Iterative schemes for obtaining dominant alpha-modes of the neutron diffusion equation

• Published in: Annals of nuclear energy 2009-08, Vol.36 (8), p.1086-1092
• Main Authors: Singh, K.P., Modak, R.S., Degweker, S.B., Singh, Kanchhi
• Format: Article
• Language: English
• ISSN: 0306-4549; 1873-2100
• DOI: 10.1016/j.anucene.2009.05.006

Two new methods of obtaining dominant prompt alpha-modes (sometimes referred to as time-eigenfunctions) of the multigroup neutron diffusion equation are discussed. In the first of these, we initially compute the dominant K-eigenfunctions and K-eigenvalues (denoted by λ 1, λ 2, λ 3 … etc.; λ 1 being equal to the K eff) for the given nuclear reactor model, by existing method based on sub-space iteration (SSI) which is an improved version of power iteration method. Subsequently, a uniformly distributed (positive or negative) 1/ v absorber of sufficient concentration is added so as to make a particular eigenvalue λ i equal to unity. This gives ith alpha-mode. This procedure is repeated to find all the required alpha-modes. In the second method, we solve the alpha-eigenvalue problem directly by SSI method. This is clearly possible for a sub-critical reactor for which the inverse of the dominant alpha-eigenvalues are also the largest in magnitude as required by the SSI method. Here, the procedure is made applicable even to a super-critical reactor by making the reactor model sub-critical by the addition of a 1/ v absorber. Results of these calculations for a 3-D two group PHWR test-case are given. These results are validated against the results as obtained by a completely different approach based on Orthomin(1) algorithm published earlier. The direct method based on the sub-space iteration strategy is found to be a simple and reliable method for obtaining any number of alpha-modes. Also comments have been made on the relationship between fundamental α and k values.