244
5 Numerical Models for Pebble-Bed Heat Transfer
It means that the radiative heat transfer in the nuclear pebble bed is driven by the
difference of T
4
(x) and the local temperature can be obtained by the mean of all
possible surrounding particles weighted by the radiative kernel function.
5.2.1.3 One-Dimensional Case
In the one-dimensional (1D) case, the continuum equation is
T
n
(x)
+∞
−∞
K 1 (x
)dx
=
+∞
−∞
K 1 (x − x
)T
n
(x
)dx
+ Q v (x),
(5.30)
where the 1D kernel function is
K 1 (x) =
+∞
−∞
+∞
−∞
K
x 2 + y 2 + z 2
dydz
(5.31)
for a thick plate with two walls at x = 0 and x = L. The boundary conditions can
be given as T (x) = T L at x > L and T (x) = T 0 at x < 0. The conductive and radiative kernel functions are shown in Fig. 5.3 at α f = 0.39. In particular, 1D general
conductive kernel function is
K 1,c (x) =
H 1
2d
, −d ≤ x ≤ d
0, else
(5.32)
For example, the packing structure at α f = 0.39 is simulated by the Discrete
Element Method (DEM) (as seen in Fig. 5.4a at L = 1.0 m and d = 0.1 m). The
length in y- and z-axis is 3m and the wall temperatures are fixed at T 0 = 800 K, and
T L = 810 K. Figure 5.4b shows the numerical results obtained by the discrete particle
simulation using Eq. (5.2) with −0.2 ≤ y, z ≤ 0.2, which are in good agreement with
the solution obtained by the current model (herein the integral equation of Eq. (5.30)
is solved). The solutions of the current 1D continuum model at different size factors
are shown in Fig. 5.5. The size factor of the pebble bed is defined as μ = d/L here.
It can be concluded that the conductive or radiative continuum equation converges
to the heat conduction in continuum mechanics at μ 1.
5.2.2 Approximation Function Method
Although radiative heat transfer is significant, it is considerably complicated in the
bed packed with lots of high-temperature particles. It cannot use the traditional
Radiative Transfer Equation (RTE) to predict thermal radiation between particles in
the packed beds efficiently. With a continuum assumption, an Approximation Function Model (AFM) is proposed here. A generic physical equation of the radiation
5 Numerical Models for Pebble-Bed Heat Transfer
It means that the radiative heat transfer in the nuclear pebble bed is driven by the
difference of T
4
(x) and the local temperature can be obtained by the mean of all
possible surrounding particles weighted by the radiative kernel function.
5.2.1.3 One-Dimensional Case
In the one-dimensional (1D) case, the continuum equation is
T
n
(x)
+∞
−∞
K 1 (x
)dx
=
+∞
−∞
K 1 (x − x
)T
n
(x
)dx
+ Q v (x),
(5.30)
where the 1D kernel function is
K 1 (x) =
+∞
−∞
+∞
−∞
K
x 2 + y 2 + z 2
dydz
(5.31)
for a thick plate with two walls at x = 0 and x = L. The boundary conditions can
be given as T (x) = T L at x > L and T (x) = T 0 at x < 0. The conductive and radiative kernel functions are shown in Fig. 5.3 at α f = 0.39. In particular, 1D general
conductive kernel function is
K 1,c (x) =
H 1
2d
, −d ≤ x ≤ d
0, else
(5.32)
For example, the packing structure at α f = 0.39 is simulated by the Discrete
Element Method (DEM) (as seen in Fig. 5.4a at L = 1.0 m and d = 0.1 m). The
length in y- and z-axis is 3m and the wall temperatures are fixed at T 0 = 800 K, and
T L = 810 K. Figure 5.4b shows the numerical results obtained by the discrete particle
simulation using Eq. (5.2) with −0.2 ≤ y, z ≤ 0.2, which are in good agreement with
the solution obtained by the current model (herein the integral equation of Eq. (5.30)
is solved). The solutions of the current 1D continuum model at different size factors
are shown in Fig. 5.5. The size factor of the pebble bed is defined as μ = d/L here.
It can be concluded that the conductive or radiative continuum equation converges
to the heat conduction in continuum mechanics at μ 1.
5.2.2 Approximation Function Method
Although radiative heat transfer is significant, it is considerably complicated in the
bed packed with lots of high-temperature particles. It cannot use the traditional
Radiative Transfer Equation (RTE) to predict thermal radiation between particles in
the packed beds efficiently. With a continuum assumption, an Approximation Function Model (AFM) is proposed here. A generic physical equation of the radiation