5.3 Discrete Modeling of Pebble Radiation
261
in modeling thermal radiation of packed pebble beds. The long-range model (full
integral scale), short-range model (partial integral scale), and microscopic models
(sub-particle scale) are compared and analyzed about existing correlations. In high
temperature packed pebble beds, the long-range model takes into account all possible
radiation between surrounding spheres, even those that are not direct Voronoï neighbors. In contract, the short-range model considers only a portion nearby. It is found
that when the solid conductivity is much larger than the effective thermal conductivity of radiation (k s k r or Λ > 10), the long-range model provides better results
than the short-range model in predicting the radiative heat exchange. The short-range
model overestimates the solid conductivity at low temperatures (lower than 1215
◦ C)
when k s ∼ k r (or Λ < 10) while underestimating the radiative heat exchange. It,
therefore, still provides predictions for total heat exchange that is in good agreement
with experimental data in cases where the errors cancel out. Moreover, the shortrange radiation model is more computationally efficient than the long-range model
and microscopic model to compute the view factors between particles of Voronoï
neighbors.
For improving the modeling of radiative heat transfer, the thermal radiation heat
exchanges are computed by three models to analyze the effect of spatial scales based
on the DEM data of particle packing. The scales include the full integral scale of
the thermal radiation, the partial integral scale, and the sub-particle scale. In the
sub-particle scale, the non-uniform temperature distribution over the particle surface is considered. Also, the existing correlations obtained from experimental data
statistically are used here for a referential comparison.
In the Lagrangian approach, the thermal radiation between particles is considered
individually. It is assumed that every particle in the packed pebble bed is an isothermal
body, and the solid conductivity within the particle is infinity. When one particle only
exchanges the radiation heat with the nearby neighbors and neglects the radiative
heat transfer with far particles, it is the so-called short-range radiation model. On
the contrary, it is extended to the long-range radiation model when all the peripheral
particles are taken into account for radiative heat exchanges.
When the particle flow in the bed is quite dense and slow or even quasi-static
(e.g., the packed pebble bed in HTGR), the porosity is about 0.4 in a stagnant state.
It is reasonable to assume that only neighboring particles, which may be technically
separated by Voronoï tessellations [36], can exchange radiative energies. Voronoï
tessellation is a numerical method to discretize the spatial domain into a group
of independent Voronoï cells with only one particle staying inside each cell. An
example is the discretization of the packed pebble bed of SANA-I (Fig. 5.18a). The
bed in Fig. 5.18a is 1.0 m in height, and 6.5 and 750 cm in inner and outer radii,
respectively. Every cell is a polyhedron and an interface exists between each pair of
neighboring particles (Fig. 5.18b). For ensuring energy conservation, it is assumed
that all the radiative heat passed through the interface of polyhedrons is obtained by
the neighboring particle. Hence, the view factor from one sphere to its neighboring
particle is equivalent to the interface of Voronoï tessellations.
Then, the view factor from one sphere to its enclosing polygon (Fig. 5.19a) can
be decomposed into the sphere to some right-angled triangles (Fig. 5.19b), where
261
in modeling thermal radiation of packed pebble beds. The long-range model (full
integral scale), short-range model (partial integral scale), and microscopic models
(sub-particle scale) are compared and analyzed about existing correlations. In high
temperature packed pebble beds, the long-range model takes into account all possible
radiation between surrounding spheres, even those that are not direct Voronoï neighbors. In contract, the short-range model considers only a portion nearby. It is found
that when the solid conductivity is much larger than the effective thermal conductivity of radiation (k s k r or Λ > 10), the long-range model provides better results
than the short-range model in predicting the radiative heat exchange. The short-range
model overestimates the solid conductivity at low temperatures (lower than 1215
◦ C)
when k s ∼ k r (or Λ < 10) while underestimating the radiative heat exchange. It,
therefore, still provides predictions for total heat exchange that is in good agreement
with experimental data in cases where the errors cancel out. Moreover, the shortrange radiation model is more computationally efficient than the long-range model
and microscopic model to compute the view factors between particles of Voronoï
neighbors.
For improving the modeling of radiative heat transfer, the thermal radiation heat
exchanges are computed by three models to analyze the effect of spatial scales based
on the DEM data of particle packing. The scales include the full integral scale of
the thermal radiation, the partial integral scale, and the sub-particle scale. In the
sub-particle scale, the non-uniform temperature distribution over the particle surface is considered. Also, the existing correlations obtained from experimental data
statistically are used here for a referential comparison.
In the Lagrangian approach, the thermal radiation between particles is considered
individually. It is assumed that every particle in the packed pebble bed is an isothermal
body, and the solid conductivity within the particle is infinity. When one particle only
exchanges the radiation heat with the nearby neighbors and neglects the radiative
heat transfer with far particles, it is the so-called short-range radiation model. On
the contrary, it is extended to the long-range radiation model when all the peripheral
particles are taken into account for radiative heat exchanges.
When the particle flow in the bed is quite dense and slow or even quasi-static
(e.g., the packed pebble bed in HTGR), the porosity is about 0.4 in a stagnant state.
It is reasonable to assume that only neighboring particles, which may be technically
separated by Voronoï tessellations [36], can exchange radiative energies. Voronoï
tessellation is a numerical method to discretize the spatial domain into a group
of independent Voronoï cells with only one particle staying inside each cell. An
example is the discretization of the packed pebble bed of SANA-I (Fig. 5.18a). The
bed in Fig. 5.18a is 1.0 m in height, and 6.5 and 750 cm in inner and outer radii,
respectively. Every cell is a polyhedron and an interface exists between each pair of
neighboring particles (Fig. 5.18b). For ensuring energy conservation, it is assumed
that all the radiative heat passed through the interface of polyhedrons is obtained by
the neighboring particle. Hence, the view factor from one sphere to its neighboring
particle is equivalent to the interface of Voronoï tessellations.
Then, the view factor from one sphere to its enclosing polygon (Fig. 5.19a) can
be decomposed into the sphere to some right-angled triangles (Fig. 5.19b), where