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Theoretical investigations on two-phase flow instability in parallel

channels under axial non-uniform heating

Xiaodong Lu a, Yingwei Wu a,, Linglan Zhou b, Wenxi Tian a, Guanghui Su a, Suizheng Qiu a, Hong Zhang b

a School of Nuclear Science and Technology, Xian Jiaotong University, Xian, Shaanxi 710049, PR Chinab Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China, Chengdu, Sichuan 610041, PR China

a r t i c l e i n f o

Article history:

Received 2 April 2013

Received in revised form 16 July 2013

Accepted 19 July 2013

Keywords:

Two-phase flow instability

Parallel channels

Axial uniform heating

Three-dimensional instability space

a b s t r a c t

Two-phase flow instability in parallel channels heated by axial non-uniform heat flux has been theoret-

ically studied in this paper. The system control equations of parallel channels were established based on

the homogeneous flow model in two-phase region. Semi-implicit finite-difference scheme and staggered

mesh method were used to discretize the equations, and the difference equations were solved by chasing

method. Cosine, bottom-peaked and top-peaked heat fluxes were used to study the influence of non-uni-

form heating on two-phase flow instability of the parallel channels system. The marginal stability bound-

aries (MSB) of parallel channels and three-dimensional instability spaces (or instability reefs) under

different heat flux conditions have been obtained. Compared with axial uniform heating, axial non-uni-

form heating will affect the system stability. Cosine and bottom-peaked heat fluxes can destabilize the

system stability in high inlet subcooling region, while the opposite effect can be found in low inlet sub-

cooling region. However, top-peaked heat flux can enhance the system stability in the whole region. In

addition, for cosine heat flux, increasing the systempressure or inlet resistance coefficient can strengthen

the system stability, and increasing the heating power will destabilize the system stability. The influence

of inlet subcooling number on the system stability is multi-valued under cosine heat flux.

2013 Elsevier Ltd. All rights reserved.

1. Introduction

The phenomenon of two-phase flow instability has been ob-

served in many industrial domains like refrigeration systems,

steam generators, boiling water reactors and reboilers. It has very

adverse influences on thermalhydraulic system, since the oscilla-

tions of the mass flow rate and system pressure induced by two-

phase flow instability can cause structural vibrations of compo-

nents, problems of system control, transient burn-out of the heat

transfer surface and degradation of the heat transfer performance.

It is obvious that the flow instabilities must be avoided and there

should be an adequate margin to ensure the system stability. Flow

instability plays an important role in water-cooled and water-

moderated nuclear reactors. Therefore, predicting the thresholds

of flow instabilities is an important work in the design and opera-

tion of nuclear reactors. In the past few decades, a considerable

amount of numerical and experimental investigations on the

two-phase flow instability have been carried out all over the world.

After the two-phase flow instability was introduced by Ledin-

egg (1938), plenty of subsequent researches (Boure et al., 1973; La-

hey, 1980; Su et al., 2002; Papini et al., 2012) on the two-phase

flow instability in heating channel system have been conducted.

In recent years, the two-phase flow instability in parallel channels

has attracted extensive attention since it is particularly difficult to

be detected. In parallel channels system, an interaction between

the channels can be established due to common boundary condi-

tions. It is well known that the density wave oscillation (DWO)

in parallel channels occurs when the slope of the system pres-

sure-drop versus flow rate curve is positive. When one channel is

disturbed, the inlet velocity of this channel is reduced which

resulting in a decrease of the pressure-drop in this channel. After

a time t, which is the time taken by a particle to reach the outlet

of the channel, the inlet velocity will increase because of the con-

stant pressure-drop boundary. An increased inlet velocity in turn

causes the residence time of the particle to go up and a lesser pres-

sure-drop. When the particle reaches the outlet of the channel, a

decrease in inlet velocity will be caused and this starts the cycle

again. At the same time, an opposite behavior can be observed in

another channel for common boundary conditions. Finally, the

oscillation of the mass flow between parallel channels is triggered,

while the total mass flow of the system remains constant. There

are two general approaches to analyze the two-phase flow instabil-

ity: frequency domain analysis method and time domain analysis

method. For the frequency domain (Lahey and Moody, 1977;

Fukuda, 1979), the system stability is evaluated with classic

control-theory techniques in which the transfer functions are

0306-4549/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.anucene.2013.07.030

Corresponding author.

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

Annals of Nuclear Energy 63 (2014) 7582

Contents lists available at ScienceDirect

Annals of Nuclear Energy

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 / a n u c e n e
http://dx.doi.org/10.1016/j.anucene.2013.07.030mailto:[email protected]://dx.doi.org/10.1016/j.anucene.2013.07.030http://www.sciencedirect.com/science/journal/03064549http://www.elsevier.com/locate/anucenehttp://www.elsevier.com/locate/anucenehttp://www.sciencedirect.com/science/journal/03064549http://dx.doi.org/10.1016/j.anucene.2013.07.030mailto:[email protected]://dx.doi.org/10.1016/j.anucene.2013.07.030http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://crossmark.dyndns.org/dialog/?doi=10.1016/j.anucene.2013.07.030&domain=pdfhttp://-/?-

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obtained from linearization and Laplace-transformation of the gov-

erning equations. However, some nonlinear problems cant besolved by the frequency domain analysis method, since it omits

some nonlinear information. The two-phase flow instability in par-

allel channels is a nonlinear problem. Hence, the models built in

time domain are applied to analyze the two-phase flow instability.

0D analysis models (Munoz-Cobo et al., 2002; Schlichting et al.,

2010) based on the analytical integration of conservation equa-

tions in the computing region have been built. In addition, more

complex but accurate 1D analysis methods have been developed

by some researches (Lee and Pan, 1999; Guo et al., 2008b; Zhang

et al., 2009) to study the stability of multiple-parallels system

using suited numerical solution techniques (finite differences, fi-

nite volumes or finite elements).

In addition, modern methods of nonlinear dynamics were

developed by Dokhane et al. (2005, 2007) and Rizwan-Uddin(2006) to investigate the stability analysis of boiling water reactors

(BWRs). In their studies, a reduced order model in conjunction

with the bifurcation code BIFDD was used to perform the stability

and semi-analytical bifurcation analyses of BWRs. Lange et al.

(2011) have made great achievements and they developed a

RAMROM method to study the nonlinear stability analysis of

BWRs, where RAM is a synonym for system code and ROM stands

for a reduced order model.

Unfortunately, most of them mentioned above have made a

hypothesis that the axial heat flux profile on the parallel channels

is uniform. In fact, the axial heat flux of fuel channels in the reactor

is non-uniform. Hence, it is unsuitable to use the uniform heat flux

to analyze the system stability in the reactor cores. Some experi-

mental and numerical works have been carried out on two-phaseflow instability under cosine heat flux. Djikam and Sluiter (1971)

found that cosine heat flux could stabilize the flow, while Bergles

(1976) pointed out that cosine heat flux had a destabilizing effect.

Dutta and Doshi (2008) have studied the effect of the axial heat

profile on different BWRs and found that a sinusoidal axial heat

profile enlarged the stability region. Contradictory results have

been obtained by these reports. Therefore, it is necessary to go fur-

ther on the study of this problem. In this paper, semi-implicit fi-

nite-difference scheme and staggered mesh method were

adopted to analyze the influence of non-uniform heating on two-

phase flow instability in parallel channels. Different axial heat flux

profiles such as cosine and bottom-peaked heat fluxes have been

studied. The marginal stability boundary (MSB) and the three-

dimensional instability space have been obtained under differentoperation conditions.

2. Theoretical model and numerical method

For our studies, the parallel channels system consists of two

plenums and two parallel channels as shown in Fig. 1. The two-

phase flow instability in parallel channels will be disturbed by

the riser section and inlet section of the channel (Guo et al.,

2008a). In order to study the effect of axial non-uniform heating

on two-phase flow instability alone, the riser and inlet sections

are neglected in this paper. The heating section is composed of

two parts which are single-phase section and two-phase section,

respectively. The assumptions made in this study are as follows:

(1) The homogeneous flow model is used for two-phase flow

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

(3) The two phases are in thermodynamic equilibrium.

(4) One-dimensional conservation equations in the axial (z)direction are used.

(5) Only bulk boiling is considered and subcooled boiling is

neglected.

Nomenclature

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

B matrixDe equivalent diameter (m)f fluid or friction pressure drop coefficient

g vapor or gravitational acceleration (m/s2)h enthalpy (kJ/kg)k loss coefficientNpch phase change number, Npch Q=Wvfg=hfgvfNsub inlet subcooling number, Nsub Dhin=hfgvfg=vfp pressure (Pa)Q heating power (W)t time (s)u velocity (m/s)W mass flow rate (kg/s)

z axial coordinate

Greek symbolsq density (kg/m3)q

tp

mixture density of two-phase fluidl dynamic viscosity (Pas)/2 two-phase multiplier coefficientD difference

Subscriptsi size class1u single-phase region2u two-phase region

Heating sectionLH

Single phase zone

Two phase zone

LN

Lower Plenum

Upper Plenum

Fig. 1. Schematic of parallel channels system.

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2.1. Theoretical model

The thermal hydraulic model is established based on a set of

field and constitutive equations and boundary conditions. By

adopting a homogeneous flow model, the basic one-dimensional

conservation equations for both single- and two-phase regions

and the state equation can be written as follows:

Mass conservation equation:

@q@t

@qu

@z 0 1

Momentum conservation equation:

@qu@t

@qu2

@z

@p

@z

f

DeXNi1

kidz zi

" #qu2

2 qg 2

Energy conservation equation:

@qh@t

@quh

@z

qlA

@p

@t3

State equation:

q qp; h 4

where f is friction pressure drop coefficient, De is equivalent diam-

eter (m), ki is loss coefficient, d is Diracs delta function, ql is linear

heating power (W/m), A is cross-sectional area of the control vol-

ume (m2).

The derivative of the state equation with respect to time t can

be written in the form:

@q@t

@q@h

@h

@t

@q@p

@p

@t5

Substituting Eq. (5) into Eqs. (1) and (3), one can obtain:

@q@h

@h

@t

@q@p

@p

@t

@

@zqu 0 6

h@q@h

q

@h

@t h

@q@p

1

@p

@t

@

@zquh

qlA

7

In order to solve the momentum conservation equation, the

friction pressure drop coefficient f consisting of the single-phase

friction pressure drop coefficient f1u and the two-phase friction

pressure drop coefficient f2u and two-phase multiplier coefficient

/2 should be calculated by appropriate empirical equations. In this

study, the f1u in the single-phase region can be obtained from Ta-

ble 1, and the f2u in two-phase region is defined as:

f2u f1u qtpqf

/2 8

MaAdams model is used to calculate the two-phase multiplier

coefficient /2 which is

/2 1 xqfqg

1

" #1 x

lflg

1

" #0:259

where qtp is the mixture density of the two-phase fluid, qf is thedensity of fluid (kg/m3), qg is the density of vapor (kg/m

3), lf isthe dynamic viscosity of fluid andlg is the dynamic viscosity of va-por (Pas).

2.2. Numerical method

In the present study, the convective terms of all conservation

equations are discretized by a first-order upwind difference

scheme. The semi-implicit finite-difference scheme and the stag-

gered mesh method have been used to ensure the stability of the

computation method and improve the computational efficiency.

And the characteristic of the staggered mesh method is that the

momentum control volumes are centered at the boundaries of

other control volumes. The spatial discretization is illustrated on

Fig. 2 for the flow in a constant cross-sectional area duct. The scalar

variables (p, h, q) are evaluated at the centers of control volumes,while the velocity vector u is located at the boundary between

adjoining control volumes.

The difference equations from Eqs. (6) and (7) at cell i can be

written as follows:

@q@h

ni

hn1i h

ni

Dt

@q@p

ni

pn1i pni

Dtqni1=2u

n1i1=2

qni1=2un1i1=2

Dz 0

10

hni @q@h

ni

qni !

hn1i h

ni

Dt hni @q@p

ni

1 !

pn

1

i pniDt

qni1=2h

ni1=2u

n1i1=2

qni1=2hni1=2u

n1i1=2

Dz

qnlA

11

The difference of momentum conservation equation at junction

i 1=2 can be expressed as:

qni1=2un1i1=2

qni1=2uni1=2

Dtqni1u

ni1

2 qni u

ni

2

Dz

pn1i1 p

n1i

Dz

f

2Deqni1=2u

ni1=2

2 qni1=2g 0

12

where the density of mixture qi1=2 is equal to the density of up-ward flow.

Eqs. (10) and (11) can be rewritten in the following form:

Ah

n1i h

ni

pn1i pni

b f1u

n1i1=2 f2u

n1i1=2 13

where

A h

ni

@q@h

ni

qni

=Dt h

ni

@q@p

n

i 1

=Dt

@q@h

ni=Dt @q

@p

ni=Dt

264

375

b qlA

0 ; f1 qn

i1=2h

ni1=2

Dz

qni1=2

Dz

0@

1A; f2

qni1=2

hni1=2

Dz

qni1=2

Dz

0@

1A

Eqs. (12) and (13) can be obtained in every node of the parallel

channels and two plenums. The difference equations of the parallel

Table 1

Correlations of friction pressure drop coefficient for single-phase.

Regions Correlations

Laminar Re < 1000 Darcy

Transition 1000 < Re < 2000 Linear interpolation

Turbulence Re > 2000 BlasiusFig. 2. Difference equation discretization schematic.

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channels and plenums can be solved directly to avoid the calcula-

tion of the flow distribution in different channels.

The momentum equations couple the velocities to the pressure

field and are applied at junctions i 1=2 and i 1=2 for the control

volume i. The velocities at junctions i 1=2 and i 1=2 are onlythe

functions of the pressures at the n 1 time step level, which can be

derived from Eq. (12) as follows:

un1i1=2 ~uni1=2

Dtqni1=2Dz

pn1i pni

pn1i1 p

ni1

14

un1i1=2 ~uni1=2

Dt

qni1=2Dzpn1i1 p

ni1

pn1i p

ni

15

where

~uni1=2 uni1=2

Dt

Dz

qni1uni1

2 qni uni

2

qni1=2

Dt

qni1=2Dzpni1 p

ni

Dt

qni1=2Dpf

n

i1=2 qni1=2g

~uni1=2 uni1=2

DtDz

qni uni 2 qni1uni12qni1=2

Dtqni1=2Dz

pni pni1

Dt

qni1=2Dpf

n

i1=2 qni1=2g

Supposing the inverse matrix of A is written as:

A1 B

B11 B12

B21 B22

!16

Eq. (13) is premultiplied by the inverse matrix of A, as shown

below:

hn1i h

ni

p

n1

i p

n

i A

1b A1f1u

n1i1=2 A

1f2u

n1i1=2 17

Substituting Eqs. (14) and (15) in to Eq. (13), a pressure field

equation can be obtained as follows:

Ci1dpi1 C

i2dpi C

i3dpi1 C

i4

18

where dpi pn1i p

ni , C

i1, C

i2, C

i3 and C

i4 calculated with variables at

the n time step are known in this equation.

The channel is divided into Nnodes. As an equation like Eq. (18)

can be obtained in every node, there are Nequations in every chan-

nel constructing a tridiagonal matrix calculated by chasing method

to get the value of pressure in every node at n 1 time step. Then

other variables can be acquired from Eqs. (4) and (10).

3. Validation

3.1. Qualification of nodalization

It is noted that a qualified model may predict unrealistic results

when the nodalization is not properly qualified. Hence, before

applying the present model to study the two-phase flow instability

of parallel channels, the number of nodes in each heated channel

needs to be evaluated by a comparison of MSBs obtained by this

model. The MSBs with different node numbers are shown in

Fig. 3. It is clear that the MSB is shifted to an asymptotically stable

line on the left with the increase of node number. Thus we can

regard this stable line as the correct solution. However, it will take

more computing time when the number of the node is too large.

Therefore, a node number of 40 is chosen as a compromise be-tween accuracy and efficiency.

3.2. Verification of the model

A comparison of the calculation data obtained from the model

evaluation with the experimental data (Lu et al., 2011) is shown

in Fig. 4. The system pressures are 6 and 10 MPa, respectively.

The mass velocity is 200800 kg/(m2 s). The length of heated sec-

tions is 1.0 m and the cross section of channel is 25 mm 2 mm.

The errors between the calculation data and the experiment data

are within 15%, which shows that the calculated results agree

well with the experiment data and the accuracy of the present

model is verified. Furthermore, Fig. 4 illustrates that the present

model is better for high pressure system since the model is a

homogenous flow model. Previous studies (such as Guo et al.,

2008a; Xia et al., 2012) and the results in the present work indicate

that the homogeneous flow model can accurately predict the MSBwith reasonable agreement with experiment data.

4. Results and discussion

In this research, the detailed information of the parallel chan-

nels system can be seen from Table 2. A plenty of calculations have

been carried out in this parallel channels system. Three kinds of ax-

ial heat flux profiles including cosine heat flux, bottom-peaked

Fig. 3. Comparison of MSBs under different node numbers (7 MPa).

Fig. 4. Comparison of experiment data and calculation data.


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heat flux and top-peaked heat flux (similar to Leung (2008)) were

used to analyze the effect of non-uniform heating on the system

stability. In addition, uniform heat flux was also studied as the ref-

erence case as shown in Fig. 5. A two-dimensional plane consisting

of phase change number Npch and subcooling number Nsub was

used to represent the stability map of the system (Ishii and Zuber,

1970). The phase change number takes into account the power to

flow ratio and heat of vaporization, and hence indicates the phase

change due to heat addition. The subcooling number indicates the

degree of subcooling at the inlet of heater. The two nondimen-

sional numbers are defined as below:

Npch Q

W

vfg

hfgvf19

Nsub Dhinhfg

vfg

vf

20

where Qis heating power (W), Wis mass flow rate (kg/s), vfg is dif-

ference in specific volume of saturated liquid and vapor (m3/kg), vfis specific volume of saturated liquid (m3/kg), hfg is latent heat of

evaporation (kJ/kg), Dhin is the subcooling (kJ/kg).

4.1. The effect of axial non-uniform heating

As shown in Fig. 5, four different axial heat flux profiles were

used to analyze the stability of the parallel channels system, under

which the MSBs of the system are presented in Fig. 6. It is obvious

that the effect of the four axial heat fluxes on the systemstability isparticularly different. A detailed description and explanation will

be given in the following part.

In Fig. 6, two crossing points (point D and E) can be found be-

tween the MSB line for uniform heat flux and that for bottom-

peaked heat flux or cosine heat flux. We define that the inlet

subcooling region above point D or point E is high inlet subcooling

region, and the region below the two points is low inlet subcooling

region. For bottom-peaked heat flux, the MSB line is on the left side

of that for uniform heat flux in high inlet subcooling region which

suggests that bottom-peaked heat flux can destabilize the stability

Table 2

Detailed parameters of parallel channels.

Parameter Value

Heating length (m) 1.0

Diameter of channel (m) 0.012

System pressure (MPa) 715

Total mass flow rate (kg/s) 0.2

Fig. 5. Different heat flux profiles.

Fig. 6. The MSBs of system under different heat fluxes (7 MPa).

Fig. 7. The single-phase pressure drop ratios at different inlet temperature (7 MPa).

Fig. 8. The MSBs of system.

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of the parallel channels systemin this region. However, in lowinlet

subcooling region, bottom-peaked heat flux has a stabilizing effect

on the systemand its MSBis shifted to the right side of the MSB for

uniform heat flux. In addition, the turning point of the MSB for bot-

tom-peaked heat flux is higher than that for uniform heat flux.

For cosine heat flux, a similar conclusion can be obtained as fol-

lows: the system under cosine heat flux will be destabilized in high

inlet subcooling region and strengthened in low inlet subcooling

region. Whats more, the MSB line for cosine heat flux is on the

right side of that for bottom-peaked heat flux, which suggests that

the system under cosine heat flux is more stable than that under

bottom-peaked heat flux.

For top-peaked heat flux, the MSB line is shifted to the right side

of that for uniform heat flux. Thus the stability of the systemunder

top-peaked heat flux is enhanced in the whole region. Therefore,

the system under top-peaked heat flux is the most stable system.

In order to explain this phenomenon, the boiling boundary and

the corresponding pressure drop distribution were studied in this

paper. The boiling boundary position is related to the inlet subco-

oling and the axial heat flux profile. In high inlet subcooling region,

the boiling boundary takes place in the top half of the channel.

However, in lowinlet subcooling region, the boiling boundary is lo-

cated in the bottom half of the channel. In addition, the boiling

boundary positions for the four axial heat fluxes are also different,

resulting in different effects on the system stability. For example,

in low inlet subcooling region, the boiling boundary for uniform

heat flux occurs relatively earlier than cosine heat flux in the bot-

tom half of the channel, causing larger two-phase friction length,

which means that a uniform heat flux is more destabilizing than

the cosine heat flux (Dutta and Doshi, 2008). In this study, the dif-

ference of the boiling boundary positions for different heat fluxes is

described by a single-phase pressure drop ratio which was defined

as the ratio of single-phase pressure drop to two-phase pressure

drop. Lower single-phase pressure drop ratio means larger

two-phase zone length. It is also well known that the single-phase

pressure drop plays a significant role on the system stability.

Therefore, the single-phase pressure drop ratio was used to com-pare the stability of the systems under bottom-peaked, cosine,

top-peaked and uniform heat fluxes. When the heating power is

equal to that of point A, B and C on the MSB line for uniform heat

flux, the single-phase pressure drop ratios at the points on the line

1, 2 and 3 in Fig. 6 are carried out as shown in Fig. 7. It can be seen

that the system stability is more stable when the single-phase

pressure drop ratio is larger. Hence, it can be concluded that the

single-phase pressure drop can stabilize the system.

4.2. Influences of various factors on system instability under cosine

heat flux

In the reactor thermalhydraulic and safety analysis, cosine

heat flux is always selected as the heating power distribution on

heated channels. Therefore, the system stability under cosine heat

flux has been drawn more attention in this paper. The influences of

various factors (such as system pressure, heating power, inlet

subcooling number, inlet resistance coefficient) on system instabil-

ity under cosine heat flux have been studied.

4.2.1. The influence of system pressure

In order to study the influence of systempressure on the system

stability under cosine heat flux, the MSBs of different system

Fig. 9. The instability space (uniform heat flux).

Fig. 10. The instability space (cosine heat flux).

Fig. 11. The MSB of system under cosine heat flux (11 MPa).


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pressures (i.e. 7 MPa, 11 MPa and 15 MPa) have been obtained as

shown in Fig. 8. The influence of cosine heat flux on the system

stability under 11 MPa and 15 MPa are the same as that under

7 MPa mentioned above. However, the influence of system pres-

sure cant be described clearly in the two-dimensional graph.

Guo et al. (2008a) proposed a concept of instability space (or insta-

bility reef) to solve this problem. In this study, the three-dimen-

sional space was used to evaluate the system instability withcosine and uniform heat fluxes and under different system pres-

sures. The three-dimensional space consists of phase change num-

ber (Npch), inlet subcooling number (Nsub) and nondimensionalpressure (p p=10patmosphere). The boundaries under different sys-

tem pressures from 7 MPa to 15 MPa with different heat flux pro-

files have been calculated. Three-dimensional instability spaces

under uniform heat flux and cosine heat flux are drawn in Fig. 9

and Fig. 10.

In Figs. 9 and 10, the instability space decreases with the in-

crease of system pressure, which demonstrates that the stability

of parallel channels under cosine heat flux increases with the in-

crease of the system pressure as well as that under uniform heat

flux. The shape of the instability space is just like a reef in the

whole two-phase zone. Hence, the instability space can be called

instability reef.

4.2.2. The influence of heating power

The influence of heating power on the system instability under

cosine heat flux was analyzed. The MSB for cosine heat flux is pre-

sented in Fig. 11, which illustrates that when the inlet subcooling

number is constant, the system goes across the MSB from the sta-

ble region to the unstable region as the heating power increases. It

is obvious that the system will be less stable with higher heating

power. When the system changes from stable state to the unstable

condition, the process is generally in the following order: damped

oscillation (Fig. 12(A)), periodic oscillation (Fig. 12(B)) and diver-

gent oscillation (Fig. 12(C)), as presented in Fig. 12.

4.2.3. The influence of inlet subcooling number

As shown in Fig. 11, inlet subcooling number has two kinds ofopposite effects on the system stability when the heating power

Fig. 12. Inlet mass flow rate curve (p = 11MPa, Tin = 180 C). (A) Q= 186.00 kW. (B) Q= 187.64 kW. (C) Q= 189.00 kW.

Fig. 13. Effect of inlet resistance coefficient on the system stability (cosine heat

flux).

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is constant. The system moves from stable state to unstable state

with the increase of inlet subcooling number in low inlet subcool-

ing region while the system changes from unstable state to stable

condition when inlet subcooling number increases in high inlet

subcooling region.

4.2.4. The influence of inlet resistance coefficient

For uniform heat flux, much research has been carried out about

the influence of inlet resistance coefficient on two-phase flow

instability. However, there has been few research on this subject

under cosine heat flux. In this study, the two-phase flow instability

with different inlet resistance coefficients for cosine heat flux has

been studied as shown in Fig. 13. The lines of MSB are gradually

shifted to the right side with the increase of inlet resistance coeffi-

cient, which indicates that the system stability can be strength-

ened by inlet resistance coefficient. Inlet resistance coefficient

promotes the single-phase pressure drop which can enhance the

system stability. In addition, the turning point of the MSB is mov-

ing to higher inlet subcooling number with the increase of inlet

resistance coefficient.

5. Conclusions

In this paper, the two-phase flow instability in parallel channels

under axial non-uniform heating has been investigated. The sys-

tem stabilities under uniform, bottom-peaked, cosine and top-

peaked heat fluxes were compared with each other. Some detailed

results under cosine heat flux have been obtained. Main conclu-

sions obtained are summarized as follows:

(1) Compared with uniform heat flux, cosine heat flux and bot-

tom-peaked heat flux have two special effects on the system

stability. In high inlet subcooling region, the system stability

is destabilized by the two heat fluxes. However, the system

stability is stabilized in low inlet subcooling region. Top-

peaked heat flux can strengthen the stability of parallel

channels system in the whole region.

(2) The system stability under top-peaked heat flux is more sta-

ble than that under cosine and bottom-peaked heat fluxes,

and the system under bottom-peaked heat flux is the most

unstable system for the three different axial non-uniform

heating.

(3) Under cosine heat flux, the increase of system pressure and

inlet resistance coefficient can stabilize the system. The

system stability will be destabilized with the increase of

heating power. The influence of inlet subcooling number

on the system stability is not single-valued under cosine

heat flux. The system stability will be stabilized with the

increase of inlet subcooling in high inlet subcooling region,

while the opposite effect can be found in low inlet subcool-

ing region.

Acknowledgements

The authors would like to thank Science and Technology on

Reactor System Design Technology Laboratory of Nuclear Power

Institute of China (KZZJJ-A-201101) and National Magnetic Con-

finement Fusion Science Program (2010GB111007) for financial

support of the present work.

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