In order to determine the feasibility of a continuous withdrawal of uranium isotopes from a thorium-based reactor, it is necessary to have a detailed time evolution of the densities of all nuclides. The model has to include both sources and sinks of the uranium isotopes, along with the necessary control procedures for the neutron flux, aimed to keep the reactor critical and affording the desired power output. After establishing the form of all the rate coefficients, the equations for the time evolution can be formulated in terms of all the relevant densities. In particular, the neutron density (or equivalently the flux) needs to be evaluated by detailing all its source, sink, and transport terms. The neutron diffusion term (D is the diffusion coefficient and the Laplacian operator), describing the changes in neutron density as a function of time and space, can be simplified by approximating the neutron depletion in the reactor volume as homogeneous, and due to the neutron flux across the surface. In this way, the time derivative of the neutron density, determined by the neutron flux at the reactor boundary, can in turn be evaluated by solving the spatial eigenvalue equation for neutron density along with the Neumann boundary condition. This procedure allows the problem to be formulated as a system of ordinary differential equations.
Considering a system of nuclei and neutrons (with energy
, velocity
, in the center of mass reference frame, and volume
V) with reduced mass
m and giving a reaction with energy-dependent cross section
, the corresponding rate coefficient
for the process involving thermal neutrons would have the form
, that is
where the second equality is obtained by defining
(
k being the Boltzmann constant and
T the temperature). The quantities
and
represent the density of translational states and the DeBroglie wavelength, respectively,
(
h is the Planck constant). We take into consideration the following reactions and decays,
plus the elastic scattering of thermal neutrons by
and
. All rate coefficients for
decay are indicated with the letter
with either an ordering subscript or the subscript “
” for the emission of delayed neutrons. The rate coefficients for neutron-nuclei interactions are indicated by
, with the subscript
f for “fission”,
c for “capture” by heavy nuclei, and
x for the neutron capture by either the fission products or the control neutron absorber. We use the reference density of metallic thorium
to define the reduced densities as
,
being the number density. We follow the convention used in the Manhattan project, indicating the reduced density with the first lowercase letter of the element symbol, its subscript being the last digit of the atomic number (
i) and the last digit of the mass number (
j) [
18]. For example, the reduced density of
,
, and
are noted as
,
, and
, respectively. The same indexes label the various sources and sinks
s. We use for the reduced density of neutrons the same symbol used for neutrons,
n, and the fission fragments in Equations (
4)–(
16) are indicated as FF. Since the rate coefficients in processes (
4)–(
16) can be calculated from known data, we may write a set of differential equations for the time evolution of the species, once the initial conditions for the number density for all species are specified. In the non-dimensional time unit
, the evolution of the homogeneous system for reactions (
4)–(
16) is given by the equations
with the dot indicating the derivative with respect to
, and
, and the parameters given in
Table 1. The variable
h indicates the reduced density of
and the variable
f indicates the reduced number density of the fission fragments, while the same letter as a subscript indicates fission.
The parameters
and
represent the net number of the prompt and delayed neutrons emitted after fission, respectively [
19]. In Equation (25) we summarize the effect of the neutron absorber
(obeying the equation
, the source term
being subject to a feedback control to keep the power output at the desired level) with the neutron poison fission products
y, exhibiting a high cross section for neutron capture (as
and
, having total a fission yield of 0.074 and obeying the equation
). Representing all neutron absorbers by the single variable
x and the absorption cross section
, and summing these two equations we obtain Equation (25).