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Article

Solving Multi-Group Reflected Spherical Reactor System of Equations Using the Homotopy Perturbation Method

1
Physics Department, Faculty of Science and Humanities, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
2
Physics Department, Faculty of Sciences and Arts, Jouf University, Tabarjal 75764, Saudi Arabia
3
College of Commerce and Business, Lusail University, Doha 122104, Qatar
4
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 20550, United Arab Emirates
*
Author to whom correspondence should be addressed.
Submission received: 6 March 2022 / Revised: 19 May 2022 / Accepted: 20 May 2022 / Published: 23 May 2022
(This article belongs to the Topic Multi-Energy Systems)

Abstract

:
The solution of the complex neutron diffusion equations system of equations in a spherical nuclear reactor is presented using the homotopy perturbation method (HPM); the HPM is a remarkable approximation method that successfully solves different systems of diffusion equations, and in this work, the system is solved for the first time using the approximation method. The considered system of neutron diffusion equations consists of two consistent subsystems, where the first studies the reactor and the multi-group subsystem of equations in the reactor core, and the other studies the multi-group subsystem of equations in the reactor reflector; each subsystem can deal with any finite number of neutron energy groups. The system is simplified numerically to a one-group bare and reflected reactor, which is compared with the modified differential transform method; a two-group bare reactor, which is compared with the residual power series method; a two-group reflected reactor, which is compared with the classical method; and a four-group bare reactor compared with the residual power series.

1. Introduction

The approximation methods have been used to solve system of differential equations; these methods can deal with complicated systems that never have been solved in classical methods, and HPM is one of the most important approximation methods.
The proposed system can be considered as two subsystems of neutron diffusion equations; each subsystem will be solved separately, where the core reflector boundary conditions are valid. This system represents two parts of a nuclear spherical reactor: the first part consists of the nuclear fuel called the core part, and the other has a material that reflects the neutrons to the core part, which improves the efficiency of the fission inside the reactor.
The general system of this work will be simplified as special cases to compare with other works [1,2] that have studied these cases using approximation and classical methods [3].
To solve the neutron diffusion equation, HPM gives the achieved result, and it is chosen because it has succeeded in solving simpler cases of neutron diffusion equations [4,5,6,7], and it is hoped to solve this general system. Solving nuclear reactor equations using other methods is also presented [1,2], and many works that solve different cases of the neutron diffusion equation and some other works [8,9,10,11,12,13,14] where another related approximation method studying diffusion equation is accomplished [15] are cited.
HPM was created firstly by He JH in 1999 [16] and performed a success in solving different fields [17,18,19], and soon, many works were accomplished using it [6,7]; the creator of HPM still depends on it in dealing with new physical problems [20,21].
The methodology of HPM is constructed on the combination of the topology concept (homotopy) and perturbation theory; this method depends on continuously changing the original difficult problem to a simple one that can be easily solved as the embedding parameter changes from unity to zero.
The full description of HPM, which is described in detail previously [6,16], will be clarified as we study this system.
The theoretical study of this work will be studied in Section 2, while the special case studies and their numerical examples are given in Section 3.

2. Theory

The basic concept of the neutron diffusion equation comes from applying Fick’s law to simplify the transport neutron equation; this simplification is fair where the behavior of the neutrons in the reactor is still reasonable. The simple neutron diffusion equation is used to express the neutrons moving in one velocity, and this is known as one-group case; for more reality, the neutrons will be divided in many velocities; this is known as the multi-group case. This study presents two multi-group subsystems where the reflector neutron diffusion equations subsystem is added to improve the work. This system consists of the fuel surrounded by a reflector, which saves the core and improves the fission inside it [22,23,24,25]. The mathematical manipulation for this system is studied next.

2.1. The Reactor Core Part

Here, the multi-group neutrons diffusion equations subsystem in the reactor core will be studied. Buckling in the core part is not unique, such as occurs in the bare reactor, there are two buckling named principal and alternate buckling, and each neutron flux is a linear combination of both principal and alternate buckling fluxes.
In HPM, fluxes depend on intersecting between constants connecting the fluxes, while the classical method depends on the buckling calculation.
Buckling is one of the most essential concepts in the nuclear reactor theory, as it represents the leaks of the neutrons in the reactor, which means it has an essential role in the stability of the nuclear reactor [24].
The multi-group neutron diffusion equations for principal buckling in the reactor core are:
2 1 ( r ) + N 11   1 ( r ) + N 12 2 ( r ) + N 13 3 ( r ) + + N 1 n n ( r ) = 0 , 2 2 ( r ) + N 21 1 ( r ) + N 22 2 ( r ) + N 23 3 ( r ) + + N 2 n n ( r ) = 0 , 2 3 ( r ) + N 31 1 ( r ) + N 32 2 ( r ) + N 33 3 ( r ) + + N 3 n n ( r ) = 0 , 2 n ( r ) + N n 1 1 ( r ) + N n 2 2 ( r ) + N n 3 3 ( r ) + + N n n n ( r ) = 0 ,  
where D i is the ith group diffusion coefficient, and N i j is a constant that connects between fluxes in different energy groups of neutrons [1], which is defined as follows:
{ N i i = χ i ν i f i ( γ i + Σ s i j ) D i ,                       N i j = Σ s j i + χ i ν j f j D i ,                                                     D i = 1 3 ( f i + s i i + Σ s i j + γ i ) .    
Constants in Equation (2) have been defined in terms of different macroscopic cross-sections; the number of neutrons produced per fission for each group ν i , , and the fraction of fission neutrons emitted with energies in the i th group is χ i .
The time-independent diffusion system of the multi-group at the core of the spherical reactor, after substituting the Laplacian in the radial part dependent on spherical coordinates, can be written as:
r 1 ( r ) + 2 1   ( r ) + r ( N 11   1 ( r ) + N 12 2 ( r ) + N 13 3 ( r ) + + N 1 n n ( r ) ) = 0 ,     r 2 ( r ) + 2 2   ( r ) + r ( N 21 1 ( r ) + N 22 2 ( r ) + N 23 3 ( r ) + + N 2 n n ( r ) ) = 0 ,   r 3 ( r ) + 2 3   ( r ) + r ( N 31 1 ( r ) + N 32 2 ( r ) + N 33 3 ( r ) + + N 3 n n ( r ) ) = 0 ,   r n ( r ) + n   ( r ) + r ( N n 1 1 ( r ) + N n 2 2 ( r ) + N n 3 3 ( r ) + + N n n n ( r ) ) = 0 .  
This system of equations describes the behavior of the neutrons in the nuclear reactor where each flux i expresses the neutron flux with a specific energy. Each flux has a maximum value at the center of the reactor, while its derivative vanishes there, so the initial conditions can be written in the mathematical form as:
i ( 0 ) = I , i   ( 0 ) = 0 , i = 1 , 2 , , n .
In order to solve these equations using HPM, we construct the homotopy [6,16] as:
H ( ϕ 1 ( r ) ,   ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) =   ( 1 p )   F 1 ( ϕ 1 ( r ) ,   ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) ) + p L 1 ( ϕ 1 ( r ) ,   ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) ) .
where H is the homotopy, L i is the original problem, and F i is the simple problem.
Thus,
H 1 ( ϕ 1 ( r ) , ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) = r 2 ϕ 1 ( r ) + 2 r ϕ 1   ( r ) + p r 2 ( N 11   1 ( r ) + N 12 2 ( r ) + N 13 3 ( r ) + + N 1 n n ( r ) ) = 0 , H 2 ( ϕ 1 ( r ) , ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) = r 2 ϕ 2 ( r ) + 2 r ϕ 2   ( r ) + p r 2 ( N 21   1 ( r ) + N 22 2 ( r ) + N 23 3 ( r ) + + N 2 n n ( r ) ) = 0 , H 3 ( ϕ 1 ( r ) , ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) = r 2 ϕ 3 ( r ) + 2 r ϕ 3   ( r ) + p r 2 ( N 31   1 ( r ) + N 32 2 ( r ) + N 33 3 ( r ) + + N 3 n n ( r ) ) = 0 , H 1 ( ϕ 1 ( r ) , ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) = r 2 ϕ n ( r ) + 2 r ϕ n   ( r ) + p r 2 ( N n 1   1 ( r ) + N n 2 2 ( r ) + N n 3 3 ( r ) + + N n n n ( r ) ) = 0 .
while the identical powers of p are:
p 0   :   r 2 ϕ i 0 ( r ) + 2 r ϕ i 0 ( r ) = 0 , ϕ i 0   ( 0 ) : finite , p 1   :   r 2 ϕ i 1 ; 1 ( r )   + 2 r ϕ i 1   ( r )   =   r 2   N 11   i 0 ( r ) + N 12 i 2 ( r ) + N i 3 3 ( r ) + + N 1 , n n ( r ) ) ,   ϕ i , 1   ( 0 )   =   0 , p 2   :   r 2 ϕ i 2   ( r ) +   2 r ϕ i 2   ( r ) = r 2   N 11 i 1 ( r ) + N 22 i 2 ( r ) + N i 3 i 3 ( r ) + + N 2 n n ( r ) ) ,   ϕ i 2   ( 0 )   =   0 , p k   :   r 2 ϕ i k   ( r )   +   2 r ϕ i k   ( r )   =   r 2   N k 1 1 ( r ) + N k 2 2 ( r ) + + N k k k ( r ) + + N k n n ( r ) ) ,   ϕ i k   ( 0 )   =   0
Then,
T i , 0 = I i , T i , n = N i i   T i , n 2 + N i 1 T 1 , n 2 + N i 2 T 2 , n 2 + + N i 3 T 3 , n 2 + + . . . + N i n T n , n 2 .
The solution of the first component of Equation (7) is:
ϕ i , 0   ( r ) = I i ,
Similarly, we obtain:
ϕ i , 1   ( r )   = T i . 2 3 ! r 2   , ϕ i , 2   ( r )   = T i . 4 5 ! r 4   , ϕ i , k   ( r ) = ( 1 ) k T i . 2 k ( 2 k + 1 ) ! r 2 k ,
In summary, the fluxes, in this case, are given by:
ϕ i   ( r )   = k = 0 ( 1 ) k T i . 2 k ( 2 k + 1 ) ! r 2 k
Now, multi-group neutron diffusion equations for alternate buckling in the reactor core part will be:
2 1 ( r ) L 11   1 ( r ) L 12 2 ( r ) L 13 3 ( r ) L 1 n n ( r ) = 0 , 2 2 ( r ) L 21 1 ( r ) L 22 2 ( r ) L 23 3 ( r ) L 2 n n ( r ) = 0 , 2 3 ( r ) L 31 1 ( r ) L 32 2 ( r ) L 33 3 ( r ) L 3 n n ( r ) = 0 , 2 n ( r ) L n 1 1 ( r ) L n 2 2 ( r ) L n 3 3 ( r ) L n n n ( r ) = 0 ,  
where L i j is a constant connects between fluxes in different energy groups of neutrons. Respectively, L i i , L i j , and D i are defined as:
{ L i i = χ i ν i f i ( γ i + Σ s i j ) D i ,                       L i j = Σ s j i + χ i ν j f j D i ,                                                     D i = 1 3 ( f i + s i i + Σ s i j + γ i ) .  
Now, the diffusion system of multi energy groups of neutrons at spherical reactor can be written as:
r 1 ( r ) + 2 1 ' ( r ) + ( r L 11   1 ( r ) r L 12 2 ( r ) r L 13 3 ( r ) r L 1 n n ( r ) ) = 0 ,     r 2 ( r ) + 2 2 ' ( r ) + ( r L 21 1 ( r ) r L 22 2 ( r ) r L 23 3 ( r ) r L 2 n n ( r ) ) = 0 ,   r 3 ( r ) + 2 3 ' ( r ) + ( r L 31 1 ( r ) r L 32 2 ( r ) r L 33 3 ( r ) r L 3 n n ( r ) ) = 0 ,   r n ( r ) + n ' ( r ) + ( L n 1 1 ( r ) r L n 2 2 ( r ) r L n 3 3 ( r ) r L n n n ( r ) ) = 0 .
One more time, the mathematical form of initial conditions can be written as:
i ( 0 ) = I , i   ( 0 ) = 0 , i = 1 , 2 , , n .
In order to solve these equations using HPM, we construct the homotopy as:
p L 1 ( ϕ 1 ( r ) ,   ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) ) .
To obtain a solution for this subsystem using HPM, the homotopy will be:
Thus,
  H 1 ( ϕ 1 ( r ) , ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) = r 2 ϕ 1 ( r ) + 2 r ϕ 1   ( r ) + p r 2 ( L 11   1 ( r ) L 12 2 ( r ) L 13 3 ( r ) L 1 n n ( r ) ) = 0 , H 2 ( ϕ 1 ( r ) , ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) = r 2 ϕ 2 ( r ) + 2 r ϕ 2   ( r ) + p r 2 ( L 21   1 ( r ) L 22 2 ( r ) L 23 3 ( r ) L 2 n n ( r ) ) = 0 , H 3 ( ϕ 1 ( r ) , ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) = r 2 ϕ 3 ( r ) + 2 r ϕ 3   ( r ) + p r 2 ( L 31   1 ( r ) L 32 2 ( r ) L 33 3 ( r ) L 3 n n ( r ) ) = 0 , H 1 ( ϕ 1 ( r ) , ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) = r 2 ϕ n ( r ) + 2 r ϕ n   ( r ) + p r 2 ( L n 1   1 ( r ) L n 2 2 ( r ) L n 3 3 ( r ) L n n n ( r ) ) = 0 .
Taking identical powers of p as:
p 0 :   r 2 ϕ i 0 ( r ) + 2 r ϕ i 0   ( r ) ,   ϕ i 0   ( 0 ) :   finite , p 1 :   r 2 ϕ i 1 ; 1 ( r )   +   2 r ϕ i 1     ( r )   =   r 2   ( L 11   i 0 ( r ) + L 12 i 2 ( r ) + L 13 3 ( r ) + + L 1 n n ( r ) ) ,   ϕ i 1   ( 0 )   =   0 , p 2 :   r 2 ϕ i 2   ( r )   +   2 r ϕ i 2     ( r )   = r 2   ( L i 0 i 1 ( r ) + L 22 i 2 ( r ) + L i 3 i 3 ( r ) + + L 2 n n ( r ) ) ,   ϕ i 2   ( 0 )   =   0 , p k :   r 2 ϕ i k   ( r )   +   2 r ϕ i k   ( r ) =   r 2   ( L k 1 1 ( r ) + L k 2 2 ( r ) + + L k k   k ( r ) + + L k n n ( r ) ) ,   ϕ i k   ( 0 )   =   0 ,
Now,
T i , 0 = I i , T i , n = N i i   T i , n 2 + N i 1 T 1 , n 2 + N i 2 T 2 , n 2 + + N i 3 T 3 , n 2 + + . . . + N i n T n , n 2 + . ,
The solution of first component of Equation (18) is given by:
ϕ i , 0   ( r ) = I i ,
The other components are:
ϕ i , 1   ( r ) = T i . 2 3 ! r 2 , ϕ i , 2   ( r ) = T i . 4 5 ! r 4 , ϕ i , k   ( r ) = T i . 2 k ( 2 k + 1 ) ! r 2 k ,
In summary, the fluxes of this part are:
ϕ i   ( r ) = k = 0 T i . 2 k ( 2 k + 1 ) ! r 2 k
As is mentioned, the total flux of any group in the core part is linear combination of the two cases as:
ϕ i   ( r )   = U k = 0 T i . 2 k ( 2 k + 1 ) ! r 2 k   + V k = 0 ( 1 ) k T i . 2 k ( 2 k + 1 ) ! r 2 k
Clearly, all fluxes should satisfy the needed boundary conditions.

2.2. The Reactor Reflected Part

After the core part, we study the reactor reflector; the multi-group neutron diffusion equations of this subsystem [3] is:
2 1 ( r ) M 11   1 ( r ) M 12 2 ( r ) M 13 3 ( r ) M 1 n n ( r ) = 0 , 2 2 ( r ) M 21 1 ( r ) M 22 2 ( r ) M 23 3 ( r ) M 2 n n ( r ) = 0 , 2 3 ( r ) M 31 1 ( r ) M 32 2 ( r ) M 33 3 ( r ) M 3 n n ( r ) = 0 , 2 n ( r ) M n 1 1 ( r ) M n 2 2 ( r ) M n 3 3 ( r ) M n n n ( r ) = 0 ,  
Each M i , j is a real (positive, zero, or negative) constant:
{ M i , i = ( γ i + Σ s i j ) D i , M i j = Σ s j i D i , D i = 1 3 ( s i i + Σ s i j + γ i ) .  
Applying Laplacian in spherical coordinates:
r 1 ( r ) + 2 1   ( r ) r ( M 11 1 ( r ) + M 12 ( r ) 2 + M 13 ( r ) 3 + + M 1 n n ( r ) ) = 0 , r 2 ( r ) + 2 2   ( r ) r ( M 21 1 ( r ) + M 22 2 ( r ) + M 23 3 ( r ) + + M 2 n n ( r ) ) = 0 ,   r 3 ( r ) + 2 3   ( r ) r ( M 31 1 ( r ) + M 32 2 ( r ) + M 33 3 ( r ) + + M 3 n n ( r ) ) = 0 ,   r n ( r ) + n   ( r ) r ( M n 1 1 ( r ) + M n 2 2 ( r ) + M n 3 3 ( r ) + + M n n   n ( r ) ) = 0 .
Now, the homotopy [6,16] will be:
H 1 ( ϕ 1 ( r ) ,   ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) = r 2 ϕ 1 ( r ) + 2 r ϕ 1   ( r ) + p r 2 ( M 11 1 ( r ) M 12 2 ( r ) M 13 3 ( r ) M 1 n n ( r ) ) =   0 , H 2 ( ϕ 1 ( r ) ,   ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) = r 2 ϕ 2 ( r ) + 2 r ϕ 1   ( r ) + p r 2 ( M 21 1 ( r ) M 22 2 ( r ) M 23 3 ( r ) M 2 n n ( r ) ) =   0 , H 3 ( ϕ 1 ( r ) ,   ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) = r 2 ϕ 3 ( r ) + 2 r ϕ 3   ( r ) + p r 2 ( M 31 1 ( r ) M 32 2 ( r ) M 33 3 ( r ) M 3 n n ( r ) ) =   0 , H n ( ϕ 1 ( r ) ,   ϕ 2 ( r ) , ϕ 3 ( r ) ϕ n ( r ) , p ) = r 2 ϕ n ( r ) + 2 r ϕ n   ( r ) + p r 2 ( M n 1 1 ( r ) M n 2 2 ( r ) M n 3 3 ( r ) M n n n ( r ) ) =   0
Here, we obtain identical powers of p terms as a set of equations:
p 0   :   r 2 ϕ i 0 ( r ) + 2 r ϕ i 0 ( r ) ,   p 1   :   r 2 ϕ i 1   ( r )   +   2 r ϕ i 1   ( r )   =   r 2   ( M 11     1 ( r ) M 12 2 ( r ) M 13 3 ( r ) M 1 n n ( r ) ) ,   p 2   :   r 2 ϕ i 2   ( r )   +   2 r ϕ i 2   ( r )   = r 2   ( M 21 1 ( r ) M 22 2 ( r ) M 23 3 ( r ) M 2 n n ( r ) ) ,   p k   :   r 2 ϕ i k   ( r )   +   2 r ϕ i k   ( r )   =   r 2   ( M k 1 1 ( r ) + M k 2 2 ( r ) + + M k k   k ( r ) + + M k n   n ( r ) ) ,  
The solution of the first component of Equation (28) is:
φ i , 0   ( r ) = A i , 0   +   B i , 0 r ,
The following constants ( A i , n ) depend on ( A i , 0 ) as:
A i , k = j = 1 n M i j   A j , k 1 ,  
In addition, ( B i , n ) depends on ( B i , 0 ) as:
B i , k = j = 1 n M i , j   A j , k 1 ,  
Then,
φ i , 0   ( r ) = A i , 0 + B i , 0   1 r φ i , 1   ( r ) = A i , 1 r 2 3 ! + B i , 1 r 2 + C i , 0 + D i , 0 1 r φ i , 2   ( r ) = A i , 2 r 4 5 ! + B i , 2 r 3 4 ! + C i , 1 r 2 3 ! + D i , 1 r 2 + E i , 0 + F i , 0 1 r φ i , 3   ( r ) = A i , 3 r 6 7 ! + B i , 3 r 5 6 ! + C i , 2 r 4 5 ! + D i , 2 r 3 4 ! + E i , 1 r 2 3 ! + F i , 0 r 2 + + G i , 0 + H i , 0 1 r
We can define ( C i , n , D i , n ,   F i , n ,…) in the same way as ( A i , n , and B i , n ), so:
φ i ( r ) = 1 r   [ ( A i , 0 r 1 ! +   A i , 1 r 3 3 ! + A i , 2 r 5 5 ! + A i , 3 r 7 7 ! + ) + ( B i , 0 +   B i , 1 r 2 2 +   B i , 2 r 4 4 ! +   B i , 3 r 6 6 ! + ) +   ( C i , 0 r 1 ! +   C i , 1 r 3 3 ! + C i , 2 r 5 5 ! + C r 7 7 ! + ) + ( D i , 0 +   D r 2 2 +   D i , 2 r 4 4 ! +   D i , 3 r 6 6 ! + ) + ]
On the other hand,
φ i ( r ) = 1 r   [ ( { A i , 0 + C i , 0 + E i , 0 + } r 1 ! + { A i , 1 + C i , 1 + E i , 1 + } r 3 3 ! + { A i , 2 + C i , 2 + E i , 2 + } r 5 5 ! + { A i , 3 + C i , 3 + E i , 3 + } r 7 7 ! + ) + ( { B i , 0 + D i , 0 + F i , 0 + } + { B i , 1 + D i , 1 + F i , 1 + } r 2 2 + { B i , 2 + D i , 2 + F i , 2 + } r 4 4 ! + { B i , 3 + D i , 3 + F i , 3 + } r 6 6 ! + ) + ]
Let
α i , k = A i , k + C i , k + E i , k + β i , k = B i , k + D i , k + F i , k +
Its consequence is that:
α i , k = j = 1 n M i , j   α i , k 1 ,   β i , k = j = 1 n M i , j   β i , k 1 .
The final solution of Equation (34) is given by:
φ i ( r ) = 1 r [ k = 0 α i , k r 2 k + 1 ( 2 k + 1 ) ! + k = 0 β i , k r 2 k ( 2 k ) ! ]

2.3. The Core-Reflector Boundary Conditions

After finding the solution of neutron diffusion equations in the reactor core and reflector parts, it is essential to apply the boundary condition, the point R , on the surface between them.
The neutron fluxes ( φ c i ( r ) ) must be continuous as well as their currents ( J i (r)) [24], which mathematically can be expressed as:
φ c i   ( R ) = φ r i   ( R ) ,   J c i   ( R ) = J r i   ( R ) ,  
Neutron currents in Equation (38) will be defined as:
J i   ( r ) ) = D i φ i     ( r ) .
Therefore, after inserting the values of fluxes and currents in Equation (38), this system of equations can be written as:
U k = 0 T i . 2 k ( 2 k + 1 ) ! R 2 k   + V k = 0 ( 1 ) k T i . 2 k ( 2 k + 1 ) ! R 2 k = 1 R [ k = 0 α i , k R 2 k + 1 ( 2 k + 1 ) ! + k = 0 β i , k R 2 k ( 2 k ) ! ] , D i c d d r ( U k = 0 T i . 2 k ( 2 k + 1 ) ! R 2 k   + V k = 0 ( 1 ) k T i . 2 k ( 2 k + 1 ) ! R 2 k ) = D i r d d r ( 1 R [ k = 0 α i , k R 2 k + 1 ( 2 k + 1 ) ! + k = 0 β i , k R 2 k ( 2 k ) ! ] ) .
By solving this system of equations computationally, we can find R.

3. Special Cases Numerical Study

The theoretical results reached in Section 2 will be simplified to compare numerically with special cases that were obtained [1,2,3]; this comparison will assure the theory. Special cases start with one-group and two-group, and finally, the multi-group is taken as a general case.

3.1. One-Group Nuclear Reactor

Here, a one-group bare reactor (which has fuel only) is studied; then, we focus on the reflector reactor (core and reflected parts). The needed cross-section data are originally taken from [25], which clarified them also, and [2] used cross-section data only. Theoretical and numerical results for both bare and reflected reactors will be compared with modified differential transform method (MDTM).

3.1.1. One-Group Bare Nuclear Reactor

Now, the neutron diffusion equation of a one-group bare reactor [24] can be written as:
2   ( r ) +   B 2   ( r ) = 0 ,
where B 2 = ( ν Σ f     ( Σ f   + Σ γ ) ) / D is the buckling, and D = 1 / 3 ( Σ f   + Σ s + Σ γ ) is the diffusion coefficient [24].
After applying HPM, the flux will be:
  ( r ) = A r k = 0 ( 1 ) k ( B r ) 2 k + 1 ( 2 k + 1 ) ! = A r s i n ( B r )
One-group bare spherical reactor is taken numerically when 1-MeV neutrons diffuse in pure 235U [2] the MDTM, which will be compared with this studied case.
At this point, necessary nuclear reactor data [2,25] are found in Table 1.
The essential critical radius result using Mathematica software is tabulated in Table 2.
Critical radius calculation using HPM, where zero flux boundary condition is used, gives the same results as MDTM, with the flux behavior tabulated in Table 3 graphed in Figure 1.
After applying HPM, the flux has its normalization value at the sphere center, and it has its zero value at the critical radius, and this is the expected behavior; here, MDTM results are reproduced.

3.1.2. One-Group Reflected Nuclear Reactor

The one-group reflected reactor [6] is taken after a bare reactor, and the neutron diffusion equations for core and reflected parts are:
2 c ( r ) +   B 2 c ( r ) =   0 , 2 r ( r ) 1 L 2 r ( r ) = 0 ,
where   B 2 is buckling, and L 2 is called diffusion area, and L is diffusion length [24]
The core and reflected parts fluxes, after applying HPM, will be:
c ( r ) =   A c r k = 0 ( 1 ) k ( B r ) 2 k + 1 ( 2 k + 1 ) ! = A c r s i n B r , r ( r ) = A r r k = 0 ( r L ) 2 k + 1 ( 2 k + 1 ) ! = A r r s i n h r L .
The solution of the neutron diffusion equation is obtained to study this numerical example, where U 235 is the core part, and H 2 O is the reflector of this reactor. More needed cross-sections [2] are given in Table 4.
For large reflector, the flux r ( r ) will be
  ( r ) = A r e r L
The critical radius of the reflected spherical reactor is compared with both MDTM and transport theory data, and the transport theory can be admitted as a benchmark where the diffusion theory is approximation when Fick’s law is used; after using of Mathematica software, the result is tabulated in Table 5.
The critical radius of the one-group reflected reactor, when zero flux boundary condition is used, is reproduced as the results of MDTM and has a good agreement with transport theory data. After this, flux behavior will be studied in both core and reflected parts, which are described in Table 6 and Figure 2.
The flux distribution using HPM, which has the same MDTM results, using core reflector boundary conditions, and it is clear that flux behavior is in good agreement with classical calculations [3,24].

3.2. Two-Group Nuclear Reactor

After one-group study, the two-group case for bare and reflected reactors are considered; neutrons are divided in two-groups: fast and thermal groups. Numerical results for the bare reactor are compared with the residual power series method (RPSM) [1], while the reflected reactor is compared with the classical method [3].

3.2.1. Two-Group Bare Nuclear Reactor

Now, a bare reactor will be obtained when neutrons move in fast and thermal velocities, and neutron diffusion equations [3] will be:
2 1 ( r ) + N 11 1 ( r ) + N 12 2 ( r ) = 0 ,     2 2 ( r ) + N 21 1 ( r ) + N 22 2 ( r ) = 0 .  
Fluxes, after using HPM, can be written as:
i ( r ) = k = 0 ( 1 ) k 1 ( 2 k + 1 ) ! T i , 2 k r 2 k ,   i = 1 , 2 .
Computational determination of analytical results is useful in comparing them with RPSM [1]; where Mathematica software is used, the important cross-section is taken from [1], where neutrons undergo fast and thermal diffusion in uranium with enrichment ratio is 93%. Two-group bare reactor cross-sections data in Table 7 are taken from [25], and [1] used them, while [25] gave meaning to each.
Values of N 11 , N 12 , N 21 ,   and   N 22 are determined depending on these cross-sections as shown in Table 8.
For a two-group bare reactor, the critical radius is obtained using both zero flux (ZF) and extrapolated boundary conditions (EBC), and it is found that fluxes vanish before a certain distance, which is called extrapolated distance, where EBC is named; next, data are compared with RPSM, and transport theory data are taken as a benchmark.
Here, 93% enriched uranium is taken as a numerical example, and the critical radius is tabulated in Table 9.
This critical radius of two-group bare reactor of HPM shows that when RPSM is used, both of them have logical results with the benchmark (Transport Theory).
The behavior of fluxes in the two-group bare reactor is found in Table 10 and Figure 3.
HPM has the same performance as RPSM in tabulated and graphical representation of the two-group bare reactor, where both fluxes and their sum converge at the same point.
As it is explained, the real critical dimension calculated by transport theory data is less than the HPM calculated point, and this is reasonable after applying Fick’s law approximation.

3.2.2. Two-Group Reflected Nuclear Reactor

A two-group reflected reactor will be studied here; unlike the bare reactor, the buckling is not unique in its principal and alternate buckling for the core part of the reactor.
The neutron diffusion equations [3] corresponding to principal buckling are:
2 1 ( r ) + N 11 1 ( r ) + N 12 2 ( r ) = 0 ,     2 2 ( r ) + N 21 1 ( r ) + N 22 2 ( r ) = 0 .
Similarly, this for alternate buckling will be:
2 1 ( r ) L 11   1 ( r ) L 12 2 ( r ) = 0 , 2 2 ( r ) L 21 1 ( r ) L 22 2 ( r ) = 0 .
After applying HPM for each case, the solution is a linear combination of both cases, which can be written as:
ϕ i c   ( r )   = U k = 0 T i . 2 k ( 2 k + 1 ) ! r 2 k   + V k = 0 ( 1 ) k T i . 2 k ( 2 k + 1 ) ! r 2 k ,   i = 1 , 2 .
Furthermore, the reflector of neutron diffusion equations will be:
2 1 ( r ) M 11     1 ( r ) M 12 2 ( r ) = 0 , 2 2 ( r ) M 21 1 ( r ) M 22 2 ( r ) = 0 .
After taking into consideration that there is no fission in the reflector and no upper scattering case, this is a known case that will be studied numerically: constant M 12   = 0 .
The fast-group flux will be:
φ 1 r ( r ) = 1 r [ k = 0   r 2 k + 1 ( 2 k + 1 ) ! ] .
while thermal flux is:
φ 2 r ( r ) = 1 r [ k = 0 α i , k r 2 k + 1 ( 2 k + 1 ) ! + k = 0 β i , k r 2 k ( 2 k ) ! ]
After finding the solution of neutron diffusion equations in core and reflector parts, it is essential to apply the boundary conditions in the surface between reactor core and reflector, where fluxes ( φ 1 (x), φ 2 (x)) and their currents ( J 1 (r),   J 2 (r)) are continuous [3], which mathematically can be expressed as:
φ c 1   ( R )   =   φ r 1   ( R ) ,   J c 1   ( R )   =   J r 1   ( R ) ,   φ c 2   ( R )   =   φ r 2   ( R ) ,   J c 2   ( R )   =   J r 2   ( R ) ,
Hence, the neutron currents are J 1 (r)) = − D 1   φ 1   (r),   J 2 (r)) = − D 2   φ 2   (r).
The derived analytical formalism is computationally determined to verify the theory, while cross-section data are taken from [3] which given in Table 11.
Values of N i j   and   L i j   i , j = 1 , 2 are shown in Table 12.
Classical calculations completely depend on finding the buckling, principal, and alternate buckling; for a more than two-group reactor, this calculation will be more complicated.
HPM studies the two-group reflected reactor and, in general, any number of group reactor uses more advance and accurate method for determination of critical radius, and clarification of the fluxes distribution depends on the combination between fluxes’ constants; these constants depend on all groups’ cross sections.
Necessary Mathematica software is used to numerically reach the critical radius for the two-group reflected reactor; critical radius of the reactor core can be found in Table 13.
After finding the critical radius for this system, fast and thermal fluxes and their sum are considered in Table 14 and Figure 4.
Table 14 gives fast and thermal fluxes and their total flux values, and thus, it is obvious that total flux decreases when the reactor radius increases and vanishes at reflector radius.
As we compare HPM results and classical results, it is seen that the HPM critical dimension is less than that of the classical method, and the total flux converges faster, which reduces the fuel and improves the reactor fission. This can be one step forward, using HPM, in accuracy of critical dimension calculation and flux distribution determination.

3.3. Multi-Group Nuclear Reactor

The four-group of neutrons diffusion equations case [1], which is an example of a multi-group reactor, is discussed as one step forward, which is represented by the following system:
2 1 ( r ) + N 11 1 ( r ) + N 12 2 ( r ) + N 13 3 ( r ) + N 14 4 ( r ) = 0 , 2 2 ( r ) + N 21 1 ( r ) + N 22 2 ( r ) + N 23 3 ( r ) + N 24 4 ( r ) = 0 ,  
with initial conditions:
i ( 0 ) = I i ,   i   ( 0 ) = 0 , i = 1 , 2 , 3 , 4 .
Hence, the solutions of i th four-group reactor flux according to HPM are given by
i ( r ) = k = 0 ( 1 ) k 1 ( 2 k + 1 ) ! T i , 2 k r 2 k .
Mathematica software is used in numerical solutions; the solution is obtained numerically for cross sections related to interactions of the four-group system. The following data obtained from [1] are correspondingly used in Table 15.
Values of N i j ,   i , j = 1 , 2 , 3 , 4 are shown in Table 16 is derived from Table 15 data.
The critical radius of a four-group reactor is calculated depending on ZF and EBC boundary conditions, and generated data are listed in Table 17.
Fluxes values of the four-group reactor are given in Table 18 and Figure 5.
Both tabulated and graphical representation show that all fluxes decrease when the reactor radius increases and vanishes at the critical radius, and this expected behavior reproduces RPSM.

4. Conclusions

The application of HPM, as an approximation method, in the reflected spherical reactor to solve diffusion equations of a multi-group system is accomplished in this work. To assure the theory, the solutions are simplified and compared with RPSM, MDTM, and classical theory. The results can be easily reproduced with the approximation methods, while flux converged faster when compared with the classical calculations; this improves the efficiency of the reactor by reducing the critical mass and dimensions using HPM. We can assure that the utilities of HPM in this work and previous studies still has the capability to solve different branches of science problems.

Author Contributions

Conceptualization, M.S.; methodology, M.A.-S.; software, E.A.M.F.; validation, M.S., E.A.M.F. and M.A.-S.; investigation, M.S. and E.A.M.F., writing—original draft preparation, M.S.; writing—review and editing, M.S.; visualization, E.A.M.F.; supervision, M.A.-S.; project administration, M.S.; funding acquisition, M.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University, grant number 2021/01/17993 and The APC was funded by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University.

Data Availability Statement

Not applicable.

Acknowledgments

The authors extend their appreciation to the Deputyship for Research and innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number (IF-PSAU/2021/01/17993).

Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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Figure 1. Flux distribution in one-group bare reactor.
Figure 1. Flux distribution in one-group bare reactor.
Mathematics 10 01784 g001
Figure 2. Flux distribution of one-group reflected reactor. Blue, core flux; Red, reflector flux.
Figure 2. Flux distribution of one-group reflected reactor. Blue, core flux; Red, reflector flux.
Mathematics 10 01784 g002
Figure 3. Fluxes distribution in two-group bare reactor. Blue, thermal flux; Red, fast flux; Green, total flux.
Figure 3. Fluxes distribution in two-group bare reactor. Blue, thermal flux; Red, fast flux; Green, total flux.
Mathematics 10 01784 g003
Figure 4. Flux distribution in the core and reflected parts of two-group reflected reactor. Black, 1 c ( r ) ; Blue, 2 c ( r ) ; Red, 1 r ( r ) ; Yellow, 2 r ( r ) ; Green, tot . c ( r ) ; Gray, tot . r ( r ) .
Figure 4. Flux distribution in the core and reflected parts of two-group reflected reactor. Black, 1 c ( r ) ; Blue, 2 c ( r ) ; Red, 1 r ( r ) ; Yellow, 2 r ( r ) ; Green, tot . c ( r ) ; Gray, tot . r ( r ) .
Mathematics 10 01784 g004
Figure 5. Four-group fluxes and total flux. Blue ,   1 ( r ) ; Red, 2 ( r ) ; Green, 3 ( r ) ; Pink, 4 ( r ) ; Black, Total flux.
Figure 5. Four-group fluxes and total flux. Blue ,   1 ( r ) ; Red, 2 ( r ) ; Green, 3 ( r ) ; Pink, 4 ( r ) ; Black, Total flux.
Mathematics 10 01784 g005
Table 1. One-group bare reactor cross-sections data.
Table 1. One-group bare reactor cross-sections data.
ν σ f σ s σ γ
2.421.3365.9590.153
N = 0.0478 × 1024 atoms cm−3
Table 2. The critical radius of one-group bare reactor.
Table 2. The critical radius of one-group bare reactor.
HPMMDTM
Critical radius10.52810.528
Table 3. Normalized flux in one-group bare reactor.
Table 3. Normalized flux in one-group bare reactor.
Method r / a c 0.0 0.25 0.50 0.75 1.0
ClassicalFlux 1.0000 0.9069 0.6563 0.3309 0.03109
RPSMFlux 1.0000 0.9069 0.6563 0.3309 0.03109
Table 4. One-group reflected reactor cross-sections data.
Table 4. One-group reflected reactor cross-sections data.
ν c Σ c f Σ c s Σ c γ
2.79710.065280.248060.01306
ν r Σ r f Σ r s Σ r γ
0.00000.00000.249370.03264
Table 5. Critical radius of one-group reflected reactor.
Table 5. Critical radius of one-group reflected reactor.
HPMMDTMTransport Theory Data
Critical radius6.51286.51286.1275
Table 6. Normalized flux in one-group reflected reactor.
Table 6. Normalized flux in one-group reflected reactor.
Method r / a c 0.0 0.25 0.50 0.75 1.0
ClassicalFlux 1.0000 0.9567 0.8301 0.6415 0.4199
RPSMFlux 1.0000 0.9567 0.8301 0.6415 0.4199
Table 7. Two-group bare reactor cross-sections data.
Table 7. Two-group bare reactor cross-sections data.
f 1 = 0.0010484   cm 1 γ 1 = 0.0010046   cm 1 S 11 = 0.62568   cm 1
S 12 = 0.029227 cm 1 ν 1 = 2.5 χ 1 = 1.0
f 2 = 0.050632   cm 1 γ 2 = 0.025788   cm 1 S 22 = 2.44383   cm 1
S 21 = 0.0000   cm 1 ν 2 = 2.5 χ 2 = 0.0
Table 8. Two-group bare reactor fluxes coefficients.
Table 8. Two-group bare reactor fluxes coefficients.
N 11 = 0.0564834 N 12 = 0.249474 N 21 = 0.220978 N 22 = 0.577793
Table 9. Critical radius of two-group bare reactor.
Table 9. Critical radius of two-group bare reactor.
BCHPMRPSM Transport Theory Date
ZF 17.120 17.120 -
EBC 16.251 16.251 16.049836
Table 10. Normalized thermal, fast, and total fluxes in two-group bare reactor.
Table 10. Normalized thermal, fast, and total fluxes in two-group bare reactor.
FluxMethod r / a c 0.0 0.25 0.50 0.75 1.0
FastClassicalFlux 2.7671 2.5238 1.7802 1.0063 0.1835
RPSMFlux 2.7671 2.5238 1.7802 1.0063 0.1835
ThermalClassicalFlux 1.0000 0.9121 0.6759 0.3637 0.0663
RPSMFlux 1.0000 0.9121 0.6759 0.3637 0.0663
TotalClassicalFlux 3.7671 3.4359 2.5460 . 1.36990.2498
RPSMFlux 3.7671 3.4359 2.5460 1.36990.2498
Table 11. Two-group reflected reactor cross-sections data.
Table 11. Two-group reflected reactor cross-sections data.
Core data
τ C = 45.3   cm 2 R 1 = 0.03179   cm 1 a 2 = 0.1104   cm 1
p C = 1.0 D 1 = 1.44   cm   D 2 = 0.229   cm  
Reflector data
τ R = 40.0   cm 2 a = 1.85   cm 1 a 2 = 0.01226 cm 1
p R = 1.0 D 1 = 1.85   cm   D 2 = 0.204   cm  
Table 12. Two-group reflected reactor fluxes coefficients.
Table 12. Two-group reflected reactor fluxes coefficients.
N 11 =   0.022076. N 12 = 0.100893 N 21 = 0 . 1388210 N 22 =   0.482096
L 11 = 0.025000 L 12 = 0.000000 L 21 = 0.226716 L 22 =   0.060098
Table 13. Critical radius of two-group core part of reflected reactor.
Table 13. Critical radius of two-group core part of reflected reactor.
BCHPMClassical Method
Critical radius30.71331.9
Table 14. Two-group reflected reactor fluxes and total flux.
Table 14. Two-group reflected reactor fluxes and total flux.
FluxMethod r / a c 0.0 0.25 0.50 0.75 1.0
Group 1HPMFlux 1.00 00 0.94312 0.765910.514920.23573
Classical methodFlux 1.0000 0.93730 0.763600.516980.24140
Group 2HPMFlux0.2860 0.26791 0.217570.146540.11478
Classical methodFlux0.28410.266380.217020.147170.11286
Total HPMFlux1.28601.211030.983480.661460.35051
Classical methodFlux1.28411.203680.980620.664150.35426
Table 15. Four-group reflected reactor cross-sections data.
Table 15. Four-group reflected reactor cross-sections data.
Group 1
(1.35 Mev–10 Mev)
ν 1 f 1 = 0.0096   cm 1 a = 0.0049   cm 1 S 12 = 0.0831   cm 1
S 13 = 0.00 S 14 = 0.00 D 1 = 2.162   cm  
χ 1 = 0.575
Group 2
(9.1 keV–1.35 Mev)
ν 2 f 2 = 0.0012   cm 1 a 2 = 0.0028   cm 1 S 21 = 0.00   cm 1
S 23 = 0.0.0585   c m 1 S 24 = 0.00   cm 1 D 2 = 1.087   cm
χ 2 = 0.425
Group 3
(0.4 ev–9.1 kev)
ν 3 f 3 = 0.0.0177   cm 1 a 3 = 0.0.0305   cm 1 S 31 = 0.00   cm 1
S 32 = 0.00 cm 1 S 34 = 0.0651   cm 1 D 3 = 0.632   cm
χ 3 = 0.0
Group 4
(0.0 ev–0.4 ev)
ν 4 f 4 = 0.1851   cm 1 a 4 = 0.1210   cm 1 S 41 = 0.00   cm 1
S 42 = 0.00 cm 1 S 43 = 0.00 cm 1 D 4 = 0.354   cm
χ 4 = 0.0
Table 16. Four-group fluxes coefficients.
Table 16. Four-group fluxes coefficients.
N 11 = 0.038150 N 12 = 0.000319 N 13 = 0.004707 N 14 = 0.049229
N 21 = 0 .080202 N 22 = 0.055925 N 23 = 0.083370 N 24 = 0.148820
N 31 = 0.092563 N 32 = 0.092563 N 33 = 0.151266 N 34 = 0.092563
N 41 = 0.183898 N 42 = 0.183898 N 43 = 0.183898 N 44 = 0.341808
Table 17. Critical radius of four-group reactor.
Table 17. Critical radius of four-group reactor.
BCHPMRPSM
ZF8.7708.770
EBC7.9057.905
Table 18. Four-group fluxes and total flux reactor.
Table 18. Four-group fluxes and total flux reactor.
FluxMethod r / a c 0.0 0.25 0.50 0.75 1.0
Group 1HPMFlux 1.00 00 0.918535 0.697804 0.400643 0.107646
RPSMFlux 1.0000 0.918535 0.697804 0.400643 0.107646
Group 2HPMFlux4.1716 3.83172 2.91093 1.67131 0.449052
RPSMFlux4.1716 3.83172 2.91093 1.67131 0.449052
Group 3HPMFlux2.7361 2.5132 1.90926 1.0962 0.294531
RPSMFlux2.7361 2.5132 1.90926 1.0962 0.294531
Group 4HPMFlux3.0931 2.84115 2.1584 1.23924 0.332964
RPSMFlux3.0931 2.84115 2.1584 1.23924 0.332964
TotalHPMFlux11.0008 10.1046 7.67639 4.40739 1.18419
RPSMFlux11.0008 10.1046 7.67639 4.40739 1.18419
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Shqair, M.; Farrag, E.A.M.; Al-Smadi, M. Solving Multi-Group Reflected Spherical Reactor System of Equations Using the Homotopy Perturbation Method. Mathematics 2022, 10, 1784. https://0-doi-org.brum.beds.ac.uk/10.3390/math10101784

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Shqair M, Farrag EAM, Al-Smadi M. Solving Multi-Group Reflected Spherical Reactor System of Equations Using the Homotopy Perturbation Method. Mathematics. 2022; 10(10):1784. https://0-doi-org.brum.beds.ac.uk/10.3390/math10101784

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Shqair, Mohammad, Emad A. M. Farrag, and Mohammed Al-Smadi. 2022. "Solving Multi-Group Reflected Spherical Reactor System of Equations Using the Homotopy Perturbation Method" Mathematics 10, no. 10: 1784. https://0-doi-org.brum.beds.ac.uk/10.3390/math10101784

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