# Uncertain One-Group Model

The scattering of neutron collision inside a reactor depends on the geometry of the reactor, the diffusion coefficient and the absorption coefficient. In general, these parameters are not crisp (exact) and hence we may get an uncertain neutron diffusion equation. In this chapter, the uncertain neutron diffusion equation for a bare square homogeneous reactor is discussed. The uncertain governing differential equation is modelled by a modified fuzzy finite element method (FFEM). Using the modified FFEM, the obtained eigenvalues and effective multiplication factors are studied. The corresponding results are compared with the classical finite element method (FEM) in special cases and various uncertain results are explained.

## Background

Uncertainty plays a vital role in various fields of engineering and science. These uncertainties occur due to incomplete data, impreciseness, vagueness, experimental error and different operating conditions influenced by the system. Different authors have proposed various methods to handle this uncertainty. They have used the probabilistic or statistical method as a tool to handle uncertain parameters. In this context, the Monte Carlo method is an alternative method, which is based on the statistical simulation of the random numbers generated on the basis of a specific sampling distribution. Monte Carlo methods have been used to solve the neutron diffusion equation with variable parameters. As such, Nagaya et al. (2010) implemented the Monte Carlo method to estimate the effective delayed neutron fraction Pf. Furthermore, Nagaya and Mori (2011) proposed a new method to estimate the effective delayed neutron fraction Pf in Monte Carlo calculations. In that article, the eigenvalue method was jointly used with the differential operator and correlated sampling techniques, whereas Shi and Petrovic (2011) used Monte Carlo methods to solve 1D two-group problems and then they proved its validity for these problems. Sjenitzer and Hoogenboom (2011) proposed an analytical procedure to compute the variance of the neutron flux in a simple model of a fixed- source calculation. Recently, Yamamoto (2012) investigated the neutron leakage effect specified by buckling to generate group constants for use in reactor core designs using the Monte Carlo method.

As such, in this process, we need a good number of observed data or experimental results to analyze the problem. Sometimes, it may not be possible to get a large number of data. As regards this process, Zadeh (1965) proposed an alternate idea, viz. the fuzzy approach, to handle uncertain and imprecise variables. Accordingly, we may use interval or fuzzy parameters to take care of the uncertainty. In general, traditional interval/fuzzy arithmetic is complicated to investigate the problem. In this context, a new technique for fuzzy arithmetic is developed to overcome such difficulty, which is proposed by Chakraverty and Nayak (2012). A few authors have investigated the mentioned problem. In this respect, Biswas et al. (1976) have given a method of generating stiffness matrices for the solution of the multigroup diffusion equation by a natural coordinate system. Azekura (1980) has also proposed a new representation of the finite element solution technique for neutron diffusion equations. The author has applied the technique to two types of one-group neutron diffusion equations to test its accuracy. Furthermore, Cavdar and Ozgener (2004) developed a finite element-boundary element-hybrid method for one- or two-group neutron diffusion calculations. In their article, a linear or bilinear finite element formulation for the reactor core and a linear boundary element technique for the reflector, which are combined through interface continuity conditions, constitute the basis of the developed method. Dababneh et al. (2011) formulated an alternative analytical solution of the neutron diffusion equation for both infinite and finite cylinders of fissile material using the homotopy perturbation method, whereas Rokrok et al. (2012) applied the element-free Galerkin (EFG) method to solve the neutron diffusion equation in X-Y geometry. From this, it is clear that neutron diffusion equations are solved using the FEM in the presence of crisp parameters only.

But the presence of uncertain parameters makes the system uncertain, and we get uncertain governing differential equations. In this context, uncertain fuzzy parameters are considered to solve heat conduction problems using the FEM, and we call it the FFEM. Bart et al. (2011) solved the uncertain solution of the heat conduction problem. In this article, authors made a good comparison between the response surface method and other methods. Recently, Chakraverty and Nayak (2012) also solved the interval/fuzzy distribution of temperature along a cylindrical rod. Here, they have presented a modified form of FFEM. The involved fuzzy numbers are changed into intervals through а-cut. Then, the intervals are transformed into crisp form by using some transformations. Crisp representations of intervals are defined by a symbolic parameterization. Traditional interval arithmetic is modified using the crisp representation of intervals. The interval arithmetic is extended for fuzzy numbers, and the developed fuzzy arithmetic is used as a tool for the uncertain FFEM. Consequently, this method is used to solve the one-group neutron diffusion equation, and the critical eigenvalues and effective multiplication factors are studied in detail. Hence, it may be used as a tool to solve different types of neutron diffusion problems for various types of nuclear reactors.

It is already mentioned that the uncertainty is considered here in terms of fuzzy or interval. In order to handle the uncertainty while using the FEM, we must formulate the FEM in uncertainty term (fuzzy or interval) using the fuzzy or interval computation. We have first formulated the following problem as traditional FEM for the sake of completeness. Then, the problem has been handled considering the uncertainty, and the example problem has been taken as a square homogeneous bare reactor.