A three-dimensional analytical thermal particle model with dispersed internal heat source distributions based on the Green's function method

A three-dimensional analytical thermal particle model with dispersed internal heat source distributions based on the Green's function method

A novel analytical method for solving the temperature distribution of a sphere with arbitrarily distributed internal heat sources is presented in this paper, based on the Green's function in three-dimensional spherical coordinates. Addressing the heat conduction problem of spherical small particles with a large number of dispersed internal heat sources, a point source approximation is employed to propose a three-dimensional point source temperature distribution calculation model. Strategies for calculating the average temperature and the maximum temperature based on secondary calculations are provided. The theoretical results align well with those obtained from finite volume method (FVM) simulations. The effects of particle numbers and distributions on the temperature field are explored in the context of high-temperature gas-cooled reactor (HTGR) fuel pebbles containing dispersed internal heat source particles. The deviation between the analytical model and FVM results is only 0.22 % at a particle radius of 400 μm, and multiple comparisons indicate that the use of point source approximation does not significantly affect temperature calculations. In contrast, adopting a one-dimensional homogenized internal heat source approximation severely underestimates the maximum temperature of the pebble. The particle packing fraction affects both the average and maximum temperatures of the pebble. While higher packing fractions or more uniform power distributions tend to reduce these temperatures, they simultaneously decrease the overall thermal conductivity of the composite material, which conversely elevates temperatures. This dual effect necessitates careful consideration and balanced optimization. When the radial distribution of heat source particles concentrates towards the boundary, both the maximum and minimum temperatures from the three-dimensional model are significantly reduced. Conversely, when heat source particles concentrate towards the poles, the temperature is higher in the vertical direction compared to uniform distribution, but lower in the horizontal direction. The distribution curves of temperature show fluctuations influenced by heat source particles, which can be rapidly calculated using the three-dimensional analytical model.