Nuclear Engineering and Design 206 (2001) 91–96

Flow excursion instability in downward flow systemsPart I. Single-phase instability

Ibrahim Babelli a,*, Mamoru Ishii b

a King Abdulaziz City for Science and Technology, P.O. Box 6086, Riyadh-11442, Saudi Arabiab Purdue Uni6ersity, School of Nuclear Engineering, Purdue Uni6ersity, West Lafayette, IN 47907-1290, USA

Received 3 May 2000; received in revised form 31 October 2000; accepted 31 October 2000

Abstract

A procedure for predicting the onset of flow excursion instability in downward flows at low-pressure and low-flowconditions without boiling is presented. It is generally accepted that the onset of significant void in subcooled boilingprecedes, and is a precondition to, the occurrence of static flow instability. A detailed analysis of the pressure dropcomponents for a downward flow in a heated channel reveals the possibility of unstable transition from single-phaseflow to high-quality two-phase flow, i.e. flow excursion. Low flow rate and high subcooling are the two importantconditions for the occurrence of this type of instability. The unstable transition occurs when the resistance to thedownward flow caused by local (orifice), frictional, and thermal expansion pressure drops equalizes the driving forceof the gravitational pressure drop. The inclusion of the thermal expansion pressure drop is essential to account forthis type of transition. Experimental data have still to be produced to verify the prediction of the present analysis.© 2001 Elsevier Science B.V. All rights reserved.

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1. Introduction

A flow is subject to a static instability if, whenthe flow conditions change by a small step fromthe original steady state, another steady state isnot possible in the vicinity of the original state(Boure et al., 1973). The resultant state could beeither steady or periodic. The cause of the phe-nomenon lies in the steady-state laws; hence,the threshold of the instability can be predicted

only by using steady-state laws (Boure et al.,1973).

Ledinegg (1938) was the first to successfullyanalyze excursive instabilities. The flow excursion,also named the Ledinegg instability, involves asudden change in the flow rate to a lower value(Boure et al., 1973). Under certain conditions, thecurve of steady-state system pressure drop versusflow (internal characteristic of the heated channel)has a negative slope; hence, since the flow rate isnot a single-valued function of the pressure drop,a flow excursion may occur (Ishii, 1982).

If the slope of the external pressure drop-ver-sus-flow curve (pump characteristic) is greaterthan the slope of the internal pressure drop-ver-

* Corresponding author. Tel.: +966-1-4813637; fax: +966-1-4813887.

E-mail addresses: ibrahim@babelli.com (I. Babelli),ishii@ecn.purdue.edu (M. Ishii).

0029-5493/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.

PII: S0 029 -5493 (00 )00403 -9

I. Babelli, M. Ishii / Nuclear Engineering and Design 206 (2001) 91–9692

sus-flow curve, the system undergoes a static flowtransition, i.e. flow excursion. This condition isexpressed as:

(Dpext

(uB(Dpint

(u(1)

The steady state internal pressure drop is com-posed of the following components:

Dpint=Dporif+Dpc+Dpfr+Dpg (2)

where the components on the right-hand side ofEq. (2) are the local (orifice) pressure drop, theconvective pressure drop, the frictional pressuredrop, and the gravitational pressure drop,respectively.

The intersection of the external (supply) pres-sure drop curve and the internal (demand) pres-sure drop curve determine the operationalconditions of the system in question (Fig. 1).

Curves S1, S2, Sc, and S4 represent the demand(internal pressure drop) curves at different chan-nel power values. The curve A–A is the supply(external pressure drop) curve. The equilibriumflow rate is determined by the intersection of thesupply and demand curves. For a channel powerless than that corresponding to Sc, the operatingpoint is at L, giving flow rate M. Increasing thepower from zero to a value corresponding to Sc

has little effect on the channel flow rate. Thecurve Sc is tangential to the supply curve A–A.Any increase in power input above Sc results in asituation in which the only intersection of thedemand curve with the supply curve is at F. The‘operating point’ of the channel therefore movesalong A–A from L to F as the power is increasedslightly from its critical value, and the flow ratedrops abruptly from M to Mc (Whittle and For-gan, 1967). The onset of flow instability may bethe result of increased channel power or decreasedchannel flow rate.

Yadigaroglu (1999) presented a derivation ofthe necessary condition for the Ledinegg flowexcursion. Lahey and Yadigaroglu (1982) showedthat by examining the zero frequency limit of thelinear model used for the analysis of density waveoscillation, one is able to determine if the systemis stable with regard to excursive instabilities.

Flow excursion instability might lead toburnout of the heated surface due to the suddenevaporation of the cooling liquid. This type ofburnout takes place at heat flux values muchlower than enthalpy burnout values.

It was generally concluded that the onset ofsignificant void in the heated channel is sufficientto predict the onset of flow excursion instability.Babelli and Ishii (1998) showed that, for flow

Fig. 1. Supply (external) and demand (internal) pressure drop curves (Whittle and Forgan, 1967).

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excursion in two-phase flow, the onset of signifi-cant void precedes the onset of flow excursioninstability but does not necessarily coincide withit. It was further proven by Babelli and Ishii(1998) that the flow excursion instability is possi-ble, under low flow and high subcooling condi-tions, even without any boiling in the heatedchannel.

This paper is the first of two on flow excursioninstability in downward flow systems. The firstpaper details the analysis of the flow excursioninstability in single-phase systems, i.e. withoutboiling. The second paper presents the analysis ofthe flow excursion instability in two-phase flowsystems and develops a criterion for the predic-tion of the onset of flow instability in such sys-tems.

2. Theoretical background

The stable operating condition for flow in aheated channel is given by:

DPint=DPext (3)

Under certain conditions, the slope of the in-ternal pressure drop curve (plotted versus inletliquid velocity) becomes algebraically smaller thatof the external pressure drop curve. The externalpressure drop is provided either by a hydrostatichead in natural circulation or by a pump inforced flow. It suffices, however, to analyze theinternal pressure drop (demand) curve for itsminimum point in the process of predicting flowexcursion. The condition for the minimum pointis given by:

((DPint)(6in

=0 (4)

The internal pressure drop equation is the axialmomentum equation. The usual form of thisequation is obtained after averaging over the pipecross-sectional area and integrating over the pipelength. Consider a heated channel with a sub-cooled flow at the inlet and subject to a constant,parallel channel type pressure drop boundarycondition. The internal pressure drop equation

for the downward flow may be written as follows(Ishii and Fauske, 1983):

DPint=DPfr+DPorif+DPte+DPg (5)

Notice that this equation is different from Eq.(2). The convection pressure drop term disap-pears, since there is no phase change, and a newterm appears in Eq. (5). This new term is due tothermal expansion of the heated liquid. This termbecomes important at low flow rates and highsubcooling. And in downward flow of in theheated channel, it contributes to flow resistance.

The frictional pressure drop is given by:

DPfr=f

2Drf6 in

2 l (6)

The orifice (minor) pressure drop is given by:

DPorif=K2

rf6 in2 (7)

The thermal expansion pressure drop is givenby:

DPte=Drfgl

2(8)

Drf=bQ

Cp6inA

is the fluid density difference due to thermal ex-pansion (Ishii and Fauske, 1983). The change inthe fluid density is considered important only inthe gravitational term, which is commonlyknown as the Boussinesq approximation, (Incrop-era and De Witt, 1990). In contrast to downwardflow orientation, thermal expansion aids the flowin upward flow orientation. The heat added tothe flow, inlet subcooling, and the liquid flowrate are the parameters with the strongest influ-ence on this type of flow instability. The effect ofliquid subcooling is reflected in the liquid proper-ties.

The gravitational pressure drop is given by:

DPg= −rfgl (9)

Notice that gravity actually aids the flow indownward flow orientation, whereas it is a pres-sure drop component (against flow direction) inthe upward flow orientation.

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Fig. 2. Frictional, orifice and thermal expansion pressure dropcomponents.

Fig. 3. Gravitational head and total pressure drop.

resulting from frictional and local losses. This isthe point of minimum pressure drop shown inFig. 3.

Fig. 3 shows the calculated gravitational driv-ing head and the total internal pressure drop forthe Freon-113 test facility plotted against inletvelocity. The gravitational pressure drop compo-nent is not a function of inlet velocity. The inter-nal pressure drop curve clearly shows a minimum,which is a characteristic of static instability. Theminimum pressure drop point occurs always atlow inlet velocities for this type of flow instabili-ties. It is important to notice that the flow ratestypical of the instability reported in this paper areoutside the operational envelope of the Freon-113test facility (Babelli, 1996), and hence the lack ofexperimental findings.

3. Analysis and discussion

The first three terms on the right-hand side ofEq. (5) are functions of inlet velocity. Frictionaland orifice pressure drop are proportional to thesquare of the inlet velocity, whereas the thermalexpansion pressure drop is inversely proportionalto inlet velocity.

A theoretical parametric study for the stabilityof liquid flowing downward in a heated channelhas been performed on the Freon-113 test facilityreported by Babelli (1996). The effects of inletsubcooling, heater power, and test section size arediscussed below.

Fig. 2 shows the calculated frictional, orifice,and thermal expansion pressure drop componentsfor the Freon-113 test facility. If the inlet velocityto the heated channel decreases, the internal pres-sure drop across the channel caused by pipe fric-tion and orifices decreases. In contrast, theinternal pressure drop across the channel causedby thermal expansion increases with inlet velocitydecrease. This latter effect is due to the competingforces of momentum and buoyancy at low flowrates. By decreasing the inlet velocity to theheated test section, the pressure drop due to losses(friction and orifice) decrease rapidly (propor-tional to the square of the inlet velocity), whereasthe increase in the thermal expansion pressuredrop is slow. At lower values of the inlet velocity,the increase in the thermal expansion pressuredrop becomes large and matches the decrease Fig. 4. Effect of inlet subcooling on the total pressure drop.

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Fig. 5. Effect of heater power on the total pressure drop.

Fig. 6 shows the effect of increasing test sec-tion size (reported in terms of the hydraulic di-ameter of the annular test section of theFreon-113 test facility, namely, 6.35, 12.7, 19.05,and 25.4 mm). Increasing test section size (forthe same inlet velocity) is, in effect, increasingliquid flow rate, which makes the system morestable.

4. Conclusions

A theoretical parametric study for the stabilityof a liquid flowing downward in a heated chan-nel has been performed on the Freon-113 testfacility reported by Babelli (1996). The effects ofinlet subcooling, heater power, and test sectionsize have been reported.

Generally speaking, significant void generationhas been considered a prerequisite for flow ex-cursion. It was shown, theoretically, that a sys-tem of parallel channels where the floworientation is downward might undergo flow ex-cursion without boiling in the test section. Thistype of flow instability is important only at lowflow rates where the effect of buoyancy becomescomparable to the fluid momentum. The analy-sis of flow excursion in a downward flow systemshould, therefore, account for the effect of ther-mal expansion.

Experimental data are needed to verify the ex-istence of this type of instability and to establisha criterion for the prediction of its onset

Appendix A. Nomenclature

cross-sectional area (m2)AD Hydraulic diameter (m)Cp heat capacity (J kg−1 K−1)f friction factor

Gravitational acceleration (m s−2)gK local loss coefficient

test section length (m)lheater power (W)Qinlet velocity (m s−1)6inthermal expansion coefficient (K−1)b

liquid density (kg m−3)rf

DP pressure drop (Pa)

Fig. 4 shows the calculated total pressuredrop for the Freon-113 test facility for four dif-ferent values of inlet temperature, namely, 5, 20,30, and 40°C. This figure shows that by decreas-ing the liquid subcooling, the total pressuredrop decreases. The location of the minimumpressure drop is not affected by varying inletsubcooling (roughly around 0.15 m s−1 for theFreon-113 test facility).

Fig. 5 shows that by increasing heater power(250, 500, 750, and 1000 W), the system be-comes less stable and the location of the mini-mum pressure drop shifts towards higher valuesof liquid flow rates.

Fig. 6. Effect of test section hydraulic diameter on the totalpressure drop.

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Acknowledgements

Part of this research was supported by the U.S.Department of Energy, Office of Energy Re-search, Nuclear Engineering Research Program.The authors would like to express their apprecia-tion for this support.

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