390
5 Numerical Models for Pebble-Bed Heat Transfer
In addition, SRM can be coupled with the CFD technique efficiently and used
for local and overall radiation predictions in a variety of gas–particle systems,
which is significantly beyond the capability of empirical correlations, since the
latter are derived from experimental data fitting and restricted greatly by empirical
parameters for particular applications.
• The Long-range Radiation Model (LRM) could be considered to correct the underestimation of the short-range model. It is an accurate approach since all surrounding
spheres with possible thermal radiative heat exchange are considered. It is usually
enough to consider three peripheral layers of Voronoï neighbors (about 200 particles), as its cumulative sum of the view factors is 0.9991. However, the long-range
model is subject to the condition of Λ >10 or k s k r .
• The Sub-Cell radiation Model (SCM) is a general theoretical approach for modeling the ETC of particle radiation. The SCM computes the theoretically precise view
factor and thermal radiation of the ordered structure of the particle system. It uses
a modification factor to obtain the thermal radiation of random packing. Besides,
as SCM considered solid conductivity, it applies to very high temperatures with
good accuracy. Notably, no empirical parameters are required in SCM since it is
based on the analytical theory of modeling the ETC of thermal radiation. Despite
requiring high computational costs, SCM can extend the application range of the
temperature from < 1, 000
◦ C in ZBS model (obtained from the experimental data
less than 1,000
◦ C) to > 1, 500
◦ C and the voids from the range of 0.260–0.476 in
the Kunii–Smith correlation (also obtained from the experimental data with voids
less than 0.476) to 0.26–0.6.
• Based on SCM and radial porosity distribution, the ETC of the radiation decreases
significantly in the near-wall region. A non-dimensional parameter called the
radiation-to-conduction ratio ξ was proposed to quantify the relative contribution
of thermal radiation to conduction. It indicates that thermal radiation becomes an
essential part when ξ > 0.1 for an accurate evaluation of radiations. As an efficient
analytical approach, the results obtained by SCM are in good agreement with the
experiments for mono-sized spheres of different materials with its size ranging
from 1.2 to 60 mm and temperature ranging from 0 to 1,200
◦ C.
In addition, a particle-scale investigation is performed for the nuclear-packed pebble beds filled with mono-sized or multi-component pebbles. When the radial porosity
distribution is considered, the ETC of the particle radiation decreases significantly in
the near-wall region. As a consequence, the discrete heat transfer model in particle
scale is presented, which combines the DEM and particle radiation model, and the
transient experimental results validate it. The radiation exchange factor increases
with surface emissivity. The SCM with the effective particle diameter can be used to
analyze the behavior of the radiation in polydisperse beds.
For the beds of non-overlapping clumped-sphere particles, the application of the
DEM-SCM for simulating the radiative and conductive heat transfer behaviors is
demonstrated, including the effects of particle shape, emissivity distribution, and
pebble flow on the transient heat transfer. By comparison of numerical results of
three kinds of clumped-shaped particles, it indicates that the radiative heat trans-
5 Numerical Models for Pebble-Bed Heat Transfer
In addition, SRM can be coupled with the CFD technique efficiently and used
for local and overall radiation predictions in a variety of gas–particle systems,
which is significantly beyond the capability of empirical correlations, since the
latter are derived from experimental data fitting and restricted greatly by empirical
parameters for particular applications.
• The Long-range Radiation Model (LRM) could be considered to correct the underestimation of the short-range model. It is an accurate approach since all surrounding
spheres with possible thermal radiative heat exchange are considered. It is usually
enough to consider three peripheral layers of Voronoï neighbors (about 200 particles), as its cumulative sum of the view factors is 0.9991. However, the long-range
model is subject to the condition of Λ >10 or k s k r .
• The Sub-Cell radiation Model (SCM) is a general theoretical approach for modeling the ETC of particle radiation. The SCM computes the theoretically precise view
factor and thermal radiation of the ordered structure of the particle system. It uses
a modification factor to obtain the thermal radiation of random packing. Besides,
as SCM considered solid conductivity, it applies to very high temperatures with
good accuracy. Notably, no empirical parameters are required in SCM since it is
based on the analytical theory of modeling the ETC of thermal radiation. Despite
requiring high computational costs, SCM can extend the application range of the
temperature from < 1, 000
◦ C in ZBS model (obtained from the experimental data
less than 1,000
◦ C) to > 1, 500
◦ C and the voids from the range of 0.260–0.476 in
the Kunii–Smith correlation (also obtained from the experimental data with voids
less than 0.476) to 0.26–0.6.
• Based on SCM and radial porosity distribution, the ETC of the radiation decreases
significantly in the near-wall region. A non-dimensional parameter called the
radiation-to-conduction ratio ξ was proposed to quantify the relative contribution
of thermal radiation to conduction. It indicates that thermal radiation becomes an
essential part when ξ > 0.1 for an accurate evaluation of radiations. As an efficient
analytical approach, the results obtained by SCM are in good agreement with the
experiments for mono-sized spheres of different materials with its size ranging
from 1.2 to 60 mm and temperature ranging from 0 to 1,200
◦ C.
In addition, a particle-scale investigation is performed for the nuclear-packed pebble beds filled with mono-sized or multi-component pebbles. When the radial porosity
distribution is considered, the ETC of the particle radiation decreases significantly in
the near-wall region. As a consequence, the discrete heat transfer model in particle
scale is presented, which combines the DEM and particle radiation model, and the
transient experimental results validate it. The radiation exchange factor increases
with surface emissivity. The SCM with the effective particle diameter can be used to
analyze the behavior of the radiation in polydisperse beds.
For the beds of non-overlapping clumped-sphere particles, the application of the
DEM-SCM for simulating the radiative and conductive heat transfer behaviors is
demonstrated, including the effects of particle shape, emissivity distribution, and
pebble flow on the transient heat transfer. By comparison of numerical results of
three kinds of clumped-shaped particles, it indicates that the radiative heat trans-