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PWR Initialization operations

System initialization

It aims at generating a complete initial steady-state condition consistent with a minimal set of design or operational data, namely:

For each SG "n", the "ini" data are: net thermal power qgvn transmitted to tertiary system, steam pressure p3n, feed-water flow wyan and enthalpy hyan, SG down-comer water level zan (or water volume v2an) and the recirculation ratio (ratio of input w2an / wyan), which is here considered as a design characteristic of SG.
For the primary side, the ini data are: primary pressure p3, bottom outlet enthalpy hsb and boron bsb, core power f2c or q2c, volume flow of primary pumps (or mass flows wel at loop_l inlet), spray flow in pressu and its initial water mass.
For the dome the flow wed injected from the cold legs or the fraction xwld of core outlet flow directly injected into the dome;
these two parameters must be adjusted to reach the desired initial dome water temperature between the cold and hot leg values.

By back solving the conservation equations, the program then computes successively:

For the flows in loops and in reactor vessel, a multiplying factor correcting the friction pressure loss coefficients in such a way that, with the input manometric head, we reproduce the input volumetric flow.
The initial mass, enthalpy and boron of each node are in turn calculated.
For the pressu, the power qyp of electric heaters necessary to close the pressu enthalpic balance, assuming that reactor outlet receives saturated water from pressu expansion line.

Core initialization

The neutronic tables correlate, for each configuration, the neutronic properties with the physical effects actually modelled in SAFPWR (coolant density, boron, effective fuel temp .

However, the slow "historical" effects, such as fuel depletion, Xenon and Samarium build-up, which are supposed invariant in the course of the transient, do not need to be explicitly represented:
Assume that, at initial steady state, is available a reference solution "0" accounting for all the effects not explicitly modelled by the program.
This reference solution must satisfy the stationary form (22.1a) of neutronic balance, where λ is an eigen-value accounting for the fact that the model selected for generating the neutronic tables is not necessarily critical.
The initialization operations (ini, core) will thereafter determine mesh-wise corrections which force the core model to reproduce the values of the reference data for reactivity, neutron flux and neutron source and power distribution.
The reference solution could possibly be obtained for meshing finer than that adopted for transient calculation, in which case the corrections will also account for meshing effects. Effects of fuel assembly mixing grids are also introduced in this manner.
Axial reflector characteristics are entered via log derivative conditions j/φ equally extracted from the reference input data.
Note that, as the correction apply to all the components of the effective 1-group model, they can be expected to remain invariant during the transient, as long as the 1-group model remains valid.
Extending the neutronic model to the same 2-group structure as the reference core is feasible and would improve the invariance of initial corrections.

Flow sheet of (do, core) successive calculations

The following simplified description should help the user better understanding the functions of the various operations and iterations involved in processing do, core.

A Flow entering c
(1): xwac= fraction of flow bypassed from a to c and escaping mixing in b.
w{,h,s}ec: floW of {mass, entHalpy, Boron} at inlEt of Core
(2): if osplit=t, same flows in each core sector inlet

B CORE CONFIGURATION
(3) position of successive configurations as interpolated in the interpolator zgri(sec) until a possible emergency rods drop occurring at time sec_drop with a drop displacement interpolated from dgri(sec).


C CORE PARTITION
(4): core confiGuRation Index igr attached to the successive core configuration level Numbers ncf
(5): core ConFiguration Number attached to the successive elements of the kn partition

D DSEC DEPENDENT DELAYED PRECURSORS VARIABLES
(6): set of variables for delayed precursors cj;
amcj is the decay constant (lAMda) for family j;
epsatc is the criterion for calculating from series development.
(7): idem for residual power (CY) decayed fission products precursors Concentration of family J
E RESIDUAL POWER
(9): decay FP source of family (J) at bos (1) in Interval ci.

F Sxkn and derivatives

In order to avoid repeating neutronic tables look-up at each feedback iteration, the neutronic model is linearized about the bos physical state and for the bos configuration.
If there is a dsec adjustment in the course of the step, F is then updated, as well as all the program segments depending from dsec; this allows, amongst other, to correct for displacement of moving rods.

(10): confIGuRation index assigned to each configuration level ncf; the group interpolators are looked up with the osec time sec=osec + dsec where osec [Old] it the time at bos
(11): the BEta's are only confiGuRation-dependent.
(12:16): bos (1) amplitude of the various the various feedback effects
(17,19): bos ("1") neutronic values and their derivatives assigned to the successive kn partitions (but for eos configuration)
(20): radial sector form factors
(21): bos reactivity calculated from bos (1) nodal (ck) values of volumic mass vm, fuel effective over temperature (u), boron (b), pressure (p), and enthalpy distribution parameter xe at core inlet.
(22): v-, u- and -b derivatives to be used for feedback iterations
(23): total ci-Fission Source by adding tabulated fission cross-section and the correction sn0ci calculated in ini, core.
(24): ci-κ/ν from tabulated value + correction calculated by ini,core to force the power distribution f2ci to the input values.
(25): first guess of eos ("2") total, delayed and actual power distribution.
It is estimated from bos flux phci (or at eos of previous step), but already accounting for the possible changes in core configuration and in physical conditions (boron, inlet flows, pressure). Their values may thus differ from the eos (converged) values

(26): total values for the core
F Sxkn and derivatives
(27): U2rck is the fuel over temp (over t1ck) radial distribution vector.
u2ck is the (scalar) effective value;
f2ck is the driving heat source coming from F.
ufck, qfck are the f2ck derivatives of u2ck and
q2ck, which use will be made for searching converged eos solution

G Core {m,h,b} balances
(28): standard eulerian balances on successive ck;
provide eos {m,h,b}, vm derivatives also utilized in J for stabilizing eos conditions;
vpck is the isentropic pressure derivative utilized in (do, pressu);
vmfck and bfck are local derivatives of vmck and bck in relation to total power f2c at constant power profile, also used in J
(29): core outlet flows, isentropic pressure derivative, and core outlet and average water temperatures.
(30): values needed for flow kinetics balances

H: invariable source terms in neutron kinetic equation;
(33):the beta's are not feedback dependent

I: Prompt jump neutronic kn properties at eos.

At the first execution the neutronic coefficients are calculated under prompt-jump approximation which is acceptable over most of the transient.
If, after later monitoring the approximation cannot be accepted, the set of operations is then iterated

(34): current step changes (from 1 to 2) of nodal physical properties from bos to eos
(35): updating of eos reactivity rkn by means of reactivity derivatives borrowed from F.
(36); idem for macro sections
(37): coefficient of ci-flux equation. Note the correction r0ci calculated in ini, core, and the reactivity - rau superimposed to all the meshes and representing a forced emergency (au) reactivity data to simulate point kinetics. Likewise, sn0ci is the correction to tabulated νΣf
(38): and their derivatives vs u, vm and b

J: Reduction to point kinetics
(39): flux factor for feed-back converged eos (level 3) solution with unaltered flux profile (phci3/phci2 = xph23).
It is calculated as the positive solution from second degree algebraic equation; it implies a quadratic evolution over the time step.
(40): updates the potential power ft
(41): and actual power if residual decay power model is active;
(42): otherwise, the individual contributions of prompt and delayed power are summed.
(43): corrections performed at a later iteration of J if the PJA cannot be retained.

K Adjusting dsec
(44): dsec must be altered in case of excessive value of power ratio f3c/f1c (with crit. epsfc) or excessive change of power derivative log (under crit epsom). IF change is too large/small, dsec is halved/doubled. However dsec is forced to remain in (dsecmin, dsecmax) interval. It is also halved if second degree equation has imaginary solution, which occasionally occurs when the flux peak is too violent.
(45): in case of dsec change, all the subprograms sensitive to dsec must be iterated. Return of B accounts also for change of expected eos rods positions.

L: Updating the physical and neutronic properties to the level 3 conditions by making use of derivatives.
(46): the potential total power ft is just scaled with xph23, but
(47): the prompt part only is scaled
(48): updating of physical nodal properties by means of the f-derivatives.
(49,50): updating of kn-neutronic properties
(51): with core at level 3 condition, the change of flux profile is obtained by solving the flux equation with Doppler feed-back only.
(52 to 56): a new estimation of level 2 physical and neutronic conditions are next obtained from the solution phci.

M: the nodal {m,b,b} are updated in turn.

N: Controlling power convergence
(57): Solution convergence at eos is tested by monitoring the level 2 to 3 power change with crit. epfc. If convergence is not satisfied, the execution returns to I. The operations upstream of I are only dsec-sensitive and must not be repeated.
Note to terminate that the feedback iteration scheme accommodates with non-negligible power profile deformation in the course of the step. The only simplification is the linearization of the physical and neutronic models at the bos.

end_step, core

At eos, the final neutronic condition must be updated from the lastly converged physical condition and a set of summary values are calculated for editing:
omfc= (2 /dsec) (f2c-f1c)/(f1c+f2c), the core log derivative
u2ack: average nodal fuel temperature
hvu2ck: volumic fuel rod enthalpy (J/m3) calculated from 0 K.
r2kn: kn-reactivity
fci: normalized power profile.
rci: ci-reactivity.
ph2c: average core flux.
omph2c: core flux log derivative.
fzic: core axial form factor.
ifzc: mesh of peak position.
aoc: core power axial offset
rc: core reactivity.
u2k9c = MAX(u2ck).
u2ac: core average fuel temperature.
hvu2c: core average volumic fuel enthalpy.
u2c: core average effective temperature.