300
5 Numerical Models for Pebble-Bed Heat Transfer
and for the Face Center Cubic (FCC) packing (α
f,i = 0.26), it is
Q e,i j =
T j − T i
R t
=
T j − T i
2
√
2
k
i d p
(5.143)
Equations (5.142) and (5.143) are the same as discussed in Bahrami et al. [11].
5.3.10.5 Effect of the Particle Shape
The equivalent structure of the pebble bed of HTR-10 [1] is a cylinder of 1.8 m in
diameter and 1.97 m in height, which is used to study the effect of the particle shape
on the thermal radiation here. Three particle shapes of the same volume are shown in
Fig. 5.52. For shape A, the pebble is a sphere of 60 mm in diameter. For shape B, the
pebble diameters are d 1 = 28.845 mm and d 2 = 57.69 mm. For shape C, the pebble
size of the bonded particle is d 1 = d 2 = d 3 = 41.6 mm. The packing density is 0.61
for the packed beds of the graphite pebbles with shape A, shape B, or shape C.
For calculating the effective thermal conductivity of radiation, the convection
and conduction are neglected here, i.e., Q c,ik = 0, Q f = 0 in Eq. (5.9). The pebbles
in the packed bed are heated by the heat source and cooled by the radiation. The
emissivity of graphite pebbles is set at a constant of 0.81. The top and bottom are
adiabatic, and the outer wall is set at a fixed temperature. The distribution of heat
source is uniform, and k r is almost constant in the bed at a low value of Q s . In the
pseudo-porous media, the radial temperature distribution at a steady state at radial
position r is formulated as
T (r ) = T w +
1 − α f,b
4V p
Q s
k r
(R
2
− r
2
),
(5.144)
where α f,b is the average porosity of the bed. T w is the outer wall temperature. R
and V p are the radius of the outer wall and the particle volume. From the numerical
results at T w = 997 K and Q s = 0.05 W in Fig. 5.53, it is found that the radial particle
temperature distribution is almost linear to r
2 for different particle shapes at steady
Fig. 5.52 Particle shapes of non-overlapping clumped-sphere method in the pebble bed: a Shape A:
d p = 60 mm, b Shape B: d 1 = 28.845 mm, d 2 = 57.69 mm, and c Shape C: d 1 = d 2 = d 3 = 41.6 mm
5 Numerical Models for Pebble-Bed Heat Transfer
and for the Face Center Cubic (FCC) packing (α
f,i = 0.26), it is
Q e,i j =
T j − T i
R t
=
T j − T i
2
√
2
k
i d p
(5.143)
Equations (5.142) and (5.143) are the same as discussed in Bahrami et al. [11].
5.3.10.5 Effect of the Particle Shape
The equivalent structure of the pebble bed of HTR-10 [1] is a cylinder of 1.8 m in
diameter and 1.97 m in height, which is used to study the effect of the particle shape
on the thermal radiation here. Three particle shapes of the same volume are shown in
Fig. 5.52. For shape A, the pebble is a sphere of 60 mm in diameter. For shape B, the
pebble diameters are d 1 = 28.845 mm and d 2 = 57.69 mm. For shape C, the pebble
size of the bonded particle is d 1 = d 2 = d 3 = 41.6 mm. The packing density is 0.61
for the packed beds of the graphite pebbles with shape A, shape B, or shape C.
For calculating the effective thermal conductivity of radiation, the convection
and conduction are neglected here, i.e., Q c,ik = 0, Q f = 0 in Eq. (5.9). The pebbles
in the packed bed are heated by the heat source and cooled by the radiation. The
emissivity of graphite pebbles is set at a constant of 0.81. The top and bottom are
adiabatic, and the outer wall is set at a fixed temperature. The distribution of heat
source is uniform, and k r is almost constant in the bed at a low value of Q s . In the
pseudo-porous media, the radial temperature distribution at a steady state at radial
position r is formulated as
T (r ) = T w +
1 − α f,b
4V p
Q s
k r
(R
2
− r
2
),
(5.144)
where α f,b is the average porosity of the bed. T w is the outer wall temperature. R
and V p are the radius of the outer wall and the particle volume. From the numerical
results at T w = 997 K and Q s = 0.05 W in Fig. 5.53, it is found that the radial particle
temperature distribution is almost linear to r
2 for different particle shapes at steady
Fig. 5.52 Particle shapes of non-overlapping clumped-sphere method in the pebble bed: a Shape A:
d p = 60 mm, b Shape B: d 1 = 28.845 mm, d 2 = 57.69 mm, and c Shape C: d 1 = d 2 = d 3 = 41.6 mm