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Tabulation of neutronic properties

The neutronic properties (reactivity, macroscopic cross-sections, delayed neutron beta's) must be assigned to each element of the partition kn, in relation with the local confIGuration ig and the physical properties (vmck, uck, bck) of the node ck including kn.
They are specified as follows: (see also rdinput.dat/core/&Lstg )

The representation of neutronic properties is hybrid: the effects of core confiGuration (g) and water density (or volumic mass vm or "v") are explicitly modelled by means of linear interpolation (1) in a set of i9gr confiGuRations and iv9g water density states (iv=1,iv9g).
These are the major effects influencing all the neutronic properties.
The remaining effects are translated by means of analytical correlations.

Delayed precursors data

They are ig-dependent only.
beg: total effective beta(ig=1,j9gr)
bejg: beta(j,g), (j=1,j9g , ig=1,i9gr)
j9g is a global value ( &Lstg/i9g); default=6

Reactivity data

Are rendered (1) as a 3-d array
rvg(ir, iv=1,iv9g, ig=1,i9g) where iv and ig are the density and configuration indexes for interpolation and ir is the component index in the "reactivity-vector" (2). (ir=1 for ro, 2 for rou,..10 for reuu.)
If several configurations are present in a particular kn (transitional kn), they are assumed to contribute each to the reactivity in proportion to their relative volume fraction they occupy in kn.
As the result of interpolating between the iv states and volume fraction averaging of the ig states, the reactivity vector may thus be regarded as a polygonal function (3) of vmck, with a symbolic parameter gkn representing the configuration mixture in kn. (gkn=ig if a single configuration ig is present in kn)
Thus, the vm-derivative Rv of the reactivity-vector is simply the slope (4) of the segment iv of the interpolation polygon. The components of the v-derivative Rv of R are named in the array (5).
The physical variables retained for analytical correlation are (6):
the deviation p of the (global) eos pressure p2 from a reference value pog, unique for all configurations.
The boron ppm deviation b of the nodal bck value from a single reference bog.
In t= t(vm,p), vm is the principal contributor to moderator temperature reactivity effect and the p effect is minor and represented by a linear correction.
u translates the "fuel pellet over-temperature effect" or "Doppler" effect associated with the deviation of the nodal pellet effective temperature uck from the nodal coolant temperature tck.
u-effect is associated with nuclear power deposition in pellet, but may also result from rapid water cooling at zero power (steam line break).
Quadratic representation (7) of the (smooth) u and b effects was found to be adequate.
Actually, √ukc is used instead because it enters a such in resonance integral calculations
h translates the effect of entHalpy turbulent mixture between adjacent reactor coolant channels as water rises in the core.
This effect is linearly correlated (6) with hck - hec, the nodal enthalpy accretion above the inlet enthalpy hec.
In absence of an explicit mixing model, this effect can be simulated, in the generating neutronic diffusion code, by comparing two cases: one with closed assembly channel (default) and one with perfectly open channels providing uniform h distribution. The degree of mixing is adjusted by an input tetag.
In the process of simulating a steam-line-break (SLB) accident this h-effect representation may by used to close the channels during the sub-critical cooling phase in oder to maximize the reactivity injection.
As the core returns to power, maintaining the channels closed would maximize action of the water density feedback to reduce the power level.
Actually, inter-channel mixing occurs as the inter-channel enthalpy difference increases due to return to power and the channels must be reopened for conservatism.
For unsymmetrical accidents like SLB, the non-uniform radial enthalpy distribution at core inlet, resulting from loops unbalance, has a significant effect on the accident consequences.
In order maximize this effect, the flows in the downcomer will be supposed separated ( osplit= t) and the loops unbalance is monitored by ehec [Ecart; entHalpy; Entry; Core], deviation of hsal(1) at outlet of sector 1 of downcomer over the average hsa for all the sectors.
In the steady-state diffusion calculations generating the neutronic tables, the effect is simulated by imposing, at core inlet, two reference non-uniform enthalpy distribution (of unbalance eh0g and eheg for each configuration g) obtained from 3-d hydraulic calculations or from experimental tests and linearly interpolating between them with the current loop enthalpy unbalance ehec resulting from thermal calculations.
Of course, the inter-channel mixing in the core must be deactivated to evaluate the impact of non-uniform h-distribution on the neutronic properties.
In the safpwr model, the inlet h-unbalance is supposed to affect the u-effect only, and is modelled by introducing an additional term (re + (reu + reuu * u ) * u) * xe to rkn (7).
xe= (ehec - ehog) / (eheg - ehog) .
The vm-derivative (8) of rkn is also needed for modeling the reactivity density feedback.

The procedures for generating the neutronic tables by running a set of 2-d stationary diffusion calculations will be described later.

Macroscopic cross-sections

"Macro data" (9) are tabulated in the same way:
sd: diffusion coefficient (cm-1 )
sv: inverse of average neutron speed (s/cm)
sn: neutron fission source coefficient (1/cm)
skn: κ/ν: nuclear power/neutron source (W cm³)
The sections "kn" are calculated at kn level and are scaled by hkn (cm), the thickness of the meshes contained in kn, in order to obtain by multiplying by phici the ci reaction rate.
As (13) indicates, sn is linearly b- and u- dependent and (14:16) sd, sv and skn are only b-dependent.
The justification for the this analytical representation of the u and b dependence can only justified a posteriori by recalculating the properties along the physical conditions actually reached during the transients.

Hot sector radial tilt factor

As explained in the introduction, if osplit=t, the core h,m,b-balances are carried out for each of the radial core sectors cl associated to the loops l.

The radial tilt factor xqcl assigned to cl may receive a constant value at input, or correlated with calculated physical conditions.
The radial tilt Xq is correlated, like the previous neutronic properties, by means of an array XQ_VG(ixq,iv,ig) (17)
The components (18) of the XQ vector allow an analytical correlation of the form (19,21):
the u, ehec, and h-effects correlations are the same form as for reactivity, but the p- and b- effects are neglected.
The quadratic u-effect accounts for radial tilt attenuation with power production (return to power in SLB). The ehec effect is attenuated the same way.
The h-effect correction allows also to attenuate Xq by inter-channel enthalpy mixing due to coolant heating.

Radial hot channel factor fxy

It is correlated exactly like Xq by entering the array FXY_iv,ig