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Pressurizer System

The pressu system includes the Outlet, the eXpansion line, the Pressu vessel itself and the Spray system.

If splito= f, O (not zero "0" !) covers the core outlet itself;
if splito= t, O represents the part o1 outlet associated to the loop_1 which is linked to the pressu.
In any case, O collects, as net inlet flow weo, the expansion rate of the rest (loops + reactor vessel - O) of the primary.
The expansion flow w(,h,b)eo is forced into the pressu system whose function is to update the pressure in order to accommodate for the total primary expansion rate.
In "normal" pressu operating condition, the vapor region of the pressu contains a saturated mixture of vapor and spray droplets and the liquid region contains saturated mixture of water and bubbles generated by the pressu heater.
Below the heater the water is either sub cooled in case of out surge (weo > 0), or saturated in case of in surge (wso < 0).
In such "normal" operating conditions the pressu is capable of accommodating the in/out-surges with a minimum pressure change (because vvp0 ≡ (d/dp)vm along the vapor saturation line is weak at normal primary pressure).

In steady state nominal conditions, the primary water expansion rate is null, but the spray and heater are nevertheless active to insure that those "normal" conditions keep prevailing for normal operational transients around the steady state.

In such circumstances, a simple "homogeneous" pressu model will suffice.
The phase separation does not need to be explicitly represented: the whole pressu vessel contains an uniform, saturated liquid-vapor mixture, and the expansion line convey either saturated (out surge) or sub cooled water (in surge).

Homogeneous pressu model overview

This model is activated by the keyword "pressu" and the pressu sub-system noted oxp [Outlet+eXpansion line+Pressu vessel]; input data are entered separately for the Outlet, the eXpansion line and the Pressure vessel as follows:

pressu
Lsto/cf []
Lstx
m9x= [index M of node; max 9; of eXpansion line]
it9x= max number of inverse m,h,b-balances sweepings of X, if needed

When using the oxp model, the heat exchange with wall is generally neglected.

Lstxm
azxm= [Accretion; elevation Z; node XM]
cfrxm= [Coeff; of Friction; XM]

Although the water volumic mass is calculated at the global pressure p3, Euler balance of X can calculate the nodal pressure drops for editing purpose.

Lstp
m2p= [total fluid Mass;initial 2;in P]
initial conditions correspond to eos (2) of fictitious 0th step of zero duration.
vp= [geometric Volume;P]
ini, pressu
(1): (m,h,b) flow entering O.
Water delivered by X to O is saturated.
(2): filling O
(3): filling X
(4): filling P
x2p: pressu quality and h2p from input m2p
(4) simply expresses the pressu stationary heat balance relating the heater power qyp to spray flow and enthalpy
do, pressu
Net flow entering O
If splito= f
(5): total flow entering the loops, and
(6): net flow into O, representing the primary fluid expansion which must be absorbed by the pressu system.
hel is the value used in last do, loop operations.

Once w{,h,b}eo obtained, an ordinary eulerian balance is carried out on the outlet node and on the successive nodes xm of X.
If the outlet flow wsxm is positive for all the nodes xm of X, the process is stable and a single nodal balance sweep suffice.

In case of out surge, the flow reverses in all or some of the X nodes and the Euler balance must then be performed in "reverse mode" where outlet enthalpy of each node needs to be specified.
For the last node m9x it is the current pressu liquid region enthalpy; for each of the other nodes it is the current enthalpy of the donor node above.
As the reverse balances process may by instable, a few (max: it9x) balances sweep iterations are carried out to get a converged solution. Note that a partially converged solution does not impair exact closure of mass and energy balances.
Pressu (m,h,b) balances
(9): net flow w(,h,b)eo into p.
wsx is the flow at X outlet.
The vapor discharged (wyp) is supposed saturated.
(10): classical (m,h,b,v) variable volume balances .
(11): water properties.
v2p is the free eos volume taken by P as result of the isobaric balances conducted at the current "guess" value
p3 of the courant guess of the searched eos pressure p2.
vm2_p is the isentropic pressure derivative of vm at eos;
vp_p ≡ (d/dp)p3v2p (12) is the expansion coefficient of the pressu considered as freely expandable.
(13,14) accumulate the expansion coefficients of the whole primary (a+b+c+loop_l) and OXP; xwl is the loop multiplier.
(15): total volume balance updating p3 to p2 in order to nullify the P volume error vp - v2p.
(16): updating of all pressure-only dependent properties for the primary domain 0.
General description of multi-regions pressu model

For severe transient like the Steam Line Break (SLB), the strong water out surge from the pressu system caused by the rapid cooling of primary water may result in complete collapsing of the liquid region in the pressu and expansion line.
The water in dome may even enter boiling because its temperature remains warmer and the dome, hence, behaves then as a second pressu!
For non-isolated SLB the reactor vessel bottom may also experiment moderate bulk boiling.

In order to tackle those extreme situations, a multi-nodal, multi-elements, non-equilibrium, Lagrangian model had to be developed for xp considered as a whole.
This model is noted "oxrp" because it includes an additional tRansition region "R" between the eXpansion region and the Pressu region; cf pic_25.2. The data specific to this model are identified with oxpr on Rdoxp model data file.

In the course of SLB transients, the possible return to power causes a rapid water in surge into the emptied pressu.
The pressu behaves then more as a water vapor gas pressurizer and this will result in a rapid pressure surge, which is beneficial for opposing to boiling crisis, but at the same time, reduces the boron flow injected by Safety Injection water.

As spray is not active (pressure still too low) at time of water surge, the main mechanism for checking the pressure surge is vapor condensation to wall.
But this can only take place in so far as the wall edge temperature remains below the vapor temperature and the condensation latent heat can be absorbed by the wall.
Thus, a correct analysis of this process necessitates a concomitant calculation of transient heat conduction in the wall.

The processing of the OXRP sub-system will be conveniently explained by commenting the successive calculations triggered by read, ini, and do, npressu (!the n stands of "New" pressu) read, npressu

npressu: "new" pressu
Lsto Normally, the heat exchange with metal is neglected in the outlet node.
However an interpolator qo(sec) [heat rate Q; added of Outlet] has been made available for testing purpose. It makes it possible to setup a stand-alone OXRP system, simulating the primary water expansion by means of heat added to a large, single, outlet volume.
Lstx
m9x: the fixed node partition extends here to the whole XRP domain.
oxq is set to true to activate heat exchange with wall.
The interpolator zxr(vxr) conveniently specifies the water volume accumulated under the elevation zxr, both counted from X bottom.
It allows to accurately define the flow area shrinking at pressu vessel bottom and head.
Lstxm
swxm [Section; Flow; XM] combined with vxm allows representing inclined parts of expansion line.
Lstux: generic thermal data for all xm nodes
Lstuxm :individual data for each xm.
vuxm is the wall metal volume exchanging heat with xm.
Film heat exchange coefficient is not internally calculated. Its value is, however, not critical because the film resistance is only a small part the wall conduction resistance.
i8xm is the number of conduction rings and fdxim is the fraction
Lstp applies here to the vapor region P of the pressu vessel, not to the whole pressu vessel.
v2p is the initial volume allocated to the P region.
mqyp: [QY: Ynjected Q] node xm containing the heater.
mx node data geometry definition must insure that the heater is totally included in a single node.
zqyp [elevation Z;QYP] water elevation for cutting off heater in order to prevent its burn-out.
hfpir= [H; Film; PIR] heat transfer coefficient for inter facial heat exchange at P/R Interface.
evap_fact= (see later)
Itp_y_x / Itp_qyp_p /qyp(p) interpolator.
Pressure is regulated by the combined action of spray and heating. Cf (typical regulation diagram)
Lsts
wss is the initial spray flow.
Its enthalpy is normally set to the cold leg enthalpy (b2o).
Direct entry has been made available for isolated OXRP testing.
Note that Spray is also designated by "a" for Aspersion in the OXRP coding.
wss(p) interpolator defines the aspersion branch of the pressure regulation diagram.
ini, npressu
Filling P
(17): P contains a volume v2p of saturated vapor of massic volume vv0.
(18): global XP heat balance
(19): mqyp heat balance.
(20): bubble flow wrbp [floW; from R; carried by Bubbles; for P] is the same as the flow wpa [W;from P; to Aspersion] condensed to aspersion.
As P does not content liquid, the water drop raining flow wpgr [W; from P; by Gouttes (droplets); to R] is null.
Likewise, wpur=0 [W; from P; associated with wall temperature U; to R] is zero.
R/P interface position
(21,22): v2x_r is the volume up to interface converted to level zxr by the interpolator.

Outlet initialization Same as for ini,oxp
Filling X
(23): the nodes xm under mqyx are filled with saturated water.
Bubbles generated in mqyx node by heater are normally not dragged downwards with the wss flow or wvs [floW; Vap. bubbles; at Sortie] of mqyx < - wss
(24): vapor quality of the heater node can be obtained by solving for x the stationary form of Euler balance with drift. The same quality is also assigned to the following elements, including R
do, npressu
(25): processing of do, npressu starts with calculating heat-exchange quxm with wall, for all nodes xm, including those in P, by means of the pipe transient conduction model.
u1xm is the bos temperature vector u1xm(1:9)
(26): qup is associated with wall condensation.
qp is the total heat received by P (if the heater node is uncovered)
(27): expresses the heat balance of the pressu flow condensed on wall.
Its stationary form implies that the condensed water reaches the rip interface at saturation condition without any mass accumulation along the wall. This expression applies only if qup < 0 (qup: wall heat rate supplied to the nodes).
(28): in the pressu region the continuous phase is vapor. The droplet fall is calculated from correlation [].
(29): gives the vapor condensation on aspersion droplets if the vapor region is saturated ( 0 < x2p < 1.
Condensation occurs in so far as the spray droplets temperature remains lower than the saturation tl0 temperature.
The expression translates the heat balance of wss + wpa flow. It implies that the droplets are supposed saturated (hl0) when reaching the PiR interface and that the process is instantaneous (no accumulation along the wall).
If the vapor region is overheated, a part x_evp of the wss flow may evaporate into the region and the remaining fraction x_cond is also supposed to reach the interface with a temperature tl0 (tl0 = tv0 at saturation).
In that case (30) represents the heat balance of the flow wpa + x_cdx wss falling on R.
The evaporated fraction is proportional to the overheating x2p - 1 and is, tentatively, evaluated trough the input factor evap_fact.
(31): total liquid flow falling on R.
Note that different contributions of the liquid flow are evaluated with the current properties of P whose defaults are the bos values.
On the contrary, the bubble flow wrbp escaping R will not be calculated until a converged x2p has been reached as a result of the OXR balance.
We start with O eulerian balance (32).
If wso > 0, hso = h2o and the balance provides the outlet flows wso, whso, wbso which will be the inlet flows wexj, whexj, wbxj of the first element xj=1 of X
If, on the contrary, wso is negative, the (reverse mode) balance uses the, still unknown, hso as input.
As a guess for hso, we group together (see fig) as a single composite element jex enough elements xj=1,jex from X to prepare the water mass -wso dsec that X will deliver to O.
The coding covers the case where all the elements (xj=1,jrx) must be spent and O then becomes exposed directly to P for a part of dsec. The process must possibly be iterated until a feasible solution is reached.
(33) Next the balance sweeping of the xj=jex,jrx elements is undertaken.
Remind that, for the Lagrangian balances, the in- and out- control surfaces follow the liquid and they are crossed upwards by the drifting bubbles. Thus, the outlet flow is wvs, the vapor flow, which becomes the inlet flow we for the following element.
For the first element jex, which may regroup several element xj=1,jex in case of outsurge, the inlet control surface is fixed and the inlet flow w(,h,b)e is the outlet flow {wso, wso hso, wso bso} from O.
For the last element jrx, the outlet control surface "floats" on the R/P separation interface which is crossed upwards by the vapor flow wvs resulting from R element balance, and downwards by total liquid wlpr flowing from P and the spray.
(34) In processing the R balance, we must include the interfacial heat and mass exchange, which is also tentatively estimated via an input hfpir [Heat exchange coeff; through Film; P/R Interface]
Presently, we have at hand all the components for carrying out the balance of the region P, which includes the liquid/vapor mixture in the pressu region, but excludes the water flowing from the spray and condensed on it or on the wall. In accordance with to our modeling, the volume occupied by that water is neglected.
P balances
(36) the net flow wep entering P includes the fraction of wss evaporated in P if x2p > 0.
(36) implies that possible flow discharged from P escapes with its enthalpy.
(37) is simply the variable volume (m,h) balance.
Note that there is no boron in P: all the boron coming from the spray is collected by R.
(38) Whenever O is exposed to P in the course of time step dsec, the (m,h) balance is done on the total XRP component.
The contribution of the possible water still present in the first element jex is accounted for.

Once P properties obtained, the transition from the last pressure p3 acitve in the course of the (m,h,b) balances to the new pressure p2 necessary to accommodate for the total primary volume, is carried out in the same way as for the homogeneous pressu.

At eos, the eos element content m{,h,b}2xj(jex:jrx) is rehomogeneized (39) into a new series of nodes xm(1,mrx) to get a configuration similar as that prevailing at initial conditions.
Note that, in the course of that elements-to-nodes rearrangement, the volume v2xj(jrx) is monitored.
If it becomes too small (< .2 vxm(mrx)), it is merged with the preceding node; if it becomes too large (< 1.5 vxm(mrx)) it is spit to create a new complete element below it.
Let us also remind that the last element occupies only a part of the last node mrx of the R region.

If the pressu processing is repeated (redo, pressu) in the course of the time-step, the nodal wall heat rate quxm, which was evaluated from the bos fluid temperature t1xm, is not recalculated.
The inter elements and inter region exchanges (wrbp, wpgr,..) are instead calculated from the eos properties h2xj, h2p, t2p. Thus, the redo, pressu operations provides the possibility to stabilize the pressu condition at the same time as that of the other primary components (for ex, wel, whel)