The P ij matrix and flux calculation of one-dimensional neutron transport in the slab geometry of nuclear fuel cell using collision probability method

Calculation of Pij matrix of one-dimensional neutron transport in the slab geometry of the nuclear fuel cell using Collision Probability (CP) method has been done. Pij matrix is one of important parameters within the distribution of neutron flux in the nuclear fuel cell. The CP method is the most efficient methods to solve the neutron transport equation in the reactor core. The study is focused on neutron interaction with nuclear fuel cell of U-235 and U-238 for homogeneous condition. The parameters to calculate the Pij matrix are the cross section of nuclear fuel, width of the region and number of regions. A lattice of slabs have been constructed using void boundary conditions for model of finite system to calculate the collision probabilities. If the Pij matrix has been calculated then neutron flux can be determined. The results show that total value of Pij matrix using CP method for U-235 and U-238 is less than one, respectively. This is in accordance with the definition of void boundary conditions for finite slab geometry. Along with Pij matrix, neutron flux is also appropriate with the reference.


Introduction
The fundamental problem in the nuclear reactor analysis is solving the neutron transport equation.The neutron distribution system is described by the integral transport equation [1].The integral transport equation is obtained from the integro-differential equation by using some basic assuming.The solution of the neutron transport equation determines the distribution of neutrons in the reactor to predict neutron flux at all points directions and energies for any given geometry and material distribution [2].The numerical solution of neutron transport equation solving should be able to explain the neutron behavior in the reactor [3].For small geometries, the neutron transport theory is more appropriate to calculate the neutron distribution than the diffusion theory [4].
There are two classification methods that used to solve the neutron transport equation: stochastic and deterministic methods.Stochastic method is the random movement of neutrons in phase space governing the probability of neutron interaction with material.Deterministic methods are based on solving the neutron transport equation which governs the average behavior of neutrons.There are several methods in the class of deterministic methods such as Method of Characteristic (MOC), Collision Probability (CP) method, Discrete Ordinates method (SN) and the spherical harmonics (PN) method.The most popular method in deterministic class is CP method.The Collision Probability method is frequently used for the construction of relevant matrices.It can handle complex geometry.It is often employed to solve small sized problems such as those over pin cell or fuel assembly in a reactor.This method is, however, unsuitable for large-sized problems such as whole core, because the matrices are large as well as dense and relevant equations are difficult to solve [5].
In previous research [1,6,7], the CP method was used to solve the transport of neutrons in a 1D cylindrical nuclear fuel cell.The integral transport is solved using CP method with non flat flux approach which is usually solved by the flat flux approach.In this research, CP method was implemented to simple geometry, with flat flux approach.The research is focused on the calculation of Pij matrix using CP method for U-235 and U-238 of 1D slab of nuclear fuell cell with void boundary condition.If the Pij matrix has been calculated then neutron flux can be determined.

Theoretical backgrounds
The CP method is one of the solutions the integral form of the neutron transport equation.The main idea behind the integral transport method is to integrate out the angular dependence and to solve the neutron transport equation for the scalar flux directly [5].The neutron transport equation is difficult to resolve analytically, unless it is simplified according to the assumptions made.An independent of time neutron transport equation is [8], The basic assumptions of the CP method are the flat flux approximation and the isotropic scattering.By defining ˆd dR   , the angular neutron flux as a solution of Eg.
(1) at position r r in the angle of space  and energy E follows the form of the integral transport equation where R is the distance between r r and ' r r , vector space is the scattering cross section at the point ' r r from Ω' with the energy E' to Ω with the energi E, and ( ', , ) S r E  r is a neutron source.If the neutron scattering and neutron sources are considered isotropic, then Eq. ( 2) becomes to Equation ( 3) can be rewritten by using a relation where is the probability of a neutron having energy E coming from a point ' r will experience a collision at a point r r in the volume element .dr Equation ( 4) can be simplified form as follows with the CP matrix of Eq. ( 5) is defined by The collision probabilities matrix Pij given by Eq. ( 6) are the first flight collision probabilities from region i to region j as they give the probability for a neutron born isotropically and uniformly distributed within region i to have its first collision in region j [5].
To simplify computing calculations, Eq. ( 5) can be rewritten as multi group energy [1,6] 1 where index i or j indicates the spatial region i or j and index g denotes the energy group, is jg  the total macroskopic cross section in region j, Sig is source, Pij is CP matrix and 2 2 1 ( ).
The special treatment that only one group one energy, Eq.( 7) can be modified into 1 In the case of one dimensional of slab geometry for one speed one energy neutron in x axis, the neutron born in the point having its first collision is shown in the Figure 1.Based on the discretization of slab, equation ( 6) can be written as [5] ' ' exp( ( , ') 2 where 1 cos ' Eq.( 9) can be expressed as The optical distance between the two points x and x' is Equation ( 13) is the expression of the first flight collision probabilities that a neutron introduced in cell i has its next collision in cell j on a slab geometry for i j  [5].On the other hand, for i = j, an expression for self-collision probability, the probability that a neutron introduced in cell i has its next collision in cell i, is [9]   The neutron flux in the slab geometry can be derived form Eq. (3 For isotropical beutron source.

Computational Method
For limitation case, the research have the following assumptions: the cross sections of U-235 and U-238 are constant in each region, the flux is constant in each volume and the source is constant.The boundary conditions that used is finite system that composed of meshes with void boundary.Void boundary conditions is implemented on both sides of slab as shown in the Figure 2.

Fig. 2. Slab with void boundary conditions that discretizied into six regions
To evaluate the first flight collision probabilities in this system is assumed that there are no contributions from outside.The consequence of this boundary condition is the sum of Pij matrix must be less than one  for all mesh due to leakages from the system.This is different with an infinite system modelled as having N meshes surrounded by reflective boundary conditions that the sum of Pij matrix must be equal to one 1 . The slab is discretized into 6 meshes.
It follows that for a domain with 6 meshes, the resulting collision probabilities matrix is of size 6 6  .In general, a computational procedure to calculate the collision probability using MATLAB R2013b follows : a. Determine the void boundary conditions.Lattice of slabs are constructed using void boundary conditions to an finite model system to calculate the collision probabilities.b.Determine the total and scattering macroscopic cross section of U-235 and U-238, source of neutron, width and number of mesh as an input of the program.c.Calculates the optical path length using Eq.(12).d.Calculates the exponential integral using Eq.( 10).e.The collision probabilities matrix is executed by using Eq. ( 13) simultaneously with Eq.( 14).f.After that, neutron flux is calculated using Eq.(15).

Results and discussion
In the case of fuel cells in a homogeneous slab geometry composed of U-235 and U-238, the value of the Pij matrix is presented in Table 1 and Table 2.It appears that diagonal of Pij values (P11, P22, P33, P44, P55 and P66) are equal and greater than the other.Beside the Pij located on the diagonal, there are a pair of Pij that have the same value (e.g.P12 with P21, P13 with P31, ect.), provided that i j  .
Table 1.Pij matrix for a finite slab composed of U-235.
The main difference is only seen in the total number of the Pij matrix.It occurs because of the results in Table 1 and Table 2 use the void boundary conditions, whereas in Table 3 it uses the reflective boundary condition.In the void boundary conditions, the total number of the Pij matrix must be less than 1 in each https://doi.org/10.1051/matecconf/201819702006AASEC 2018 regions, on the other hand the total number of the Pij matrix must be equal to 1 in each regions.In other words, the calculation of the CP matrix has been in accordance with the theory.The greater the removal cross sectional, the smaller the scalar flux in the region, since the removal cross section indicates the probability that neutrons will be transferred to the energy other due to a collision.While scattering cross section shows the amount of neutrons scattered without being absorbed or out of the slab.The greater the scattering cross section value, the greater also the scalar neutron flux.The profile of the neutron distribution of slab fuel U-235 and U-238 is simetry that have maximum value occurred in midle of slab region.
Figure 4 is the neutron flux distribution data from a study of Mtsetfwa [5] for a homogeneous slab geometry of fuel cell composed of U-235 for multigroup enrgy.There is no different pattern result of one dimensional flux neutron distribution using one group and multigroup energy.This indicates that the results of this study have been in accordance with the reference.

Conclusions
A one dimensional neutron transport theory based on CP method is developed for slab geometry.The calculation requires the cross sections U-235 and U-238 and void boundary condition as input.Total value of Pij matrix using CP method for U-235 and U-238 is less than one, respectively.This is in accordance with the definition of void boundary conditions for finite slab geometry.Along with Pij matrix, neutron flux is also appropriate with the reference.In general, implementation of neutron transport using CP method with applied to the geometry slab of fuel cell yields fairly good results comparing with reference.

Fig. 3 .
Fig. 3. Neutron Flux in the 1 D slab of fuel cell.Neutron flux distribution in the slab geometry nuclear fuel cells composed of U-235 and U-238 is shown in the Figure 3.It appears that the scalar neutron flux of U-235 is greater than U-238.This event occurs due to the scattering and removal cross section of U-235 are also greater than U-238.

Table 2 .
Pij matrix for a finite slab composed of U-238.