Computation accuracy and efficiency of a power series analytic method for two- and three- space-dependent transient problems Ahmed E. Aboanber * , Yasser M. Hamada Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt Keywords: Neutron diffusion equation Analytic solution Power series Control rod withdrawal Power density Doppler feedback abstract The establishment of solutions to large-scale three-dimensional (3-D) reactor benchmark problems is needed to serve as standards for the verification of design codes and for the detailed error analysis of calculational methods. A number of partially and fully inserted control rods, represented by absorber added to certain subassemblies, cause a strong nonseparable power distribution. In addition, the exis- tence of a very large thermal flux peak in the reflector makes this a very difficult and challenging problem to solve. PWS code has been developed to include a numerical solution for the time-dependent neutron diffusion equations for the nuclear reactor analysis. The new technique employs a new parameter (a) which can reduce the rapid increase in magnitude of the power series coefficients. These coefficients, in turn, are determined by back substitutions in the non-linear canonical diffusion equations and treating terms of the same degree to obtain a modified recurrence relation which is valid for any type of the stiff non- linear kinetic diffusion equations. The validity of the algorithm was tested with three kinds of well-known two-group benchmark prob- lems. The first one is the two-dimensional TWIGL seed-blanket reactor kinetics problem. The second is the two- and three-dimensional LAR BWR benchmark problem simulating a rod drop accident of a BWR core. The third is the three-dimensional LMW LRA transient problem which simulates an operational transient involving rod movements. The obtained results with the proposed PWS code are compared with those provided by other reference codes, indicating an overall agreement and excellent performance. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction Nuclear safety requires precise knowledge of the behavior of reactor cores in accidental situations, such as, for instance, an ejection of control rods from the core and the resultant chain reaction runaway. In order to define the core refueling patterns, it is necessary to solve analytically and/or numerically the neutron- kinetics equations which describe and predict the temporal evolution of the neutron population throughout the reactor. Reactor kinetics has developed along two paths based on the point and the space–time models. The governing equations and solution methods for the models have been analyzed and reviewed exten- sively by many workers (e. g. Weinberg and Wigner, 1958; Clark and Hansen, 1964; Ash, 1979; Stacey, 1969; Bell and Glasstone, 1970; Hetrick, 1971; Henry, 1975; Lewins, 1978; Gehin, 1992; Stacey, 2001). The accurate prediction of reactor behavior and power is diffi- cult because it is necessary to calculate the three-dimensional power distribution in large and geometrically complicated cores. Furthermore, rapid transients of reactor power caused by a reac- tivity insertion due to a postulated drop or abnormal withdrawal of the control rod from the core have strong space-dependent feed- back associated with them. Therefore, a fast time-dependent three- dimensional transient analysis code is needed for simulating such phenomena. This paper gives a description of the main features, capabilities and development of the power series (PWS) kinetics code (Aboanber and Hamada, 2002, 2003) system for the analytic solution of the two- and three-dimensional kinetics diffusion equations with reactivity feedback due to a postulated drop or abnormal withdrawal of control rods from the core and an adiabatic fuel heatup thermal model. The code is used for reload safety analysis and all kinds of transients in which the power distribution is significantly affected. * Corresponding author. Tel.: þ20 040 3450408; fax: þ20 040 3350804. E-mail addresses: aboanber@tanta.edu.eg, aaboanber@yahoo.com (A.E. Aboanber), hoooorgrk@yahoo.com (Y.M. Hamada). Contents lists available at ScienceDirect Progress in Nuclear Energy journal homepage: www.elsevier.com/locate/pnucene 0149-1970/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pnucene.2008.10.003 Progress in Nuclear Energy 51 (2009) 451–464