24
1 Introduction
particle-scale method called Short-Range radiation Model (SRM) is developed [211].
The SRM was efficient in the calculation and the numerical results were in good
agreement with the experimental data [211]. With SRM in packed pebble beds, the
framework of CFD-DEM coupling with particle radiation was discussed and the
simulations of HTR-10 benchmark were performed [33, 257]. Moreover, when the
particle temperature exceeds 1200
◦ C, the SRM will overestimate the radiation flux
in the packed pebble beds [33, 211]. Thus, under very high-temperature ranges (over
1500
◦ C), Sub-Cell radiation Model (SCM) was recommended [258], in which the
porosity, surface emissivity, and solid conductivity are taken into account. With the
aid of space discretization realized by Voronoï tessellation [259], it is also possible
to use SCM for analyzing the heat transfer mechanisms [256, 260, 261], of monoor multi-sized particles. Thus, with some extensions if necessary, it is an excellent
approach to fulfill the purpose of radiative heat transfer modeling of the packed
pebble bed.
For the theoretical evaluation, the two-flux radiation model [242] and the ray
tracing method [262], are reported. The pebble beds are modeled as the isotropic
homogeneous medium of absorption, scattering, and emitting. However, it is the
scattering coefficient and absorption coefficient of the packed pebble beds rather
than the physical properties of the pebbles, such as surface emissivity and solid
conductivity applied in the numerical model. At low surface emissivity, the two-flux
radiation model and ray tracing method are no longer applicable [263].
It has long-range effects which are closely dependent on the local structure and
material property of the bed and packed particles. All of the two-flux method, the
discrete ordinate method [264] and the homogenization method [265], are not suitable
for computing the radiation of diffuse surfaces in the densely packed pebble system.
The numerical results are approximately independent of the surface emissivity of
particles in those models, even if the emissivity is very close to 0. In general, the
effect of emissivity on thermal radiation is a monotonically increasing function in
the granular system and the flux decreases to 0 exactly when the emissivity goes to
0 [231, 243, 266].
When the solid conductivity of particle is far greater than the effective thermal
conductivity (ETC) of radiation, the surface temperature distribution of the particle
will be uniform, and ETC is directly proportional to the third power of the average
temperature in Kelvin [232, 267, 268]. For particle radiation in large-scale granular
systems, many numerical models have been developed, such as the sub-cell radiation
model (SCM) [258], the boundary element method (BEM) [28], the Monte Carlo
method [269, 270], and the empirical correlations based on the experimental data
[240, 271, 272]. In those models, besides particle temperature, the particle properties
(diameter, emissivity, and solid conductivity) and the porosity of the bed are often
taken into account to evaluate the ETC of thermal radiation and the radiation exchange
factor. Moreover, the local average model [207] and the short-range radiation model
(SRM) [33], are efficient approaches for predicting particle radiation flux in CFDDEM simulations of granular systems.
However, only the particle-particle radiation in partial integral scales is considered. That means only the surrounding particles within distances of 1.5 times of the
1 Introduction
particle-scale method called Short-Range radiation Model (SRM) is developed [211].
The SRM was efficient in the calculation and the numerical results were in good
agreement with the experimental data [211]. With SRM in packed pebble beds, the
framework of CFD-DEM coupling with particle radiation was discussed and the
simulations of HTR-10 benchmark were performed [33, 257]. Moreover, when the
particle temperature exceeds 1200
◦ C, the SRM will overestimate the radiation flux
in the packed pebble beds [33, 211]. Thus, under very high-temperature ranges (over
1500
◦ C), Sub-Cell radiation Model (SCM) was recommended [258], in which the
porosity, surface emissivity, and solid conductivity are taken into account. With the
aid of space discretization realized by Voronoï tessellation [259], it is also possible
to use SCM for analyzing the heat transfer mechanisms [256, 260, 261], of monoor multi-sized particles. Thus, with some extensions if necessary, it is an excellent
approach to fulfill the purpose of radiative heat transfer modeling of the packed
pebble bed.
For the theoretical evaluation, the two-flux radiation model [242] and the ray
tracing method [262], are reported. The pebble beds are modeled as the isotropic
homogeneous medium of absorption, scattering, and emitting. However, it is the
scattering coefficient and absorption coefficient of the packed pebble beds rather
than the physical properties of the pebbles, such as surface emissivity and solid
conductivity applied in the numerical model. At low surface emissivity, the two-flux
radiation model and ray tracing method are no longer applicable [263].
It has long-range effects which are closely dependent on the local structure and
material property of the bed and packed particles. All of the two-flux method, the
discrete ordinate method [264] and the homogenization method [265], are not suitable
for computing the radiation of diffuse surfaces in the densely packed pebble system.
The numerical results are approximately independent of the surface emissivity of
particles in those models, even if the emissivity is very close to 0. In general, the
effect of emissivity on thermal radiation is a monotonically increasing function in
the granular system and the flux decreases to 0 exactly when the emissivity goes to
0 [231, 243, 266].
When the solid conductivity of particle is far greater than the effective thermal
conductivity (ETC) of radiation, the surface temperature distribution of the particle
will be uniform, and ETC is directly proportional to the third power of the average
temperature in Kelvin [232, 267, 268]. For particle radiation in large-scale granular
systems, many numerical models have been developed, such as the sub-cell radiation
model (SCM) [258], the boundary element method (BEM) [28], the Monte Carlo
method [269, 270], and the empirical correlations based on the experimental data
[240, 271, 272]. In those models, besides particle temperature, the particle properties
(diameter, emissivity, and solid conductivity) and the porosity of the bed are often
taken into account to evaluate the ETC of thermal radiation and the radiation exchange
factor. Moreover, the local average model [207] and the short-range radiation model
(SRM) [33], are efficient approaches for predicting particle radiation flux in CFDDEM simulations of granular systems.
However, only the particle-particle radiation in partial integral scales is considered. That means only the surrounding particles within distances of 1.5 times of the