346
5 Numerical Models for Pebble-Bed Heat Transfer
the diffusion function is a better candidate for the spatial distribution function.
When the cell size is less than the particle diameter (ΔL < dp or χ <1.91), the
void fraction of the particle-Divided Finite Volume Method (DFVM) will be no
longer applicable to the CFD-DEM simulations. In this case, the Smoothed Void
Fraction Method (SVFM) is suggested to be used in the CFD-DEM simulations,
and the cell-to-particle volume ratio χ = 0.2472 is recommended. By comparing
the void fraction distribution of Voronoï tessellation, the smoothing degree in
the Smoothed Void Fraction Method (SVFM) is recommended to be within 0.5–
0.7. For the Smoothed Void Fraction Method (SVFM), the wall effect on the
spatial distribution function should be considered when particles in the packed
pebble beds are near physical boundaries. The practical approach to compute the
void fraction field based on the particle positions are applied in the CFD-DEM
simulations, which is in good agreement with the virtual symmetric point method.
The CFD-DEM simulation with the smoothed void fraction method is performed
for an experimental spout fluidized bed at Δx = 0.83d p . The numerical results
of time-averaged vertical particle velocities are in general agreement with the
PEPT measurements. By sub-particle scale meshes (the cell-to-particle volume
ratio χ = 0.2472), the Smoothed Void Fraction Method (SVFM) is utilized in CFDDEM simulation for the HTR-10 benchmark problem. The results at the smoothing
degree η = 0.5 are in good agreement with the empirical code of THERMIX.
5.5 Further Issues
5.5.1 Evaluation of Emissivity Effects in Four Radiation
Models
In this section, multiscale modeling of particle radiation based on the gray-body
assumption was performed in full integral scales for large-scale granular systems of
random packing. Moreover, the black radiation model, the uniform radiation model,
and the local radiation model were discussed for the particle–particle and particle–
wall radiation. Based on the fundamental physical mechanism of thermal radiation,
the resistance network method was applied to derive the particle-scale radiation
model from being combined with the Discrete Element Method (DEM).
Four models of multiple-body radiation within the packed granular bed were
proposed and discussed in full integral scales, i.e., black radiation model, uniform
radiation model, local radiation model, and particle-scale radiation model. For particle radiation with black surfaces, the black radiation model was proven to be valid
strictly, and it agreed with existing correlations and models. The radiative flux in
the packed bed increased significantly with the particle emissivity. However, the
existing Asakuma radiation model, two-flux model, and the newly proposed local
radiation model were almost independent of the particle emissivity since the uniform
assumption was inappropriate for the gray radiation in the packed bed. The effect of
5 Numerical Models for Pebble-Bed Heat Transfer
the diffusion function is a better candidate for the spatial distribution function.
When the cell size is less than the particle diameter (ΔL < dp or χ <1.91), the
void fraction of the particle-Divided Finite Volume Method (DFVM) will be no
longer applicable to the CFD-DEM simulations. In this case, the Smoothed Void
Fraction Method (SVFM) is suggested to be used in the CFD-DEM simulations,
and the cell-to-particle volume ratio χ = 0.2472 is recommended. By comparing
the void fraction distribution of Voronoï tessellation, the smoothing degree in
the Smoothed Void Fraction Method (SVFM) is recommended to be within 0.5–
0.7. For the Smoothed Void Fraction Method (SVFM), the wall effect on the
spatial distribution function should be considered when particles in the packed
pebble beds are near physical boundaries. The practical approach to compute the
void fraction field based on the particle positions are applied in the CFD-DEM
simulations, which is in good agreement with the virtual symmetric point method.
The CFD-DEM simulation with the smoothed void fraction method is performed
for an experimental spout fluidized bed at Δx = 0.83d p . The numerical results
of time-averaged vertical particle velocities are in general agreement with the
PEPT measurements. By sub-particle scale meshes (the cell-to-particle volume
ratio χ = 0.2472), the Smoothed Void Fraction Method (SVFM) is utilized in CFDDEM simulation for the HTR-10 benchmark problem. The results at the smoothing
degree η = 0.5 are in good agreement with the empirical code of THERMIX.
5.5 Further Issues
5.5.1 Evaluation of Emissivity Effects in Four Radiation
Models
In this section, multiscale modeling of particle radiation based on the gray-body
assumption was performed in full integral scales for large-scale granular systems of
random packing. Moreover, the black radiation model, the uniform radiation model,
and the local radiation model were discussed for the particle–particle and particle–
wall radiation. Based on the fundamental physical mechanism of thermal radiation,
the resistance network method was applied to derive the particle-scale radiation
model from being combined with the Discrete Element Method (DEM).
Four models of multiple-body radiation within the packed granular bed were
proposed and discussed in full integral scales, i.e., black radiation model, uniform
radiation model, local radiation model, and particle-scale radiation model. For particle radiation with black surfaces, the black radiation model was proven to be valid
strictly, and it agreed with existing correlations and models. The radiative flux in
the packed bed increased significantly with the particle emissivity. However, the
existing Asakuma radiation model, two-flux model, and the newly proposed local
radiation model were almost independent of the particle emissivity since the uniform
assumption was inappropriate for the gray radiation in the packed bed. The effect of