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Nuclear Engineering and Design 215 (2002) 187198

Theoretical and experimental study on density waveoscillation of two-phase natural circulation of low

equilibrium quality

Su Guanghui a,b,*, Jia Dounan a, Kenji Fukuda b,1, Guo Yujun a

a Xian Jiaotong Uni6ersity, 710049, Peoples Republic of Chinab Kyushu Uni6ersity, Hakozaki 6-10-1, Higashi-ku, Fukuoka, Japan

Received 14 March 2001; received in revised form 20 March 2001; accepted 19 October 2001

Abstract

A theoretical and experimental study of density wave oscillation (DWO) in natural circulation is presented in this

paper. Experiments were performed on a natural circulation test facility. The influences of mass flow rate, pressure,

inlet subcooling, heat flux and exit quality on DWO were analyzed. The marginal stability boundary (MSB) of DWO

was obtained. A criterion of two-phase natural circulation, which predicts the stability thresholds, was developed by

lumped parameter method. It is a function of non-dimensional parameters, such as phase change number Npch,

subcooling number Nsub, Froude number, Fr, geometry number Nl and friction number ~. The geometry number and

friction number are first defined in this paper. A correlation of DWO period was also obtained, it is also a functionof the above non-dimensional parameters. The results of the present criterion and period correlation were compared

with those of the experimental data and references. It is shown that they agree very well. 2002 Elsevier Science B.V.

All rights reserved.

www.elsevier.com/locate/nucengdes

1. Introduction

The flow instabilities and thermo-hydraulic os-

cillations have long been studied and many works

have been performed by many researchers since

flow excursion was first discovered by Ledinegg in1938, Ledinegg (1938). These studies were

prompted by potential harmfulness caused by

instabilities in large-scale nuclear reactor systems.

The 1979 accident at Three Mile Island (TMI)

reactor proved the importance of the concept of

inherent reactor safety. Passive residual heat re-

moval systems (PRHRS) have been adopted such

as in JPSR (Murao et al., 1995) and AP600(Mcintyre and Beck, 1992) reactors. Natural cir-

culation is used in some PRHRSs such as in

AC600 (Su et al., 1994) provided by China. The

natural circulation can also be used in the primary

side of such reactors as the low temperature heat-

ing reactor (THR; Wang, 1993). Boiling natural

circulation is the major cooling mechanism in the

* Corresponding author. Tel.: +86-29-266-7082; fax: +86-

29-266-7802.

E-mail addresses: [email protected] (S. Guanghui), sugh@

nucl.kyushu-u.ac.jp (K. Fukuda).1 Tel./fax: +81-92-6423789.

0029-5493/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 0 2 9 - 5 4 9 3 ( 0 1 ) 0 0 4 5 6 - 3
mailto:[email protected]

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S. Guanghui et al. /Nuclear Engineering and Design 215 (2002) 187198188

SBWR (Barbucci, 1995). Flow in natural circula-

tion systems is induced by difference in fluid

densities between the hot leg (riser) and the cold

leg (downcomer). For a two-phase natural circula-

tion loop, heat input and its removal induces a

large volumetric change owing to phase changes-

boiling and condensation-, thus the system easily

becomes unstable. In such a two-phase naturalcirculation loop, various types of flow instabilities

occur depending on the system geometry, fluid

properties and the operating conditions etc. One

of the most important phenomena in natural cir-

culation loop is density wave oscillation (DWO)

which can affect the operation and safety features

of the system. That is, the self-sustained DWO

may cause many undesired problems such as me-

chanical and thermal fatigue of components by

mechanical vibration and thermal waves to re-

quire the system control considerations.There are many studies dealing with two-phase

oscillations. A large part of the studies is on

forced circulations and relatively less works

(Wissler et al., 1956; Chexal and Bergles, 1973;

Ardron, 1984; Lee and Ishii, 1998; Balunov, 1990;

Delmastro et al., 1991; Yun, 1994 etc.) have been

done on natural circulation. Boure et al. (1973)

made a clear classification of flow instabilities.

But most of the instabilities mentioned by Boure

concerned forced circulation. Fukuda and Kobori

(1979) classified DWO into two types, namely,

type I and II. Type I is the DWO occurring at low

quality conditions, for which gravitational pres-

sure drop in unheated riser plays a dominant role;

and Type II is one occurring at high quality

conditions, for which frictional pressure drop is

dominant. Lee (1991), studied the flow instabili-

ties in an open two-phase natural circulation loop

and constructed typical instability map. Kyung

and Lee (1994) investigated flow characteristics in

an open two-phase natural circulation loop with

Freon-113 as the working fluid and observed

three basic circulation modes (periodic circulation

A, continuous circulation and periodic circulation

B) with variation of the heat flux and constant

inlet subcooling. Further, Kim and Lee (2000)

classified natural circulations into six different

modes. Instability map in the plane of the heat

flux and the heater-inlet subcooling was plotted.

Rohatgi and Duffey (1998) presented critical sub-

cooling number as a function of Froude number,

Fr in natural circulation two-phase flow and

showed the fundamental parametric dependencies

on the loop loss coefficient and the ratio of the

heated channel to downcomer heights. Inada et al.

(1995, 1997, 2000) studied two-phase flow insta-

bility in a boiling natural circulation loop atrelatively high system pressure and analyzed the

inlet throttling effect on flow stability. Zhou

(1997), Jiang et al. (2000) studied stability struc-

ture of low quality DWO in a natural circulation

system at heating reactor conditions by the fre-

quency domain method. Su (1997), Su et al.

(2001) presented a non-linear dynamic model in

time domain of two-phase natural circulation.

But unfortunately, there is not a perfect crite-

rion to predict the stability of two-phase natural

circulation, neither a period correlation to calcu-late DWO period. The main purpose of this paper

is to derive the stability criterion and period cor-

relation of two-phase natural circulation. Experi-

mental study was also done to verify the criterion

and correlation.

2. Experimental investigation

2.1. Test facility

The schematic diagram of the test system is

shown in Fig. 1. Basically it consists of the pri-

mary loop, secondary loop (cooling loop), electri-

cally heating system and measuring system. The

primary loop is a two-phase natural circulation

one, which consists of pre-heater, test section,

riser, condenser, downcomer, pressurizer, throttle

valve, and connection tubes. All of the compo-

nents and tubes were made of stainless steel.

Table 1 shows the main parameters of the natural

circulation loop and test section.

Distilled water was used as working fluid. It

enters the heated test section with the required

conditions; exits as two-phase mixture and flows

through the riser into the condenser, through the

downcomer, the throttle valve and pre-heater

where it reaches the specified subcooling, then it

flows back into the inlet of the test section.

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S. Guanghui et al. /Nuclear Engineering and Design 215 (2002) 187198 189

Fig. 1. The natural circulation loop.

Table 2

Ranges of main parameters of experiment

Parameter UnitRange

Pressure 0.22 MPa

Inlet subcooling C540

Mass flow rate kg s100.4

03540 kg m2

s1

Mass velocity230 kWHeat power of test section

ference (drop) of the test section was detected by

a pressure transducer. The fluid temperatures of

the test section inlet and outlet, and the outside

wall temperatures were detected by thermocou-

ples. The heated power was calculated from the

coefficient of heat balance, the voltage and the

current of power supply. All of the experimental

data were recorded by a data acquisition system.

The mass flow rate, pressure difference between

the test section inlet and outlet, the fluid tempera-

ture of the test section inlet, the pressures of the

test section inlet and outlet and the wall tempera-

ture were recorded by a multi-channel data

recorder. The ranges of the main experiment

parameters are listed in Table 2.

2.2. Results

Under the specified run conditions, the DWO

will occur if the heated power increases step by

step. The state of the system is very sensitive to

the fluctuations of heated power when the system

run conditions are near to the unstable region.

Although the power increment is very small, it

can cause the self-sustained oscillations in the

system mass flow rate, test section pressure differ-

ence, wall temperatures and fluid temperature etc.

The oscillation periods are short (240 s), there-

fore, the frequencies are high. The shapes of these

oscillations are quite regular, the amplitudes are

identical. The wall temperature and the mass flow

rate oscillate out of phase, i.e. with a 180 phase

lag, the pressure difference and the mass flow rate

oscillate in phase. For the same conditions, if the

power is relatively small, the periods of oscillation

are long, and in the same time, there are smaller

fluctuations in which the period is shorter than 1

The mass flow rate G, system pressure P, the

temperatures of test section inlet Tin and outlet

Tex, the temperatures of test section outside wall

Tw, heated power Q, and the pressure difference

between test section inlet and outlet were mea-

sured and recorded during experiments. The natu-

ral circulation mass flow rate was measured by a

orifice flow meter and the differential pressure

transmitter. The pressure of the system was mea-

sured by a pressure transducer. The pressure dif-

Table 1

Main parameters of natural circulation loop

UnitParameters Value

mTotal height of loop 12

9.5 mHeight between the center of hot and

cold sources

Size of riser tube 342.5 mm

Height of riser 7.5 m

11Height of downcomer m

Size of test section tube 162 mmLength of test section m3

mmLength of heated section 670

Power of condenser 350 kW

Power of test section 180 kW

150Power of pre-heater kW

Mass flow rate 5 kg s1

16Pressure of system MPa

250Temperature of fluid C

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s. When the power increases, the period of fluctu-

ation decreases, the oscillations increase, e.g. the

amplitude and the mean magnitude of mass flow

rate also increase.

2.2.1. Effect of system pressure

System pressure can influence the system insta-

bility, as shown in Fig. 2. Keeping other parame-

ters such as mass flow rate, inlet subcooling etc.

constant, the critical power (heat flux) will in-

crease if system pressure increases. Critical power

is defined as the heated power when the system

begins to oscillate. The system stability will be

improved if the critical power increases. For the

same conditions, and identical increase of heated

power, the disturbance of gravitational pressure

drop under the conditions of high pressure is less

than that of low pressure because the density

difference between the liquid phase and vapor

phase will decrease if the system pressure in-

creases. In this case, the self-sustained oscillation

of mass flow rate can not be continued, or it can

not be produced, so the system tends to be stable.

The system pressure has no obvious influence on

the DWO period, but the DWO amplitude will

decrease and the DWO period will increase a little

if the system pressure increases, as shown in Fig.

3. Also, the critical exit equilibrium quality, which

is defined as the exit equilibrium quality when

oscillation occurs, will increase with an increase of

pressure, as shown in Fig. 4.

2.2.2. Effect of inlet subcooling

The inlet subcooling has a complex, nonlinear

influence on DWO, as shown in Fig. 5. On the

Fig. 3. The influence of system pressure on the period of

oscillations.

one hand, if the inlet subcooling increases, the

length of single phase of liquid (or single phase

region) in the test section will increase. If theincreasing of inlet subcooling equals to the in-

creasing of the inlet resistance coefficient, the

stability of the system will increase. On the other

hand, for constant heat flux, if the inlet subcool-

ing increases, the average quality of the test sec-

tion will decrease, the period of bubble producing

will increase, evaporated time will increase, the

responding time that inlet mass flow rate responds

to change of pressure difference due to the evapo-

ration will decrease, and a DWO may occur.Thus, if the first mechanism is predominant, the

system tends to be stable if the inlet subcooling

Fig. 4. The influence of system pressure on critical exit equi-

librium quality.Fig. 2. The influence of system pressure on stability.

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Fig. 5. The influence of inlet subcooling on stability.

Fig. 7. The influence of mass flow rate on critical exit equi-

librium quality.

mass flow rate has strong cooling action to the

wall of the test section, it is difficult for bubbles

agglutination. But the increasing of mass flow rate

will result in the decreasing of the exit equilibriumquality, as shown in Fig. 7. So the DWO will

occur at the low exit equilibrium quality under

high mass flow rate. Under these conditions, the

average density of fluid increases, the circulation

time of the fluid through the test section will

become longer, so the DWO period will increase,

as shown in Fig. 8. With the same enthalpy

increment per unit fluid, the increase of mass flow

rate will result in the increase of oscillation ampli-

tude of parameters such as mass flow rate, pres-

sure difference of the test section, and tempera-ture. Fig. 9 shows the oscillation amplitude

changes with mass flow rate.

increases. If the second one is predominant, the

system tends to be unstable if the inlet subcooling

increases. Indeed, there is a critical inlet subcool-ing. When the inlet subcooling is smaller than this

critical inlet subcooling, the system stability will

decrease when the inlet subcooling increases. And

on the other hand, when the inlet subcooling is

bigger than the critical inlet subcooling, the sys-

tem stability will be increased if the inlet subcool-

ing increases. The critical inlet subcooling is about

20 C in the experiments.

2.2.3. Effect of mass flow rate

If mass flow rate increases, the critical powerwill increase, so the system will tend to become

stable, as shown in Fig. 6, since the fluid at high

Fig. 6. The influence of mass flow rate on stability.

Fig. 8. The influence of mass flow rate on the period of

oscillations.

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Fig. 9. The influence of mass flow rate on oscillation ampli-

tude. Fig. 11. The influence of exit equilibrium quality on the period

of oscillations.

2.2.4. Effect of heat flux and exit quality

Keeping the main parameters such as system

pressure, mass flow rate and inlet subcooling etc.

constant, the factor which decides whether the

system is stable or not will be the heat flux. Thesystem tends to be unstable with an increasing

heat flux. In the steady-state, the increase of heat

flux will make the natural circulation mass flow

rate increase. The averaged value of mass flow

rate will also increase with increasing heat flux

when DWO occurs. Moreover, the oscillation am-

plitude will also increase. The amplitude of wall

temperature is not big but its frequency is the

same as that of mass flow rate fluctuation. So the

characteristics of wall temperature oscillation are

low amplitude and high frequency. The oscillation

period will become short with an increase of heat

flux, as shown in Fig. 10. It also will become short

with an increase of exit equilibrium quality, as

shown in Fig. 11. With the increasing heat flux,

the exit equilibrium quality will increase, the aver-

aged fluid density will decrease, the mass velocity

in the test section will increase, the time when the

fluid stays in the test section will become short, sothe period of oscillation will also be reduced. But

in high heat flux region, the change of period will

be small and almost tends to be flat with an

increase of heat flux.

2.2.5. Marginal stability boundary

The above analyses were done under the condi-

tions of keeping all of the parameters constant

except the one whose effect on the stability will be

studied. If all of the parameters charge it will be

difficulty to analyze the influence effects of

parameters on stability because there are many

parameters which can affect the stability. So non-

dimensional parameters, which include all of the

parameters that can affect the stability, are helpful

to analyze the influence effects of parameters. In

this paper, phase charge number Npch, subcooling

number Nsub, and Fr are employed, and their

definition are given by Eqs. (18), (17) and (19))

respectively. The marginal stability boundary

(MSB) can be shown by the above non-dimen-

sional parameters. The MSB shown by Npch ver-

sus Nsub plane is plotted in Fig. 12. In general,

Nsub will increase with an increase of Npch. When

Npch\Nsub, net vapor will be produced in the

system, and the quality will increase with the

increasing (NpchNsub). The MSB in Fig. 12 is

for type I DWO because the experiments were

done under low equilibrium quality. In Fig. 12,Fig. 10. The influence of heat flux on the period of oscillations.

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the results of references are also shown. The

difference between this paper and references may

be due to the differences of test loops. The MSB

as a function of Npch, Nsub and Fr is plotted in

Fig. 16.

3. Stability criterion

The most different aspect of the natural circula-

tion mode from the forced circulation one is that

the mass flow rate is determined by the heated

power. The bigger the mass flow rate is, the

stronger the natural circulation ability is. Thus, a

riser, which is also called a chimney, is always

employed in a natural circulation as shown in Fig.

1 to increase the mass flow rate. So consider the

boiling channel of the natural circulation loop,

which is divided into three control volumes asshown in Fig. 13. The above control volume is the

riser (chimney), the bottom one is single phase

flow region and the middle one is two-phase flow

region. The interface between the single phase

flow and two-phase flow region will be changeable

with the change of run conditions, so the length

of these two control volumes will also be change-

able. To simplify the analysis, the governing equa-

tions, which describe the interaction between the

liquid and vapor phase, are based the following

hypothesis (Guido et al., 1991; Su, 1997):Fig. 13. Boiling channel of natural circulation.

1. the subcooling region is neglected because its

length is very short;

2. the heat flux of the channel is constant;

3. the friction pressure drop for single phase flow

is concentrated on the inlet of this channel and

the friction pressure drop for two-phase flow is

concentrated on the outlet of this channel;

4. the homogeneous phase flow is used as two-

phase flow model;

5. the temperature distribution in single phase

flow region is linear.

The basic conservation equations of mass and

energy can be written as following.

For single phase (liquid) flow mass conserva-

tion equation:

AzsdZsp

dt=WinWsp (1)

Fig. 12. MSB of DWO.

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Energy conservation equation:

Azsd

dt(Zsphsp)=WinhinWsphs+Q

Zsp

Z(2)

For two-phase flow mass conservation

equation:

A ddt

(ztpZtp)=WspWout (3)

Energy conservation equation:

Ad

dt(ztphtpZtp)=WsphsWouthtp+Q

Ztp

Z(4)

The action of gravitation is very important in

natural circulation, so momentum conservation

equation can be written as:

(Kin2)sWin2 +(Kout+2) tp

2 Wout2 +2zgZA2

=2A2DP (5)

in which the equation of state, that is, the rela-

tionship between enthalpy and density will be as

follow when system pressure is constant(Guido et

al., 1991; Su, 1997).

z=s+hhs

hfg

fgn1

h\hs (6)

z=zs h5hs (7)

Furthermore, the following parameters are

defined to obtain the non-dimensional parame-

ters:

Zd=Z (8)

td=zsZA

Wo(9)

Wd=Wo (10)

Hd=Q

Wo(11)

Eq. (1) can be written in non-dimensional form:

AzsdZsp

dt=Azs

dZsp

d(t/td)

1

td=Wo

dZ(tp

dt(=WinWsp

(12)

dZ(sp

dt(=W( inW( sp (13)

Similarly, Eqs. (2)(4) are rewritten as follows:

d

dt((Z(sph(sp)=W( inh(inW( sph(s+Q(Z(sp (14)

d

dt((ztpZ(tp)=W( spW(out (15)

d

dt((ztpZ(tph(tp)=W( sph(sW(outh(tp+Q(Z(tp (16)

Furthermore, the following non-dimensional

parameters are defined:

Subcooling number:

Nsub=Dhinfg

hfgf(17)

Phase change number:

Npch=Qfg

Wfhfg(18)

Friction number:

~=2(Kin+Kout)

Kout+2(19)

Froude number, Fr:

Fr=V2

gL(20)

Geometry number:

Nl=Zr

Z(21)

The above equations are perturbed around

steady state and applying these non-dimensional

parameters, one finds:

d

dtlZsp=

4

~

Npch2

Npch

2(1+NpchNsub)Fr(Kout+2)

nlhtp

+4Npch

~ 2

Npch

NsublZsp (22)

d

dtlhtp=

Npch(1+NpchNsub)

(1Nl)NpchNsub

1~

(NpchNsub)

11

(1+NpchNsub)Fr(Kout+2)

1

nlhtp

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+Npch(1+NpchNsub)

(1Nl)NpchNsub

12

(NpchNsub)4~

1

Nsub

Npch

Nsub

nlZsp

+((2/~)(NpchNsub)1)NlNpch(1+NpchNsub)

(1Nl)NpchNsub

(23)

A system of nonlinear partial differential equa-

tions is constituted by Eqs. (22) and (23), and its

general solution is as:

X=Xoexp(ut)+X1 (24)

Where X=

lZsp

lhtp

is a determinant; Xo is a

constant matrix, and X1=

f1(t)

f2(t)

is its charac-

teristic solution.

Introducing the following parameters in order

to convenience for writing:

A1=4

~

Npch2

Npch

2(1+NpchNsub)Fr(Kout+2)

(25)

A2=4Npch

~

2Npch

Nsub(26)

A3=Npch(1+NpchNsub)

(1Nl)NpchNsub 1~ (NpchNsub)

11

(1+NpchNsub)Fr(Kout+2)

1

n(27)

A4=Npch(1+NpchNsub)

(1Nl)NpchNsub

12

(NpchNsub)

4~

1

Nsub

Npch

Nsub

n(28)

The characteristic equation of homogeneousequation of inhomogeneous differential equations

is:

uA2 A1

A4 uA3

= (uA2)(uA3)A1A4

=0 (29)

That is:

u2+Cu+D=0 (30)

C= (A2+A3) (31)

D=A2A3A1A4 (32)

If:C\0 and DB0 (33)

is satisfied, the system will be unstable under the

conditions of low quality, that is, the MSB ob-

tained by Eq. (33) will be located in the low

quality region. So the oscillation will be type I in

this case. That is, the DWO (Type I) will occur if

C\0 and DB0.

And if:

CB0 and D\0 (34)

is satisfied, the system will also be unstable, but it

is under the conditions of high quality, that is to

say, the MSB obtained by Eq. (34) will be located

in the region of high quality. So the oscillation

will be type II in this case. That is to say, DWO

(Type II) will occur if CB0 and D\0.

The nondimensional frequency is:

f(=D (35)Thus, the period of DWO is (Guido et al., 1991;

Su, 1997):

T=2y

f(td (36)

We can obtain the MSB of DWO using this

criterion expressed by Eqs. (33) and (34). The

results of Eq. (33) were compared with those of

experiments and RETRAN02 (Gao and Li, 1989),

see Figs. 1417. Figs. 14 and 15 are plotted in

Npch versus Nsub, Fig. 16 is plotted in the NpchFr0.5 versus Nsub, in which both the MSBs of type

I DWO obtained from experiments and criterion

and the MSBs of type II DWO obtained from

criterion were shown. Fig. 17 are also plotted in

the Npch Fr0.5 versus Nsub which shows a compari-

son between MSBs obtained by criterion and

RETRAN02. The figures show that the MSBs

obtained by criterion agreed well with the results

of experiments and reference (Gao and Li, 1989).

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Fig. 14. Comparison between MSBs obtained by criterion and

experiments (A).


experiments (C).

The period obtained analytically by Eq. (36)

was compared with the results of experiments,

and it was in good agreement with the results of

experiments, as shown in Fig. 18.

4. Conclusions

DWO is quite an important phenomenon in

natural circulation and it can be classified into

two types. The type I DWO is experimentally

studied in this paper. Experiments were per-

formed on a natural circulation test facility.

The system will tend to be stable if the systempressure increases. The system pressure has no

obvious influence on DWO period, but the

DWO amplitude will decrease and the DWO

period will increase a little if the system pres-

sure increases. The critical exit equilibrium

quality will increase with an increase of

pressure.

The inlet subcooling of the test section hasobvious, complex, nonlinear influence on

DWO. There is a critical inlet subcooling which

is about 20 C. When the inlet subcooling is

smaller than this critical inlet subcooling, the

system stability will decrease when the inlet

subcooling increases. And when the inlet sub-

cooling is bigger than the critical inlet subcool-

ing, the system stability will be increased if the

inlet subcooling increases.

If mass flow rate increases, the system will tend

to be stable. The critical exit equilibrium qual-ity will decrease, the DWO period will increase

and DWO amplitude will increase with the

increase of mass flow rate.

With the increasing of heat flux, system tendsto be unstable and the period of DWO be-


experiments (B).


RETRAN02.

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Fig. 18. Comparison between results obtained by correlation and experiments.

comes short. But in the conditions of high heat

flux, the change of period will be small and

almost tends to be flat.

The maps of MSB for type I DWO in theplane of Nsub and Npch, and of Npch Fr

0.5 and

Nsub were obtained.

The theoretical study was also presented in this

paper.

The boiling channel of natural circulation isdivided into three control volumes. The non-

dimensional changes are done to the basic

mass, energy and momentum conservation

equations.

A criterion for predicting stability and a corre-lation of DWO period are obtained. The re-

sults obtained by the criterion and the

correlation agreed well with those of experi-

ments and reference.

The MSB maps for both type I DWO and typeII DWO can be obtained by the criterion.

5. Nomenclature

A cross sectional area

gravitational accelerationg

mass flow velocityG

specific enthalpyh

friction coefficientK

heat lengthL

pressureP

heat fluxq

Q heated power

timet

velocityV

W mass flow rate

axial coordinatez

void fractionh

z density

specific volume

Subscripts

f liquid

transfer from liquid to vaporfg

g vaporinletin

outletout

steady-stateo

riserr

s saturated

single-phase flowsp

two-phase flowtp

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