3.2 Experimental Facility and Methodology
127
If the effective thermal diffusivity and boundary conditions are known in advance,
the interior temperature of the pebble bed can be calculated through the above method.
In addition, grid dependence should be checked to ensure a sufficiently accurate
solution.
Moreover, according to Eq. (3.5), the relationship between diffusivity and conductivity in a vacuum-atmosphere pebble bed should be
λ = (1 − ε)ρ g c g α.
(3.10)
The average density of a pebble bed is defined as ¯
ρ = (1 − ε)ρ g . The ε, ρ g and c g
are known in advance through other approaches, and λ can be given by Eq. (3.10).
Objective Function
An objective function S is a measurement of the error between calculated and experimental interior temperatures. Therefore, it can be regarded as an error norm. The
optimization method used to estimate the α, namely vector p, depends on the minimization of the objective function S, which means a least squares method [6, 7].
That is,
S( P) =
[Y − T ( P)] T [Y − T ( P)],
(3.11)
[Y − T ( P)]
T
= [Y 1 − T 1 ( P), . . . , Y i − T i ( P), . . . , Y N − T N ( P)]. (3.12)
p is the independent variable of S, and Y and T ( P) are the measured temperatures
by middle-level sensors between the two boundary sensors and the estimated temperatures by a direct problem inside the pebble bed, respectively. It should be noted
that subscript i indicates the measuring time, i = 1, . . . , N . N is the total number
of measuring-time points used in the inverse method.
In Fig. (3.2a), if the middle sensors mean four points T2–T5, correspondingly N
includes four sets of time points data of T2–T5. It can also be chosen as the only
T4 used in Eq. (3.12). The two boundary temperature sensors can also be chosen
flexibly, such as T3 and T5 with one middle measured point T4.
S( P) is bounded and real-positive residual, and it will reach zero if P can retrieve
the practical α with respect to temperature ideally. In fact, it can’t be zero due to the
experimental error generally, and only the global optimal value, a small real value,
in fact, can be obtained.
Optimization Method
For the minimization of S( P), the Levenberg-Marquardt (LM) method is used to
search the optimal value. The LM method was firstly derived by Levenberg [8] and
Marquardt [9], by modifying the ordinary least squares norm with a combination of
the Gauss and Steepest Descent methods. The LM is a local optimization algorithm
that is deterministic essentially. The iteration of P through the LM method can search
an optimal value for S( P), as follows:
P
k+1
= P
k
+ [( J
k
)
T J
k
+ μ
k
Ω
k
]
−1
( J
k
)
T
[Y − T ( P
k
)],
(3.13)
127
If the effective thermal diffusivity and boundary conditions are known in advance,
the interior temperature of the pebble bed can be calculated through the above method.
In addition, grid dependence should be checked to ensure a sufficiently accurate
solution.
Moreover, according to Eq. (3.5), the relationship between diffusivity and conductivity in a vacuum-atmosphere pebble bed should be
λ = (1 − ε)ρ g c g α.
(3.10)
The average density of a pebble bed is defined as ¯
ρ = (1 − ε)ρ g . The ε, ρ g and c g
are known in advance through other approaches, and λ can be given by Eq. (3.10).
Objective Function
An objective function S is a measurement of the error between calculated and experimental interior temperatures. Therefore, it can be regarded as an error norm. The
optimization method used to estimate the α, namely vector p, depends on the minimization of the objective function S, which means a least squares method [6, 7].
That is,
S( P) =
[Y − T ( P)] T [Y − T ( P)],
(3.11)
[Y − T ( P)]
T
= [Y 1 − T 1 ( P), . . . , Y i − T i ( P), . . . , Y N − T N ( P)]. (3.12)
p is the independent variable of S, and Y and T ( P) are the measured temperatures
by middle-level sensors between the two boundary sensors and the estimated temperatures by a direct problem inside the pebble bed, respectively. It should be noted
that subscript i indicates the measuring time, i = 1, . . . , N . N is the total number
of measuring-time points used in the inverse method.
In Fig. (3.2a), if the middle sensors mean four points T2–T5, correspondingly N
includes four sets of time points data of T2–T5. It can also be chosen as the only
T4 used in Eq. (3.12). The two boundary temperature sensors can also be chosen
flexibly, such as T3 and T5 with one middle measured point T4.
S( P) is bounded and real-positive residual, and it will reach zero if P can retrieve
the practical α with respect to temperature ideally. In fact, it can’t be zero due to the
experimental error generally, and only the global optimal value, a small real value,
in fact, can be obtained.
Optimization Method
For the minimization of S( P), the Levenberg-Marquardt (LM) method is used to
search the optimal value. The LM method was firstly derived by Levenberg [8] and
Marquardt [9], by modifying the ordinary least squares norm with a combination of
the Gauss and Steepest Descent methods. The LM is a local optimization algorithm
that is deterministic essentially. The iteration of P through the LM method can search
an optimal value for S( P), as follows:
P
k+1
= P
k
+ [( J
k
)
T J
k
+ μ
k
Ω
k
]
−1
( J
k
)
T
[Y − T ( P
k
)],
(3.13)