3.3 Effective Thermal Diffusivity and Conductivity
141
problem in this experiment is the heat conduction diffusion equation derived from
Eq. (3.40) to Eq. (3.43), as follows:
1
r
∂
∂r
r α(T )
∂ T
∂r
=
∂ T
∂t
, r ∈ [R in , R out ], t ∈ [0, +∞),
(3.45)
with initial condition and boundary conditions of
I.C.: T (r, 0) = I (r ),
B.C.: T (R in , t) = f in (t), T (R out , t) = f out (t),
(3.46)
where f in and f out are boundary temperature functions of R in and R out , respectively.
I (r ) is the initial temperature distribution. Here, the initial condition and boundary
conditions are given by experimental data. If the α is given in advance, the direct
problem can be solved readily through the finite volume method. The time step and
grid independence should be checked in advance. Therefore, computed temperatures
of a pebble bed can be obtained by the direct problem with given conditions and
physical properties
3.3.2.3 Objective Function of IHCP
In the inverse method, an objective function is a criterion to estimate error between
computed and experimental temperatures, and it is also used to verify whether the estimated parameters are close to the real parameters. Generally, the objective function
S( P) is a least squares method with an independent variable P (Eqs. (3.11)–(3.12)).
In this experiment, since there are six thermocouples in the radial direction, the
middle sensors in the inverse method can be chosen flexibly. It should be noted
that two boundary positions, namely R in and R out , are used to calculate the direct
problem, and the positions of middle sensors are used to interpolate for the T ( P) at
each step of the direct problem computation.
3.3.2.4 Optimization Method of IHCP
The optimization method of IHCP is used to minimize the nonlinear objective function S, and the optimal variable P is regarded as the estimated parameter closest
to real parameters. The Levenberg-Marquardt (LM) method is used to search the
optimal P in this study. The LM method was first derived by Levenberg [8] and
Marquardt [9], by modifying the ordinary least squares method with a combination
of the Gauss and Steepest Descent methods. The LM method repeats the iteration to
search an optimal parameter for the objective function S( P) by Eqs. (3.13–3.15).
In this nonlinear optimization problem, the specific iteration control was shown
in the previous paper, and a special integration equivalence principle should be used
141
problem in this experiment is the heat conduction diffusion equation derived from
Eq. (3.40) to Eq. (3.43), as follows:
1
r
∂
∂r
r α(T )
∂ T
∂r
=
∂ T
∂t
, r ∈ [R in , R out ], t ∈ [0, +∞),
(3.45)
with initial condition and boundary conditions of
I.C.: T (r, 0) = I (r ),
B.C.: T (R in , t) = f in (t), T (R out , t) = f out (t),
(3.46)
where f in and f out are boundary temperature functions of R in and R out , respectively.
I (r ) is the initial temperature distribution. Here, the initial condition and boundary
conditions are given by experimental data. If the α is given in advance, the direct
problem can be solved readily through the finite volume method. The time step and
grid independence should be checked in advance. Therefore, computed temperatures
of a pebble bed can be obtained by the direct problem with given conditions and
physical properties
3.3.2.3 Objective Function of IHCP
In the inverse method, an objective function is a criterion to estimate error between
computed and experimental temperatures, and it is also used to verify whether the estimated parameters are close to the real parameters. Generally, the objective function
S( P) is a least squares method with an independent variable P (Eqs. (3.11)–(3.12)).
In this experiment, since there are six thermocouples in the radial direction, the
middle sensors in the inverse method can be chosen flexibly. It should be noted
that two boundary positions, namely R in and R out , are used to calculate the direct
problem, and the positions of middle sensors are used to interpolate for the T ( P) at
each step of the direct problem computation.
3.3.2.4 Optimization Method of IHCP
The optimization method of IHCP is used to minimize the nonlinear objective function S, and the optimal variable P is regarded as the estimated parameter closest
to real parameters. The Levenberg-Marquardt (LM) method is used to search the
optimal P in this study. The LM method was first derived by Levenberg [8] and
Marquardt [9], by modifying the ordinary least squares method with a combination
of the Gauss and Steepest Descent methods. The LM method repeats the iteration to
search an optimal parameter for the objective function S( P) by Eqs. (3.13–3.15).
In this nonlinear optimization problem, the specific iteration control was shown
in the previous paper, and a special integration equivalence principle should be used