# Inverse Heat Conduction Problem (IHCP) Applied to a High by AOrJoU

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Inverse Heat Conduction Problem (IHCP) Applied to a
High Pressure Gas Turbine Vane

David Wasserman
MEAE 6630
Conduction Heat Transfer
Prof. Ernesto Gutierrez-Miravete
•Analyzing the entire geometry too difficult
•Simplify by analyzing as 1-Dimensional Transient Slab Problem
*non-homogeneous boundary conditions
*no heat generation
*no coatings
Problem:
•Determine amount of heat flux necessary to keep vane below maximum operating
temperature of superalloy to avoid microstructural damage
Boundary Conditions and Initial Condition:
•Keep gaspath side fixed at maximum operating temperature, cool side subjected to
heat flux
•Assume cool side is initially at maximum operating temperature before cooling air is
turned on (not true but simplified analysis)
Solution Method:
•Assumed constant thermal properties obtained as the average of properties at
maximum operating temperature and steady state temperature
•Originally wanted to analyze as steady state problem but it was a limiting
case and proved to be too complicated
•Changed to transient analysis
-problem: did not have transient temperature measurements
-assume: cool side starts at maximum operating temperature and
drops linearly to the steady state temperature
•Inverse problems are ill-posed in that the solution does not satisfy requirements of
-existence
-uniqueness
-stability
•Use minimization of least squares norm to guarantee existence of solution

S q          T j q  ˆ
ˆ ˆ
ˆ
                             q
ˆ
M                                  M
 2             T j q   Y j  2  q j j  0
ˆ             
ˆ
qi
ˆ       j 1  qiˆ                         j 1   qi
ˆ

•After some computation and substitution we get the formal solution for IHCP

q  XT X   I XT Y
1

•Alpha* - regularization parameter : helps with stability
•X - matrix of sensitivity coefficients : change in temp wrt change in heat flux
•Y - vector of measured temperatures from sensor at x = 0 surface
•I - identity matrix
•q - estimated heat flux at the surface
•In order to calculate sensitivity coefficients, need to solve auxiliary problem
 2 1 
         in    0 x L
x 2
 t

k     1     @       x0
x
 1       @       xL
 1       for     t0

•Nonhomogeneous auxiliary problem can be split into steady state problem and
homogeneous problem
Steady State                                          Homogeneous B.C.’s
d 2                                                 2 1 
 0 in 0  x  L                                                          in     0 x L
dx 2                                                x 2  t

d                                                 k        0               @        x0
k     1 @    x0                                         x
dx                                                     0                   @        xL
 1    @    xL                              h    x    s  f *  x    for t  0,0  x  L

Solution

2  mt cos  m x
  x, t   1  L  x    e
1                  2

k          kL m 0     m2
Results and Conclusions:
•Inverse problem solved using MATLAB program
•Heat Flux is decreasing with time ---> might be correct trend
-Need large amount of flux at beginning, less at steady state
•Steady state value obtained from program does not agree with steady state hand calc
•Solved ex 5-3 to see if MATLAB program was correct--->Results were reassuring
•Heat flux vector depends on method used to calculate temperature vector
•Experiment with different “measured” temperature vectors
•Substitute heat fluxes back into direct problem to get resulting temperatures as a check
•Possibly try ANSYS

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