Lie symmetry Analysis and Invariant Solutions for Multiregion Neutron Diffusion Equation

In this paper, an approach of determining analytical solutions of the mono-kinetic multiregion neutron diffusion equation from two-dimensional Cartesian geometry is presented. The technical approach is based on the Lie symmetry group for partial differential equation. The local symmetry groups to the one-parameter transformation are obtained. The invariant solutions spanned of an expansion of neutron fluxes with respect to the space, time and material regions are reported.


Introduction
The mathematical description of neutron diffusion for regions is depended on time , space and media. This description is provided by the neutron diffusion theory (Fick's law), necessary to the comprehension of many properties of the nuclear reactor physics.
Neutron diffusion is performed on a region towards other regions. Indeed, the diffusion theory is used to define the neutron's behavior. The neutron flux is then a combination of time, space and media.
The flux and the current are continuous through the system conditions after the change of region. The flux and the current are continuous through the system conditions after the change of region. The fluctuation varies along the direction and this one can be applied by coupling flux and homogeneous regions 1,2, … . A region is considered by the material which it is composed (fuel, control rod,...). The homogeneous mono-kinetic multiregion neutron diffusion equation (MNDE) for multiplying media is, assumed without delayed neutrons, external source and feedback mechanisms and other influences, written as where 0, , ". $ . % $ , & . % $ , , Σ $ , Σ $ and ". ()%%)*" $ are denoted the neutron flux, neutron speed, diffusion coefficient, macroscopic absorption cross section, macroscopic fission cross section and the fraction of neutrons emitted by fission, respectively [1][2][3][4][5], [14]. The model of analytic solutions to the diffusion equation is appropriated for the conditions of prompt neutron behavior. The Lie symmetry group is established to the variable (dependent or independent) transformations leaving partial differential equation (PDE) invariant [6][7][8], [11]. The key to find symmetry groups is the group related to the parameter + ≪ 1 . Using Lie symmetry group allow to determine also the invariant solutions to the mono-kinetic MNDE.
The objective of the paper is to obtain symmetry groups and invariant solutions for the MNDE using the Lie symmetry group. The outline is as follows. The Section 2 considers the material. The section 3 presents the Lie symmetry analysis of the MNDE. The Section 4 may obtain invariant solutions of the equation. Finally, in Section 5, conclusions are made.
The seven different compositions of material regions are presented in the core and surrounded by reflector of width -= . = 23.1226 7 .
The solutions of diffusion equation will be illustrated from physical parameters relating = 8 to regions of BIBLIS benchmark. The fraction of neutrons emitted by fission is = 2.47.
The neutron speed is assumed to & = 3 × 10 B . % $ and the initial flux 0 = 10 C ". $ % $ [5]. The coefficients 1 and 4 of system are: The boundary conditions at the surface, the continuity conditions within each interface and the initial condition for time-dependent problem are defined to the diffusion equation resolution with the geometrical description and physical environments.
Within time-dependent situation, spatial distribution of neutron at = 0 is: so that the initial flux is constant within each region and considered as the initial state. The interface flux continuity condition is (-, ., ) = I (-, ., ), while that for the current is These conditions materialize the limit of absorbent material at the core edge, including reflector.

Invariant Solutions
In the previous section, the symmetry groups are obtained.
The invariant function is obtained by integrating the characteristic equation: The case of ( . or (reduces the Eq. (2) to the ordinary differential equation (ODE) where 7 , 7 " 7 and 7 " are the integration constants.  The solutions (46) and (47) correspond to the first type of steady-state solution of Eq. (2). The plot of each region has the same shape evolving in space. It means that as the width of each region is reached, then the fluxes with respect toand . increase their descriptive values.
The case of ( reduces the Eq. (2) to the following ODE ( " 4 ( 0, with (′ '(/' . By solving the ODE, the invariant solution is -, ., where 7 C is the integration constant. The solution (49) presents the property of the kinetic model without delayed neutrons and external source. The plot shows the evolution of fluxes by regions with respect to time. The fluxes decrease as soon as the time increases in value and become null in every point of the regions. If 4 takes a positive value, the output of the neutron flux will be greater than the value of the initial flux 0 average, so there will be a bifurcation at the level of the concerned region.
2. The characteristic equation of the generator ^l corresponding to the rotation group is: The similarity variable is " oe-. ; ( " . The Eq. (2) is reduced to the ODE in the form of Bessel's equation with ( " '(/'". By solving the Bessel's equation, the invariant solution is where 7 l and 7 l " are the integration constants.
where 7 k and 7 m are the integration constants.
Where ª j and ª j " are the integration constants. š .
The similarity variable is " -.
To the steady-state solution, the case for 7 where the neutrons flux is more remarkable having a higher value than the other regions. Whereas for 3, the value of the flux is small compared to the other regions, so has a neutron flux easily absorbed, well 3 / 4 / 1 / 2 / 6 / 5 / 8 / 7.
To the kinetic model and its influence, 3 represents the maximum value per region, the regional flux corresponding to the fast decay of reactor relative to other regions. There is no possibility of fission but only absorption. The value decreases by region and beyond this period ( 10 $l %) the flux tends to 0. The linear plots become flat.
On the other hand, the region 7 has a slow decay having a lower value than the other. In this case, a large possibility of fission reaction and a weak absorption rate are remarked. Intermediate regions in both regions ( 3,7) retain the appearance of Figure 2 = 3 > 4 > 1 > 2 > 6 > 5 > 8 > 7.
The distribution of neutrons produced by collisions occurs in the media. Its dependence on space is determined by the properties of the media, except for the normal distribution, and its dependence on time shows the evolution. The system therefore represents a situation in which the collisional state exists.
On the other part, the distribution gives the departure of the collisional state flux caused by the fissile material. As a result, the material and the properties of the media affect the flux.

Conclusions
The symmetry groups related to one-parameter are obtained and the MNDE are solved from Lie symmetry group in order to have several invariant solutions representing each of them the sense of diffusion. These explicit solutions for Cartesian geometry are examined that are given for the first time in this work and had characteristics that might be important for the nuclear reactor physics.
However, the case of the generator ^B gives a trivial solution at = 0 (-, ., ) = 7 B = 0 , which illustrates the initiator flux of the chain reaction. The neutron coefficients of the 2D BIBLIS Benchmark have been used in order to illustrate behavior of the system that occur significant branch.
The 2D BIBLIS benchmark has demonstrated the ability of solutions to multiple regions. The neutron's behavior is affected by the diffusion effects, boundary conditions and surface to determine the integration constants. The neutron flux profiles are plotted through several graphs for various values of the emerging parameters. Each plot has shown the integrated system fluxes of the 8 regions. In most of solutions, we found that the kinetic model is a very important part for any form of solutions generated by symmetries groups. The solutions were averaged at times = 10 $m %, 10 $k %, 10 $B % and 10 $l %.
The observations concerning the validity of the multiregion solution in relation to the relevance of diffusion in the characterization of neutron fluxes behavior for reactor conditions lead to contributions to mentioned solutions.
The two-dimensional mono-kinetic MNDE has been used to explore the neutron behavior in Cartesian model.