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

286
CHAP. 8
O NE-DIMENSIONAL INVERSE H EAT C ONDUCTION PROBLEM
the temperature data, can the X sensitivity coefficient be replaced by
Z in Eq. (6.3.8)? Why?
6 .4. Write an efficient computer subroutine. t o solve systems o f linear
algebraic equations that are tridiagonal in form; see Eqs. (6.3.10) and
(6.3.11).
6 .5. Verify the algebra for &is. (6.3.12) - (6.3.14), the algorithm for calculating
the sensitivity coefficients.
6.6. Calculate the sensitivity coefficients Zl.1 a nd Z l.2 for x 1/L=0.5 and
1 for L1t+ = 0.05 a nd 0.1 respectively and for planar geometry with a
perfectly insulated surface. Use the pure implicit method. Compare your
results with the tabular values given in Table 1.1.
CHAPTER
7
M ULTIPLE H EAT
FLUX E STIMATION
N
6.7. Prove that
I
W1 = 1 in Eq. (6.4.11).
1 =1
6 .S. Repeat Example 6.1 for x 1/L= 1 a nd for M + =0.05 a nd 0.5.
6 .9 . I n t he analysis of Section 6.6, the Z (step) sensitivity coefficient was used
in the Taylor series expansion Eq. (6.6.4). Why Z and not X ?
7.1
I NTRODUCTION
6 .10. Repeat Example 6.2 for L1t+ =0.1.
6 .11. Extend the analysis o f Section 6.9 t o include multiple temperature
sensors.
6 .12. C ompare the computational molecules for all the space marching
methods o f Section 6.10.
6 .13. As a h eat transfer consultant, you have been asked t o comment on the
d ata quality from the following inverse problem : A thermocouple is
located 0.001 m below the surface o f a 0 .005 m thick copper slab. The
known back face b oundary condition is a specified heat flux of 50 kW/
m 2 • T he person describing the experiment to you " thinks" the unknown
heat flux you are trying to estimate is o f the order o f magnitude of
1 kW/m2. Will this experiment yield meaningful information?
In the previous chapters the case o f the single unknown surface heat flux history
was considered. I n this chapter the case o fthe multiple heat flux I HCP is treated.
A multiple heat flux case arises when both surfaces of a one-dimensional
body are exposed t o unknown heat flux histories. See Figure 7.1a for a plate
heated o n b oth sides. The unknown heat flux histories are q1(t) a nd Q2(t).
Figure 7.1b depicts either a hollow cylinder o r sphere. Again there are two
unknown heat flux histories. In both Figures 7.1a a nd 7.1b there must be at
6 .14. a.
F or t he D'Souza procedure, derive a two-node expression for the
surface heat flux for a solid cylinder with the sensor a t the center.
Derive the difference equations from an energy balance.
b. Repeat p art (a) for three nodes.
c. C ompare the expressions with that obtained using the exact ex pression given by Burggraf.
(a)
6 .15. Solve Problems 6.14 for a solid sphere.
6 .16. F or the Weber procedure, derive a two-node expression for the surface
f iGURE 7 .1 Some one-dimensional cases o f
bodies exposed t o two unknown heat lIux histories,
q I (t) and qa(/). (a), plate; (b), hollow cylinder o r sphere.
heat flux for a solid cylinder with the sensor a t the center. The wave
speed is infinite. Derive the difference equations from an energy balance.
Compare the equation with the exact expression given by Burggraf.
6 .17. Solve Problem 6.16 for a solid sphere.
(b)
287
268
C AHP.7
MULTIPLE HEAT FLUX ESTIMATION
SEC. 7 .2
289
l WO I NDEPENDENT HEAT FLUXES CASE
(7.2.1)
q (y, t )
The symbol Tlq _o means the calculated temperature vector T with q, given
by Eq. (7.2 .3a), set equal to O [See Eq. (3.2.22).] Fo~ the case oft~o q components
.
at each time, J temperature sensors, a nd r future times, the various components
o fT are
F IGURE 7 .2 Two-dimensional body heated by spaceand time-variable surface heat lIux.
least two temperature sensors that are not a t t he same location. The heat
flows are one-dimensional, and the heat flux histories qt(t) a nd qz(t) a re independent. Hence any time variation in q l(t) has no effect o n q2(t). T hus this
problem is little different from those covered in previous chapters.
I t is n ot necessary that the two heat fluxes are restricted t o one-dimensional
heat flow. F or example, the plate o f Figure 7.1a might not have parallel faces ;
also the radii in Figure 7. 1b might n ot be concentric. Another case would be a
s quare with two adjacent sides each with a different heat flux history.
A more typical multiple heat flux I HCP is for the case o f a body exposed to a
heat ftux that is both space- and time-variable (see Figure 7.2). Thus the temperature distribution is two-dimensional. Any coordinate system can be used and the
analysis is n ot restricted to two-dimensional cases. The body can be composed
o f several materials a nd c an be irregular. The only requirement is that a method
o f solving the direct problem (known surface heat ftux time a nd space variation)
is available.
In this chapter the method o f solving the direct problem is n ot o f interest.
Rather a method of solution is assumed such as finite differences, finite elements,
o r D uhamel's theorem. F or simplicity the problems are considered to be linear
but nonlinear cases can be treated by using the modifications discussed in
C hapter 6.
Few papers o n t he two-dimensional inverse heat conduction problem have
been written. O ne by Bass et at. I gives a description o f a c omputer program
for a solid cylinder with radial a nd a ngular dependence; finite elements were
used. An analytical solution was presented by Imber. 2 Some two-dimensional
inverse problems for which the geometry is found (and thus are not the IHCP)
have been examined by Hsieh and c o-workers.3.4
T he scope o f this chapter is t o discuss the two heat flux case illustrated in
Figure 7.1 a nd t o cover the multiple heat flux case. Both the sequential function
specification and sequential regularization methods are employed. There is
less emphasis o n displaying numerical results than in previous chapters.
7 .2
T WO I NDEPENDENT HEAT FLUXES CASE
T he temperatures in a one-, two-, o r three-dimensional body with temperatureindependent thermal properties can be given in the standard form o f
T(M)
]
T = T (M+l)
[
T (M + r -l)
,T(i)= [TI(i)]
Tz(i)
(7.2.2a,b)
1J(i)
where T is a Jr x 1 matrix; that is, Jr vector,
q=[:Z~I)
q(i)=[::~~G
],
(7.2.3a, b)
q (M+r-l)
where q is a rp x 1 matrix with p = 2 in this case,
[
:~~~
a (l)
X = a~3) a(2)
a(1)
(7 .2.4)
a (r-1) . ..
a(r)
where X is a J r x p r matrix and
a \2(i)]
a21(i)
:
'
(7.2.5a, b)
aJ2(i)
where a li) is a J x p matrix with p =2 in this case. F or the case of p components
of q at each time, q (i) given by Eq. (7 .2.3b) is a vector of p elements, and a li)
given by Eq. (7 .2.5a) is a matrix with p columns. F orthe special case of one sen~r
(J = 1) a nd one heat flux history ( p= I), a 1\(i) is simply Aq, i -I which is used 10
Chapters 3, 4, a nd 5.
The components o f X in the most general sense can be considered to be
sensitivity coefficients. See Eqs. (6.5.18) and (6.5.19). The X components can be
generated using finite differences, Duhamel'S theorem, o r o ther methods.
F or t he first two times (associated with 1M and I M+ I) a partially expanded
form o f Eq. (7 .2.1) is
T(M)=T(M)lq(M) _ o+ a(l)q(M)
T(M + 1 )=T(M + l)q(M) =q(M+ 1 .=o+a(l)q(M + 1 )+ a(2)q(M)
(7.2.6a)
(7 .2.6b)
2 70
C HAP.7
MULTIPLE HEAT FLUX ESTIMATION
Both these matrix equations represent J scalar equations, some of which are
Tl (M)~ T1(M)l f .(M);' " ;4J(M);O + all(l)ql(M)+ alz(l)qz(M)
1 J(M) = 1J(M)lf .(M); . .. ;4J(M);O+aJ1(l)ql(M)+aJ2(l)qz(M)
T1(M + 1 )= T1(M + 1)14.(M); " · ;fJ(M) ; f.(M+ I ) ;
'"
(7.2.7)
; fJ (M+ l );o+a l1 (l)ql(M + I )
+ alz(l)Q2(M + l )+al1(2)ql(M)+a 1 2(2)q2(M)
1 J(M + 1 )= 1 J(M + 1)lf.(M); " · ;fJ(M+ 1);O+aJ1(l)ql(M + l )+an(I)q2(M + I)
+ aJ1(2)ql(M)+ ad2)q2(M)
(7.2.8)
Notice that in the I HCP there are two unknown q components at the first
time ( t M ); namely, q l(M) a nd Q2(M). There are J measurements at that time
and J must be equal to o r greater than the number o f q components (2 in this
case). For the case o f J = 2 it is possible to solve Eq. (7 .2.7) simultaneously
for q l(M) and q 2(M) with 7 ;(M) set equal to Y1(M), and T2(M) equal to Yz(M).
Knowing q l(M) and q z(M), Eq. (7.2.8) can be solved for q l(M + I) and q z(M + I).
Such a procedure is analogous to the Stolz method discussed in Sections 4.3
and 5.3.2. This procedure is usually unacceptable due to its extreme sensitivity
to measurement errors. Instead the function specification and regularization
methods are used in this chapter.
The trial function method of Section 4.6 includes both the function specification and regularization methods. It also allows a smooth transition between
the two methods. The trial function criterion is used here in an effort to unify
the methods but the two methods of function specification and regularization
represent two extreme cases. The trial function criterion given by Eq. (4.6.1)
is written as
s =(y - T)T " ,-I(y - T)+a[H(q-iiWW[H(q-q)]
S EC. 7.2
271
T WO INDEPENDENT HEAT FLUXES CASE
(Section 1.4.2) are valid, the expected value of Eq. (7 .2.12) is equal to the number
o f measurements minus the number o f q components. (See Reference 5, p. 268.)
7.2.1
S equential F unction S pecification M ethod
F or the function specification method the q function is temporarily set equal
to q as indicated by Eq. (7.2.10). The simplest function to specify is t hat o f q
independent of time. This is a t emporary assumption that is used to obtain
q(M). (See Section 4.4.3 .) Using this assumption gives
q (M)=q(M+I)= '" = q(M+r-l)
(7.2.13)
which is analogous to Eq. (4.4.20). T o be more explicit, for q(M) with two
components, E q . (7 .2.13) gives
q l(M)=ql(M + 1 )= . , . = ql(M + r-l)
(7 .2.14a)
(7.2.14b)
See Figure 7.3. The objective is to estimate q l(M) and q z(M) a t the M th time
t--
ql(M-l) - -- -- ---
---r-
-- --:-. , --:- -1'-'
.,
~-
~-
I
:
(7 .2.9)
I
'
,
I
, ',
.
:
I
'
,
I
, ql(M)
,
•
I
where '" is the covariance matrix of the random measurement errors in Y;
a is the regularization parameter. a scalar; H is the matrix for a zeroth o r first
o r other order regularization; and q is the trial function.
F or the function specification method. ex is made so large that
(7.2.10)
and for the regularization method
-------~
q=O
(7.2.11)
r -i-r-
and ex is adjusted to cause (see Sect. 4.5 .3.3)
(Y _ t )T " ,-l(y _ t)~expected value
(7.2.12)
The t vector in Eq. (7 .2.12) is the estimated value o fT as a result o f minimizing
S. When the first four. sixth, and seventh standard statistical assumptions
I --
- -y--r - Ii - T - -,-I
I
I
I
I
I
I
I
I
I
I
I q.2
I
I
I
I
I
I
I
( M)
1
FIGURE 7 .3 Two independent heat flux histories showing the temporary constant heat flux
assumption.
C HAP.1
212
M ULTIPLE H EAT FLUX E STIMATION
'=[::~Z~J
(7.2.22)
(7.2.15)
where
J
A(r)
C ij=
(7.2.16)
T hen using
(7.2.17)
Z =XA
L
=
,= ~ T",-IZ)-IZT",-I(Y-TI,=o)
" ,-I = (1-21
(7.2.18)
J
(7.2.19)
L
[au(1) + aIJ(2)][Yi(M + 1 )-7;(M + l)~.(AI)= " ' =.2(11+ 1 '=0]
Finally q l(M) a nd q2(M) are, in algebraic form,
(7.2.26a)
(7.2.26b)
(7.2.20)
o
EXAMPLE 7.1. F or the case o f two independent heat flux histories, q l(/) a nd q2(/),
use the sequential function estimation procedure with the constant heat flux approximation t o give explicit forms o f the algorithm. Assume that the standard statistical
assumptions apply. Let there be three measurements taken a t a time and use two
future temperature measurement times.
So/ulion. T he e quation t o use is Eq. (7.2.19) with " ,-I canceling, ) =3, a nd r =2. T he
Z matrix is
O ][A(1)]
a(l) A(2)
[a(I)A(1)
]
= a(2)A(1)+ a(1)A(2)
a \l(l)
~
V1
(7.2.25)
i -1
a nd then " ,-I cancels in Eq. (7.2.19).
a dl)
a21(1)
a dl)
a u(1)
a l2(1)+au(2)
a ll(l)+ad2)
a l2(1)+ad2)
a ll(1)
a \l(l)+a\l(2)
a ll (1) + a ll (2)
a ll(l)+all(2)
(7.2 .24)
J
which gives the q(M) vector as indicated by Eq. (7.2.15). After it is obtained, M
is increased by one a nd Eq. (7.2.19) is used again.
I f the first four standard assumptions a re valid. '" simplifies t o
' -..
D2
L a /)(I)[Yi(M)-7;(M)I•• (M)=.,(M)=o
The matrix derivative o f Eq. (7.2.18) with respect t o , yields the estimator
Z -XA- [a(l)
a(2)
=[DI]
where
+
",=(121,
(7 .2.23)
[ a..I(l)+a.,1(2)][a.,il)+a.,j(2)]
1 =1
S =(Y - TI,=o-Z,)T " ,-I(y - TI,=o-Z,)
'2 _
J
a ..ma"'j(1)+
T he ZT(y - TI,=o) matrix is
T
T
Z T(y _ TI =0) [ A(I)a (I)[Y(M)- T(M)I,=o] + A(I)a (2) + A(2)a(1)]
,
(Y(M + 1 )- T(M + 1)1,-0)
Dj =
the function t o minimize with respect t o , is
L
", - I
F or the constant q assumption indicated by Eq. (7.2.l4a,b), the A(i) matrix is
A(i)=[~ ~]
273
T WO INDEPENDENT H EAT FLUXES CASE
T he ZTZ matrix is
step. I n o rder t o permit greater generality, q is set equal t o
q =A" A =[A(lIJ'
SEC. 7.2
(7.2.21)
7 .2.2
S equential R egularization M ethod
T he sequential regularization method can start with the criterion given by
Eq. (7.2.9) with ii set equal to the zero vector. Introducing the model given by
Eq. (7.2.1) for T, taking the matrix derivative with respect to q, a nd setting the
matrix equation equal to zero, gives the matrix normal equation,
EXT " ,-IX+cxHTWH]q=X T. p-I(y - Tlq=o)
(7.2.27)
This represents a set o f pr simultaneous algebraic equations. F or the case o f two
unknown heat flux histories, p = 2 a nd the number o f unknowns is 2r.
I f b oth B a nd i i' a re zero matrices in Eq. (4.6.7), then Eq. (7.2.27) is o btained
because HTWH can represent any o r all o f the H [WiH i terms in Eq. (4.6.7).
T he difference is in the interpretation o f the various terms to allow for both
multiple measurements and also multiple heat flux components a t each time.
In the present case, there are J(~2) measurements a t each time (t M , t M + I , . .. ,
l M+r-l) a nd t here are p =2 c omponents o f t he heat flux at each time. The X
matrix is given by Eqs. (7.2.4, 5) a nd q is given by Eq. (7 .2.3) for p =2.
T he main quantity in Eq. (7.2 .27) t hat has not been discussed is H. As pointed
2 74
C HAP.7
M ULTIPLE H EAT FLUX E STIMATION
o ut there are a number o f forms that it can take. F or a zeroth-order regularization, the simplest form o fH is I . See Eq. (4.5.16a). Such a choice o f" does not
distinguish between independent (or unrelated) heat flux histories such as
q t(t) and qz(t) o f Figure 7.3 o r the continuous ones o f Figure 7.4.
Another H is for the first-order regularization. I t usually involves first
differences in time and n o differences in space; that is, differences are not taken
between qt(t) a nd qz(t). The first differences can be represented by " given by
H{~ -~ -1
!]
S EC.7.3
EXAMPLE 7.2. C onstruct the partitioned matrices for the matrix in the brackets o n the
left o f Eq. (7.2.27) for X with r =2 a nd p =2; ';=1111; a nd W =1. Use first-order time
differences.
Solution.
a nd t hen
(7·=1
T he H 1WH m atrix becomes
(7.2.29)
H TWH=HTH=[-:
H~~H~~l-~
0
-I
21
-I
:I-~ ~]=[-:
- :]
a nd thus
T.I.-I
TWU [ [aT(l)a(l)+ a T
(2)a(2)]11- 1 +d a T(2)a(1)u-1-d]
X .,. X+ocH
=T
a (1)a(2)u-1-d
aT
(1)a(1)u-l+d
HTWH for r = 4 becomes
-I
21
-I
0
F or r =2, t he X matrix from Eq. (7.2.4) is
. [ a(1) 0 ]
X = a(2) a(1)
which is for r =4 a nd 1 has the dimensions o f p x p which is 2 x 2 in the present
case. F or the case o f
W =I
2 76
M ULTIPLE H EAT FLUX CASE
-!]
(7.2.30)
N otice t hat t he effect o f t he regularization procedure is t o increase the diagonal terms
and decrease the off-diagonal terms. This results in improved conditioning o f t he set o f
e quations-it becomes less sensitive t o measurement errors.
0
where for p =2 [i.e., qt(t) a nd qz(t)], 1 is
I=[~ ~J
7.3
M ULTIPLE H EAT FLUX CASE
(7.2.31)
F or this case o f two heat flux histories, p =2, and there are four future time
steps, that is, r =4; there are p r=8 unknown components o f the heat flux. In
the sequential procedure only the first components o f q t (t) a nd qz(t) are retained,
namely q t(M) a nd qz(M). Then the next time step is begun with M increased
by one.
F or the case o f a heat flux t hat is a function o f b oth time and position as shown
in Figure 7.2, the variation over the surface is controlled in a manner similar to
that for time. There are many ways to accomplish this. In this section examples
are given for both the sequential function specification and sequential regularization methods.
In Chapter 1 sensitivity coefficients were plotted for both one-dimensional
and two-dimensional cases. T o determine the heat flux qM for the onedimensional case it was found that a few future time steps were needed. Measurements over the whole time domain were not needed. This point was also discussed in Chapter 5 with reference to the whole domain regularization method.
See Section 5.4. This characteristic o f needing only a limited number of future
time steps is a consequence o fthe nature o fthe heat conduction equation which
for constant thermal properties and rectangular coordinates is
iPT aZT)
k ( axl + ayZ = pc
FIGURE 7 .4
Multiple heat ftux function specification case.
aT
at
(7.3.1)
There is a first derivative with time. There is no propagation backward in time.
Any change in the surface heating condition at any time t greater than tM does
not influence the temperatures for t less than tM' This is illustrated by the
278
C HAP.7
MULTIPLE HEAT FLUX ESTIMATION
sensitivity coefficients for various one-dimensional cases in Figures 1.10- 1.14.
T he sensitivity coefficient for the heat flux component qM (which is constant
between 1M - I and 1M , and zero otherwise) is zero for time I less than 1M ,
T he character o f Eq. (7 .3.1) for a heat flux variation across the surface in the
y-direction (see Figure 7.2) is different from that for a time variation. Notice
th~ second ~erivative in Eq. (7.3.1) with respect to y. T he temperature at any
~01nt y >O IS affected by the heating a t both smaller and larger y values. This is
dlustrated by the sensitivity coefficients shown in Figure 1.17 and implies that
all the surface heat flux components a t a given time must be found simultaneously. F or this reason the estimation procedures for the multiple heat flux
case require a solution of a set of simultaneous, full-matrix equations. (A
full-matrix set o f algebraic equations Ax = b has a full square matrix A whereas
a sp~se set o f equations may have an A matrix which is mainly zeros and a
relatively small number o f nonzero elements. An example is the tridiagonal set
o f equations.)
Bc:cause. o f th.e necessity for treating the complete space variation o f q(r, t),
t~e dimenSionality o f the problem is considerably increased over the case o f a
stngle heat flux I HCP. The importance o f the sequential-in-time algorithms
becomes more apparent also. Suppose that the surface heat flux is divided in
space so that there are 10 spatial components. A discrete variation in time could
easily have 100 time co~ponents. I n a whole domain method alllOoo( = 10 x 100)
components ~ust be simultaneously estimated. This is computationally much
more expe~slve than the seq.uential methods where as few as 10 components
are found simultaneously. This means that only 10 o r so simultaneous equations
need .be solv~ r ather than 1000. Since the number o f the computations in the
solution of Simultaneous, full-matrix equations varies as the number o f the
equations cubed, the reduction in computation time using sequential methods
is very large indeed.
.
S EC.7.3
which can be represented by q(M). The temporary assumption o f q independent
in time is used, which is given by Eq. (7.2.13). With this assumption and the
relationships implied by Figure 7.4, Eqs. (7.2.15) and (7.2.16) are replaced by
[~ll] , IJ=
q=AIJ, A =:
A(r)
7/8
5/8
3/8
A(i)= 1/8
0
0
n
(7.3.2)
P2(M)
P2(M)
0
0
0
0
1/6
1/2
5/6
(7.3.3)
7.3.2 S equential R egularization M ethod f or M ultiple
H eat F luxes
In order to describe clearly the sequential regularization multiple heat flux
method, the minimization criterion is first expressed in a continuous rather than
a discrete form. The continuous form is
1'='111
7.3.1 S equential F unction S pecification M ethod f or
M ultiple H eat F lux C omponents
q l(M), q2(M), . .. , q7(M)
1/8
3/8
5/8
7/8
5/6
1/2
1/6
l,(Ml]
because the q(M) components are related, the A(i) matrix is n ot diagonal as is
Eq. (7.2.16) which is for two unrelated heat fluxes.
The solution proceeds in the same manner as for the case discussed in
Section 7.2.1. T he sequential procedure implied by Eq. (7.2.19) is employed.
s= '111.,-.
The case ~f multiple connected (i.e., continuous) heat flux components can be
trea~ed UStng the function specification method in a manner similar t o t hat in
Section 7.2.1 which is for two unrelated heat flux segments. The method is
described with an example.
. The spatial v~riation o f the surface heat flux can be described by many
different expr~slons; for example, constant segments, Iinear-in-position segments, parabohc segments, and splines. Figure 7.4 illustrates the case o f linear
segm~nts with two segments. Three parameters, P I, P2 a nd P3, a re utilized to
descnbe the two segments; these parameters can change with time. The linear
segments shown in Figure 7.4 represent the heat fluxes at time t . T he linear
segments are divided into seven equal increments o f Ay and t he; are seven q
components. The heat fluxes to be found are
271
MULTIPLE HEAT FLUX CASE
I"
[ Y(x,y,t)-T(x,y,tWdydt+<l'
y eO
1'111.,-.
' III
I" (02q)2
-0
0
0
d ydt
Yt
(7.3.4)
Notice that this expression involves a first derivative of the surface heat flux
with respect to t and also to y ; it is a first-order regularization criterion in
both t a nd y. Fluctuations in the estimated heat flux in both t a nd y are reduced
by the regularization term. The integral in Eq. (7.3 .4) involving the measured
temperatures Y (x, y, t) is over y from zero to L . (Experimental temperatures
may not be uniformly spaced in y, however.)
A summation form analogous to Eq. (7.3.4) is
q
,
S=
J
L L [Yj(M + i-l)-1j(M + i_l)]2
i =1 j =1
, - I J -I
+CX
LL
[ qj+I(M+i)_qj+I(M+i-l)-qj(M+i)+qj(M+i-l)]2
i =1 j = 1
(7.3.5)
2 78
C HAP.1
M ULTIPLE HEAT FLUX ESTIMATION
T he second double summation in Eq. (7 .3.5) represents the cross-derivative
integral in Eq. (7.3.4). The expression given by Eq. (7.3.5) can be generalized
t o the matrix expression given by Eq. (7.2.9) with q set equal to zero. Equation
(7.2 .9) is m ore general since it includes the matrices ", - I a nd W.
T he sequential regularization method for multiple heat fluxes is developed
from Eq. (7.2.9) with q= 0. T he first differences in space a nd time are obtained
by setting H, a p r x p r matrix, equal t o
219
P ROBLEMS
T he HTWH matrix portion o f Eq. (7 .2.27) with W = 1 is
p
-p
-p
2P
-P
H TWH=HTH=
0
-P
2P
-P
0
H =H,H,
H ,=
-I
0
0
(7.3.6)
1
0
-I
0
-I
0
0
1
0
0
-I
0
0
0
H ,=diag[h
h
...
-1
P =hTh=
(7.3.7)
(7.3.8)
where H, a nd H , a re also p r x p r matrices. T he identity matrices in H, have
dimensions o f p x p ; the H , matrix affects the first time difference. I t is the same
as Eq. (7.2.28), with - 1 a long the main diagonal (except the last which is zero)
and 1 o n the diagonal j ust a bove the main diagonal. The H , matrix is p artitioned
into elements such that the only nonzeros are the matrices, h, which are along
the diagonal o f H , . The h matrix is p x p in dimensions a nd simulates first
differences in space,
(7.3 .9)
When Eqs. (7.3.7) and (7.3.8) are used in Eq. (7.3.6), H becomes
H = [ -;
o
o
....
-:
~
..
-h
:J
h
00
-1
2
-1
-P
P
0
-1
2
-1
0
h]
2P
-P
(7.3.11)
where P is
0
0
0
-P
-1
2
-1
(7.3.12)
-1
The structures o f HTH and P are similar. See also Eq. (4.5 .23). They are tridiagonal with diagonal, positive, a nd e qual elements except the first a nd last
terms are one-half in magnitude. The off-diagonal terms are all negative a nd
equal. Except for the first a nd last rows, the rows represent the coefficients for
second differences.
REFERENCES
Bass, B. R., D rake, J. B., a nd O tt, L., O RMDlM : A Finite Element F ormulation for TwoDimensional Nonlinear Inverse H eat C onduction Analysis, N U REG/CR-I 709, O RNL/
N UREG/CSDfTM-17, U.S. Nuclear Regulatory Commission, Washington, D.C., December
1980.
2. Imber, M., Two-Dimcnsional Inverse Conduction P roblem-Further O bservations, A IAA J.
13, 114-115 (1975).
3. Hsieh, C. K. a nd Su, K . c., A Mcthodology o f Predicting Cavity Geometry Based o n the Scanned Surface Temperature D ata- Prescribed Surface Temperature a t t he Cavity Side, J . Heat
Transfer 101, 324- 329 (1980).
4. Hsieh, C. K. a nd Su, K. C. A M ethodology o f Predicting Cavity Geometry Based o n t he Scanned Surface T emperature D ata- Prescribed H cat Flux a t t he Cavity side, J . Heat Transfer
1 03,42- 46 (1981).
5. Beck, J. V. a nd Arnold, K. J ., Parameter Estimation in Engineering and Science, Wiley, New
Y ork,1977.
I.
P ROBLEMS
(7.3.10)
7 .1.
Use t he sequential function specification method for two heat fluxes
and r = 3 to obtain algorithms for I lt(M) a nd q l(M) in the gain function
280
C HAP.7
M ULTIPLE H EAT F LUX E STIMATION
form. Give the gain coef1icients K I j a nd K 2j ( forj= I , 2. 3) in terms o f t he
influence functions cPj; w herej = I. 2 refers to the sensors a nd; = I. 2. ....
M + r-I refers to time.
7 .2. W rite a c omputer p rogram for the algorithms o f P roblem 7.1.
7 .3. a.
b.
7 .4. a.
b.
Using exact d ata from Table 1.1 a t x /L = 0 a nd I with dimensionless
time steps o f 0.01, lind estimates o f t he heat fluxes at x /L=O a nd I
using the c omputer p rogram for Problem 7.1.
Repeat part (a) for the sensors a t x /L=0.25 a nd 0.75. Also solve
p arts for i ll + = I.
C HAPTER
8
H EAT TRANSFER
C OEFFICIENT E STIMATION
F or P roblem 7.1 d erive digital lilter algorithms to o btain e stimates
o f q l(M) a nd ( /2(M). Let the tilter coefticients be .Ij;. j = I. 2; ; =
- 2. - 1.0.1 . ....
O btain n umeral values for the filter coefficients• .Ij;. for the flat
plate o f T able 1.1. Use i ll + = 0.05 a nd s ensors a t x =O a nd L.
7 .5. Use the z eroth-order s equential regularization m ethod with r =3 to
~erive a lgorithms for estimating two heat flux histories. q l(M) a nd
2
( Xu
be a n i nput.
Q 2(M). Let
7 .6. W rite a c omputer p rogram for the algorithms o f P roblem 7.5.
7.7.
a.
b.
Using the exact d ata from Table 4.1 a tx/L=Oand I w ithill+=O.OI.
find estimates o f t he h eat fluxes a t x /L=O a nd l ;use t he c omputer
p rogram from Problem 7.5.
Repeat p art (a) for the sensors a t x /L=0.25 a nd 0.75.
8.1
I NTRODUCTION
T he e stimation o f t he heat transfer coefficient, h, from transient temperature
measurements has aspects o f b oth t he inverse heat conduction problem a nd
p arameter e stimation.
An example o f t he t reatment a s a n I HCP is t hat o f a o ne-dimensional case
with a known ambient temperature, Too(t), such a s t he transient determination
o f boiling heat transfer coefficients using an initially h ot spherical c opper solid
suddenly immersed in water a t its saturation temperature. F rom t ransient
temperatures measured inside o r a t t he surface o f t he copper body, the methods
o f t he I HCP c an be used t o e stimate the surface heat flux, qM, a nd t he surface
temperature, tOM; t he definition o f t he heat transfer coefficient, h, c an be used
t o o btain t he estimate o f h given by
~M
qM
TooM -O.5(tOM + o•M
t
(8.1.1)
- 1)
In this expression tOM is t he estimated surface temperature a t time t M; qM, TooM,
a nd ~M a re usually most accurately evaluated a t t M - 1/ 2 •
An example o f a case t hat c an be treated a s a p arameter e stimation problem
is a flat plate over which a fluid is flowing a t a c onstant t emperature, Too; see
Figure 8.1. If t he plate is s uddenly heated by some electric heaters inside the
plate, the plate temperature begins to rise a nd t he heat transfer coefficient is a
s trong function o f p osition from the leading edge o fthe plate. I n s ome cases the
time variation is small a nd t he basic form o f h is a function o f x; t hat is, h = hex),
is k nown, such as
(8.1.2)
281