dwomode1

7/29/2019 dwomode1

1/10

Nuclear Engineering and Design 241 (2011) 47044713

Contents lists available at ScienceDirect

Nuclear Engineering and Design

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / n u c e n g d e s

Numerical research on oscillation of two-phase flow in multichannels under

rolling motion

Youjia Zhang a,, Guanghui Su a, Suizheng Qiu a, Wenxi Tian a, Hua Li a, Libo Qian a, Yong Li b,Xiao Yan b, Yanping Huang b

a State Key laboratory of Multiphase Flow in Power Engineering, Department of Nuclear Science and Technology, Xian Jiaotong University, Xianning West Road 28,

Xian City 710049, Chinab Key Laboratory of Bubble Physics and Natural Circulation, Nuclear Power Institute of China, 610041 Chendu, China

a r t i c l e i n f o

Article history:

Received 19 January 2011

Received in revised form 16 April 2011

Accepted 16 April 2011

a b s t r a c t

Two-phase flow instability and dynamics of a parallel multichannels system has been theoretically

studied under periodic excitationinduced by rolling motion in the present research. Basedon thehomo-

geneous flow model considering the rolling motion, the parallel multichannelsmodel and system control

equations are established by using the control volume integrating method. Gear method is used to solve

the system control equations. The influences of the inlet, upward sections, heating power and rolling

amplitudeson theflow instability under rolling motion havebeen analyzed.The marginalstability bound-

ary (MSB) under the rolling motion condition is obtained. The unstable regions occur in both low and

high equilibrium quality and inlet subcooling regions. The multiplied period phenomenon occurs in the

high equilibrium quality region and the chaos phenomenon appears on the right of MSB. The concept of

stability space is presented.

2011 Elsevier B.V. All rights reserved.

1. Introduction

The phenomenon of two-phase flow instability occurred widely

in manyindustrial systems and equipments, such as nuclear reactor

steam generators, reboilers, and cooling plants etc. It is undesir-

able since the oscillation caused by two-phase flow instability can

affect the operation and safety features of the systems. Therefore,

analysis of two-phase flow instability is very important for the

safety of nuclear reactors and other two-phase flow equipments.

Su et al. (1998, 2001, 2002) conducted both theoretical and exper-

imental studies on density wave oscillation of two-phase natural

circulation. The MSB of density wave oscillation was obtained. A

criterion of two-phase natural circulation, which predicts the sta-

bility thresholds, was developed by lumped parameter method.

Yun et al. (2005) performed a theoretical study on two-phase

instability in natural circulation loops of China advanced researchreactor (CARR). The phase-space trajectory of the boiling boundary

and the mass flow rate were discussed. Because of the inherent

defect of frequency-domain analysis method, many researchers

(Clausse and Lahey, 1990; Zhou, 1994; Chang and Lahey, 1997; Lin

and Pan, 1994; Lin et al., 1998; Lee and Pan, 1999, 2005a,b; Durga

Prasad and Pandey, 2008a; Durga Prasad et al., 2008b) adopted the

time-domain analysis method to investigate the characteristics of

Corresponding author.

E-mail address: [email protected] (Y. Zhang).

two-phase flow instability. In addition, there are several extensive

reviews of instabilities in two-phase flow system in recent years,

notably by Tadrist (2007), Durga Prasad et al.(2007) and Kakac and

Bon (2008).

Recently, the two-phase flow instability among parallel mul-

tichannels has received extensive attention. In multichannels

system, the instability of natural circulation is very complicated

because of the interaction among channels. Satoh et al. (1998)

conducted a numerical investigation on the effects of multiple

loops on the natural circulation instability. They considered that

the instability of natural circulation in multiple loops system is

more complicated than that of a single loop system because of

the interaction among the loops. Lee and Pan (1999) investigated

the nonlinear dynamics of multiple parallel boiling channels with

forced flows by using the Galerkin nodal approximation method.

They found thesystem becomes unstabledue to channel to channelinteraction with the increase of the channel number. The oscilla-

tions in the pressure drop and in the mass flux to the channels

have been investigatedwhen outof phase oscillations occur. Durga

Prasad et al. (2008b) investigated the oscillation modes under dif-

ferent operating conditions and channel to channel interaction

during power fluctuations and on-power refueling in a double

channels natural circulation boiling system. The system is mod-

eled usinga lumped parametermathematicalmodel. At high power

levels, disturbances in one channel significantly affect the stability

of the other channel. Yun et al. (2008a) conducted a theoretical

research on the behavior of two-phase flow instability in multi-

0029-5493/$ see front matter 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.nucengdes.2011.04.037
http://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.nucengdes.2011.04.037http://www.sciencedirect.com/science/journal/00295493http://www.elsevier.com/locate/nucengdesmailto:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.nucengdes.2011.04.037http://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.nucengdes.2011.04.037mailto:[email protected]://www.elsevier.com/locate/nucengdeshttp://www.sciencedirect.com/science/journal/00295493http://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.nucengdes.2011.04.037

7/29/2019 dwomode1

2/10

Y. Zhang et al. / Nuclear Engineering and Design 241 (2011) 47044713 4705

Nomenclature

A cross-sectional area of the control volume (m2) or

amplitude (m)

A+ nondimensional cross-sectional area of the channel

Eu Euler number, Eu = Ps/fu2s

f frictional coefficient

f+ nondimensional frequency, f+ =f/(us/LH)

Fr Froude number, Fr = u2s /gLHh enthalpy (J/kg)

h+ nondimensional enthalpy, h+ = (h hf)/[Q0/(fAxsus)]

k loss coefficient

L length of the control volume (m)

L+ nondimensional channel length, L+ = L/LHM+ nondimensional channel mass, M+ = M/fLHAHNpch phase change number, Npch = (Q/W)(vfg/(hfgvf))

Nsub subcooling number, Nsub = ((hf hi)/hfg)(vfg/vf)

Ns node number in single-phase region

P pressure (Pa)

PH heater perimeter (m)

P pressure drop (Pa)

P+ nondimensional pressure drop, P+ = P/fu2sP+

fnondimensional frictional pressure drop

P+a nondimensional acceleration pressure dropP+g nondimensional gravitational pressure dropP+

Inondimensional inertial pressure drop

P+add

nondimensional additional pressure drop

Q heating power (kW)

q power density (W/m2)

t time (s)

t+ nondimensional time, t+ = t/(LH/us)

u velocity (m/s)

u+ nondimensional velocity, u+ = u/usus characteristic velocity, us = ui,nvf specific volume of saturated liquid (m

3/kg)

z axial coordinate (m)z+ nondimensional length, z+ =z/LH

Subscripts

E entrance of the channel

f fluid or frictional pressure drop

H heated section

i channel inlet

j jth channel

n nth node in single-phase region

o stable state

R upward section

s standard value (characteristic value)

tot total

1 single phase2 two-phase

Greek symbols

+ nondimensional boiling boundary, + = /LH+ nondimensional coefficient of viscosity density (kg/m3)

f the density of fluid (kg/m3)

+ nondimensional density, + = /f

channels system. A physics model that includes the inlet, heated,

and upward sections was built. The instability boundaries of the

multichannels system were obtained in the parameter plane of

subcooling and phase change numbers. The instability space or

instability reef had been found. The present work in this paper is

an extension of the work conducted by Lee and Pan(1999) and Yun

et al. (2008a,b).

The marine reactor plant is influenced thermal-hydraulically by

variousshipmotions(Ishida andTomiai,1992). Therefore,the influ-

ence of ocean motions (such as rolling, pitching, heaving, swaying,

yawing and surging) on the two-phase flow instability among par-

allel multichannels must be taken into account. Rolling motion can

alter the relative positions of the equipments, which have an effect

on the natural circulation and heat transfer. All the ocean motions

can influence thebubble distribution andpressuredifference. These

motions can affect the flow instability of parallel multichannels

system. Thus, it is necessary to have some research in this field

for actual engineering application and the safety of nuclear reac-

tors. There were many papers on the two-phase flow instability

among parallel multichannels system under rolling motion con-

dition such as by Pang et al. (1995), Kim et al. (2001), Ishida and

Yortisune (2002), Murata et al. (2002). In China, many researchers

and scholars (Gao et al., 1997, 1999; Yang et al., 2002) conducted

many theoretical investigations on the characteristics of coolant

system under ocean conditions. Tan and Pang (2005a) and Tan

et al. (2005b, 2006, 2007, 2009) carried out a series of theoretical

and experimental researches on the characteristics of natural cir-

culation under rolling motion condition. Yun et al. (2007, 2008b)

investigated the two-phase flow instability of parallel multichan-

nels system under ocean conditions. The MSB of 9-channel system

was obtained.

The present study extends Lee and Pans method (1999), com-

binedwith Gaos model(1997) for pressure drop in a multichannels

system under the ocean conditions. In Lee and Pans model, there

is only a heated section, and the subcooled boiling is not con-

sidered. The present model is developed by Yun et al. (2008a),

including the inlet and upward sections besides the heated sec-

tion. The subcooled boiling is also considered. On this basis, the

ocean conditions were added into this model (Yun et al., 2008b).

Therefore, the present model can be used to estimate the stability

of parallel multichannels system under ocean conditions.

The influence of rolling motion on the two-phase flow insta-bility among parallel multichannels is analyzed. The MSBs of

multichannels (i.e. 4, 9-channel) systems are obtained under

different pressures. The accurate stability spaces under rolling

motion are presented.

2. System and physical model

2.1. Parallel multichannels model

The parallel multichannels model which is made up of two

plenums and many parallel channels (as shown in Fig. 1) is based

on the model of Lee and Pan (1999). The channel is divided into

three parts from bottom to top: inlet, heated and upward sec-

tions, respectively. The heated section is composed of two parts,

which are single-phase and two-phase sections, separately. In themodel of Lee and Pan (1999) there was only heated section. And

in the present study, the models of inlet and upward sections are

also included. This is based on the previous work of Clausse and

Lahey (1990). In our study, the manifestation of flow instability is

intertube pulse. It belongs to the dynamic instability.

The nondimensional system conservation equations can be

written as follows:

Mass conservation equation:

+

t++

z+(+u+) = 0 (1)

Energy conservation equation:

t+

(+h+) +

z+

(+h+u+) = 1 (2)

7/29/2019 dwomode1

3/10

4706 Y. Zhang et al. / Nuclear Engineering and Design 241 (2011) 47044713

Upper Plenum

Lower Plenum

heating section

upward section

inlet sectionthrottle valve

LR

LH

LE

single phase zone

two-phase zone

subcooling boiling

saturated boiling

Fig. 1. Schematic of parallel multichannels system.

Momentum conservation equation:

t+(+h+) +

z+(+u+

2) = +u+

2

Ni=1

ki(z+

z+i

)++

2

2

1

Fr+ Eu

P+

z+(3)

where = (1 + Npchh+)1, h+ > 0; + = 1, h+ 0; = 1 , in single-

phase region, frictional factor; = 2 , in two-phase region,

frictional factor.

The present theoretical model is basedon the following assump-

tions:

1. Uniform heat flux in the axial direction.

2. The fluid is in subcooled state at the channel inlet.

3. The homogeneous flow model for two-phase flow.

4. Constant properties at a given system pressure.

5. The two phases are in thermodynamic equilibrium.

6. The flow is one-dimensional flow.

Based on the conservation equations and assumptions above,the system equations are as follows:

dL+n,j

dt+= 2ui,j

+ 2Ns

Npch,jNsub,j

(L+n,j

L+n1,j

) dL+

n1,j

dt+

dM+H,j

dt+= u+

i,j +

e,ju+

e,j

d+e,j

dt+=

1 +

+e,j

ln(+e,j

)

1 +e,j

d+

j

dt++ +

e,ju+

e,j u+

i,j

(1 +e,j

)2

(1 +j

)[1 +e,j

+ ln(+e,j

)]

dM+R

dt+=

ARAH

u+e (+

e +

R )

du+edt+

=du+

i

dt+

d+

dt+

du+i,jdt+

=Ajdu+i,1dt+

+ Bj, j = 2, 3, . . . , M

du+i,1

dt+=

(dW+tot/dt+)

Mj=2

A+H,j

Bj

1 +M

j=2A+

H,jAj

(4)

where = (qpHvfg)/(AHhfg), calculating coefficient of the flowvelocity change at the inlet; Aj and Bj, calculating coefficient of

velocity, dependent on the pressure drop and mass of the two-

phase flow in the channel; vfg, difference in specific volume ofsaturated liquid and vapor (m3/kg); hfg, latent heat of evaporation

(J/kg).

Detailed contents about the system equations were presented

in the previous study ofYun et al. (2008a) and Zhang et al. (2009).

7/29/2019 dwomode1

4/10


2.2. Rolling motion model

The pressure drop of channels can be written as follows:

P+j

= P+f,j

+ P+a,j

+ P+g,j

+ P+I,j

+ P+add,j

(5)

where P+add

is the additional pressure drop, which is caused by

rolling motion and its specific expression can be obtained by inte-

grating along thelengthof channel. We consider the rolling motion

means solving the additional force caused by the additional pres-sure drop. In the progress of system calculation, considering the

rolling motion means putting the additional pressure drop reckon

in the momentum equation.

When the ship is under the rolling motion condition, the influ-

ence of inclination must be taken into account at the same time

since the channels are tilted too. Noticeably, when the ship is in

periodic fluctuation or translation of varying speed all the channels

have the same P+add,j

. Therefore, such motions have no influence

on the local multichannels system. However, these motions will

affect the pressure drop of the whole loop. If the ship is in periodi-

calfluctuation, the gravity pressure drop of thewhole flow channel

will be changed, and then the total mass flow will be changed too.

Hence, periodical total mass flow changing was used to simulate

the two motions. Thereby, dW+tot/dt+ is used to analyze the effectof these motions (periodical fluctuation or translation of varying

speed).

W+tot = W+

tot,0(1 +A sin(2f+t+)) (6)

where W+tot,0 =

Wtot,0/fAHusThen,

dW+totdt+

= 2f+AW+tot,0

cos(2f+t+) (7)

3. Mathematical solving method

Based on the all facts mentioned above, Eqs. (4), (5) and (7) are

combined to solve the rolling motion. Thus, the thermodynamiccharacteristics of the multichannels system under a given ocean

condition can be obtained by solving the set of non-linear ordi-

nary differential equations. The set of equations can be written as

follows:

dy

dt=

f(t, y, y, x) (8)

y(t0) = y0 (9)

where y is the state vector, [L+, M+H, M+

R , +e , u

+

i,j, W+

total, u+e ]

T. and

x is the input vector.

The input vector consists of the heat input power, inlet loss

coefficient, inlet subcooling, mass flow rate and ocean conditions

etc. We adopted the Gear multi-value method (Gear, 1974), whichdesigned especially for stiff equation system to solve the set of

equations. In the calculation the nodes of single-phase are 3 and

the upward section is looked as one control volume. The set of

equations includes 70 equations. The original step length is 10 3,

the precision of solution is 104 and the relative error is 5 105

that provided the best trade-off between calculation accuracy and

computation time.

4. Results and discussion

As shown in Fig. 2, 9-channel system is arranged symmetrically.

Four channels (channel 1, 2, 6 and 7) are located at the rolling axis

(x-axis). The other four channels (channel 3, 4, 8 and 9) are located

at the radial position (y-axis). Channel 5(CH 5) is located at the

x

z

y

13

6

8

5 2

4

7

9 channel

Fig. 2. Schematic of 9-channel system.

Table 1

The effect of inlet section length on the system stability.

Heating powerQ(kW)

Inlet sectionlength (m)

Upward sectionlength (m)

Stability

120 0.3 0 Stable

120 0.4 0 Stable

120 0.5 0 Stable

120 0.6 0 Unstable

120 0.7 0 Stable

120 0.8 0 Unstable

120 0.9 0 Unstable

origin of coordinate axis. The lengths of channel, inlet and upward

sections are 1, 00.9 and 00.3m, respectively.

In general, there are four kinds of flow oscillations in the

multichannels system, including damped oscillation, limit cycle

oscillation, chaotic oscillation and divergent oscillation. When thedamped oscillation occurs, the system is stable. However, when

the chaotic oscillation or divergent oscillation occurs, the system

becomes unstable. In the previous study, if the amplitude of limit

cycle oscillation is not great than 510%, the system is considered

to be stable. In the present study, we define the system is stable if

the amplitude of limit cycle oscillation is less than or equal to 8%,

otherwise the system is unstable.

4.1. The influence of inlet and upward sections on the system

stability

When the inlet resistance coefficient kin is 0, inlet temperature

Tin is 270C, system pressure P is 15 MPa, inlet flow rate uin is

2.75m/s, swing angular velocity Rv is 0.1 rad/s, rolling amplitude

Ra is 20, the effect of inlet and upward sections on the stability of

8-channel system had been analyzed. Specific calculation results

are shown in Tables 1 and 2.

There are seven operating conditions to analyze theinfluence of

inlet section length on the system stability in Table 1. The heating

Table 2

The effect of upward section length on the system stability.

Heating power

Q(kW)

Inlet section

length (m)

Upward section

length (m)

Stability

100 0.2 0 Stable

100 0.2 0.1 Stable

100 0.2 0.2 Unstable

100 0.2 0.3 Unstable

7/29/2019 dwomode1

5/10


Fig. 3. Inlet velocity oscillation curves of 9-channel system under rolling motion condition. (a) Q= 65kW, (b) Q=100 kW, (c) Q= 140kW, and (d) Q=145kW.

power Q is 120 kW, the upward section length is 0 m. The length

of inlet section increases gradually from 0.3 to 0.9 m. In Table 1, it

reveals that the influence of inlet section length on the system sta-

bilityis variable.On theone hand, theinherentstability of systemis

intensified with the increase of the single-phase section length. On

the other hand, the quotient of additional pressure drop has been

enhanced under the rolling motion condition with the increase of

the single-phase section length. The differences of additional pres-

sure drop on each channel are more obvious. Because the position

of each channel is different, which results in the flow oscillation

increasedand thesystembecome unstable.So, theinfluenceof inlet

section lengthon thesystem stabilityis variable.The data in Table 2

shows that increasing thelength of upwardsection will weakenthesystem stability.

4.2. The influence of heating power on the system instability

Fig. 3(a)(d) is the inlet velocity oscillation curves of 9-channel

system when the inlettemperature Tin is 260C (P= 13MPa) chang-

ing the heating power under the rolling motion condition. Fig. 3(a)

shows that the system is unstable when the heating power is low.

With the increase of the heating power (Fig. 3(b) and (c)), the

system becomes stable. While continuing to increase the heating

power, the system is in the unstable state of divergent oscillation

developedfrom thestable state of intertubepulse. Hence,the influ-

ence of heating power on the system instability is also variable.

The increase of heating power strengthens the system stability in

low equilibrium quality region, whereas it weakens the system

stabilityin high equilibriumquality region. Interestingly, the diver-

gent oscillation has a counterflow phenomenon when the heating

power Q is 145kW as shown is Fig. 3(d). This is attributed to the

Fig. 4. Inlet velocity oscillation curves of CH 1 under rolling motion condition (4-

channel system, P=15MPa, Tin =270

C).

7/29/2019 dwomode1

6/10


Fig. 5. Inlet velocity oscillation curves (a) and phase space trajectory (b) under rolling motion.

large amplitude of the oscillation, some flow oscillation back to theprevious channel.

4.3. Multiplied period phenomenon

In high equilibrium quality region, the inlet velocity oscillation

curves of CH 1 with different oscillation periods are shown in Fig. 4.

The black, red, blue and dark cyan lines are inlet velocity oscil-

lations curves of CH 1 when the heating power are 70, 90, 110

and 140 kW, respectively. (For interpretation of the references to

color in this sentence, the reader is referred to the web version of

the article.) In Fig. 4, the multiplied period phenomenon appeared

with the increase of heating power gradually. This is caused by the

nonlinear characteristics of system. Because Eqs. (4) are nonlinear

equations, there may has a periodic solution. Owing to the rollingmotion, which is a periodic effort added into the system, the multi-

pliedperiodphenomenon occurs. The blackline (Q=70kW)isaunit

flow oscillation curves. It is merely caused by the rolling motion.

The characteristic of intertube pulse of parallel multichannels sys-

temitself occurs with the increase of heating power, and these two

aspects superimpose together. The blue line is the double periodic

flowoscillation curvesandthe dark cyan line is thefourfold periodic

flow oscillation curves.

The multiplied period phenomenon usually occurs at the right

side of the MSB in high equilibrium quality region, namely, the

unstable zone. The emergence of multiplied period enhances the

flow fluctuation among channels. Thus, the interaction among

channels is strengthened resulting in the anomalistic oscillation

appears. So, that is the chaos phenomenon exactly.

4.4. Chaos phenomenon

Chaos is a non-linear phenomenon. It is impossible to use the

traditional approach (such as solving the nonlinear differential

equation) to analyze the chaos. The feasible method is analyzing

the geometry image in the space. Nearby the marginal stability

boundary (MSB) where the equilibrium quality is high, it is easy to

discover the chaos phenomenon. As shown in Fig.5 (9-channel sys-

tem, Tin =280C, Q=130kW, P= 13MPa),the flowoscillationcurves

(Fig. 5(a)) are interlaced. There are at least two chaos attractors

emerged in the phase-space trajectory (Fig. 5(b)). Usually, judging

a system whether it is chaotic or not can be based upon the chaos

attractors in the phase plane trajectory. If there are many chaos

attractors appear in the phase space trajectory, this work conditionis the chaos operating condition.

4.5. The marginal stability boundary under rolling motion

condition

The MSB of 9-channel system is shown in Fig. 6 (P=11MPa,

Rv = 0.1rad/s, Ra = 20). The system is stable inside the MSB. The

systembecomes unstable outside the MSB at the two phase region.

In the low equilibrium quality region, the instability is caused by

the inlet section. While in the high equilibrium quality region,

the instability results from the inherent characteristic of the sys-

tem, which means the true intertube pulse led to the instability.

Moreover, even without rolling condition the intertube pulse will

also come into being. But under this circumstance, the flow oscil-lation results from both the intertube pulse of the system and

the flow fluctuation, which is caused by rolling motion at the

same time. In the low subcooling region, the subcooling has great

affects on the system instability and it is not single-value. On the

one hand, the length of single-phase region is increased at the

inlet section with the increase of inlet subcooling number. The

Fig. 6. The MSB of 9-channel system (P= 11MPa).

7/29/2019 dwomode1

7/10


Fig. 7. The MSBs of 4-channel system under rolling motion condition.

single-phase liquid has a well stability, which increased the inletresistance. Thereby, the system stability is enhanced. On the other

hand, the increasing of inlet subcooling number decreases the quo-

tient of equilibrium quality at the inlet section. It increases the

growth circulate period of bubble. Therefore, the time of evap-

orating grows is prolonged. This induces intertube pulse occurs

and weakens the system stability. The influence of low subcool-

ing on the system instability is multi-valued because the two sides

mentioned above are added together. In the low inlet subcooling

region, because the inlet temperature is close to the saturation

temperature of water under 11 MPa, therefore the length of two-

phase section is increased and the unstable probability of system

is high.

In the high inlet subcooling region, the influence of subcooling

on the system stability is also multi-valued. When the subcool-ing is constant, there is a critical value either. At this region,

the extremely high subcooling makes the equilibrium quality

decrease sharply, the growth circulate period of bubble has been

prolonged. Therefore, the unstable probability of system is also

high.

4.6. The MSB of multichannels system under rolling condition

As shown in Fig. 7, the shape of MSBs for 4-channel system

(under 7, 9, 11, 13 and 15 MPa) is a polygon. They seem similar

and the area of each MSB is nearly identical. The MSBs shift to left

with the increase of pressure. It means the system can get into

the stable region more quickly from the unstable region when the

pressure is increased. But we cannot conclude that the system ismore stable when the pressure is increased. This is different from

the non-rolling condition. Usually, the system will become more

stable when the pressure is increased under the non-rolling condi-

tion. However, due to the influence of rolling motion, the effect of

pressure on the system stability is weakened. When the pressures

are 13 and 15 MPa, respectively, some operating condition points

on the MSBs cross the equilibrium quality line (Xe = 0). It means

because of the rolling motion, even at the single-phase region, the

system may be unstable.

The MSBs of 9-channel system (under 7, 9, 11, 13 and 15 MPa)

are shown in Fig. 8. It is evident that the shape of MSBs for 9-

channel system is a quadrilateral. It is notably that there are some

differences between the MSBs for 4-channel system and 9-channel

system. Firstly, in the high equilibrium quality region when the

Fig. 8. The MSBs of 9-channel system under rolling motion condition.

inlet subcooling number is about 1.5, the MSB of 4-channel system

has a salient compared with that of 9-channel system. We sup-

pose that the flow rate (amplitude) for 9-channel can be averaged

into other more channels compared with 4-channel system. So, the

influence of rolling motion is weakened. Therefore, the MSBs of

9-channel system are much slippier than that of 4-channel sys-

tem. Secondly, the area of MSB for 4-channel system is larger than

that of 9-channel system (as shown in Fig. 9). Because the influ-

ence of rolling motion on the system stability can be replaced by

the influence of pressure drop of channels on the system stabil-

ity. Fig. 2 shows 9-channel system has more channels (CH 5-9)

compared with 4-channel system. Thus, the total pressure drop of

9-channelsystem is largerthan that of 4-channelsystem. Theinflu-

ence of rolling motion on the system stabilityfor 9-channel systemis greater than that of 4-channel system. So, the 4-channel system

is more stable than 9-channel system under this condition. Some

researchers said the system with more channels is more stable.

Nevertheless, this rule is notapplies to thesystem under therolling

condition.

Fig. 9. The MSBs of 4 and 9-channel system (P=9 MPa).

7/29/2019 dwomode1

8/10


Fig. 10. The stability space of 4-channel system.

4.7. Three-dimensional stability space

In almost all the investigations of flow instability under rolling

motion condition, the MSB is only a simple curve at two-

dimensional coordinate. A concept of stability space (the stability

island) is presentedin thepresent work. Thesystem stabilitycan be

described in the three-dimensional space. The three-dimensional

stability space is made up of phase change number (Npch), subcool-

ing number (Nsub) and nondimensional pressure (P+). We defined

the nondimensionalpressureP+=P/(10Patm) forconvenience, where

Patm is atmospheric pressure.

The stability spaces for 4 and 9-channel system are shown inFigs. 10 and 11, respectively. The stability space is made up of the

MSBs under different system pressures (such as 7, 9, 11, 13 and

15 MPa).

Fig. 11. The stability space of 9-channel system.

Fig. 12. The MSBs of 4-channel system with different rolling amplitudes

(P= 13MPa).

4.8. The Influence of rolling amplitudes

Fig. 12 is the MSBs of 4-channel system with different rolling

amplitudes (P= 13 MPa). The system is stable when the operating

condition point is inside the MSB. When the operating condition

point is outside the MSB at the two-phase region, the system is

unstable. This figure illustrates that the increase of rolling ampli-

tudes will destabilize the system stability. Since the area of the

MSBs are decreased with the increase of rolling amplitudes.

4.9. The influence of rolling condition on the system stability

The inlet velocity oscillation curves of rolling and non-rolling

conditions are shown in Fig. 13(a) and (b), respectively. These

two instances are under the same operating condition (P=11MPa,

Tin =220 C). The inlet velocity oscillation curve of 4-channel sys-tem under rolling motion condition is shown in Fig. 13(a). In this

figure the flow oscillation curves are moved up compared with the

equilibrium position. Based on the datum line (ui+ = 1), the flow

oscillation amplitudes of upper part are greater than that of lower

part. So, the system is unstable. The inlet velocity oscillation curve

of non-rolling motion condition is shown in Fig. 13(b). It is obvious

that thesystemcomesbackto itsequilibrium positionafterbe given

a sudden disturbance at the 50th second. Therefore, the system

is stable. For these two operating conditions, the system is sta-

ble when the heating power below 170 kW under the non-rolling

motion condition. While, the system stable region is between 105

and 150 kW for the rolling motion condition. So, we can conclude

the rolling motion causes the system instability occurs in advance,

which results in the system become more unstable.

4.10. The influence of channel number on the system stability

The homogeneous flow model is adopted to deal with the two-

phaseflow of the multichannels system underthe rolling condition.

The previous studies (such as Yun et al., 2008b) and the results in

this work indicated that the homogeneous flow model can accu-

rately predict the MSB and the major characteristics of limit cycle

oscillations under the middle andhigh systempressure conditions.

The effect of interaction among parallel channels on the system

stability can be analyzed to extrapolate the model in parallel mul-

tichannels system with different channel numbers. The MSBs of

different channel number systems can also be obtained. Never-

theless, the homogeneous flow model neglects the slip, which

7/29/2019 dwomode1

9/10


Fig. 13. Inlet velocity oscillation curves of rolling condition (a) and non-rolling condition (b).

may strengthen the system stability (Lahey and Podowski, 1989),between two phases. Therefore, the homogeneous flow model is

conservativein terms of predicting theMSB. Moreover,the increas-

ing channel number strengthens the interaction among channels.

It will interfere in the analysis on the system instability.

5. Conclusions

The present research in this paper extended the models of Lee

and Pan (1999) to analyze the multichannels system considering

the inlet and upward sections (Clausse and Lahey, 1990) under

the rolling motion. In the present model, the homogeneous flow

assumption is adopted for two-phase flow. Generally speaking, the

homogeneousflowmodelcandescribethetwo-phaseflowinstabil-ity of parallel multichannels system under middle or high pressure

condition. But the drawback of this model is neglect the possi-

ble thermodynamic and hydraulic disequilibrium between phases.

Because of the slip effect between the two phases is not consid-

ered in this model, and the slip effect can stabilize the system

stability. Therefore, the predicted results of the characteristics of

two-phase flow will leans conservative. Especially at the condi-

tion of low pressure, high equilibrium quality and heat transfer

rate or fast transient process, the forecast error is obvious. More-

over, the interaction among channels will be enhanced with the

increase of channel number. Thus, the model predicted results will

be influenced, andthe credibility of systemstability which is calcu-

lated by this model will be reduced. The two-phase flow instability

in the parallel multichannels system under rolling motion con-

dition had been investigated. Some conclusions obtained are as

follows:

(1) The parallel multichannels system has four unstable regions,

namely low andhigh equilibrium quality regions and inlet sub-

cooling regions under the rolling motion condition. Increasing

the length of upward section can weaken the system stability.

(2) The influence of heating power on thesystem instability is vari-

able. In the low equilibrium quality region, the system stability

is intensified with the increase of heating power. While in the

high equilibrium quality region, increasing the heating power

can weaken the system stability.

(3) The multiplied period phenomenon appears with the increase

of heating power. Nearby the MSB of high equilibrium quality

region, the chaos phenomenon emerges under some operatingconditions.

(4) Because of the rolling motion, the effect of pressure on the sys-

tem stability is not obvious. The MSBs shift to left with the

increase of pressure on the plane ofNpch and Nsub.

(5) The three-dimensional stability spaces of parallel multichan-

nels systems are obtained under the rolling motion condition.

(6) The system will become more unstable with the increase of

rolling amplitudes.

(7) The critical power and the system instability will appear ahead

of time since the rolling motion.

Acknowledgments

This work has been performed in the frameworkof the program

of the New Century Excellent Talents in University (NCET-06-0837)

andsupported by theNationalNatural Science Foundation of China

(10675096).

References

Clausse, A., Lahey, R.T., 1990. An investigation of periodic and strange attractors inboiling flows using chaos theory.In: Proceedings of theNinth International HeatTransfer Conference, vol. 2 , Jerusalem, pp. 38.

Chang, C.J., Lahey Jr., R.T., 1997. Analysis of chaotic instabilities in boiling systems.Nuclear Engineering and Design 167, 307334.

Durga Prasad, G.V., Pandey, M., Kalra, M.S., 2007. Review of research on flow insta-bilities in natural circulation boiling systems. Progress in Nuclear Energy 49,429451.

Durga Prasad, G.V., Pandey, M., 2008a. Stability analysis and nonlinear dynamics ofnatural circulation boiling water reactors. Nuclear Engineering and Design 238,229240.

Durga Prasad,G.V., et al.,2008b.Studyof flowinstabilities in double-channel naturalcirculation boiling systems. Nuclear Engineering and Design 238, 17501761.

Gao, P.Z., Pang, F.G., Wang, Z.X., 1997. Mathematical model of primary coolant innuclear power plant influenced by ocean conditions. Journal of Harbin Engi-neering University 18, 2629.

Gao, P.Z., Liu, S.L., Wang, Z.X., 1999. Effects of pitching and rolling upon naturalcirculation. Nuclear Power Engineering 20, 3639.

Gear, A.C., 1974. OrdinaryDifferential Equation System Solver. Lawrence LivermoreLaboratory, Report UCID-30001, Revision 3.

Ishida,T., Tomiai,I., 1992.Developmentof analysiscodefor thermalhydro-dynamicsof marine reactor under multi-dimensional ship motions. Retran-02/Grav,

JAERI-M, 91-226.Ishida, T., Yortisune, T., 2002. Effects of ship motions on natural circulation of deep

sea research reactor DRX. Nuclear Engineering and Design 215, 5167.Kakac, S., Bon, B., 2008. A review of two-phase flow dynamic instabilities in tube

boiling systems. International Journal of Heat and Mass Transfer 51, 399433.

7/29/2019 dwomode1

10/10


Kim, J.H., et al., 2001. Study on the natural circulation characteristics of the integraltypereactor forvertical andinclinedconditions.NuclearEngineeringand Design207, 2131.

Lahey Jr., R.T., Podowski, M.Z., 1989. On the analysis of various instabilities in two-phase flows. In: Hewitt, G.F., Delhaye, J.M., Zuber, N. (Eds.), Multiphase Scienceand Technology, vol. 4. Hemisphere, New York (Chapter 3).

Lee, J.D., Pan, C., 1999. Dynamics of multiple parallel boiling channel systems withforced flows. Nuclear Engineering and Design 192, 3144.

Lee, J.D., Pan, C., 2005a. Nonlinear analysis for a nuclear-coupled two-phase naturalcirculation loop. Nuclear Engineering and Design 235, 613626.

Lee,J.D., Pan,C., 2005b. Dynamic analysisof multiple nuclear-coupled boiling chan-

nelsbased on a multi-point reactor model. Nuclear Engineeringand Design 235,23582374.

Lin, Y.N., Pan, C., 1994. Non-linear analysis for a natural circulation boiling channel.Nuclear Engineering and Design 152, 349360.

Lin, Y.N., Lee, J.D., Pan, C., 1998. Nonlinear dynamics of a nuclear-coupled boilingchannel with forced flows. Nuclear Engineering and Design 179, 3149.

Murata, H., Sawada, K., Kobayashi, M., 2002. Natural circulation characteristics ofa marine reactor in rolling motion and heat transfer in the core. Nuclear Engi-neering and Design 215, 6985.

Pang,F.G., et al., 1995. Theoretical research foreffect of ocean conditions on naturalcirculation. Nuclear Power Engineering 16, 330335.

Satoh, A., Okamoto, K., Madarame, H., 1998. Instability of single phase natural cir-culation under double loop system. Chaos, Solitons & Fractals 9, 15751585.

Su, G.H., et al., 1998. An experimental investigation of density wave oscillation intwo phase natural circulation system. Chinese Journal of Nuclear Science andEngineering 18, 1924.

Su, G.H., et al., 2001. Theoretical study on density wave oscillation of two-phasenatural circulation under low quality conditions. Journal of Nuclear Science andTechnology 38, 607613.

Su,G.H., et al.,2002. Theoretical andexperimentalstudy on density wave oscillationof two-phasenaturalcirculationof lowequilibriumquality.NuclearEngineeringand Design 215, 187198.

Tadrist, L., 2007. Review on two-phase flow instabilities in narrow spaces. Interna-tional Journal of Heat and Fluid Flow 28, 5462.

Tan, S.C., Pang, F.G., 2005a. Overlapped flow of flow oscillation caused by rollingmotion and density waveoscillation of natural circulation. Nuclear Power Engi-neering 26, 140143.

Tan, S.C., et al., 2005b. Characteristics of single-phase natural circulation underrolling. Nuclear Power Engineering 26, 554558.

Tan, S.C., Pang, F.G., Gao, P.Z., 2006. Experimental research of effect of rolling uponheat transfer characteristic of natural circulation. Nuclear Power Engineering27, 3336.

Tan,S.C.,et al.,2007.Effectof rollingmotionon flowinstabilityof naturalcirculation.

Nuclear Power Engineering 28, 4245.Tan, S.C., et al., 2009. Experimental study on two-phase flow instability of natu-

ral circulation under rolling motion condition. Annals of Nuclear Energy 36,103113.

Yang, J., Jia, B.S., Yu, J.Y., 2002. Analysis of natural circulation ability in PWR coolantsystem under ocean condition.Chinese Journalof Nuclear Science andEngineer-ing 22, 199203.

Yun, G., et al., 2005. Two-phase instability analysis in natural circulation loops ofChina advanced research reactor. Annals of Nuclear Energy 32, 379397.

Yun, G., et al., 2007. Effect of inlet and riser sections on two-phase flow instabilityunder rolling. Nuclear Power Engineering 28, 5861.

Yun, G., et al., 2008a. Theoretical investigations on two-phase flow instability inparallel multichannel system. Annals of Nuclear Energy 35, 665676.

Yun, G., et al., 2008b. The influence of ocean conditions on two-phase flow insta-bility in a parallel multi-channel system. Annals of Nuclear Energy 35, 15981605.

Zhang, Y.J., et al.,2009. Theoretical researchon two-phase flowinstabilityin parallelchannels. Nuclear Engineering and Design 239, 12941303.

Zhou, Z.W.,1994. Stabilityanalysisof two-phase flow in a low quality natural circu-

lation loop with a lumped parameter model. Nuclear Power Engineering 15 (3),222229.

dwomode1

Documents