Estimation of Boundary Heat Flux with Conjugate …
349
of final time. This visible pitfall of the method can be easily removed by using a final
time considered as large compared to interested time due to this the deviation is not
present in output at final time for particular initial value.
4.3 Stopping Criteria
The algorithm will terminate when value of an objective function S
ˆ
q
i+1
(t)
will be
less than ε. Thus, the stopping criterion for current case is [16]:
S
ˆ
q
i+1
(t)
< ε
(12)
where ε is a small number, and for current case, its value (ε) is taken as 10
-9 .
5 Computational Procedure for CGM
The step by step procedure for CGM with the adjoint problem is outlined as given
below. Suppose we initiate iteration with guess q
0 (t). For i = 0 and after that:
Step 1. From q
i (t), solve the Eq. (1a) for obtaining temperature distribution.
Step 2. From Y (t) and T (x meas , t), check the stopping criterion from the Eq. (12).
Continue the next iteration if stopping criterion is not satisfied
Step 3. After getting Y (t) and T (x meas , t), calculate the adjoint problem from the
Eq. (9) and calculate λ(0, t).
Step 4. After getting λ(0, t), calculate the S
[q(t)] from the Eq. (11).
Step 5. After getting the gradient S
[q(t)], calculate the ϒ
i from the Eq. (3c) and
the d
i
(t) from the Eq. (3b).
Step 6. Put q = d
i
(t) in sensitivity problem and calculate the sensitivity problem
from Eq. (4) to get T [x meas , t;d
i
(t)].
Step 7. After getting T [x meas , t;d
i
(t)], calculate the β
i from the Eq. (6).
Step 8. After getting β
i and the d
i
(t), calculate the next estimate q
i+1 (t) from the
Eq. (3a), and again start with first step.
6 Experimental Setup Description and Results
The experimental setup is shown in Fig. 1 by schematic diagram. The experiment was
performed at the Institute for Plasma Research (IPR) Bhat, Gandhinagar. The test
facility was originally developed at IPR to measure the effective thermal conductivity
of lithium metatitanate pebble beds [17].
The SS rod of length 40 mm and diameter 45 mm is used in this experiment.
For maintaining the one-dimensional heat conduction, the SS rod is insulated by the
349
of final time. This visible pitfall of the method can be easily removed by using a final
time considered as large compared to interested time due to this the deviation is not
present in output at final time for particular initial value.
4.3 Stopping Criteria
The algorithm will terminate when value of an objective function S
ˆ
q
i+1
(t)
will be
less than ε. Thus, the stopping criterion for current case is [16]:
S
ˆ
q
i+1
(t)
< ε
(12)
where ε is a small number, and for current case, its value (ε) is taken as 10
-9 .
5 Computational Procedure for CGM
The step by step procedure for CGM with the adjoint problem is outlined as given
below. Suppose we initiate iteration with guess q
0 (t). For i = 0 and after that:
Step 1. From q
i (t), solve the Eq. (1a) for obtaining temperature distribution.
Step 2. From Y (t) and T (x meas , t), check the stopping criterion from the Eq. (12).
Continue the next iteration if stopping criterion is not satisfied
Step 3. After getting Y (t) and T (x meas , t), calculate the adjoint problem from the
Eq. (9) and calculate λ(0, t).
Step 4. After getting λ(0, t), calculate the S
[q(t)] from the Eq. (11).
Step 5. After getting the gradient S
[q(t)], calculate the ϒ
i from the Eq. (3c) and
the d
i
(t) from the Eq. (3b).
Step 6. Put q = d
i
(t) in sensitivity problem and calculate the sensitivity problem
from Eq. (4) to get T [x meas , t;d
i
(t)].
Step 7. After getting T [x meas , t;d
i
(t)], calculate the β
i from the Eq. (6).
Step 8. After getting β
i and the d
i
(t), calculate the next estimate q
i+1 (t) from the
Eq. (3a), and again start with first step.
6 Experimental Setup Description and Results
The experimental setup is shown in Fig. 1 by schematic diagram. The experiment was
performed at the Institute for Plasma Research (IPR) Bhat, Gandhinagar. The test
facility was originally developed at IPR to measure the effective thermal conductivity
of lithium metatitanate pebble beds [17].
The SS rod of length 40 mm and diameter 45 mm is used in this experiment.
For maintaining the one-dimensional heat conduction, the SS rod is insulated by the