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Nuclear Engineering and Design 235 (2005) 12671282

Influence of subcooled boiling on out-of-phase oscillationsin boiling water reactors

J.L. Munoz-Cobo a,, S. Chiva b, A. Escriva a

a Department of Chemical and Nuclear Engineering, Polytechnic University of Valencia, P.O. Box 22 012, 46071 Valencia, Spainb Department of Technology, Fluid Mechanics Area, Jaume I University, Campus del Riu Sec, 12080 Castellon, Spain

Received 13 January 2005; received in revised form 14 January 2005; accepted 31 January 2005

Abstract

In this paper, we develop a reduced order model with modal kinetics for the study of the dynamic behavior of boiling water

reactors. This model includes the subcooled boiling in the lower part of the reactor channels. New additional equations have been

obtained for the following dynamics magnitudes: the effective inception length for subcooled boiling, the average void fraction

in the subcooled boiling region, the average void fraction in the bulk-boiling region, the mass fluxes at the boiling boundary

and the channel exit, respectively, and so on. Each channel has three nodes, one of liquid, one with subcooled boiling, and one

with bulk boiling. The reduced order model includes also a modal kinetics with the fundamental mode and the first subcritical

one, and two channels representing both halves of the reactor core. Also, in this paper, we perform a detailed study of the wayto calculate the feedback reactivity parameters. The model displays out-of-phase oscillations when enough feedback gain is

provided. The feedback gain that is necessary to self-sustain these oscillations is approximately one-half the gain that is needed

when the subcooled boiling node is not included.

2005 Elsevier B.V. All rights reserved.

1. Introduction

The most advanced thermalhydraulic codes for

boiling water reactors, like RAMONA, TRAC-B, and

TRAC-M, use a two fluid model, and solve the con-servation equations of mass, energy, and momentum

for each phase of the two-phase mixture (Wulff et al.,

1984). These equations must be solved at each channel

Corresponding author. Tel.: +34 963 877 631;

fax: +34 963 877 639.

E-mail address: [email protected] (J.L. Munoz-Cobo).

of the reactor, and these reactor channels are coupled to

the reactor vessel by proper boundary conditions. The

channel thermalhydraulics is modelled in 1D and ne-

glecting transversal effects, however, the vessel is usu-

ally modelled in 3D. As a consequence, these codesare extremely time consuming, even with a small num-

ber of thermalhydraulic nodes, and the application of

these tools is very limited.

A complementary effort has focused on the devel-

opment of reduced order models, often consisting of a

limited number of ordinary differential equations that

represent the most important dynamical processes of a

0029-5493/$ see front matter 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.nucengdes.2005.01.018

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1268 J.L. Munoz-Cobo et al. / Nuclear Engineering and Design 235 (2005) 12671282

Nomenclature

A channel areacpl specific heat of the liquidcpf specific heat of the liquid at saturation

Dh hydraulic diameter

f(r)Pm power distribution factor for the mth

mode and the rth core regionFs fraction of the energy transferred to the

channel that is invested in steam produc-

tion

Gin,r(t) mass flux at the inlet of channel r

Gbb,r(t) mass flux at the bulk-boiling boundary

of channel r

Gex,r(t) mass flux at the exit of channel rG2 average mass flux in the two-phase re-

gionhin specific enthalpy of the liquid at the en-

trance of the channelhl,z1 specific enthalpy of the liquid at the boil-

ing inception point

hl,sb average specific enthalpy in the sub-

cooled boiling region

hf specific enthalpy of the liquid at satura-

tion conditions

(hA)ch,r fuel to coolant heat transfer coefficienttimes the heat transfer area at channel r

H1 single-phase heat transfer coefficientHc length of the channels in the reactor core

kl heat conductivity of the liquid

K1,r form loss coefficient at the channel inlet

plus form loss coefficients of the fuel rod

spacers in the mono-phasic regionK2,r form loss coefficient of the fuel rod spac-

ers in the bi-phasic regionKex,r form loss coefficients at the channel exit

nm(t) oscillating normalized components ofthe neutron flux expansion in lambda

modes

p pressure

P power

P0 steady-state powerqf,r heat generation rate fluctuation at core

region r

Q heat flux

Qf,r heat generation rate at fuel core region r

Qch,r heat transfer rate to the fluid of channel r

Tlinc liquid temperature at the inception point

for boilingTwinc wall temperature at the inception point

for boilingV volume of the core

V(r) volume of the region rof the reactor core

W(r)nm reactivity weighting distribution factors

WPK square power weighting factorsx dynamic quality of the two-phase flow

mixturexbb(t) dynamic quality at the boiling boundary

xex,r(t) dynamic quality at the exit of channel r

Z1,r(t) effective boiling inception length at

channel rZbb,r(t) bulk-boiling boundary length at

channel r

Greek letters

bb(t) void fraction at the boiling boundarybbR average void fraction at the bulk-boiling

region

sb average void fraction in the subcooledboiling region

fraction of delayed neutron precursors

p pressure drop2 two-phase pressure drop multipliers bubble decay constant due to bubbles

collapsing in the subcooled core of the

channel

disintegration constant of delayed neu-

tron precursors

neutron generation time

f,r lumped temperature fluctuations in the

fuel of channel r

g steam density

fg

difference between gas and liquid den-

sity at saturation conditions (g f)lg difference between gas and liquid den-

sity at subcooled conditions (g l)l liquid density

f liquid density at saturation conditions

Fnm feedback reactivity for the nth and mth

modesl,sb average liquid density in the subcooled

boiling region

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J.L. Munoz-Cobo et al. / Nuclear Engineering and Design 235 (2005) 12671282 1269

BWR. These fast running codes give, in general, worse

predictions than the 3D codes, but allow performing

sensitivity analysis and give qualitatively correct pre-

dictions for a wide range of operating and design pa-rameters. These reduced order models use generally

(Van Bragt and Van der Hagen, 1998; Van Braght, 1998;

Munoz-Cobo et al., 2000; Lee and Onyemaechi, 1989)

a two nodes model for each channel of the reactor core.

The first node is a single-phase node, where the liq-

uid is uniformly heated from the channel entrance up

to the boiling boundary where saturation is reached.

The following node is a two-phase node, where the

two phases are in thermodynamic equilibrium and all

the heat transferred to the channel is invested in steam

production.

However, in real cases there is a region called the

subcooled boiling region, where there is not thermody-

namic equilibrium between the phases, and part of the

power transferred to the channel is invested to heat the

liquid, while the rest is invested in steam production. In

this subcooled boiling region, the bubbles are formed

at the walls (clad surface), and detach from the walls

to travel to the subcooled region of the channel, where

they collapse. Because reactivity feedback is very sen-

sitive to void variations, and is axially weighted by a

distribution factor that depends on the square of the

power distribution, then, the void fraction variations inthe subcooled boiling region will have a big contribu-

tion to the void feedback reactivity.

In this paper, we have developed a reduced or-

der model where each channel is represented by three

nodes: one subcooled liquid node from the channel en-

trance to the boiling inception point, where subcooled

boiling starts; the second node starts at the inception

of subcooled boiling and ends at the boiling bound-

ary, where the average enthalpy of the two-phase mix-

ture attains saturation conditions; finally, the last node,

or bulk-boiling region, extends from the bulk-boilingboundary to the channel exit. The goal of this paper is

to study the influence of the subcooled boiling on the

feedback mechanisms that lead to the development of

out-of-phase instabilities in boiling water reactors, and

to make a consistent lumped model that includes the

subcooled boiling in a realistic way.

The studies performed on the void feedback reactiv-

ity (Wulff et al., 1984; Munoz-Cobo et al., 1994) have

concluded that the typical functionalization of the void

reactivity as a second-degree polynomial in the void

fraction, must be multiplied by a weighting factor that

depends on the square of the power distribution. There-

fore, the void feedback reactivity is enhanced in the

reactor regions where the power is higher.The new fuel designs tend to make the power dis-

tribution more peaked at the reactor bottom, or lower

part of thecore.Therefore,the reactivity feedbackat the

subcooled boiling region will become more important

with these new designs because is enhanced by the bot-

tom peaked axial power distribution. This is the main

reason to add a subcooling boiling node to the classical

reduced order models with only two nodes, and where

the reactivity feedback in the subcooled boiling region

is not considered.

As it is well known, theout-of-phase instabilities ap-

pear because one or two subcritical modes are excited.

The mechanism to excite these modes is to provide

enough reactivity feedback to overcome the eigenvalue

separation between the fundamental mode and the

subcritical one (Munoz-Cobo et al., 2000). For out-of-

phase oscillations, the inlet mass flow rate to the reactor

core remains constant, and the two oscillating core re-

gions adjust their flows to maintain approximately con-

stant the pressure drop across the core (March-Leuba

and Rey, 1993).

The paper has been organized as follows: Section 2

is devoted to the development of the model equations.Section 3 is devoted to the results of the model and the

discussion about the influence of subcooling boiling

on the onset of the instabilities. Finally, Section 4 is

devoted to analyze the main conclusions of this paper.

2. The reduced order model of a BWR with

subcooled boiling

In this section, we explain the main characteristics

of the DWOS M SU (density wave oscillation withmodal kinetics and subcooled boiling) lumped param-

eter model.

One of the defects of the previous lumped models

is that they do not include the subcooling void frac-

tion. Therefore, the void fraction along the channel is

smaller in previous lumped models than the real one,

mainly at the beginning of the channel. This effect has a

strong influence on the void reactivity feedback, which

depends on the void fraction and the power distribution

through the square power distribution factor. Now, in

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Fig. 1. DWOS lumped parameter model. This model is a two channel model, with three regions at each channel, the liquid region from Z= 0 to

the boiling inception height Z1 the subcooled boiling region from Z1, to the boiling boundary length Zbb, and the bulk-boiling region from Zbbto the channel exit.

the lower part of the channel, the power is higher due to

the fact that the density of thermal neutrons is higher.

This is a direct consequence of the bigger moderation

of fast neutrons produced by the higher density of the

water (less voids) in this region. Therefore, the square

power distribution factor is higher in the lower part ofthe channel and, therefore, the consideration of the sub-

cooled boiling will have a strong influence on the void

reactivity feedback.

The core fuel assemblies are grouped into two core

regions that are represented by two averaged channels.

The DWOS lumped parameter model of each channel

has three regions or nodes, as it is displayed in Fig. 1.

2.1. Calculation of the effective inception

temperature Tlinc, and the dynamic equation of the

inception length Z1 for subcooled boiling

The most important thing, in any effective subcool-

ing boiling model, is the ability of the model to be

able to predict accurately where significant void frac-

tion appears, this location of the void departure will be

denoted by Z1. Several criteria can be used to predict

the inception point (Lahey and Moody, 1993). How-

ever, we have checked several criteria for the effective

inception point and the differences were very small, so

we have chosen a criterion that is obtained equating the

single-phase forced convection heat flux, at Z=Z1, to a

JensLottes type heat flux. This criterion has been suc-

cessfully used by the LAPUR code (Otaduy, 1979), for

many years. This criterion lead to the following expres-

sion for the liquid temperature at the inception point,

Tlinc, in terms of the heat flux, Q

, at the fuel walls:

Tlinc = C0.252 Q

0.25 + Tsat Q

H1, (1)

where H1 is the single-phase forced convection heat

transfer coefficient, given by the DittusBoelter for-

mula. Tsat is the liquid saturation temperature, and the

constant C2 is given by:

C2 = 2.5454 exp

4p

6, 207, 385

. (2)

Therefore, the liquid enthalpy at Z1 is given by:

hlz1 = hf cpl(Tsat Tlinc). (3)

Therefore, at steady-state conditions, the effective in-

ception boundary length can be computed by means of

the following obvious formula, deduced assuming the

uniform heating of the channel:

Z1,0 =(hlz1 hin)Gin,0A

Qch,0/Hc, (4)

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where hin is the enthalpy of the subcooled liquid at

the entrance of the channel, A the channel flow area,

Gin,0 the mass flux at steady-state conditions, Qch,0 the

power transferred to the channel fluid at steady-stateconditions, and finally, Hc is the total channel length.

To get the dynamical equation governing the evolu-

tion ofZ1,r, with time at the rth channel, we integrate

the energy conservation equation in the single-phase

region. This calculation yields:

dZ1,r

dt=

2Qch,r

AHc

Z1,r

l,inhin l,z1hl,z1

+ 2Gin,rhin hl,z1

l,inhin l,z1hl,z1(5)

where Qch,r is the heat transferred per unit time to thechannel fluid, l,z1 the liquid density at the inception

point Z1, Gin,r the mass flux at the entrance of the rth

channel, and finally, hl,z1 is the liquid enthalpy at the

boundary of the effective inception point for subcooled

boiling.

2.2. Dynamic equation governing the average

void fraction in the subcooled boiling region

To obtain the dynamic equation for the average void

fraction in the subcooled boiling region, we start fromthe mass conservation equation of the steam in this

region, given by:

t(gr) = sgr

z(xrGr) +

FsQch,r

Ahfg(6)

where r is the void fraction in the rth channel, Grthe mass flux in the rth channel, s the bubble decay

constant due to the bubble collapse in the subcooled

region of the channel, and finally, Fs is the fraction of

the energy transferred to the channel that is invested in

steam formation.Integration of Eq. (6), between the subcooled boil-

ing inception length Z1 and the bulk-boiling boundary

length Zbb of the rth channel, yields:Zbb,rZ1,r

dz

t(g)

= sg(Zbb,r Z1,r)sb,r xbb,rGbb,r(t)

+FsQch,r

AhfgHc(Zbb,r Z1,r), (7)

where sb,r is the average void fraction in the sub-cooled boiling region of the rth channel, xbb,r the

dynamic quality at the boiling boundary of the rth

channel, and finally, Fs is the average fraction of theenergy transferred to the channel that is invested insteam formation in the subcooled boiling region.

Now, we apply the Leibnitz rule to the left-hand

side of Eq. (7), and on account of the definition of the

average void fraction in the subcooled-boiling region,

given by:

sb,r =1

Zbb,r Z1,r

Zbb,rZ1,r

dz . (8)

It is obtained, from Eq. (7), the following equation

for the evolution of the average void fraction, sb,r,in the subcooled boiling region of the rth channel:

d

dtsb,r =

sb,r

Zbb,r Z1,r

d

dtZ1,r

+bb,r sb,r

Zbb,r Z1,r

d

dtZbb,r ssb,r

xbb,rGbb,r

(Zbb,r Z1,r)g+

FsQch,r

AhfgHcg. (9)

We observe that the evolution of the average void

fraction in the subcooled region depends positively onthe average of the transferred energy invested in steam

formation, and negatively of the collapsing of bubbles

term that is proportional to the average void fraction in

this region.

2.3. Bulk-boiling boundary dynamics

The equation for the dynamic behavior of the boil-

ing boundary length, Zbb,r(t), at the rth representative

channel is given by (Van Bragt and Van der Hagen,

1998; Lee and Onyemaechi, 1989):

dZbb,r(t)

dt=

2Gin,r(t)

f

2Qch,rZbb,r(t)

(hf hin)AHcf(10)

where Qch,r is the fuel to coolant heat transfer rate per

fuel assembly of the rth core region, which can be writ-

ten as:

Qch,r = P0,chr(t) + (hA)ch,rf,r, (11)

where P0,chris the steady-state power perfuel assembly

in the rth channel, (hA)ch,rthe fuel to coolant heat trans-

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fer coefficient times the heat transfer area in a typical

or average channel of the rth region, and f,r denotes

the lumped temperature fluctuation in the fuel of the

rth channel.

2.4. The equation for the mass flux at the boiling

boundary

The continuity equation in the subcooled boiling

region is given by:

t(rg + (1 r)l) =

zGr. (12)

The integration of Eq. (12), between the subcooled

boiling inception boundary length,Z1,r, and the boiling

boundary length, Zbb,r, yields:

Zbb,r

Z1,r

dz

t=

Gin,r Gbb,r

g l,sb, (13)

where l,sb is the average density of the liquid in the

subcooling boiling region, and we have made the app-

roximation Gz1,r= Gin,r. Finally, Gbb,r is the total massflux at the boiling boundary length of the rth channel.

The next step is to integrate the energy equation

from Z1,r to Zbb,r. The two-phase energy equation is

given by:

t((1 )lhl + ghg)

=

z ((1 x)Ghl + xGhg) +

Qch

AHc (14)

Assuming that that the channel is uniformly heated

and integrating Eq. (14) fromZ1,rtoZbb,rit is obtained:

(ghg l,sbhl,sb)

Zbb,rZ1,r

dz

t

= (Gz1hl,z1)r {(1 xbb)hl,bb + xbbhg}rGbb,r(t)

+Qch,r

ArHc(Zbb,r Z1,r) (15)

where Gz1,r is the liquid mass flow rate at the inception

subcooling boiling boundary, hl,z1 the liquid enthalpy

at the inception point for subcooled boiling, hl,bb the

enthalpy of the liquid at the boiling boundary. We haveassumed that l,sbhl,sb is approximately time indepen-

dent.

Next, we eliminate the integral in Eq. (15), with

the help of Eq. (13), and we make the approximation

Gin,r= Gz1,r. From the resulting equation, and aftersome algebra, it is obtained the following expression

for the mass flux at the boiling boundary:

Gbb,r(t) =[(ghg l,sbhl,sb) (lg)sbhl,z1]r

[ghg l,sbhl,sb lg,sb(hl,bb + xbbhlg,bb)]rGin,r(t)

[lg,sbQch(Zbb Z1)]r

AHc[ghg l,sbhl,sb lg,sb(hl,bb + xbbhlg,bb)]r(16)

where we have introduced the following definitions:

lg,sb = g l,sb, (17)

lg,bb = g l,bb, (18)

hlg,bb = hg hl,bb. (19)

Eq. (16) relates the mass fluxat the boiling boundary

with the mass flux at the channel inlet and the rate of

heat transfer to the subcooled boiling region.

2.5. The equation of the mass flux at the channel

exit

First, we integrate the mass conservation Eq. (12)

from the boiling boundary length Zbb,r to the total

height Hc of the channel, this calculation yields:HcZbb,r

dz

t=

Gbb,r Gex,r

g f. (20)

Then, we integrate the energy conservation equationbetween the same limits; this calculation yields:

(ghg fhf)

Zbb,rZ1,r

dz

t

= {(1 xbb)hl,bb + xbbhg}rGbb,r(t)

{(1 x)exhf + xexhg}rGex,r

+Qch,r

ArHc(Hc Zbb,r). (21)

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Now, we eliminate the integral in Eq. (21), with the

help of Eq. (20). Then, after some algebra, it was ob-

tained the following expression for the mass flux at the

channel exit:

Gex,r(t) =[((1 xbb)hl,bb + xbbhg)fg (ghg fhf)]r

{[(1 x)exhf + xexhg]fg (ghg fhf)}rGbb,r(t)

+[fgQch(Hc Zbb)]r

AHc{[(1 x)exhf + xexhg]fg (ghg fhf)}r(22)

Eq. (22) gives the mass flux at the exit of the channel

in terms of the mass flux at the boiling boundary and

the heat transfer rate to the bulk-boiling region.

2.6. Equation for the evolution of the average void

fraction in the bulk-boiling region (bbR)

The equation for the evolution of the average void

fraction, in the bulk-boiling region, is easily obtained

by integrating the mass conservation equation betweenZbb,r and Hc. This calculation yields after some ar-

rangements:

d

dtbbR,r =

Gbb,r(t) Gex,r(t)

fg(Hc Zbb,r)

(bb bbR)rHc Zbb,r

dZbb,r

dt, (23)

where the mass fluxes Gbb,r and Gex,r, at the region

boundaries, are easily calculated by means of Eqs. (16)

and (22), respectively.

2.7. Momentum conservation equation

A key assumption in many thermal-hydraulic mod-

els of out-of-phase oscillations is that the pressure drop

across the channels remains approximately constantduring the out-of-phase oscillation transient (Van Bragt

and Van der Hagen, 1998; Van Braght, 1998; Munoz-

Cobo et al., 2000; Lee and Onyemaechi, 1989; March-

Leuba and Rey, 1993). Assuming that the channel area

is constant, we get, integrating the momentum equation

along the channel, the following result:H0

Gr(z, t)

tdz = p pacc,r pg,r pf,r,

(24)

wherep =pin pex is the difference between the inletand outlet channel pressures, that is channel indepen-

dent, because all the channels have commons lower and

upper plena.

pacc,r represents the pressure drop due to the fluid

acceleration in the channel:

pacc,r

= x2exG2exexg

+(1 xex)

2G2ex

(1 ex)fr

Gin,r

l,in

(25)

pg,r represents the gravitational pressure drop,

which is given by the following expression:

pg,r = glZ1,r(t) + [g(1 sb,r)l,sb

+ gsb,rg](Zbb,r(t) Z1(t))

+ [g(1 bbR,r)f + gbbR,rg]

(Hc ZbbR,r(t)), (26)

where sb,r and bbR,r are the average void frac-tions, in the subcooled boiling region and in the bulk-

boiling region of the rth channel, respectively. Finally,pf,r gives the friction pressure drop, which is calcu-

lated by means of the following expression:

pf,r =

K1,r + fr

Z1,r(t)

DH

G2in,r(t)

2l

+ K2,r2r Jr

G22r

2f

+ frHc Z1,r(t)

DH2r Jr

G22r2f

+ Kex,r2ex,rJr

G2ex,r(t)

2f, (27)

where the first term of expression (27) gives the pres-

sure drop in the single-phase region of the rth channel,

with K1,r being the form loss coefficient due to the

losses at the channel inlet and the fuel rod spacers, lo-

cated in the single-phase region. The second term gives

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the channel pressure drop due to the rod spacers located

in the biphasic region. The third term gives the pres-

sure drop due to friction with the walls. Finally, the

last term gives the pressure drop at the channel exit. Inthese terms, 2 is the two-phase pressure drop multi-

plier, and r is the Jones and Dight (1963) correction

factor.

To compute the pressure drop in the channels, we

neglect the time derivative term (Lee and Onyemaechi,

1989), and we assumethatthe average flux in the bipha-

sic region at a given time can be approximated by:

Gr = 0.5(Gin,r(t) + Gex,r(t)). (28)

Therefore, we write

p0 = pacc,r + pg,r + pf,r. (29)

2.8. Constitutive equations and calculation

procedures

2.8.1. Fraction of power invested in steam

production in the subcooled boiling region

According to Lahey and Moody (1993) and Otaduy

(1979), the energy transferred from the fuel to the sub-

cooled boiling fluid can be split into three components:

(i) formation of steam bubbles near the heating sur-face which may detach into the main flow stream, (ii)

pumping of the liquid mass out of the control volume

by the expanding action of the steam bubble formation,

(iii) single-phase convective heating through the parts

of the heating surface not generating bubbles. The in-

vestigations performed have concluded that the steam

formation and the pumping process are predominant

over the normal convective process. Therefore, we can

write:

Fs

=qevap

qevap + qpump=

1

1 + qpumpqevap

, (30)

where qevap is the part of the energy flux at the sur-

face associated to the steam formation, while qpump is

the part of the energy flux associated to the pumping

process.

According to Rouhani and Axelsson (1970), the ra-

tio of the heat fluxes, due to pumping and to steam for-

mation, can be calculated with the approximation that

the liquid that leaves the control volume is at saturation

and therefore we have:

=qpump

qevap=

l(hf hl)

ghfg. (31)

On account of Eqs. (30) and (31), and the LAPUR

code (Otaduy, 1979), we use for Fs the following ex-

pression:

Fs =1

1 + fpHl

1

, (32)

where Hl =hfhlhfg

gives the degree of subcooling of

the liquid, and = fsf

. Finally, fp is an adjustable

factor, equal to 1.3, that will allow for better to correlate

predictions with measurements.

2.8.2. Steam decay constant in the subcooled

boiling region

The model used is based in the model of Jones and

Dight (1963) and LAPUR (Otaduy, 1979), the results

of this model have been compared with the expression

used by the RELAP5 code that gives similar results.

The Jones model uses the following expression for the

bubble decay ratio:

s = c0H2l , (33)

where Hl is the degree of subcooling of the liquid,

and 0 are given by the following expressions:

=

hfg

cpf(Tc0 Tsat)

2and 0 =

H2w

kflcpf, (34)

where Tc0 is the clad temperature at the inception of

subcooled boiling, Hw the single-phase heat transfer

coefficient, and cpf is the specific heat of water at sat-

uration.

Finally, we have checked that assuming the parame-

ter c equal to 0.005 gives values of the subcooled boil-ingvoid fraction that areclose to theexperimental ones,

when only one subcooled node is used. The problem is

that when we have many nodes in the subcooled boil-

ing region, for instance, 50, then in the three or four

nodes near the inception point, s is very big and then

becomes much more smaller in the rest of nodes. We

have checked that when using c equal to 0.005, we get

a value for the average subcooling void fraction that is

very close to the average value obtained using 30 nodes

in this region.

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2.8.3. Relation between the quality and the void

fraction

To obtain the value of the quality at the boiling

boundary,xbb(t), and at the channel exit,xex(t), we startfrom the formula that relates the void fraction with the

quality (Thom, 1964):

=x

1 + ( 1)x(35)

where in the HEM model (Todreas and Kazimi, 1990)

= vg/vl.The average void fraction in the subcooled boiling

region of the rth channel can be obtained as follows:

(t)sb,r =1

xbb,r(t)

xbb,r(t)

0

dx. (36)

Direct substitution of Eq. (35) in (36) yields:

(t)sb,r =

1

1

1

( 1)xbb,r(t)

ln(1 + (1)xbb,r(t))

. (37)

The average void fraction in the bulk-boiling region

can be easily obtained as follows:

(t)bbR,r = 1xex,r(t) xbb,r(t)

xex,r

(t)

xbb,r(t) dx. (38)

Direct substitution of the expression (35), which re-

lates the void fraction with the quality, followed by

integration yields:

(t)bbR,r =

1

1

1

( 1)(xexit,r(t) xbb,r(t))

ln1 + ( 1)xex,r

1 + ( 1)xbb,r

. (39)

Eq. (39) gives the average void fraction in the bulk-

boiling region, in terms of the quality at the channel

exit and the quality at the boiling boundary.

When solving the set of dynamics equations, we get

at each time step the average void fractions in the sub-

cooled boiling region, and in the bulk-boiling region.

To get the mass fluxes Gbb,r(t) and Gexit,r(t), we need to

know the qualities at the boiling boundary, xbb(t), and

at the channel exit, xexit(t). We get these qualities by

iteration in Eqs. (37) and (39).

2.9. Neutronic model and feedback reactivity

The normalized components nm(t) of the neutron

flux expansion inmodes obey, for the case ofonlytwomodes, the following set of coupled differential equa-

tions (Munoz-Cobo et al., 2000; Hashimoto, 1993):

dn0

dt=

F00(t)

0n0(t) +

F00(t)

0

+F01(t)

0n1(t) + c0(t)

dn1

dt=

s1 + F11(t)

1n1(t) +

F10(t)

1

+F

10

(t)

1n0(t) + c1(t)

dc0

dt=

0n0(t) c0(t)

dc1

dt=

1n1(t) c1(t)

, (40)

where cm(t) are the oscillating normalized components

of the expansion of the delayed neutron precursor con-

centrations in terms of harmonic -modes, i the neu-

tron generation time for the ith mode,s1 the subcritical

reactivity of the first subcritical mode. Finally,Fmn are

the feedback reactivities fora given configurationof thereactor core, and a given position of the reactor control

rods.

The main feedback reactivity contributions are the

void and Doppler feedback reactivities, so we write:

Fmn(t) = Vmn(t) +

Dmn(t). (41)

The modal void feedback reactivities Vmn(t) are

computed adding up the void reactivity contributions

of the various reactor core regions:

Vmn(t) =r

V,rmn (t), (42)

where the void reactivity contribution V,rmn (t) from the

rth core region can be obtained from the reactivity

weighting distribution factors W(r)m,n (Munoz-Cobo et

al., 2000) as follows:

V,rmn (t) = RV00(

r)W(r)m,n, (43)

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where RV00(r)is the scaled void reactivity for the rth

subcore region defined by:

RV00(r) =

0, M

L0

(r)0,M00

(r)=K

(C1 + C2rk + C3(

rk)

2)WPKrK, (44)

where Mand L are the production and removal opera-

tors, and 0 and 0 are the direct and adjoint fluxes,

means integration over a given region. We have as-sumed that the ratio of Eq. (44) for the rth core region

scales with the void fraction rk as in the full core, i.e.

with the same polynomial fit constants C1, C2, and C3,

the index k denotes the axial nodes in the fitting. WPKare the typical square power weighting factors:

WPK =P2KKP

2K

(45)

The modal reactivity weighting factors can be approx-

imated by (Munoz-Cobo et al., 2000):

W(r)mn =

m, n

(r)m,m

(46)

The Doppler reactivity contribution of the rth core

region can be obtained by a similar method (Munoz-

Cobo et al., 2000). The reactivity coefficients C1,C2, C3, displayed in Table 1, have been obtained by

means of a consistent multidimensional methodology

(Munoz-Cobo et al., 1994).

The modal reactivity weighting factors, defined

at expression (46) were computed with the code

LAMBDA (Miro, 2002), and the results are displayed

in Table 2.

The LAMBDA code permits us to solve the 3D-

eigenvalue equation:

Ln = nMn, (47)

Table 1

Void feedback reactivity coefficients for Cofrentes event January

1991

C1 0.23256

C2 0.64433

C3 0.92516

Table 2

Reactivity weightingfactors for a tworegions model with twomodes

the fundamental one and the first harmonic

Cofrentes case 1 instability event 1991Ringhals test

W100 = 0.4995 W200 = 0.5005 W

100 = 0.4985 W

200 = 0.5014

W101 = 0.4529W201 = 0.4529 W

101 = 0.4299W

201 = 0.4312

W110 = 0.4541W210 = 0.4541 W

110 = 0.4321W

210 = 0.4298

W111 = 0.4918 W211 = 0.4901 W

111 = 0.4748 W

211 = 0.4783

where L is the differential operator for the absorption

and leakage of neutrons given by:

L = (D1 ) + a1 + 12 0

12 (D1

) +a2

;(48)

D1 and D2 are the diffusion coefficients for the fast

and thermal groups, respectively; a1 and a2 are

the absorption cross-sections of the fast and thermal

group, respectively; 12 =r is the removal or down-

scattering cross-section from the fast to thermal group;Mis the two-group production operator

M=

f1 f2

0 0

; (49)

= column[1, 2] and n = 1/kn are the nth eigen-vector and the nth Lambda eigenvalue, respectively.

The method used by the LAMBDA code is to trans-

form the eigenvalue problem, associated to the differ-

ential operatorL, into an algebraic eigenvalue problem.

This step is performed by discretizing the space in par-

allelepiped cells, followed by a Legendre expansion of

the fluxes in the cells. This method was originally de-

veloped by Herbert (1987), and applied successfully by

Ginestar et al. (2002), and belongs to the class of nodal

collocation methods.The previous method allows us to compute the

eigenvalues and eigenvectors of the two group diffusion

equation in three dimensions. In the Case of Cofrentes

NPP, the application of this method for the determina-

tion of theeigenvalues of the out-of-phase modesyields

the results displayed at Table 3.

Finally, in Fig. 2, we display the first subcritical

harmonic mode of Cofrentes Reactor computed for the

conditions of the instability event of 1991, or case 1 of

the report D16f (Escriva et al., 2003).

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Table 3

Eigenvalues of the Cofrentes nuclear reactor and residual errors for

the case 1 of report D16f (Escriva et al., 2003), with 30 30 27

nodes and a Legendre polynomial expansion of second order

Eigenvalues Residual error

k0 0.99724 2.23 107

k1 0.99090 1.43 104

k2 0.98850

This result shows us that if the first harmonic mode

is excited one-half of the reactor core will increase its

power while the other half will decrease its power.

In Table 2, we display the reactivity weighting fac-

tors of Cofrentes NPP, when the reactor core is divided

in two regions and only the first harmonic mode is ex-cited. We display also in Table 2, the values of the re-

activity weighting factors computed for Ringhals test

(Munoz-Cobo et al., 2000). We can see that in both nu-

clear plants the reactivity weighting factors have simi-

lar values.

Letusanalyzefirstthephysicalmeaningofthisreac-

tivity weighting factors. The physical meaning of the

reactivity weighting factors is obvious from Table 2.

For case 1, the core is split in two regions, approxi-

mately of the same size. In region (1), the first har-

monic mode is negative, while in region (2), is positive.

Because both region have the same size and are prac-tically symmetrical, the contribution of each region to

the total reactivity of the fundamental mode is approx-

imately one-half. Therefore, the reactivity weighting

factors, for the fundamental mode should be approx-

imately equal to 0.5. This is the value obtained for

Cofrentes and Ringhals, using Eq. (46), and the funda-mental eigen-modes for both reactors computed with

the LAMBDA code.

For a cubic reactor core with two regions, an ele-

mental calculation shows that the reactivity weighting

factors W(1)01 and W

(2)01 are equal to 4/(3) = 0.424

and 4/(3) = 0.424. Obviously, the reactors are not cu-

bic but these values are very close to the values dis-

played in Table 2, with the true geometry and the true

eigen-modes.

Because for a given rth region the model only com-

putes the average void fraction in that particular region,

then we make the approximation:

rk(t) = k0 + r(t), (50)

i.e. we have assumed that the void fraction perturba-

tions are the same in all the axial nodes belonging to

the same region. k0 is the axial steady-state void frac-

tion distribution, computed with a 1D average channel

model for LAPURX (Otaduy, 1979). We have com-

puted this distribution with an average channel model

with 25 axial nodes.

Finally, the totalreactivitymn iswrittenintheform:

Fmn = Kgmnr

(RV00(r) + RD00(

r, Trf ))Wrmn, (51)

Fig. 2. First subcritical harmonic mode for Cofrentes NPP. Case 1 of the Nacusp Project (Escriva et al., 2003).

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where Kgmn is a bifurcation parameter to increase the

feedback reactivity of the system, and RD00(r, Trf )

is the scaled Doppler reactivity (Munoz-Cobo et al.,

2000).

2.10. Power generation and fuel dynamics

The power generated in a given region can be ob-

tained integrating the reactor heat generation rate per

unit volume over the volume of this region. This cal-

culation yields (Munoz-Cobo et al., 2000):

Qf,r(t) = P0f(r)P0 +

m=0

P0f(r)

Pmnm(t)

= Q(r)f,0(t) + qf,r(t), (52)

where P0 is the steady-state power, and f(r)Pm are the

power distribution factors for the mth mode, which are

defined as follows:

f(r)Pm =

V(r)(f1m1 + f2m2) dVV(f101 + f202) dV

, (53)

where V(r) is the volume of the rth region of the reac-

tor core and Vis the total volume of the core, f1 and

f2

are the fission cross-sections for the fast and ther-

mal group, respectively. m1 and m2 are the fast and

thermal components of the mth lambda eigenfunction.

These coefficients have been computed with the code

LAMDA (Miro, 2002), and the result of the calculation

is displayed in Table 4.

Dividing the power generated in this region by the

number of fuel assemblies of this region we get the

power generated per fuel channel assembly of the rth

core region, this magnitude is denoted by Qfch,r. Then,

the governing equation for the average fueltemperature

fluctuation f,r, in a fuel channel assembly of the rth

core region, is given by thefollowing equation (Munoz-

Table 4

Power distribution factors of Cofrentes NPP for case 1, when the

core is divided in two region and we consider a model with only two

modes

Region 1 Region 2

Fundamental mode f(1)P0 = 0.50042 f

(2)P0 = 0.4995

First harmonic mode f(1)P1 = 0.5039 f

(2)P1 = 0.4960

Cobo et al., 2000):

cf,rMf,rd

dtf,r(t) = qfch,r(t) (hA)ch,rf,r(t), (54)

where Mf,r is the fuel mass contained in a fuel channel

assembly of the rth region, qfch,r the fluctuation in the

heat generation rate in the fuel of one fuel channel as-

semblyof therth region, cf,rthe specific heat of thefuel,

and (hA)ch,r is the effective heat transfer coefficient

times the heat transfer area. The fluctuation qfch,r(t)

in the heat generation rate for one fuel channel assem-

bly can be obtained from expression (52), dividing this

expression by the number of fuel channel assemblies

Nch(r), of a given region r. This calculation yields:

qfch,r(t) =

m=0P0f(r)Pmnm(t)Nchan(r)

(55)

3. Results and discussion

3.1. Steady-state results of DWOS model for

Cofrentes NPP

The steady-state and the dynamic equations de-

scribed in this paper were implemented in a FORTRAN

code denoted: DWOS M SU. The input data for theCofrentes model were obtained from (Escriva et al.,

2003). In Table 5, we display the main parameter val-

ues of this model.

Then, we solve the steady-state equations with the

parameter values given in Table 5, the main results of

the DWOS steady-state calculations are displayed in

Table 6. Also, in this table, we display some results of

the SIMULATE code for the same case.

Table 5

Parameter values used for Cofrentes instability event

Parameter Value Parameter Value

p (Pa) 6.6478 106 Kex 0.9

hin (J/kg) 0.111 107 f 0.0266

Hc (m) 3.81 Dh 0.01306

A (m2) 0.009783 s1 0.006398

P0 (W) 1.121 109 (s) 3.591 104

Gin0 (kg/m2 s) 515.97 6.636 102

Nch 624 0.556 102

Tl,in 529 (s) 4.11

K1 57.43 cfMf 82270

K2 2.81

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Table 6

Steady-state effective inception length Z1, bulk-boiling boundary

length Zbb, average void fraction in the subcooled boiling region

sb , average void fraction in the bulk-boiling region bbR, aver-

age void fraction in the biphasic region 2, average void fractionin the channel c, quality at the exit of the channel, xex, computed

with DWOS, quality in the upper plenum computed with SIMU-

LATE, xUP(SIMULATE)

Z1,0 0.345

Zbb,0 1.479

sb,0 0.216

bbR,0 0.67

c 0.47

2 0.522

xex 0.169

xUP(SIMULATE) 0.146

We observe that the quality at the exit of the channel

computed with DWOS is 0.169. We have displayed the

quality at the upper plenum computed by the SIMU-

LATE code for this same case; however, this quality

is smaller due to the fact that, in the upper plenum,

the two-phase mixture coming from the fuel channel

assemblies mixes with the water coming from the by-

pass, this fact reduces the channel exit quality about a

10%, so the DWOS result and the SIMULATE result

agrees.

Then, in Table 7, we display the pressure drop inthe channel computed with DWOS and SIMULATE at

steady-state conditions. We observe that the pressure

drops computed with both codes are very similar in

spite of the simplifications of the DWOS code.

Also, we observe that the effective inception point

for subcooled boiling starts at 0.345 m from the begin-

ning of the channel, with an average void fraction of

0.216. This means that the subcooled boiling region

has an average void fraction that cannot be neglected

in reduced order models.

Table 7

Total pressure drop in the channel computed with DWOS, p,

total pressure drop in the channel computed with SIMULATE,

pSIMULATE, acceleration pressure drop, pacc, gravity pressure

drop, pg, friction pressure drop, pf

p (Pa) 0.380 105

pSIMULATE (Pa) 0.392 105

pacc (Pa) 0.127 104

pg (Pa) 0.155 105

pf (Pa) 0.211 105

3.2. Transient results

To simplify the model, we have reduced the number

of feedback gain parameters to only two (Munoz-Coboet al., 2000), the feedback gain for the fundamental

mode denoted by Kg0:

Kg0 = Kg00 = Kg01 (56)

And the feedback gain for the first harmonic reac-

tivity denoted by Kg1:

Kg1 = Kg11 = Kg10 (57)

The feedback reactivity coefficients ofTable 4 were

calculated for a model with 25 axial nodes, following

a consistent methodology (Munoz-Cobo et al., 1994).

Because in this particular model, the number of axial

nodes is three (one for the subcooled liquid region,

one for the subcooled boiling region, and one for the

bulk-boiling region), it is expected that the feedback

gainnecessaryto achieve limitcycle oscillations should

be bigger than one. These results have been recently

proved by Ginestar et al. (1999); these authors have

proved that with the reduction of the number of nodes,

the feedback gain must be increased to get the critical

value.

To get self-sustained out-of-phase nuclear-coupleddensity wave oscillations, we fix the feedback gain of

thefundamentalmode at a valueof one, andwe increase

the reactivity feedback gain Kg1 of the first harmonic

mode. The critical value is attained at Kg1 = 3.1. In this

case, the model displays out-of-phase oscillations with

a period of 2.95 s, and a frequency of 0.34 Hz. At this

point, we must remark that the feedback gain necessary

to achieve out-of-phase oscillations, when the subcool-

ing boiling is not included in the DWOS model, is more

than two times the gain that it is necessary when the

subcooling boiling is included.Let us analyze now some results of the DWOS code.

Fig. 3 displays the bulk-boiling boundary lengths Zbb,1and Zbb,2, when out-of-phase oscillations are excited

with a feedback gain of 3.2. We observe that when in

one-half of the reactor core the bulk-boiling boundary

lengthZbb,1 attains its maximum value, in the other half

of the rector corethe bulk-boilingboundary lengthZbb,2reaches its minimum value.

Other important parameters are the mass fluxes

(kg/m2 s) at channel 1, we have observed that the mass

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Fig. 3. Bulk-boiling boundary lengths, Zbb,1 and Zbb,2, vs. time when out-of-phase oscillation are excited with a feedback gain of 3.2.

fluxes Gbb,1(t) and Gex,1(t), at the bulk-boiling bound-

ary of channel 1 and at the exit of channel 1, are delayed

with respect to the mass flux at the entrance of channel

1. Also, we have noticed that the mass flux Gex,1, at the

exit of channel 1, oscillates with a delay of 180 with

respect to the mass flux at the entrance of this same

channel. This behavior is typical of density wave oscil-

lations in unstable channels. This behavior is displayedin Fig. 4, which shows the mass flux at the entrance and

exit of channel 1 versus time.

Next, we must remark that the average void fraction

in the bulk-boiling region of channel 1 oscillates out-

of-phase of the average void fraction in thebulk-boiling

region of channel 2. This behavior is typical of out-of-

phase instabilities.

Finally, the powers transferred to channels 1 and 2

are out-of-phase, i.e. while in one-half of the reactor

core the power attains its maximum value, at theother half of the reactor core the power attains its

minimum value. This is a direct consequence of the

Fig. 4. Mass fluxes at the inlet of channel 1, Gin,1(t), and the exit of channel 1, Gex,1(t), vs. time.

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oscillation of the normalized amplitude of the first

subcritical mode n1(t) with time and a period of 2.95 s,

equivalent to a frequency of 0.34 Hz. Because for

out-of-phase oscillations n1(t) is bigger than n0(t), thenthe power in both halves of the reactor core oscillates

out-of-phase. However, the fundamental mode still has

oscillations of small amplitude due to the driven termF01

n1(t).

4. Conclusions

The fact that parallel channels of a BWR have com-

mon lower and upper plena impose the same pressure

drop p to all the reactor fuel channel assemblies, but

this pressure drop can change with time if the number

of channels is low (Munoz-Cobo et al., 2002). How-

ever, when the number of fuel channel assemblies is

very large, as in BWR, the pressure drop is practically

constant and the oscillations are negligible (Hennig,

2001). The second boundaryconditionfor out-of-phase

oscillations is that the mass flow rate coming from the

downcomer, and entering into the reactor core, is prac-

tically constant but it oscillates at the inlet of each half

of reactor core.

In this paper, we have improved the DWOS model,adding one additional node to each channel, this

extra node contains the subcooling boiling model.

Therefore, the new DWOS model is a two chan-

nel thermalhydraulic model of the reactor core cou-

pled to high order modal kinetics, with three nodes

at each channel. These nodes are: the liquid node,

where all the heat is invested in the heating of the

liquid, the subcooling boiling node, where a fraction

of the heat is invested in steam production, and the

bulk-boiling region where all the heat is invested in

steam production. The coupling, between the neu-tronic and the thermalhydraulic parts of the model,

is performed through the modal feedback reactivities

and the power distribution factors (Munoz-Cobo et al.,

2000).

The new DWOS model contains four additional dif-

ferential equations in comparison with the old one

(Munoz-Cobo et al., 2000), two for the dynamics of

the inception boundary lengths Z1,1 and Z1,2, and two

for the dynamics of the average void fraction in the

subcooled boiling region of each channel.

The steady-state results of the DWOS code and the

SIMULATE code are very similar for the pressure drop

along the channels and the exit qualities, as we display

in Section 3.Increasing the feedback gain of the first harmonic

mode beyond the critical value, the model displays out-

of-phase oscillations. We have found that the frequency

of these oscillations is 0.34 Hz, which is lower than the

experimental value found from the signal analysis of

0.47 Hz, and the value found with the LAPUR 5.2 code

of 0.41 Hz, this is due to use a homogeneous model in

this paper (Aguirre et al., 2005). Also, it is interesting

to point out that the mass flux oscillations at the bulk-

boiling boundary are delayed with respect to the mass

flux oscillations at the channel inlet. Also, the mass flux

oscillations at the channel exit are delayed with respect

to the mass flux oscillations at the bulk-boiling bound-

ary and with respect to the oscillations at the channel

inlet. The delay between the mass flux oscillation at

the channel exit and the channel inlet is half a period or

180, this fact is typical of density wave oscillations,

and suggests that the density wave mechanism is also

at the root of out-of-phase oscillations.

The oscillations of the power transferred to the

channels are out-of-phase. Nevertheless, when we have

out-of-phase instabilities, the normalized oscillating

component n1(t) displays a behavior practically sym-metric with time, and therefore as a consequence the

power at each half of the reactor core also shows this

behavior. This fact has been confirmed experimentally

and numerically with 3D calculations performed with

the codes RAMONA, MODKIN, and LAMBDA

REACT. Therefore, some of the characteristics of

the out-of-phase oscillations can be studied with the

DWOS code.

Acknowledgments

The authors of this paper are indebted with the mem-

bers of the European project NACUSP, and with the

MCYT by their support under the Contract BMF2001-

2690.

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