Using the Invariance Principle to Solve Neutron Diffusion Problems on Analog Computers
Abstract Generated abstract
This paper applies Ambartsumian’s invariance principle to a one-dimensional neutron diffusion problem in a nuclear reactor with fast and thermal neutron groups. By separating neutron fluxes into leftward and rightward components, the authors derive coupled balance equations and introduce a reflection matrix relating oppositely directed fluxes. They obtain a nonlinear matrix differential equation expressing invariant imbedding, which transforms the original boundary-value diffusion problem into a sequence of initial-value problems. The formulation is presented as suitable for solution on an analog computer, allowing the neutron diffusion problem to be computed in a single operation.
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Use of the Invariance Principle for Solving Neutron Diffusion Problems on Analog Computers
B. S. Kostin, I. G. Vintizenko
(Presented by the scientific seminar of the Computing Laboratory of TPI)
In 1943 V. A. Ambartsumian introduced a new methodology for studying problems of atmospheric scattering. The method was developed in subsequent works by Ambartsumian, Chandrasekhar [2], and Sobolev. It can be successfully applied in the theory of neutron diffusion. A number of such problems are considered in the works of Bellman [3]. In the present work we shall show the possibility of applying Ambartsumian’s methodology to solving the problem of neutron diffusion in a one-dimensional nuclear reactor.
Let us consider a one-dimensional nuclear reactor in which all neutrons are divided into two groups—fast and thermal.
In the reactor the following processes take place:
a) fission of nuclei by thermal neutrons,
b) formation of fast neutrons due to fission,
c) absorption of thermal neutrons,
d) slowing down of fast neutrons to thermal energies.
Fig. 1. One-dimensional nuclear reactor
We shall divide the fluxes of thermal and fast neutrons in the reactor into fluxes going to the left and fluxes going to the right through a point of the reactor (Fig. 1). Denote, respectively:
$U_2(x)$ — the expected number of thermal neutrons that have passed through the point $x$ to the left over all the time after the beginning of the process;
$U_1(x)$ — the expected number of thermal neutrons moving to the right through the point $x$;
$V_2(x)$ — the expected number of fast neutrons that have passed through the point “$x$” to the left over all the time after the beginning of the process;
$V_1(x)$ — the expected number of fast neutrons that have passed through the point “$x$” to the right.
Let us assume the following character of the processes taking place in the nuclear reactor:
1) fission — when a thermal neutron is absorbed by the fuel nuclei of the active zone, and a fast neutron is emitted at once;
2) slowing down—when a fast neutron is absorbed by the nuclei of the reactor material, and in doing so a thermal neutron is emitted.
The probabilities of the emitted thermal and fast neutrons moving to the left or to the right are assumed equal. We shall further denote:
\(\Sigma_a\) — the absorption cross section of thermal neutrons,
\(\Sigma_f\) — the slowing-down cross section of fast neutrons,
\(\Sigma_x = k\Sigma_a\) — the cross section for the “appearance” of fast neutrons.
According to the assumptions made above about the nature of the processes, and the notation introduced, for the flux of thermal neutrons moving to the left one can write the neutron-balance condition:
\[ U_2(x)-U_2(x+\Delta x) = -\Sigma_a \Delta x\, U_2(x) +\frac{1}{2}\Sigma_f \Delta x\, V_2(x) + \frac{1}{2}\Sigma_f \Delta x\, V_1(x), \tag{1a} \]
or, as \(\Delta x \to 0\),
\[ -\frac{dU_2(x)}{dx} = -\Sigma_a U_2(x) +\frac{1}{2}\Sigma_f V_2(x) +\frac{1}{2}\Sigma_f V_1(x). \tag{1} \]
Similarly, one can write the equations for the fluxes of fast and thermal neutrons directed to the right:
\[ \frac{dV_1(x)}{dx} = -\Sigma_f V_1(x) +\frac{1}{2}\Sigma_x U_1(x) +\frac{1}{2}\Sigma_x U_2(x); \tag{2} \]
\[ \frac{dU_1(x)}{dx} = -\Sigma_a U_1(x) +\frac{1}{2}\Sigma_f V_1(x) +\frac{1}{2}\Sigma_f V_2(x). \tag{3} \]
For the fluxes of fast neutrons directed to the left,
\[ -\frac{dV_2(x)}{dx} = -\Sigma_f V_2(x) +\frac{1}{2}\Sigma_x U_2(x) +\frac{1}{2}\Sigma_x U_1(x). \tag{4} \]
The system of equations (1–4) in matrix form has the form
\[ \frac{dY}{dx}=AY+BZ; \tag{5} \]
\[ \frac{dZ}{dx}=AZ+BY, \tag{6} \]
where \(Y\) is the fluxes of fast and thermal neutrons directed to the right;
\(Z\) is the fluxes directed to the left;
\[ Y= \begin{pmatrix} U_1\\ V_1 \end{pmatrix}, \qquad Z= \begin{pmatrix} U_2\\ V_2 \end{pmatrix}, \]
\[ A= \begin{pmatrix} -\Sigma_a & \dfrac{1}{2}\Sigma_f\\ \dfrac{1}{2}\Sigma_x & -\Sigma_f \end{pmatrix}, \qquad B= \begin{pmatrix} 0 & \dfrac{1}{2}\Sigma_f\\ \dfrac{1}{2}\Sigma_x & 0 \end{pmatrix}. \]
Following the principle of V. A. Ambartsumian, set forth in [1], one can introduce a matrix \(R\), relating the fluxes of opposite directions,
\[ Y=RZ, \tag{7} \]
\[ R= \begin{pmatrix} r_{11} & r_{12}\\ r_{21} & r_{22} \end{pmatrix}. \]
Considering in greater detail the neutron processes occurring in the layer \(\Delta x\) (Fig. 1), one can obtain an equation for \(R\).
The neutron flux \(Z_1\), coming from the point \(x+\Delta x\), has probability \(B\cdot \Delta x\) of causing a change in the neutron flux directed to the right (due to fission and due to the slowing down of fast neutrons). Part of this flux \(Z_1\) reaches the interval \((0-x)\) and there produces a flux of the opposite direction (a flux to the right).
If, however, the flux \(Z_1\) does not cause, in the layer \(\Delta x\), a change in the neutron flux to the right, then it completely reaches the interval \((0-x)\) and there produces a flux \(Y_1\), directed to the right. In both cases the neutron flux emerging from the interval \((0\div x)\) and directed to the right reaches the layer \(\Delta x\). In doing so it may pass through this layer without change, but there is a probability that in the layer \(\Delta x\) thermal neutrons will cause fission, and fast ones may slow down to thermal energies.
Part of the neutrons produced as a result of fissions and of the slowing down of fast neutrons again falls on the interval \((0\div x)\) and there produces a flux directed to the right.
Summarizing the above, one may write:
\[ R(x+\Delta x)=B\cdot\Delta x+(1+A\cdot\Delta x)R(x)(1+A\Delta x)+R(x)B\Delta xR(x), \tag{8} \]
as \(\Delta x\to 0\) and, neglecting terms with \((\Delta x)^2\), we obtain
\[ -\frac{dR(x)}{dx}=B+AR(x)+R(x)A+R(x)BR(x), \tag{9} \]
\[ R(0)=1. \tag{10} \]
Now the problem of neutron diffusion can be solved in the following sequence:
1) first, equation (9) is solved with the initial condition (10);
2) then the equation is solved
\[ -\frac{dZ(x)}{dx}=AZ+BRZ, \tag{11} \]
obtained by substituting (7) into (6);
3) from relation (7), \(Y(x)\) is found.
Thus, the separation of neutron fluxes into fluxes directed to the left and fluxes directed to the right makes it possible to represent the boundary-value problem of neutron diffusion in the form of the system of equations (5) and (6). Using equation (9), which serves as the mathematical expression of the invariance principle, the system (5)--(6) can be solved as an initial-value problem. The use of an analog computer makes it possible to find the solution of the neutron-diffusion problem in one operation.
REFERENCES
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V. A. Ambartsumyan. On the question of diffuse reflection of light by a turbid medium. Dokl. Akad. Nauk SSSR, 1943, 38, issue 8.
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S. Chandrasekhar. Radiative Transfer. IIL, Moscow, 1953.
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R. Bellman and R. Kalaba. “On the principle of invariant imbedding and Propogation trough inhomogeneous media.” Proc. Nat. Acad. Sci, USA, v—42, N 8, 1956.