instabilities in parallel channel

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Journal of NUCLEAR SCIENCE and TECHNOLOGY, 16[5], pp 343-355 (May 1979). 343

Instabilities in Parallel Channel of Forced-Convectionoiling Upflow System, (III)

System with Different Flow Conditions between Two Channels

asanori ARITOMI, Shigebumi AOKI and Akira INOUEesearch Laboratory for Nuclear Reactors,

okyo Institute of Technology*eceived October 20, 1978

evised February 16, 1979he instabilities observed in a parallel-channel system carrying boiling fluid in forcedupward flow have been studied theoretically and experimentally, using water as test fluid.he system, where the thermo-hydraulic flow conditions in two channels are differentfrom each other, is studied theoretically and experimentally, in order to make clear the

ffects of the different conditions on the flow instabilities. The different conditions be-tween parallel channels are made artificially by changing the entrance throttlings, theeater lengthes and the heat fluxes.onsequently, the instability in the system where the own characteristic frequencies

are approximatelly equal in two channels almost agree with the one obtained under theuniform condition equivalent to the average operating condition in two channels, suchas the different entrance throttlings and the different heater lengthes. On the otherhand, the system, where the characteristic frequencies differ from two channels, becomesmore stable with increasing the difference of flow condition, such as the different heatfluxes.

KEYWORDS: boiling flow instability, parallel-channel system, stable region,nstable boundary, different flow condition, two phase flow, flow oscillation, insta-

ility, stability, eigenfreguency, heat fluxI. INTRODUCTION

he thermo-hydraulic instabilities in parallel boiling channels have gained momentumin the last decade for their importance to the safety of fossil-fuel boilers, boiling waterreactors, as well as steam generators of fast breeder reactors and advanced thermaleactors(1)~(5). Especially, the development of LMFBR becomes the pressing problem forthe following energy source taking the place of oil energy, and the development of steam

generators for LMFBR becomes one of the most important problem.he plants comprising parallel boiling channels are multifarious, and their operating

conditions differ from each other, such as operating fluid, heating method, exit flowcondition and system pressure etc. It becomes hence difficult to apply a common approachto the understanding of the thermo-hydraulic instabilities in every systems. The presentwork attempts to classify and to simplify the phenomena and mechanisms of theirinstabilities to faciliate their understanding.

n companion paper(6)(7), the authors have been shown that the instabilities observedin boiling fluid in forced upward flow through twin parallel channels can be classified* Ookayama, Meguro-ku, Tokyo 152.

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into four types. The instability being occurred through the interaction between thechannels have been taken up in particular. A mathematical model well explained thisphenomenon was proposed, and the instabilities appeared in a system comprising morethan three channels showed a behavior quite simular to these in a two-channel system(6).The experimental results in two parallel channels with the uniform flow condition con-firmed the analytical ones.

n the plant with parallel boiling channels, the flow conditions between channels arenot exactly same, and it is not necessarily a good policy to design the plant with simu-lar channel condition. Studies on the instability under the different flow conditionsbetween channels are only found out in Veziroglus' work(8). In their paper, the abovementioned instabilities of four types were discussed using a common approach. Con-sequently, the mechanisms of these instabilities were not sufficiently understood, and theeffects of the difference of flow condition on the instabilities generated through theinteraction between parallel channels have not been made clear.

he purpose of this paper is to understand these effects. The instabilities underthe hydraulic or the thermal different condition are studied experimentally and theo-retically, as (1) the entrance throttlings differ between two channels and (2) the heaterlengthes or the heat fluxes differ.

I. EXPERITENTAL A PPARATUSn outline of the experimental apparatus is mentioned here, since the details were

reported in a companion paper(7).igure 1 shows the schamatic diagram of the forced-convection water loop operated

at atmospheric pressure. It has been deviced to suppress flow oscillations in the loopitself, so as to bring out only the instability caused by the interaction between parallelchannels.

Figure 2 is the geometry and the di-mension of two parallel channels betweenthe inlet and the outlet plenums. Eachchannel has a heater,electrodes and a ro-tary flowmeter formeasuring flow oscil-lation. The test sec-tion has been devicedto suppress the staticinstability, that is,the head flow oscilla-tion of Ledinegg'stype, so that only thedynamic instabilitycaused by the inter-action between chan-nels has been ob-served.

Fig. 1 Arrangement of flow loop Fig. 2 Test section

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An orifice (not shown in Fig. 2) was installed at the upstream of the flowineter inchannel (2) to observe the effects of the different entrance throttlings.

he heated tube is made of thin stainless steel. The heat fluxes in two channelsare identified, since the resistance per unit length is uniform and two heater are arrangedin a series. In the second condition, the heater in channel (2) is shortened as comparedwith the one in channel (1), so that the heat inputs differ between two channels. Theeffects of the different heater lengthes on the instability are observed.

urther, the variable resistence (8) and a magnetic switch (9) (shown in Fig. 1) areprovided in parallel to the heater of channel (2), in order to get the different heat fluxesbetween two channels. The effects of the different heat fluxes on the instability areobserved.

III I ANALYSIShe analytical model used in this paper is essentially the same as the one in a com-

panion paper(6). A revised point and its process are shown below.. Initial Conditionhe continuity and energy equations are solved in each channel and the momentum

equations are integrated with the proposal procedure. c114,',/dt is derived from

Two procedures are examined to get the initial conditions.(1) The characteristic curves of the static pressure drop Pin-Pout to the inlet velo-

city are different between two channels. Solving the equiliblium inlet velocity u(1',nd uin), where Pin Pout agrees in two channels through static analysis, the initialonditions are solved statically by substituting UinI), and Uin into the continuity andnergy equations in each channel. We selected -5, -2, -1, 1, 2 and 5 m/s2 asu2,) 0)1 ldd

(2) The initial conditions are obtained in the same way as the proposal one usinghe average inlet velocity air, as the initial inlet velocities in two channels. Theesulting velocities re;,) and u)' differ from each other under the different flow con-

ions. Making use of this process, du;,),0,1dt is assumed by substituting the initialonditions into Eq. (1),

From the analytical results of procedures (1) and (2), it was found that the insta-bilities are not dependent on the initial conditions and the magnitude of initial distur-bance like the results under the same parallel channel condition(6)(7).

ater initial conditions were adopted in this paper.. Phase Analysisrom Eq. (1),

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Expressing perturbation terms, Eq. ( 3 ) becomes

As the pressure drops from the inlet to the outlet in two channels are equal to thatfor the equilibrium inlet velocity C' and u12,) n the steady state

D,PT=DPTV). ( 5 )Equation ( 4 ) becomes

Substituting for the first term of Eq. ( 6 )

We obtain

The frequency response of Eq. ( 8 ) agrees with the one of the following equations :

because DPT(i-p/o) is occurred by it'/-;,), nd i;6';',' s shifted by 180- for up,,. The pulsatingflow with very small amplitude, where the linear is kept, is given z-i;,) n Eq. (14) ofRef. (6). The amplitude ratio and shift of DPT for m(d/dt)itl,' are calculated, and theformer is called by gain |1G(i)| and the latter the shift in phase tT). The followingequation agrees with the results of Eq.( 8 )

We call "characteristic frequency" fc that which tT of Eq. (10) is equal to 180- and"characteristic gain" |c| that which corresponds to the characteristic freque ncy.he stable boundary and the period of flow oscillation in the unstable region are

agree with the results obtained by the analytical model. That is exemplified in APPENDIX.IVEXPERIMENTAL RESULTS

. Effects of Different Entrance Throttlings between Two Channelsorifice was inserted at the upstream of flowmeter in channel (2) only, to observe

different entrance throttlings between two channels on the instability. Now, since thepressure drop of the section Lt(in Fig. 2) was proportional to the square of the inletflow rate, the proportional constant would represent the coefficient of entrance throttlepressure drop. The throttling coefficient in channel (1) was 110 kg,s2/m4, since a orificewas not inserted but the flowmeter and the electrode etc. throttled flow.

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The experimental and analytical resultsof the stable boundary are shown in Fig. 3.The quantity adopted for the abscissa isthe throttling coefficient in channel (2).The experimental results are agree withthe analytical ones. The results in thesystem with the identical entrance trottl-ing in two channels are reploted from acompanion paper(7) in the figure, that is,a dashed line (1) represents the results inthe system where the throttling coefficientsin two channels are equal and the valuesadopted for the abscissa and a dashed line(2) represents the one in the system wherethe throttling coefficients are equal and110kg,s2/m4 in two channels.

o compare with the result of Fig. 3,the average throttling coefficient in twochannels is introduced as

In this case, the stable boundary representsas a dashed line in Fig. 4. It becomesvery close to solid line which representsthe experimental results.

ith the view of gaining a better in-sight into the characteristics of flow oscil-lation, further runs were conducted in theunstable region indicated by the stabilitycurve. The flow oscillations were recorded

Fig. 3 Stable boundary in reference toifferent entrance throttlings

Fig. 4 Stable boundary in reference toifferent entrance throttlings

Fig. 5 Variation of nature of flowscillation in reference toifferent entrance throttlings

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by oscillograph and the averages of 20~30 cycles were retained as data.he typical experimental results are shown in Fig. 5 at C,P=110kg,s2/m4 and C7=

298 kg,s2/m4. The results(7) in the system with the same entrance throttling in twochannels are replotted. The amplitudes and periods of flow oscillations lie between theones at CR=CR'=110 kg,s2/m4 and =298 kg,2s/m4.

rom the results in Figs. 4 and 5, the instabilities in the system with the differententrance throttlings between two channels show a behavior simular to equal parallelchannels with the average throttling coefficient of two channels.

. Effects of Different Heater Lengthes between Two Channelshortening the heater in channel (2) only, the different input powers were artificially

created between two channels in this case.he stable boundaries obtained experimentally and analytically are shown in Fig. 6, in

which the average inlet velocity of both channels are used and LB)=0.4 m and LB)=0.3 mmean the heater lengthes of channels (1) and (2) respectively. Close agreement is ob-tained between the experimental results and the analytical ones. The stable boundariesobtained in the system with the same heater length are shown in the figure at LB=0.3 mand =0.4 m.

o consider the results shown in Fig. 6,the average heater length is introduced to

The analytical results in the system, wherethe heater lengthes of both channels areequal to 0.35 m, are shown in Fig. 7. Itshows good agreement with the results ofFig. 6.

uns, further, were conducted in theunstable region indicated by the stabilitycurve. The amplitudes and the periodsobserved in the region are shown in Fig.8 at LB=0.4 m and LB)=0.3 m as well asthe results in the system with the sameheater lengthes. The oscillation period wasalmost independent on the heater lengthin equal channel condition(7) The presentresults confirm it again in the differentlengths. The oscillation amplitude lies be-tween the ones in the system of LB=LB=0.3 m and 0 .4 m.

Both results of the different heater lengthes and the different entrance throttlingsshow the same trend to flow oscillation. Even if there are different flow conditionbetween two channels, the instabilities in the system where the own characteristicfrequencies of each channel are nearly equal, almost agree with the one in the systemwith the same flow condition equivalent to the average operating condition in twochannels.

Fig. 6 Stable boundary in referenceo different heater lengthes

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3. Effects of Different Heat Fluxes between Two Channelsy contacting a magnet switch (9) (shown in Fig. 1) arranged in parallel to the heaterin channel (2), the heat flux of channel (2) differed from the one of channel (1). Thedifference of thermal conditions was articially created between two channels. As param-eter expressing the magnitude of the different heat fluxes between two channels, a rateRq of heat flux was introduced, which is defined by

This value was varied with the variable resistence (8) (in Fig. 1).he stable boundary obtained experimentally and analytically are shown in Fig. 9.

The experimental results almost agree with the analytical ones. In this case, the aver-age heat flux at the stable boundary increases and the system becomes more stable withincreasing the ratio of heat flux. That fact may be very important and useful fordesign. The stable boundary in the condition in Table 1 was obtained analytically toascertain the aforementioned results and is shown in Fig. 10. The figure shows thesame tendency as the one in Fig. 9. The stabilization with the ratio of heat flux in-creases in this condition.

uns were conducted in the unstable region and the amplitudes and periods of flowoscillations were observed in various average heat fluxes and the ratios of heat flux.The typical experimental results of oscillation amplitude are shown in Fig. 11. It be-comes clear from the figure that the amplitude ratio decreases with the average heatflux and the ratio of heat flux. These are the same trends as the stable boundary.

Fig. 7 Stable boundary in referenceo different heater lengthes

Fig. 8 Variation of nature of flowscillation in reference toifferent heater lengthes

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All experimental results of oscillation period fall along a single line when arrangedusing the average time required for passage through preheating region in two channels,as shown in Fig. 12. The results induce an account that the time required for passagethrough preheating region affects mainly the characteristic frequency of the systemwith the different heat fluxes between both channels.

he times required for passage through preheating region in two channels differfrom each other in the cases of the different heat fluxes between two channels. Thisis significant difference from the afore-mentioned cases with other different conditions.

Fig. 9 Stable boundary in referenceo different heat fluxes

Fig. 10 Analytical results of stableoundary in reference toifferent heat fluxes

Fig. 11 Flow oscillation with changen ratio of heat flux Fig. 12 Period of flow oscillation in refer-nce to time required for passage

hrough predheating region

Table 1 Standard conditions of loop operation

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4. Transient Experimentased on the stability curve obtained in Fig. 9, two kinds of transient response

experiment as shown in Fig. 13 were examined by using magnet switch, in order tostudy the effects of the history of the inlet velocity on the instability. The condi-tion, when the heat fluxes are equal in two channels, is called "Condition A" and theone, when the heat fluxes differ between twochannels, is called "Condition B". The transientresponse experiments were carried out at Ft,,,=0.23 m/s, Rq=0.305 and the other conditions inTable 1. The following became clear throughtwo kinds of the transient experiment.

1) The figure of transient inlet velocity changeoes not reappear, owning to the phase oflow oscillation when step change added ando the heterogeneity of two-phase flow.

2) Transient time is within 5 s.3) The instability depends not on the history

f the inlet velocity but on the opperatingonditions, that is, the stability curve agreesell with the one in Fig. 9.

4) The amplitudes and periods of flow oscil-ation agree with the ones in Figs. 11 and 12

ithin the experimental errors after a shortnterval from the step change added.

The above-mentioned results coincide with the conclusion obtained in companionpapers(6)(7) and the instability in the system depends not on the disturbance and thehistory of the inlet velocity but only on the opperating conditions.

V. DISCUSSIONs a result of studying the effects of three kinds of different thermo-hydraulic con-

ditions between two channels on the instability, the generated phenomena are classifiedinto two types :

1) The generated instability is simular to the one in the system with the same con-itions equivalent to the average conditions in two channels.

2) The system becomes more stable as increasing the difference of the operatingonditions between two channels.

The system with the different entrance throttlings or with the different heater lengthescorresponds to type (1), and the system with the different heat fluxes does to type (2).

o study the system with the different heater lengthes concerning type (1) phaseanalysis was applied to two operating conditions in Table 2, situated in the unstableregion. The characteristic gain and frequency of the system and the shifts in phaseof each channel are shown in Table 3. The difference of flow distribution between twochannels are about 6%. The gain curves move parallel and the shift curves of phasealmost agree for this extent of flow distribution, as shown in Fig. 14 replotted from acompanion paper(7). The effects of the heater length on the instability show almostthe same trend as the ones of flow distribution, as had discussed in it. Consequenly,

Fig. 13 Operating condition ofransient experiment

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the difference of the shift in phase is slightand the characteristic gain of the systemalmost agree with the arithmetic sum ofthe gains of two channels.

or this reason, the instability in thesystem with the different entrance throt-tlings or with the different heater lengthesalmost agrees with the one in the systemwith the same condition equivalent to theaverage condition in two channels.

o study the system with the differentheat fluxes concerning type (2), phase an-alysis is applied to two operating condi-tions in Table 4 situated in the unstableregion. The characteristic gains and fre-quencies of the system and the shiftswere obtained in Table 5. The charac-teristic gain is 1.053 in case (i) and 1.138in case (ii). The system with the differentheat fluxes is more stable than the onewith the same heat fluxes.

The reasons for decreasing of the characteristic gain was investigated . Regardingthe characteristic frequency in the system, the following equation is established :

'" Isin .61)+1 Gm sin 71)=0 . (14)The gain and the own characteristic frequency in channel (2) with lower heat flux arelower than these in channel (1). The shift in phase increases sharply above 180- andthe gain decreases with increasing the frequency. Consequently,|1G(2)| becomes fairlylower than the own characteristic gain of channel (2). On the other hand, the shift inphase decreases slightly from 180- in channel (1), because of Eq. (14) and |1G("I>G(2)|.

|(1)| increases slightly in comparison with the own characteristic gain of channel (1).

Table 2 Operating conditions in referenceo different heater lengthes Table 3 Analytical results adoped for Table 2

Table 4 Operating conditions in referenceo differents heat fluxes Table 5 Analytical results adopted for Table 4

Fig. 14 Frequency response showny gain and shift in phase

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For these reason, the decrease in |G(2)| surpasses the increase in |G(1)| , and the systembecomes more stable.ince the following equation is established concerning the characteristic gain

cl-=2I G'" Icos z-'71'+ G(2' Icos r7(15)As increasing the difference of shift in phase between two channels, IGe I becomes lowerthan the arithmetic average of |G(1)| and |G(2)|, and the system becomes stable.

The gains |G(1)| shown on Table 5were obtained by using the average massof inertia term m (Eq. (9) ), whereas thegains |G'(i)| were obtained by using m(i)at fc, as shown in Table 6.

he arithmetic average of |G(i)| is1.073 and the one of |G'(i)|1.113. The dif-ference of boiling boundary in two channels causes the different masses of inertiaterm.

Both results are almost equal each other, and ZB becomes lower with increasing theheat flux.

rom the afore-mentioned discussion, it was found that the different characteristicfrequencies and the different masses of inertia term between two channels stabilize thesystem.

Table 6 Analytical results adopted for Table 4

VI. CONCLUSIONSlose agreement between the experimental results and the analytical ones is obtained

in regard to the stable boundary. The reliability of the analytical model and the gener-ality of the experimental results are confirmed.

he results through phase analysis agree with the analytical ones of proposal modelas for the stable boundary and the oscillation periods in the unstable region. It is madeclear that the phase analysis is applicable to the instability in the system with the dif-ferent thermo-hydraulic conditions between two channels.

n parallel-channel boiling system, a characteristic oscillation is independent of themagnitude and nature of initial disturbance and of the history of inlet velocity withoutreference to the existence of the different conditions between two channels.

n the different flow conditions where the own characteristic frequencies are appro-ximatelly equal in two channel, such as in the different entrance throttlings and heaterlengthes, the instability almost agrees with the one in the system with the same flowcondition equivalent to the average operating condition of two channels.

n the other hand, the system, where the flow conditions as well as the characteristicfrequencies and the masses of inertia term differ from two channels, becomes morestable with increasing the difference of flow condition, and the system is most unstablewhen the flow conditions are equal in two channels, such as the system with the dif-ferent heat fluxes. This conclusion may be interesting for designing the system to bestable, particularly possessing two parallel channels.

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[NOMENCLATURE]R: Throttling coefficient (kgs2/m4) q": Heat flux (kcal/m2h): Acceleration due to gravity (m/s2) u : Velocity (m/s)|| : Gain ZE: Boiling boundary (m)|c|: Characteristic gain gl: Density of liquid (kg/m3)B: Heater length (m) tT Shift in phase: Mass of inertia term (kgs2/m3), ee Eq.14) in Ref. (6) Subscripts: Pressure (kg/m') in: Inlet, out: OutletDT : Total pressure drop (kg/m2), see Eq. (14)n Ref.(6)

- REFERENCES -(1) STENNING, A.H.: Trans. ASME, Ser. D, 86, 213~217 (1964).(2) MEYER, J.E., ROSE, R.P.: ibid., Ser. C, 85, 1~9 (1963).(3) DAVIES, A.L., POTTER, R.: EURATOM Rep., Proc. Symp. on Two-Phase Flow Dynamics, Eind-

oven, p. 1255~1266 (1967).(4) D'ARCY, D.F.: ibid., p. 1173~1223.(5) CROWLEY, J.D., et al,: ibid., p. 1132~1171.(6) ARITONII, M., et al.: J. Nucl. Sci. Technol., 14(1), 22~30 (1977).(7) ARITOMI, M., et al.: ibid., 14(2), 88-96 (1977).(8) STENNING, A.H., VEZIROGLC, T.N. : ASME Pap. 64 WA/FE-28, (1964).

APPENDIX]he results obtained through the phase analysis are compared with the ones obtained

through the analytical model, in order to examine the reliability of phase analysis.he characteristic gain of the system corresponds to the characteristic frequency,

which is shifted by 180- through phase analysis, and were obtained using the averageheat flux in two channels as parameter. The results are shown in Fig. A1 as comparedwith the ones obtained through the analytical model. The characteristic gain increases

with increasing the average heat flux, andthe average heat flux, where the charac-teristic gain becomes unity, is 0.34x106

Fig. A1 Characteristic gain and frequencyn reference to heat flux

Fig. A2 Comparison between phasenalysis and mathematicalodel for values of oscil-

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kcal/m2,h. On the other hand, the results obtained through the analytical model shownin Fig. 10 indicate that the average heat flux above 0.35 x 10' kcal/m2,h is lain in theunstable region and the one under 0.325 x10' kcal/m2 h in the stable region. The stableboundary obtained through the phase analysis agrees thus with the one obtained throughthe analytical model.

he oscillation periods obtained through the analytical model are compared withthe reciprocals of the characteristic frequency based on phase analysis. Figure A2shows both results and they are agree within 2%.

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instabilities in parallel channel

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