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2-phase drift lagrangian balances for pressurizer

In pressurizer, for non equilibrium condition, the eulerian drift model developed for the SG core does not easily apply because the flow distribution may rapidly change in value and even direction in the course of a time-step.
Furthermore, the vapor region must be explicitly represented, as well as the bubble rise from liquid region.
Therefore, a lagrangian drift model is more appropriate.

The liquid region is decomposed into variable volume elements j whose representation is the same a in pict.
The inlet flow values w*e (we, whe, wbe) are given, as resulting from the primary and outlet balances. For the first element j=1 of the eXpansion line, these flow are the exit flows w*so of the Outlet. For the following elements of the liquid region, the inlet section is transported by the liquid and is only crossed by the flow wv of the saturated bubbles drifting from the previous element.
Similarly, the exit section is only crossed by the flow wvs of saturated bubbles drifting from the core of the element.
In the last element jrx (tRansition) of the liquid column however, the outlet control surface is supposed to float on the liquid, so that it is also crossed by a non-zero flow wls of the droplets falling from the vapor region (plus any water collected directly from spray or condensed on the wall of the vapor region).
Interface condensation-evaporation mass and heat exchange may also take place at jr outlet.
The resolution scheme is basically the same as for the eulerian model:
liquid celerity cl is eliminated from the generic planar drift relations (1) and (2), to result in (3).
defining (4:5) the liquid and vapor masses ml and mv crossing the plane in δs, and
(6) an asymptotic (at zero α) bubbles-to-liquid flow mlv0 [Mass; from Liquid; to Vapor; in 0 α condition] leads
(8) to the generic planar relation giving the vapor mass mv crossing the plane in terms of the liquid mass ml, local quality x and
vapor volume fraction (7).
mlv0 (6) is only pressure-dependent
This relation (8) holds for any plane and, particularly, at outlet plane s of the elements.
Let us remindl that mls is null except for the last, transition element jr, where its value is supposed known, as calculated from the current conditions in the vapor region.
(9:10) are the generic "elemental" (per element) balance equations.
Collecting the known terms in m1e (11) and mh1e (12), and eliminating m2 gives
(13) h2 in terms of vapor mass msv escaping element outlet
an "implicit" outlet extraction model (15:16) is assumed as usual, which allows expressing
(17) mvs in terms of outlet mls and elemental eos vapor quality.
Thus, x2 becomes the root of the implicit equation (18)
The resolution method for x2 is illustrated by the picture 2 and by relations (18:32). (x2,h2) is the intersection point of the curve (13) with the line hl + x2 hlv.
The guess h1_a corresponds to zero drift.

The resolution algorithm has been experimented on a simple case with the following data:

p3=155e-5, for which clv0=.063 m/s
we=q=0, m1=150, h1=1.8e6 --> (x1=.176, αa=.55)
The chart shows that the solution h2 is very close to the intersection point even if the x_a guess (.176) departs sensibly from the converged solution at x2=.11. This can be explained by the weak variation of h(x).
It is probable that the first guess solution h2_a = 1.6955e6 would be quite acceptable and the we could make the economy of the Gaussian correction calculation.