5.2 Continuum Modeling of Pebble Radiation
243
at steady state for the conductive or radiative heat transfer. K (x) is the general kernel
function given as
K (x) = K (r ) = ρ
2
0 H (r )g(r ),
(5.22)
where r =
x 2 + y 2 + z 2 .
For conduction, Eq. (5.21) becomes
ρ 0 N c
R 3
H (|x − x |)
4π |x − x |
δ(|x − x | − d)T (x )dx dy dz − ρ 0 H 1 N c T (x) + Q v (x) = 0.
(5.23)
The delta function is approximately given as
δ(x) = δ (x) =
1
, 0 ≤ x ≤
0, else
(5.24)
at → 0. For spherical particle conduction, the governing equation (Eq. (5.21)) is
again reduced to
ρ 0 N c
H 1
4π d 2
|x−x |=d
T (x
)d S
− ρ 0 H 1 N c T (x) + Q v (x) = 0.
(5.25)
The equation without heat source becomes
T (x) =
1
4π d 2
|x−x |=d
T (x
)d S
(5.26)
It is also the result of the mean value theorem of the Laplace equation. Thus, the
discrete conduction in the bulk region of the nuclear pebble bed is equivalent to that
in the continuum of the pseudo-porous media.
On the other hand, for particle radiation, Eq. (5.21) is re-written as
ρ 0 ε r A i σ
R 3
K r (|x − x
|)T
4
(x
)dx
dy
dz
− T
4
(x)
+ Q v (x) = 0, (5.27)
where the radiative kernel function is
K r (x) = K r (r ) =
X (r )g(r )
R 3 X (r )g(r )dxdydz
(5.28)
In particular, if there is no heat source inside the pebbles, the equation is
T
4
(x) =
R 3
K r (|x − x
|)T
4
(x
)dx
dy
dz
(5.29)