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$: International Journal of Heat and Mass Transfer · PDF fileCorrelations of two-phase frictional pressure drop and void fraction in mini-channel W. Zhanga,1, T. Hibikib, K. Mishimac,*$
International Journal of Heat and Mass Transfer 53 (2010) 453–465

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Correlations of two-phase frictional pressure drop and void fraction in mini-channel

W. Zhang a,1, T. Hibiki b, K. Mishima c,*

a Graduate School of Energy Science, Kyoto University, Kyoto 606-8317, Japanb School of Nuclear Engineering, Purdue University West Lafayette, IN 47907-2017, USAc Research Reactor Institute, Kyoto University, Kumatori, Sennan, Osaka 590-0494, Japan

a r t i c l e i n f o

Article history:Received 12 March 2008Accepted 15 June 2008

Keywords:Frictional pressure dropTwo phaseVoid fractionFlow boilingMini-channelSmall diameter

0017-9310/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2009.09.011

* Corresponding author. Present address: Institutetute of Nuclear Safety System, Incorporated, 64 SatFukui 919-1205, Japan. Tel.: +81 770 37 9100; fax: +8

E-mail addresses: [email protected] (T. Hibiki), [email protected] (K. M

1 Present address: Department of General Technologing Research and Design Institute, 29 Hong Cao Road,

a b s t r a c t

Alternative correlations of two-phase friction pressure drop and void fraction are explored for mini-chan-nels based on the separated flow model and drift-flux model. By applying the artificial neural network,dominant parameters to correlate the two-phase friction multiplier and void fraction are picked out. Itis found that in mini-channels the non-dimensional Laplace constant is a main parameter to correlatethe Chisholm parameter as well as the distribution parameter. Both previous correlations and the newlydeveloped correlations are extensively evaluated with a variety of data sets collected from the literature.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In relation to many cutting-edge electronic chips, avionics, com-pact heat exchangers and bioengineering devices, mini-channelcooling technologies have attracted considerable attention in re-cent years. In comparison with single-phase flow, flow boiling isdeemed as an optimum option to be applied in mini-channels inview of its extremely high heat transfer efficiency at the cost ofsmall wall temperature rises. However, a penalty of flow boilingis the increased pressure drop and pressure fluctuation, which lim-it the applicable range of flow boiling in such devices. Therefore, acomprehensive understanding of pressure drop and void fractionduring two-phase flow in mini-channel is of considerable practicalimportance for the design and performance evaluation of suchcooling devices.

Starting from studies on adiabatic two-phase flow, extensiveexperimental and analytical efforts have been accumulated oncharacteristics of two-phase flow and/or flow boiling pressure dropin mini-channel. However, in regard to the applicability of existingcorrelations to mini-channel, there exist some discrepancies in theliterature. Mishima and co-workers [1,2] extensively studied

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of Nuclear Technology Insti-a, Mihama-cho, Mikata-gun,1 770 37 2008.(W. Zhang), [email protected]).

y, Shanghai Nuclear Engineer-Shanghai 200233, China.

air/water two-phase flow pressure drop in rectangular and circularmini-channels with diameters ranging from 1 to 5 mm, and foundthat the separated flow model could well predict their data and theChisholm’s parameter C [3] was successfully correlated by thehydraulic diameter of channel. However, Triplett et al. [4] reportedthat for bubbly and slug flows at high Reynolds numbers the exper-imental two-phase frictional pressure drop data agreed reasonablywell with the predictions of a homogeneous model with a mixtureviscosity, whereas at low Reynolds numbers or for annular flow,both the homogeneous mixture model and Friedel’s correlation[5] predicted the data poorly. In addition, Tran [6] measuredtwo-phase flow pressure drop during a phase-change heat transferprocess with three refrigerants (R-134a, R-12, and R-113) underpressures ranging from 138 to 856 kPa, and in two round tubes(i.d.: 2.46, 2.92 mm) as well as a rectangular channel (i.d.: 4.06mm). They reported that correlations for conventional channelsfailed to predict their experimental data. In contrast, Kawaharaet al. [7] investigated nitrogen/water two-phase flow in a quartzcapillary with the inner diameter of 100 lm. They showed that thetwo-phase friction multiplier data were in good agreement withexisting correlations for conventional channels.

To date, studies on void fraction in mini-channels are still lim-ited. Experimental investigations on void fraction in mini-channelwith diameters in the order of 1 mm, or smaller than that, were ad-dressed in the literature [1,2,4,8–11]. Among them, Kariyasaki et al.[8] correlated their data in terms of the gas volumetric quality (orhomogeneous void fraction), b. Moriyama et al. [9] measured voidfractions during N2–R113 adiabatic gas–liquid two-phase flow inextremely narrow channels with a clearance of 5–100 lm between

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Nomenclature

C Chisholm parameterC0 distribution parameterDh hydraulic equivalent diameter of flow channeldp/dz friction pressure gradient along channel axisG mass fluxj superficial velocityjT mixture volumetric flux, jg + jfLo* non-dimensional Laplace constant, {r/[g(qf � qg)]}0.5/Dh

p pressurepcr critical pressureRe Reynolds number, GDh/lf

Ref liquid Reynolds number, G(1 � xeq)Dh/lf

Refo all-liquid Reynolds number, GDh/lf

Reg gas Reynolds number, G�xeqDh/lg

v velocityVgj drift velocityWefo Weber number, G2Dh/(rqf)X Martinelli parameterxeq thermodynamic equilibrium quality

Greek symbolsa void fractionq densityr surface tension/2 two-phase friction multiplier

Subscriptscal calculational valueexp experimental valueF frictionf saturated liquid or liquidfo all flow taken as liquidg saturated vapor or gastp two-phase

Mathematical symbolexp exponential function

454 W. Zhang et al. / International Journal of Heat and Mass Transfer 53 (2010) 453–465

horizontal parallel plates, and proposed a drift-flux type correla-tion to correlate their data. Later, Mishima and Hibiki [2], and Haz-uku et al. [11] also reported that the measured void fractions couldbe successfully reproduced by the drift-flux type correlations. Incontrast to this, Triplett et al. [4] claimed that homogeneous mod-els provided the best predictions for their data in bubbly as well asslug flow regimes, and existing correlations significantly over-pre-dicted void fractions in annular flow regimes.

Therefore, it is evident that although existing experimentalworks have revealed some unique phenomena in mini-channels,there is still no general theory or correlation available, and somediscrepancies existing among the experimental results are not clar-ified yet. In view of this, as an extension of efforts by Mishima andco-workers [1,2] and a part of study dedicating to the developmentof a series of correlations for flow boiling in mini-channel [12,13],the purpose of this study is to summarize the previous studies ontwo-phase frictional pressure drop and void fraction in mini-chan-nels, evaluate their applicability, and then propose alternativecorrelations for mini-channels.

2. Correlation development

2.1. Application of neural network

The artificial neural network (ANN) is an advance informationprocessing techniques. The ANN is composed of elements analo-gous to the elementary functions of biological neurons [14,15].One of the most important characteristics of the ANN is its capabil-ity to learn from trained data and to predict for new data. The back-propagation neural network (BPN) is one of the simple butpowerful ANNs. Owing to its objectivity of judgment, the BPN is re-garded as a powerful alternative to current techniques for the pre-diction of CHF [16] and the classification of flow regimes [17]. Sinceany functional relationship can be approximated by a BPN if thesigmoid layer has enough neurons [14], an architecture-fixedBPN (layer and neuron numbers fixed) can be utilized to carryout the input sensitivity (trial and error) analysis in order to selecta set of non-dimensional numbers which could well correlate thetwo-phase frictional multiplier or void fraction. The BPN wastrained under MATLAB environment. The TRAINCGP algorithm inthe NEURAL NETWORK TOOLBOX of MATLAB was employed totrain the network in this study.

2.2. Frictional pressure drop

In order to correlate the data for two-phase friction pressuredrop, it is necessary to find out dominant experimental parametersrelated to two-phase friction pressure drop. The hydraulic diame-ter of channel, Dh, mass flux, G, pressure, p, and thermal equilib-rium quality, xeq (or their alternatives) are often considered asthe related correlating parameters of local two-phase friction pres-sure drop (or its equivalence, the two-phase frictional multiplier).By combining physical properties with these experimental param-eters, two-phase friction pressure gradient can be expressed inmany non-dimensional forms. For instance, the correlations ofLockhart and Martinelli [18], Friedel et al. [5], and Zhang and Webb[19] employed different non-dimensional numbers. However, asdemonstrated by numerous studies on two-phase friction pressuredrop since the pioneering work by Lockhart and Martinelli in1940s, the separated flow model is the most commonly used meth-od [20]. The success of this model to correlate the existing data hasdemonstrated the usefulness of the Martinelli parameter, X, whichis a combination of the inertial and viscous forces of both phases.Therefore, it may be deemed as one of the most dominant param-eters to correlate two-phase friction pressure gradient for mini-channels. A widely used correlation to calculate the two-phasefrictional multiplier is that proposed by Chisholm and Laird [21],

/2f ��ðdp=dzÞtp�ðdp=dzÞf

¼ 1þ CXþ 1

X2 ; ð1Þ

where Chisholm parameter C ranges in value from 5 to 20 for con-ventional channels, depending on whether the liquid and gas flowsare laminar or turbulent. This successful performance of Eq. (1) hasbeen shown in the literature [1,7,22,23]. However, the functionalform of Chisholm parameter C needs to be clarified for mini/mi-cro-channels. The dependence of the Chisholm parameter C onexperimental parameters was made clear by the application of theBNP to the collected database. The output of the BNP is set to bethe two-phase frictional multiplier, /2

f . The Martinelli parameteris set as one of inputs to the BNP. The mean deviation is used as ameasure of predictive accuracy, defined as

Mean deviation ¼ 1N

Xjð/exp � /calÞ=/expj � 100%; ð2Þ

where N is the data number. Based on the analysis of input sensitiv-ity (trial and error method), it was found that the introduction of

10-2

10-1

100

101

102

103

0

5

10

15

20

25

C =21(1-e-0.142/Lo )

C =21(1-e-0.674/Lo )

C=21(1-e-0.358/ Lo)

Chi

shol

m P

aram

eter

, C

[-]

Non-D. Laplace Constant, Lo [-]

Fig. 1. Chisholm parameter C as a function of non-dimensional Laplace constant.

W. Zhang et al. / International Journal of Heat and Mass Transfer 53 (2010) 453–465 455

the hydraulic diameter of channel as an input of the BNP can signif-icantly improve the prediction accuracy by the BNP. The mass flux,G, quality, xeq, and pressure, p, have minor effects on the results.These findings are accordant with what were reported in previousstudies [1,2,7,24]. Further analysis by the BNP disclosed that thenon-dimensional Laplace constant, Lo*, defined as

Lo� � rgðqf � qgÞ

" #0:5,Dh; ð3Þ

if introduced into inputs, can work as well as the hydraulic diameterof channel, and Weber number, Wefo, all-liquid Reynolds number,Refo, the reduced pressure, p/pcr, and quality, xeq, have the secondaryeffects on the predictive accuracy of the BNP (within ±5%) for col-lected databases.

Based on the above discussion, this study attempts to modifythe correlation of Mishima and Hibiki [2] for the Chisholm param-eter C in mini-channel:

C ¼ 21½1� expð�0:319DhÞ�; ð4Þ

where Dh in mm, is the hydraulic diameter of channel. Since thiscorrelation is dimensional and may be difficult to scale physicalphenomena, a non-dimensional one would be desirable. Thereforethe hydraulic diameter of channel is replaced by the non-dimen-sional Laplace constant (or confinement number) Lo* in theMishima–Hibiki correlation for the Chisholm parameter, C. The rea-sons for choosing the Laplace constant as the best choice may bestated as follows. First, this results from the application of theANN in this study. Second, theoretically the Laplace constant scalesthe wave length of the Rayleigh–Taylor instability. When the bub-bles are squeezed in the mini-channel, the formation of bubblesand the bubble movement are limited by interfacial stability. TheRayleigh–Taylor instability is the interfacial instability betweentwo fluids of different densities that are stratified in the gravity fieldor accelerated normal to the interface. It is commonly observed thatthe boundary between two stratified fluid layers at rest is not stableif the upper fluid density is larger than the lower-fluid density.Since the Rayleigh–Taylor instability can lead to the destruction ofthe single common interface, it is important in the formation ofbubbles or droplets. In particular, the critical wavelength predictedby the related stability analysis is one of the most significant lengthscales for two-phase flow. Note that this instability is not limited tothe gravitational field. Any interface, and fluids that are acceleratednormal to the interface, can exhibit the same instability. In addition,the Laplace constant is a nominal bubble size related to the capillaryparameter. Table 8 lists the average values of the non-dimensionalLaplace, Lo*, and the Chisholm parameter, C for each data set. Fig. 1illustrates the relationship between Lo* and C. According to Sad-atomi et al. [25] and Moriyama et al. [9], the friction pressure canbe correlated by the equation proposed by Chisholm and Laird[21], Eq. (1), with C = 21 for conventional channels, and C = 0 for ex-tremely narrow gaps. By noting these asymptotic values, the Chis-holm parameter may be correlated by the following equation:

C ¼ 21½1� expð�0:358=Lo�Þ�: ð5Þ

The applicable ranges of this correlation are as follows: 0.014 6Dh 6 6.25 mm, Ref 6 2000, Reg 6 2000.

2.3. Void fraction

Void fractions can be successfully correlated by the drift-fluxmodel for conventional channels [26]. According to Moriyamaet al. [9], Mishima and co-workers [1,2], Takamasa et al. [27], Hazukuet al. [11], and Hibiki and Ishii [28], the drift-flux model is applicableto correlate the data of void fraction in mini-channels. Therefore,based on it, a correlation will be proposed in this study.

According to the drift-flux model, the relationship between thegas velocity, vg, and the mixture volumetric flux, jT, can be ex-pressed by the following equation:

vg � jg=a ¼ C0jT þ Vgj; ð6Þ

where C0 is the distribution parameter, and Vgj is the drift velocity.In a conventional round tube, the distribution parameter and thedrift velocity are given by equations corresponding to flow regimes[26]. Since the flow regime maps for mini-channels are still underdevelopment [2,29], for simplicity, this study tends to develop a sin-gle equation for either the distribution parameter or the drift veloc-ity. According to the existing studies [2,4,8], the behavior of voidfraction for the annular flow regime may be different with thosefor the bubbly and churn flows. Therefore, the discussion here islimited to bubbly and churn flows. Following our previous study[2], the drift velocity is assumed to be zero in mini-channels. Then,Eq. (6) can be simplified as

jg=a ¼ C0jT: ð7Þ

And then the development of a correlation for void fraction turns tothe development of an equation for the distribution parameter.

In order to develop an equation for the distribution parameter,dominant non-dimensional numbers should be determined first.The liquid superficial velocity, jf, gas superficial velocity, jg, hydrau-lic diameter of channel, Dh, and pressure, p (or their alternatives)are often considered the related correlating parameters. Manynon-dimensional numbers can be made if the above experimentalparameters are combined with physical properties of gas or liquid,and thus the distribution parameter can be expressed in manynon-dimensional forms. For instance, Ishii [26] assumed the distri-bution parameter depends on the density ratio, qg/qf, and the Rey-nolds number, Ref, defined by jfqfDh/lf. In consideration of variousvoid distributions such as wall void peak in bubbly flow systemsand the bubble size being one of key parameters to govern the voiddistribution, Hibiki and Ishii [30] introduced into the correlation ofthe distribution parameter the bubble Sauter mean diameter,which can be predicted from a correlation for the non-dimension-alized bubble Sauter mean diameter, the non-dimensional Laplacelength, and the Reynolds number. The relationships between thedistribution parameter and non-dimensional numbers were madeclear as well by the application of the ANN to the collected data-base, in the way as shown in the pressure drop part. It is shown

0.2 1 10 400.4

0.8

1.2

1.6

2.0

C0 =1.20+0.380 e-1.39/Lo

Data Source Kariyasaki et al. Moriyama et al. Mishima et al. Mishima and Hibiki Triplett et al. Hazuku et al.

Dis

tribu

tion

Para

met

er, C

0 [-]

Non-D. Laplace Constant, Lo [-]

Fig. 2. Distribution parameter as a function of non-dimensional Laplace constant.


that only using the non-dimensional Laplace constant, Lo*, definedin Eq. (3), can predict the distribution parameter with the smallestmean deviation of 9.56%. Other non-dimensional numbers have thesecondary effects on the predictive accuracy of the BNP.

Therefore, as an extension of Mishima and Hibiki’s work in1996, this study attempts to modify the correlation of the distribu-tion parameter proposed by Mishima and Hibiki [2]:

C0 ¼ 1:2þ 0:510 expð�0:691DhÞ; ð8Þ

where the unit of Dh is in mm. Since this correlation is dimensional,it is also desirable to develop a non-dimensional correlation. There-fore the non-dimensional Laplace constant is determined to replacethe hydraulic diameter in the Mishima–Hibiki correlation. The rela-tionship between the average values of the non-dimensional La-place, Lo*, and those of the distribution parameter, C0 for eachdata set is illustrated in Fig. 2. The asymptotic value of the distribu-

Table 1Databases for two-phase frictional pressure drop in mini-channels.

Symbols Reference Adiabaticor diabatic

Geometry Diameter orgap �width (mm

s Moriyama et al. [9] Adiabatic Rectangular duct (0.007, 0.025,0.052, 0.098) � 3

4 Mishima et al. [1] Rectangular duct (1.07, 2.45, 5.00)~ Mishima and

Hibiki [2]Round tube 1.05, 2.05, 3.12, 4

O Triplett et al. [4] Round tubesemi-triangular

1.10, 1.45, 1.09, 1

. Lee and Lee [23] Rectangular duct (0.4, 1.0, 2.0, 4.0)} Kawahara et al. [7] Round duct 0.10/ Chung and

Kawaji [24]Round duct 0.0495, 0.0996,

0.250, 0.526} Liu et al. [35] Round tubes,

square ducts0.91, 2, 3.02, 0.99� 0.99, 2.89 � 2.8

g Ungar andCornwell [31]

Adiabatic Round tube 1.46, 1.78, 2.58, 3

Zhang and Webb [19] Round tubeRound multi-port

2.13, 3.25, 6.20

� Cavallini et al. [32] Rectangularmulti-port

1.4

f Tran [6] Diabatic Round tube 2.46Yu et al. [33] Round tube 2.98

Total (13 databases) Adiabatic,diabatic

Round tube,Rectangular duct

0.007–6.25

tion parameter at small non-dimensional Laplace constants (i.e. forconventional channels) may be approximated to be 1.2 according toIshii [26]. Therefore, the following equation for the distributionparameter could be obtained by using the least square method:

C0 ¼ 1:20þ 0:380 expð�1:39=Lo�Þ: ð9Þ

The applicable conditions of this correlation are as follows:0.200 6 Dh 6 4.90 mm, Ref 6 2000, Reg 6 1000, air–water two-phase flow, atmospheric pressure.

3. Results and discussion

3.1. Pressure drop

3.1.1. Collected databaseAvailable 13 data sets for two-phase friction pressure drop in

mini/micro-channels are tabulated in Table 1. These data sets canbe classified into three groups, that is, adiabatic liquid–gas flow,adiabatic liquid–vapor flow, and flow boiling groups. Adiabatic li-quid–gas flow involves two fluids, between which there are nomass and heat transfers. It is the simplest two-phase flow, andhas been extensively investigated by many researchers. Adiabaticliquid–vapor flow involves only one type of fluid with two states,i.e. liquid and vapor. Although the experimental condition is keptto be adiabatic, mass and heat transfers between two states mayoccur due to variation of pressure drop during flow. The pressuredrop gradient in mini-channels is a little bit higher than in conven-tional channels. Taking this point into account, existing correla-tions may be unsuitable for mini-channels. Experimental studiesby Ungar and Cornwell [31], Zhang and Webb [19], and Cavalliniet al. [32] deal with adiabatic liquid–vapor flow. Tran [6] and Yuet al. [33] conducted experiments on flow boiling in mini-channels,which is the most complex two-phase flow, involving thermal andhydraulic coupling. Our collected database includes all the threegroups of two-phase flow, containing several working fluids. Thecovered hydraulic diameters of channel range from 0.07 to6.25 mm. It should be noted that although the data bases ofMoriyama et al. [9] and Kawahara et al. [7] are deemed to be for

)Working fluids Flow

directionChannelmaterial

Datanumber

0R113–N2 Horizontal Nickel + Pyrex

glass plates104

� 40 Water–air Vertical upward Acrylic resin 306.08 Water–air Vertical upward Pyrex glass 299

.49 Water–air Horizontal Pyrex, acrylic,poly-carbonate

192

� 20 Water–air Horizontal Acrylic 42Water–N2 Horizontal Fused silica 64Water–N2 Horizontal Fused silica 0

9(Water, ethanol,oil)–air,

Vertical Pyrex glass 205

.15 Ammonia–vapor Horizontal Unknown 133

R134a, R22,R404a–vapor

Horizontal Aluminum, copper 51

(R134a, R236ea,R410A)–vapor

Horizontal Aluminum 38

(R134a, R12)–vapor Horizontal Brass, stainless steel 440Water–vapor Horizontal Stainless steel 327(Water, R12, R113,R22, R134a, R404a,ammonia)–(air, N2, vapor)

Vertical,horizontal

Nickel, Pyrex, acrylic,poly-carbonate, silica,aluminum, copper,brass, stainless steel

2201

10-1

100

101

102

103

104

105

10610

-1

100

101

102

103

104

105

106

VV VT

TTTVLi

quid

Rey

nold

s N

umbe

r, Re f

[-]

Gas Reynolds Number, Reg [-]

Fig. 3. Distribution of all adiabatic data in plot of Ref versus Reg.


micro-channel according to the channel definitions of Kandlikar[34], they are included in this collected database since their corre-sponding flow regimes are suitable for evaluation of the existingmodels. Moreover, the flow patterns observed in the experimentof Kawahara et al. [7] were liquid film flows along the channel wallwith gas flowing in the channel core, and thus the separated flowmodel is deemed to hold for prediction of their data. No data werecollected from Chung and Kawaji [24] for lack of enough parame-ters. It should be mentioned that Liu et al. [35] recently presentedan extensive data base for two-phase friction pressure drop in ver-tical capillaries of circular and square cross-sections using air asthe gas phase and water, ethanol, or an oil mixture as the liquidphase. A negative friction pressure drop was observed for the bulkof experimental data at very low liquid velocities, which meansthat the total pressure drop is less than the hydrostatic pressuredrop and local down flows of liquid film may occur. All negativefriction pressure drop data were excluded here. Data of Tran [6]using R113 as a test fluid were also excluded since these data werenot verified by us with their own correlation. Since some proper-ties of fluid are not available in hand, a part of data by Zhangand Webb [19] and Cavallini et al. [32] were not included in thisdatabase. All collected data are obtained from the open literatureand verified by the correlations developed by their authors.

Fig. 3 shows the distribution of all adiabatic data in the plot ofRef versus Reg. Most of collected data fall into the flow conditions ofliquid and/or gas being laminar, and few data in the turbulent–tur-bulent (TT) condition, if flow conditions are divided in the way ofLockhart and Martinelli [18]. Therefore, low Reynolds number flowconditions (VV, TV, and VT) are often encountered in mini-channel,and flow at such conditions may have features of laminar flow. Thisis a new characteristic of two-phase flow in mini-channel. Fromthe definition of Reynolds number it is evident that low Reynoldsnumber results from the limitation on flow rate due to high pres-sure drop gradient in the experiment, and the inherent smalldimension of mini-channel. However, most of existing correlationswere developed for turbulent flow, so they may be unsuitable inprinciple for such flow conditions in mini-channel.

3.1.2. Evaluation of correlations3.1.2.1. Existing correlations based on homogeneous model. Accord-ing to some previous studies by Ungar and Cornwell [31], Triplettet al. [4], and Kawahara et al. [7] on two-phase frictional pressure

drop, the homogeneous model was shown to be capable tocorrelate experimental data within an acceptable margin of error,although the observed flow patterns were much less homogeneousin mini/micro-channels [4,7]. Here an extensive evaluation of sixexiting correlations based on the homogeneous model was made.The mean deviation is used as a measure of predictive accuracy.For the convenience of comparison, the two-phase friction pres-sure gradient predicted by the homogeneous model is also ex-pressed as a product of the single-phase pressure gradient for theliquid phase being considered to flow alone, and the two-phasefriction multiplier. Table 2 shows the comparison of six correla-tions with the seven databases for adiabatic gas–liquid two-phaseflow. As reported by Kawahara et al. [7], the correlation of Dukleret al. [36] presented reasonably good predictions for seven dat-abases for adiabatic gas–liquid two-phase flow. Seventeen datasets among 26 in all were predicted within the mean deviationof 20%. For all data in Table 2, the correlation of Dukler et al. gavethe smallest total mean deviation of 18.0%, followed subsequentlyby the Beattie–Whalery correlation [37] with 22.9%. Table 3showed the evaluation with databases for liquid–vapor two-phaseflow. As reported by Ungar and Cornwell [31], and Triplett et al. [4],all of six correlations based on the homogeneous model give satis-factory predictions for three databases. The best performance ispresented by the correlation of Ackers [38] with the smallest totalmean deviation of 9.47%. Table 4 lists the comparison of the sixcorrelations with the data for flow boiling. Here the total two-phase frictional pressure drops predicted by the correlations werecompared to the experimental values. The agreement is relativelypoor. Fig. 4(a) shows the behavior of the correlation of Dukler[36] with the data obtained by Mishima et al. [1] and Mishimaand Hibiki [2] for adiabatic air–water flow in rectangular and circu-lar channels, respectively. There exist some deviations of predic-tion at the low experimental multiplier. Further examination ofTable 2 found that the Dukler correlation tends to behave poorfor channels with relatively large hydraulic diameter. Fig. 4(b)demonstrated that the correlation works well for the data of Ungarand Cornwell [31]. However, it systematically under-predicts thedata of Tran [6], as shown in Fig. 4(c). The similar performancecan be observed for the Beattie–Whalery correlation [37].

3.1.2.2. Existing correlations based on separated flow model. It wasreported that the Lockhart–Martinelli type correlation can repre-sent the experimental data of two-phase friction pressure dropreasonably well for many flow conditions, see Mishima and co-workers [1,2] as well as Zhao and Bi [22]. An extensive evaluationof nine existing correlations is presented here. Among these corre-lations, seven (correlations of Mishima and Hibiki [2], Tran [6], Leeand Lee [23], Yu et al. [33], Zhang and Webb [19], Qu and Mudawar[39], Lee and Mudawar [40]) are developed for mini/micro-chan-nels. Since the Lockhart and Martinelli’s classical correlation [18]is the foundation of most of recently developed correlations, andthat of Friedel [5] is one of the most accurate correlations for con-ventional channels [41], thus both correlations are included here aswell. The evaluation results of each correlation with databases foradiabatic liquid–gas flow, adiabatic liquid–vapor and flow boilingare tabulated in Tables 5–7, respectively. Since the correlations ofYu et al., Qu and Mudawar, and Lee and Mudawar were developedjust for one or two flow regimes (laminar or turbulent flow of gasor liquid), therefore only a part of data falling into their respectiveapplicable ranges were used to evaluate the behaviors of these cor-relations. From Table 5, correlations of Mishima–Hibiki, and Lee–Lee generally predict the seven databases satisfactorily. Fourteenamong 26 data sets are predicted within the mean deviation of20% by Mishima–Hibiki correlation, 16 by that of Lee and Lee. Inall, Mishima–Hibiki correlation presents the smallest total meandeviation of 16.6%, followed subsequently by that of Lee and Lee

Table 2Evaluation of correlations of frictional pressure drop based on homogeneous model with data for liquid–gas flow.

Reference Diameter or gap �width(mm)

Workingfluids

Geometry Mean deviationa

McAdams[45]

Ackers[38]

Cicchitti[46]

Dukler[36]

Beattie–Whalery[37]

Lin et al.[47]

Moriyama et al. [9] 0.025 � 300.098 � 30

R113–N2 Rectangular duct 153108

158114

��

5.64

11.8

38.726.4

216157

Mishima et al. [1] 1.07 � 402.45 � 405.00 � 40

Water–air Rectangular duct 13.314.915.7

15.415.315.6

21.718.315.6

10.513.524.2

8.98

11.715.6

17.716.7

15.5

Mishima and Hibiki[2]

1.052.053.124.08

Water–air Round tube 14.9

8.2513.5

19.8

29.110.114.720.2

63.019.720.742.4

12.511.416.828.7

7.508.33

11.020.8

39.714.014.521.6

Triplett et al. [4] 1.101.451.09

Water–air Round tube, roundtubeSemi-triangular

18.913.929.3

25.920.644.8

56.756.7115

16.622.114.9

11.8

11.8

10.9

28.223.850.1

Lee and Lee [23] 0.4 � 204.0 � 20

Water–air Rectangular duct 106

17.6

13018.2

16738.9

21.522.7

68.217.8

14819.4

Kawahara et al. [7] 0.10 Water–N2 Round duct 66.9 127 � 15.7 60.1 110

Liu et al. [35] 0.91 Water–air, Round tube, 15.4 15.0 14.9 22.2 24.3 14.72.00 Water–air, Round tube, 15.3 15.8 16.4 13.0 22.0 16.3

3.02 Water–air, Round tube, 25.9 26.9 28.6 32.3 21.2 27.9

2.89 � 2.89 Water–air, Square duct, 18.3 17.4 17.6 32.7 17.1 17.2

0.91 Ethanol–air, Round tube, 85.8 99.2 109 17.9 87.5 106

2.00 Ethanol–air, Round tube, 28.2 30.0 31.2 17.9 37.8 30.9

3.02 Ethanol–air, Round tube, 7.33 7.35 7.36 7.19 7.38 7.36

0.99 � 0.99 Ethanol–air, Square duct, 54.4 60.9 66.0 12.8 63.1 64.2

2.89 � 2.89 Ethanol–air, Square duct, 18.7 20.6 22.0 13.3 23.2 21.7

3.02 Oil–air, Round tube, 21.3 116 134 24.5 83.9 76.9

2.89 � 2.89 Oil–air, Square duct 11.0 62.8 69.9 12.2 56.7 47.5

Total 30.4 41.9 71.7 18.0 22.9 46.5

The symbol � denoting that the mean deviation is larger than 200%.a Mean deviation defined as (1/N)

P|(/f,exp � /f,cal)//f,exp| � 100%, an underlined value denoting the smallest of mean deviations by existing correlations for each data set.


with 20.2%. The underlined value in the Tables 2–7 denotes thesmallest mean deviation among those by existing correlations foreach data set. The smallest mean deviations predicted by Lee andLee’s correlation are the greatest in number. From Table 5, the cor-relations of Qu–Mudawar and Lee–Mudawar predict the data wellwithin their applicable ranges. Table 6 lists the evaluation resultsof nine existing correlations with three databases for liquid–vaporflow. Lee and Mudawar’s correlation works well for their applica-ble ranges. For all flow conditions, the Mishima–Hibiki correlationbehaves best in total. Table 7 shows the evaluation with data forflow boiling. The classical Lockhart–Martinelli correlation has thesmallest total mean deviation of 35.1%. The second is the Mishi-ma–Hibiki correlation. The correlations of Tran, and Zhang and

Table 3Evaluation of correlations of frictional pressure drop based on homogeneous model with d

Reference Diameter orgap �width (mm)

Working fluids Geometry

Ungar andCornwell [31]

1.461.782.583.15

Ammonia-Vapor

Round tube

Zhang and Webb[19]

2.133.256.203.25

R134a–vapor,R134a–vapor,R134a–vapor,R22–vapor

Multi-port, routube,Round tube,Round tube,

Cavallini et al. [32] 1.4 R134a–vapor,R236ea–vapor

Rectangularmulti-port

Total

Webb work poorly for databases obtained by other authors. Thecorrelations proposed by Mishima and Hibiki, and Lee and Lee havesimilar performance for data of liquid–gas flow. However the for-mer correlation works a little bit better for both liquid–vapor flowand flow boiling. Fig. 5 further illustrates the behavior of predictionby Mishima–Hibiki correlation. However, Mishima–Hibiki correla-tion over-predicted the data of Ungar and Cornwell by 30%, asshown in Fig. 5(b). The similar performance of prediction by theLee–Lee correlation can be observed.

3.1.2.3. Newly developed correlation. This section shows the evalua-tion of the newly developed correlation, Eq. (5). The results are tab-ulated in Tables 5–7. The bold value in the tables denotes the

ata for liquid–vapor flow.

Mean deviation

McAdams[45]

Ackers[38]

Cicchitti[46]

Dukler[36]

Beattie–Whalery [37]

Lin et al.[47]

8.8714.76.63

7.38

10.9

11.47.1811.0

36.416.724.225.5

8.8819.18.127.61

8.8816.1

6.237.75

11.114.37.389.15

nd 13.818.515.917.2

4.959.567.1210.1

2.31

8.29

5.84

8.11

15.619.818.119.0

9.5014.111.714.2

11.416.513.415.4

18.322.4

9.0810.4

6.08

5.20

20.425.0

14.320.2

15.920.1

12.6 9.47 18.4 14.3 11.4 12.4

Table 4Evaluation of correlations of frictional pressure drop based on homogeneous model with data for flow boiling.

Reference Diameter or gap �width(mm)

Workingfluids


McAdams[45]

Ackers[38] Cicchitti[46]

Dukler[36]

Beattie–Whalery[37]

Lin et al.[47]

Tran et al.[6]

2.462.46

R12,R134a

Roundtube

49.059.0

40.350.6

35.4

42.9

53.162.4

44.255.7

45.055.9

Yu et al.[33]

2.98 Water Roundtube

76.8 64.1 162 40.8 45.5 91.6

Total 64.8 54.5 92.5 51.5 49.3 69.2

a Mean deviation defined as (1/N)P

|(DpF,tp,exp � DpF,tp,cal)/DpF,tp,exp| � 100%.

10-1 100 101 10210-1

100

101

102

-30%

+30%

Dukler Correlation

Data Source Mishima et al. Mishima and Hibiki Pr

edic

ted

Mul

tiplie

r, φ

f,cal

[-]

Experimental Multiplier, φ f,exp [-]

100 101 102100

101

102

+30%

-30%

Dukler Correlation

Data Source Ungar and Cornwell

(a)

(c)

(b)

Pred

icte

d M

ultip

lier,

φf,c

al [-]


101 102 103 104 105 106101

102

103

104

105

106

+30%

-30%

Dukler Correlation

Data Source Tran

Pred

icte

d Pr

essu

re D

rop,

Δp F,

cal [-]

Experimental Pressure Drop, Δ pF,exp

[-]

Fig. 4. Evaluation of Dukler correlation (1964) with data for (a) liquid–gas flow, (b)liquid–vapor flow, and (c) flow boiling.


smallest mean deviation for each data set among those by all cor-relations including the new one. From Table 5, 13 data sets among26 in all were predicted within 20%. The total mean deviation forall data of liquid–gas flow was 17.9%, a little bit higher than thatpredicted by Mishima–Hibiki correlation. The table shows for allflow conditions the newly developed correlation works satisfacto-rily for liquid–vapor two-phase flow with the smallest total meandeviation of 21.7%, about 6% less than that of Mishima–Hibiki cor-relation. For flow boiling data, the newly developed correlationworks best among the 10 correlations, as shown in Table 7. Fig. 6further illustrates its performance.

3.1.3. RecommendationIt should be noted that the application of the newly developed

correlation out of the verified range is not recommended. Sincemost of data in this collected database fall into the flow conditionof either liquid flow or gas flow being laminar, it can be expectedthat the newly developed correlation would be able to predictthe friction pressure drop in such flow conditions. For other flowconditions such as both liquid and gas flow being turbulent, theReynolds number, Ref, may be an important non-dimensional num-ber to correlate the Chisholm parameter. Furthermore, althoughthe newly developed correlation could predict all the data for theflows of three groups, i.e. adiabatic liquid–gas flow, adiabatic li-quid–vapor flow and flow boiling, within an acceptable margin oferror, it was found from Fig. 1 that most of the averages of the Chis-holm parameter, C, for liquid–gas flow are above the curve de-picted by the newly developed equation, however, those forliquid–vapor flow are below the curve. For liquid–gas two-phaseflow, Eq. (5) may work better if the constant of �0.358 is replacedwith �0.674. For liquid–vapor flow, however, the constant of�0.142 would be better.For adiabatic liquid–gas two-phase flow:

C ¼ 21½1� expð�0:674=Lo�Þ�: ð10Þ

For adiabatic liquid–vapor two-phase flow:

C ¼ 21½1� expð�0:142=Lo�Þ�: ð11Þ

On this point, future experimental investigation is recommended toaccumulate more data sets for clarifying the reasons why the differ-ence occurs between adiabatic liquid–gas flow and adiabatic liquid–vapor flow.

3.2. Void fraction

3.2.1. Collected databaseSix available datasets in the literature for void fraction of two-

phase flow in mini-channels were collected in this study and tab-ulated in Table 9. The hydraulic diameters range from 0.025 mm to4.90 mm, and channel geometry are circular tube or rectangularduct. The working fluids contain water and R113 as liquid phases,with air and nitrogen as gas phases. The flow orientation is vertical

Table 5Evaluation of correlations of frictional pressure drop based on separated flow model with data for liquid–gas flow.

Reference Diameter orgap �width(mm)

Workingfluids


Lockhart–Martinelli [18]

Friedel [5] Mishima–Hibiki [2]

Tran [6] Lee–Lee [23] Yu et al. [33] Zhang andWebb [19]

Qu andMudawar [39]

Lee andMudawar [40]

Thisstudy

Moriyama et al. [9] 0.025 � 300.098 � 30

R113–N2 Rectangular duct 65.947.0

��

3.318.77

��

5.6211.6

//

��

3.25

7.89

3.669.59

3.389.17

Mishima et al. [1] 1.07 � 402.45 � 405.00 � 40

Water–air Rectangular duct 9.7013.418.0

77.590.165.7

11.4

11.46.95

15610829.2

19.018.1

11.4

51.0//

��

46.927.3/

15.642.1/

7.7310.314.5

Mishima and Hibiki [2] 1.052.053.124.08

Water–air Round tube 15.913.117.725.6

18310898.0147

14.4

8.648.24

16.8

�192145123

36.422.321.030.9

50.757.764.754.5

��

16.412.7

4.8218.5

11.321.528.226.8

27.122.423.520.9

Triplett et al. [4] 1.101.451.09

Water–air Round tube, round tubeSemi-triangular

13.916.321.8

70.969.9133

14.3

15.222.4

81.571.8127

22.228.917.2

35.143.338.0

��

23.916.814.1

20.123.7

13.3

16.421.514.7

Lee and Lee [23] 0.4 � 204.0 � 20

Water–air Rectangular duct 59.319.3

�48.5

55.623.7

�31.8

12.5

15.6

/56.5

��

110/

71.922.9

27.925.8

Kawahara et al. [7] 0.10 Water–N2 Round duct 75.9 � 17.4 � 10.7 34.1 � 38.3 47.2 11.7

Liu et al. [35] 0.91 Water–air, Round tube, 20.3 52.4 20.6 37.5 22.1 / � 25.5 23.4 17.9

2.00 Water–air, Round tube, 15.0 49.5 27.0 31.6 12.5 / � 25.0 11.4 14.6

3.02 Water–air, Round tube, 23.8 28.4 19.5 12.9 33.6 / � 21.1 28.7 22.8

2.89 � 2.89 Water–air, Square duct, 21.3 18.9 11.6 11.8 32.2 / � 17.0 25.7 18.7

0.91 Ethanol–air,

Round tube, 60.8 199 63.0 67.3 18.0 / � 25.8 30.0 50.6

2.00 Ethanol–air,

Round tube, 26.0 68.6 41.2 28.5 17.8 / 139 36.0 31.5 32.6

3.02 Ethanol–air,

Round tube, 6.36 11.0 12.7 6.17 7.11 / 32.6 11.5 8.72 8.77

0.99 � 0.99 Ethanol–air,

Square duct, 41.2 132 44.7 55.3 12.9 / � 30.7 26.3 35.5

2.89 � 2.89 Ethanol–air,

Square duct, 15.2 51.2 31.5 15.7 13.1 / 114 21.5 15.3 23.7

3.02 Oil–air, Round tube, 35.7 197 54.8 97.1 24.5 / � 30.4 24.5 25.3

2.89 � 2.89 Oil–air, Square duct 19.1 126 31.3 45.2 12.2 / 170 14.7 12.2 12.5

Total 25.7 138 16.6 181 20.2 49.3 � 24.3 23.9 17.9

The symbol � denoting that the mean deviation is larger than 200%.The symbol / meaning that data are out of applicable parametric ranges of a correlation.

a Mean deviation is defined as (1/N)P

|(/fo,exp � /fo,cal)//fo,exp| � 100% for correlations of Friedel [5], Tran [6] and Zhang and Webb [19], however, it is defined as (1/N)P

|(/f,exp � /f,cal)//f,exp| � 100% for other correlations.

460W

.Zhanget

al./InternationalJournal

ofH

eatand

Mass

Transfer53

(2010)453–

465

Table 6Evaluation of correlations of frictional pressure drop based on separated flow model with data for two-phase liquid–vapor flow.


Workingfluids

Geometry Mean deviation

Lockhart–Martinelli[18]

Friedel[5]

Mishima–Hibiki [2]

Tran[6]

Lee–Lee[23]

Yuet al.[33]

Zhang andWebb [19]

Qu andMudawar[39]

Lee andMudawar[40]

Thisstudy

Ungar andCornwell[31]

1.461.782.583.15

Ammonia-Vapor

Round tube 42.015.536.042.9

10230.210191.4

30.312.338.649.5

13482.696.996.4

19.0

5.8216.025.1

36.442.532.932.6

11165.410294.9

6.3323.716.936.0

13.313.0

6.44

12.1

15.18.4821.729.8

Zhang andWebb[19]

2.133.256.203.25

R134a–vapor,R134a–vapor,R134a–vapor,R22–vapor

Multi-port,round tube,Round tube,Round tube,

52.645.747.851.1

14.88.9611.110.5

20.527.642.030.5

45.435.129.434.2

73.577.894.083.3

29.0///

4.25

5.23

3.95

5.22

////

45.9///

35.838.747.540.4

Cavalliniet al. [32]

1.4 R134a–vapor,R236ea–vapor

Rectangularmulti-port

39.7

6.81

7.4214.2

4.7812.2

47.746.3

51.67.96

38.242.4

5.7127.5

//

37.815.8

11.210.1

Total 37.5 58.8 27.6 79.3 34.3 36.1 60.9 19.1 11.7 21.7

Table 7Evaluation of correlations of frictional pressure drop based on separated flow model with data for flow boiling.


Workingfluids

Geometry Mean deviation

Lockhart–Martinelli[18]

Friedel[5]

Mishima–Hibiki [2]

Tran[6]

Lee–Lee[23]

Yuet al.[33]

Zhang andWebb [19]

Qu andMudawar[39]

Lee andMudawar[40]

Thisstudy

Tran [6] 2.462.46

R12,R134a

Roundtube

47.823.1

36.135.1

38.029.0

25.9

13.2

40.325.2

56.965.1

31.133.5

//

56.9119

46.528.1

Yu et al.[33]

2.98 Water Roundtube

39.0 181 48.3 � 84.6 28.8 � / 51.9 31.4

Total 35.1 97.4 39.2 � 55.0 40.6 � / 92.9 32.8

Table 8Averages of parameter C for each datasets in mini-channels.

Reference Adiabatic or diabatic Geometry Diameter or gap �width (mm) Workingfluids

Flow Direction Lo* Parameter C

Moriyama et al. [9] Adiabatic Rectangular duct 0.007 � 300.025 � 300.052 � 300.098 � 30

R113–N2 Horizontal /21.50/5.50

/0.22/0.69

Mishima et al. [1] Adiabatic Rectangular duct 1.07 � 402.45 � 405.00 � 40

Water–air Verticalupward

1.300.580.30

6.6611.2822.69

Mishima and Hibiki [2] Adiabatic Round tube 1.052.053.124.08

Water–air Verticalupward

2.611.340.880.67

7.5712.3815.0713.16

Triplett et al. [4] Adiabatic Round tubeSemi-triangular

1.101.451.091.49

Water–air Horizontal 2.491.892.51/

4.956.792.87/

Lee and Lee [23] Adiabatic Rectangular duct 0.4 � 201.0 � 202.0 � 204.0 � 20

Water–air Horizontal 3.491.430.750.41

2.037.5611.5913.04

Kawahara et al. [7] Adiabatic Round duct 0.10 Water–N2 Horizontal 27.38 0.24Chung and Kawaji [24] Adiabatic Round duct 0.0495

0.09960.2500.526

Water–N2 Horizontal 54.8927.2810.875.17

0.150.221.743.18

Ungar and Cornwell [31] Adiabatic Round tube 1.461.782.583.15

Ammonia-Vapor

Horizontal 1.421.160.800.66

2.856.743.492.67

Zhang and Webb [19] Adiabatic Round tube,Round multi-port

2.133.256.203.25/

R134a–vaporR134a–vaporR134a–vaporR22–vaporR404a–vapor

Horizontal 0.280.230.120.26/

6.8910.308.719.01/

Cavallini et al. [32] Adiabatic Rectangular multi-port

1.4 R134a–vaporR236ea–vaporR410A–vapor

Horizontal 0.540.62/

8.9210.45/


10-1 100 101 10210-1

100

101

102

-30%

Mishima-Hibiki Correlation

Data Source Mishima et al. Mishima and Hibiki Pr

edic

ted

Mul

tiplie

r, φ f,c

al [-]

Experimental Multiplier, φf,exp [-]

100 101 102100

101

102

-30%


Data Source Ungar and Cornwell

(a)

(c)

(b)

Pred

icte

d M

ultip

lier,

φ f,cal

[-]


101 102 103 104 105 106101

102

103

104

105

106

-30%


Data Source Tran

Pred

icte

d Pr

essu

re D

rop,

Δp F

,cal

[-]

Experimental Pressure Drop, ΔpF,exp [-]

+30%

+30%

+30%

Fig. 5. Evaluation of Mishima–Hibiki correlation (1996) with data for (a) liquid–gasflow, (b) liquid–vapor flow, and (c) flow boiling.

10-1 100 101 10210-1

100

101

102

-30%

+30%

Newly Developed Correlation

Data Source Mishima et al. Mishima and HibikiPr

edic

ted

Mul

tiplie

r, φ

f,cal

[-]


100 101 102100

101

102


-30%

+30% Data Source Ungar and Cornwell

(a)

(c)

(b)

Pred

icte

d M

ultip

lier,

φf,c

al [-]


101 102 103 104 105 106101

102

103

104

105

106


-30%

+30% Data Source Tran

Pred

icte

d Pr

essu

re D

rop,

Δp F

,cal

[-]

Experimental Pressure Drop, ΔpF,exp [-]

Fig. 6. Evaluation of newly developed correlation with data for (a) liquid–gas flow,(b) liquid–vapor flow, and (c) flow boiling.


upward or horizontal. All data were obtained under atmosphericpressure.

Fig. 7 shows the distribution of collected data in the plot of Ref

versus Reg. It is evident that most of data fall into the flow condi-tions of liquid and/or gas being laminar, and few data into the tur-bulent–turbulent condition, similar to that shown in Fig. 3 forpressure drop. It can be concluded that low Reynolds number flowconditions are often encountered in mini-channel, and flow mayhave features of laminar flow under such conditions.

3.2.2. Evaluation of correlations3.2.2.1. Existing correlations. In what follows, an evaluation is pre-sented of four existing correlations for mini-channels: correlationsof Kariyasaki et al. [8], Moriyama et al. [9], Mishima–Hibiki [2], and

Hibiki–Ishii [28]. The corresponding relations between the symbolsutilized in Figs. 7–10 and the datasets are listed in Table 9. Theevaluation results are illustrated in Fig. 8. It indicates that theMishima–Hibiki correlation presents the best predictive accura-cies, with the smallest mean deviation of 12.9%. Most of predic-tions are well within the error band of ±30%. The second is 26.6%,given by the Hibiki–Ishii correlation. As pointed out by Hibikiand Ishii [28], this correlation is applicable only to bubbly flow ina mini-channel. It under-predicts the data at high void fractionas shown in Fig. 8(d). The correlation of Kariyasaki et al. over-pre-dicts the data about 30% for the data of low and intermediate voidfractions. This correlation tends to predict the data well within theacceptable margins of errors at high void fractions over 60%. The

Table 9Collected databases for void fraction of two-phase flow in mini-channels.

Symbols Reference Geometry Diameter orgap �width (mm)

Working fluids Channel material Flow orientation

s Kariyasaki et al. [8] Circular tube 1.00,2.40,4.90

Water–air Acrylic Vertical upward

4 Moriyama et al. [9]) Rectangular duct 0.025 � 30,0.052 � 30,0.096 � 30

R113–N2 Nickel + Pyrexglass plates

Horizontal

. Mishima et al. [1] Rectangular duct 1.07 � 40,2.45 � 40

Water–air Aluminum Vertical upward

} Mishima and Hibiki [2] Circular tube 1.09,2.15,3.08,3.90

Water–air Aluminum Vertical upward

/ Triplett et al. [4] Circular tube 1.10 Water–air Pyrex Horizontal} Hazuku et al. [11] Circular tube 1.02 Water–air Acrylic Vertical upwardTotal (6 databases) Circular tube,

Rectangular duct0.025–4.90 Water/R113-

Air/N2

Acrylic, nickel,Pyrex, aluminum

Vertical, horizontal

10-1

100

101

102

103

104

10510

-1

100

101

102

103

104

105

VV VT

TTTV

Liqu

id R

eyno

lds

Num

ber,

Re f

[-]

Gas Reynolds Number, Reg [-]

Fig. 7. Distribution of collected data in plot of Ref versus Reg.


correlation of Moriyama et al. over-predicts the data about 40% forlow void fractions less than 30%. At high void fractions, this corre-lation fails to correlate the data.

3.2.2.2. Newly developed correlation. Fig. 9 illustrates the totalbehavior of the newly developed correlation, Eq. (9). The meandeviation is 12.7%, slightly smaller than that of the Mishima–Hibikicorrelation. From the figure, most of data are well predicted withinthe error band of ±30%. Fig. 10 shows the comparison of the newlydeveloped correlation and the existing correlations with the dataobtained by Hazuku et al. [11]. The values of gas velocity are plot-ted versus those of mixture volumetric flux. The figure indicatesthat the newly developed one works as well as the Mishima–Hibikicorrelation. The Hibiki–Ishii correlation [28] predicts the data sat-isfactorily at the low values of mixture volumetric flux, however,under-predicts the data at the high values. The predictions of cor-relations proposed by Kariyasaki et al. and Moriyama et al. largelydeviate from the data. It should be noted that since Kariyasaki et al.uses several equations to correlate the void fraction in terms of thegas volumetric quality, b (= jg/(jf + jg)), instead of the drift-fluxmodel, and the Hibiki–Ishii correlation also consists of severalequations, thus, their predictions are not on straight lines.

3.2.3. RecommendationIt should be mentioned that Serizawa et al. [42] calculated the

cross-sectional averaged void fraction for air–water two-phaseflow in a 20 lm i.d. silica tube from high-speed video pictures byassuming symmetrical shape of bubbles and gas slugs for bubblyflow and slug flow, respectively. Their data were correlated withthe Armand correlation [43]. In contrast, Kawahara and co-workers[7,44] measured the void fractions by analyzing the recordedimages of the gas–liquid interface in the observation windows ofseveral mini-channels (with diameters ranging from 250 lm to526 lm) as well as micro-channels (with diameters below100 lm), and reported that for micro-channels a strong deviationof the void fraction–volumetric quality relationship from the Ar-mand correlation was observed. The reasons for the contradictoryconclusions obtained by Serizawa et al. and Kawahara et al. may bedue to the surface conditions of channel wall and the design of themixing chamber. It was reported that the two-phase flow struc-tures in micro-channels (with hydraulic diameter of 20–100 lm)is more seriously affected by the wettability between the tubeand the fluids, and a formation of dry area between gas slug andthe tube wall can be observed in the experiment when the flowis very low [42]. The unique void fraction–volumetric quality rela-tionship observed by Kawahara and co-workers may arise from thedesign of the mixing chamber, which may be prone to producelarge bubbles resulting in the flow patterns of liquid film flowingalong the channel wall with gas flow in the core, and difficult toyield fine bubbles for bubbly flow pattern. In this study it was as-sumed that the channel wall of mini-channels would be all wettedand no dry area would exist, and consequently the existing theo-ries might be extendedly applied. For the above reasons, the col-lected databases in this study do not contain the data sets ofSerizawa et al. and Kawahara and co-workers, and are tailoredfor mini-channel according to the channel definitions of Kandlikar[34], instead of micro-channel.

As mentioned in the friction pressure drop part, since most ofdata in this collected data base fall into the flow condition of eitherliquid or gas flows being laminar, it can be expected that the newlydeveloped correlation would be able to predict the void fraction inthis flow condition for mini-channels. For other flow conditionssuch as both liquid and gas flow being turbulent, the Reynoldsnumber, Ref, may be an important non-dimensional number to cor-relate the distribution parameter. In addition, the density ratio, qg/qf may be also a significant non-dimensional number to correlatethe distribution parameter since the distribution parameter shouldbecome unity as the density ratio approaches unity [26]. However,since existing available data sets are for air–water two-phase flow

0 20 40 60 80 1000

20

40

60

80

100

+30%

-30%

Mean Dev.:27.1%

Correl. ofKariyasaki et al.

Pred

icte

d Vo

id F

ract

ion,

αca

l [%

]

Exp. Void Fraction, αexp [%]

0

20

40

60

80

100

-30%

+30%

Mean Dev.:63.8%

Correl. ofMoriyama et al.

(d)

(b)

(c)

(a)

Pred

icte

d Vo

id F

ract

ion,

αca

l [%

]


0

20

40

60

80

100

-30%

+30%

Mean Dev.: 12.9%

Correl. of Mishima-Hibiki

Pred

icte

d Vo

id F

ract

ion,

αca

l [%

]


0 20 40 60 80 100

0 20 40 60 80 100 0 20 40 60 80 1000

20

40

60

80

100

-30%

+30%

Mean Dev.: 26.6%

Correl. of Hibiki-Ishii

Pred

icte

d Vo

id F

ract

ion,

αca

l [%

]


Fig. 8. Evaluation of existing correlations with collected data.

0 20 40 60 80 1000

20

40

60

80

100

-30%

+30%

Mean Dev.: 12.7 %


Pred

icte

d Vo

id F

ract

ion,

αca

l [%

]

Exp. Void Fraction, αexp

[%]

Fig. 9. Evaluation of newly developed correlation with collected data.

0 1 2 3 4 5 6 70

2

4

6

8

10Prediction of Correlations

Present Study Kariyasaki et al. Moriyama et al. Mishima-Hibiki Hibiki-Ishii

Dh=1.02 mm

Data of Hazuku et al.

Gas

Vel

ocity

, vg

[m/s

]

Mixture Volumetric Flux, jT [m/s]

Fig. 10. Evaluation of correlations with data obtained by Hazuku et al. [11].


under atmospheric pressure, the effect of the density ratio on thedistribution parameter cannot be reflected in this newly developedcorrelation. Therefore, for other flow conditions, pressures, and flu-ids, further study is needed. Moreover, the asymptotic value of thedistribution parameter was assumed to be 1.2 for simplicity in thenewly developed correlation. This may be valid for circular chan-nel. For rectangular channels, however, its value may be different.

4. Conclusions

Accurate prediction of two-phase frictional pressure drop andvoid fraction in mini-channels is essential to have a good under-standing of two-phase flow as well as the successful modeling oftwo-phase flow in such channels. This study gave an extensiveevaluation of existing correlations with a collected database cover-


ing a wide range of running parameters, and proposed alternativecorrelations for two-phase friction pressure drop and void fractionin mini-channels. The detailed conclusions could be drawn asfollows:

(1) An extensive evaluation shows that the Dukler correlationbased on the homogeneous model works satisfactorily fordata sets of adiabatic liquid–gas flow as well as liquid–vaporflow, however, systematically under-predicts the data offlow boiling. All the tested correlations based on the homo-geneous model predict well the data for liquid–vapor flow.

(2) The correlations proposed by Mishima and Hibiki, and Leeand Lee, based on the separated flow model generally pre-dict all of the collected data within an acceptable marginof error. The good behaviors of both correlations for the pre-diction of liquid–gas two-phase frictional pressure drop areconfirmed.

(3) By applying the ANN, the main non-dimensional number tocorrelate the two-phase friction multiplier is picked out. Acorrelation of the Chisholm parameter C for mini-channelis newly developed using the non-dimensional Laplaceconstant.

(4) An extensive evaluation of the existing void fraction correla-tions indicates that the Mishima–Hibiki correlation also pre-sents the best performance.

(5) By applying the ANN, the non-dimensional Laplace constantnumber is found to successfully correlate the distributionparameter. An alternative correlation of void fraction formini-channel is proposed using this non-dimensionalLaplace constant.

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