ORIGINAL

Francisco J. Collado Æ Carlos Monne Æ Antonio Pascau

Changes of enthalpy slope in subcooled flow boiling

Received: 24 February 2005 / Accepted: 25 February 2005 / Published online: 16 December 2005� Springer-Verlag 2005

Abstract Void fraction data in subcooled flow boiling ofwater at low pressure measured by General Electric inthe 1960s are analyzed following the classical model ofGriffith et al. (in Proceedings of ASME-AIChE heattransfer conference, #58-HT-19, 1958). In addition, anew proposal for analyzing one-dimensional steady flowboiling is used. This is based on the physical fact that ifthe two phases have different velocities, they cannotcover the same distance—the control volume length—inthe same time. So a slight modification of the heat bal-ance is suggested, i.e., the explicit inclusion of the vapor–liquid velocity ratio or slip ratio as scaling time factorbetween the phases, which is successfully checkedagainst the data. Finally, the prediction of void fractionusing correlations of the net rate of change of vaporenthalpy in the fully developed regime of subcooled flowboiling is explored.

1 Introduction

Subcooled flow boiling, which takes place when a sub-cooled liquid enters into a heated channel with the walltemperature exceeding the saturation temperature in acertain amount, has been extensively studied with ref-erence to power and process industries (Griffith et al.1958; Zuber and Findlay 1965; Staub et al. 1969; Ishii

1977; Bergles et al. 1981; Collier 1981). In this kind ofboiling flow, the saturated vapor bubbles formed at theheated wall can steadily coexist with the subcooled bulkliquid in thermal nonequilibrium. The vapor volumefraction or void fraction, here denoted by e, is defined asthe fraction of vapor volume at any point of the channel.The accurate prediction of the void fraction axial profile,which logically will depend on inlet flow condi-tions—velocity, pressure, and subcooling—and on theapplied heat flux, has clear implications on the reactivityand stability of nuclear reactors as well as on pressuredrop and critical heat flux estimations in a large amountof thermal processes.

Till now, many empirical correlations of the cross-sectional averaged void fraction have been published,see for example a recent and extensive review of Codd-ington and Macian (2002), most of them based on thedrift-flux model of Zuber and Findlay (1965) and Ishii(1977). Yet, as those authors recognize, due to the lackof theoretical knowledge and/or the complexity of thephysical processes involved, such correlations usuallysubstitute physical fundamentals by several empiricallyderived constants.

However, it is necessary to highlight the extremedifficulty of measuring the local temperature of thesubcooled liquid with accuracy in thermal nonequilib-rium with vapor bubbles. So the most important phe-nomenological models for the subcooled flow boilinghave been recently analyzed by Bartel et al. (2001)showing that only three researchers had made localmeasurements (transversal profiles) in the subcooledregion until now. Perhaps, among the most accuratelocal measurements ever taken in the subcooled flow arethose of General Electric (GE) in the 1960s (Staub et al.1969) under task I of an experimental program con-ducted for the joint US–Euratom Research and Devel-opment Program. This set of measurements, taken forlow-pressure water, will center the analysis in this workwith the final objective of exploring the feasibility ofaccurate predictions of the axial void fraction profile inthis complex flow.

F. J. Collado (&) Æ C. MonneDepartamento de Ingenierıa Mecanica-Motores Termicos,Universidad de Zaragoza-CPS, Maria de Luna 3,50018 Zaragoza, SpainE-mail: fjk@unizar.esTel.: +34-976-762551Fax: +34-976-762616

A. PascauDepartamento de Ciencia de los Materiales y Fluidos-Mecanicade Fluidos, Universidad de Zaragoza-CPS, Zaragoza, Spain

Heat Mass Transfer (2006) 42: 437–448DOI 10.1007/s00231-005-0653-6

In this work, first with the help of the phenomeno-logical and analytical model developed by Griffith et al.(1958), we will review the general behavior of the sub-cooled flow boiling tests taken by GE (1969). We willlook for identifying the two regions proposed in thatmodel: a first part with negligible rate of increase inenthalpy flux in the vapor and with the channel surfaceonly partially covered by vapor bubbles, and a secondregion in which the heated channel wall is already totallycovered by several layers of bubbles and there is anappreciable rate of increase in enthalpy flux in the vapor.This second region is called the ‘‘fully developed’’ regimeof subcooled flow boiling. As will be shown later, theability to calculate the void fraction correctly in the firstregion and where is the transition from the first to thesecond region, i.e., the critical point, are essential for anyaccurate prediction of the void fraction profile.

The calculation of the void fraction in the first regionand its extent will be based on the hydrodynamic con-siderations proposed by Griffith et al. (1958) whereas forthe second region, we will use the classical heat balancewith a slight modification, namely the explicit inclusionof the vapor–liquid velocity ratio or slip ratio, whichappears dividing to the linear heat flux per unit inletmass. To justify this, a change of the classic heat balanceis necessary to take into account that we are treating aone-dimensional two-phase flow in the same space—thecontrol volume length. Then if the phases have differentvelocities, it is impossible that they cover the same dis-tance in the same time. We will show that the need oftime scaling between the phases arises in a natural wayfollowing the classical definitions of the thermodynamicproperties of the mixture, the slip factor being the timescale factor between the phases. To advance that theaccurate measurements of GE would confirm this newpoint of view.

Finally, under the limited amount of data available,some attempts are made to predict the void fraction infunction of some new correlations of the net rate ofchange of vapor enthalpy in the fully developed regionof the subcooled flow.

2 Griffith et al. (1958) model of subcooled flow firstregion

2.1 Calculation of the void fraction in the first region

For the sake of convenience, a brief review of this modelis commented here. The objective of the experimentalprogram of these authors from the M.I.T—with water at3.5, 6.9, and 10.3 MPa, inlet velocities of 6.1 and 9.1 m/s, inlet subcooling from 5 to 83.3�C and a heat flux rangeof 0.8–8.5 MW/m2—was to determine the void volumein a subcooled boiling system. Basically, the procedureused was to fix the heat flux, velocity, and pressure onthe surface of the one-heated-wall channel in boiling andphotograph it. The test section was vertically oriented

and the flow was up. The bubbles on the photographwere measured, counted and then the void volume wascalculated.

For given value of pressure, velocity, and heat flux,there was a certain value of the bulk temperature atwhich boiling begun. This value depended on the filmcoefficient at the surface. The equation which gave val-ues of incipient boiling heat flux closest to those actuallyobserved was

Nuz ¼ 0:036Re0:8z Pr1=3: ð1Þ

It appeared that the appropriate length to use in thelocal Nusselt (Nu) and Reynolds (Re) numbers was thelength from entrance to the area in question (z coordi-nate), as the area of interest was so close to the entrance.

This photographic study showed that in the first re-gion the vapor on the surface is in the form of a numberof small bubbles, which do not penetrate far into thesubcooled liquid flowing over the surface. In this region,it is also apparent that the local heat flux is larger, but ofthe same order of magnitude, as the heat flux that wouldexist if no boiling were present for the same conditions.It was also found that the total heat flux applied couldbe divided into a nonboiling (subindex nb) and a boiling(subindex b) heat flux:

q ¼ qnb þ qb ¼ hz TS � TLð Þ þ qb: ð2ÞThe nonboiling heat flux was obtained from the localfilm convective coefficient, hz, from Eq. 1. The physicaljustification of this separation was indicated by thephotographs, in which it was apparent that part of theheated surface was bare. In Eq. 2, these authors rec-ommended using as surface temperature, Ts, the satu-ration temperature, Tsat, rather than the true walltemperature, Tw, as, in general, the true wall tempera-ture is not known with any precision and is only slightlyhigher than the saturation one.

We can say that in the GE tests, the two possibletemperatures have been checked. Of course, this adds tothe model the difficulty of predicting the average walltemperature.

To relate void fraction with the boiling heat flux, theauthors assumed that, in the first region, this boilingheat flux is nearly equal to the condensing heat flux, qc,which is a function of the area fraction covered bybubbles

qb � qc ¼ BohzAc

ATS � TLð Þ: ð3Þ

in which Ac/A is the condensing area per unit area due tothe bubbles and Bo is a dimensionless constant. Clearly,Ac/A is a function of the vapor volume on the surfacedepending on the geometrical configuration (Bo).

Pictures of the bubbles on the surface showed that thevapor was in the form of strands of bubbles. Thenthe condensing heat flux would be proportional to thestrand dimensions, closely related with the bubblediameter.

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For the conditions of the investigation, the authorssuggested that the limiting mechanism of bubble size wasa fluid dynamical process, which might cause the re-moval of the bubble from the surface rather than a heattransfer process. So they relate the velocity boundarylayer height with the bubble size attained. The velocitylayer thickness is approximately proportional to thethermal layer thickness times the Prandtl number, so

height / khzPr : ð4Þ

The void volume per unit area, a, is proportional to theheight times width times the length of strand per unitarea or,

a / length

AkhzPr

� �2

: ð5Þ

The condensing area would be proportional to

Ac

A/ k

hzPr

� �length

A

� �; ð6Þ

with Eq. 5, putting the former condensing area fractionin function of a,

Ac

A/ a

hz

kPr

� �: ð7Þ

Substituting Eq. 7 in Eq. 3, the void volume per unitarea is

qb � qc ¼Boah2 TS � TLð Þ

kPr) a ¼ qbkPr

1:07h2z TS � TLð Þ ;

ð8Þwhere the constant Bo was evaluated from the measureddata, the best value being 1.07.

The void fraction, e, for this first region will be thevoid volume per unit area, a, divided by the channelspacing, s, for one heated wall (half-channel spacing fortwo heated walls)

e ¼ as¼ qbkPr

1:07h2z TS � TLð Þs : ð9Þ

2.2 Extent of the first region

Griffith et al. (1958) also suggested a procedure todetermine the extent of the first region, i.e., the distancefrom the entry where the ‘‘fully developed boiling’’ be-gins. Based on the examination of experimental data ofdifferent sources, they proposed that when the boilingheat flux is approximately five times the forced convectiveheat flux, the ln(q) versus ln(DT) no longer changes slope,assuming that this is also the region in which the surface isvirtually covered with bubbles. Then when the surface isat the transition point, the liquid temperature will reach acritical value, Tct, which can be derived from Eq. 2

qb � 4qnb ) Tct ¼ TS �q5hz

)DTct ¼ Tct � TL zð Þ:

ð10Þ

Finally, the beginning of subcooled flow boiling (the firstregion beginning) could also be worked out from Eq. 2when the boiling heat flux is zero

Tbegin ¼ TS �qhz: ð11Þ

3 Thermodynamic properties and heat balancein subcooled flow boiling

3.1 Thermodynamic properties and classicalheat balance

Previously, to the analysis of the GE data, we will brieflyestablish the thermodynamic properties used. First, thevapor mass quality, x, is strictly defined as:

x ¼ dmvapor

dm¼ qGe

qm

: ð12Þ

So, it cannot take negative values at all. qG is the vapordensity function of the pressure, p, at the position z fromthe inlet. And qm is the vapor–liquid mixture densitywhich is defined as:

qm ¼ qGeþ ð1� eÞqL; ð13Þin which qL is the subcooled liquid density, qL=qL(p, TL),with TL cross-sectional average of the subcooled liquidtemperature at a distance z from the entrance.

The inverse of the mixture density is the mixturespecific volume, vm:

vm ¼ x vG þ ð1� xÞ vL; ð14Þwith vG and vL the specific volume of the saturated vaporand the subcooled liquid, respectively.

For the sake of convenience, we will solve the vaporvoid fraction in Eq. 12, e, in function of the vapor massquality, x:

e ¼ x vGvm¼ x vG pð Þ

x vG pð Þ þ ð1� xÞ vL p; TLð Þ : ð15Þ

As the vapor bubbles and the subcooled liquid are inthermal nonequilibrium, we need three independentvariables to solve for void fraction, namely, mass qual-ity, pressure (or inlet pressure if the pressure drop is nothigh) and subcooled liquid temperature.

The mixture enthalpy, hm, will be a combination ofthe subcooled liquid enthalpy, hL(p, TL) and of that ofsaturated vapor, hG(p)

hm ¼ xhG þ ð1� xÞhL: ð16ÞFinally, to comment that the classical heat balance forthe subcooled flow boiling, neglecting potential and

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kinetic terms, and the net generation of vapor (Collier1981), would be

hLðzÞ ¼ hLi þqPmLi

z ¼ hLi þ q0z ) dhL

dz¼ q0; ð17Þ

where q is the uniform heat flux applied (kW/m2), P isthe heated perimeter of the channel, mLi is the inlet massflow rate of liquid (kg/s), and q¢ is the linear heat per unitinlet mass (kJ/m kg). Evidently, under these assump-tions, the slope of the liquid enthalpy should be equal tothe linear heat q¢. Of course, if we had also included thevapor enthalpy in the classical heat balance the liquidenthalpy slope should be lower than the linear heat q¢.

3.2 Modification of the heat balance

In a previous and rather preliminar work by Collado(2000), working with the same GE–Task I da-ta—although using a rounded hydraulic diameter of0.5 in. instead of the actual one (0.5454 in.)—showedthat the slope of the above defined mixture enthalpy wasgreater than the specific linear heat q¢. So, it was alreadysuggested to introduce the slip ratio in the heat balanceas a scale time factor between the phases.

The justification was already mentioned in Sect. 1: ifthe phases have different velocities and we treat themin the same space, it is impossible that the two phasescover the same distance in the same time. Based on theclassical expressions of the mass flow rate, it is easy toshow that the time scale factor between the phasesshould be the vapor–liquid velocity ratio or slip ratio, S.

Defining as usual the vapor mass flow rate, mG, andthe liquid mass flow rate, mL:

mG ¼ qG e uG Ac ð18ÞmL ¼ qL 1� eð Þ uL Ac; ð19Þwhere uG and uL are the vapor and liquid velocities,respectively, and Ac the cross-sectional area of the duct.To relate the above mass flow rates with the thermo-dynamic (actual) vapor mass content, Eq. 12, we mul-tiply this mass quality by uGAc/uGAc

x ¼ qG euGAc

qG euG Ac þ qL 1� eð ÞuG Ac

¼ mG

mG þ uG=uLð ÞqL 1� eð ÞuLAc¼ mG

mG þ S mL; ð20Þ

where S is the slip ratio defined as

S ¼ uGuL: ð21Þ

Although Eq. 20 is not new at all, it would suggest theneed of scaling the time-dependent variables of onephase—in this case mass flow rate—before combiningthem with that of the other phase. The time scale factorwould be the slip ratio, S.

Hence, it seems reasonable to think that a physicalfact as the slip between the phases should have some

implication on the heat balance. The suggested energybalance, neglecting kinetic and gravity terms, assumingonly liquid water at the inlet with a mass flow rate ofmLi, and exclusively using thermodynamic propertieswould be:

q00P ¼ d mLiSð Þhm½ �dz

: ð22Þ

The key modification is that the inlet mass flow rate ofwater appears multiplied by the slip ratio to convert thetime scale of the water to the time scale of the vapor.We are assuming that heat enters into our controlvolume exclusively through vapor bubbles, which con-dense in the bulk subcooled fluid. This would becoherent with the formerly commented model ofGriffith et al. (1958) if the wall was completely coveredwith bubble layers.

Assuming that S is constant along the channel, wehave

q00PmLiS

¼ q0

S¼ dhm

dz¼ d xhGð Þ

dzþ d 1� xð ÞhL½ �

dz¼ q0

SGþ q0

SL;

ð23Þwhere SG and SL have been defined by conveniencefor the analysis of GE data. Of course, the interest willbe to distinguish how the absorbed heat is shared be-tween the net growing rate of vapor-first addend, andbasically the liquid heating-second one. Moreover, SG

opens the possibility of correlating the mass vaporcontent closely connected to the void fraction throughEq. 15.

SG will be obtained from the comparison of the slopeof xhG (in the fully developed region) with the linearspecific heat applied in

d xhGð Þdz

¼ q0

SG) SG ¼

d xhGð Þ=dzq0

: ð24Þ

In the first region we will use the Griffith et al. (1958)model in combination with the liquid heating suppliedby SL.

In conclusion, for the second region we suggest, asan approximation, a linear growing of the vaporcontent. It will start in the transition or critical pointwith a value defined from the void fraction value in thispoint:

xhGð Þfd ¼q0

SGz� zctð Þ þ xhGð Þct: ð25Þ

4 Task I, GE data (1969)

4.1 Conditions of the Task I (tests without additives)

Under Task I of an experimental program conducted byGE for the join US–Euratom Research and Develop-ment Program (Staub et al. 1969), the measurement of

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the initiation and early development of the subcooledvoid profile was carried out, including all required pro-file measurements and observations, with low-pressurewater (0.12–0.31 MPa). A vertical, one-side uniformilyheated, ten-to-one aspect ratio, rectangular cross sectionof Dh=0.01385 m (0.5454 in.) was employed to permitthe accurate measurement of transverse and axial voidfraction profiles (with an accuracy of ±3% voids), andliquid-phase temperature profiles (accuracy ±0.1�C).The test section was 0.0762 m·0.00762 m·0.3429 m(3 in.·0.3 in.·13.5 in.). It was used to satisfy a maxi-mum heat flux capability of 3.15 MW/m2 (106 Btu/h ft2), and a maximum mass velocity of 3,391 kg/s m2

(2.5·106 lb/h ft2).In Table 13 of the final report, Staub et al. (1969)

gave the following general data for each run of task I:inlet conditions, uniform heat flux applied, averageheater surface temperature and the average pressuregradient in the first and the second half of the channel.Also for each run, the cross-sectional averages of thesaturation temperature (so pressure), subcooling (so li-quid temperature), and vapor void fraction along theheight of the channel—normally four to six posi-tions—were supplied. These cross-sectional averages ofvoid fraction and liquid temperature were worked out bythe authors through numerical integration of the mea-sured transversal profiles at each cross section, whichwere also presented in the report. The liquid temperaturetransversal profile normally included ten points, whereasfor the vapor void fraction about seven transversalmeasurements were taken. Both were nonsymmetric dueto the one-side heating.

After a preliminary analysis, it was decided to sepa-rate the tests in which water reached saturation or wasvery close to it (see Table 1; Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9,10, 11) from the tests in which water at the outlet was farenough from it (Table 2; Figs. 12, 13, 14, 15, 16, 17, 18,19, 20, 21, 22, 23).

4.2 First region

Figure 1a graphically shows the Griffith et al. (1958)procedure for defining the transition point, i.e., the pointof vapor net generation (PVNG). As we have com-mented, the two possible temperatures, wall temperatureand inlet saturation temperature, have been checked.Then it is also logic to use these two temperatures in thecalculation of the void fraction in the first region, ei, seeFig. 2b. We first localized, by visual inspection of thevoid fraction profile, some point that marked a clearchange of slope. Then we checked on the temperaturegraphs, such as Figs. 1a or 2a, which of the two tem-perature options best positioned that point. In Tables 1and 2, the last column indicates what temperature hasgiven the best approximation to that point, and the DTct

column gives the separation of this point from thecrossing of the liquid and surface temperatures, i.e., theseparation from the model, see Fig. 2a.

Figure 1d shows a typical void fraction profile whereit is not possible to detect a clear change of behavior. Weassume in such cases that the ‘‘fully developed’’ subco-oling region is established practically from the beginning(in Tables 1 and 2, z=0). Hence, in Fig. 1a, the Tct

function of Tsati would give the best approximation tothe void fraction profile behavior.

In general, and given the extreme complexity of thiskind of flow, the Griffith et al. (1958) model works quitewell (Tables 1 and 2) becoming an essential tool of theanalysis. It is important to notice that the great geo-metrical similarity (rectangular cross section, upwardsflow and one-side-heated channel) between GE experi-mental rig and that of MIT one would favor the wellbehavior of the model.

In general, if the wall temperature marks the criticalpoint, the first region void fraction calculated withsuch temperature is which best fits the experimentaldata, see for example Figs. 2a, b and 3a, b. And, in

Table 1 Subcooled tests near or reaching saturation in Task 1 (Staub et al. 1969)

No. oftests

uLi(m/s)

pi(bar)

DTsubi

(�C)q(kW/m2)

q¢(kJ/m-kg)

Tsati

(�C)Twall

(�C)DTct

(�C)DTsub0(�C)

SL SG eo eo-cala zct

36 0.18 1.18 19.3 331.1 242.0 104.3 121.1 �1.8 0.0 0.91 13.3 0.595 – Twall

55 0.36 1.19 11.3 753.7 276.5 104.5 130.6 z=0 0.01 0.99 12 0.822 0.78 Tsati

66 0.36 1.20 11.4 728.5 264.1 104.8 136.1 z=0 0.01 0.96 12 0.863 0.77 Tsati

70 0.36 1.14 9.4 375.3 137.7 103.4 127.2 z=0 0.002 1.03 12 0.632 0.65 Tsati

63 0.36 1.14 9.3 372.1 136.5 103.3 128.3 z=0 0.002 1.04 12 0.667 0.64 Tsati

29 0.36 1.12 9.1 372.1 137.0 102.9 117.2 �1.2 0.0 0.98 12 0.688 0.65 Twall

48 0.36 1.13 6.0 167.1 60.8 103.1 116.1 - 0.5 0.89 12 0.189 0.15 Twall

34 0.36 1.19 8.4 198.7 72.9 104.4 121.1 �1.0 2.1 0.98 12 0.344 0.29 TwTst

56 0.72 1.16 10.7 760.0 139.1 103.8 131.7 z=0 0.0008 0.94 10.5 0.701 0.67 Tsati

65 0.71 1.17 10.7 734.8 136.4 103.9 134.4 z=0 0.0009 0.97 10.5 0.692 0.67 Tsati

71 0.96 1.4 7.8 725.3 99.8 108.4 138.3 z=0 0.0012 0.98 6.1 0.703 0.69 Tsati

33 0.96 1.3 6.1 346.9 48.1 107.8 123.9 �3.9 1.7 0.86 7.2 0.488 0.48 Tsati

43 0.96 1.16 3.6 157.7 21.8 103.8 114.4 �1 1.1 0.93 6.1 0.356 0.36 Tsati

53 1.4 1.32 7.8 750.6 70.6 107.5 134.4 z=0 0.8 0.92 7.8 0.600 0.56 Tsati

76 0.37 3.09 12.1 737.9 272.8 134.5 161.1 z=0 0.009 1 5.5 0.779 0.75 Tsati

72 0.37 3.08 9.2 369.0 136.4 134.4 151.7 �3.8 0.0017 1.05 5.5 0.536 0.54 TwTst

84 0.36 3.05 9.2 422.6 158.1 134.1 160.6 z=0 0.003 1.1 5.5 0.636 0.64 Tsati

awith SL=1

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Fig. 1 a Test 1-T-55. Griffithet al. (1958) procedure for zct.b Test 1-T-55. Behavior ofmixture enthalpy nearsaturation. c Test 1-T-55.Correlation of measured vaporenthalpy. d Test 1-T-55.Measured and calculated voidfraction

Fig. 2 a Test 1-T-29. Griffith et al. (1958) procedure for zct. b Test1-T-29. Measured and calculated void fraction

Fig. 3 a Test 1-T-48. Griffith et al. (1958) procedure for zct. b Test1-T-48. Measured and calculated void fraction

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general, the same happens for saturation temperature,Fig. 7a, b.

For other tests, see Fig. 11a, b, it has been thearithmetic mean of the wall and the saturation tem-perature that has best fitted the critical point and thevoid fraction in the first region. Although in test 1-T-34, see Figs. 4a, b, the arithmetic mean marks thetransition point but the void fraction in the first regionis better calculated with the wall temperature(Figs. 5, 6).

It is evident that to work with the wall tempera-ture—actually the average along the heated wallchannel (Staub et al. 1969)—implies to be able toestimate in some manner this temperature. Quite re-cently, the author has shown, for the same set ofmeasurements, a new thermodynamic procedure basedon the entropy to calculate the average wall tempera-ture (Collado 2002, 2003), which has supplied quitepromising results.

For the tests far from saturation gathered in Table 2,in which there have been transition, the first region voidfraction has been, in general, better calculated with thewall temperature (see Figs. 12, 13, 14, 15, 16, 17, 18, 19,20, 21, 22, 23). Perhaps due to the fact, experimentallyverified by Griffith et al. (1958), that before transitionthe wall is partially bare of bubbles. Indeed this ideawould also justify why we have had to use the saturationtemperature to place transition in the Table 1 tests at

0.36 m/s in which the fully developed region startedfrom the beginning.

4.3 Liquid enthalpy and heat balance in the secondregion

Figure 1b shows the liquid enthalpy behavior before andreaching saturation obtained from the measurements ofGE: it seems that far from saturation but already in thesubcooled regime, see Eq. 11, the liquid enthalpy, con-tinuous bold line, is greater than the value calculated withthe classical heat balance, slashed bold line, see Eq. 17.

Then the ‘‘liquid slip’’, SL, defined by Eq. 23, resultsless than one: in Fig. 1b for test 1-T-55, it is 0.92excluding the saturation point, and for the majority ofthe tests far from saturation, see Table 2, its value rangefrom 0.88 to 0.92.

However, as the liquid is reaching saturation orthermal equilibrium with vapor, it is confirmed for alltests in such situation that the enthalpy liquid slopedrops trying to coincide with the classical or equilibriumbalance at saturation point. So, in Fig. 1b, the correla-tion of the liquid enthalpy slope now including the sat-uration point gives a SL equal to 0.98.

This situation is clearly confirmed in Table 1, for testsnear or reaching saturation the ‘‘liquid slip’’ is near one.

Fig. 4 a Test 1-T-34. Griffith et al. (1958) procedure for zct. b Test1-T-34. Measured and calculated void fraction Fig. 5 a Test 1-T-56. Griffith et al. (1958) procedure for zct. b Test

1-T-56. Measured and calculated void fraction

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This would mean that the saturation point, indepen-dently of the vapor content, is definitely placed at theposition defined by the classic heat balance, Eq. 17.

Figure 1c explains the procedure to correlate thevapor enthalpy net growing rate, Eq. 24, for a test suchas the 1-T-55, in which the fully developed region isestablished nearly from the beginning. In the linearcorrelation, they have been considered all the measuredpoints included saturation, in a compromise between thesubcooled and the saturation zones.

The growing rate of vapor is definitely nonlinear;although the void fraction results are quite acceptable,see Fig. 1d. Furthermore, it would seems that the ‘‘va-por slip’’, SG, in the fully developed region could bequite regular in function with the inlet velocity andpressure, see Table 1, although they are not tested en-ough to establish conclusions.

It would also seem that the ‘‘vapor slip’’ obtained fora test in which the fully developed regime is establishedfrom the beginning could also be used for the secondregion in those tests in which do there is a first region,see Figs. 2b, 4b, 7b, 8b, and 11b. These tests are rela-tively near saturation.

For the tests in Table 2, relatively far from saturation,the ‘‘vapor slip’’ seems to coincide with that of fullydeveloped tests at atmospheric pressure and inlet velocityof 0.36 m/s, SG=12 (Figs. 12, 13). Although this value

appears again for other inlet velocities and pressures(Fig. 14, 0.97 m/s, Figs. 15, 17 and 18, 1.4 m/s, andFigs. 21, 22, 2.9 m/s). However, for other tests inTable 2, see Figs. 16, 19, 20, and 23, SG is not equal to12. Again it would be necessary much more data to at-tempt to give some conclusion about SG values (Figs. 9,10, 14).

Evidently, the linear growing of vapor in the secondregion is a rather simplified model and closer the boilingwater is to transition point higher the ‘‘vapor slip’’ willbe, meaning that the average vapor slope will be lower.

The combination of SG and SL would give the globalslip, S, Eq. 23, which with the values of Table 1, hasclearly values less than one, confirming that the classicalequilibrium balance, Eq. 17 is not verified.

Of course, it is not possible that S be constantthrough the duct: although SL is quite regular, SG can bemuch more irregular and variable. Indeed its changesallow that the nonequilibrium balance adapts to theactual enthalpy behavior—for example, the classic heatbalance could not have mean of respond to the abruptchange at saturation.

However, the assumptions that the ‘‘vapor slip’’ is aconstant in the second region and that the liquid slip isconstant along the whole duct (SL=1) have given quiteacceptable approximations to the actual void fractionprofiles.

Fig. 7 a Test 1-T-33. Griffith et al. (1958) procedure for zct. b Test1-T-33. Measured and calculated void fractionFig. 6 a Test 1-T-71. Griffith et al. (1958) procedure for zct. b Test

1-T-71. Measured and calculated void fraction

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Fig. 8 a Test 1-T-43. Griffith et al. (1958) procedure for zct. b Test1-T-43. Measured and calculated void fraction

Fig. 9 a Test 1-T-53. Griffith et al. (1958) procedure for zct. b Test1-T-53. Measured and calculated void fraction

Fig. 11 a Test 1-T-72. Griffith et al. (1958) procedure fro zct. b Test1-T-72. Measured and calculated void fraction

Fig. 10 a Test 1-T-76. Griffith et al. (1958) procedure for zct. b Test1-T-76. Measured and calculated void fraction

445

Table 2 Subcooled tests far from saturation in Task 1 (Staub et al. 1969)

Test uLi(m/s)

pi(bar)

DTsubi

(�C)q(kW/m2)

q¢(kJ/m-kg)

Tsati

(�C)Twall

(�C)DTct

(�C)DTsub0(�C)

SL eo eo-cala �(ei-Tx) zct

1-T-23 0.36 1.12 21.9 321.7 118.0 102.8 120 – 12.1 0.91 0.035 0.139 Twall

1-T-24 0.37 1.13 18 384.7 136.3 103.1 122.8 – 6.9 0.92 0.156 0.186 Twall

1-T-26 0.36 1.15 19.8 369 134.3 103.5 113.9 – 8.7 0.92 0.083 0.213 Twall

1-T-41 0.36 1.15 10.7 148.2 53.5 103.6 117.8 – 5.8 0.9 0.041 0.072 Twall

1-T-51 0.36 1.15 8.7 170.3 62.0 103.5 121.1 – 3.2 0.9 0.067 0.086 Twall

1-T-52 0.36 1.15 23.2 611.8 225.3 103.5 127.2 �7.7 5.1 0.92 0.421 0.49 Twall

1-T-61 0.36 1.14 24.3 599.2 218.9 103.4 131.1 �1.1 6.8 0.92 0.303 0.40 Twall

1-T-67 0.36 1.16 17.4 359.5 130.8 103.7 130.6 – 6.6 0.92 0.155 0.130 Twall

1-T-69 0.36 1.15 19.9 372.1 136.5 103.7 128.9 – 8.6 0.92 0.108 0.136 Twall

1-T-73 0.36 3.08 17.6 372.1 137.6 134.4 160.6 – 6.7 0.91 0.076 0.081 Twall

1-T-32 0.96 1.16 15.4 208.1 28.6 103.8 115.0 – 12.5 0.91 0.05 0.008 Tsati

1-T-42 0.97 1.12 10.2 309.1 42.1 102.7 120 �1.5 6.4 0.92 0.129 0.126 Tsati

1-T-59 0.95 1.18 15.9 290.1 40.1 104.4 122.8 – 12.0 0.9 0.015 0.011 Tsati

1-T-28 1.4 1.19 8.7 369 34.5 104.5 113.9 �1.6 4.8 0.91 0.102 0.105 Tsati

1-T-54 1.4 1.19 14.0 715.9 67.0 104.4 132.2 �1.1 7.2 0.91 0.317 0.316 Tsati

1-T-57 1.4 1.18 21.8 731.6 68.4 104.3 132.2 – 15.8 0.91 0.034 0.015 Tsati

1-T-60 1.4 1.19 22.6 731.6 68.1 104.4 131.7 – 16.4 0.91 0.023 0.015 Tsati

1-T-64 1.4 1.19 8.9 372.1 34.1 104.7 115.0 �0.75 5.3 0.91 0.108 0.116 Tsati

1-T-74 1.4 3.08 9.4 378.4 35.9 134.4 152.2 +2.2 6.2 0.89 0.106 0.109 Tsati

1-T-82 1.4 3.09 9.5 378.4 36.3 134.5 153.3 +0.7 6.3 0.89 0.116 0.103b Tsati

1-T-46 2.0 1.34 5.3 334.3 22.3 108.1 123.3 +0.11 2.4 0.89 0.175 0.172 Tsati

1-T-47 2.9 1.48 10.9 775.8 34.9 111.0 114.4 +0.6 6.5 0.9 0.221 0.207b Tsati

1-T-62 2.9 1.18 13.1 744.2 33.3 104.3 128.3 – 8.8 0.9 0.019 0.004 Tsati

1-T-75 2.9 3.08 11.4 785.2 36.0 134.4 156.1 +1.7 6.9 0.89 0.072 0.075 Tsati

1-T-83 2.9 3.08 11.4 791.6 36.6 134.4 162.8 +1.5 7.7 0.88 0.123 0.09b Tsati

aWith SL=1bei with Tsati

Fig. 12 Test 1-T-52. Measured and calculated void fraction

Fig. 13 Test 1-T-61. Measured and calculated void fraction

Fig. 14 Test 1-T-42. Measured and calculated void fraction

Fig. 15 Test 1-T-28. Measured and calculated void fraction

446

Finally to highlight that perhaps one of the mostinteresting results of this analysis would be the practicalimplications of the former assumption: that the liquidenthalpy used in the correlations of ‘‘vapor slip’’ and inthe void fraction calculations has been that of classic

heat balance, Eq. 17, i.e., SL=1. It was checked thatchanging SL from 0.9 to 1 nearly did not affect to thecalculated void fraction profile. The separation of vaporgrowing (SG) from liquid heating (SL), Eq. 23, wouldhave this clear advantage on the formerly proposedprocedure (Collado 2000) of working with the mixtureenthalpy as a whole.

Fig. 16 Test 1-T-54. Measured and calculated void fraction

Fig. 20 Test 1-T-47. Measured and calculated void fraction

Fig. 18 Test 1-T-82. Measured and calculated void fraction

Fig. 17 Test 1-T-74. Measured and calculated void fraction

Fig. 19 Test 1-T-46. Measured and calculated void fraction

Fig. 21 Test 1-T-75. Measured and calculated void fraction

447

Then the great amount of void fraction data takenuntil now without taking the subcooled liquid temper-ature could be quite useful following the analysis pro-posed.

Acknowledgements The authors thank to the Spanish Minister ofEducation and Science (MEC) the funding of this research throughthe special action ENE2004-0279-E and the research project DPI2005-08654-CO4-04.

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Fig. 23 Test 1-T-64. Measured and calculated void fraction

Fig. 22 Test 1-T-83. Measured and calculated void fraction

448