1.4 Pebble Bed Heat Transfer
21
systems are usually of small-size (0.1 ∼ 6.0 mm in diameter) within low temperature ranges (20 ∼ 400
◦ C) [217–219]. By contrast, the particles in the packed pebble
beds of HTGRs are 60 mm in diameter at high-temperature. Concerning heat transfer, numerical models for particle-particle conduction in contact and particle-fluid
convection [216, 220], are applicable for the high-temperature packed pebble beds.
But the particle thermal radiation, which is a non-contact and long-range interaction,
increases significantly with temperature and cannot be neglected at high-temperature.
The models for particle radiation of the packed pebble beds have been developed in
different spatial scales and validated by experimental results [211]. The Short-range
Radiation Model (SRM) was added in the CFD-DEM framework for HTGR [33],
where the average cell length was about 2.3 times of particle diameter. Therefore,
the sub-particle-scale simulation of particle radiation is still a challenging work that
should be reasonably and carefully handled.
To date, for almost all cases of packed or fluidized beds, only coarse meshes (larger
than particle scale) are widely applied in the CFD-DEM simulations, which means
that the average cell volume of the CFD mesh is about 15–100 times of the threedimensional particle volume [214, 218, 221]. Some simulations and discussions are
performed only in two dimensions [207, 222]. It is difficult to apply particle scale or
sub-particle-scale meshes in the CFD-DEM simulations, since the void fraction is 0
or 1 when the grid node of particle- or sub-particle scale is totally inside or outside
the particle. The so-called Enlarged Porous Particle Method (EPPM) is one of the
primary solutions to solve the discontinuous spatial distribution of voids, where the
particle is assumed to be enlarged. In contrast, the particle mass is kept the same [223].
The region affected by the particle is extended artificially to neighboring cells [220].
Another solution is also based on the EPPM, where the uniform density distribution
in EPPM is replaced by a spatial distribution function [224, 225]. The Gaussian
function and other monotonically decreasing and continuous functions are typical
candidates [224, 226, 227]. The function based on the diffusion equation [225, 228],
is also recommended, which makes the smoothing degree adjustable for void fraction
computation. The smoothed void fraction method provides the flexibility to discuss
the flow and heat transfer in packed pebble beds on particle-scale mesh size or even
sub-particle-scale mesh size.
1.4.1 Gas-Pebble Heat Transfer
The thermal radiation and conduction play essential roles in heat transfer, especially
for the decay heat removal [229], since the highest temperature may reach 1620
◦ C
under severe conditions [230]. The heat transfer experiments of packed beds were
reported by many researchers to measure the total effective Thermal Conductivity
(ETC), which mainly includes the radiative part and the conductive part [231, 232,
252].
The effective thermal conductivity is a macroscopic lumped parameter of packed
beds, in which the fluid-particle system is modeled as a continuous porous media.
21
systems are usually of small-size (0.1 ∼ 6.0 mm in diameter) within low temperature ranges (20 ∼ 400
◦ C) [217–219]. By contrast, the particles in the packed pebble
beds of HTGRs are 60 mm in diameter at high-temperature. Concerning heat transfer, numerical models for particle-particle conduction in contact and particle-fluid
convection [216, 220], are applicable for the high-temperature packed pebble beds.
But the particle thermal radiation, which is a non-contact and long-range interaction,
increases significantly with temperature and cannot be neglected at high-temperature.
The models for particle radiation of the packed pebble beds have been developed in
different spatial scales and validated by experimental results [211]. The Short-range
Radiation Model (SRM) was added in the CFD-DEM framework for HTGR [33],
where the average cell length was about 2.3 times of particle diameter. Therefore,
the sub-particle-scale simulation of particle radiation is still a challenging work that
should be reasonably and carefully handled.
To date, for almost all cases of packed or fluidized beds, only coarse meshes (larger
than particle scale) are widely applied in the CFD-DEM simulations, which means
that the average cell volume of the CFD mesh is about 15–100 times of the threedimensional particle volume [214, 218, 221]. Some simulations and discussions are
performed only in two dimensions [207, 222]. It is difficult to apply particle scale or
sub-particle-scale meshes in the CFD-DEM simulations, since the void fraction is 0
or 1 when the grid node of particle- or sub-particle scale is totally inside or outside
the particle. The so-called Enlarged Porous Particle Method (EPPM) is one of the
primary solutions to solve the discontinuous spatial distribution of voids, where the
particle is assumed to be enlarged. In contrast, the particle mass is kept the same [223].
The region affected by the particle is extended artificially to neighboring cells [220].
Another solution is also based on the EPPM, where the uniform density distribution
in EPPM is replaced by a spatial distribution function [224, 225]. The Gaussian
function and other monotonically decreasing and continuous functions are typical
candidates [224, 226, 227]. The function based on the diffusion equation [225, 228],
is also recommended, which makes the smoothing degree adjustable for void fraction
computation. The smoothed void fraction method provides the flexibility to discuss
the flow and heat transfer in packed pebble beds on particle-scale mesh size or even
sub-particle-scale mesh size.
1.4.1 Gas-Pebble Heat Transfer
The thermal radiation and conduction play essential roles in heat transfer, especially
for the decay heat removal [229], since the highest temperature may reach 1620
◦ C
under severe conditions [230]. The heat transfer experiments of packed beds were
reported by many researchers to measure the total effective Thermal Conductivity
(ETC), which mainly includes the radiative part and the conductive part [231, 232,
252].
The effective thermal conductivity is a macroscopic lumped parameter of packed
beds, in which the fluid-particle system is modeled as a continuous porous media.