The following applications illustrates SAFPWR flexibilty for sensibilty studies:

The data changes in **input.dat** are:

om1= 1|0

It improves (chart 02) the prediction but suffers from the well know Courant flow restriction: the results become more and more instable as

For

an additional small node

For **om2= < 1** the results (chart 03) become totally unstable!

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

Alternatively, the element volume

The nodes

Actually it is obseved (chart 03) that

In order to confirm this interpretation, the base case 001_dx.dat
(**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`

)
is repeated in Lagrange mode, but by keeping `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

We take a case with

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!

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

Presently, the pulse wave progresses (chart 06) at constant speed without any deformation.

It starts crossing the oulet section at

`download`

)
node 11 volume is reduced from .2 to .05 and the euler mode is totaly explicit (

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.

For

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.

If

Remind that the best solution for

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.