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Solving the flux equation

The method for obtaining a feed-back converged solution for the flux equation (20.1a) involves 4 successive calculation stages which must be repeated until convergence.

Stage 1
We, in the first place, search for an approximate solution (20.1b) differing from the current guess φ2ci by its amplitude only.
Substituting (20.1b) into the flux equation, updating the coefficients by the linear developments (20.1e:f) leads to an algebraic equation (20.1c) for φ3.
As the axial profile of φ3 is not yet the correct converged one, (c) would generates a different solution φ3 for each mesh ci.
Therefore, in order to obtain a single equation for f3c, the ci balances are summed over the core and f3c is thus obtained as the positive root of the resulting second degree algebraic equation.
The total derivatives of σ and d in relation to fc are expressed via (20.2a,b); the partial derivatives representing the various physical and neutronic effects:
∂σ/∂√u translating the Doppler effect is calculated from the neutronic tables where the neutronic data are tabulated versus √u.
∂u/∂fci results from the fuel rod thermal model .
∂fci/∂Φ relates the flux with power ().
φ2ci/f2c is just the ∂φ3ci/∂f3c resulting from (b).
The second term in (20.2a) represents the water density effect; ∂σ/∂v comes from the neutronic tables
∂v/∂h from the water properties correlations ,
∂h/∂fc from the core enthalpy balance and relates the variations of local enthalpy to thermal power for frozen fixed f2ci profile.
∂d/∂fc is the corresponding derivatives for diffusion coefficient, which is assumed to be water density sensitive only .
Stage B
Once the amplitude f3c, thus obtained, the corresponding physical and neutronic conditions are updated accordingly (20.3), making use of the same derivatives.
Stage C
the hydraulic distribution v2ck remaining now frozen, the power profile is corrected by solving the system (20.4a), assuming, this time, that Doppler effect only is active.
In view of linearizing this system, the approximation (20.4c) is accepted. We thus finally obtain a tridiagonal linear system whose direct solution is classical.
The stages a to c must now be repeated until a eos, feed-back converged solution is reached.
Convergence is monitored by comparing f3c with f2c (convergence criterion epsfc).
Stage c: updating of u2ck.
We take profit of the particularly simple form of the algebraic equation solved at stage a to carry two controls:
A control of acceptability of the prompt-jump approximation, on the basis of the relative contribution of Λ ω2.
This approximation is generally always valid, except, in fast reactivity transients, during the very short periods when the reactor approaches or crosses prompt criticality (?). Whenever acceptable the PJ approximation is very effective in so far as it allows, without impairing the solution quality, to tolerate time steps dsec very large compared to prompt neutron life-time.
On the basis of monitoring (f3c-f1c)/f1c and log derivative, the time step duration dsec may be automatically adjusted.
This allows tightening the time stepping solely whenever necessary, during the nervous parts of the transient.
Furthermore, dsec and PJ monitoring are only performed once at each iteration cycle, on the simple, scalar representation of the neutron balance.

The described algorithm accommodates for marked deformation of the axial profile in the course of the time-step.
It provides an accurate, converged solution at eos, even if the water reactivity temperature coefficient is positive.