5.6 Summary
389
in continuum and discrete frameworks. Various models of radiation have been proposed in different scales, from the particle or sub-particle scale to the integral scales
throughout the bed. Further deep in issues and mechanisms have been analyzed too.
A new analytical expression of ETC is derived based on the continuum model to
solve conduction and radiation between pebbles in the packed bed. In this model, the
governing equation of energy is presented in a uniform framework to solve the conduction and thermal radiation of the nuclear pebble bed. It is a physics-based equation
determined by the temperature, number density, heat transfer coefficient, and radial
distribution function. The results predicted by this model are in good agreement with
existing models and correlations, which indicates that the local temperature in the
radiation case without internal heat sources is determined by all possible surrounding
pebbles weighted by a radiative kernel function.
Based on the continuum assumption, an Approximation Function Model (AFM) is
also proposed, serving as a suitable replacement of the traditional Radiative Transfer
Equation (RTE), which cannot provide a good prediction of radiative heat transfer
in densely packed bed of particles. Besides, the AFM is also feasible to apply the
current equation to analyze other radiation models. A generic physical expression
for the radiation exchange factor is given for the packed bed. The radiation in the
AFM is equivalent to that of the heat conduction at size parameter ξ 1, and
Effective Thermal Conductivity (ETC) in the central region of the packed bed is
higher than that of the whole bed. The AFM with radiative and conductive heat
transfer is developed, and its demonstrative application to the pebble-bed experiments
(TF-PBEC and HTTU) shows that the solutions by the current model are in a good
agreement with the experimental measurements.
For heat transfer of HTGR, the particle thermal radiation is an essential part of
engineering and research purposes, and it is necessary to model the ETC of thermal
radiation. The effect of spatial scale in modeling thermal radiation of packed pebble
beds is analyzed. 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, which can be summarized as follows:
• The Short-range Radiation Model (SRM) is not suitable for predicting the hightemperature particle radiation, because it underestimates the effective thermal
conductivity and radiation exchange factor at high temperatures and high surface emissivity. And it is not suitable for high conductivity materials (k s k r ).
Alternatively, it can be used for k s ∼ k r or Λ < 10. 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
overestimate the particle radiation flux when the temperature exceeds 1,200
◦ C.
389
in continuum and discrete frameworks. Various models of radiation have been proposed in different scales, from the particle or sub-particle scale to the integral scales
throughout the bed. Further deep in issues and mechanisms have been analyzed too.
A new analytical expression of ETC is derived based on the continuum model to
solve conduction and radiation between pebbles in the packed bed. In this model, the
governing equation of energy is presented in a uniform framework to solve the conduction and thermal radiation of the nuclear pebble bed. It is a physics-based equation
determined by the temperature, number density, heat transfer coefficient, and radial
distribution function. The results predicted by this model are in good agreement with
existing models and correlations, which indicates that the local temperature in the
radiation case without internal heat sources is determined by all possible surrounding
pebbles weighted by a radiative kernel function.
Based on the continuum assumption, an Approximation Function Model (AFM) is
also proposed, serving as a suitable replacement of the traditional Radiative Transfer
Equation (RTE), which cannot provide a good prediction of radiative heat transfer
in densely packed bed of particles. Besides, the AFM is also feasible to apply the
current equation to analyze other radiation models. A generic physical expression
for the radiation exchange factor is given for the packed bed. The radiation in the
AFM is equivalent to that of the heat conduction at size parameter ξ 1, and
Effective Thermal Conductivity (ETC) in the central region of the packed bed is
higher than that of the whole bed. The AFM with radiative and conductive heat
transfer is developed, and its demonstrative application to the pebble-bed experiments
(TF-PBEC and HTTU) shows that the solutions by the current model are in a good
agreement with the experimental measurements.
For heat transfer of HTGR, the particle thermal radiation is an essential part of
engineering and research purposes, and it is necessary to model the ETC of thermal
radiation. The effect of spatial scale in modeling thermal radiation of packed pebble
beds is analyzed. 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, which can be summarized as follows:
• The Short-range Radiation Model (SRM) is not suitable for predicting the hightemperature particle radiation, because it underestimates the effective thermal
conductivity and radiation exchange factor at high temperatures and high surface emissivity. And it is not suitable for high conductivity materials (k s k r ).
Alternatively, it can be used for k s ∼ k r or Λ < 10. 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
overestimate the particle radiation flux when the temperature exceeds 1,200
◦ C.