5.3 Discrete Modeling of Pebble Radiation
307
overestimate the particle radiation flux when the temperature exceeds 1,200
◦ C. 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.
Although the microscopic model is the most accurate model even for k s ∼ k r , it
is computationally unacceptable for the practical use of thermal heat transfer prediction of the overall packed pebble bed. However, it can be used for improving the
prediction of the short-range and long-range models. Correction of the long-range
model prediction is suggested in the present section.
The Sub-Cell radiation Model (SCM) is a general theoretical approach for modeling Effective Thermal Conductivity (ETC) of particle radiation. It computes the
theoretically precise view factor and thermal radiation of the ordered structure of the
particle system and uses a modification factor to compute the thermal radiation of
random packing. It can calculate the Effective Thermal Conduction (ETC) of thermal radiation. In addition, as SCM considered solid conductivity, it is applicable to
very high temperatures with good accuracy. For example, the results obtained by
SCM are in good agreement with the Kunii–Smith correlation, especially in very
high-temperature ranges (over 1,500
◦ C). Using SCM, a one-dimensional radial heat
transfer model was applied for analysis of the HTTU experiments. The results of
effective thermal conductivity and radial temperature distribution are in good agreement with the experimental data. More importantly, no empirical parameters are
required in SCM since it is based on the analytical theory of modeling the ETC
of thermal radiation. Therefore, despite requiring large 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.
The limitations of the SRM and SCM are (1) They are suitable for dense packing
only. If there are only several spheres in a large space, then wall-wall radiation will be
an essential part to be considered. (2) They are mainly suitable for spherical particles,
and consequently, only spherical pebbles are considered in this work. Therefore, the
most applicable conditions of the present work are the packed pebble beds with large
quantities of mono-sized spherical pebbles of dense random packing.
In applications, the particle-scale radiation model was used to discuss the effect
of particle surface emissivity. It is shown that the radiation exchange factor increases
307
overestimate the particle radiation flux when the temperature exceeds 1,200
◦ C. 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.
Although the microscopic model is the most accurate model even for k s ∼ k r , it
is computationally unacceptable for the practical use of thermal heat transfer prediction of the overall packed pebble bed. However, it can be used for improving the
prediction of the short-range and long-range models. Correction of the long-range
model prediction is suggested in the present section.
The Sub-Cell radiation Model (SCM) is a general theoretical approach for modeling Effective Thermal Conductivity (ETC) of particle radiation. It computes the
theoretically precise view factor and thermal radiation of the ordered structure of the
particle system and uses a modification factor to compute the thermal radiation of
random packing. It can calculate the Effective Thermal Conduction (ETC) of thermal radiation. In addition, as SCM considered solid conductivity, it is applicable to
very high temperatures with good accuracy. For example, the results obtained by
SCM are in good agreement with the Kunii–Smith correlation, especially in very
high-temperature ranges (over 1,500
◦ C). Using SCM, a one-dimensional radial heat
transfer model was applied for analysis of the HTTU experiments. The results of
effective thermal conductivity and radial temperature distribution are in good agreement with the experimental data. More importantly, no empirical parameters are
required in SCM since it is based on the analytical theory of modeling the ETC
of thermal radiation. Therefore, despite requiring large 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.
The limitations of the SRM and SCM are (1) They are suitable for dense packing
only. If there are only several spheres in a large space, then wall-wall radiation will be
an essential part to be considered. (2) They are mainly suitable for spherical particles,
and consequently, only spherical pebbles are considered in this work. Therefore, the
most applicable conditions of the present work are the packed pebble beds with large
quantities of mono-sized spherical pebbles of dense random packing.
In applications, the particle-scale radiation model was used to discuss the effect
of particle surface emissivity. It is shown that the radiation exchange factor increases