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Neutron kinetics model

Multi-group 3-D kinetics

General form
We start with the multi-group kinetic equations in diffusion approximation, which are written as (1) in order to single-out the neutron axial leakage.
n(x,V)(2): neutron density (neutrons/cm3) at point
x(x,y,z) and velocity V;
φ ≡ n V (3 ) is the corresponding scalar flux (neutr/cm²/s);

fiφ (7) is the total neutronic source generated by fission of fissile isotope "i";
a fraction (1-βi) of this source is emitted at fission time (prompt source), the complement βi is delayed.
βji fiφ represents the production rate of family "j" delayed fission source which release their neutron at a probability rate of λj per sec.
cj(x) is the concentration at x of family j delayed emitters.
Prompt neutrons are emitted into a "prompt" velocity distribution spectrum χpi(V), so the first term in (1) represents the generation rate (source) of prompts neutrons of velocity V at point x.
The delayed neutrons are emitted into another spectrum χj(V) characteristic of each family j, and softer than prompt neutron emission spectrum.
As notations imply, χpi(V) depend on fissile isotope "i", and delayed spectra are characteristic of each family j.
In practice, the number of j families is limited to 6 (default value). The data (βji, λj) are part of the fuel assembly neutronic program library.
Rigorously the λj are also i-dependent, but that dependence is neglected here.
r(x,V) φ(x,V) (8) is the net removal rate from (x,V) by in- and out-scattering, absorption, and radial leakage, the axial leakage being explicitly accounted for by the term d Δz φ.
The time variable t (actually noted "s" or "sec" in the program to avoid confusion with water temperature "t") is implicit in n, φ, as well as in the operators d, f, r.
In summary, (1) renders the neutron density balance:
∂n/∂t= prompt source - removal - axiall leakage + source by delayed emitters decay;
(1b) represents the balance at x of family j delayed emitters density :
The first term in (1b): production rate of emitters of family j at x
Second term: removal rate by radio-active decay
Radial transport effects are normally modeled, in the design neutronic program, by means of discontinuity factors.

Stationary form

Setting (9a,b) to 0 the time derivatives in (1) leads to the classical form (9c) of n(x,V) balance, where the core eigen-value λc needs to be inserted to insure criticality and
(9d) defines, for fissile isotope i, the stationary, "composite", average emission spectrum for prompt and delayed neutrons. This spectrum is part of the fuel assembly neutronic library.
With (9e) we obtain a compact representation (10a) of the criticality equation, which may, by making use of the core "static reactivity" (10b), be rewritten as (10c)