5.2 Continuum Modeling of Pebble Radiation
253
cubes) is still in general agreement with the view factor between cubical particles in
a packed bed.
5.2.2.3 Approximation Function Model (AFM)
In the Approximation Function Model (AFM), the governing energy equation for
the bed packed with mono-sized spheres with conduction, convection, and thermal
radiation can be formulated as
ρ b C p f
∂ T (x, t)
∂t
+ ρ b C p f V · ∇T (x, t) = ∇ · (k c ∇T (x, t) − Q r ) + Q v , (5.51)
where ρ b and C p f are the density and specific heat, respectively. V is the fluid
velocity and k c is effective thermal conductivity for particle–particle conduction.
The radiation term from Eqs. (5.37)–(5.39) is re-written as
∇ · Q r = ρ 0 Q r,i = πσ ε r ρ 0 d
2
T
4
(x, t) −
K (|x − x
|)T
4
(x
, t)d x
,
(5.52)
where K (x) is the kernel function given by
K (x, y, z) = K
r
d
=
h
0
(η)g(η)
π d 3
+∞
1
h 0 (η)g(η)η 2 dη
(5.53)
where r =
x 2 + y 2 + z 2 and η =
r
d
.
If only thermal radiation and heat source terms are considered, the 3D energy
equation is reduced to
T
4
(x) =
R 3
K (|x − x
|)T
4
(x
)d x
+
Q v (x)
πσ ε r ρ 0 d 2
(5.54)
For a spherical packed bed of radius R, if the temperature at the outer wall is fixed,
the boundary condition can be given by T (r ) = T w at r > R. Then, the AFM can be
solved by the Monte Carlo Method (MCM) [32, 33]. In this case, the parameters are
R = 1 m, d = 60 mm, α f = 0.39, and ε r = 1. The radial temperature distribution at
T w = 900 K and Q v = 100 W/m
3 is shown in Fig. 5.11, which is in good agreement
with the conduction equation of Eq. (5.40). Thus, the behavior of thermal radiation in
the packed bed is similar as to the conduction in the continuum region. Furthermore,
the general formula for the radiative approximation equation without a heat source
is given by
p(x) =
κ(x, x
) p(x
)d x
(5.55)
253
cubes) is still in general agreement with the view factor between cubical particles in
a packed bed.
5.2.2.3 Approximation Function Model (AFM)
In the Approximation Function Model (AFM), the governing energy equation for
the bed packed with mono-sized spheres with conduction, convection, and thermal
radiation can be formulated as
ρ b C p f
∂ T (x, t)
∂t
+ ρ b C p f V · ∇T (x, t) = ∇ · (k c ∇T (x, t) − Q r ) + Q v , (5.51)
where ρ b and C p f are the density and specific heat, respectively. V is the fluid
velocity and k c is effective thermal conductivity for particle–particle conduction.
The radiation term from Eqs. (5.37)–(5.39) is re-written as
∇ · Q r = ρ 0 Q r,i = πσ ε r ρ 0 d
2
T
4
(x, t) −
K (|x − x
|)T
4
(x
, t)d x
,
(5.52)
where K (x) is the kernel function given by
K (x, y, z) = K
r
d
=
h
0
(η)g(η)
π d 3
+∞
1
h 0 (η)g(η)η 2 dη
(5.53)
where r =
x 2 + y 2 + z 2 and η =
r
d
.
If only thermal radiation and heat source terms are considered, the 3D energy
equation is reduced to
T
4
(x) =
R 3
K (|x − x
|)T
4
(x
)d x
+
Q v (x)
πσ ε r ρ 0 d 2
(5.54)
For a spherical packed bed of radius R, if the temperature at the outer wall is fixed,
the boundary condition can be given by T (r ) = T w at r > R. Then, the AFM can be
solved by the Monte Carlo Method (MCM) [32, 33]. In this case, the parameters are
R = 1 m, d = 60 mm, α f = 0.39, and ε r = 1. The radial temperature distribution at
T w = 900 K and Q v = 100 W/m
3 is shown in Fig. 5.11, which is in good agreement
with the conduction equation of Eq. (5.40). Thus, the behavior of thermal radiation in
the packed bed is similar as to the conduction in the continuum region. Furthermore,
the general formula for the radiative approximation equation without a heat source
is given by
p(x) =
κ(x, x
) p(x
)d x
(5.55)