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A transient solution to the initial core eigen-value problem

One-group representation of the initial critical core diffusion equation may be expressed as wel as (4) where the scalar core reactivity ρc needs be substracted from the planar reactivity ρ in order to get a critical core for imposed values of initial power f0c, core inlet conditions wec, hsb, bsb and primary pressure p3.

This core "eigen reactivity" is related with the core "eigen value" λc by means of (5).
ρc may as wel be interpreted as a scalar "numerical" feedback correction which must be substracted from the local planar reactivity ρ and adjusted till the current core transient reactvity rc vanishes to zero.

As SAFPWR provides, at each time step end ( end_step, core) an accurate estimation of current net core reactiviy rc,
it is tempting to try capturing the critical "eigen value" core condition as the outcome of a pseudo transient "controlled" by an additional "numerical" reactivity feedback term r0g correcting (6) the planar reactivity r
In order to force the core to criticality, r0g will be incremented, (7) after each step end, with the current estimation rc of core (residual) dynamic reactivity, where amec is an input relaxation factor, introduced to control possible feedback instability
If this "numerical" feedback process converges, rc will eventually vanish to 0, and r0g tend to the actual core static (eigen) reactivity ρc.

In order to obtain a correct monitoring of rc, all the neutronic distibutions (flux, power, delayed neutron precurors,...) must be allowed to "float" freely, in such a manner that the converged level of f2c may actually differ from the imposed f0c.
However, the physical feedbacks must actually be calculted from a properly renormalized power after each step: more specifically, the nodal power injected into the fuel pellets in each node ck for calculating the fuel temperature transitient field must be forced to f2ck*f0c/f2c and the nodal thermal power subsequently used in the core enthalpic balances needs to renormalized as q2ck*f0c/f2c.

This non conventional method performs perfectly well, as will be illustrated in the following tests

Test exemples

cos_600_1_ini.dat is an elementary application exemple modelling a pure reactivty (no physical feedback), uniform core problem with zero flux boundary conditions.
The critical flux is cos and the theoretical critical core reactivity, for a critical (planar reactivity rog= 0) fuel loading is
r0g= sd/sn (π/azc)2 = -3.45717e-3

The sequences are the same as for a normal transient run and dsec can be adjusted for best convergence.

wrt.pl0 monitors the evolution of the floating power f2c, the current deviation rc of dynamic reactivity from zero and the build-up of core static reactivity r0g.
core / &Lstc/
In this so called "trans/ini" mode of executing the program, the initial power will not specified, in ini_core, by the distribution of mesh power (Lstc/f2ci=..) but by the total power f2c with a default flat profile.
The input f2c will be saved as f0c before it is allowed to float in the course of subsequent do, core steps.
In this application without any physical feedback, f2c is arbitrary.
The default value 1 for amec appears to be the best choice (no relaxation) for all the tests.
oqc= 1 skips thermal calculation in pellets, (q2ck=f2ck),
it9fc=it9dsc=0: no power or dsec iteration are needed.
Prompt jump approximation is never deactivated in this "trans/ini" run
Lstci The trans/ini mode is enabled by simply NOT entering any f2ci=.. record.
ini, core starts initialising all the fieds with a flat profile (f2ci= f2c * azci / azc)
In trans/ini mode, the default swsknug= t should be retained as no initial distribution is imposed. It implies that sknog= f must be specified for linking power to the fission source.
At application's end, end_step, core renormalizes the last floating f2ci iteration to f0c:
f2ci = f2ci*f0c/f2c
and saves it in a f2ci.dat file appearing in the working directory.
In a subsequent "normal" programme run, entering the filename "f2ci.dat" replaces the f2ci= data record.
&Lstg / r0g= 0, unless a better guess value is available. It is the initial value of the "numerical core reactivity " correction.
As, at outcome of the converged trans/ini run, the delayed neutron fraction has nof effect on the critical core condition, any convenient set of beg, bejg may be used. A single family j9g= 1 is OK.
amjg and beg are adjusted to get optimal convergence towards critical core condition.
Preliminary numerical experience indicates that a low beg results in faster convergence, but if the value is too low (.001 for ex.) the floating power may become negative (which is acceptable for an homegeneous system!), but the transient still converges towards the correct critical condition.
beg plays here the role of the upper bound guess for the largest (fundamental) eigenvalue, which is used for efficiently accelerating the power iterations by means the Wieland method in classical stationary core diffusion programs.

Application results

cos_600_1_ini_wrt.pl0 is copy of wrt.pl0 generated by the trans/ini run: 5 one-sec steps are sufficient for capturing the critical eigenvalue condition. The current (residual) core reactivity rc ,controled by the r0g numerical feedback tends to 0, while rOg builds up to the estimation of the "eigen" core reactivity ρc≡ r0g and f2c floats towards a level 2.3 without significance.
r0g = -3.4535e-3 is close to the exact theoretical value -3.45717e-3.
Chart 1 plots the power profile fci (from plot1, plot2) at sec= 0, 1, and 2, compared with the exact cos profile: the correct profile is already reached after 3 steps.
Of course, this is an easy case because no odd unsymetrical harmonic modes are present in an uniform core, so that dominance ratio is favorable.
Repeating the application with a finer meshing (i9c= 90) results in r0g= -3.4568E-03, which indicates that r0g ---> exact value as meshing is finer.
Repeating the application with entering the converged r0g= -3.4535E-03, accelerates the convergence and replicates the same value as r0g output.

Stationnarity check

cos_600_1_trans.dat application will check the stationnarity of the core just initialysed in trans_ini mode.
The data file differs from the first one by:
entering f2ci.dat name under Lstci,
entering the trans/ini result r0g= -0.0034535 under &Lstg,
beg= here the normal values are used.

Results

wrt.pl0confirms that the core remains perfectly stationnary.
The distributions are identical as for the trans/ini case, except those depending on the bejg , which are initialysed by ini, core with the correct f2ci and bejg

Testing the trans/ini mode for non uniform cores and with physical retroactions will be examined later.