theoretical calculation of annular upward ﬂow in a narrow ......on single- and two-phase ﬂow...

Nuclear Engineering and Design 225 (2003) 219–247

Theoretical calculation of annular upward flow in anarrow annuli with bilateral heating

Guanghui Su∗, Junli Gou, Suizheng Qiu, Xiaoqiang Yang, Dounan JiaState Key Laboratory of Multiphase Flow in Power Engineering, School of Energy and Power Engineering,

Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an City 710049, PR China

Received 12 August 2002; received in revised form 30 April 2003; accepted 12 June 2003

Abstract

Based on separated flow, a theoretical three-fluids model predicting for annular upward flow in a vertical narrow annuli withbilateral heating has been developed in present paper. The theoretical model is based on fundamental conservation principles:the mass, momentum, and energy conservation equations of liquid films and the momentum conservation equation of vapor core.Through numerically solving the equations, liquid film thickness, radial velocity, and temperature distribution in liquid films,heat transfer coefficient of inner and outer tubes and axial pressure gradient are obtained. The predicted results are compared withthe experimental data and good agreements between them are found. With same mass flow rate and heat flux, the thickness ofliquid film in the annular narrow channel will decrease with decreasing the annular gap. The two-phase heat transfer coefficientwill increase with the increase of heat flux and the decrease of the annular gap. That is, the heat transfer will be enhanced withsmall annular gap. The effects of outer wall heat flux on velocity and temperature in the outer liquid layer, thickness of outerliquid film and outer wall heat transfer coefficient are clear and obvious. The effects of outer wall heat flux on velocity andtemperature in the inner liquid layer, thickness of inner liquid film and the inner wall heat transfer coefficient are very small; thesimilar effects of the inner wall heat flux are found. As the applications of the present model, the critical heat flux and criticalquality are calculated.© 2003 Elsevier B.V. All rights reserved.

1. Introduction

Two-phase annular flow with heat transfer is preva-lent in many processes such as industrial and energytransformation processes. Recently, advances in highperformance electronic chips and the miniaturizationof electronic circuits in which high heat flux will becreated and other compact systems such as the re-frigeration/air conditioning, automobile environmentcontrol systems have resulted in a great demand for

∗ Corresponding author. Tel.:+86-29-2667082;fax: +86-29-2667802.

E-mail address: [email protected] (G.H. Su).

developing efficient heat transfer techniques to ac-commodate these high heat fluxes (Warrier et al.,2002). One of the simplest heat transfer systems maybe using single-phase forced convection or two-phaseflow boiling in small tubes or rectangular channelsor annular gaps or lunate channels. Experiments haveshowed that annular flow are main flow pattern in nar-row annular gaps (Sun et al., 2001). It is easy to formvapor slug in the narrow channel due to the accumula-tion of bubbles. Thus, vapor slug is longer than liquidcolumn. And then, annular flow is formed. Annularflow is a particularly important flow pattern since itfrequently occurs in a wide range of such parametersas quality, pressure and flow conditions. In this regime,

0029-5493/$ – see front matter © 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0029-5493(03)00160-2

220 G.H. Su et al. / Nuclear Engineering and Design 225 (2003) 219–247

Nomenclature

A cross sectional area (m2)Bo boiling numberDep liquid droplet deposition rate

per unit area (kg m−2 s−1)En liquid droplet entrainment

rate per unit area (kg m−2 s−1)f friction coefficientg gravitational acceleration

(m s−2)h heat transfer coefficient

(W m−2 C−1)hfg = hg − hf latent heat of vaporization

(kJ kg−1)kf thermal conductivity

(kW m−1 C−1)K Von Karman’s constantP pressure (MPa)Pr periphery of liquid film (m)Prt turbulent Prandtl numberq heat flux (kW m−2)ri outer radius of inner tube (m)Re Reynolds numbers gap size (mm)T temperature (C)u velocity (m s−1)Wl mass flow rate of liquid

film (kg m−2 s−1)We Weber numberx qualityy distance from wall (m)y+ non-dimensional distanceZ axial distance (m)

Greek lettersα void fractionδ thickness of liquid film (m)ε eddy viscosity (m2 s−1)ηl thermal diffusivity (m2 s−1)µ dynamic viscosity

(kg m−1 s−1)ρ density (kg m−3)σ surface tension (N m−1)τ shear stress (N m−2)ΛK modified factor

Subscriptsc pertains to vapor coreCAL pertains to calculated valueEXP pertains to experiment valuef pertains to liquidg pertains to vapori pertains to inner liquid film,

inner tubel pertains to liquid filmo pertains to outer liquid film,

outer tubesat saturated state

an inherently unsteady liquid film covers the surfaceof the heated tube while vapor with entrained dropletsflows in the central core. The liquid film structure isinfluenced by such variables as system geometry, floworientation, liquid/vapor velocity difference, etc. A lotof literatures of annular flow have been presented upto now. Entrainment and deposition are very importantfor developing a theoretical model of annular flow.Thus, there exist a lot of reports of entrainment anddeposition in annular flow.Jr et al. (2002)studied theliquid entrainment, droplet concentration, and pres-sure gradient at the onset of annular flow in a verticalpipe. Okawa et al. (2002)presented an entrainmentrate correlation in annular flow and declared that theircorrelation may be applicable to wide range of flowconditions. Pan and Hanratty (2002a,b)developedcorrelations of entrainment for annular flow in verticaland horizontal pipes.Bertodano et al. (2001)scaled theentrainment rate mechanism by the Kelvin–Helmholtzinstability at the liquid film surface and developed anew correlation based on their experimental data ob-tained in 10 mm tubes for low-viscosity fluids in theripple annular flow.Kataoka et al. (2000)presented asimple approximate correlation obtained from equilib-rium state where entrainment rate and deposition ratebecome equal.Binder and Hanratty (1991, 1992)mod-eled the droplet deposition in vertical gas/liquid annu-lar flow by considering dispersion from a ring sourceon the wall and described droplet using Lagrangianmethod in horizontal gas–liquid annular flows. A rel-atively simple model was proposed for predicting de-position rate based on local measurements of dropletfluxes in the core of horizontal annular flow (Paras

G.H. Su et al. / Nuclear Engineering and Design 225 (2003) 219–247 221

and Karabelas, 1991). Experimental data of annularflow have been obtained by many workers and nu-merous correlations have been developed. Examplesinclude those of:Hurlburt and Newell (1999), Holtet al. (1999), Fu and Klausner (1997), Klausner et al.(1991)andJung and Radermacher (1989)for pressuredrop, andNozu and Honda (2000), Fossa et al. (1995)and Kattan et al. (1998)for heat transfer. However,most of these studies are based on data obtained fromrelatively large diameter conduits. Due to the natureof the geometries of narrow annular channel, the heattransfer performances and the associated pressure dropcan be very different from that in large tubes or chan-nels. Only a few studies in the open literatures reporton single- and two-phase flow boiling heat transfer inannular gap.Hasan et al. (1990)reported that exper-iments on subcooled flow boiling heat transfer werecarried out in a vertical annular channel, the innerwall of which is heated and the outer wall insulated.Fron-113 was the working fluid. Flow boiling heattransfer data were reported at three mass velocities(579, 801, and 1102 kg m−2 s), three pressure (312,277, and 243 kPa), and three inlet subcoolings (20.0,30.0, and 36.5C). A multiple-hysteresis phenomenonis identified. The measured wall heat transfer coeffi-cients were compared with those predicted by variouscorrelations and an improvement of the Shah correla-tion for annuli was suggested.Bibeau and Salcudean(1994)studied the bubble ebullition in forced convec-tive subcooled nucleate boiling using a vertical circularannulus at atmospheric pressure, for mean flow veloc-ities of 0.08–1.2 m s−1 and subcoolings of 10–60C.The maximum bubble diameter and condensation timewere shown to be influenced by the location relative tothe onset of significant void.Shah (1976, 1983)pre-sented that a general correlation for heat transfer dur-ing subcooled and saturated boiling in tubes had beencompared with a large amount of data for boiling in an-nuli. The data included the fluids water, Fron-113, andmethanol; mass flow rate from 278 to 7774 kg m−2 s;heated fluxes from 0.03 to 4.4 kW m−2, liquid sub-coolings from 2 to 109C; and reduced pressures from0.009 to 0.89. The inner tube diameter varied from4.5 to 42.2 mm and the annular gap from 1 to 6.6 mm.The data included heating of the inner, outer, and bothtubes of the annular channel.Yin et al. (2000)experi-mentally studied the subcooled flow boiling heat trans-fer and visualized the associated bubble characteristics

for refrigeration R-134a flowing in a horizontal annu-lar duct having inside diameter of 6.35 mm and out-side diameter of 16.66 mm. The effects of the imposedwall heat flux, mass flow rate, liquid subcooling, andsaturated temperature of R-134a on the resulting nu-cleate boiling heat transfer and bubble characteristicswere examined in detail.Sun et al. (2001)numericallycalculated the liquid nitrogen film thickness, distri-bution of velocities and temperatures in the liquidlayer for vertical two-phase flow in an annular gapin the range of heat flux from 6000 to 12000 W m−2;mass flow rate from 500 to 1100 kg m−2 s. Su et al.(2003)presented a theoretical model which describesthe model of vertical upward annular flow in nar-row annular channel with bilateral heating under lowmass flow rate. However, the profiles of velocity inthe liquid layers are linear due to that the number ofcalculation grids in liquid layers is not large enoughwhen the model was numerically solved. It is obvi-ous that the linear solution will departure from thetrue solution. Thus, there is need to further study theannular flow in a vertical upward annular gap.

2. Analytical model

2.1. Basic equations

The flow in the bilaterally heated annular gap is pre-sented inFig. 1. In the two-phase annular flow regime,there exist the inner and the outer liquid films due tothe bilateral heating. The inner liquid film covers theouter surface of inner tube, and the outer liquid filmcovers the inner surface of outer tube while the va-por with entrained droplets flows in the central core.The following assumptions are necessary in order toanalyze the annular flow under these conditions:

(1) The flow is steady and incompressible;(2) The liquid film is uniform around the tube periph-

ery;(3) The pressure is uniform in the radial direction;(4) Vapor is produced by the evaporation at the liq-

uid/vapor interface;(5) Liquid droplets entrained in the vapor core are

uniformly distributed;(6) There is no slip between the vapor and the

droplets.


Fig. 1. Two-phase flow patterns in the bilateral heated gap.

In the annular region, there exists the mass transferbetween the liquid droplets in the vapor core and theliquid films. On the one hand, the liquid droplets inthe vapor core will be deposited on the liquid filmsurfaces. On the other hand, the liquid droplets will becontinuously entrained from the liquid films into thevapor core and thus there will be more droplets in thevapor core. The vapor core is supplied with the liquidevaporated at the liquid/vapor interface and with thevapor bubbles due to the boiling process in the liquidfilms. Based on these observations, the mass flow rateof the liquid films will be changed as a function of thedistanceZ along the channel. The mass conservationequations of the inner and outer liquid films are asfollows, respectively.

For the inner film:

(dWli )f

dZ= Pri

(Depi − Eni − qi

hfg

)(1)

(dWlo)f

dZ= Pro

(Depo− Eno − qo

hfg

)(2)

where,Wli andWlo are the mass flow rates of the in-ner and outer liquid films, respectively;Pri andPro arethe peripheries of the inner and outer liquid films, re-spectively;Depi andDepo are the liquid droplets depo-sition rates per unit area of the inner and outer liquid

films, respectively;Eni andEno are the liquid dropletsentrainment rates per unit area of the inner and outerliquid films, respectively;qi andqo are the heat fluxesof the inner and outer tubes, respectively;hfg is thelatent heat of vaporization.

A cylindrical element of the inner liquid film isconsidered now. Its height isZ, its inner radius isr,and its outer radius is (ri + δi ). According to the forcebalance on this element, the momentum conservationequation of the inner film will be

2πrτZ= 2π(ri + δi)τiZ

− ρfgπ(ri + δi)2 − r2Z

− dP

dZπ[(ri + δi)

2 − r2]Z (3)

where,ri is the outer radius of the inner tube;δi isthe thickness of the inner liquid film;τ is shear stressat radiusr in the inner liquid film;τ i is shear stressat the liquid/vapor interface of inner liquid film;ρf isthe liquid density;g is the gravitational acceleration.

The shear stress in the inner liquid film can be ob-tained fromEq. (3), that is

τ = (ri + δi)

rτi − 1

2

(ρfg + dP

dZ

)[(ri + δi)

2 − r2

r

](4)

The shear stressτ at radiusr in the outer liquidfilm can be obtained using the same method. It can bewritten as:

τ = (ro − δo)

rτo − 1

2

(ρfg + dP

dZ

)[r2 − (ro − δo)

2

r

](5)

where,ro is the inner diameter of the outer tube;δois the thickness of the outer liquid film;τo is shearstress at the liquid/vapor interface of the outer liquidfilm; dP/dZ is axial pressure gradient, and can be cal-culated by the momentum conservation equation ofvapor core. Thus, it can be expressed by

−dP

dZ= τiPri + τoPro

Aαc+ 1

αc

d

dZ(ρcαcu

2c) + ρcg (6)

where,ρc, αc, and uc are the density, void fraction,and mean velocity of vapor core, respectively. See theAppendix A for their calculating equations.

Knowing the shear stress in the liquid film, the ve-locity in the liquid film can be obtained according to


the Newton viscosity law:

u =∫ δ

0

τ

µEdy (7)

The boundary conditions ofEq. (7)are as follows:y = 0, u = 0

y = δ,du

dy= τint

µE

where,y is the distance from the wall;u is the localliquid velocity in the liquid film;τ int is the shear stressat the liquid/vapor interface;µE is the effective viscos-ity of the fluid.µE = µf for laminar flow, andµE =µf + ερf for turbulence flow;ε is the eddy viscosity.

The mass flow rate of liquid film can be obtainedby integrating the velocity:

(Wl)f =∫ δ

02πrρfudy (8)

The thickness of the liquid film is very small, thus theconvection can be ignored. The axial conduction canalso be ignored if the flow is in one-dimension. Thus,the energy equation is as follows:

d

dy

[(ηl + εH)

dT

dy

]= 0 (9)

where,ηl is the thermal diffusivity;εH is the eddythermal diffusivity, its value will be zero when theliquid film flow is laminar flow.

The boundary conditions ofEq. (9)are as follows: qw = −kf

dT

dy, y = 0

T = Tsat, y = δ

where,qw is the wall heat flux;T is the temperature;Tsat is the saturated temperature of fluid;kf is thethermal conductivity.

There are a lot of boiling heat transfer correlationsof two-phase flow that can be found in the open lit-eratures. Such correlations normally attempt to pre-dict heat transfer coefficient both in the convective andnucleate flow boiling regions. However, these corre-lations are only satisfactory under certain conditions.

The present study calculates the heat transfer coeffi-cient of annular gap according to its basic concept.The heat transfer coefficient of two-phase flow is de-fined by

h = q

Tw − Tsat(10)

where,q is heat flux;Tw is the wall temperature.

2.2. Constitutive equations

The variablesε, εH, τ i and En must be specifiedin order to simulate the annular flow using the abovemodel because the relations above are not yet closed.The interfacial shear stress is one of the most importantvariables that effect the solutions of the present modeland can be calculated by the correlation proposed byWallis (1969)

τi = 1

2fiρgu

2g (11)

where,fi/fgo = (1+300δ)/De; De = Do−Di ; fgo =0.079Re−0.25

g ; andReg = GxDe/µg.Detailed turbulence measurements for the liquid

films of annular flow in the vertical annular gap arenot available. The usual practice in treating the liquidfilm eddy viscosity is to assume the wall turbulenceis similar to that of single-phase flow and employ thesingle-phase flow formulas for liquid film eddy viscos-ity. Although such an assumption has yet to be exper-imentally validated, modified single-phase flow eddyviscosity and turbulent Prandtl number can be used(Fu and Klausner, 1997). Thus,ε can be calculated bythe following equations:

ε

υ= 0.001y+3, y+ < 5

ε =

Ky

[1 − exp

(−y+

25

)]2 ∣∣∣∣dudy

∣∣∣∣ (1.0 − y

δ

)nφ, y+ ≥ 5

(12)

where, υ = (µ/ρ), µ is dynamic viscosity.K isVon Karman’s constant and equals 0.41. The fac-tor φ which has been introduced inEq. (12) is toaccount for enhanced turbulence in the liquid filmcaused by the disturbance created by the evapora-tion radial momentum flux and is expressed byφ =B0.3

o [(1 − x)/x]0.1, where the boiling number is de-fined asBo = q/(Ghfg). The wall unitsy+ = yu∗/υ,andu∗ = (τw/ρf )

1/2.


The relationship between the liquid film eddy vis-cosity and eddy thermal diffusivity is as follows (Fuand Klausner, 1997; Celata et al., 2001):

Prt = ε

εH(13)

where,Prt is the turbulent Prandtl number and can becalculated by

Prt = 1.07, y+ < 5

Prt = 1.0 + 0.855− tanh[0.2(y+−7.5)], y+ ≥ 5

(14)

The deposition can be calculated with the equationgiven byKataoka, Ishii and Nakayama (2000)basedon fully developed annular flow with heat insulation

DepDe

µf= 0.022Re0.74

f

(µg

µf

)0.26

ϕ0.74 (15)

where,ϕ is the fraction of liquid flowing as dropletsin the vapor core and is defined as (Ishii and Mishima,1984; Celata et al., 2001): ϕ = (jf − jff )/jf , where,jf is the volumetric flux which pertains to the liquidphase, andjff is the volumetric flux which pertainsto the liquid film.Ref is the liquid Reynolds numberwhen the liquid singly flows through the channel.

In the annular gap with bilateral heating, on theone hand, the deposition rate strongly depends on theconcentration of droplet (number of droplet per unitvolume) in the vapor core. The area of the interfacebetween the outer liquid film and the vapor core islarger than that between the inner liquid film and thevapor core. Therefore, the rate of deposition on theouter liquid film is larger than that on the inner liquidfilm. Thus, theEq. (15)must be modified in order tobe used to calculate the deposition rate in a bilateralheating annuli by a dimensionless factorΛK (Doerfferet al., 1997)

Dep,K = DepΛK (16)

where,ΛK is a modified factor and is defined as:ΛK =rK/(ro + ri), K = i,o (i denotes inner liquid film ando denotes outer liquid film).

On the other hand, the deposition may be blockedby the evaporative effect on the interface between theliquid film and vapor core. Thus, the deposition ratecaused by this effect can be given by (Milashenko andNigmatulin, 1989)

Dep,e,K = −CΩK (17)

where,ΩK is the block coefficient, and is defined byΩK = (ρg/0.065ρf )Vg,K, andVg,K = (qK/hfgρg) isthe evaporative velocity of liquid film, K= i,o asabove.C is the concentration of droplet in the vaporcore and can be calculated byC = Wle/(Wle + Wg),Wle andWg is mass flow rate of liquid droplet and ofvapor in the vapor core, respectively. Thus, the depo-sition rate may be finally calculated by

Dep,tol,K = Dep,K + Dep,e,K (18)

The droplet entrainment rate,En, can be calculatedconsidering the contribution of two different mecha-nisms of droplets formation: breakup of disturbancewaves and boiling in the liquid film (Mishima et al.,1989). According to the analysis ofKataoka and Ishii(1983), droplet entrainment is mainly attributed to thebreakup of large disturbance waves on the liquid/vaporinterface. Regarding the second mechanism of dropletformation, according to the analysis ofMishima et al.(1989) and Kay (1994), burst of boiling bubbles inthe liquid film may cause droplet entrainment at suffi-ciently high heat flux. Thus, entrainment rate may beobtained by the following equation:

En = Enw + Ene (19)

where,Enw is wave droplet entrainment rate (entrain-ment due to disturbance waves);Ene is boiling dropletentrainment rate (entrainment due to boiling in the liq-uid film).

The Enw is evaluated by the following equations(Kataoka et al., 2000; Mishima et al., 1989; Ishii andMishima, 1984):

EnwDe

µf= 0.72× 10−9Re1.75

f We(1 − ϕ∞)0.25(

1 − ϕ

ϕ∞

)2

,ϕ

ϕ∞≤ 1

EnwDe

µf= 0.022Re0.74

f

(µg

µf

)ϕ0.74

∞ ,ϕ

ϕ∞> 1

Enw = 0, Reff < Reff c

(20)


where, ϕ∞ is the equilibrium entrainment fraction(Ueda et al., 1981), and can be given byϕ∞ =tanh(7.25 × 10−7We1.25Re0.25

f ), and We is Webernumber for entrainment:We = (ρgj

2gDe/σ)[(ρf −

ρg)/ρg]1/3. σ is the surface tension.Ref is liquidReynolds number, and can be given byRef = (G(1−x)De)/µf , where, Reff is the liquid film Reynoldsnumber, and can be calculated byReff = G(1−x)(1−ϕ)De/µf . Reff c is the minimum liquid film Reynoldsnumber, below which no droplet entrainments ex-ist due to superficial waves occurs (Celata et al.,2001), it can be calculated byReffc = (y++/0.347)1.5

(ρf /ρg)0.75(µg/µf )

1.5, wherey++ is the non-dimen-sional distance associated with the onset of the en-trainment. According toCelata et al. (2001), y++ =10.0.

The entrainment due to boiling in the liquid film,Ene, may be given by the correlation ofUeda et al.(1981):

Enehfg

q= 4.77× 102

(2qδ

h2fgσρg

)0.75

. (21)

The above correlations for entrainment are onlysuitable to circular tubes. They must be modified inorder to be used to calculate the entrainment rates ina bilateral heating annuli. That is:

Enw,K = EnwΛK . (22)

Fig. 2. N-S flowchart of solution procedure.

3. Solution procedure

The relations above are closed now. Knowing theproperties of fluid, heat fluxes, mass flow rate, qual-ity, and geometric parameters, the above model canbe solved.Fig. 2 is a flowchart describing the solu-tion procedure to the above equations, which is alsosummarized as:

(1) An initial guess is made for the film thickness:δi and δo (δi denotes the film thickness of in-ner liquid film andδo denotes that of outer liquidfilm).

(2) Eq. (6) is used to compute the pressure gradientdP/dZ.

(3) Eq. (7)is used in conjunction withEqs. (4) and (5)to compute the inner and outer liquid film velocityprofiles:ui (y) anduo(y), andEq. (8) is employedto calculate the inner and outer liquid film massflow rates. The inner liquid film mass flow rate ishereby denoted byWli ,3, and the outer liquid filmmass flow rate is denoted byWlo,3.

(4) Eqs. (1) and (2)are also used to compute the innerand outer liquid film mass flow rates. To distin-guish between the results obtained by steps (3) and(4), the inner liquid film mass flow rate obtainedby present step is hereby denoted byWli ,4, andthe outer liquid film mass flow rate is denoted byWlo,4.


(5) Check ifWli ,3 equals toWli ,4, andWlo,3 equalsto Wlo,4. If not, guess the initial values of thefilm thicknessδi and δo again. Steps (2)–(5) arerepeated.

(6) The heat transfer coefficient defined inEq. (10)isevaluated.

It is noted that variable-step integration methodshould be used in the procedure of solving the veloc-ity Eq. (7) in order to ensure that the step is smallenough or the number of the grid is large enough toobtain high accuracy. If the fixed step is employed,the velocity profiles in the liquid film will be linear(Su et al., 2003).

4. Verifications of present model

To validate the correction of proposed model, anattempt has been made to put together a databasethat covers a wide range of flow and thermal condi-tions. However, No database with bilateral heating inthe annular gap has been found in the open literatures.Therefore, the present model has been verified by ourexperimental data.Fig. 3is a sketch of the experimen-tal apparatus used in this study to investigate the flow

Fig. 3. Schematic diagram of experimental apparatus.

boiling in a vertical annular gap. It consists of twomain loops, namely, the primary and the secondaryloops. The secondary loop, which is connected to theprimary one through the condenser, is used to con-dense the primary loop and is not shown in detail inFig. 3. The primary loop contains a pump, a pressur-izer, an electrical pre-heater, a mass flow meter, a testsection, a condenser, and connected pipes. The tem-perature and mass flow rate in the secondary loop mustbe controlled to have enough cooling capacity forcondensing the fluid in the primary loop and subcool-ing the water to a preset subcooled temperature. Twopower supplies were used to heat the inner and outertubes in the test section. Distilled water is circulated inthe primary loop. By adjusting charge of N2 throughthe nitrogen (N2) charging line, the system pressurecan be regulated. By adjusting the gate opening of theby-pass valve, the mass flow rate of the test section canbe regulated. Note that the mass flow rate and systempressure should be further adjusted simultaneouslyby the needle valve at the upstream of the flow metersection and the power of the heater in the pressurizer,respectively, in order to control them at the requiredlevels. The mass flow rate of test section is measuredby a precise mass flow meter which consists of


orifice, three valves and differential pressure trans-mitter. After subcooled, the distilled water flows backto the main pump. The signals from the thermo-couples, pressure transducers and other transducerswere recorded by a data acquisition system and thenanalyzed by a computer immediately. The absolute

Fig. 4. Comparison of heat transfer and axial pressure gradient between the proposed model and experiment. (a) One example of comparisonof two-phase heat transfer coefficient between the experiment and the present model withs = 0.95 mm in the axial direction. (b) Oneexample of comparison of two-phase heat transfer coefficient between the experiment and the present model withs = 2 mm in the axialdirection. (c) Comparison of two-phase heat transfer coefficient between the experiment and the present model. (d) Comparison of axialpressure gradient between the experiment and the present model. (e) Wall temperature and saturated temperature curves obtained by theproposed model.

pressure at the entrance of the test section is mea-sured with two absolute pressure transducers. Thepressure drop is measured with a differential pressuretransducer. The temperatures of fluid are measured by1.0 mm K-type NiCr–NiSi thermocouples. The innerwall temperatures of inner tube are measured with 15


Fig. 4. (Continued ).


Table 1Accuracy of measurement system

Parameters Uncertainty (%)

Temperature 0.75Mass flow rate 0.251Pressure 0.251Pressure drop 0.251Heated power 1.619

0.5-mm o.d. K-type thermocouples. The outer walltemperatures of the outer tube are measured with0.1-mm o.d. T-type thermocouples, which are attachedat the four locations of the circumference with 90apart. The accuracy of the measurements is listed inTable 1.

The two-phase heat transfer coefficients and axialpressure gradients obtained by the proposed modelare compared with those of experiments, as shownin Fig. 4. TheFig. 4a and bshow the comparison oftwo-phase heat transfer coefficient between proposedmodel and experiments in different gap (s denotesgap size). A good agreement has been found.Fig. 4cshows the deviation between the experiment andpresent model is±25%. Its root mean square (RMS)error is 15.11%. However, about 70% data of heattransfer coefficient were over-predicted (Fig. 4a–c)within above error range. That is to say, there are about70% two-phase heat transfer coefficients are largerthan those of experimental data. The reason may bethat the wall temperature obtained by the proposedmodel is lower than that of experiment. The outerwall temperatures were measured by thermocouplesin our experiments. The accuracy of thermocouples is0.75%. The outer wall temperatures vary very smallalong the axis. The thermocouples may not measurethese small changes along the axis. Thus, the walltemperatures did not change along the axis. How-ever, the pressure decreases slightly along the axis.Thus, the saturated temperature decreases slightly.The thickness of the liquid film also decreases alongthe axis. Therefore, the wall temperatures obtainedby the energy equation of the present model alsodecrease along the axis, as shown inFig. 4e. Thus,(Tw − Tsat) obtained by proposed model is lower thanthat obtained by experiment, and the two-phase heattransfer coefficient obtained by proposed model islarger than that obtained by experiment.

Fig. 4dshows the comparison of axial pressure gra-dient between the present model and experiment. Thedeviation between them is±30%, the RMS is 18.86%.The present paper selects quite a number of correla-tions to close the set of equations for the annular flowmodel, which are developed for large tube. Althoughthe modified factor is employed to apply them for thecase of annular gap, but the errors for the axial pres-sure gradient seem quite large. Therefore, the furthertheoretical study must be done to develop a model fornarrow annular gap. As the first step, the error (±30%)is acceptable.

5. Results and discussions

Fig. 5 shows the temperature profiles in the liquidfilms obtained by the proposed model with six differ-ent annular gap and for three values of wall heat fluxof inner tube or of outer tube under the mass flowrate is 120 kg m−2 s−1 conditions. Abscissas denotethe non-dimensional distance from wall andy+ = y/δ

in Figs. 5 and 6, wherey is the distance from wallandδ is the thickness of liquid films.y+ = 0 is at thewall, andy+ = 1 is at the liquid/vapor interface. They-axis denotes non-dimensional temperatureTi /Tsat orTo/Tsat in Fig. 5, whereTi is the temperature in theinner liquid layer,To is the temperature in the outerliquid layer, andTsat is the saturated temperature. Twofigures in which the gap size is different are shown un-der the same conditions inFigs. 5–8in order to clearlyexpress the general parametric trends of gap size. Asobserved inFig. 5, the temperature in the inner orouter liquid layer will almost decrease linearly fromthe wall temperature to saturated temperature whilethe distance changes from the wall to the liquid/vaporinterface. The liquid film mass flow rate is very low(120 kg m−2 s−1), thus the flow is laminar one. Theenergy equation is a conductive one. Therefore, thetemperature in the liquid layer changes almost linearly.Fig. 5a shows the effects of outer wall heat flux onthe temperature in the inner liquid layer with differentannular gap with a fixed heat flux of inner tube,qiw =50 kW m−1. The effects are very small. With increaseof outer wall heat flux, the temperature in the innerliquid layer decreases very slightly. The more liquidwill evaporate with increasing outer wall heat flux,the velocity of vapor core will increase slightly. The

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Fig. 5. The temperature curves in the liquid films. (a) Effects of outer wall heat flux on the temperature profiles in the inner liquid layer. (b) Effects ofouter wall heat fluxon the temperature profiles in the outer liquid layer. (c) Effects of inner wall heat flux on the temperature profiles in the inner liquid layer. (d) Effects of inner wall heatflux on the temperature profiles in the outer liquid layer.

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Fig. 6. The velocity profiles in the liquid layers. (a) The velocity profiles in the inner liquid layer with different outer wall heat flux. (b) The velocity profiles in the outerliquid layer with different outer wall heat flux. (c) The velocity profiles in the inner liquid layer with different inner wall heat flux. (d)The velocity profiles in the outerliquid layer with different inner wall heat flux.

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Fig. 7. The liquid layer thickness. (a) Effects of outer wall heat flux on the inner liquid layer thickness. (b) Effects of outer wall heat flux on the outer liquid layer thickness.(c) Effects of inner wall heat flux on the inner liquid layer thickness. (d) Effects of inner wall heat flux on the outer liquid layer thickness.

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Fig. 8. The two-phase heat transfer coefficient in annular gap. (a) Effects of outer wall heat flux on the two-phase heat transfer coefficients of inner tube. (b) Effects ofouter wall heat flux on the two-phase heat transfer coefficients of outer tube. (c) Effects of inner wall heat flux on the two-phase heat transfer coefficients of inner tube. (d)Effects of inner wall heat flux on the two-phase heat transfer coefficients of outer tube.

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inner liquid film will be entrained and become thin.Thus, its temperature gradient will decrease slightly.Fig. 5b shows the effects of outer wall heat flux onthe temperature in the outer liquid layer with dif-ferent annular gap. The effects are very clear andobvious. The temperature gradient in the outer liquidfilm will increase with an increase of outer wall heatflux. Although the liquid layer will become thin withincreasing outer wall heat flux, the effects on temper-ature gradient in outer liquid layer is larger than thoseon liquid film thickness. Fig. 5c and d show the ef-fects of inner wall heat flux on the temperature in theliquid layer. The effects are similar to those of Fig. 5aand b. There are very slight influences of inner wallheat flux on the temperature in the outer liquid layer.The effects on the temperature in the inner layer arevery large. It can also be found from Fig. 5 that thetemperature gradient will increase with increasingthe gap with same mass flow rate and heat flux. Thethickness of liquid layer will increase with an increaseof gap, thus the temperature gradient will increase.

Fig. 6 shows the velocity profiles in the liquid films.As described above, y+ is the non-dimensional dis-tance from the wall. The velocities of liquid films atthe wall are zero due to the effect of viscosity. The ef-fects of heat flux on velocity profiles are very small.With increasing outer wall heat flux, the velocity inthe inner liquid layer will increase slightly (Fig. 6a),the velocity in the outer liquid film will also increaseslightly (Fig. 6b). The evaporation of the outer liq-uid film will increase with an increase of outer heatflux, thus more liquid evaporates, the vapor velocityin the core will increase, and the shear stress on theliquid/vapor interface will increase, the drag force toliquid layer will increase. Therefore, the inner liquidfilm velocity will increase. The thickness of outer liq-uid layer will decrease obviously with increasing outerwall heat flux. This effect is smaller than that of liquidfilm acceleration, because the wall heat flux (the max-imum value is 50 kW m−2) is low relatively. Thus thevelocity in the outer liquid layer will increase slightly.The effects of inner wall heat flux on the velocity pro-files of the inner and outer liquid films are very similarto those of outer wall heat flux on the velocity profilesof outer and inner liquid films (Fig. 6c and d). Thatis, the change of inner wall heat flux will almost haveno effects on the liquid layer velocity. As observed inFig. 6, the velocity gradient of the outer liquid layer

is larger than that of the inner liquid layer, and thevelocity gradient in the liquid layer will increase withan increase of gap. The influence of wall heat flux onvelocities becomes small with increasing the gap size.

Fig. 7 shows the liquid layer thickness will decreasealong axial length. The inner liquid layer is thinnerthan the outer liquid layer under same conditions. Be-cause: (1) in single-phase flow, both in the laminar andturbulent flow regimes, τi > τo (Knudsen and Katz,1958). This is also assumed to be true in two-phase an-nular flow at the liquid/vapor interface (Doerffer et al.,1997). (2) Dep,o > Dep,i (Saito et al., 1978; Doerfferet al., 1997). The thickness will increase with increas-ing annular gap with same mass flow rate, pressure,and heat flux. It will decrease with increasing heatflux with same mass flow rate. The influences of outerwall heat flux or inner wall heat flux on the inner liq-uid layer thickness or outer liquid layer thickness arevery slight. These are similar to the influences of heatflux on temperature and velocity in the liquid layers.With same mass flow rate and an increase of wall heatflux, the liquid layer which covers on the correspon-dent wall will clearly become thin due to the increaseof evaporation. At the same time, the velocity of vaporcore will increase, the drag force to liquid film willbecome large, therefore the liquid film covers on theanother wall will become thin.

Fig. 8 shows the effects of heat flux on two-phaseheat transfer coefficient and the curves of heat trans-fer coefficient along axial length. The two-phase heattransfer coefficient increases along the axial length.The liquid layer will become thin and the tempera-ture gradient in the liquid layer will decrease along theaxial length, thus the heat transfer coefficient will in-crease continuously with same heat flux. As describedabove, the thickness of liquid layer and temperaturegradients in the liquid layer will decrease with in-crease of heat flux, thus, the heat transfer coefficientwill increase with increasing heat flux. The two-phaseheat transfer coefficient of outer wall will increaseobviously with increasing outer wall heat flux, whilethe amplitude of increase will increase with increas-ing heated length. The effect of outer wall heat fluxon inner wall heat transfer coefficient is very small.Similarly, the inner wall heat transfer coefficient willincrease with increasing inner wall heat flux. And itseffect on outer wall heat transfer coefficient is alsoslight. The heat transfer coefficient will increase with

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Fig. 9. Effects of critical quality on critical heat flux. (a) Effects of critical quality on CHF occurs at inner tube with P = 3 MPa. (b) Effects of critical quality on CHFoccurs at outer tube with P = 3 MPa. (c) Effects of critical quality on CHF occurs at inner tube with P = 5 MPa. (d) Effects of critical quality on CHF occurs at outer tubewith P = 5 MPa. (e) Effects of critical quality on CHF occurs at inner tube with P = 10 MPa. (f) Effects of critical quality on CHF occurs at outer tube with P = 10 MPa.

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decreasing annular gap with same mass flow rate andheat flux. Therefore, the smaller the annular gap, thestronger heat transfer.

6. Predicting for critical heat flux (CHF)

The present model may be used to predict CHFin the narrow annuli with bilateral heating. The heatflux at δ = 0 will be the CHF. Although the CHFoccurs when the liquid film is enough thin and breaksdown easily around the dry patches, it is reasonableto assume that the CHF occurs when the liquid filmvanishes, that is, δ = 0. The local liquid film mass flowrate will be zero at δ = 0. Based on this conception,The CHF may be predicted.

Fig. 9 plots the CHF as a function of critical qualityin five different annuli gap and for three values ofpressure. At a fixed value of qo, the CHF of the innerwall increases with the decrease of the critical quality(Fig. 9a, c, and e). At the same time, Fig. 9a, c, and eshow the effects of the outer wall heat flux on theCHF of the inner wall. At a fixed CHF of the innerwall, the critical quality increases with the increaseof the outer wall heat flux. With the increase of theouter wall heat flux, the more liquid in the liquid filmswill evaporate. Thus, the critical quality will increase.

Fig. 10. Effects of mass flow rate on critical quality.

Fig. 9b, d, and f show the profiles of the outer wall CHFwith the change of the critical quality. The results aresimilar to those of the inner wall. As shown in Fig. 9,the critical quality decreases with the increase of thegap width. In the same annular channel, at a certainCHF, the critical quality when CHF occurs at the outerwall is larger than that when CHF occurs at the innerwall.

Fig. 10 plots the critical quality as a function ofmass flow rate in five different annuli gap and for twovalues of pressure. For the same outer wall heat fluxand inner wall heat flux, the critical quality decreaseswith the increase of the mass flow rate. The velocityof the vapor core will increase with the increase of themass flow rate, and the shear stress on the liquid/vaporinterface will increase, the more liquid droplets willbe entrained into the vapor core. Therefore, the criticalquality will decrease. Fig. 10 also describes the effectof the annular gap width on the critical quality. Forthe same mass flow rate, the critical quality decreaseswith increase of the annular gap width. The larger thegap width of the annular channel, the thicker the liquidfilms on the channel walls. Under the same conditions,the liquid film in the large gap is thicker than that inthe small gap, the vapor core can more easily entrainthe liquid droplets from the liquid film into the vaporcore. Thus, the critical quality will decrease.


Fig. 11. Effects of pressure on critical quality with different mass flow rate.

Fig. 11 shows the effects of the pressure on the crit-ical quality. Under lower mass flow rate conditions,the critical quality increases slowly with the increaseof pressure. While for higher mass flow rate, the criti-cal quality increases slightly with the increase of pres-sure and then has a maximum value at about 5 MPa.Above this pressure, critical quality decreases slowlywith the further increase of pressure.

Fig. 12 shows the effects of tube radius (cur-vature of heating surface) on CHF with the same

annular gap size (2 mm) under the same pressureand same mass flow rate conditions, in which DINand DOUT are the outer diameter of inner tube andthe inner diameter of outer tube, respectively. Un-der the same CHF conditions, the larger the tuberadius, the smaller the critical quality. Curvature ofthe heating surface usually has a significant effecton distribution of liquid film. Curvature radius is itsradius for circular tube. Concave surface which hassmaller radius keep liquid film on the surface much


Fig. 12. Effects of tube radius on CHF.

better than that which has larger radius or convexsurface.

7. Conclusions

A separated flow model of annular upward flow in avertical annuli gap with bilateral heating is presentedin this study. By numerically solving the proposedmodel, the liquid film thickness, velocity, and temper-ature fields in the liquid layer, two-phase heat trans-fer coefficient and pressure gradient can be obtained.The following conclusions have been found throughthe analysis of the results of this model:

(1) With same mass flow rate and heat flux, the thick-ness of liquid film in the annular narrow channelwill decrease with decreasing the annular gap, thetwo-phase heat transfer coefficient will increasewith the increase of heat flux and the decrease ofthe annular gap. That is, the heat transfer will beenhanced with small annular gap.

(2) The effects of outer wall heat flux on velocity andtemperature in the outer liquid layer, thickness

of outer liquid film and outer wall heat transfercoefficient are clear and obvious. The effects ofouter wall heat flux on velocity and temperaturein the inner liquid layer, thickness of inner liquidfilm and the inner wall heat transfer coefficient arevery small; the similar effects of the inner wallheat flux are found.

(3) Under same conditions, the outer liquid film isthicker than the inner liquid film.

(4) The present model can be used to calculate theCHF and critical quality in the narrow annular gap.

Appendix A. ρc, αc, and uc

The momentum conservation equation for vaporcore can be written as:

−dP

dZ= −

(dP

dZ

)frict

−(

dP

dZ

)acc

−(

dP

dZ

)elev

= τiPri + τoPro

Aαc+ 1

αc

d

dZ(ρcαcu

2c) + ρcg

(A.1)


where, −dP/dZ is the total pressure drop;−(dP/dZ)frict is the frictional pressure drop due tothe frictional loss; −(dP/dZ)acc is the accelerationpressure drop due to the momentum change; and−(dP/dZ)elev is the elevation pressure drop due tothe effect of gravitational force field. ρc, αc, and uc

are the density, void fraction, and average velocity ofvapor core, respectively.

The frictional pressure drop can be obtained by:

−(

dP

dZ

)frict

= 2π(ri + δi)τi + 2π(ro − δo)τo

Aαc

= 2[(ri + δi)τi + (ro − δo)τo]

(ro − δo)2 − (ri + δi)2(A.2)

The elevation pressure drop can be calculated by:

−(

dP

dZ

)elev

= ρcg = ρg[ϕ(1 − x) + x]

[x + ϕ(1 − x)ρg]/ρ× g

(A.3)

where, x is equilibrium quality.

ρc = (Wl)E + Wg

(Wg/ρg) + [(Wl)E/ρf)]

= W(1 − x)ϕ + Wx

(Wx/ρg) + [W(1 − x)ϕ/ρf ]

= (1 − x)ϕ + x

(1 − x)ϕ(ρg/ρf) + x× ρg

(A.4)

where, (Wl)E is the mass flow rate or droplets in thevapor core.

The acceleration pressure drop can be expressed by:

−(

dP

dZ

)acc

= 1

αc

d

dZ(ρcαcu

2c) (A.5)

whereuc = W

x + ϕ(1 − x)

ρcAc= G

x + ϕ(1 − x)

ρcαc

. (A.6)

Eqs. (A.4) and (A.6) are substituted into Eq. (A.5), weget:

−(

dP

dZ

)acc

= G2

αc

d

dZ

x2 + ϕ(1 − x)

ρgαc︸︷︷︸A

+ ϕ2(1 − x)2 + x(1 − x)ϕ

ρfαc︸︷︷︸B

(A.7)

where, G is mass flow velocity.dA

dz= d

dZ

[x2 + ϕ(1 − x)x

ρgαc

]

= 1

ρgαc[ϕ + 2(1 − ϕ)x]

dx

dZ

−ϕx + (1 − ϕ)x2

ρgα2c

dαc

dx

dx

dz

(A.8)

dB

dz= −2ϕ2(1 − x) + ϕ(1 − 2x)

ρfαc

dx

dZ

−ϕ2(1 − x)2 + x(1 − x)ϕ

ρfα2c

dαc

dx

dx

dz

(A.9)

Eqs. (A.8) and (A.9) are substituted into Eq. (A.7), weget:

−(

dP

dZ

)acc

= G2

α2c

1

ρg[ϕ + 2(1 − ϕ)x] + −2ϕ2(1 − x) + ϕ(1 − 2x)

ρf

dx

dZ

− G2

α3c

ϕx + (1 − ϕ)x2

ρg+ ϕ2(1 − x)2 + x(1 − x)ϕ

ρf

dαc

dx

dx

dz(A.10)

The following assumptions are made to calculatethe void fraction in the vapor core:

(1) There is no slip between the vapor and thedroplets, as described in Section 2;

(2) Keep thermodynamic balance between the vaporand the droplets. The mass quality can be obtainedby energy balance;

(3) The dynamic pressure of liquid phase is equal tothat of vapor core. This assumption is also calledas equivalent velocity head model. Then,

(ul)2f

2ρf = u2

c

2ρc (A.11)


Based on its definition, ϕ can be expressed as:

ϕ = (Wl)E

(Wl)f + (Wl)E(A.12)

where, (Wl)f is the liquid film mass flow rate.Also, based on its definition, the mass quality can be written as:

x = Wg

Wg + (Wl)f + (Wl)E(A.13)

Eq. (A.4) is substituted into Eq. (A.11), then

uc

(ul)f=(ρf

ρc

)1/2

=[(ρf/ρg)x + (1 − x)ϕ

x + (1 − x)ϕ

]1/2

(A.14)

If the total flow area is denoted as A, it may be the sum of the flow areas of every part: liquid film, droplets,and vapor. That is:

A = Wg

ρguc+ ϕ[(Wl)f + (Wl)E]

ρfuc+ (1 − ϕ)[(Wl)f + (Wl)E]

ρf(ul)f(A.15)

Based on its definition of void fraction, it can be written as:

α= Ag

A= Wg/(ρguc)

A=[

1 + ρg

ρfϕ

(1 − x

x

)+ (1 − ϕ)

ρg

ρf

(1 − x

x

)uc

(ul)f

]−1

=

1 + ρg

ρfϕ

(1 − x

x

)+ (1 − ϕ)

ρg

ρf

(1 − x

x

)[(ρf/ρg)x + (1 − x)ϕ

x + (1 − x)ϕ

]1/2−1

(A.16)

The void fraction in vapor core can be obtained by:

αc = Ag + (Al)E

A= 1 + (ρg/ρf)ϕ[(1 − x)/x]

1 + (ρg/ρf)ϕ[(1 − x)/x] + (1 − ϕ)(ρg/ρf)[(1 − x)/x](uc/(ul)f)

= 1 + (ρg/ρf)ϕ[(1 − x)/x]

1 + (ρg/ρf)ϕ[(1 − x)/x]︸︷︷︸AA

+ (1 − ϕ)(ρg/ρf)[(1 − x)/x]︸︷︷︸EE

[(ρf/ρg)x + (1 − x)ϕ

x + (1 − x)ϕ

]︸︷︷︸

BB︸︷︷︸CC

1/2

︸︷︷︸DD

(A.17)

αc is differentiated with respect to x, then dαc/dx can be obtained by

dαc

dx= d(AA/DD)

dx= d(AA)/dx

DD+ (AA)

d(1/DD)

dx

= d(AA)/dx

DD− (AA)

(DD)2d(DD)

dx(A.18)

where,d(AA)

dx= −ρg

ρfϕ

1

x2(A.19)


d(CC)

dx= (EE)

d(BB)

dx+ (BB)

d(EE)

dx(A.20)

d(EE)

dx= −(1 − ϕ)

ρg

ρf

1

x2(A.21)

d(BB)

dx= 1

2

[(ρf/ρg) − ϕ]/[x + (1 − x)ϕ] − [(ρf/ρg)x + (1 − x)ϕ](1 − ϕ)/[x + (1 − x)ϕ]2[(ρf/ρg)x + (1 − x)ϕ]/[x + (1 − x)ϕ]1/2

(A.22)

Eqs. (A.21) and (A.22) are substituted into Eq. (A.20),then we get

d(CC)

dx

= 1

2(1 − ϕ)

ρg

ρf

(1 − x

x

) [(ρf/ρg) − ϕ]/[x + (1 − x)ϕ] − [(ρf/ρg)x + (1 − x)ϕ](1 − ϕ)/[x + (1 − x)ϕ]2[(ρf/ρg)x + (1 − x)ϕ]/[x + (1 − x)ϕ]1/2

−[(ρf/ρg)x + (1 − x)ϕ

x + (1 − x)ϕ

]1/2

(1 − ϕ)ρg

ρf

1

x2(A.23)

d(DD)

dx= d(AA)

dx+ d(CC)

dx(A.24)

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