Safpwr core model: tabulation of neutronic properties

As explained earlier, the neutronic properties (rog, betg, fxy,..) are derived from a set of static, XY, diffusion representation of the neutronic state of the various configurations ig=1, i9g, for a series of independent variables vm (massic volume), u (fuel effective temp, measured from the water temp tw), bo (boron ppm), ... selected to "map" the domain of variation of these variables in the course of the transient.

In order to facilitate the preparation of the diffusion codes input values, an ad-hoc utility program tab_ccc has been imbedded into SAFPWR.
This program models the ini state of the average heat channel as a heat-generator with a single configuration and a single thermal node.
[tab_ccc.dat (download)] file is a commented example of input.dat enabling execution of this utility program.

heat-gen group specifies average channel thermal characteristics, just like in a normal application, except that tab_ccc is invoked by option oqc= -2.

The data have been extracted from the input file of the NEA rod withdrawal benchmark.
The input.dat file must be complemented by additional data entered into the file output.02 file located along input.dat in the SAFPWR directory.
Are specified there:

number ix9fnc of calculated core power levels, specified by the fraction of nominal xfnc values.

Do not forget to leave at least one space after the "=" sign.
Double cote "" must also surround input_ name= so that the input strings such as wec= will be interpreted as a label in a delimited file to be imported in a spread-sheet.

The set xfnc must generate enough effective temperature u2c values for performing quadratic fitting of the Doppler effect.
In principle, 3 values suffice, but entering more values will allow checking the precision of quadratic fitting of the neutronic data in relation to [√(tw + u2c) - √tw].

Outcome of tab_ccc execution will be appended to the same output.02 file, as a continuation of the input data. Output data are "coma delimited" formatted, suitable for readily importing into the spread sheet program for further examination and processing.

For each tabulated vmc(iv) tab_ccc determines the corresponding enthalpy hc(vmc, p3) and water temp twc, which are properties of the average core plane modelled by the diffusion code
Thereafter, for each (iv,ixfcn) combination:

Core inlet enthalpy hec, obtained from the balance relation f2c=wec(hc-hec), and the corresponding water temp
twec, which must be entered to the diffusion code to reach the desired planar enthalpy hc.

Next the au2c (Accretion of effective fuel temp over twc) resulting from f2c, the linear power fluc = xfuc f2c / luc injected into the pellet stack.
Its derivative ufluc vs fluc.

For info, follows the accreted (over twc) average pellet temperature au2ac.

It is observed that curvature of the plots due to variation of thermal constants with fuel temp is weak.
By construction (cf ), the derivative ufluc at a point xfnc is the angular coefficient of the cord joining the origin to this point.
The graph represents such a cord for xfc=1. It is observed that the cord constitutes a fair approximation of the u2c values around that point.

It also appears that the effect of vm on u2c is small enough to be neglected.

tab_ccc generates the dependency relation (hec,u2c,ufluc)= f(vmc,f2c,wec,p3)
for the average channel.
In general the xy calculation fed by hec and f2c will not result in the same average effective temperature as the SAFPWR value u2c unless the thermal models are made consistent.

The above observations justifies using, in the xy generation code, a simple linear cord fitting relation
u(flu)= uflu(fluc)*flu
for modulating u2c over the various cells of the core in relation to the local flu.
This relation may be unique for all vmc (graph 1).
In addition, this linear representation insures that both SAFPWR and its xy code generator provide generate the same average planar effective temp.

It may be argued that, for given average value of u2c, the Doppler feedback will also depends on the radial u profile and on the manner the core is heated:
for slow heating the profile will depend on current power profile whilst for fast (adiabatic heating) it is rather the "historical" profile.
In SAFPWR, it is postulated that this profile effect is neglected. However, it does not make much sense to worry about this distribution effect, considering the probably greater uncertainty in specifying the weighing factors xur allowing to calculate the effective temp from its radial distribution, the effect of internal power profile in the pellet (for depleted, MOX, poisoned pellets).

In order to illustrate this point, we have repeated the nea benchmark gwzp_b case with forcing the effective temp to be identical to the average (i.e. by entering xu2c=0,4*.25,0 instead of .3,4*0,.7.
As expected the Doppler feedback is lower, as depicted on graph [] comparing f2c and u2c7. The fuel temp is about 50 K lower.

If several types of fuel are present , tab_ccc should be repeated to generate the uflu of each of them.

The effect of channel opening is evaluated by comparing runs with closed and open channels.
For info output.03 provides detailed temperature profile in the fuel rod as a function of enclosed area sri.
The area is normalized to the pellet area.

chart 2 confirms that, at steady state and nominal power, the temperature profile is quasi linear (vs xsu) over the pellet.
At higher power, the curvature increases because of the fuel conductivity increases with temperature.
Point at .125, .375, .625, .875 give the average temp of the 4 rings. The other points are at center and border and in clad.

Tabulations of neutronic properties.