5.2 Continuum Modeling of Pebble Radiation
257
Fig. 5.14 Effective thermal conductivity of pebble-bed experiments in central region and the whole
bed
5.2.2.5 2D Application and Validation by TF-PBEC and HTTU
Experiments
A two-dimensional (2D) Approximation Function Model (AFM) with both radiative
and conductive heat transfers is formulated as
πσ ε r ρ 0 d
2
T
4 (x, y) −
R 2
K 2 (x − x
, y − y
)T
4 (x
, y
)d A
= k c ∇
2 T (x, y) + Q v (x, y),
(5.64)
where d A
= dx
dy
. It is a nonlinear integro-differential equation and the 2D kernel
function is
K 2 (x, y) = K 2 (ρ) =
∞
−∞
K (x, y, z)dz =
∞
−∞
K (
ρ 2 + z 2 )dz.
(5.65)
In the experiment of packed beds [5, 21], the axial and azimuthal temperature gradients are neglected. The governing equation of radial temperature distribution in
cylindrical coordinate for the packed bed is formulated as
k c
d
2 T (ρ)
dρ 2 +
1
ρ
dT (ρ)
dρ
= πσ ˆ
ε r ρ 0 d
2
(T
4
(ρ) − I (ρ)),
I (ρ) =
2π
0
∞
0
K 2 (
ρ 2 − 2ρρ cos θ + ρ 2 )T
4
(ρ
)ρ
dρ
dθ.
(5.66)