388
5 Numerical Models for Pebble-Bed Heat Transfer
in the uniform radiation model was inappropriate for the gray radiation in a packed
bed. In the local radiation model, the effect of emissivity was handled correctly, and
the boundary condition was re-written as
A i
q r,i (r i )d A i = 0 for all particles without
a heat source in granular systems. But the computation cost for the local radiation
model is enormous, and it is not recommended for large-scale granular systems.
In the derivation of radiation resistance of the gray body, the particle-scale radiation model was proposed in this section, which was efficient in simulation and feasible
to be coupled with the Discrete Element Method (DEM). The particle-scale radiation
model is a good approximation of the local radiation model and applicable to the
prediction of radiative heat transfer in a densely packed bed. The contact thermal
resistance with stagnant fluid is discussed for the thermal discrete element method.
The expressions of the thermal resistance and the effective thermal conductivity are
derived. It is feasible to add the contact thermal resistance in the particle simulation code LIGGGHTS. In the large-scale pebble bed, the macroscopic conduction
equation is presented, and it agrees with the thermal discrete element method.
In the packed bed, where particle phase is randomly dispersed in fluid phase,
the effective thermal conductivity is determined by the coordination number and the
contact thermal resistance, which includes the solid–solid conduction through the
contact area and the fluid near the contact point. The proposed model in this section for
the effective thermal conductivity is in general agreement with the experimental data
of various conductivity ratio and void fraction. It is shown that when the conductivity
ratios of particle to fluid is higher than 10
4 , the conduction through the contact area
is the dominant part. The conduction is in the point contact mode at κ < 2000.
Additionally, a correlation is proposed to predict the radiation exchange factor under
different void fractions of a packed bed. The radiative effective thermal conductivity
increases significantly with the temperature and slightly with the void fraction.
For large-scale nuclear-packed pebble beds, the full-range radiation model based
on deep learning is introduced. The radiative heat transfer in a packed bed is a longrange interaction, and all possible particles determine local heat transfer. The heat
flux and effective thermal conductivity are derived in a matrix form for a packed
bed of structured and random packing. The key to performing numerical simulation
of the full-range radiation is to calculate the view factor matrix. The deep neural
network model is trained by 1,240,404 view factor cases to learn the rules to calculate
the complex view factor function efficiently. With the GPU acceleration, the realtime simulations of the full-range radiation model are performed for the packed bed
in HTR-PM. In the decay heat removal process, where only thermal radiation is
considered, the highest temperature with the time is under the design limitation.
5.6 Summary
It is a fundamental task to model the Effective Thermal Conductivity (ETC) of the
pebble bed, which considers the convection and radiation in analogous formulations of the conduction. In this chapter, the effective heat transfer is modeled both
5 Numerical Models for Pebble-Bed Heat Transfer
in the uniform radiation model was inappropriate for the gray radiation in a packed
bed. In the local radiation model, the effect of emissivity was handled correctly, and
the boundary condition was re-written as
A i
q r,i (r i )d A i = 0 for all particles without
a heat source in granular systems. But the computation cost for the local radiation
model is enormous, and it is not recommended for large-scale granular systems.
In the derivation of radiation resistance of the gray body, the particle-scale radiation model was proposed in this section, which was efficient in simulation and feasible
to be coupled with the Discrete Element Method (DEM). The particle-scale radiation
model is a good approximation of the local radiation model and applicable to the
prediction of radiative heat transfer in a densely packed bed. The contact thermal
resistance with stagnant fluid is discussed for the thermal discrete element method.
The expressions of the thermal resistance and the effective thermal conductivity are
derived. It is feasible to add the contact thermal resistance in the particle simulation code LIGGGHTS. In the large-scale pebble bed, the macroscopic conduction
equation is presented, and it agrees with the thermal discrete element method.
In the packed bed, where particle phase is randomly dispersed in fluid phase,
the effective thermal conductivity is determined by the coordination number and the
contact thermal resistance, which includes the solid–solid conduction through the
contact area and the fluid near the contact point. The proposed model in this section for
the effective thermal conductivity is in general agreement with the experimental data
of various conductivity ratio and void fraction. It is shown that when the conductivity
ratios of particle to fluid is higher than 10
4 , the conduction through the contact area
is the dominant part. The conduction is in the point contact mode at κ < 2000.
Additionally, a correlation is proposed to predict the radiation exchange factor under
different void fractions of a packed bed. The radiative effective thermal conductivity
increases significantly with the temperature and slightly with the void fraction.
For large-scale nuclear-packed pebble beds, the full-range radiation model based
on deep learning is introduced. The radiative heat transfer in a packed bed is a longrange interaction, and all possible particles determine local heat transfer. The heat
flux and effective thermal conductivity are derived in a matrix form for a packed
bed of structured and random packing. The key to performing numerical simulation
of the full-range radiation is to calculate the view factor matrix. The deep neural
network model is trained by 1,240,404 view factor cases to learn the rules to calculate
the complex view factor function efficiently. With the GPU acceleration, the realtime simulations of the full-range radiation model are performed for the packed bed
in HTR-PM. In the decay heat removal process, where only thermal radiation is
considered, the highest temperature with the time is under the design limitation.
5.6 Summary
It is a fundamental task to model the Effective Thermal Conductivity (ETC) of the
pebble bed, which considers the convection and radiation in analogous formulations of the conduction. In this chapter, the effective heat transfer is modeled both