annular upward flow

8/12/2019 annular upward flow

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Nuclear Engineering and Design 225 (2003) 219247

Theoretical calculation of annular upward ow 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,

Xian J iaotong University, No. 28, Xianning West Road, Xian City 710049, PR China

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

Abstract

Based on separated ow, a theoretical three-uids model predicting for annular upward ow 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 liquidlms and the momentum conservation equation of vapor core.Through numerically solving the equations, liquid lm thickness, radial velocity, and temperature distribution in liquid lms,heat transfer coefcient of inner andouter tubes andaxial pressure gradient are obtained. Thepredicted results are compared withthe experimental data and good agreements between them are found. With same mass ow rate and heat ux, the thickness of liquid lm in the annular narrow channel will decrease with decreasing the annular gap. The two-phase heat transfer coefcientwill increase with the increase of heat ux and the decrease of the annular gap. That is, the heat transfer will be enhanced with

small annular gap. The effects of outer wall heat ux on velocity and temperature in the outer liquid layer, thickness of outerliquid lm and outer wall heat transfer coefcient are clear and obvious. The effects of outer wall heat ux on velocity andtemperature in the inner liquid layer, thickness of inner liquid lm and the inner wall heat transfer coefcient are very small; thesimilar effects of the inner wall heat ux are found. As the applications of the present model, the critical heat ux and criticalquality are calculated. 2003 Elsevier B.V. All rights reserved.

1. Introduction

Two-phase annular ow 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 ux 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 efcient heat transfer techniques to ac-commodate these high heat uxes ( Warrier et al.,2002 ). One of the simplest heat transfer systems may

be using single-phase forced convection or two-phaseow boiling in small tubes or rectangular channelsor annular gaps or lunate channels. Experiments haveshowed that annular ow are main ow 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 ow is formed. Annularow is a particularly important ow pattern since itfrequently occurs in a wide range of such parametersas quality, pressure and ow conditions. In this regime,

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


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G.H. Su et al. / Nuclear Engineering and Design 225 (2003) 219247 221

and Karabelas, 1991 ). Experimental data of annularow have been obtained by many workers and nu-merous correlations have been developed. Examples

include those of: Hurlburt and Newell (1999) , Holtet al. (1999) , Fu and Klausner (1997) , Klausner et al.(1991) and Jung and Radermacher (1989) f or pressuredrop, and Nozu 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 ow boiling heat transfer inannular gap. Hasan et al. (1990) reported that exper-iments on subcooled ow 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 uid. Flow boiling heattransfer data were reported at three mass velocities(579, 801, and 1102kg m 2 s), three pressure (312,277, and 243 kPa), and three inlet subcoolings (20.0,30.0, and 36.5 C). A multiple-hysteresis phenomenonis identied. The measured wall heat transfer coef-cients were compared with those predicted by various

correlations 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 ow veloc-ities of 0.081.2 m s 1 and subcoolings of 1060 C.The maximum bubble diameter and condensation timewere shown to be inuenced by the location relative tothe onset of signicant void. Shah (1976, 1983) pre-sented that a general correlation for heat transfer dur-ing subcooled and saturated boiling in tubes had been

compared with a large amount of data for boiling in an-nuli. The data included the uids water, Fron-113, andmethanol; mass ow rate from 278 to 7774 kg m 2 s;heated uxes from 0.03 to 4.4 kW m 2 , liquid sub-coolings from 2 to 109 C; and reduced pressures from0.009 to 0.89. The inner tube diameter varied from4.5 to 42.2mm 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 ow boiling heat trans-fer and visualized the associated bubble characteristics

for refrigeration R-134a owing 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 imposed

wall heat ux, mass ow 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 lm thickness, distri-bution of velocities and temperatures in the liquidlayer for vertical two-phase ow in an annular gapin the range of heat ux from 6000 to 12000 W m 2;mass ow rate from 500 to 1100 kg m 2 s. Su et al.(2003) presented a theoretical model which describesthe model of vertical upward annular ow in nar-row annular channel with bilateral heating under lowmass ow rate. However, the proles of velocity inthe liquid layers are linear due to that the number of calculation 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 ow in a vertical upward annular gap.

2. Analytical model

2.1. Basic equations

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

(1) The ow is steady and incompressible;(2) The liquid lm 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.


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222 G.H. Su et al. / Nuclear Engineering and Design 225 (2003) 219247

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

In the annular region, there exists the mass transferbetween the liquid droplets in the vapor core and theliquid lms. On the one hand, the liquid droplets inthe vapor core will be deposited on the liquid lmsurfaces. On the other hand, the liquid droplets will be

continuously entrained from the liquid lms 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 liquidlms. Based on these observations, the mass ow rateof the liquid lms will be changed as a function of thedistance Z along the channel. The mass conservationequations of the inner and outer liquid lms are asfollows, respectively.

For the inner lm:

(dW li) f dZ

= P ri D epi E ni qih fg (1)

(dW lo) f dZ

= P ro D epo E no qoh fg

(2)

where, W li and W lo are the mass ow rates of the in-ner and outer liquid lms, respectively; P ri and P ro arethe peripheries of the inner and outer liquid lms, re-spectively; Depi and Depo are the liquid droplets depo-sition rates per unit area of the inner and outer liquid

lms, respectively; E ni and E no are the liquid dropletsentrainment rates per unit area of the inner and outerliquid lms, respectively; qi and qo are the heat uxes

of the inner and outer tubes, respectively; hfg is thelatent heat of vaporization.A cylindrical element of the inner liquid lm is

considered now. Its height is Z , its inner radius is r ,and its outer radius is ( r i + i). According to the forcebalance on this element, the momentum conservationequation of the inner lm will be

2r Z = 2(r i + i) i Z

f g (r i + i)2 r 2 Z

dP

dZ

[(r i + i)2 r 2] Z (3)

where, r i is the outer radius of the inner tube; i isthe thickness of the inner liquid lm; is shear stressat radius r in the inner liquid lm; i is shear stressat the liquid/vapor interface of inner liquid lm; f isthe liquid density; g is the gravitational acceleration.

The shear stress in the inner liquid lm can be ob-tained from Eq. (3), that is

=(r i + i)

r i

12

f g +dP dZ

(r i + i)2 r 2

r(4)

The shear stress at radius r in the outer liquidlm can be obtained using the same method. It can bewritten as:

=(r o o)

r o

12

f g +dP dZ

r 2 (r o o)2

r(5)

where, r o is the inner diameter of the outer tube; ois the thickness of the outer liquid lm; o is shearstress at the liquid/vapor interface of the outer liquid

lm; dP /d Z is axial pressure gradient, and can be cal-culated by the momentum conservation equation of vapor core. Thus, it can be expressed by

dP dZ

= iP ri + oP ro

A c+

1 c

ddZ

( c cu 2c ) + cg (6)

where, c, c, 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 lm, the ve-locity in the liquid lm can be obtained according to


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the Newton viscosity law:

u =

0

E

dy (7)

The boundary conditions of Eq. (7) are as follows:

y = 0, u = 0

y = ,dudy

= int E

where, y is the distance from the wall; u is the localliquid velocity in the liquid lm; int is the shear stressat the liquid/vapor interface; E is the effective viscos-ity of the uid. E = f for laminar ow, and E = f + f for turbulence ow; is the eddy viscosity.

The mass ow rate of liquid lm can be obtained

by integrating the velocity:

(W l)f = 0 2r f u dy (8)The thickness of the liquid lm is very small, thus theconvection can be ignored. The axial conduction canalso be ignored if the ow is in one-dimension. Thus,the energy equation is as follows:

ddy

( l + H)dT dy

= 0 (9)

where, l is the thermal diffusivity; H is the eddythermal diffusivity, its value will be zero when theliquid lm ow is laminar ow.

The boundary conditions of Eq. (9) are as follows:

qw = kf dT dy

, y = 0

T = T sat , y =

where, qw is the wall heat ux; T is the temperature;T sat is the saturated temperature of uid; k f is thethermal conductivity.

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

The present study calculates the heat transfer coef-cient of annular gap according to its basic concept.The heat transfer coefcient of two-phase ow is de-

ned byh =

qT w T sat

(10)

where, q is heat ux; T w is the wall temperature.

2.2. Constitutive equations

The variables , H , i and E n must be speciedin order to simulate the annular ow 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)

i =12

f igu 2g (11)

where, f i/f go = (1+ 300)/D e; D e = D o D i; f go =0.079 Re 0.25g ; and Reg = GxD e/ g.

Detailed turbulence measurements for the liquidlms of annular ow in the vertical annular gap arenot available. The usual practice in treating the liquid

lm eddy viscosity is to assume the wall turbulenceis similar to that of single-phase ow and employ thesingle-phase ow formulas for liquid lm eddy viscos-ity. Although such an assumption has yet to be exper-imentally validated, modied single-phase ow eddyviscosity and turbulent Prandtl number can be used(Fu and Klausner, 1997 ). Thus, can be calculated bythe following equations:

= 0.001y+ 3 , y + < 5

= Ky 1 exp y+

25

2 du

dy1.0

y

n, y + 5

(12)

where, = (/) , is dynamic viscosity. K isVon Karmans constant and equals 0.41. The fac-tor which has been introduced in Eq. (12) is toaccount for enhanced turbulence in the liquid lmcaused by the disturbance created by the evapora-tion radial momentum ux and is expressed by =B0.3o [(1 x)/x ]

0.1, where the boiling number is de-ned as Bo = q/( Gh fg) . The wall units y+ = yu/ ,and u = ( w / f )1/ 2 .


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

Pr t = H

(13)

where, Pr t is the turbulent Prandtl number and can becalculated by

Pr t = 1.07, y + < 5

Pr t = 1.0 + 0.855 tanh[0 .2(y + 7.5)], y + 5(14)

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

D epD e f

= 0.022 Re0.74f g f

0.26

0.74 (15)

where, is the fraction of liquid owing as dropletsin the vapor core and is dened as ( Ishii and Mishima,1984; Celata et al., 2001 ): = (j f j ff )/j f , where, jf is the volumetric ux which pertains to the liquidphase, and jff is the volumetric ux which pertainsto the liquid lm. Ref is the liquid Reynolds number

when the liquid singly ows 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 lm and the vapor core islarger than that between the inner liquid lm and thevapor core. Therefore, the rate of deposition on theouter liquid lm is larger than that on the inner liquidlm. Thus, the Eq. (15) must be modied in order tobe used to calculate the deposition rate in a bilateralheating annuli by a dimensionless factor K (Doerfferet al., 1997 )

D ep,K = D ep K (16)

where, K is a modied factor and is dened as: K =rK/(r o + r i) , K = i, o (i denotes inner liquid lm ando denotes outer liquid lm).

On the other hand, the deposition may be blockedby the evaporative effect on the interface between theliquid lm and vapor core. Thus, the deposition ratecaused by this effect can be given by ( Milashenko andNigmatulin, 1989 )D ep,e,K = C K (17)

where, K is the block coefcient, and is dened by K = ( g/ 0.065 f )V g,K, and V g,K = (qK /h fgg) isthe evaporative velocity of liquid lm, K = i, o asabove. C is the concentration of droplet in the vaporcore and can be calculated by C = W le /(W le + W g) ,

W le and W g is mass ow rate of liquid droplet and of vapor in the vapor core, respectively. Thus, the depo-sition rate may be nally calculated byD ep, tol,K = D ep,K + D ep,e,K (18)

The droplet entrainment rate, E n , can be calculatedconsidering the contribution of two different mecha-nisms of droplets formation: breakup of disturbancewaves and boiling in the liquid lm ( Mishima et al.,1989 ). According to the analysis of Kataoka 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 of Mishima et al.(1989) and Kay (1994) , burst of boiling bubbles inthe liquid lm may cause droplet entrainment at suf-ciently high heat ux. Thus, entrainment rate may beobtained by the following equation:E n = E nw + E ne (19)

where, E nw is wave droplet entrainment rate (entrain-ment due to disturbance waves); E ne is boiling dropletentrainment rate (entrainment due to boiling in the liq-

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

E nwD e f

= 0.72 10 9 Re1.75f We(1 )0.25 1

2,

1

E nwD e f

= 0.022 Re0.74f g f

0.74 ,

> 1

E nw = 0, Reff < Reff c

(20)


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where, is the equilibrium entrainment fraction(Ueda et al., 1981 ), and can be given by =tanh(7.25 10 7We1.25 Re0.25f ), and We is Weber

number for entrainment: We = ( gj 2g D e/) [( f

g)/ g]1/ 3. is the surface tension. Ref is liquidReynolds number, and can be given by Ref = (G( 1 x)D e)/ f , where, Reff is the liquid lm Reynoldsnumber, and can be calculated by Reff = G( 1 x)( 1)D e / f . Re ff c is the minimum liquid lm Reynoldsnumber, below which no droplet entrainments ex-ist due to supercial waves occurs ( Celata et al.,2001 ), it can be calculated by Reffc = (y ++ / 0.347)1.5

( f / g)0.75( g/ f )1.5 , where y++ is the non-dimen-sional distance associated with the onset of the en-

trainment. According to Celata et al. (2001) , y++

=10.0.The entrainment due to boiling in the liquid lm,

E ne , may be given by the correlation of Ueda et al.(1981) :

E neh fgq

= 4.77 102 2qh 2fg g

0.75

. (21)

The above correlations for entrainment are onlysuitable to circular tubes. They must be modied inorder to be used to calculate the entrainment rates in

a bilateral heating annuli. That is:E nw,K = E nw K . (22)

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

3. Solution procedure

The relations above are closed now. Knowing the

properties of uid, heat uxes, mass ow rate, qual-ity, and geometric parameters, the above model canbe solved. Fig. 2 is a owchart describing the solu-tion procedure to the above equations, which is alsosummarized as:

(1) An initial guess is made for the lm thickness:i and o (i denotes the lm thickness of in-ner liquid lm and o denotes that of outer liquidlm).

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

(3) Eq. (7) is used in conjunction with Eqs. (4) and (5)to compute the inner and outer liquid lm velocityproles: u i( y) and uo( y), and Eq. (8) is employedto calculate the inner and outer liquid lm massow rates. The inner liquid lm mass ow rate ishereby denoted by W li,3 , and the outer liquid lmmass ow rate is denoted by W lo,3.

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

li,4, and

the outer liquid lm mass ow rate is denoted byW lo,4.


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(5) Check if W li,3 equals to W li,4, and W lo,3 equalsto W lo,4 . If not, guess the initial values of thelm thickness i and o again. Steps (2)(5) are

repeated.(6) The heat transfer coefcient dened in Eq. (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 xed step is employed,the velocity proles in the liquid lm will be linear(Su et al., 2003 ).

4. Verications of present model

To validate the correction of proposed model, anattempt has been made to put together a databasethat covers a wide range of ow 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 veried by ourexperimental data. Fig. 3 is a sketch of the experimen-tal apparatus used in this study to investigate the ow

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 the

primary 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 ow meter, a testsection, a condenser, and connected pipes. The tem-perature and mass ow rate in the secondary loop mustbe controlled to have enough cooling capacity forcondensing the uid 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 N 2 throughthe nitrogen (N 2) charging line, the system pressurecan be regulated. By adjusting the gate opening of theby-pass valve, the mass ow rate of the test section canbe regulated. Note that the mass ow rate and systempressure should be further adjusted simultaneouslyby the needle valve at the upstream of the ow metersection and the power of the heater in the pressurizer,respectively, in order to control them at the requiredlevels. The mass ow rate of test section is measuredby a precise mass ow meter which consists of


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orice, three valves and differential pressure trans-mitter. After subcooled, the distilled water ows back to 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 coefcient between the experiment and the present model with s = 0 .95 mm in the axial direction. (b) Oneexample of comparison of two-phase heat transfer coefcient between the experiment and the present model with s = 2 mm in the axialdirection. (c) Comparison of two-phase heat transfer coefcient 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 pressure

transducer. The temperatures of uid are measured by1.0 mm K-type NiCrNiSi thermocouples. The innerwall temperatures of inner tube are measured with 15


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Fig. 4. (Continued ).


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Table 1Accuracy of measurement system

Parameters Uncertainty (%)

Temperature 0.75Mass ow 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 90

apart. The accuracy of the measurements is listed inTable 1 .

The two-phase heat transfer coefcients and axialpressure gradients obtained by the proposed modelare compared with those of experiments, as shownin Fig. 4. The Fig. 4a and b show the comparison of two-phase heat transfer coefcient 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 heat

transfer coefcient were over-predicted ( Fig. 4ac )within above error range. That is to say, there are about70% two-phase heat transfer coefcients 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 wall

temperatures 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 lm also decreases alongthe axis. Therefore, the wall temperatures obtainedby the energy equation of the present model alsodecrease along the axis, as shown in Fig. 4e. Thus,(T w T sat) obtained by proposed model is lower thanthat obtained by experiment, and the two-phase heattransfer coefcient obtained by proposed model islarger than that obtained by experiment.

Fig. 4d shows 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 owmodel, which are developed for large tube. Althoughthe modied 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 rst step, the error ( 30%)is acceptable.

5. Results and discussions

Fig. 5 shows the temperature proles in the liquidlms obtained by the proposed model with six differ-ent annular gap and for three values of wall heat uxof inner tube or of outer tube under the mass owrate is 120kgm 2 s 1 conditions. Abscissas denotethe non-dimensional distance from wall and y+ = y/in Figs. 5 and 6, where y is the distance from walland is the thickness of liquid lms. y+ = 0 is at thewall, and y+ = 1 is at the liquid/vapor interface. The y-axis denotes non-dimensional temperature T i / T sat or

T o / T sat in Fig. 5, where T i is the temperature in theinner liquid layer, T o is the temperature in the outerliquid layer, and T sat is the saturated temperature. Twogures in which the gap size is different are shown un-der the same conditions in Figs. 58 in order to clearlyexpress the general parametric trends of gap size. Asobserved in Fig. 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 lm mass ow rate is very low

(120kgm 2

s 1

), thus the ow 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 ux onthe temperature in the inner liquid layer with differentannular gap with a xed heat ux of inner tube, qiw =50kWm 1. The effects are very small. With increaseof outer wall heat ux, the temperature in the innerliquid layer decreases very slightly. The more liquidwill evaporate with increasing outer wall heat ux,the velocity of vapor core will increase slightly. The


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Fig. 5. The temperature curves in the liquid lms. (a) Effects of outer wall heat ux on the temperature proles in the inner liquid layer. (b) Eon the temperature proles in the outer liquid layer. (c) Effects of inner wall heat ux on the temperature proles in the inner liquid layer. (ux on the temperature proles in the outer liquid layer.


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Fig. 5. ( Continued ).


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Fig. 6. The velocity pro les in the liquid layers. (a) The velocity pro les in the inner liquid layer with different outer wall heat ux. (b) liquid layer with different outer wall heat ux. (c) The velocity pro les in the inner liquid layer with different inner wall heat ux. (d)liquid layer with different inner wall heat ux.


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


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Fig. 8. The two-phase heat transfer coef cient in annular gap. (a) Effects of outer wall heat ux on the two-phase heat transfer coef couter wall heat ux on the two-phase heat transfer coef cients of outer tube. (c) Effects of inner wall heat ux on the two-phase heat transfeEffects of inner wall heat ux on the two-phase heat transfer coef cients of outer tube.


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inner liquid lm will be entrained and become thin.Thus, its temperature gradient will decrease slightly.Fig. 5b shows the effects of outer wall heat ux on

the temperature in the outer liquid layer with dif-ferent annular gap. The effects are very clear andobvious. The temperature gradient in the outer liquidlm will increase with an increase of outer wall heatux. Although the liquid layer will become thin withincreasing outer wall heat ux, the effects on temper-ature gradient in outer liquid layer is larger than thoseon liquid lm thickness. Fig. 5c and d show the ef-fects of inner wall heat ux on the temperature in theliquid layer. The effects are similar to those of Fig. 5aand b . There are very slight in uences of inner wallheat ux 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 ow rate and heat ux. Thethickness of liquid layer will increase with an increaseof gap, thus the temperature gradient will increase.

Fig. 6 shows the velocity pro les in the liquid lms.As described above, y+ is the non-dimensional dis-tance from the wall. The velocities of liquid lms atthe wall are zero due to the effect of viscosity. The ef-fects of heat ux on velocity pro les are very small.

With increasing outer wall heat ux, the velocity inthe inner liquid layer will increase slightly ( Fig. 6a ),the velocity in the outer liquid lm will also increaseslightly ( Fig. 6b ). The evaporation of the outer liq-uid lm will increase with an increase of outer heatux, 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 liquidlm velocity will increase. The thickness of outer liq-uid layer will decrease obviously with increasing outer

wall heat ux. This effect is smaller than that of liquidlm acceleration, because the wall heat ux (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 ux on the velocity pro-les of the inner and outer liquid lms are very similarto those of outer wall heat ux on the velocity pro lesof outer and inner liquid lms ( Fig. 6c and d ). Thatis, the change of inner wall heat ux 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 in uence of wall heat ux on

velocities 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 ow, both in the laminar andturbulent ow regimes, i > o (Knudsen and Katz,1958 ). This is also assumed to be true in two-phase an-nular ow at the liquid/vapor interface ( Doerffer et al.,1997 ). (2) D ep ,o > D ep , i (Saito et al., 1978; Doerfferet al., 1997 ). The thickness will increase with increas-ing annular gap with same mass ow rate, pressure,and heat ux. It will decrease with increasing heatux with same mass ow rate. The in uences of outerwall heat ux or inner wall heat ux on the inner liq-uid layer thickness or outer liquid layer thickness arevery slight. These are similar to the in uences of heatux on temperature and velocity in the liquid layers.With same mass ow rate and an increase of wall heatux, 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 lm willbecome large, therefore the liquid lm covers on the

another wall will become thin.Fig. 8 shows the effects of heat ux on two-phase

heat transfer coef cient and the curves of heat trans-fer coef cient along axial length. The two-phase heattransfer coef cient 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 coef cient will in-crease continuously with same heat ux. As describedabove, the thickness of liquid layer and temperaturegradients in the liquid layer will decrease with in-

crease of heat ux, thus, the heat transfer coef cientwill increase with increasing heat ux. The two-phaseheat transfer coef cient of outer wall will increaseobviously with increasing outer wall heat ux, whilethe amplitude of increase will increase with increas-ing heated length. The effect of outer wall heat uxon inner wall heat transfer coef cient is very small.Similarly, the inner wall heat transfer coef cient willincrease with increasing inner wall heat ux. And itseffect on outer wall heat transfer coef cient is alsoslight. The heat transfer coef cient will increase with


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Fig. 9. Effects of critical quality on critical heat ux. (a) Effects of critical quality on CHF occurs at inner tube with P = 3MPa. (b) Eoccurs 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 qualiwith 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


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

6. Predicting for critical heat ux (CHF)

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

Fig. 9 plots the CHF as a function of critical qualityin ve different annuli gap and for three values of pressure. At a xed 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 ux on theCHF of the inner wall. At a xed CHF of the innerwall, the critical quality increases with the increaseof the outer wall heat ux. With the increase of theouter wall heat ux, the more liquid in the liquid lms

will evaporate. Thus, the critical quality will increase.

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

Fig.9b,d, and f show thepro les 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 of mass ow rate in ve different annuli gap and for twovalues of pressure. For the same outer wall heat uxand inner wall heat ux, the critical quality decreaseswith the increase of the mass ow rate. The velocityof the vapor core will increase with the increase of themass ow 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 ow rate, the critical quality decreaseswith increase of the annular gap width. The larger thegap width of the annular channel, the thicker the liquidlms on the channel walls. Under the same conditions,the liquid lm in the large gap is thicker than that inthe small gap, the vapor core can more easily entrainthe liquid droplets from the liquid lm into the vapor

core. Thus, the critical quality will decrease.


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

Fig. 11 shows the effects of the pressure on the crit-ical quality. Under lower mass ow rate conditions,the critical quality increases slowly with the increaseof pressure. While for higher mass ow 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 ow 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 of the heating surface usually has a signi cant effecton distribution of liquid lm. Curvature radius is itsradius for circular tube. Concave surface which hassmaller radius keep liquid lm on the surface much


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Fig. 12. Effects of tube radius on CHF.

better than that which has larger radius or convexsurface.

7. Conclusions

A separated ow model of annular upward ow in avertical annuli gap with bilateral heating is presentedin this study. By numerically solving the proposedmodel, the liquid lm thickness, velocity, and temper-ature elds in the liquid layer, two-phase heat trans-fer coef cient and pressure gradient can be obtained.The following conclusions have been found through

the analysis of the results of this model:

(1) With same mass ow rate and heat ux, the thick-ness of liquid lm in the annular narrow channelwill decrease with decreasing the annular gap, thetwo-phase heat transfer coef cient will increasewith the increase of heat ux and the decrease of the annular gap. That is, the heat transfer will beenhanced with small annular gap.

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

of outer liquid lm and outer wall heat transfercoef cient are clear and obvious. The effects of outer wall heat ux on velocity and temperaturein the inner liquid layer, thickness of inner liquidlm and the inner wall heat transfer coef cient arevery small; the similar effects of the inner wallheat ux are found.

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

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

Appendix A. c , c , and u c

The momentum conservation equation for vaporcore can be written as:

dP dZ

= dP dZ frict

dP dZ acc

dP dZ elev

= iP ri + oP ro

A c+

1 c

ddZ

( c cu2c ) + cg

(A.1)


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Based on its de nition, can be expressed as:

=(W l)E

(W l)f + (W l)E(A.12)

where, ( W l)f is the liquid lm mass ow rate.Also, based on its de nition, the mass quality can be written as:

x =W g

W g + (W l) f + (W l)E(A.13)

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

u c(u l) f

= f c

1/ 2

=( f / g )x + (1 x)

x + (1 x)

1/ 2

(A.14)

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

A = W g

gu c+ [(W l)f + (W l)E ]

f u c+ (1 )[(W l) f + (W l)E ]

f (u l)f (A.15)

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

=AgA

=W g /( gu c)

A= 1 +

g f

1 xx

+ (1 )g f

1 xx

uc(u l) f

1

= 1 +g f

1 xx

+ (1 )g f

1 xx

( f / g )x + (1 x)x + (1 x)

1/ 2 1

(A.16)

The void fraction in vapor core can be obtained by:

c =Ag + (A l)E

A= 1 + ( g / f )[(1 x)/x ]

1 + ( g / f )[(1 x)/x ] + (1 )( g / f )[(1 x)/x ](u c /(u l) f )

=1 + ( g / f )[(1 x)/x ]

1 + ( g / f )[(1 x)/x ]

AA+ (1 )( g / f )[(1 x)/x ]

EE( f / g )x + (1 x)

x + (1 x)

BB

CC

1/ 2

DD

(A.17)

c is differentiated with respect to x, then d c /d x can be obtained by

d cdx

=d(AA / DD )

dx=

d(AA )/ dxDD

+ (AA )d(1/ DD )

dx

=d(AA )/ dx

DD

(AA )(DD )2

d(DD )dx

(A.18)

where,d(AA )

dx=

g f

1x2

(A.19)


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d(CC )dx

= (EE )d(BB )

dx+ (BB )

d(EE )dx

(A.20)

d(EE )dx = (1 )

g f

1x2 (A.21)

d(BB )dx

=12

{[( f / g ) ]/ [x + (1 x) ]} { [( f / g )x + (1 x) ](1 )/ [x + (1 x) ]2}{[( f / g )x + (1 x) ]/ [x + (1 x) ]}1/ 2

(A.22)

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

d(CC )dx

=12

(1 )g

f

1 xx

{[( f / g ) ]/ [x + (1 x) ] [( f / g)x + (1 x) ](1 )/ [x + (1 x) ]2}{[(

f /

g)x + (1 x) ]/ [x + (1 x) ]}1/ 2

( f / g )x + (1 x)

x + (1 x)

1/ 2

(1 )g f

1x2

(A.23)

d(DD )dx

=d(AA )

dx+

d(CC )dx

(A.24)

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annular upward flow

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