• Nuclear Engineering and Technology
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• v.52 no.2
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• pp.223-229
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• 2020

The present work proposes a solution to the static Boltzmann transport equation approximated by the simplified P₃ (SP₃) on angular, and the analytic coarse mesh finite difference (ACMFD) for spatial variables. Multi-group SP₃-ACMFD equations in 3D rectangular geometry are solved using the GMRES solution technique. As the core time dependent analysis necessitates the solution of an eigenvalue problem for an initial condition, this work is hence devoted to development and verification of the proposed static SP₃-ACMFD solver. A 3D multi-group static diffusion solver is also developed as a byproduct of this work to assess the improvement achieved using the SP₃ technique. Static results are then compared against transport benchmarks to assess the proximity of SP₃-ACMFD solutions to their full transport peers. Results prove that the approach can be considered as an acceptable interim approximation with outputs superior to the diffusion method, close to the transport results, and with the computational costs less than the full transport approach. The work would be further generalized to time dependent solutions in Part II.

• Nuclear Engineering and Technology
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• v.52 no.2
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• pp.230-237
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• 2020

In this part, an implicit time dependent solution is presented for the Boltzmann transport equation discretized by the analytic coarse mesh finite difference method (ACMFD) over the spatial domain as well as the simplified P₃ (SP₃) for the angular variable. In the first part of this work we proposed a SP₃-ACMFD approach to solve the static eigenvalue equations which provide the initial conditions for temp dependent equations. Having solved the 3D multi-group SP₃-ACMFD static equations, an implicit approach is resorted to ensure stability of time steps. An exponential behavior is assumed in transverse integrated equations to establish a relationship between flux moments and currents. Also, analytic integration is benefited for the time-dependent solution of precursor concentration equations. Finally, a multi-channel one-phase thermal hydraulic model is coupled to the proposed methodology. Transient equations are then solved at each step using the GMRES technique. To show the sufficiency of proposed transient SP₃-ACMFD approximation for a full core analysis, a comparison is made using transport peers as the reference. To further demonstrate superiority, results are compared with a 3D multi-group transient diffusion solver developed as a byproduct of this work. Outcomes confirm that the idea can be considered as an economic interim approach which is superior to the diffusion approximation, and comparable with transport in results.