kntco_applic

kntco_7a_trans_ini.dat (download) illustrates the safpwr flexibility for obtaining, by means a single trans/ini application, the moderator temperature coefficient
rctec[(bec= 0, 200, ...,2800), (hec= 1.33e6, 1.27e6), (f2c= 1)] , on a complete set of bec values from 0 to 2800 ppm, by 200 ppm increments, for 2 levels of hec, and a single values f2c= 1 W.

With dsec= 2 we get 5 steps every 10 s. Note the stair-jump at 10.1 s.
bsb= is incremented by 200 ppm on every "period" of 20 s (or 10 time-steps) in order to cover the complete range of bsb from 0 to 2800 ppm.
In order to allow rctec calculation hsb= alternates from 1.27e6 to 1.33e6 and back every 10 s "cycle" of the 20 s "period" .

Interpretation of the results.

chart 43 and chart 44 details the core net reactivity rc, core eigenreactivity r0g and analytical rctec calculated on the same (100 s, 120s) period. It confirms that 5 time-steps of 2 s duration are adequate for reaching criticality (constant r0g, zero rc ) at the end of each cycle of 5 steps.
A direct approximation of rctec on the period is represented as the cord slope: [r0g(120))-r0g(110)] /11.09 were 11.09 K= tec(120 s) - tec(110 s)is the constant temperature jump increment.
The "cord " slope rcbec is monitored as (r0g(120)-r0g(110))/200 ppm for hec= 1.33 e6 or r0g(110)-r0g(90) for hec= 1.26e6. The difference illustrates the effect of tec on rctec.

Chart 25 compares analytical and cord evaluation of rctec on a set of f2c values.
For the sake of well reading that chart, Chart 45 details the plot around 2200 ppm:
For f2c= 1W (pink plot) the converged analytical value is 24.1 e-5/K whilst the cord value (large round pink marker) is 22.5e-5.
As anticipated, rctec increases linearly vs bsb, with a slope augmenting weakly with f2c.
rctec starts becoming positive from about 500 ppm. At first sight, our proportional water expansion boron correction is exaggerated because we should expect rctec turning positive from about 1000 ppm only?
At first sight the cord rctec is reasonably well approximated by its analytical value, with however a tendency to overestimate it as the power increases.

Chart 45 shows the details for 2200 ppm:
For f2c= 1, (violet curve) the analytical value is 24.1 e-5/K vs 22.2 for the cord value; the overestimation is about 2 e-5/K.
For f2c= 1e9 (blue), the overestimation increases to 2.5e-5/K. For f2c= 2e9, it reaches 6 e-5/K and 10e-5 for 3e9W.

Boron reactivity coefficient rcbec.

Chart 35 compares analytical vs cord (markers) values for f2c= 1 (0 W, red) and f2c= (2e9 W. blue).
At zero power, the boron coefficient is very well approximated by its analytical values: -7.01 e-5/ppm at 559 K and -6.87 e-5/K at 570 K.
At f2c= 2e9, the deviation (analytic - direct) is weak at low ppm but increases with bsb, reaching .1 e-5/ppm at bsb 2800 ppm.
The square vs circle markers positions reflect the effect of moderator temperature on boron coefficient.
Altogether, the analytical rcbec can be relied upon.

Explaining the deviation of analytic vs cord values for rctec.

We have not been able to resolve the issue of systematic overestimation analytic over cord values.
A possible explanation: the cord value accounts for the (second order) effect of temperature increase on the axial flux profile, whilst the analytical values, in accordance with first order perturbation theory, neglects it.

Effect of boron and power on axial power profile.

This effect is well illustrated on Chart 36 plotting aoc vs bec, for f2c= (0, 1e9 , 2e9, 3e9W).
As anticipated, at zero power (red stars), aoc remains close to zero regardless of the level of bec.
This is normal since core neutronic properties are not yet affected by power. The minor deviation from zero can be explained by incomplete convergence at end of period.
As power increases, the slope of the aoc(bec) quasi linear plot increases slowly with power but all the plots intersect the aoc= 0 base line at bec = 500 where rctec is zero.
I suspect that with a more realistic boron model the intersection would be closer to 1000 ppm.

Chart 9 exhibits the axial power profile at full power for bec= 0, 1000, 2000 and 3000 ppm. and chart 32 shows the situation at 2000 ppm for 4 power levels.
It hghlights the strong direct correlation between rctec (via boron effect) and upward power shifting.
Chart 9 suggests that, at 3000 ppm boron, the upward power shifting would be too penalizing as far as axial peaking factors (and dnbrc) are concerned.

Power reactivity coefficient rcfc

Application bild_2_ini.dat (download) illustrates how it is possible to generate, by means of a single run, the analytic and cord values of
rcfc[hsb=1.27e6, bsb=0, (f0c=0,1*2.775e8,2*2.775e8,...10*2.775e8)]
f0c specify here the thermal power level imposed for converging the thermal feedback's.
The application is next repeated for bsb= 1000, 2000, 3000 ppm, considered as a parameter.

Chart 48 details the f0c stepping process on period of 100 s, made of 20 time-steps of 5 s duration.
Results are plotted vs q2c on charts 46, 50, 51 , 52 for bsb= 0, 1000, 2000, 3000 ppm, respectively.
At 0 ppm, (chart 46) analytic rcfc is about -33 e-13/W and increases slowly with q2c.
Cord values of rcfc, referred here as the cord slope d r0g/ d qc is about 4 e-13/W lower.

Point kinetics

In order to assess the role of power profile, it is instructive recalculating the reactivity coefficients on a point-kinetics model consistent with our spatial representation.

ini_cintpt_1.dat (download) application is the point kinetics counterpart of bild_2_ini.dat
PLots: Of course, plot1 and plot2 are not of interest in point kinetics but any acceptable data must be entered, because the program falls in error if the lines are not completed.

At 2000 ppm, where (chart 25) the cord rctec is well positive, around +25 e-5 /K, whilst the cord value of rcfc at full power (chart 51) is still well negative (-15 e-13/W) with an analytical value -7 e-13/W.
On the other hand the cord, point-kinetic value starts getting positive: the point-kinetics model , which does not profit from the spatial redistribution effects appears therefore overly pessimistic .

Kinetic reactivity coefficients and core stability.

Stability verification on Stationarity tests.

lof_c1_ini_check.dat (download) application aims at checking that on a spatial kinetics representation, the core at 2000 ppm and nominal power, initialized by trans/ini procedure which predicted a negative cord rcfc of -15 e-5/W, will remain stationary when let alone without any control.
This application differs from the corresponding lof_c1_ini.dat (download) initialization on the following data:

In Lstci, we must now insert data file lof_c1_f2ci.dat (download) containing the converged (critical) f2ci distribution saved from the previous trans/ini run lof_c1_ini.dat application.

Chart 51 showed that, at bec=2000, point_kinetic represention predicted a cord rcfc slightly positive (about +1e-13/W). Does it mean that the uncontrolled core would run away?
This is actually the case, as confirmed by the lof_c1_cinpt_ini_check.dat (download) application, which shows (fine red plot f2c_c1_cntpt) that the core appears to remain stationary for about 250 s, then diverges thereafter because (chart bd20) the power coefficient rcfc, initially just positive (cord value 1e-13; analytic value 2.7e-13) keeps increasing whilst the spatial representation (with analytic rcfc= -8e-13) remains stationary.
The divergence is more apparent when (chart bd35), f2 c1 is plotted on log scale.

On chart 51, we showed that at bec=2000 and power q2c= 1.94e9, slightly lower than full power, the cord point-kinetic rcfc is still slighly negative (-1.48e-13).
lof_c_.6_cinpt_ini_check.dat (download) application confirms (chart bd2, blue plot) that the core remains stationary, as expected.
The same type of verifications have been pursued on the core spatial representation, with increasing boron values. They indicate that boron must be pushed to more than 4000 ppm before reaching the instability threshhold (cord rcfc>0).
It is expected however that such an unrealistic, high boron level, would anyway be unsustainable for other raisons: solubility limits, excessive axial-offset and corresponding peaking factors,....

Core stability in the course of boron dilutions transients.

We will presently test the validity of our (cord-based) rcfc < 0 stability criterion on reactivity transients resulting from driving the core from 0 to nominal power by means of boron dilutions. Initial zero power critical boron is taken as a parameter.

lof_c0_bdil.dat (download) is the model input file used for such power escalation transients.
The sequences are always the same.
bsb variations are specified by means of bottom/Lstb/ Itp_sec interpolator specifying a linear boron dilution from 2000 to 1900 ppm in 100 s.
r0g and f2ci.dat are retrieved, as usual, from the corresponding ini run lof_c0_ini.dat (download) cases.

chart bd28 summarizes the various dilution cases analyzed. It plots (thick lines) core power f2c on left y axis and the corresponding reactivity rc (thin lines) on right y axis.
Plot d0_200>100 (light blue thick line) shows that diluting from 200 (zero power) to 100 ppm in 100 s will lead to a final, stable power of 2.3e9 W, at the outcome of a 300 s transient duration.
Plot b0_600>500 (red thick line) models the same 100 ppm dilution, but departing from 600 ppm. It reaches a power at 2.5e9 W after 500 s.
As rctec becomes less and less negative as boron ppm increases, the water warming will release additional reactivity which "reinforce" the boron dilution effect: it takes less and less boron dilution for reaching the same final power.
Diluting from 2000 ppm (blue plots) necessitates only 65 ppm for escalating power to 2.7e9 W (just above nominal power).
From 3000 ppm (black plots), it takes 45 ppm for reaching 3.2e9 W.
From 4000 ppm (hypothetic case!) it would take only 19 ppm for reaching 3e9 W! For this case we have assumed a longer dilution duration (400 s instead of 100 s). Dilution rate affects the time pace of the transient, but not its final power level!

Diluting 45 ppm from same boron 4000 ppm level (chart bd30) would carry the reactor to an excessively high power although the power coefficient remains just negative: core reactivity reaches rapidly a level 300 e-5 at 200 s and keeps running away afterwards.
We are actually facing here an unchecked (atws: "anticipated transient without scram) fast reactivity accident which is not or little mitigated by the natural reactivity feedback's action.

Effect of high boron, positive mtc core on power spatial distribution.

For the c0_65_bdil.dat (download) case (diluting 65 ppm from 2000 ppm in 100 s) chart bd34 illustrates well the progresse upwards axial-offset shifting as boron ppm increases.
for the same c0-65 case, chart bd33 graphs the trend of fqc, the hot channel factor.

Some tentative conclusions

Within the frame of our simplified model, we can only attempt here some general qualitative remarks pertaining to tendency behavior.

a) Increasing nominall power boron concentrations to 2000 ppm does result in positive moderator temperature rctec of about +10 e-5/K without, however, jeopardizing core stability. This does not result in power runaway thanks to the power coefficient rcfc which still remains well negative because of the negative contribution of the Doppler rcfc_u component overriding the positive rcfc_v density effect component.

b) It must however be checked if, in absence (or mitigation) of burnable poisons, the consequences of the thereof resulting upward power shifting and the possible possibly adverse effect on radial distributions can be sustained.

c) It must be investigated how the larger axial offset swing over the core cycle can be managed.
It is also clear that the axial power profile history over the cycle would be significantly affected: we should expect, in absence of axial shaping rods action, a monotonic decreasing axial-offset history more or less balanced around zero?

Consequences of positive MTC on design accidents.

Negativity of power reactivity coefficient is not the sole criterion of importance because it characterizes core transient response close to stationarity when both rcfc_u and rcfc_v are synchronously active.
Actually, there may indeed exist accidental transients wherein reactivity injection initiated by moderator warming (in case rctec_v > 0) precedes the corrective doppler feed-back.
The best example I think of is the loss-of-flow-accident analyzed earlier. LOF accidents are analyzed at boc in order to minimize the mitigating action of the moderator reactivity feedback.

lof_core_bsb.dat (download) application is selected as reference for our exercises.
It is based on the core-only accident model: core boron inlet bsb and primary pressure p3 are maintained at their initial value. These assumptions were shown to be (no too overly) conservative. cf charts lof11 and lof12. (lof|lof_safpwr.123)
Main data modification pertaining of boron effect :

bottom/Lstb/ bsb= 0, 1000, 2000 ppm considered as parameter.

In core/&Lstc we add kntco= t allowing analytic reactivity coefficients to be calculated in processing end_step, core.

Initialization by imposing initial Lstci/ f2ci distribution is retained for the purpose of allowing comparison with our old analyses.
It exhibits (chart lof18) an upward peaked power profile of the sort anticipated in case of high boron, positive mtc operation.

Lstrvg/ robg data must now emulate the water expansion boron effect: we retain the pessimistic linear water expansion boron model which resulted, for 1000 ppm, in a slightly positive (cord) rc_tec = +8 e-5/K at nominal power.

hot_channel/Lstfcyvg/ fxyog values have been kept unchanged, although they should, just like robg, be adapted to reflect a possible adverse effect of positive MTC on radial peaking factor. The necessary coding modifications have not been implemented yet.

Results:

Chart lof16 plots f2c, q2c, and rc for bsb= 0, 1000 and 2000 ppm.
At 2000 ppm the cord value for rctec which was previously calculated as 23e-5/K (about 10e-5/K should be expected for a more realistic boron model ?) and 32e-5/K for the analytic value.
The cord value of the power coefficient rcfc was estimated at a negative value of -15e-13/W (chart 51) and about -8e-13/W for the analytic one.

Chart lof16 confirms that, at 0 ppm (red plots and dots), where rctec is still negative, the moderator heating (tavc plot in chart lof19) immediately resulting from the loss of flow reduces rc to about -15e-5 at the moment rods start dropping.

At 1000 ppm boron (black plots and dots) the positive rctec has the effect of adding to rc about +15e-5 at the same time, which results in a slight increase of f2c.
However, the thermal power q2c (black thin) is barely affected thanks to pellet thermal inertia.

At 2000 ppm (pink plots and dots) , rc addition reaches 50 e-5, which results in peaking f2c at 3e9 and q2c at 2.8e9.
As soon as the safety banks start dropping, at 2 s, thermal power goes back again.

Chart lof14 illustrates the impact on dnbrc: min dnbrc reached at about .4 s after rods start dropping, is degraded from 2.88 to 2.68 as consequence of adding 2000 ppm.
The loss of dnbr margins does not appear that dramatic, however

At last, Chart lof15 indicates that, at critical moment of min dnbrc (about 2.4 s), aoc is still sightly positive: dnbrc does not benefit much from the downward shifting due to rod drop.

Kinetic coefficients application examples