SAFPWR home page

Numerical Integration of reactor kinetic equations

The equations may be presented as (19.1), recast in lower case to indicate that they are attached to each small mesh Iinterval "i" of the core partition.
The effective β~ is renamed here as β^ to avoid confusion with the "tilted beta" to be introduced later.

As the λj are constant, the delayed emitters balance equation (19.1b), can be integrated analytically, under the assumption that fφ changes quadratically in the course of the time-step (fig 19), so that its time trajectory is completely determined by 3 parameters: the initial and final values of fφ, and the bos value Ω1 of the logarithmic derivative.

This analytical integration of (19.1b) over the time step may be found as (19.2), with coefficients given by the recurrence expressions (19.3).
For vanishingly small values of δt, those expressions become 0/0 undetermined and are, therefore, approximated by series developments (19.4). The switching to (19.4) is controlled by the criterion epsf.
With the notations (19.5a:c), the eos decay source becomes (19.5d).
In SAFPWR, for integrating (19.1a) we depart from the classical method for time-discretization: instead of simply integrating the equation over the time-step, we consider it at eos (19.6a).
Inserting (19.5d) into the neutron balance (19.6a), taken at eos, and introducing a "tilted" beta β~ defined as (19.6b) leads to a neutron density balance equation (19.6c) where the eos "source term" S2 (16.6d) is explicitly known from the current bos quantities.
"tilted" β~ remains close to the effective β^ so long as δt < .1 s. For δt = 1 s, the ratio β~/β^ is approx. 0.87.
It is instructive to introduce by (19.6e) the log derivative ω of n and the "neutron life-time" Λ from (19.6f), which allows writing the kinetic balance as the simple form (19.7a) whose coefficients, taken at eos, (subscript 2) depend on the, still unknown, eos values of ρ, f, d through the Doppler, density and boron feedbacks.
Likewise, Λ ω2 may be viewed as an additional feed-back: if time evolution of n in the course of δt is assumed, ω2 will be fully determined from n1, n2 and ω1 (see fig). For example a quadratic trend will imply (19.7c), the group (19.7d) as a whole becoming a function of n2.
With (19.7c), the neutron density balance may be simply recast as a diffusion-like equation (19.8b) with a eos "cross-section" σ2 defined as (19.8d) and a "source" (19.8c) depending on bos values only.
(19.8e) thus represents the non-linear flux equation to be solved at each time-step, iteratively, because of the feedback's.

The form of this remarkable equation remains the same for 2 or 3 dimensions. For example, for a 3-dim representation, σ2 is to be taken from the nodal reactivity and Δ becomes the classical 3-d Laplace leakage operator.

It is again observed that Λ ω2 behaves here as a "fuse correction" term which prevents that operator (19.7d) becoming singular as the reactor becomes prompt or super-prompt critical.
The "Prompt-jump approximation" can be accepted whenever the contribution of the Λ ω term in (19.7d) can be neglected.