306
5 Numerical Models for Pebble-Bed Heat Transfer
Fig. 5.59 The particle velocity profile of x = 0 plane at τ = 6 × 10 4 (a) and the temperature
response with the time for the five given points (b)
pebble bed is discussed from sub-particle scales (microscopic scale model) to full
integral scales (long-range model) through partial integral particle scales (short-range
model). For the full and partial integral scales, all the spheres in packed pebble beds
are assumed as isothermal bodies with infinite thermal conductivity, or k s k r . For
the sub-particle scale, the effect of finite conductivity, namely k s ∼ k r , is considered.
Based on the effective heat transfer cell, the Short-range Radiation Model (SRM)
and the Sub-Cell radiation Model (SCM) were compared to obtain analytical results
of ETC.
With comparison to existing data and correlations, the following has been found.
The long-range radiation model 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 .
For the short-range model, it is not suitable for large conductivity materials (k s
k r ). Alternatively, it can be used for k s ∼ k r or Λ < 10. The overestimation of solid
thermal conductivity and the underestimation of the view factors and radiative heat
transfer may lead to good predictions of the overall effective thermal conductivity in
cases that the errors cancel out. As a result, the accuracy of the short-range radiation
model is acceptable at temperatures lower than 1,215
◦ C. Under this condition, it
is efficient to use the short-range model to compute the view factors and obtain
the temperature field of the packed pebble beds with acceptable errors. Using the
short-range radiation model as the basic numerical approach for predicting thermal
radiation flux in packed pebble beds of random packing, the results of effective
heat transfer cells are in good agreement with the previous numerical simulations.
By ignoring the solid conductivity, SRM is a simplified model, and it may slightly
5 Numerical Models for Pebble-Bed Heat Transfer
Fig. 5.59 The particle velocity profile of x = 0 plane at τ = 6 × 10 4 (a) and the temperature
response with the time for the five given points (b)
pebble bed is discussed from sub-particle scales (microscopic scale model) to full
integral scales (long-range model) through partial integral particle scales (short-range
model). For the full and partial integral scales, all the spheres in packed pebble beds
are assumed as isothermal bodies with infinite thermal conductivity, or k s k r . For
the sub-particle scale, the effect of finite conductivity, namely k s ∼ k r , is considered.
Based on the effective heat transfer cell, the Short-range Radiation Model (SRM)
and the Sub-Cell radiation Model (SCM) were compared to obtain analytical results
of ETC.
With comparison to existing data and correlations, the following has been found.
The long-range radiation model 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 .
For the short-range model, it is not suitable for large conductivity materials (k s
k r ). Alternatively, it can be used for k s ∼ k r or Λ < 10. The overestimation of solid
thermal conductivity and the underestimation of the view factors and radiative heat
transfer may lead to good predictions of the overall effective thermal conductivity in
cases that the errors cancel out. As a result, the accuracy of the short-range radiation
model is acceptable at temperatures lower than 1,215
◦ C. Under this condition, it
is efficient to use the short-range model to compute the view factors and obtain
the temperature field of the packed pebble beds with acceptable errors. Using the
short-range radiation model as the basic numerical approach for predicting thermal
radiation flux in packed pebble beds of random packing, the results of effective
heat transfer cells are in good agreement with the previous numerical simulations.
By ignoring the solid conductivity, SRM is a simplified model, and it may slightly