254
5 Numerical Models for Pebble-Bed Heat Transfer
Fig. 5.11 Radial temperature profile in a spherical packed bed with uniform heat source
where p(x) = T
4
(x), κ(x, x
) = K (|x − x
|), and = R
3 . It indicates that the local
temperature T
4
(x) is determined by all possible surrounding particles weighted by
the kernel function. For heat conduction, it is p(x) = T (x) and the integral domain
is B(x, r ), which is a spherical domain centered at x of radius r . The kernel
function is a constant, i.e., κ(x, x
) =
1
V B
and V B is the volume of the domain. The
approximation function model becomes
T (x) =
1
V B
B(x,r )
T (x
)d x
=
3
4πr 3
B(x,r )
T (x
)d x
(5.56)
It is the mean value theorem of the Laplace equation [33]. Thus, temperature T (x)
is the mean of neighboring ones inside the sphere in the steady conduction.
5.2.2.4 One-Dimensional Application and Validation
For one-dimensional (1D) case, the Approximation Function Model (AFM) of thermal radiation is
p(x) =
+∞
−∞
K 1 (x − x
) p(x
)dx
+
Q v (x)
πσ ε r ρ 0 d 2
(5.57)
where p(x) = T
4
(x). The one-dimensional kernel function is K 1 (x, x
) =
+∞
−∞ K (x, y, z)dydz, which is shown in Fig. 5.12 at α f = 0.39, and K 1 (−x) =