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Control Rods Group Withdrawal Accident at Power

In this application, (rgp4.dat) (download)the core and its configurations are represented explicitly.
Otherwise, the data are the same as for the previous odt protection application .
(chart 01) 3 configurations are used: ig=1 for ARO (all rods out), ig=2 for the group withdrawing from 2.3 m elevation, and ig=3 for the emergency bank.
Configurations 2 and 3 fall together at same drop velocity from time sec_drop.

In S3, dnb is called after hot_channel in order to check the ability of odt trip system to insure protection against nucleate boiling crisis.
Description of input data

In this (rgp4.dat) (download)application, water transport in down_comer and loop_1 is modelled in Euler mode.
Withdrawal speed is "fast" (full rise in 5 s) after a stabilization period of 25 s allocated, for allowing, in case Lagrange mode is enabled, proper transition from Euler to Lagrange initial steady state conditions).
As for the odt application examples, the secondary temperature field is established at initial state and kept unchanged during the transient.

core_config
Lstau/&Auf2c/ f2c_au: 1.18 * nominal power. Protection by over nuclear power trip.
&Auopdt and &Auotdt: cf odt_test application
core
&Lstc
k9c= 7: 7 nodes of 4 meshes each.
It is reminded that the t&h variables are uniform within each node, but the dnb is nevertheless calculated on the (finer) mesh partition in order to better account for the possible grid perturbation effect on dnbr.
Lstci
(chart 13) f2ci: the initial aro profile is slightly top-peaked , close to the profile of f2ck at 3.5 s.
The actual initial profile (of f2ck=q2ck=f2ci) at sec=0 accounts for configuration ig=2 partial insertion.
This initial profile is one of the important key parameters for dnb.
&Lstg
swsknug= f: for the present test all the core ini corrections are supported by the sole reactivity correction r0ci.
For actual applications, it is advised to enter critical initial properties (swsknug= t).
Lstgg: for relatively weak reactivity release, delayed neutron properties are not critical.
The reactivity feedback will slow down the reactivity release rate, but will also delay tripping, so that most pessimistic set of data for release rate and reactivity feedback's can only be obtained by means sensitivity runs.
Fission products residual power effect is not activated. This effect should slightly delay thermal power release. Neglecting it is (not overly) conservative.
hot_channel: By default, the hot_channel thermal properties are the same as those of the average channel.
Lstfxyvg/fxyog: are taken identical for all configurations, because the reference dnb program generally accept only single value for the fxy.

Xrgp4.dat(download) repeats the same application in Lagrange mode.

Results

chart 02 confirms that changes of wec and hec (hsb) are small, and exhibits that transition from Euler to Lagrange initial conditions takes about 25 s

Trip results, written on console and possibly redirected to safpwr.lis:

au_f2c:
sec_au_f2c, sec_drop, f1c, f2c_au, f2c
26.6674290 28.7674294 3.18015642E+09 3.38140006E+09 3.48167706E+09

au_otdt: sec_au_otdt, sec_drop, dat2_odt, dat2_otdt 28.2404938 28.7674294 43.5895996 39.8018456
au_opdt:
sec_au_opdt, sec_drop, dat2_odt, dat2_opdt
28.921 28.419 44.237 44.043
zaugri= 3.2249 3.650 3.650
(chart 03) The first trip, due to overpower au_f2c (green plots) is detected at 26.67 s.
Rods (chart 04) start dropping at 28.77 s.
If the odt protection system could be directly connected at core inlet and outlet (chart 03; red plots), "ideal" otdtC [OTDT trip; Core] trip would comes second at 27.7 s when datc crosses datc_otdt.
Dynamic, real otdt (blue plot) trip comes third at 28.3 s, before sec_drop.
Dynamic compensation acts thus slightly late (1 s after the ideal opdtc).
Uncompensated otdtS [OTDT; Static] trip (intersection of dat2_odt with dat2S_otdt) would happen only later at about 29.0, thus after sec_drop: in absence of over_power trip, q2c would reach excessive value.
Consequently, dynamic compensation is indispensable.

Lagrange representation of transport effects

(Xrgp4.dat) To pin point differences vs Euler case, search for "!Lagrange" comment string in input file

Trip output lines on safpwr.lis:
au_f2c:
sec_au_f2c, sec_drop, f1c, f2c_au, f2c
26.5685539 28.6685543 3.20731546E+09 3.38140006E+09 3.51350374E+09
au_otdt:
sec_au_otdt, sec_drop, dat2_odt, dat2_otdt
28.1578178 28.6685543 44.9690552 40.1522751
au_opdt:
sec_au_opdt, sec_drop, dat2_odt, dat2_opdt
28.7788868 28.6685543 44.9690552 44.4853210
zaugri= 3.29192853 3.65001011 3.65001011
(chart 06) The
au_f2c trip (green plots)occurs at 26.57 s, like for Euler mode because the transport model has a negligible impact on reactor kinetics.
datc_otdt (red) follows at 28.7 s and
dat2_odt (blue) at 29.2: the transport model has thus little impact on trip times even if the individual temperature signal (chart 7) deviate somewhat.
As expected, the difference in terms of dnbr is insignificant: both models predict a min dnbr of 1.32 at max thermal power time.
Slow Euler RGP case

Complete Rod withdrawal would now take 20 s, instead of 5 s for the fast case (rgp5.dat) (download).
Now, the reactivity feedback's attenuates the power rise so that au_f2c trip is just avoided, and no edit appear on safpwr.lis for this trip.
Trip outputs on safpwr.lis:

au_otdt:
sec_au_otdt, sec_drop, dat2_odt, dat2_otdt
31.1055088 33.2055092 43.3078613 41.6589546
au_opdt:
sec_au_opdt, sec_drop, dat2_odt, dat2_opdt
33.9799461 33.2055092 44.1430664 44.1286163
zaugri= 2.85700154 3.65001011 3.65001011
dat2_otdt (chart 09 ) now occurs first at 31.10, followed by
dat2_opdt at 34.0 s.

The transport dynamic compensation remains adequate.
Lagrange Slow RGP (Xrgp05.dat) (download)
Trip edits in safpwr.lis:
au_otdt:
sec_au_otdt, sec_drop, dat2_odt, dat2_otdt
31.7214146 33.8214149 44.4307251 43.9053612

au_opdt:
sec_au_opdt, sec_drop, dat2_odt, dat2_opdt
32.7061195 33.8214149 44.9415894 44.6937523

au_f2c:
sec_au_f2c, sec_drop, f1c, f2c_au, f2c
33.9049377 33.8214149 3.35267763E+09 3.38140006E+09 3.38441728E+09
zaugri= 2.89841223 3.65001011 3.65001011
Overpower and otdt trips come first, at about the same time 30.28.
opdt follows 1 s later 31.28.
Compensation is satisfactory:
datccrosses datc_opdt at 27.3 s
Uncompensated otdt would comes only at 29.3 s.
dnbr (chart 08) is about the same (1.32) for euler and Lagrange, as expected.

From these preliminary results, we would be tempted to conclude that dnbr and odt safety margins are close for both euler and Lagrange predictions. If confirmed by further verifications, it is a encouraging observation, in so far as Lagrange model is much more complex.

Static verification of dnbr at mindnbr state point

It may be advisable to confirm SAFPWR transient dnbr predictions by means of a licensed t&h design program like COBRA, equipped with the fuel-dependent dnb correlations and allowing for a better inter-channel mixing grids representation.

This verification could be carried out at stationary condition, with input data extracted from SAFPWR value at the "state point" where mindnbr is observed.
The state point characteristics, which must be replicated in the stationary verification calculation, are:
system variables such as p3, wec, hec, which are still close to their initial values,
q2ci, the total power deposited in coolant for the average channel, and also
qsci, the heat flux in the hot channel film.

In SAFPWR, the transient relation relating, in the average channel, and at each mesh ci, the nuclear power f2ci generated within the pellet, the thermal power q2ci deposited in the coolant, and the heat rate qsci transferred through the film from pellet to coolant is:
q2ci = (1 - xfuc) * f2ci + qsci
where xfuc is the fraction of nuclear power directly deposited in pellet.

At state point, we must replicate, in the average channel, q2ci, and qsci.
But, in a steady state representation of the average channel, simulated by a heat_gen application, q2ci is identical to f2ci, so that we need to enter a pseudo fraction xsfuc [Stationary xfuc ], such that
q2ci = (1 - xsfuc) * q2ci + qsci.
This implies:
(1-xsfuc)*q2ci = (1-xfuc)*f2ci.
As xfuc can only be specified core-wise, the definition relation for xsfuc is actually
(1-xsfuc)*q2c = (1-xfuc)*f2c.

Steady state replication of thermal state point conditions is carried out by an ini, heat_gen representation of the core and the hot channel, where we must enter as f2ci the q2ci taken from the state point.
As, in SAFPWR, the thermal power is only calculated at the nodal level, q2ci is not available, but only q2ck.
In order to get q2ci, we need repeating the transient application with 1 node per mesh
(rgp4_test.dat, download)

If the transient is run in Euler mode, the initial stationary steady state being readily achieved by ini, core, the transition period to stationary Lagrange state is not required, and the rod withdrawal can start from the beginning.
The data differ from the previous cases in Lstck and by taking dsec= .5 in order to better delimit the dnbr minimum of 1.28 at sec= 3.5.
trip output from safpwr.lis for the rpg4_test.datrun:
au_f2c:
sec_au_f2c, sec_drop, f1c, f2c_au, f2c
1.543 3.64 3.37E+09 3.38E+09 3.509E+09
au_otdt:
sec_au_otdt, sec_drop, dat2_odt, dat2_otdt
3.20 3.64 43.22 41.67
zaugri= 3.28 3.65 3.65
(chart 11) Mindnbr= 1.28 is observed at 4.0 s, just after the rods start dropping, at 3.64 s.
(chart 13) exhibits the deformation of q2ck and f2ck from initial condition to state point at sec= 4.0 s.
q2ck distribution at 4.0 s is about half way between the initial q2ck and f2ck at 4.0 s: it would thus be overly conservative to calculate dnb margins with nuclear distribution f2ck.
Static dnb simulation at state point

The thermal representation of the core at steady state condition can conveniently be carried out by the procedure
ini, core_config, heat_gen, hot_channel, dnb
(rpg4_incc.dat) (download) with the heat_gen representation of the core, and with option oqc= 2 enabling activation of hot_channel.

In Lstcfg, enter the position
zgri= 3.2852, 3.65001, 3.65001
of the core config at state point, and under
hot_channel, the tables fxyog for calculating fxy in various configurations
In Lstci enter f2ci=q2ck, taken from plot of f2ck generated by the transient application at sec= 3.5.
wsb= 12027 and hsb= 1.2503e6 are the state point values.
At sec= 3.5,
xfuc=1-(1-xfuc)*f2c/q2c
=1-(1-.974)*3.7080/3.3167 =.9709.
Using this value does not replicate exactly the same state point qsc7i distribution in hot channel because xfuc is a core-wise value.
However, a slightly different value .963, obtained by trial and error does lead to a nearly exact duplication (see chart 14 ) of the distribution; the sum(qsc7i) is exactly the same, however).
With the state point values for wec, hec, p3, zgri and f2ci, rgp4_inccc.dat leads (chart 11) to a static dnbrc7= 1.24 somewhat lower than the transient value 1.28 , in spite of the fact that the system hydraulic values and the coolant heating by film transfer and direct deposition are exactly well replicated.
The explanation for the discrepancy is that, the transient dnb prediction of the water quality x2ci (chart 05 ) has a smaller, historical, value than the steady state value.
Tentative conclusions:
Static dnb verification with state point value of q2ck is conservative (dnbr=1.08 vs 1.158)
Static verification with f2ck at state point would be overly conservative
Note of closure

It is not obvious to justify the choice of the lead an lag constants appearing in the odt protection equations, because they apply directly on temperature differences and averages at measuring locations, rather than on each individual measurements, in which case, the lag and lead values could possibly be related with coolant transport process in primary circuit.

Nor, is it clear whether transport delay in sampling lines and thermometer electric delays are included in dsec_au or in the lags of the odt relations.
If the sampling lines are of the same length, carry the same flow, and if the heat-exchange with their walls is neglected, then their delay effect in the odt protection could well be represented by an additional delay in dsec_au.
If not, modeling those lines by a Lagrange stack could easily be contemplated for implementation in SAFPWR.