Inverse Heat Conduction Problem Applied to a High Pressure Turbine
Vane
David Wasserman
MEAE 6630
Conduction Heat Transfer
Prof. E. Gutierrez-Miravete
4/6/00
Introduction and Goals
Inverse problems are encountered in many branches of engineering and science.
In one particular branch, heat transfer, the inverse problem can be used to estimate such
conditions as temperature or surface heat flux, or can be used to determine important
thermal properties such as the thermal conductivity or heat capacity of solids. This can
be useful for certain applications where the direct measurement of the heat flux would be
difficult or impossible due to extremely high temperatures. This type of situation can be
encountered at the surface of a space reentry vehicle or in a gas turbine engine
combustion chamber or high-pressure turbine. It is important to know temperatures and
heat fluxes because one would want to know if there is a danger of approaching the
maximum operating temperature of the solid. In this project, I will be analyzing a simple
one-dimensional model of a high-pressure gas turbine vane platform, with constant
thermal properties and no heat generation within the metal. One side of the platform is
exposed to gaspath hot air and the other side is exposed to cooling air bled off of the
compressor. A sensor at the cool surface of the metal measures temperatures over a
range of time. The goal is to use the measured temperatures and the inverse analysis to
estimate the amount of heat flux required on the cool side of the platform to keep the
metal on the gaspath side below the maximum operating temperature in order to avoid
microstructural or possibly melting. Now that the problem has been stated, let us get into
the mathematics of it.
Problem Formulation
The direct formulation of the heat conduction problem is given as such:
 2T 1 T

x 2  t
T
 qt 
1)  k
x
T  Thot  Tmax
T  Tmax
in
0 x L
t0
@
x0
t0
@
for
xL
t0
t0
0 xL
In the direct formulation the surface heat flux at x = 0 is considered known. In the
inverse problem, the surface heat flux is considered unknown. The inverse formulation
of the problem is given as such:
 2T 1 T

x 2  t
T
 qt   ?
2)  k
x
T  Thot  Tmax
T  Tmax
in
0 x L
@
x0
@
for
xL
t0
t0
0 xL
Here is a picture of what we are talking about.
-k (dT/dx) = q(t) = ?
T=Tmax
sensor
x=L
x=0
Figure 1
Method of Solution
The main difficulty encountered when attempting to solve inverse problems is
that the solutions are very sensitive to changes in the input data and thus may not be
unique. This can be a result of measurement and modeling errors. This places the
inverse problem in the class of mathematical problems called the ill-posed problems.
This is because the solution of the inverse problem does not satisfy the general
requirement of existence, uniqueness, and stability with small changes to the input data.
In contrast, direct heat conduction problems are well posed because they satisfy the
requirements of existence, uniqueness, and stability of the solution. Various methods for
solving inverse heat conduction problems have been proposed and executed but many
were unstable or not useful for practical applications. To generate a successful solution
to an inverse problem one generally needs to transform the problem into a well-posed
approximate solution. One good way to do this is to transform the problem into a least
squares problem. This transformation requires that the inverse solution minimize the
least squares norm, rather than make it zero, which guarantees the existence of an inverse
solution.
To solve the inverse problem by this method we require that the estimated
temperatures match the measured temperatures as closely as possible over a specified
time domain. The estimated temperatures are computed from the solution of the direct
problem by using the estimated heat flux components, whereas the measured
temperatures are recorded using a sensor placed at a "strategic" location that will provide
the least measurement error. To ensure optimal matching between the measured and
estimated temperatures we require that the least squares norm is minimized with respect
to each of the unknown heat flux components. Here is the least squares norm modified
by the addition of a zeroth-order regularization term:
M


M
2
ˆ )   Y j  Tˆ j (q
ˆ )     qˆ 2j
3) S (q
j 1
where qˆ  qˆi
j 1
for i  1,2,..., M  and the superscript ^ denotes the estimated values.
The other quantities are defined by
S q̂
= sum of squares
qˆ j  qˆ t j 
= estimated surface heat flux at the boundary
Y j  Y t j 
= measured temperature at surface x = 0, at times tj
Tˆ j q̂ 
= estimated temperature at the surface x = 0 at times t = tj computed by
using estimated heat flux, qˆi , i  1,2,..., M   q̂

= the regularization parameter > 0
In equation 3, the first term is the traditional least squares. The second term is the
zero-order regularization term used to reduce instability or oscillations that are inherent in
the solution of ill-posed problems. If the regularization parameter goes to zero the
solution exhibits oscillatory behavior and becomes unstable if a large number of
parameters are to be estimated. If the regularization parameter is a large value, the
solution is damped and deviates from the exact solution. Studies have shown that a
relatively wide range of alpha star can be used, and depending on the value of the
standard deviation of measurement errors (I will assume zero measurement error in this
project), the optimum value of the regularization parameter ranged from 10E-2 to 10E-4.
For my solution I used the median of the optimum value range, 10E-3.
What we would like to do next is minimize the least squares equation by
differentiating it with respect to each of the unknown heat flux components and setting
the resulting expression equal to zero. Doing so yields:
4)
M T
M
ˆ qˆ 
qˆ
S qˆ 
 2 j
Tˆj qˆ   Y j  2   qˆ j j  0
qˆi
qˆi
qˆi
j 1
j 1


where i = 1,2,…,M and
5)
qˆ j 0

qˆi 1
for i  j
for i  j
Equation 4, can be rearranged as
M
Tˆj qˆ 
qˆ

ˆ
ˆ


Y

T
q


qˆ j j


j
j
qˆi
qˆi
j 1
j 1
M
6)


where i = 1,2,…,M and
7)
Tˆj qˆ  Tˆj qˆ1 , qˆ 2 ,..., qˆ M 

 X ji  sensitivity coefficients wrt q̂i
qˆi
qˆi
Equation 6 can be written in the matrix form as
8) X T Y  T   q
where the vectors are given by
 Tˆ1 
 Y1 
 qˆ1 
 ˆ 
Y 
 qˆ 
T2 
2 


, Y
, q 2
9) T 
  
  
  
ˆ 
 
ˆ 
YM 
qM 
TM 
and the sensitivity matrix X with respect to q is written explicitly as
 T1
 q
 1
T
T  2
10) X  T   q1
q
 
 T
 M
 q1
T1
q2
T2
q2
TM
q2
T1
q M
T2
q M

TM

q M










In this sensitivity matrix the terms above the diagonal must be zero because the
temperatures T̂i calculated at any instant of time ti must be independent of the future
heat fluxes, qˆ j , j  i .
In order to solve equation 8 it is desirable to express it in a more convenient form.
This is achieved by expanding the estimated temperatures in a Taylor series with respect
to an arbitrary value of the heat flux like so:
Tˆ j
qˆ  qˆ0 
ˆk k
k 1 q
M
11) Tˆ j  Tˆ0 j  
If we choose the arbitrary point to be 0, then the equation reduces to
Tˆ j
qˆ
ˆk k
k 1 q
M
12) Tˆ j  
or in matrix form as
13) T 
T
q  Xq
q T
If we substitute equation 12 into equation 6 we get
M Tˆ
M

Tˆ j qˆ  
qˆ
j

ˆ
Y

q


qˆ j j
 j 


k
ˆ k 
qˆi 
qˆi
j 1
k 1 q
j 1
M
14)
The matrix form of this equation is
15) X T Y  Xq    q
which can be rearranged as
16) q  X T X   I  X T Y
1
Equation 16 is the formal solution of the inverse heat conduction problem for the
unknown heat flux over the period 0  t  t f . Once the sensitivity coefficients, the
regularization parameter, and the measured temperatures are available, we can directly
compute the heat flux. Since I already selected the regularization parameter to be 10E-3
and the measured temperatures are known, all that is left is the calculation of the
sensitivity coefficients.
Because the direct heat conduction problem associated with the inverse problem
is linear, we can use Duhamel's theorem to solve the direct problem containing a time
dependent boundary condition. This will let us determine the sensitivity coefficients.
Duhamel's theorem states:
17) T  x, t   
t
 0
q 
  x, t   
d
t
where  x, t  is the solution to the following auxiliary problem (obtained from the direct
problem formulation):
 2 1 

x 2  t

1
18)  k
x
 1
 1
in
0 x L
@
x0
@
for
xL
t0
This auxiliary problem is a 1 dimensional linear transient problem with no heat
generation and nonhomogeneous boundary conditions. The auxiliary problem can be
split into a set of simpler problems containing a nonhomogeneous steady state problem
and a homogeneous transient problem, which can be solved by the method of separation
of variables. The steady state problem is given by:
d 2
 0 in 0  x  L
dx 2
d
19)  k
1 @
x0
dx
 1
@
xL
and the homogeneous problem is given by:
 2 1 

x 2  t

k
0
20)
x
 0
h   x    s  f * x 
in
0 x L
@
x0
@
xL
for t  0,0  x  L
Then, the solution of the original auxiliary problem is determined from
21)  x, t    s x   h ( x, t )
The steady state problem can be integrated twice , with the use of the boundary
conditions, to yield:
22)  s  x   1 
1
L  x 
k
The homogeneous problem is the familiar transient 1-D slab problem, whose solution can
be written immediately as:

23) h x, t    e mt
2
m 0
1
N  m 
X  m , x 
L
x0
X  m , x  f * x dx 
From Table 2-2 on pages 48-49 of the textbook we can see that our homogeneous
problem corresponds to that of case 6. Now we can determine the eigenfunctions, inverse
of the norm, and the eigenvalues, which are respectively:
24) X  m , x   cos  m x,
1
N  m 

2
, cos  m L  0
L
where the eigenvalues, beta, are the positive roots of the third equation. Substituting into
the homogeneous solution we get:
25)  h  x, t  
L
2  m2 t
 1

e
cos  m x  1  1  L  x  cos  m x dx 


x

0
L m 0
 k

If we perform the integrations (which are fairly lengthy and will not be shown here) we
get:
26) h x, t   
27)  m 
2  m2 t cos  m x
e
kL m0
 m2
2m  1 ,
2L
m  0,1,2,...
Thus, the overall solution to the nonhomogeneous auxiliary problem is
28)  x, t   1 

2
1
L  x   2  e mt cos 2 m x
k
kL m0
m
Now that we have solved the auxiliary problem we can return to Duhamel's theorem and
use the auxiliary solution. Duhamel's theorem can be written in the alternative form as:
29)
T  x, t   
t
 0
q 
  x, t   
d

since
30)
  x, t   
  x, t   

t

The integral in equation 29 can be discretized as:
  x, t M  n 1     x, t M  n 


n 1
M
T  x, t M   
31)
  qn  x, t M ( n 1)     x, t M n 
M
n 1
which is written more compactly as:
M
32) TM   qn  M n
n  1,2,..., M
n 1
In matrix form this is written as:
 T1   0
 T   
1
 2 
33)      2
  
    
TM   M 1
0
1


0
0



1
  q1 
 q 
 2 
  
 
  
0  qM 
If we recall the definition of X from equation 10, we can see that the coefficient matrix
must be the sensitivity matrix X. Thus,
 0
 
1

34) X  X ij    2

 
 M 1
where i  i 1  i
0
1


0
0



1






0 
and i   x, ti  . Phi represents the temperature rise in the solid
for a unit step increase in the surface heat flux, and can be computed directly from the
solution to the auxiliary problem, which is evaluated at the sensor location, x = 0, over
the range of times. Now that we have the measured temperatures, the sensitivity
coefficients, and the regularization parameter, we can determine the unknown surface
heat flux vector from equation 16.
Results
The solution to the inverse problem was executed in MATLAB (see Appendix A for the
included code). Originally, I was attempting to analyze the steady state case for the
turbine vane, which I would consider cruise conditions. At cruise the turbine
temperatures remain fairly constant, so there are no transients such as in take-off or
climb. Since steady state is a limiting case for this inverse problem, the analysis is
somewhat different than the procedure given in the textbook, and consequently was too
difficult. As was agreed upon by myself and Professor Gutierrez-Miravete, I analyzed
the transient solution so that the procedure could be followed more easily. I was able to
obtain typical metal temperatures for the hot surface and the cool surface, and thermal
properties, since they vary with temperature, were selected at the average of the hot and
cool metal temperatures. Since temperatures were not available over a range of times, a
linear temperature distribution of Tcold  at  b was assumed. Therefore, if we assume
that the temperature on the cool side of the platform is equal to the hot metal temperature
of 2000 F at the initial time t = 0, and the temperature on the cool side is equal to the
steady state temperature 1200 F at the final temperature, we get an equation for the
temperature on the cool side of the platform as a function of time:
 T  Thot 
t  Thot
35) Tcool   s. s.

t
final


I assumed that steady state temperature was reached after 60 seconds, and that
temperature measurements were taken at four evenly spaced times of 15 seconds. Using
these conditions for the starting and ending times, temperatures, and sensor
measurements, the MATLAB program produced a surface heat flux vector of
116.27
103.35

36) q  
 90.43 


 77.52 
where q is measured in BTU/sq in/sec and the four points correspond to t = 15, 30, 45, 60
seconds at cool side temperatures of 1800, 1600, 1400, and 1200 F respectively (see
Graph 1). The results are somewhat promising. They showed a heat flux vector
decreasing with time. At first this seemed counter-intuitive. I was expecting to see a heat
flux vector that was increasing with time. To determine if this was a problem with my
MATLAB code I solved example 5-3 in the textbook. This problem had a time varying
heat flux of t, which, when plotted, gives a line with a slope of one. I solved the direct
problem in MATLAB, took the temperatures and perturbed them slightly (I used 99% of
the values) for use as the measured temperatures, and solved the inverse problem for the
estimated heat flux. The resulting graph from the inverse solution showed a time varying
heat flux that was parallel to the original heat flux but offset slightly under it (see Graph
2). This proved that my MATLAB program was coded correctly. However, I knew that I
was still getting incorrect results for the project, although they were of the correct trend.
We would expect to see the heat flux vector decrease with time because more heat flux is
needed at the beginning of the problem than at the end. If we plot Temperature vs. x, we
can see that for early times the temperature stays fairly constant along the slab until we
approach the cool side, and then temperature drops off sharply. As time progresses, the
temperature in the slab approaches the straight line representing the linear temperature
distribution at steady state. Since the heat flux is the negative of the derivative of the
temperature with respect to x, the slope of the temperature profile represents the heat
flux. As described here, the slope of the temperature profile at x = 0 (cool side) becomes
less and less negative as we approach the steady state time value. The problem with the
results was that, although the trend was correct, the value of the heat flux obtained at
steady state was incorrect. If we solve the steady state problem given as:
d 2T
 0 in 0  x  L
dx 2
x0
37) T  1200 @
T  2000 @
xL
we get a solution of:
38) T  x  
800
x  1200
L
If we take the first derivative of T with respect to x and multiply by -k to obtain the heat
flux we get:
39) q   k
800
L
which, when substituting in, gives a value of -49.07 BTU/sq in/sec. This is obviously
different from the value obtained from MATLAB. Also, when the input to the problem
was changed (number of points, total time elapsed, etc), the final value for heat flux
remained the same. This check of the steady state value for heat flux tell me that
something is incorrect with my problem formulation. This is probably due to the fact that
I created the measured temperature vector (for lack of any better information) and forced
it to a fixed value at the final time. Also, the measured temperature vector is not
necessarily linear, and might be more of a parabolic or hyberbolic shape. This
assumption of a linear temperature drop could also produce incorrect results.
Graph 1
Graph 2
On another note, the regularization parameter was varied within the suggested
optimum values to determine its effect on the stability of the solution. I found that
varying this parameter had little to no effect on the results. The major factor was the
value used for the final time. If the value was large, as it was for 60 seconds, the
summation terms approached zero very rapidly due to the combined effect of the time
and the squared eigenvalue. Since this is the case, the sensitivity coefficients all
approached zero, except for the ones corresponding to t = 0, which appear on the
diagonal. Since it is desirable to have large, uncorrelated values for the sensitivity
coefficients, this poses a problem. As time was decreased, to determine its effect on the
solution, a significant change was not seen until the final time was one second or lower.
For these small times, the sensitivity coefficient matrix was zero only above the diagonal,
which is correct, but some of the values were small, which is undesirable.
Conclusions
Inverse problems can be very difficult to solve to obtain a good estimation of the
unknown heat flux. Obviously something is wrong with the problem formulation
because of the incorrect value obtained for the steady state heat flux. Also, problems
may have arisen due to the fact that the true temperatures over the time range were not
know but rather estimated linearly. Only the maximum operating temperature and the
steady state temperature on the cool side were truly known. Additionally, the solution
seemed to reach steady state values after only 1 second, which is not true of the real
world situation. Perhaps the heat flux could be estimated more accurately using software
such as ANSYS or PATRAN, or by using a different representation for the measured
temperature vector, such as a parabolic or hyperbolic shape.
Appendix A
%David Wasserman
%MEAE 6630 - Conduction Heat Transfer
%Term Project
%4/2/00
clear
%Define Constants
%k is BTU/sec/inches/F
k=132.5/60/144;
%rho is lb/in^3
rho=.323;
%Cp is BTU/lb/F
Cp=.14;
%L is inches
L=.25;
%a is in^2/sec
a=k/rho/Cp;
%astar is dimensionless regularization parameter
astar=10e-4;
%Temps in F
Th=2000;
Tss=1200;
%time in sec
tf=60;
M=4;
delt=tf/M;
%Create time vector
t=zeros(M+1,1);
for i=2:(M+1)
t(i)=t(i-1)+delt;
end
%Create measured temperatures vector
T=zeros(M,1);
for i=1:M
T(i)=((Tss-Th)/tf)*t(i+1)+Th;
end
%Calculate the phi's necessary for the sensitivity coefficients
%phi is the solution to the auxiliary problem
for i=1:M+1
sum=0;
for m=1:4
beta(m)=(2*(m-1)+1)*pi/2/L;
sum=sum+exp(-a*beta(m)^2*t(i))/beta(m)^2;
end
phi(i)=1+(L/k)-(2*sum/k/L);
end
%Calculate delta phi's
for i=1:M
dphi(i)=phi(i+1)-phi(i);
end
%Create matrix of sensitivity coefficients
X=[];
for i=1:M
for j=1:M
if j>i
X(i,j)=0;
else
X(i,j)=dphi(i-j+1);
end
end
end
%Create Identity matrix
I=eye(M);
%Calculate the unknown heat flux vector
%q is BTU/in^2/sec
A=X'*X+astar.*I;
B=inv(A);
q=B*X'*T