Nuclear Engineering and Design 190 (1999) 353–360

Technical note

An analytical investigation of role of expansion tank insemi-closed two-phase natural circulation loop

Sang Yong Lee *, Young Lyoul Kim 1

Department of Mechanical Engineering, Korea Ad6anced Institute of Science and Technology, 373-1, Kusong-Dong, Yusong-Gu,Taejon 305-701, South Korea

Received 9 November 1998; accepted 5 January 1999

Abstract

Role of the expansion tank in a semi-closed two-phase natural circulation loop was examined analytically with theemphasis placed on the flow instability. Loopwise steady circulation rate was obtained, and conditions for flowinstability were examined by using the method of the linear stability analysis with perturbations. The homogeneoustwo-phase model was adopted for the analysis. As well as the pressure at the expansion tank, the length and thecross-sectional area of the tube connected to the expansion tank appeared to be the important parameters determiningthe flow instability. The system was predicted to be stable with the longer length and the smaller cross-sectional areaof the expansion-tank line and also with the higher expansion-tank pressure. © 1999 Elsevier Science S.A. All rightsreserved.

Keywords: Two-phase natural circulation; Semi-closed loop; Expansion tank; Flow instabilities

1. Introduction

In natural circulation loop, the flow is inducedby the density difference of fluid between the riserand the downcomer sections. There is no mechan-ically driving part to induce the loopwise circula-

tion, and the only source of the driving force isthe heat input. Understanding of the two-phasenatural circulation behavior is very important inanalyzing the hypothetical loss of coolant accident(LOCA) in nuclear power plants. Also there aremany systems adopting the concept of two-phasenatural circulation, such as thermosyphon reboil-ers, waste heat recovery systems, solar water heat-ing systems, and geothermal power plants.

Fig. 1 shows the typical shape of the semi-closed loop. In this loop, the cross-sectional areaof the condenser section is equivalent to (or not

* Corresponding author. Tel.: +82-42-869-3026; fax: +82-42-869-5210.

E-mail address: e–hyunny@cais.kaist.ac.kr (S.Y. Lee)1 Present address: Korea Institute of Industrial Technology,

35-3, Hongchon-Ri, Ibjang-Myun, Chonan 330-820, Korea.

0029-5493/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved.

PII: S0029 -5493 (99 )00079 -5

S.Y. Lee, Y.L. Kim / Nuclear Engineering and Design 190 (1999) 353–360354

large enough compared to) that of the othersections. Thus, unlike the case with the openloop (Lee and Lee, 1991), the velocity and theenthalpy fluctuations at the heater exit propa-gate along the loop and directly affect the flowat the heater inlet. The time-averaged system-pressure remains constant since the excessamount of the fluid due to volume expansion byboiling is flowing out to the expansion tankwhere the constant-pressure head is maintained.However, the instantaneous system-pressure isnot being fixed because of the inertia (length) ofthe liquid column within the expansion-tank lineand the frictional resistance there. Moreover, thecondensation-induced instability may occur withthis type of the loop. Thus the flow behaviorbecomes more complicated than the open-loopcase.

The two-phase flow instabilities were well ex-plained in the excellent review paper on thetwo-phase instability by Boure et al. (1973). Ex-cept for the question on the suitability of thetwo-phase flow model adopted, most of the as-pects of the two-phase natural circulation in the

open loops have been unveiled such as byChexal and Bergles (1973), Fukuda and Kobori(1979), Lee and Lee (1991), Kyung and Lee(1994, 1996), and many others. Also, the flowcharacteristics in a semi-closed natural circula-tion loop have been examined either experimen-tally or analytically represented by Ramos et al.(1985), Chen and Chang (1988), Lee and Ishii(1990), Knaani and Zvirin (1993), and Tanimotoet al. (1998), etc. Despite those extensive works,to the authors’ knowledge, there has been noattempt to examine the role of the expansiontank in semi-closed two-phase natural circula-tion loop. For example, in the work of Lee andIshii (1990), the expansion tank was consideredto behave only as a mass sink/source to theloop maintaining the constant pressure. How-ever, the pressure fluctuations can occur due tothe flow resistance in the expansion-tank line.Therefore, in the present study, the role of theexpansion tank in the flow instability within thesemi-closed loop was examined conceptuallybased on the simplified analytical method.

2. Analytical approach

For the simplicity, it was assumed that theflow is homogeneous in the two-phase region.The purpose of the present study is to examinethe nature of the two-phase natural circulationin the semi-closed loop with an expansion tank.Hence, using of the simple, homogeneous modelin the two-phase region does not degrade themeaningfulness of the analyses, though the re-sult may not be accurate quantitatively. Exceptfor the modeling of the expansion tank, thegoverning equations and the constitutive rela-tions are basically the same with those by Leeand Ishii (1990). The simple one-dimensionalgoverning equations of mass, momentum, andenergy for each section shown in Fig. 1 can bewritten as follows:

2.1. Continuity equation

(r

(t+u(r

(z+r(u(z

=0 (1)Fig. 1. Semi-closed two-phase natural circulation loop.

S.Y. Lee, Y.L. Kim / Nuclear Engineering and Design 190 (1999) 353–360 355

2.2. Momentum equation

−(p(z

=r(u(t

+ru(u(z

+rg sin u+ f1

2Dlo

ru2

+%n

Kn

12

ru2d(z−zn) (2)

Here, the last term of Eq. (2) denotes thepressure losses across the restrictions (i.e. concen-trated losses) at the axial location zn.

2.3. Energy equation

Assuming the uniform heat flux at the heatersection, the energy equation becomes

(i(t

+u(i(z

=qh%%j

rA+

1r

(p(t

(3)

and, at the condenser section, Eq. (3) can be usedsimply by replacing qh¦ by −q c¦ which can be setto

q c%%=

d−ca−b

·qh%% (4)

At the adiabatic sections, the first term of theright-hand side of Eq. (3) does not appear. Eq. (4)assumes that the constant and uniform heat re-moval over the condenser section; by doing so,the time-averaged subcooling temperature at theadiabatic liquid region can be maintained con-stant though the instantaneous temperature mayfluctuate, and the analytical result can be exhib-ited in the form similar to the open-loop case (Leeand Lee, 1991). It should be noted that, for theopen loop case, modeling of the cooled region(condenser section) is not required, and also thetime derivative term of pressure in Eq. (3) doesnot appear.

2.4. Method of approach

In the present two-phase flow analysis, the den-sities of vapor and liquid were taken at the satura-tion condition corresponding to the time-averagedsystem pressure. However, in taking account ofthe pressure fluctuation effect, only the vaporphase was assumed to be compressible since thecompressibility of the liquid phase is much smallerthan that of the vapor phase.

Integrating the momentum equation (Eq. (2))along the loop, the following relation should besatisfied.7(p(z

dz=0 (5)

Stability of the flow can be judged either bysolving the unsteady equations directly and exam-ining the transient behavior, or by using the per-turbation technique with linearization. Use of thelatter method is the more effective and was basi-cally adopted in the present study. The derivationprocedure of this method, which is rather lengthy,is similar to that used in the previous work (Leeand Lee, 1991)), and will not be repeated here butbriefly explained.

Using the perturbation technique with lin-earization as

ui= ul+dul(t) (6)

dul(t)=o eSt (7)

where,

S=s+ jv (8)

j=−1 (9)

the left-hand side of Eq. (5) can be rewritten as7dpdz

dz=7dp

dzdz+

7d(dp)dz

dz. (10)

Here, the first term in the right-hand side de-notes the time-averaged pressure drop along theloop and always becomes zero. From this, thetime-averaged circulation rate can be readily ob-tained. The second term in the right-hand side ofEq. (10) implies the fluctuation of the loopwisepressure drop with the fluctuations of the velocity,vapor density (in the two-phase region) and theboiling and condensing boundaries as7d(dp)

dzdz=H1(S)·dul+H2(S)·drg+H3(S)·dLh

+H4(S)·dLc. (11)

From the behavior of the liquid within theexpansion-tank line, the density fluctuation of thevapor phase drg in Eq. (11) can be expressed bydul eventually. The terms dLh and dLc denote the

S.Y. Lee, Y.L. Kim / Nuclear Engineering and Design 190 (1999) 353–360356

fluctuations of the boiling and condensingboundaries, and also can be expressed by dul.Finally, Eq. (11) can be reduced to7d(dp)

dzdz=H(S)·dul. (12)

The functions H(S) and H1(S)�H4(S) arequite lengthy and the details are shown in thework of Kim (1997). It should be mentioned that,for the case of the open loop, drg and dLc of Eq.(11) do not appear from the beginning, and thefluctuation of the loopwise pressure drop is shownto be different; nevertheless, the final equationtakes the form of Eq. (12).

The perturbation equation (Eq. (12)) can berewritten as

dul=1

H(S)7d(dp)

dzdz (13)

which means the change of the flow rate inducedby the pressure drop disturbance imposed on thesystem. Therefore, the stability of the system canbe determined from the nature of the roots of thecharacteristic equation, H(S) as

H(S)=0 (14)

The velocity and pressure fluctuations withinthe loop are interrelated to the motion of theliquid column within the expansion-tank line.Therefore, the motion of the liquid within theexpansion-tank line of the semi-closed loopshould be modeled as follows. For the controlvolume shown with the dotted line in Fig. 1, thevelocity inside the expansion-tank line is ex-pressed as

dup

=D lo

2

Dp2 (ulc−ul)

=D lo

2

Dp2 (dulc−dul)

(15)

and the momentum balance becomes

rlLp

d(dup)dt

=dp−rlgDp

2

DET2

&dup dt

− fprl

Lp

Dp

(dup)2

2(16)

where dulc and dul are the perturbed liquid veloc-ities within the loop before and after the junctionto the expansion-tank line. The left-hand-sideterm of Eq. (16) implies the liquid inertia withinthe expansion-tank line, and the first term of theright-hand side of Eq. (16) is the excitation forceoriginated from the pressure fluctuations withinthe loop. The second term of the right-hand sideof Eq. (16) indicates the gravitational force by thechange of the liquid level in the expansion tank.However, DET is much larger than Dp in mostcases (i.e. the liquid level within the expansiontank remains almost stationary), and the secondterm of the right-hand side of Eq. (16) may beneglected. Therefore, the pressure inside the ex-pansion tank can be considered constant. Thethird term of the right-hand side of Eq. (16)denotes the pressure drop by the pipe-wall fric-tion. The liquid flow within the expansion-tankline is assumed to be laminar. Also the compress-ible volume (vapor) in the two-phase region of theloop may behave like an ideal gas. Then Eq. (16)becomes

rlLp

d(dup)dt

=P( sys

rg

·drg−32mtLp

Dp2 dup (18)

where, P( sys denotes the time-averaged system pres-sure during operation.

For more accurate analysis, the Clausius–Clapeyron equation should be adopted along withthe ideal gas equation to take account of thesystem pressure fluctuations. This, of course, addsthe terms of saturation temperature fluctuationthat complicates the analytical procedure consid-erably. However, the change of the saturationtemperature with the system pressure fluctuationsis relatively minor compared to the change of thevapor density. Therefore, the effect of the satura-tion temperature fluctuations was not consideredhere. In other words, the fluctuations of the ther-modynamic properties of the fluid, except for thevapor density, show only the secondary impor-tance and not considered in the present analysis.

As already noted, when boiling occurs, theproperties of the liquid phase within the loop weretaken at the saturation condition correspondingto the time-averaged system pressure. However,for the single-phase (liquid) natural circulation

S.Y. Lee, Y.L. Kim / Nuclear Engineering and Design 190 (1999) 353–360 357

Fig. 2. Variation of the liquid velocity in the semi-closed andthe open loops (water, DTsub=90°C, Ki=1000, K0=10).

Fig. 3. Effect of the length of the expansion-tank line on theflow instability (Ki=1000, K0=10, PET=0.1 MPa, Dp=0.008 m).

that occurs at the low heat-flux condition withinthe semi-closed loop, all the properties were takenat the time-averaged local temperature; otherwisethe single-phase natural circulation cannot bepredicted.

3. Results and discussions

Fig. 2 shows the variation of the time-averagedliquid velocity at the heater inlet with the changeof the heat flux when the inlet subcooling is high.The basic geometric dimensions of the loop forsample calculations are listed in Table 1. For thepresent case, water was chosen as the workingfluid. In the same figure, the result for the openloop with the same geometrical dimension exceptfor the condenser section is shown for reference.The flow rate first increases and then decreaseswith increasing of the heat flux. This is because, atthe low heat-flux range, the increase of the drivingforce predominates over the increase of the loop-

wise friction as the heat flux is increased. How-ever, at the high heat-flux range, the case becomesreversed.

Since the system is basically in a constant pres-sure condition, the amount of the fluid within theloop changes with the heater power input, and thecondensing boundary as well as the boilingboundary varies with the heater power input.Hence, unlike the case of the open loop, thedriving force changes with the heater power input,and this is one of the distinguishing features ofthe semi-closed loop from the open-loop case.Nevertheless, it can be realized that the shapes ofthe curves representing the time-averaged circula-tion rates are similar to the open-loop case exceptfor those in the low heat-flux range. In the lowheat-flux range (i.e. between points O and E), theentire space within the semi-closed loop is full ofthe liquid; therefore, the single-phase natural cir-culation is induced, which is proportional to theone third power of the heater power input asreported by Ishii and Fauske (1983). On the otherhand, for the open-loop case, the single-phasenatural circulation does not occur since the liquidlevel within the condenser section is always main-tained constant, and the fluid in the riser sectioncan be carried over through the top horizontalsection only by boiling. Point E denotes the onsetof the two-phase natural circulation. In reality,the two-phase natural circulation starts earlier

Table 1Basic dimensions of the loop for sample calculations

a 0.3 mh1.1 m0.5 m l 0.5 mb0.1 m Dlo 0.012 mc0.3 md Z 1.4 m0.2 me

S.Y. Lee, Y.L. Kim / Nuclear Engineering and Design 190 (1999) 353–360358

than at point E because of the subcooled boilingwhich is not modeled in the present study.

Fig. 3 shows an instability map in the plane ofheat flux versus inlet subcooling. Basically, thestable region appears at the lower left-hand cornerof the map, similar to the case of the open loop.(Lee and Lee, 1991) The stable region becomeswider at the lower inlet subcooling, but narrowsdown as the inlet subcooling increases, and such atrend is more pronounced with the shorter pipeline to the expansion tank. In general, the flowbecomes stabilized with the longer expansion-tankline. If the line to the expansion tank becomesvery long, the inertia of the liquid and also thefrictional pressure drop within it become verylarge and hardly react to the pressure fluctuationswithin the loop. The stability of the system ismaximized when the expansion-tank line becomesinfinitely long; thus the role of the expansion tankin this case is only to maintain the time-averagedsystem pressure to be constant but not to react tothe instantaneous pressure fluctuations. This canbe practically achieved by placing the flow resis-tance (i.e. valves and orifices) at the expansion-tank line. On the other hand, if theexpansion-tank line becomes shorter (i.e. with asmall inertia), the liquid within the line easilyreacts to the pressure fluctuations within the loop.Also, by the same reason, the stable operation

Fig. 5. Effect of the expansion-tank pressure on the flowinstability (Ki=1000, K0=10, Dp=0.008 m, Lp=200 m).

region decreases with the larger diameter of theexpansion-tank line as shown in Fig. 4. This isbecause, as shown in Eq. (16), the viscous damp-ing decreases with the larger diameter of theexpansion-tank line. The outside of the top-plateau of the stable boundary is subject to thepressure drop oscillation.

With the higher pressure at the expansion tank(i.e. with the higher system pressure), the systembecomes stabilized as shown in Fig. 5. This can beexplained as follows: The magnitude of the pres-sure-drop fluctuations within a two-phase naturalcirculation loop is basically limited by the hydro-static pressure difference between the top and thebottom of the loop. In this respect, the magnitudeof the pressure-drop fluctuations becomes rela-tively small with the higher system pressure im-posed by the expansion tank. Therefore, themotion of the liquid in the expansion tank and itsconnecting line to the loop is less susceptible tothe pressure-drop fluctuations within the loop.Accordingly, the system becomes stabilized withthe higher expansion-tank pressure.

The excursive instability regions are predictedto occur at the upper right-hand corners of Figs.3–5. As shown in Fig. 2, with the high inletsubcooling condition, there is a heat-flux rangewhere the excursive instability may occur (i.e. at acertain range of heat flux, more than one circula-tion rate exist for a given heat flux) similar to the

Fig. 4. Effect of the diameter of the expansion-tank line on theflow instability (Ki=1000, K0=10, PET=0.1 MPa, Lp=200 m).

S.Y. Lee, Y.L. Kim / Nuclear Engineering and Design 190 (1999) 353–360 359

case with the open loop. It should be noted thatthe boundary of the excursive instability is pre-dicted to be unchanged with the length and thediameter of the expansion-tank line since it isdetermined from the time-averaged equationswhere the pressure fluctuation terms disappeared.

4. Summary and conclusions

In the present study, the flow instabilities withinthe semi-closed loop were re-examined with theeffect of the expansion tank taken into account.The circulation rate and the flow instability condi-tions were predicted.

The change of the steady circulation rate withinthe semi-closed loop with the heat flux appears tobe similar to that within the open loop. The stableoperation region appeared at the lower left-handcorner of the instability map with the heat fluxand the inlet subcooling taken as its axes. Thestability of the system is very sensitive to thelength and the cross-sectional area of the expan-sion-tank line; the system becomes more unstablewith the shorter length and/or with the largercross-sectional area of the expansion-tank line.Also the system is stabilized with the higher sys-tem pressure.

Model improvements on the two-phase flowand the heat transfer boundary condition at thecondenser section, and the parametric studies forthe semi-closed loop are beyond the scope of thepresent paper and are left for the future works.Also the predicted results should be confirmedwith the experiments.

5. Nomenclature

A cross sectional arealengthsa, b, c, d

D inner diameterlengthe

f friction factorgravityg

H characteristic functionlengthh

i specific enthalpypressure loss coefficientK

l, L lengthspressurep

q %% heat fluxcomplex variableS

DT temperature differencetimet

u velocityloop heightZ

z axial coordinate

Greeksd perturbed termso amplitude of the perturbed

velocityangle of inclination from horizon-u

tal line (positive upward)position of the phase-changeLboundary

m viscosityj perimeter

densityr

s growth rate of amplitudeangular frequencyv

Subscriptscondenser sectionc

ET expansion tanksaturated vapor stateg

h heater sectioninleti

l liquid at the heater inletliquid at the condenser outletlc

lo loopintegern

p pipe-line to the expansion tanksubcoolingsub

sys system

Superscripts− time-averaged values

Acknowledgements

The authors would like to express their appreci-ation to the Center for Advanced Reactor Re-

S.Y. Lee, Y.L. Kim / Nuclear Engineering and Design 190 (1999) 353–360360

search (CARR) and the Korean Advanced Insti-tute of Science and Technology (KAIST) for thefinancial support on this subject.

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