Criticality calculations in a nuclear reactor by using the Lyapunov exponent method M. Shayesteh a , S. Behnia b,⇑ , A. Abdi Saray a a Department of Physics, Imam Hossein University, Tehran, Iran b Department of Physics, Urmia University of Technology, Urmia, Iran article info Article history: Received 19 July 2011 Received in revised form 8 December 2011 Accepted 9 December 2011 Available online 8 February 2012 Keywords: Enrichment Multiplication factor Criticality calculation Coupled map lattice Lyapunov exponent Monte Carlo method abstract This paper studies the stability of the slab reactor with respect to the enrichment. For this purpose, the coupled map lattice theory is applied to the multi-group diffusion equations. Applying mean Lyapunov exponent theory introduced by Shibata [H. Shibata, Physica A 264 (1999) 226] on the model shows that, two successive phases: subcritical and supercritical. In order to compare the performance of the selected method by using the MCNP and ANISN codes the obtained results controlled. The model, in spite of its simplicity in form, shows a greater efficiency in prediction of critical enrichment. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Considering that the safety of a nuclear system is closely relevant to its stability, predicting its instability becomes of utmost impor- tance and so, many practical and theoretical studies have been per- formed on the subject in the recent decades (Morales-Sandoval and Hernandez-Solis, 2005; Lewins and Ngcobo, 1996). On the other side, having a glance on the neutron energy cross section details (such as uranium-235 and plutonium-239, and fertile, e.g., tho- rium-232 and uranium-238), it is obvious that obtaining exact solu- tions for the energy-dependant neutron transport equation for general reactor problems seems quite impossible (Bell and Glas- stone, 1970). Having a look to all these, one comes to this conclusion that like multigrid algorithms, adaptive mesh refinement and diag- onalization, an approximate method with a reasonable accuracy should be adopted to study the matter (Mei, 2006; Aboanber and Nahla, 2007; Ozgener and Ozgener, 2001). Studying sophisticated physical systems using nonlinear analysis is a newly opened horizon adopted to a variety of scientific issues and so on the nuclear reactors surveillance (Shu, 2006; Wahi and Kumawat, 2011; Konno and Hay- ashi, 1996). Different studies are focused on finding the neutron multiplication factor (k) with a prescribed target value. Guzmán- Arriaga and Espinosa-Paredes (2010) developed a novel method for a fuel lattice design using factoring-based approximation. This method is focused on finding the radial distribution of the fuel rod having difference fissile contents to obtain a prescribed neutron multiplication factor. This method was applied for transuranic fuel design (Guzmán et al., 2010). Among the different nonlinear analysis methods presented up to now, the coupled map lattice (CML) has been known as one of the most powerful ones used to study and analyze the qualitative and fundamental nature of so many complex physical systems (Kaneko, 1993). The CML method is based on a dynamic system with continuous field variables but discrete in space and time. The purpose of this paper is to introduce the mean Lyapunov exponent approach based on CML model on stability anal- ysis of nuclear reactors. In order to control the accuracy, the Monte Carlo N-Particle (MCNP) and ANISN codes used in calculation of neu- tron diffusion critical point too. The introduced model is a starting point for a more complete, yet simple, model for the nonlinear dynamics of reactors (Shibata, 2001a; Khoda-bakhsh et al., 2008). 2. Problem formulation 2.1. Computational domain and lattices As shown in Fig. 1, an infinite slab reactor with a core thickness of 80 cm containing natural to enriched (20%) uranium-235, and a graphite reflector on both sides with 10 cm in thickness, has been chosen for the study. The properties of the core and the reflector are denoted by the subscripts c and r, respectively. The computa- tional domain is divided into 100 meshes and due to the symmet- rical nature of the model, the study has been performed in only one dimension (Fig. 2). 0306-4549/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2011.12.017 ⇑ Corresponding author. Tel.: +98 441 345 8902; fax: +98 441 3554184. E-mail address: s.behnia@sci.uut.ac.ir (S. Behnia). Annals of Nuclear Energy 43 (2012) 131–135 Contents lists available at SciVerse ScienceDirect Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene