Using M ultinode Unsteady
Surface Element Method, AIAA P apcr N o. 831437, 1983.
4. Stolz, G., J r ., Numerical Solutions t o a n Inverse P roblem o f H eat C onduction for Simple
Shapes, J . H eat Transfer 81, 2 0 26 (1960).
5. Beyer, W. H ~ Ed., Handbook o f Tables for Probability and Statistics, 2 nd ed., T he Chemical
Rubber Co., Cleveland, O H, 1968, p. 484.
6. Beck, J . V. a nd Arnold, K. J ., Parameler Estimation in Engineering and Science, Wiley,. New
York,1977.
7. Farnia, K., ComputerAssisted Expcrimental a nd Analytical Study o f T ime{fempcrature·
D epcndent Thermal Propcrties o fthe Aluminum Alloy 2024T351, P h.D . Dissertation, Dept.
o f Mechanical Engineering, Micbigan State University, 1976.
8. Twomey, S., O n t he N umerical Solution o f F redholm Integral Equations o f t he F irst Kind
by the Inversion o f t he Linear System P roduced by Quadrature, J . Assoc. Compo Mach. 10,
9 7101 (1963).
9. Twomey, S., T he Application o f N umerical Filtering t o t he Solution o f Integral Equations
Encountered in Indirect Sensing Measurements, J . Franklin Inst. 179, 95 109 (1965).
T M=To+
I Vqn4>Mn+1>
11=1
T M=To+
I qnA4>Mft
n =1
a nd
V qn=qnqftt
M
a re equivalent where M is a n arbitrary positive integer.
5 .4. A plate ~f thickne~s 2 L initially a t the constant temperature o f To is
d ropped Into a flUId a t Teo. T he boundary conditions for onehalf the
plate can be given by
oT
ox
 =0 a t x =O
'
L~x~L,

k
aTI
ox
L = h[T(L,
t ) Teo ]
where h is the heat transfer coefficient.
a.
Using the solution o f t he problem for a step change in the surface
heat flux a t x = L a nd insulated a t x = 0, d enoted cJ)(x, t), derive
T (L, t) = To +
f~ h (Teo 
T(L, A)]
OcJ)(L~: 
A) dA
where h is assumed constant. (How is </>(x, t) related to cJ)(x, t)?)
(a)
214
C HAP . 5
b.
INVERSE CONVOLUTION FOR A SINGLE SURFACE HEAT FLUX
E quation (a) can be considered an integral equation for the unknown function T IL, t). By using exact matching, derive the following:
215
PROBLEMS
show that
q =qo+oq
where qo is t he solution with OY,.=O a nd hq is the solution in Problem
5.7. U nder what conditions does a bounded value o f h Y,. result in a n
u nbounded response in qj? Show t hat your answer is equivalent to
tPz > 3tP,
c.
where TM = T (L, tM)'
T he foregoing analysis is not restricted to a flat plate. Describe
some other geometries for which this analysis applies. This problem
is related to the surface element method.'  J
5 .5. Modify the analysis o f Problem 5.4 to treat the case of T... = T... (t).
5 .6. S tarting with Eq. (a) of Problem 5.4, derive a sequential function
specification algorithm for estimating Tx ( t) given the temperature
history T (L, t). Use r = 2 and assume that Tx (t) is temporarily constant.
5 .7. T he Stolz method can be investigated by considering a single temperature error, bY,., a nd a special sensitivity matrix. The Stolz method can
be represented by the set of n algebraic equations
a
b
b
b
b
a
b q.
5.10. Repeat Problem 5.9 for the center point in a sphere. Use Table 1.4.
Answer part (a), At+ ::::0.19.
5.11. Write a computer program for Eq. (5.3.4). Let r be variable from 1 to 5
Use the program developed in Problem 5.11 to verify the results
o f Figure 5.7.
b. Use the program developed in Problem 5.11 to verify the results
o f Figure 5.10.
c. Use the program developed in Problem 5.11 to verify the results
o f Figure 5.12.
0
where by" is the r th entry in the right vector. Show t hat
5.13. a.
b Y,.,
b ' y.
u q.=  , o q.+,= 2 0.
a
a
<
b qj=(I
~)6.j"
j =r+2, r +3, . ..
5.15. Solve Problem 5.13 for a solid sphere with the sensor at the center.
i =r+l.r+2, . . .
5 .8. This problem uses the results of Problem 5.7. F or the right side of the
matrix equation equal to Y where
Y =[Y, Y2
. ..
Y,.+c'iY.
b.
c.
Use the program developed in Problem 5.11 t o investigate the
stability o f t he function specification method for the flat plate with
the sensor at the insulated surface. Let r = 2.
Repeat p art (a) for r = 3.
Repeat part (a) for r =4.
5.14. Solve Problem 5.13 for a solid cylinder with the sensor at the center.
Using these expressions, show that
b ( b)j ,''
b qj= a2bY• 1  u
1'1 a nd ,pI values are
5.12. a.
bY,.
...
Using the criterion for stability for the Stolz method. ,pz > 3,p"
given in Problem 5.8, find the stability limit for a solid cylinder with
the sensor a t the cl!nter. Use Table 1.3.
Answer: At+ ::::0.23.
b. F ind the stability limit for the Stolz m ethod for the same geometry
as in part (a) by using the temperature impulse test case.
c. Give conclusions regarding your work for parts (a) a nd (b).
5.9. a.
a nd make the maximum value of M be 40. T he
inputs.
0
0
hq,
bq:
a
ba
bba
bbb
when tPl+' ,pI ::::constant for i = 1.,; • .... Use this criterion to find the
stability limit for the Stolz procedure for the finite plate of Table 1.1
a nd the semiinfinite body of Table 1.2.
y.+,
y .]
5.16. Write a computer program for Eq. (5.3.4a). The K j values are inputs.
Write the program t o t reat r =3 a nd solve for the flat plate geometry
with the sensor at x =L a nd heated at x = 0. T he t imestepsareAr+ =0.05.
Investigate the effect of arbitrary K j values by modifying those that are
obtained for the function specification method which are K , =0.292,
216
C HAP.5
INVERSE CONVOLUTION FOR A SINGLE SURFACE HEAT FLUX
PROBLEMS
time
Consider the heat impulse test to check for conservation of energy.
W hat conclusions can you draw?
5.17. Repeat Problem 5.16 for the temperature impulse test case.
5 .18. Write a computer program for the function specification method for
the I HCP with q temporarily constant for r = 3 and two interior temperature histories.
5 .19. a . Calculate the dimensionless components of XTX for n = 10 and
at + = 0.05 for the flat plate case with the sensor a t x = L and the plate
heated a t x =O. Also display the components for XTX+cxHfH.
a nd for cx=O.OOI. P lot several rows for XTX a nd XTX+cxHfH.,
including the last row, versus the column numbers. What conclusions can you draw from your results?
b. Repeat part (a) for the case o f a solid sphere with the sensor at the
center. Use at; =0.05.
5 .20. a.
Write a computer program for the whole domain firstorder
regularization method for the I HCP .
b. Use the program to reproduce the results shown in Figure 5.15 .
c. Use the program to reproduce the results shown in Figure 5.16.
d. Use the program to reproduce Figure 5.17.
5 21. Modify the program developed in Problem 5.20 t o solve the zerothorder regularization method and repeat Problem 5.20.
5 .22. Write a computer program for the zerothorder sequential regularization method for r =5 a nd compare the gain coefficients with those
obtained using the function specification method for at + = 0.05 for a
flat plate with the sensor at x =L a nd heated a t x =O. (Use Table 4.1.)
Vary cx over a large range of values.
5 .23. An experiment for a flat plate heated at x = 0 and insulated at x = 1 in.
is discussed in Reference 6, p . 402. The plate is Armco iron with k =43 .3
Btu/hrftF and p c= 55 .6 Btu/ft 3 F. The initial temperature is 81.36"F.
Calculate the surface heat flux using the function specification method
and the following actual data.
 .j
time
0
1.8
3.6
F
Temp., O
time
Temp., O
F
81.36
81.52
81.36
16.2
18.0
19.8
85 .85
86.91
88.05
a.
b.
c.
Temp., O
F
time
Temp., OF
5.4
7.2
9.0
10.8
12.6
14.4
K2 = 8.56, a nd K3 = 31.8 . F or example, try
a. K l =0.292, K 2=8.56, and K 3= 15.9
b. K l =0.292, K 2=8.56, a nd K 3=47.4
c. K l = 0, K2=O, and K 3=31.8

2 17
81.36
81.85
82.42
83.48
84.48
85 .28
21.6
23.4
25.2
27.0
28.8
88.79
89.69
89.77
89 .93
90.09
Use r =3.
Use r =4.
Use r =5.
5 .24. Repeat Problem 5.23 using the whole domain firstorder regularization
method. Select several values of cx.
5 .25. Repeat Problem 5.23 using the sequential firstorder regularization
method with r = 5 and several cx values.
5.26. a.
F ind the digital filter coefficients for Problem 5.23 for the function
specification method for r =4 .
b. Find the digital filter coefficients for Problem 5.23 for the whole
domain regularization method. Select an optimal oc value. T he·
midrange o f times is to be used to obtain the filter coefficients.
5.27. Using a computer program for the sequential regularization method
investigate the effect of the regularization order. Consider the zeroth,
first and second orders. Let cx vary over a wide range. Investigate both
the deterministic and random errors.
5 .28. F or the whole domain regularization method generate a figure
analogous to Figure 5.22. Consider values o f cx= 1 0 6 ,0.001, and 1.
5 .29. F or the whole domain regularization method generate a figure
analogous to Figure 5.23. Consider values of cx= 1 0 6 ,0.001, and 1.