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Reactor kinetics validation against the ANL benchmarks

The test core is 1-d with 3 homogeneous regions. The reference initial solution (Lstci/f2ci, phci, sn0ci )and (&Lstg/am0g) was obtained by 2-group diffusion calculation with 90 equal meshes and 0 flux boundary condition.

In SAFPWR, the core (anl_1.dat ) (download) is modelled with 1 node ck per region, with size azck= 0.4, 1.60, .40.
At initial condition, the 2-group diffusion values for configuration ig= 2,3,4 (anl.txt) are attached to bottom, middle and top regions respectively; bottom and top compositions are identical.

The transient is initiated by applying a 10% reduction of the bottom sc2 as a linear ramp terminated at 1 s.
The corresponding perturbed diffusion data set is attached to configuration ig=1.
The transient reference solution is available in terms of time evolution (with short dsec=.01) of total power f2c/f2c0 and power ratio
f2ck(ck=1:3)/f2c.
As neutronic feedback is nonexistent (pure diffusion problem) the relation from fission source to power can be specified via an arbitrary constant value for κ/ν= skn0= 1.333e-9.

Initialization calculations

The (ini, core) procedure requires, as initial, reference critical core ("0") data the eigen value am0g and the axial distribution of f2ci, phci and sn0ci.

The benchmark reference transient solution was generated by finite difference, diffusion, axial kinetics and we must therefore use a consistent (same meshing) model for generating the stationary initial core representation.
The results, obtained by a classical finite differences 2-group neutronic program, are included into the [anl1.dat] input file for the SAFPWR treatment of the anl1 benchmark.
Next step is to calculate, by making use the generic axial collapsing relations , the 1-group SAFPWR neutronic properties for each configuration ig.
In the present example, the initial diffusion model representation is reduced to the classical system (1,2) of 2-group diffusion equations, where Δ is the axial component ∂2/∂z2 of the Laplace leakage operator.
The benchmark original macroscopic data sc1g, ...are recalled in anl.txt table. The outcome of this classical 2-group, pure diffusion, problem are mesh fluxes ph1ci and ph2ci and eigenvalue am0g.
Next we calculate (3:5) the initial distributions phci, f2ci and sn0ci to enter under lstg/
Once the neutronic 2-group critical values thus obtained, we must repeat the same diffusion, eigenvalue, problem for each configuration ig in turn, but with zero axial leakage (xy 2-d geometry) and with g-data corresponding to a critical core, ie with the sn1g and sn2g divided by am0c.
As the radial leakage is null, this radial diffusion problem collapses, for each confiGuration g, to the trivial algebraic homogeneous eigenvalue system (6,7).
kg is the corresponding eigenvalue, also referred as "planar configuration multiplication factor".
We need, in addition, to solve the adjoint form (8,9), obtained, for our 2 neutron group model, by simply permuting sc2g with sn2g/am0c.
ps1g (ψ) and ps2g are the adjoint fluxes, solution of this system.
Application of generic reduction-to-one group relations [] to our simple model generates 1-group sng (11), sdg (10) and planar reactivity rg (14).
The average sv values and the neutron life-time are calculated from (15:19).
Delayed neutron properties are the same for all the compositions g and are specified in anl1.dat .

Calculations of g-data are summarized in anl.txt file

Description of anl1_input.dat

The general structure is the same as for the point kinetics benchmarks.

Representing progressive transition of homogeneous bottom region 1 properties from homogeneous configuration ig=1, to homogeneous configuration ig=2, is not feasible in the frame of SAFPWR input possibilities, because such an homogeneous perturbation of a region neutronic properties is never encountered in practice.
We will, instead, approximate such a transition by means a linear withdrawal of the configuration ig=1, inside the bottom region, initially filled with composition 2.

SAFPWR solution of ANLa benchmark

Chart 1 compares SAFPWR results with the reference for the same dsec=.01 as used by reference.
The agreement is quite good except during the withdrawal of group 1.
We developed a special test version of the program for allowing homogeneous transition in region 1.
This version corrects the observed deviation.
rc and omc nearly reach their asymptotic level after 4 s of transient.
Examination of ypjc plots indicates that, for the present low reactivity problem, the prompt jump approximation is never deactivated. Hence, the values of sv have no effect on the results.
This is true, however, as far as lambda remains very low.
Charts (2:3) show results for dsec=.05 and .25 are just as good (or even better ?) as for .01 s.
Chart 4 also confirms the good agreement in terms of power shape, translated by the ratio f2ck(1)/f2c at sec=4.
At last, let us apply the Nordheim test to asymptotic values of rc and omfc.
For that purpose, the .05 s case was extended to 8 s (Chart 5) where asymptotic omfc=.17435 is nearly reached.
The corresponding Norheim reactivity is .003749 which is very close to the calculated value .003746.
Rigorously, Nordheim formula applies only to 1-group spatial kinetics if lambda, and the delayed neutrons properties amjg and bejg are uniform ,which is indeed the case here.
We showed also that extension to 2-group spatial kinetics is still possible on the same condition and thanks, in addition, to a redefinition of the eigenvalue problem, consisting in adding omega/v1g to sc1g and omega/v2g to sc2g. However, for omega= .17435, those corrections are negligible.

In conclusion the agreement with the anl1 benchmark is quite good, which brings, particularly, also an validation of the data reduction procedure to one-group approximation.

Anl_sin benchmark

Anl_sin differs from anl_ramp to the extend that, instead of reducing region 1 sc2 by 1% in 1 s, sc2 is obtained by a sinusoidal 1% amplitude modulation of its initial value, with a period of 1 s ().

A special test version of SAFPWR had to be prepared to allow running this problem because it is tedious to simulate a sinus variation through the zgri_sec interpolator.

Charts(6:8) compare SAFPWR with the benchmark, in terms of f2c, for dsec=.025, .05. and .1 s.
The agreement is again quite good.
For dsec= .1 s, although there are not enough points to capture the peaks, the calculated points, except for the max peaks, still fall back well on the reference curve.

This test illustrates the good stability and accuracy of the program, even for large steps.
The ability of correctly fall back on the good trend, even after missing the top peaks, is doubtless thanks to the second degree scheme used for integrating the neutron kinetics and to the analytical scheme for integrating the delayed precursors equations.

A last instructive observation: although the time average of region 1 sc2 amplitude variation is zero, the resulting rc reactivity average is nevertheless positive (about .001).
This is due to the fact that region 1 reactivity is weighted by the product of local direct and adjoint fluxes, which are larger than average when the perturbation is positive and lower when it is negative.