Exact Solutions to the Diffusion Process of Fission Products in the TRISO Fuel Particles of Pebble Bed HTRs: Case Study of ¹³⁷Cs and ¹³⁴Cs

In this paper, we discuss our study of the fission product diffusion process in TRISO fuel particles used in pebble bed high-temperature reactors (HTRs). Different from the previous numerical solution, the analytical solution of this diffusion process by variables separation was derived. It was also accessible to obtain the analytical expressions of the fission product concentration distribution C(t), the corresponding release fractions F(t), and the ratio of release and productive amounts R(t)/B(t) of fission products. Furthermore, to reduce the rounding errors, parameters mentioned in the diffusion equations were nondimensionalized, which made the result fairly reliable and credible. Since the analytical solutions are exact, many unnecessary assumptions and approximations in Booth's model are avoided. On the basis of HTR-10 design benchmark, the C(t), F(t), and R(t)/B(t) of ¹³⁷Cs and ¹³⁴Cs in TRISO fuel particles were calculated and then compared with the finite element solutions. The results show that analytical solutions are effective and consistent with the physical picture.