282
5 Numerical Models for Pebble-Bed Heat Transfer
profile predicted by the present model is also in good agreement with the experimental
measurements (see Fig. 5.36b).
5.3.9 Application of SCM for Pebble Beds
A particle-scale investigation is performed for the nuclear-packed pebble beds filled
with mono-sized or multi-component pebbles. The Sub-Cell radiation Model (SCM)
is a general theoretical approach to predict the Effective Thermal Conductivity (ETC)
of particle radiation. When the radial porosity distribution is considered, the ETC of
the particle radiation decreases significantly in the near-wall region. The discrete heat
transfer model in particle scale is presented, which combines the Discrete Element
Method (DEM) and particle radiation model, and it is validated by the transient
experimental results. It is shown that the radiation exchange factor increases with
surface emissivity. The results of the SCM under different surface emissivities are in
good agreement with existing correlations. Compared with the discrete simulation
results of polydisperse beds, it is found that the SCM with the effective particle
diameter can be used to analyze the behavior of the radiation in polydisperse beds.
In this section, the particle-scale investigation of thermal radiation is performed for
the nuclear-packed pebble beds. The radial porosity distribution of the wall effect on
the particle radiation and surface emissivity is discussed. The predicted values of SubCell radiation Model (SCM) are compared with the experimental data and empirical
correlations. Combined with SCM and the Discrete Element Method (DEM), particle
motion and transient heat transfer models are presented and applied to the analysis
of the multi-component polydisperse beds.
The sub-cell of the Face-Centered Cubic (FCC) packing for particle radiation
is shown in Fig. 5.37. The Voronoï tessellation of the FCC packing is a rhombic
dodecahedron for every particle. Point O 1 and Point O 2 are the center positions of
the particles, and surface ABCD is the sub-surface of the Voronoï cell connected with
the two particles. Based on the grey surface radiation theory and combined with the
Schotte equation [17], Effective Thermal Conductivity (ETC) of thermal radiation
(k r,SCM ) is formulated as
k r,SCM =
1 − α f
1
k s
+
1
k ∞
+ α f k ∞ ,
k ∞ =
3
2Γ
5
3
√
2π
6
1
3 (1 − α f )
1
3
α f
σ d p T
3
1−ε r
ε r
+
1
1+V 12
(5.112)
where α f and ε r are the porosity of the bed and emissivity of the particle surface,
respectively. k s is the solid conductivity of the particle material, and k ∞ is the value of
ETC when k s is infinity. d p , and T are the diameter and temperature of the particle,
respectively. σ is the Stefan–Boltzmann constant (5.670367 × 10
−8 W· m
−2
·K
−4 ).
5 Numerical Models for Pebble-Bed Heat Transfer
profile predicted by the present model is also in good agreement with the experimental
measurements (see Fig. 5.36b).
5.3.9 Application of SCM for Pebble Beds
A particle-scale investigation is performed for the nuclear-packed pebble beds filled
with mono-sized or multi-component pebbles. The Sub-Cell radiation Model (SCM)
is a general theoretical approach to predict the Effective Thermal Conductivity (ETC)
of particle radiation. When the radial porosity distribution is considered, the ETC of
the particle radiation decreases significantly in the near-wall region. The discrete heat
transfer model in particle scale is presented, which combines the Discrete Element
Method (DEM) and particle radiation model, and it is validated by the transient
experimental results. It is shown that the radiation exchange factor increases with
surface emissivity. The results of the SCM under different surface emissivities are in
good agreement with existing correlations. Compared with the discrete simulation
results of polydisperse beds, it is found that the SCM with the effective particle
diameter can be used to analyze the behavior of the radiation in polydisperse beds.
In this section, the particle-scale investigation of thermal radiation is performed for
the nuclear-packed pebble beds. The radial porosity distribution of the wall effect on
the particle radiation and surface emissivity is discussed. The predicted values of SubCell radiation Model (SCM) are compared with the experimental data and empirical
correlations. Combined with SCM and the Discrete Element Method (DEM), particle
motion and transient heat transfer models are presented and applied to the analysis
of the multi-component polydisperse beds.
The sub-cell of the Face-Centered Cubic (FCC) packing for particle radiation
is shown in Fig. 5.37. The Voronoï tessellation of the FCC packing is a rhombic
dodecahedron for every particle. Point O 1 and Point O 2 are the center positions of
the particles, and surface ABCD is the sub-surface of the Voronoï cell connected with
the two particles. Based on the grey surface radiation theory and combined with the
Schotte equation [17], Effective Thermal Conductivity (ETC) of thermal radiation
(k r,SCM ) is formulated as
k r,SCM =
1 − α f
1
k s
+
1
k ∞
+ α f k ∞ ,
k ∞ =
3
2Γ
5
3
√
2π
6
1
3 (1 − α f )
1
3
α f
σ d p T
3
1−ε r
ε r
+
1
1+V 12
(5.112)
where α f and ε r are the porosity of the bed and emissivity of the particle surface,
respectively. k s is the solid conductivity of the particle material, and k ∞ is the value of
ETC when k s is infinity. d p , and T are the diameter and temperature of the particle,
respectively. σ is the Stefan–Boltzmann constant (5.670367 × 10
−8 W· m
−2
·K
−4 ).