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First PWR accident simulation exemple: effect of water transport model

The following applications illustrates SAFPWR flexibilty for sensibilty studies:

Input.dat

001_b.dat (download) | 001_a.dat
[The "|" bar separating two data files or data records means "by comparison of" or "versus".]

The data changes in input.dat are:

om2= 0 |1;
om1= 1|0
: simulates a pure explicit eulerian scheme whereby enthalpy and boron values are extracted at node oulet with their bos values.
It improves (chart 02) the prediction but suffers from the well know Courant flow restriction: the results become more and more instable as dsec exceeds the node crossing time (1 s, in the present case).
For vli=4*1 & om2=0 , hs overshoots at sec= 5 and stabilizes later.
001_c | 001_a:
i9l= 5 | 4
vli= 3*1, .2, .8 | 4*1:
an additional small node li(4)= .2 is inserted (for the purpose, for example, of monitoring fluid temperature there).
For om2= < 1 the results (chart 03) become totally unstable!
001_dx.dat (download) | 001_a.dat:

Lagrange mode

The only way to conciliate stability and accuracy is by resorting to a Lagrange integration scheme whereby fluid balances are carried out on fluid elements moving with it, instead of on fixed nodes.
Lagrange option is enabled by simply evoking the keyword xloop_1 instead of loop_1 (X for eXtended Lagrange balance), and by entering under

Lstl1 an additional data
ndavel= 5 [Number;DeltA;element Volume; at Entry;of L) for splitting the fluid volume (here 1 m) entering loop1 in dsec= 1 s, into 5 equal elements of the same size .2 as the smallest node vli(4).
Alternatively, the element volume davel may be specified instead.
The nodes li=4 and 5 are reached and covered by "hot" elements after 4 s and loop outlet sl at sec= 5 (chart 02).
Actually it is obseved (chart 03) that li=4 is already reached by the hot front at sec=3. This effect is caused by the thermal expansion of the entering elements.
001_ex (download) | 001_dx:
ndavel=1
hel= 3*1.3e6 constant
bel=0, 1000, 1000.
In order to confirm this interpretation, the base case 001_dx.dat (download) is repeated in Lagrange mode, but by keeping hel constant in order to prevent water expansion and the front progression is now followed on the boron concentration (chart 05)
bs reaches 1000 ppm at sec=5 only because at sec=4, the borated front has just reached the outlet.
001_j.dat (download)
The effects of "numerical diffusion" and Courant restriction are better illustrated by applying a single zig-zag bel pulse, rather than a ramp step .
We take a case with vli= 20*.2 and sec=.2 so that the elements move at the same speed as for the base case.
hel remains fixed at 1.3e6 but we apply now a 1 ppm amplitude boron pulse of .8 s duration defined by the interpolator
loop_1
Itp_sec
sec= 0, .2, .6, .8, 1000/
hel= 5*1.3e6
bel= 0, 1, -1 , 0, 0
The problem is firstly solved in implicit euler mode (om2=1).
(chart 07) exhibits the plots of the zig-zag bel pulse at inlet together with eos boron ppm concentration in nodes li=1, 5 and bsl at outlet: the effect of numerical diffusion is dramatic. After only 2 s the pulse is already "diluted". In node 5, the pulse has almost completly vanished!
001jx.dat (download) | 001_j.dat
The application is now repeated in Lagrange mode with ndavel=1 so that element and nodes are now of the same size.
Presently, the pulse wave progresses (chart 06) at constant speed without any deformation.
It starts crossing the oulet section at sec= 4 exactly, as expected
001_k.dat|001j.dat: (download)
node 11 volume is reduced from .2 to .05 and the euler mode is totaly explicit (om2=0).
dsec=.2= crossing duration for all the nodes except for the small one (11) for which dsec=4*crossing duration.
The pulse is transported (chart 07) without any deformation until it hits the small node where it becomes "trapped" by the Courant stability criterium and literally "explodes".
With om2=.5 the solution is still stable but the pulse amplitude vanishes rapidly.
For om2=.1, it is damped until the small node is reached, from which time it starts again to expand indefinitely.
Finally, (chart 09) compares implicite euler solutions for the successive combinations (vli, dsec) = (4*1, 1), (4*1, .2), (20*.2, .2), (20*.2, 1).
For vli=4*1 (black plots), the improvement gained by using a smaller dsec is minor.
Using vli= 20*.2 (red curves) and dsec=.2 results in a small improvement.
If dsec is kept at 1 s, the improvement is not worthwhile.
Remind that the best solution for dsec= .2, provided with Lagrange scheme (chart 04), is a ramp of .2 s duration starting at sec=4.

Note that implicite euler solution provides also correct results in limit situations where all the nodes are of the same size and dsec=crossing duration. This situation is rarely encountered in practice.

Is numerical dispersion effect observed for Euler solution a critical safety issue?
The consequences of numerical diffusion deformation on the response of the temperature sensors in the reactor "over-temperature and over- power" protection systems will be analysed in depth in the chapter on protections.
Note finally that the water does not move strictly as a piston, because the fluid velocity is smaller near the pipe walls.