Inverse Heat Conduction Problem using TC deconvolution and IR - - PowerPoint PPT Presentation
Inverse Heat Conduction Problem using TC deconvolution and IR measurements: Application to heat flux estimation in a Tokamak Application to heat flux estimation in a Tokamak JL. Gardarein, J. Gaspar IUSTI Laboratory Provence University JL.
Inverse Heat Conduction Problem using TC deconvolution and IR measurements: Application to heat flux estimation in a Tokamak
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Application to heat flux estimation in a Tokamak
IUSTI Laboratory Provence University
Tutorial 11
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Application to a 1D Inverse Heat Conduction Problem
(If we have time)
Plasma at T ~ 100 millions of degrés
The tokamaks…
The confinement is insuring with 2 magnetic fields : – an axial field produced with the toroidal coils – a poloidal field created with the plasma current
Lawson Critera: (Density) x (T°) x (Confinement Time) > 1021 KeV.s.m-3
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– a poloidal field created with the plasma current The resulting magnetic fields are helicoïdal
Plasma Wall Interaction: Heat flux of about 10MW/m2
Limiteur Divertor
DSMF : Dernière surface magnétique fermée (Last Closed Magnetic Surface) SOL : Scrape off layer
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Heat flux of about 10MW/m2 Heat flux intercepted by the Plasma Facing Component Tore Supra (TS)
Cadarache (France)
Joint European Torus (JET)
Culham (UK)
r(m) 2.4 2.6 2.8 3.0
Z(m)
lignes de champ magnétiques
TC TC côté intérieur côté extérieur
r(m) 2.4 2.6 2.8 3.0
Z(m)
lignes de champ magnétiques
TC TC côté intérieur côté extérieur
Divertor de JET
Type K Facq = 20Hz
Z(m)
lignes de champ magnétiques
TC TC côté intérieur côté extérieur
Z(m)
lignes de champ magnétiques
TC TC côté intérieur côté extérieur
JET Divertor
Resolution ~ 8-10mm Spectral Range: [3-5 µm] Facquisition = 50 Hz Observation of each Divertor’Side
JET : Diagnostics and Components
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r(m) 2.4 2.6 2.8 3.0
r(m) 2.4 2.6 2.8 3.0
300 200 100
Example of Experimental Temperatures
IR data T (°C) x time (s)
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T ( TC (1 cm) time (s)
500 400 300 200 100
Objectives : Heat flux computation on the plasma facing components Why ?
° ° °C)
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Problem
Problems
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We have to use the thermocouple data => Solve an inverse Problem
A simplified version of our problem : 1D, Linear,Semi-Infinite Wall
k = 1 W/mK Cp = 1000 J/kg.m3 ρ = 2500 kg/m3 z
Direct Problem
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Temperature + Noise Direct Problem It’s possible to have the temperature field T(z,t) using:
Knowing the thermal properties, is it possible to estimate the heat flux with a temperature measurement ? k = 1 W/mK Cp = 1000 J/kg.m3 ρ= 2500 kg/m3
z
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e
Inverse Problem
Convolution / Deconvolution Linear Syst. e(t) s(t) INPUT OUTPUT
− − − − + + + + = = = = ⊗ ⊗ ⊗ ⊗ + + + + = = = =
t
d t i e t s t i t e t s t s ) ( ) ( ) ( ) ( ) ( ) ( ) ( τ τ τ
Theory of the Linear System
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− − − − + + + + = = = = ⊗ ⊗ ⊗ ⊗ + + + + = = = = d t i e t s t i t e t s t s ) ( ) ( ) ( ) ( ) ( ) ( ) ( τ τ τ
q(t) ? T(t)
i(t) = Impulse response of the System
) (t δ
t Linear Syst.
i(t)
z
− − − − = = = = ⊗ ⊗ ⊗ ⊗ = = = = − − − −
t
d t i Q t i t Q T t T ) ( ) ( ) ( ) ( ) ( τ τ τ
τ τ τ ∆ − − + − ≈ ∂ − ∂ = − ) ( ) 1 ( ) ( ) ( f F u f F u t t u t i ) (t u
) (t H
t
Convolution / Deconvolution
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= = = = − − − − = = = =
= = = = − − − − − − − − + + + + − − − − = = = = − − − − = = = =
F f f F F f F F
f q u f q f F u f F u T T T
1 1
) ( ) ( )) ( ) 1 ( ( ∆ ∆
τ ∆ ∂t u(t) = Step Response of the System (response to the Heaviside Function)
System
F = Number of step time
∆ ∆ ∆ ∆ ∆ ∆ = ∆ ∆ ∆
− − F F F F
q q q u u u u u u T T T
1 2 1 1 2 1
. . . . . .
F
F = Number of step time
X
Convolution
T
Q
3 1 2 2 1 3 2 1 1 2 1 1
. . u q u q u q T T u q u q T T u q T T ∆ ∆ ∆ ∆ ∆ ∆ + + + + + + + + = = = = − − − − + + + + = = = = − − − − = = = = − − − −
1 ) ( ) 1 ( − − − − ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ − − − − + + + + = = = = F k k u k u uk ∆
= = = = − − − −
= = = =
F f f F F
f q u T
1
) ( ∆ ∆
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T X Q .
1 − − − −
= = = =
Deconvolution
X
T
Q
3 3 2 2 1 1
....... . u q u q u q u q T T
F F F F F
∆ ∆ ∆ ∆ + + + + + + + + + + + + + + + + = = = = − − − −
− − − − − − − − − − − −
The matrix X can be inverted if X is well conditioned
1
) (
−
= X X X K
If K(X) is low => X is well conditioned If K(X) >> 1 => X is ill conditioned
k = 1 W/mK Cp = 1000 J/kg.m3 ρ= 2500 kg/m3 e
z
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− − − − − − − − = = = = ατ ατ π ατ π τ τ 2 2 4 ² exp 2 ) , ( x erfc x x b q x T
We can solve this problem with Excel or Matlab
Deconvolution of the temperature at z=5mm: Application with Excel and Matlab
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=> The matrix is ill conditioned because the problem is ill posed => The solution doesn’t respect the Stability condition: Q is very sensitive to measurement errors contained in deltaT.
T X Q .
1 −
=
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=> Need to use a regularization procedure => Regularization with a penalisation => We choose the Tikhonov operator
Q I T Q X R X J ~ . ~ . ) , , (
2
γ γ + + + + − − − − = = = =
J is the function to minimise R is the regularization operator γ is the regularization parameter
Q ~
2
~ . ) , , ( T Q X R X J − − − − = = = = γ
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T X X X Q . ~
1 t t t
I I
− − − −
+ + + + = = = = γ
heat flux; it is a biased value of Q
Q ~ Q ~
How to choose γ ? => γ is chosen to have the best compromise between an exact solution and a stable solution => Exact solution means that => Stable means that
~ . → → → → − − − − T Q X min ~ → → → → Q
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Application to the tiles of the JET Divertor x
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500 400 300 200 100
IR data T (°C) TC (1 cm) time (s) time (s)
Deduced from Other Diag. To compute with the TC data
Q(x,t) = f(x) . g(t)
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and a spatial shape deduced from the IR data. ⇒ FEM code ⇒ Semi-analytical method
u(x,z,t)
Step response at the TC location ) (t H
t
x z
uTC(t)
) (x f
x
Outer Side Erosion Zone
With IR
Inner Side Carbon Layer
With IR With TC
Without layer modelling
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≠ ≠ ≠ Flux TC
With IR With TC With TC
CFC RTC Carbon Deposit
Sensitive Analysis:
k k k
t Q t Z β β β ∂ ∂ = ) , ( ) (
TC IR
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Req= 3.10-4 m²K/W
(heq = 3.3 kW/m²K)
=> Impossible to identify 4 parameters
Conclusion
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= = = = ατ τ τ 2 2 ) , ( x ierfc b q x T − − − − − − − − = = = = ατ ατ π ατ π τ τ 2 2 4 ² exp 2 ) , ( x erfc x x b q x T
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