There is growing interest in developing pebble bed reactors (PBRs) as a candidate of very high temperature gas-cooled reactors (VHTRs). Until now, most existing methods of nuclear design analysis for this type of reactors are base on old finite-difference solvers or on statistical methods. But for realistic analysis of PBRs, there is strong desire of making available high fidelity nodal codes in three-dimensional (r,$\\\\\\\\\\\\\\\\theta$,z) cylindrical geometry. Recently, the Analytic Function Expansion Nodal (AFEN) method developed quite extensively in Cartesian (x,y,z) geometry and in hexagonal-z geometry was extended to two-group (r,z) cylindrical geometry, and gave very accurate results.
In this thesis, we develop a method for the full three-dimensional cylindrical (r,$\\\\\\\\\\\\\\\\theta$,z) geometry and implement the method into a code named TOPS. The AFEN methodology in this geometry as in hexagonal geometry is \\\\\\\\\\\\\\\"robust\\\\\\\\\\\\\\\" (e.g., no occurrence of singularity), due to the unique feature of the AFEN method that it does not use the transverse integration. The transverse integration in the usual nodal methods, however, leads to an impasse, that is, failure of the azimuthal term to be transverse-integrated over r-z surface. We use 13 nodal unknowns in an outer node and 7 nodal unknowns in an innermost node. The general solution of the node can be expressed in terms of that nodal unknowns, and can be updated using the nodal balance equation and the current continuity condition. For more realistic analysis of PBRs, we implemented {\\\\\\\\\\\\\\\\em Marshak} boundary condition to treat the incoming current zero boundary condition and the partial current translation (PCT) method to treat voids in the core.
The TOPS code was verified in the various numerical tests derived from Dodds problem and PBMR-400 benchmark problem. The results of the TOPS code show high accuracy and fast computing time than the VENTURE code that is based on finite difference method (FDM), demonstrating the efficacy of the AFEN method in $(r,\\\\\\\\\\\\\\\\theta,z)$ geometry.
고온가스로(VHTR)에 대한 관심이 높아지고 있는 상황에서, 새로운 타입의 원자로인 Pebble Bed Reactors(PBRs)에 대한 노심해석은 기존의 FDM 방법을 이용한 코드인 VSOP을 이용하거나 stochastic한 방법인 MCNP를 이용해야하는 상황이다. PBRs의 경우 원통형(cylindrical)과 truncated cone shape이 혼합된 형태의 원자로이므로 FDM으로 해석할 경우 cone shape의 부분이 매우 작은 사각형으로 이루어진다고 가정해야 한다. 그리고 FDM을 이용할 경우 매우 작은 mesh를 이용해야 하는 방법으로 인해 많은 계산시간이 약점이 된다. 다른 노심 해석 방법인 노달 방법의 경우 rectangular의 경우 문제가 없지만, 원통형 geometry의 경우 $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\theta$ 방향의 경우 transverse integration 문제 때문에 원통형 geometry의 경우 적용이 불가능하다. 하지만 analytic function expansion nodal(AFEN) 방법을 cylindrical geometry로 확장할 경우, 기존의 노달 방법의 장점인 매우 빠른 계산 시간의 이점과 함께 transverse integration을 이용하지 않기 때문에 기존 노달 방법이 가졌던 문제점 없이 PBMR을 노달 방법으로 해석할 수 있다. 또한 truncated cone shape에 대한 정확한 노드 구성이 가능하다.
이러한 필요성 때문에 본 연구가 행해졌다. 본 논문에서는 원통형 geometry로 AFEN 방법을 확장하였다. 우선 선행 단계로 innermost node의 경우 7개의 nodal unknowns, outer node의 경우 13개의 nodal unknowns를 정했다. 그리고 diffusion equation을 separation of variable 방법을 이용하여 풀어 sin, cos, sinh, cosh, Bessel functions 등의 analytic basis function으로 표현되는 general solution을 구하였다. 이 general solution의 계수는 앞에서 정의된 nodal unknown을 이용하여 표현되고, 이 solution을 이용하여 node average flux는 diffusion equation을 전체 노드 체적에 대해 적분함으로, 각 interface flux의 경우는 current continuity condition을 이용하여 update 할 수 있다. 앞에서 설명한 방법을 적용하여 TOPS code를 만들고 4개의 문제를 테스트하여 검증하였다.
우선, Dodds problem으로 알려진 two-group r-z benchmark 문제를 풀어서 검증해 본 결과, reference k-effective 값과 0.0004\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\% 정도의 오차를 보여주었다. 이 문제의 경우 $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\theta$ division과 무관하게 항상 일정한 값을 보여주었다. 두 번째로 $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\theta$ 방향으로 heterogeneous geometry 문제를 풀었을 경우 $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\theta$ division이 적을 경우는 오차가 크나, $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\theta$ division이 증가할수록 점점 reference k-effective 값으로 수렴하여 24개로 나누었을 경우 0.0083\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\%의 오차를 보여주었다. two-group 으로 유도된 식을 matrix function theory를 이용하여 확장하여 multigroup으로 확장된 TOPS를 기존의 문제를 이용하여 검증해 본 결과 또한 정확하게 일치하는 결과를 보여주었다.
또한 multi-group의 문제를 검증하기 위하여 PBMR-400을 이용하여 유도된 문제를 실험해 본 결과, 역시나 two-group의 경우와 마찬가지로 높은 정확도를 보여주었다. 더욱 실제적인 노심해석을 이용하여 PBR의 윗부분에 존재하는 void region 처리를 위하여 partial current translation 방법을 적용하였고, incoming current zero 경계치 조건을 적용하기 위하여 {\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\em Marshak} B.C.을 적용하였다. 개선된 TOPS코드의 검증 결과 역시 two-group의 경우와 마찬가지로 높은 정확도를 보여주었다.
더 실제적인 PBRs의 해석을 위하여 아직 개발되지 않은 truncated cone shape에 대한 확장과, 바깥쪽 노드의 더욱 정확한 해석을 위한 azimuthal 로의 세분화 와 가속기법이 추가된다면 앞으로 더욱 좋은 원통형 노심 해석 코드가 될 것이다.
