International Journal of Heat and Mass Transfer 79 (2014) 34–53

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International Journal of Heat and Mass Transfer

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

Review

A review of heat transfer and pressure drop characteristics of singleand two-phase microchannels

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.07.0900017-9310/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: xgn@nwpu.edu.cn (G. Xie), Bengt.Sunden@energy.lth.se

(B. Sunden).

Masoud Asadi a, Gongnan Xie b,⇑, Bengt Sunden c

a Department of Mechanical Engineering, Azad Islamic University, Science and Research Branch, Tehran, Iranb School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710129, PR Chinac Division of Heat Transfer, Department of Energy Sciences, Lund University, P.O. Box 118, SE-22100 Lund, Sweden

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 November 2013Received in revised form 24 June 2014Accepted 28 July 2014Available online 20 August 2014

Keywords:Pressure dropHeat transferSingle-phaseTwo-phaseMicrochannel

An impressive amount of investigations has been devoted to enhancing thermal performance of micro-channels. The small size of microchannels and their ability to dissipate heat makes them as one of thebest choices for the electronic cooling systems. In this paper, a comprehensive review of available studiesregarding single and two-phase microchannels is presented and analyzed. 219 articles are reviewed toidentify the heat transfer mechanisms and pressure drops in microchannels. This review looks into thedifferent methodologies and correlations used to predict the heat transfer and pressure drop character-istics of microchannels along the channel geometries and flow regimes. The review shows that earlierstudies (from 1982 to 2002) were largely conducted using experimental approaches, and discrepanciesbetween analytical and experimental results were large, while more recent studies (from 2003 to2013) used numerical simulations, correlations for predicting pressure drop and heat transfer coefficientswere considerably more accurate.

� 2014 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342. Single-phase flow and heat transfer in microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.1. Pressure drop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.2. Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3. Multi-phase flow and heat transfer in microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1. Pressure drop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2. Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4. Summary and conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Conflict of interest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1. Introduction

In early 1981, Tuckerman and Pease [1] first explained the con-cept of microchannel heat sinks and predicted that single-phase

forced convective cooling in microchannels could potentiallyremove heat at a rate of the order 1000 W

cm2. Forced convection inchannels and liquid injection has been used for faster and largerscale cooling in industry for decades. Microchannel heat transfer,however, has become increasingly popular and interesting toresearchers due to high heat transfer coefficients, with potentialfor record-high heat transfer coefficients and low to moderatepressure drops when compared to conventional air and liquid

Nomenclature

a channel height (m)b channel width (m)BL boiling numberBo Bond numberC Lockhart–Martinelli parameterCP specific heat (J/kg K)Co convective numberdh hydraulic diameter (m)f friction factorFr Froude numberFfl fluid-surface parameterG mass velocityh heat transfer coefficient (W/m2 K)j total mixture volumetric fluxk thermal conductivity (W/m K)Ma Mach numberNu Nusselt numberP pressure (Pa)Pr Prandtl numberRe Reynolds numberS Chen’s suppression factorT temperature (K)u velocity (m/s)Y Chisholm parameterWe Weber number

Greek symbolss shear stress (Pa)l dynamic viscosity(Pa s)q density (kg/m3)a channel aspect ratioj Hagenbach factorULO two-phase multiplier (for liquids only)r surface tension

Subscriptsapp apparentcrit criticalf frictional, forced convectionFD fully developedGn Gnielinski’s methodL liquidm momentum, means staticsat saturationTP two-phase mixturetot totalV vaporW wall

M. Asadi et al. / International Journal of Heat and Mass Transfer 79 (2014) 34–53 35

cooled systems [2–7]. For example, microchannel heat sinks havebeen demonstrated for high-power laser diode array cooling andhave achieved a heat flux removal rate of 500 W

cm2 [8–10].In most cases when the cooling requirement is over 100 W

cm2 thecooling cannot be easily met either by simple air-cooling or water-cooling systems. In many applications, where high heat flux of thecomponents has to be dissipated, the required heat sinks must belarger than the components themselves. Nevertheless, hot spotsusually appear, and non-uniform heat flux levels are observed atthe heat sink level. This has motivated researchers to developnew heat sinks that can be directly embedded on the back of theheat source for uniform heat flux removal. Such a heat sink isusually made of silicon, with a silicon oxide layer to keep the com-ponent electrically insulated. Very narrow rectangular channels areformed with fins in the micrometer range that ensure uniform heatflux removal by circulating cold fluid through the rectangularmicrochannels.

Several investigators have proposed different criteria forminichannels. Serizawa et al. [11] described one criterion forclassification of microchannels as follows:

k P dh where k and dh are the Laplace constant and channeldiameter, respectively.

Mehendale et al. [12] classified micro heat exchangers based onthe hydraulic diameter as:

Micro heat exchangers : 1 lm 6 dh 6 100 lmMacro� heat exchangers : 100 lm 6 dh 6 1 mmCompact heat exchangers : 1mm 6 dh 6 6 mmConventional heat exchangers : dh P 6 mm

8>>><>>>:

ð1Þ

Kandlikar and Grande [13] used the hydraulic diameter for classifi-cation of single-phase and two-phase heat exchangers as,

Microchannels : 10 lm 6 dh 6 200 lmMinichannels : 200 lm 6 dh 6 3 mmConventional channels : dh P 6 mm

8><>: ð2Þ

Also, Palm [14] described the microchannels as heat transferelements where the classical theories cannot correctly predict thefriction factor and heat transfer characteristics. Stefan [15] used amicroscale system as one whose typical phenomena are absent ina macro system. Therefore, it is not always suitable to differentiatemini- and microchannels by a specific diameter like other research-ers, although this definition is often used nevertheless.

Halelfadl et al. [16] focused on analytical optimization of a rect-angular microchannel heat sink using aqueous carbon nanotubesbased nanofluids as coolant. The optimized results showed thatthe use of the nanofluid as a working fluid reduces the totalthermal resistance and can enhance significantly the thermal per-formances of the working fluid at high temperatures. Warrier et al.[17] proposed and analyzed a novel two-phase microchannel cool-ing device that incorporates perforated side walls for potential useas an embedded thermal management solution for high heat fluxsemiconductor devices. Yu and Zhang [18] focused on the hydrau-lic and thermal characteristics of fractal tree-like microchannelswith different aspect ratios for Reynolds numbers ranging from150 to 1200. The experimental results showed that the fractaltree-like microchannels had a much higher heat transfer coeffi-cient than the straight microchannels. Wang and Wu [19], Wanget al. [20], Revellin et al. [21], and Senn and Poulikakos [22] per-formed similar studies on the thermal performance of tree-likemicrochannels.

The impacts of periodic reversed flow and induced boiling fluc-tuations on the performance of a microchannel evaporator used inair-conditioning system can cause some problems [23–30]. Tuoand Hrnjak [23] proposed a novel solution to reduce these effectsby venting and bypassing back flow vapor accumulated in the inletheader. Frost formation on a louvered fin microchannel heatexchanger was experimentally investigated by Moallem et al.[31]. They developed a novel methodology to measure frostthickness and frost weight at intervals during the frosting period.The experimental data showed that at a given air dry bulb temper-ature, the fin surface temperature and air humidity are the primary

36 M. Asadi et al. / International Journal of Heat and Mass Transfer 79 (2014) 34–53

parameters that influence the frost growth rates. The cycle frostingand defrosting performance of two types of microchannel heatexchangers were investigated by Xu et al. [32]. Liu et al. [33]presented an experimental study of two-phase boiling heat trans-fer of the liquid methanol in co- and counter current microchannelheat exchangers with gas heating. There are a large number ofstudies which have used the microchannel to enhance the thermalperformance of heat exchangers [34–45].

Morini et al. [46] numerically studied a heat sink consisting ofmicrochannels of rectangular cross-section through which a polarfluid was circulated by means of an electro-osmotic pump. Sarangiet al. [47] numerically modeled the boiling heat transfer in micro-channels. They studied two-phase forced convection in microchan-nels using water as the fluid medium microchannel. Aluminum flattubes have attracted more and more attention in recent years, espe-cially in the air-conditioning and refrigeration industries [48–55].Zhang et al. [48] conducted numerical and experimental studieson in-plane bending of a microchannel aluminum flat tube. Theyconcluded that the degradation of the tube channel is relativelysmall under common process conditions. The bending radius is themain factor which influences the forming quality of the flat tubes,the tool-tube clearance mainly affects the wrinkling of the flat tubes,and channel diameter has little effect on the formability of the tube.

Research on convective heat transfer and pressure drop oninternal microtube and microchannel has been extensivelyconducted in the past decade [56–59]. Szczukiewicz et al. [56]measured the heat transfer coefficient of refrigerants in multi-microchannel evaporators. A two phase microgap heat sink has alarge potential to minimize the drawbacks associated with twophase microchannel heat sinks, especially flow instabilities, flowreversal and lateral variation of flow and wall temperaturebetween channels. Alam et al. [57] conducted some experimentsto investigate the heat transfer and pressure drop characteristicsof deionized water in a microgap heat sink and compared theseexperimental results with similar data obtained for a microchannelheat sink. Fani et al. [58] studied on size effects spherical nanopar-ticles on thermal performance and pressure drop of a nanofluid in atrapezoidal microchannel heat sink. They showed that with anincrease in the nanoparticle diameter, the average Nusselt numberof the base fluid decreases more than that of the nanoparticles andthis signifies that the base fluid has more efficacy on thermal per-formance of cooper-oxide nanofluid.

Table 1Characteristic values of laminar flow in circular and noncircular channels [69].

Channel cross section Channel geometry Hydraulic diameter Constant

Circle Diameter, d dh 64Rectangular a, b, a/b = 0.1 2ab/(a + b) 85.76Rectangular a, b, a/b = 0.2 2ab/(a + b) 76.8Rectangular a, b, a/b = 0.4 2ab/(a + b) 65.28Rectangular a, b, a/b = 0.6 2ab/(a + b) 60.16Rectangular a, b, a/b = 0.8 2ab/(a + b) 57.6Square Side a a 56.96

2. Single-phase flow and heat transfer in microchannels

2.1. Pressure drop

Wu and Little [60] performed several experiments with a gasflowing instead of a liquid in a trapezoidal-shaped silicon/glassmicrochannel in order to measure the flow friction and heat trans-fer characteristics. They reported that the transition from laminarto turbulent flow occurs at the Reynolds numbers of 400–900depending on the test conditions. They suggested that reducingthe transition Reynolds number improved the heat transfer charac-teristics. Pfahler et al. [61] experimentally investigated threemicrochannels of rectangular cross-section ranging in betweenarea 80–7200 lm2 using N-propanol as the working fluid andreported the results of fluid flow and friction factor. Later, theycontinued a series of experiments to measure friction factors bydifferent liquids and gases in microchannels. Peng et al. [62], Wanget al. [63], Peng and Wang [64], and Peng and Peterson [65]performed studies on heat transfer and fluid flow characteristicsfor different microchannel structures. Yu et al. [66] presentedresults for flow of nitrogen gas and water in microtubes withdiameters of 19.52 and 102 lm.

For rectangular flow channels, the hydraulic diameter is defined

dh ¼4ab

2ðaþ bÞ ð3Þ

There are several different definitions of the channel aspect ratio.Here, the channel aspect ratio is defined as,

a ¼ ab

ð4Þ

Many researchers used the laminar theory as the most commonobservation of discrepancy.

C� ¼ðf ReÞexp

ðf ReÞtheoryð5Þ

where f Re number is non-dimensionalized, and is calculated theo-retically and experimentally. Obviously, the desired value of C⁄ is 1,but our research indicated that the majority of data falls between0:6 6 C� 6 1:4. Although an explanation for this behavior couldcome from experimental uncertainties, many authors have notdiscussed the entrance and exit losses or the flow developing lengthand reported great discrepancy. The theoretical values of thefriction factor for a circular tube are:

f ¼ 64Re : Laminar flow

f ¼ 0:316Re�0:25 : Turbulent flow

(ð6Þ

Hwang and Kim [67] studied the pressure drop characteristics inmicrotubes with inner diameters of 0.244, 0.430, and 0.792 mmusing R-134a as the working fluid in the Reynolds number rangeof 150–10,000. Yen et al. [68] conducted an experimental investiga-tion for microtubes of diameters 0.19, 0.30, and 0.51 mm usingHCFC123 and FC-72 as the working fluids and reported results offluid flow and heat transfer. They showed that the friction factorin microtubes is well matched with its analytical laminar flow valuein the Reynolds number range of 20–265. Celata et al. [69] investi-gated water flow in a microchannel both experimentally and ana-lytically, in the Reynolds number range from 20 to 4000 andhydraulic diameters from 30 to 344 lm, as listed in Table 1.

For laminar flow, the Poiseuille number, Po = f Re is constant.This number is a function of the aspect ratio of rectangular chan-nels. Shah and London [105] presented a correlation to determineit. This correlation reads

f Re¼24 1�1:3553aþ1:9467a2�1:7012a3þ0:9564a4�0:2537a5� �ð7Þ

where the channel aspect ratio must be less than unity. If it isgreater than unity, the inverse is taken.

For the turbulent regime, Nikuradse [106] presented a correla-tion, where the friction factor is seen upon a non-dimensional wallroughness as

f ¼ 3:48� 1:737 lneD

� �h i�2ð8Þ

The friction factor can also be determined using the implicitColebrook et al. [107] equation.

Fig. 1. A comparison of experimental friction factor by Wu and Little [60], Yanget al. [199], Kohl et al. [195], Morini et al. [46,196], Tang et al. [197], andVijayalakshmi et al. [198].

Fig. 2. A comparison of different correlations for the Nusselt number proposed byWu and Little [60], Yu et al. [100], Adams et al. [111], and Gnielinski et al. [112].

M. Asadi et al. / International Journal of Heat and Mass Transfer 79 (2014) 34–53 37

1ffiffiffif

p ¼ 3:48� 1:737 lneD

� �þ 9:35

Reffiffiffif

p( )

ð9Þ

Steinke and Kandlikar [101] used two components to determine thefriction factor. The first was the friction factor from the theory offully developed flow, and the second one was the Hagenbach factor.

DP ¼ 2ðf ReÞlVL

d2h

þ jðxÞq�V2

2ð10Þ

where j(x) is the Hagenbach factor given by,

jðxÞ ¼ f app � f FD

� �4 � xdh

ð11Þ

where fapp and fFD are the apparent and fully developed friction fac-tor, respectively. The experimental data shows that the proposedcorrelation just can predict the friction factor for the laminar flowregime, but for the turbulent regime there is a considerable discrep-ancy. In general, because of this correlation has been developed forfully developed flow and in microchannels due to the temperaturevariation along the duct, the temperature profile cannot be consid-ered to be fully developed, the proposed correlation may haverelatively high deviation from experimental data.

The Hagenbach factor will begin at a value of zero and increaseto some fully developed constant value j(1). For a rectangularpassage, the fully developed Hagenbach factor is:

jð1Þ¼ 0:6796þ1:2197aþ3:3089a2�9:5921a3þ8:9089a4�2:9959a5� �ð12Þ

Kandlikar and Grande [13] presented an equation for both develop-ing and fully developed turbulent flow.

f app ¼ 0:0929þ 1:01612L=dh

� �Re� �0:268�0:3293

L=dh

� �ð13Þ

where the laminar-equivalent Reynolds number, Re⁄, was proposedby Jones [108] for rectangular channels as

Re� ¼ Re23þ 11

24að2� aÞ

� �ð14Þ

It is necessary to notice that this correlation is presented forminichannels. However, Kumar et al. [202] reported that it can beused for microchannels.

Fig. 1 shows the experimental friction factor vs. Reynolds num-ber presented by different authors. The data covers a wide range ofhydraulic diameters for different gases, cross-sections as well asprocess conditions. For the laminar flow regime, there is a reason-able agreement among the results presented by Morini et al. [46],Kohl et al. [195], Morini et al. [196], Tang et al. [197] andVijayalakshmi et al. [198]. However, there is a great deviationbetween Wu and Little [60] and Yang et al. [199]. This might bedue to the usage of different working fluids, materials and surfaceroughness. It is interesting to note that both of them expressed thesame Reynolds number range for the critical Reynolds number.

The remarkable works in single-phase pressure drop of gasesand liquids from 1981 to now are shown in Tables 2 and 3, respec-tively. Celata et al. [69] reported that their experimental predic-tions in the laminar flow regime are in good agreement with theconventional theory and are independent of the relative roughnessof the microchannel (up to e

Dh< 1%). A similar statement has been

reported by Morini et al. [46,196], Tang et al. [197], and Liu et al.[200], although Tang and co-authors observed that for smallermicrotubes with the inner diameter ranging from 10 to 20 lm,there is a reduction in the friction factor values. In the turbulentregime, Morini et al. [46] stated that the experimental data arelower than the predictions of the Blasius correlation for smoothtubes, even if one considers the compressibility effects. These

results confirm the findings of other researchers like Yu et al.[66], Li et al. [86], Choi et al. [117], Vijayalakshmi et al. [198], Yanget al. [199], and Kumar et al. [202]. The difference between theexperimental friction factors and the Blasius correlation might bedue to the fact that the Blasius correlation corresponds toincompressible-turbulent flow through a circular channel, whilesome authors used compressible flow in their studies.

The compressibility effects become significant for gaseous flowwhen one of the below inequalities is satisfied:

Ma > 0:3DPPin> 0:05

(ð15Þ

where Ma is the average Mach number along the duct and Pin is thepressure at the inlet section. For the first inequality, the gas flow

Table 2Selected literature for single-phase pressure drop: liquids.

Author Year Fluid Shape dh (lm) a Re C⁄ L/dh Remarks

Tuckerman and Pease [1] 1981 Water R 92–96 0.17–0.19 291–638 0.73–1.06 104–109 –Harley and Bau [170] 1989 Isopropanol and fluorocarbon R, T 49.75–66.67 0.5–3.1 25–250 >1.0 – –Missaggia et al. [70] 1989 Water R 160 0.25 2350 33.54 6 –Pfahler et al. [171] 1990 Isopropanol, silicon oil R, T 1.6–65 0.008–0.4 0.0001–300 <1.0 – fRe decreases with the Reynolds number in the

smallest channel. The polar nature of fluid hassignificant effect of friction factor

Riddle et al. [71] 1991 Water R 86–96 0.06–0.16 96–982 0.79–4.06 156–180 –Rahman and Gui [72] 1993 Water R 299–491 3.00–6.00 275–3234 121.89–507.10 94–154 –Rahman and Gui [73] 1993 Water, R11 R 299–491 3.00–6.00 275–3234 121.89–507.10 94–154 –Urbanek et al. [172] 1993 Isopropanol T,Ti 5–25 – – 0.79–4.06 – Temperature variation affects the Poiseuille

numberGui and Scaringe [74] 1993 Water T 338–388 0.73–0.79 834–9955 1.28–5.33 119–136 Relative surface roughness affects the friction

factorPeng et al. [62] 1993 Methanol R 311–646 0.29–0.86 1530–13455 ID 70–145 In laminar regime, the Poiseuille number

depends on the Reynolds numberWilding et al. [173] 1994 Water, biological fluids T 26–63 1.0–7.5 17–126 – 185.25–438.75 –Wang and Peng [174] 1994 Water, methanol R 311–747 0.29–1.14 80–3600 – 60–145 Considered the variation of thermo–physical

properties of the fluids. Earlier laminar-to-turbulent transition.

Peng and Peterson [75] 1995 Water R 311 0.29 214–337 ID 145 Fluid temperature, velocity and microchannelsize have strong effects on flow transition

Jiang et al. [175] 1995 Water R,T 20–65 0.11–1.31 0.006–1.6 0.98–1.56 39–500 –Cuta et al. [176] 1995 Water, R124 R 425 0.27 101–578 0.39–2.04 48 –Cuta et al. [76] 1996 R124 R 425 0.27 101–578 0.39–2.04 48 –Peng and Peterson [65] 1996 Water R 133–200 0.5–1.0 136–794 13.50–27.70 25–338 Transition regime changes with aspect ratio of

channelJiang et al. [77] 1997 Water C, T 8–68 0.38–0.44 728 0.22–3.05 69–276 Flow resistance in non-circular microchannels

is smaller than the conventional predictions byclassical theory and varied with cross-sectionsizes

Harms et al. [177] 1997 Water R,T 404–1923 0.04–4.10 173–12900 – 13–6164 The channel shows better flow and heattransfer performance with decreasing channelwidth

Tso and Mahulikar [78] 1998 Water C 728 NA 16.6–37.5 ID 76–89 Reported flow transition using heat transferanalysis

Vidmar and Barker [79] 1998 Water C 131 NA 2452–7194 1.77–5.58 580 –Adams et al. [111] 1999 Water T 131 ID 3899–21429 ID 141Mala and Li [80] 1999 Water C 50–254 NA 132–2259 1.38–20.07 150–490 In low Reynolds number regime, there is a

nonlinear trend between pressure drop andflow rate. Relative surface roughness hasstrong effects on friction factor

Papautsky et al. [81] 1999 Water R 44–47 5.69–26.42 0.002–4 0.98–1.41 164–177 –Meinhart et al. [178] 1999 Water R 54.5 0.1 – – 458.33 Flow characteristics deviated from classical

theory for channel dimensions < 100 lmPfund et al. [82] 2000 Water R 253–990 19.19–78.13 55.3–3501 0.01–1.81 101–396 Critical Reynolds number decreases with

decreasing channel depth. Transition is suddenbut not discontinuous in microchannels

Qu et al. [83] 2000 Water T 51–169 1.54–14.44 6.2–1447 0.55–1.68 165–543 Friction factor depends on relative surfaceroughness in laminar flow regime

Qu et al. [84] 2000 Water T 62–169 2.16–11.53 94–1491 ID 178–482 A roughness viscosity model to interpretexperimental data has been developed

Rahman [85] 2000 Water R 299–491 3.00–6.00 275–3234 487–2028 94–154 Laminar-to-turbulent transition is somewhatgradual because of small channel dimension

Xu et al. [32] 2000 Water R 30–344 0.58–24.53 5–4620 0.53–3.18 145–1070 Experimental predictions are in goodagreement with the conventional theory when

38M

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Table 2 (continued)

Author Year Fluid Shape dh (lm) a Re C⁄ L/dh Remarks

dh > 30 lmRen et al. [179] 2001 Water R 28.1–80.3 124–335 1–60 >1.0 10.67–347 Pressure drop depends on the channel’s height

and the ionic concentration of the liquidsChung et al. [86] 2002 Water C 100 NA 1.9–3237 0.89–2.08 875 –Gao et al. [180] 2002 Water R 199.2–1923 25–250 100–8000 >1.0 42.6–411.6 The transition observed varies between 2500

and 4000 depends on the conditionsJudy et al. [87] 2002 Water, methanol, isopropyl C, R 14–149 1.00 7.6–2251 0.83–1.27 1203–5657 Friction factor predictions are in good

agreement with the conventional theory forrectangular channels. Friction factor affects bythe material of construction of microchanneland test fluid

Lee et al. [88] 2002 Water R 85 0.25 119–989 1.06–2.39 118 The discrepancy between the experimentalresults and conventional theory is mainly dueto the uncertainty involved in the pressuredrop measurement

Qu and Mudawar [89] 2002 Water R 349 0.32 137–1670 0.70–1.94 128 Variation of pressure drop with respect to Re isdepends on thermal variation of fluid viscosityand pressure losses at inlet and outlet sections

Celata et al. [181] 2002 R114 C 130 – 100–8000 >1.0 92 Friction factor is in good agreement with theconventional theory for Re < 585. Relativesurface roughness affects the friction factorand laminar-to-turbulent transition

Li et al. [86] 2003 Water C 79.9–205.3 – 300–2500 0.85–1.9 182–519 For Re < 1700 friction factor in roughmicrotubes (3–4%) is 15–37% higher than theconventional theory. Inaccurate diametermeasurement of microtubes dominates theoverall error of experiments

Bucci et al. [90] 2003 Water C 172–520 NA 2–5272 0.87–3.24 ID Friction factor depends on the surfaceroughness when flow is in the laminar regime

Jung and Kwak [91] 2003 Water R 100–200 1.00–2.00 50–325 0.69–2.15 75–150 –Lee and Garimella [92] 2003 Water R 318–903 0.17–0.22 558–3636 ID 28–80 –Park et al. [93] 2003 Water R 73 4.44 4.2–19.1 ID 654 PIV velocity measurement is used to

investigate the flow characteristics. Thermalvariation affects the velocity profiles in themicrochannel

Tu and Hrnjak [94] 2003 R134a R 69–305 4.11–11.61 112–3500 0.89–2.35 131–288 –Wu and Cheng [95] 2003 Water T, Ti 26–291 ID 11.1–3060 0.73–1.98 ID Geometrical parameters have considerable

influences on the apparent friction factorconstant of the trapezoidal microchannels

Wu and Cheng [96] 2003 Water T 169 1.54–26.20 16–1378 0.58–1.88 192–467 –Baviere et al. [97] 2004 Water R 14–593 83.33 0.1–7985 0.91–3.04 138–429 –Hsieh et al. [98] 2004 Water R 146 1.74 45–969 0.96–3.39 164 Hydrodynamic developing length is smaller in

comparison with the conventional theoryLelea et al. [99] 2004 Water C 125.4, 300, 500 – 50–800 0.92–1.1 410–857 The friction factor and the entrance effects

predictions are in good agreement with theclassical theory

Hao et al. [100] 2005 Water T 237 0.361 50–2800 – 127 f Re relationship deviates from the linearbehavior when Re > 700

Steinke et al. [101] 2006 Water R 227 0.8 14–789 1.15–3.75 45 Uncertainty in fRe is dominated by themicrochannel width and height measurements

Shen et al. [102] 2006 Water R 436 2.67 162–1257 1–2.84 16–754 In rough microchannels, surface roughness hasa considerable influences on the laminar flow.For high Reynolds number values, f Re is higherthan the conventional theory predictions andincreasing with growing Re

Hrnjak and Tu [103] 2007 R134a R 69.5–304.7 0.09–0.24 112–9180 1.02–1.09 315–691 Surface roughness results in increasing the

(continued on next page)

M.A

sadiet

al./InternationalJournal

ofH

eatand

Mass

Transfer79

(2014)34–

5339

Tabl

e2

(con

tinu

ed)

Au

thor

Yea

rFl

uid

Shap

ed h

(lm

)a

Re

C⁄L/

d hR

emar

ks

fric

tion

fact

orin

mic

roch

ann

els

asw

ell

asaf

fect

sth

ela

min

ar-t

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ansi

tion

Jun

gan

dK

wak

[183

]20

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ater

R10

0–20

01.

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.00

50–3

500.

69–2

.15

75–1

50In

trod

uce

da

corr

elat

ion

for

pres

sure

drop

inm

icro

chan

nel

sG

amra

tet

al.[

184]

2008

Wat

erR

100–

300

0.33

–1.0

065

–280

0–

107–

320

Pres

sure

drop

issi

gnifi

can

tly

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(abo

ut

20%

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ra

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ess

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out

0.15

Wib

elan

dEh

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d[1

85]

2009

Wat

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43.9

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212

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dan

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arac

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lam

inar

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ule

nt

tran

siti

onM

irm

anto

etal

.[10

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ater

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let

and

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circ

ula

r,R

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ctan

gula

r,T

=tr

apez

oid,

Ti=

tria

ngl

e;Y

=Y

es,N

=N

o.

40 M. Asadi et al. / International Journal of Heat and Mass Transfer 79 (2014) 34–53

cannot be incompressible, and the momentum and energy equa-tions have to be solved as coupled. While for the second inequality,the density variation along the microchannel cannot be ignored.The result of Vijayalakshmi et al. [198] shows that for Reynoldsnumbers up to 1600, the pressure distribution is linear as expectedin incompressible flow, but for Re beyond 1600, the pressuredistribution tends to show a non-linear behavior. Vijayalakshmiand co-authors believed that this is due to the compressibilityeffects. This is in agreement with Kumar et al. [202] who observedthat the compressibility effects are less important for Re < 2000with an inner diameter larger than 20 lm. However, Araki et al.[203] expressed that the compressibility effects in gas flows becomeimportant when the pressure drop of the duct is around 10 kPa,even if the Mach number is less than 0.1.

Hrnjak and Tu [103] stated that the roughness may not besignificant in the laminar regime, but it can have a dramatic effecton the turbulent flow friction factor in microchannels. Similarly,Tang et al. [197] and Vijayalakshmi et al. [198] concluded thatthe friction factor is independent of the roughness in the laminarregime as they found their results to have discrepancy comparedto the conventional theories. However, many authors believed thatthe friction factors depend on the relative roughness in the laminarflow regime [13,66,69,82,83,186,203,205].

The experimental data proposed by Morini et al. [46,196],Hrnjak and Tu [103], Kohl et al. [195], and Sharp and Adrian[121] indicated no evidence of early transition from the lami-nar-to-the turbulent flow regime. In contrast, some researchersexpressed an earlier transition with respect to the predictionsof the conventional theories [60,62,65,74,80,174,203,205]. Thecritical Reynolds number ranges for liquids and gases are shownin Figs. 3 and 4, respectively. In general, the laminar-to-turbu-lent transition seems to be influenced by the geometry of thecross-section, the roughness of the microchannel walls, thecompressibility effects and the length-to-hydraulic diameter( L

Dh) ratio.

2.2. Heat transfer

The Nusselt number of a fully developed laminar flow is 4.364when there is a constant heat flux boundary condition at thetube wall. Grigull and Tratz [109] considered the thermalentrance problem for laminar flow with a constant wall heat flux.They evaluated the Nusselt number as a function of the dimen-sionless axial distance, Reynolds number, and Prandtl numberand found

Nu ¼ 4:364þ 0:00668ðdh=xÞRe � Pr

1þ 0:04 ðdh=xÞRe � Pr½ �2=3 ð16Þ

Shilder et al. [110] conducted an experimental investigation for sin-gle phase flow in a microchannel that had a hydraulic diameter of0.6 mm. Admas et al. [111] conducted experimental work in theturbulent region with water flow in circular microchannels of diam-eters 0.76 and 0.109 mm. Based on their data, they proposed thefollowing equation,

Nu ¼ NuGn þ ð1þ FÞ ð17Þ

where,

NuGn ¼ðf=8Þ Re� 1000ð ÞPr

1þ 12:7 f=8ð Þ1=2 Pr2=3 � 1� � ð18Þ

is the Gnielinski for value with f as

f ¼ 1:82 logðReÞ � 1:64ð Þ�2 ð19ÞF ¼ 7:6� 10�5Reð1� ðdh=d0Þ2Þ ð20Þ

M. Asadi et al. / International Journal of Heat and Mass Transfer 79 (2014) 34–53 41

NuGn represents the Nusselt number predicted by Gnielinski’s[112] correlation. The least-squares fit to all the data sets studiedby Adams et al. [111] resulted in d0 = 1.164 mm. According to

Table 3Selected literature for single-phase pressure drop: gases.

Author Year Fluid Shape dh

(lm)a Re C⁄ L/

Wu and little[60]

1983 Ar, H2, N2 R,T 55.8–83.1

2.37–4.75

100–15000

1.1–3.9

1072

Pfahler et al.[171]

1990 Ar, H2, N2 R, T 76.12 0.008–0.4

0.0009–300

0.8–0.9

–

Choi et al.[117]

1991 N2 C 3.00–81.2

0– 30–20000

0.83 6481

Pong et al.[187]

1994 N2, He R 0.03–2.33

0.03–0.24

– – –

Arklic et al.[188]

1994 He R 2.59 39 0.0014–0.012

0.35–0.45

28

Yu et al. [66] 1995 N2 C 19–102

– 250–20,000

0.79–4.06

<1

Shih et al.[189]

1996 N2, He R 2.33 33.33 0.001–0.1

0.3–0.45

17

Stanley et al.[190]

1997 N2 R 56–256

– 50–10,000

– –

Wu et al. [191] 1998 N2 R 3.37 10.3 0.1–1.00

0.3–0.4

13

Araki et al.[192]

2000 H2, N2 T, Ti 3.92–10.3

1.41–19.71

0.0065–0.0345

0.78–1.18

1.3.

Turner et al.[193]

2001 Air, N2,He

R, T 5–96 20–435

0.1–1000

0.8–1.1

2651

Asako et al.[194]

2005 N2 C 150 – 1508–2188

0.78–1.56

2032

Kohl et al.[195]

2005 Air R 24.9–99.8

1.03–3.90

6.8–18,814

0.5–1.5

2253

Morini et al.[46]

2006 N2 C 133–730

– 100–10,000

0. 9–1.2

5737

Tang et al.[197]

2007 N2 C, R 50–300

– 3–6200 0.77–1.32

3330

Morini et al.[196]

2009 N2 C 100–300

– 100–25,000

0. 9–1.2

1650

Vijayalakshmiet al. [198]

2009 N2 T 60.5–211

2.6–3.6

0.04–0.18

<1.00 2486

Fig. 3. The critical Reynolds n

Fig. 5, the correlations are well matched with the experimentaldata for the 0.102, 0.76 and 1.09 mm microchannels of Yu et al.[66], within ±18.

dh Remarks

0–0

Relative surface roughness affects the friction factor. Earlier laminar-to-turbulent transition has been observedFriction factor depends strongly on the material of construction and thetest field

0–00

Critical Reynolds number decreases with hydraulic diameter

Pressure distribution along the microchannel is not linear mainly becauseof rarefaction and compressibility effects

92 Friction factor decreases with Knudsen number

.00 Friction factor values depend on relative surface roughness

17 Friction factor decreases with Knudsen number

For dh > 150 lm, critical Reynolds number is between 2300 and 3000

05 Results are agreement with the predictions of Arklic et al. [188]

59–83

In trapezoidal microchannel, friction factor is smaller than prediction byconventional theory due to the rarefaction effects

3–76

Mach number has dominant influence on f while the differences due tosurface roughness are likely masked by uncertainty

6–7

Developed a correlation for computing friction factor with the help ofMuch number as flow parameter

0–3

When compressibility effects become significant critical Reynolds numberdepends upon L/dh

5–59

Effects due to acceleration of the fluid become important for largeReynolds numbers

3–00

In circular tube when the Reynolds number is low, high roughnessincreases the friction factor

7–00

For Re < 1000, results are in good agreement with the conventional theory.However, for Re > 1000, the friction factor tends to deviate from thePoiseuille law

6–0

At low hydraulic diameter, compressibility affects transition from laminar-to-turbulent due to the flow acceleration

umber ranges for liquids.

Fig. 4. The critical Reynolds number ranges for gases.

Fig. 5. Comparison between the experimental results of Yu et al. [68] and thecorrelation proposed by Adams et al [111].

42 M. Asadi et al. / International Journal of Heat and Mass Transfer 79 (2014) 34–53

Garimella and Vishal [119] reported that there is little agree-ment between the results from different investigations for heattransfer in both laminar and turbulent regimes, because most ofthe experimental results deviate from theoretical predictions thatassume macroscale behavior. They concluded that transition fromthe laminar to turbulent regimes appears to occur at relatively lowReynolds numbers in many of the experimental studies relative toexpectations from conventional analysis. Mala and Li [80] reportedthat flow in microtubes experience an early transition at Re >300–900 depending on the test conditions, whereas Peng et al.[62] suggested that transition occurs at 700. Silverio and Moreira[120] showed that transition does not occur below Re = 1800, inagreement with what Sharp and Adrian [121] reported. Table 4shows a complete list of literature about single-phase heattransfer.

As can be seen from Fig. 2, there is a great deviation betweendifferent correlations. Wu and Little [60] analyzed the heat transfercharacteristics in both the laminar and turbulent flow regimes.

They found the Nusselt numbers higher than those predicted bythe conventional theories. Then Choi et al. [117] concluded thatWu and Little’s correlation is not in agreement with their experi-mental data. Admas et al. [111] tested Gnielinski’s correlation forthe prediction of the Nusselt number in the turbulent flow regime,in order to complement the data presented by Yu et al. [66]. Theirexperimental data suggested a modification factor for the Gnielin-ski’s correlation based on Re and hydraulic diameter. Goingthrough many papers conducted on heat transfer characteristicsof single-phase flow reveals that:

� The experimental Nusselt numbers are lower than those pre-dicted by conventional theories [62,64,65,75,78,83,174,180,206].� The experimental Nusselt numbers are higher than those pre-

dicted by conventional theories [13,60,72,76,77,85,90,100,111,117,148,175].� The experimental data are in good agreement with the conven-

tional correlations for laminar and turbulent flow regimes[17,89,90,95,177,207].

Despite the publication of a very large number of dedicatedpapers, considerable discrepancies between the results still exist.It may be due to the following reasons:

� In the entrance region, the temperature and velocity profiles aredeveloping, and so the Nusselt number changes along themicrochannel. Based on the classical fluid dynamics analogy,the two entry lengths are; hydrodynamic and thermal entrylengths. When the velocity profile is considered as fully devel-oped, the thermal entry length effects need to be considered.This is the most typical situation for flows with Pr > 1. The Gra-etz number is defined as a criterion for neglecting the entranceeffects. Morini [208] reported that for Graetz numbers largerthan 10, the entrance effects have to be considered, though Rosaet al. [209] suggested that the entrance effects might berelevant at moderate or higher Reynolds numbers.� The difference in fluid temperature between the inlet and outlet

sections could be relatively high in microchannels. Therefore,the variation of the thermophysical properties along the micro-channel could be one of the candidate reason for the apparentdeviations of the Nusselt numbers.

M. Asadi et al. / International Journal of Heat and Mass Transfer 79 (2014) 34–53 43

� The rarefaction effects have to be checked with the help of theKnudsen number. In fact, for rarefied gases (Kn > 0.001), slipflow and temperature jump conditions at the wall have to beconsidered. Generally, the slip flow condition tends to increasethe Nusselt number, whereas temperature jump tends todecrease it.� Because of density variations, compressibility effects may be

relevant only for gas flows. As a criterion, when the Mach num-ber is lower than 0.3, the flow is treated as incompressible.However, this is a necessary but not sufficient condition toallow the flow to be considered approximately incompressible.In some cases, strong pressure losses may lead to density vari-ations with significant influence on the velocity and tempera-ture profiles, and hence on the heat transfer.� For very small hydraulic diameters, the internal heat generation

due to the viscous forces can produce a temperature rise even ifthe flow is adiabatic. This temperature variation because of theviscous forces changes the values of the fluid thermophysicalproperties along the channel and can considerably affect theheat transfer. The Brinkman number, which is defined as theratio of the viscous heating rate to the average heat transferrate, can be employed to evaluate the viscous heating effects.� A possible explanations for the discrepancies between the

experimental data and conventional correlations might be theexperimental uncertainties. In this regard, much of the earlierpublished results may be unreliable and not useful forcomparison.

Table 4Selected literature for single-phase heat transfer.

Author Conditions Channelgeometry

Flow regime

Kays andCrawford[113]

Re < 2200 Rectangular Fully developed

Incropera andDeWitt [114]

Re < 2200 Circular Simultaneously develop

Incropera andDeWitt [114]

Re < 2200 Circular Thermally developing la(constant wall temperat

Incropera andDeWitt [114]

Re > 10000 Circular Fully developed turbule

Incropera andDeWitt [25]

Re > 10000 Circular Fully developed turbule

Stephan andPreußer [115]

0.7 < Pr < 7 or RePrD/L < 33 for Pr > 7

Circular Simultaneously developwall temperature)

Stephan andPreußer [115]

0.7 < Pr < 7 or RePrD/L < 33 for Pr > 7

Circular Simultaneously developwall temperature)

Shah and London[105]

Re < 2200 Circular Thermally developing la(constant wall heat flux

Shah and London[105]

Re < 2200 Circular Fully developed laminar

Kakac et al. [116] 2200 < Re < 10000 Circular Transitional

Gnielinski [112] 3000 < Re < 5 � 106 Circular Transitional and fully deturbulent

Wu and Little[60]

Re > 3000 Rectangular Laminar flow

Choi et al. [117] Re < 2200 Rectangular Laminar flowChoi et al. [117] 2500 < Re < 20,000 Rectangular Turbulent flowYu et al. [100] 6000 < Re < 20,000 Rectangular Turbulent flowPeng et al. [62] Re < 2200 Rectangular Laminar flowPeng et al. [62] Re > 10,000 Rectangular Turbulent flowGrigull and Tratz

[109]Re < 2200 Circular Laminar flow with cons

Admas et al [111] Re > 10,000 Circular Turbulent flow

Bejan et al [118] Re < 2200 ID Thermally and hydrodyndeveloping laminar flow

ID = insufficient data.

3. Multi-phase flow and heat transfer in microchannels

3.1. Pressure drop

For gas–liquid two-phase flow, most of the available pressuredrop correlations are based on the traditional homogenous flowmodel (HFM) or separated flow model (SFM). In HFMs, the twophases are assumed to be thoroughly mixed and thus the two-phase frictional pressure drop can be computed from the correla-tions presented in the single-phase flow case. The mean propertiesthat are weighted relative to the vapor and liquid contents, andthat only latent heat may be exchanged between the phases areassumed. Hence, a lot of two-phase mixture viscosity correlationshave been suggested. An intelligent selection of such correlationsis crucial for the successful application of this model.

The two-phase pressure drop consists of frictional, acceleration,and gravitational terms.

dPdZ

� �TP¼ dP

dZ

� �frictional

þ dPdZ

� �acceleration

þ dPdZ

� �gravitational

ð21Þ

When the flow is horizontal, the gravitational term is neglected.Also, the acceleration term is neglected when the flow is adiabatic.Therefore, the total pressure drop is the frictional term, which canbe defined using the HFM method.

dPdZ

� �TP

¼ dPdZ

� �frictional

¼ 2f TPG2

qTPDhð22Þ

Correlation

Nufd ¼ 8:235 1� 1:883=aþ 3:767=a2 � 5:814=a3

þ5:361=a4 � 2=a5

� �

ingNu ¼ 1:86 RePrD

L

� �1=3 lflw

� �0:14

minarure)

Nu ¼ 3:66þ 0:19ðRePrD=LÞ0:8

1þ0:117ðPrReD=LÞ0:467

nt Nu = 0.023Re0.8Pr1/3

nt Nu ¼ ðf=8ÞRePrKþ12:7ðf=8Þ1=2ðPr2=3�1Þ

; Nu ¼ ðf=8ÞRePrKþ12:7ðf=8Þ1=2ðPr2=3�1Þ

ing (constant Nu ¼ 3:657þ 0:0677ðRePrD=LÞ1:33

1þ0:1PrðReD=LÞ0:3

ing (constant Nu ¼ 3:657þ 0:0677ðRePrD=LÞ1:33

1þ0:1PrðReD=LÞ0:3

minar) Nu ¼ 1:953ðRePrD=LÞ1=3 : ðRePrD=LÞP 33:3

4:364þ 0:0722RePrD=L : ðRePrD=LÞ � 33:3

Nu ¼ 4:861 1� 3:656aþ 12:821a2 � 27:441a3 þ 37:373a4

�28:365a5 þ 8:888a6

� �

Nu ¼ 0:116 Re2=3 � 125� �

Pr1=3 1þ ðD=LÞ2=3h i

lf

lw

� �0:14

veloped Nu ¼ ðf=8ÞðRe�1000ÞPr1þ12:7ðf=8Þ1=2ðPr2=3�1Þ

; f ¼ 11:82 lnðReÞ�1:64ð Þ2

Nu = 0.00222Re1.09Pr0.4

Nu = 0.000972Re1.17Pr1/3

Nu = 3.82 � 10�6Re1.96Pr1/3

Nu = 0.007Re1.2Pr0.2

Nu = 0.1165(D/P)0.81(b/w)�0.79Re0.62Pr0.33

Nu = 0.072(D/P)1.15[1 � 2.421(z � 0.5)2]Re0.8Pr0.33

tant heat flux Nu ¼ 4:36þ 0:00668ðdh=xÞRe�Pr1þ0:04½ðdh=xÞRe:Pr�2=3

Nu = NuGn + (1 + F); NuGn ¼ ðf=8ÞðRe�1000ÞPr1þ12:7ðf=8Þ1=2ðPr2=3�1Þ

f = (1.82log(Re) � 1.64)�2; F = 7.6 � 10�5Re(1 � (dh/d0)2)amically

Nu ¼ C0 L=dhRePr

� ��0:5

44 M. Asadi et al. / International Journal of Heat and Mass Transfer 79 (2014) 34–53

where fTP is a two-phase friction factor, and G and qTP are the massvelocity and two-phase density, respectively.

qTP ¼xqGþ 1� x

qL

� ��1

ð23Þ

The two-phase friction factor can be defined as

f TP ¼16

ReTP: ReTP 2000

0:079Re�0:25TP : ReTP > 2000

(ð24Þ

With

ReTP ¼GDh

lTPð25Þ

where lTP is the two-phase viscosity.Numerous two-phase viscosity models have been introduced.

Recently, Cioncolini et al. [186] proposed a new correlationapproach based on the vapor core Weber number, capable of pro-viding physical insight into the flow. This new macroscale methodwas then extended to cover microscale conditions for both laminarand turbulent regimes. Costa-Patry et al. [167] conducted flowboiling experiments in 85 lm wide multi-microchannels. Theyshowed that Cioncolini’s correlation well agrees with theirpressure drop results.

f TP ¼ 0:0196We�0:372c Re0:318

L ð26Þ

Other two-phase viscosity models are as following:McAdams et al. [122]:

lTP ¼xlGþ 1� x

lL

� ��1

ð27Þ

Owens [182]:lTP ¼ lL ð28Þ

Cicchitti et al. [123]:lTP ¼ xlG þ 1� xð ÞlL ð29Þ

Dukler et al. [124]:lTP ¼ blG þ 1� bð ÞlL ð30Þ

Beattie and Whalley [201]:lTP ¼ blG þ 1� bð Þð1þ 2:5bÞlL ð31Þ

Lin et al. [210]:

lTP ¼lLlG

lG þ x1:4ðlL � lGÞð32Þ

Awad and Muzychka [211]:

lTP ¼2lG þ lL � 2ðlG � lLÞð1� xÞ2lG þ lL þ ðlG � lLÞð1� xÞ ð33Þ

According to two-phase flow patterns observed in the research byVenkatesan et al. [162], Cioncolini et al. [186], and Choi and Kim[212], indicate that the HFM may be only applicable to bubbly flowin which the liquid flow field is thought to be less disturbed by thepressure of small bubbles. Liu et al. [213] stated that this flowpattern typically takes place at relatively high liquid velocitiesand low gas velocities. However, the deviation from the HFMassumption can be observed in the Taylor flow (also known as slugflow), where gas bubbles with lengths greater than the tube diam-eter move along the capillary separated from each other by liquidslugs. However, Taylor flow cannot be considered as SFM becauseof the alternate movement of Taylor bubbles and liquid slugs downthe channel. In other words, this flow pattern has been eliminatedfrom consideration in the SFM.

Choi and Kim [212] reported that the most accurate viscositymodel is the Beattie and Whalley’s model, which is based on thevolumetric quality. Their experimental data showed that thetwo-phase pressure drop was over-predicted by Dukler’s model

and under-predicted by other models. Chung and Kawaji [151]found that the Dukler’s viscosity model predicts the data poorlyfor the 530 and 250 lm channels, but produces reasonable agree-ment with the 100 and 50 lm microchannel data. On the otherhand, the HFM with the Beattie and Whalley model roughly pre-dicts the two-phase pressure drop data for 530 and 250 lm chan-nels but significantly over-predicts the data for the 100 and 50 lmchannels. Table 5 shows the mean deviation of the two-phase pres-sure drop between experimental data and theoretical models.From Choi and Kim’s data for Dh = 490 lm, it is observed that thepredictions by Duklers’ model is poor, while for Dh ¼ 141 lm itcan predict the two-phase pressure drop reasonably well. Thisresult is in agreement with the statement of Chung and Kawaji[151]. However, no considerable difference is observed betweenthe performance of the Beattie and Whalley’s model for 490 and141 lm.

Lee and Mudawar [134] expressed that the trend prediction bythe Cicchitti two-phase viscosity model is somewhat differentfrom that of the other models because this model is qualityweighted and therefore provides significantly higher estimates ofthe two-phase mixture viscosity at low quality than the othermodels. Because of the low exit quality data are associated withsmaller viscosity, the HFM with Cicchitti over-predicts these data,but under-predicts the high exit quality data. Lee and Mudawar[134] concluded that the combination of overprediction at lowquality and underprediction at high quality yields a relativelyfavorable mean deviation for the Cicchitti method. This also canbe found from the results of Cioncolini et al. [186], Yue et al.[214], and Kawahara et al. [136]. It is noteworthy that the bestmodel for predicting the two-phase pressure drop for each studydiffers from each other. It is important to note that the hydraulicdiameter and two-phase fluids are the key notes for selection ofa successful model. From the various studies conducted by differ-ent authors, it can be found that the Dukler and Cioncolini modelscan predict the data reasonably in microchannels, while theBeattie and Whalley and McAdams can be used at mini andmacroscales.

In the separated flow model (SFM), gas and liquid are consid-ered to flow separately in the channel with each phase occupyinga sector of the channel cross-section. The SFM has been tested suc-cessfully for air–water flow in miniature triangular channels withDh ¼ 0:87� 2:89 mm [219], and for N2 � water in both a circularmicrochannel Dh ¼ 100 lm [136] and a square microchannel witha hydraulic diameter of 96 lm [151]. SFMs are based on atwo-phase multiplier, which is defined as follows,

/2L ¼

dPdZ

� �TP

dPdZ

� �L

or /2LO ¼

dPdZ

� �TP

dPdZ

� �LO

ð34Þ

where ðdPdzÞLO is the pressure drop for all liquid flow (liquid only)

defined as follows,

dPdz

� �LO

¼ f LO2G2

DqLð35Þ

and for the gas side,

dPdz

� �VO

¼ f VO2G2

Dqvð36Þ

Lockhart and Martinelli [126] also used a two-phase multiplier torelate the two-phase pressure drop to the single-phase pressuredrop for liquid flow:

U2L ¼ 1þ C

Xþ 1

X2 ð37Þ

X ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdP=dzÞLðdP=dzÞV

sð38Þ

Table 5Mean deviation between experimental data and HFM models.

Mc Adams Owens Cicchitti Dukler Beattie and Whalley Lin Awad and Myuztchka

Choi and Kim [212], Dh = 490 lm 40.81 352.1 300.7 66.84 30.69 111.99 107.87Choi and Kim [212], Dh = 141 lm 169.87 709.07 650.12 42.1 43.5 346.56 298.29Cioncolini et al. [186] 43.6 40.0 26.6 47.7 39.2 – 27.8Lee and Mudawar [134] 28.26 – 15.98 30.61 26.79 26.44 –Yue et al. [214], Dh = 528 lm 19.31 244.38 221.14 – – 67.54 –Yue et al. [214], Dh = 333 lm 18.15 201.92 187.96 – – 76.91 –Kawahara et al. [136], distilled water 28.00 40.5 40.3 33.9 49.0 39.2 –Kawahara et al. [136], ethanol 4.8 wt% 59.1 114.9 113.6 27.7 55.2 101.6 –Kawahara et al. [136], ethanol 49 wt% 46.6 99.9 99.3 10.4 95.1 90.9 –Kawahara et al. [136], ethanol 100 wt% 71.2 110.9 110.1 8.3 93.4 104.5 –Average 52.4 212.62 196.2 33.43 54.11 107.29 144.65

M. Asadi et al. / International Journal of Heat and Mass Transfer 79 (2014) 34–53 45

The two-phase multiplier, U2L is determined by the coefficient C and

the Lockhart–Martinelli parameter, X2, which is the ratio of thesingle-phase liquid and gas pressure gradients.

Friedel et al. [125] presented a correlation for the two-phasepressure gradient multiplier as

U2LO ¼ Eþ 3:24FH

Fr0:045We0:035 ð39Þ

E ¼ ð1� x2Þ þ x2 qL

qV

f VO

f LOð40Þ

where Fr and We are the Froude and Weber numbers, respectively, Fand H are also Friedel parameters. fLO and fVO are the friction factorsfor the total mass flux G flowing as gas and liquid properties,respectively.

Because most of the experiments were conducted under thecondition that ReL and ReG were lower than 1000, according toLockhart and Martinelli’s model the C value should be 5. However,the results of Yue et al. [214,215], Chung and Kawaji [151], andFukano and Kariyasaki [216] observed that the data are not wellpredicted by a single value of 5. Chung and Kawaji stated thatthe value of C is seen to decrease as the channel diameter isreduced from 530 to 50 lm. A similar behavior was observed byYue et al. [214], although they believed that the Lockhart–Martinelli model fails to describe the dependence of the C valueon mass flux as well as on gas and liquid superficial velocities.

Some researchers have suggested empirical correlations for thecoefficient C to determine the two-phase multiplier. Table 6 showsa list of these C values. Cavallini et al. [129] have recently reportedthat Mishima and Hibiki’s method can predict the two-phase pres-sure drop for flow condensation of refrigerants R-134a and R-236ain 1.4 mm tubes. The correlation of Mishima and Hibiki [127]evidently assumes that C depends on channel size. There is basedon the observation that C depends on phase mass fluxes, and usingexperimental data from several sources as well as their own datathat covered channel gaps from 0.4 to 4 mm.

Chisholm et al. [130] introduced the following equation for thetwo-phase multiplier:

U2LO ¼ 1þ ðY2 � 1Þ Bx

2�n2 ð1� xÞ

2�n2 þ x2�n

h ið41Þ

where the exponent n = 0.25 and the Chisholm parameter Y is:

Y ¼ ðdP=dzÞVO

ðdP=dzÞLOð42Þ

If the Chisholm parameter is 0 < Y 6 9:5, the parameter B is:

B ¼4:8 : G 6 500 kg=m2 s2400

G : 500 < G 6 1900 kg=m2 s55

G0:5 : G > 1900 kg=m2 s

8><>: ð43Þ

when 9.5 < Y < 28,

B ¼520

YG0:5 : G 6 600 kg=m2 s21Y : G > 600 kg=m2 s

(ð44Þ

for Y P 28,

B ¼ 15;000

Y2G0:5 ð45Þ

Zhang and Web [131] measured adiabatic two-phase flow pressuredrops for R-134a, R-22, and R-404a flowing in a multi-port extrudedaluminum tube with a hydraulic diameter of 2.13 mm, and in twocopper tubes with inside diameters of 6.25 and 3.25 mm, respec-tively. They found that the Friedel’s correlation cannot predict thetwo-phase data accurately. Therefore, they proposed a new correla-tion for the two-phase pressure gradient multiplier as follows,

U2LO ¼ ð1� x2Þ þ 2:87x2 P

Pcrit

� ��1

þ 1:68x0:8 1� xð Þ0:25 PPcrit

� ��1:64

ð46Þ

where Pcrit is the critical pressure, a constant for each fluid.Muller-Steinhagen and Heck [133] used a data bank containing

9300 measurements of frictional pressure drop to develop a newcorrelation as:

dPdz

� �f¼ Fð1� xÞ1=3 þ dP

dz

� �VO

x3 ð47Þ

F ¼ dPdz

� �LOþ 2

dPdz

� �VO� dP

dz

� �LO

� �x ð48Þ

where ðdPdzÞLO is given by Eq. (35), and ðdP

dzÞVO by Eq. (36).Concerning the flow pattern, such as annular flow and churn

flow, the SFM seems to be more realistic. In slug-annular flowregimes, small bubbles are also present in the liquid slugs (this isa feature that is not seen in Taylor flow) [159–161]. Churn flowoccurs at very high gas velocities. It includes the very long gas bub-bles and relatively small liquid slugs. The annular flow regime isobserved at excessively high gas velocities and very low liquidvelocities [154]. Venkatesan et al. [162] used a common nomencla-ture for the definition of the basic flow patterns. The intermittentflow regime is used to cover slug, slug-annular, and annular flow,while the dispersed regime consists of bubbly and dispersed bub-bly flow. Lee and Mudawar [134], Yue et al. [214,215], and Chungand Kawaji [151] also used the flow patterns to analysis the two-phase pressure drop. However, Lee and Mudawar [134] expressedthat the churn flow patterns are rarely detected in the SFM, in con-trast to the statement of Yue et al. [214].

Venkatesan et al. [162] reported that for a 3.4 mm tube, themean deviation with the HFM with Dukler’s model in the bubblyflow regime is 10%, while Chisholm’ correlation predicts the pres-sure drop with 17% mean deviation. Slug flow was predicted by the

Table 6Summary of two-phase pressure drop multiplication factors.

Author Year Fluid Shape dh (lm) Re Model used Correlation developed Remarks

Mishima and Hibiki [127] 1996 Water, air C 1000–4000

– Lockhart–Martinelli

C ¼ 21ð1� e�0:319Dh Þ It should be noted that the value of parameter Cbecomes zero when the hydraulic diameter is as small as0.2 mm

Lee and Lee [128] 2001 Water, air R 780–6670 0.303–17700 Lockhart–Martinelli

C ¼ A l2L

qLrDh

� �q lI jr

� �Res

LOThe new correlation can be used in laminar–laminar,laminar–turbulent, turbulent–laminar, and turbulent–turbulent regimes

Kawahara et al. [136] 2002 Water, N2 C 100 250–20,000 – – The agreement between the experimental data andhomogeneous flow model is generally poor, withreasonably good predictions (within_20%) obtained onlywith Dukler’s model for the mixture viscosity

Qu and Mudawar [218] 2003 Water R 349 – Lockhart–Martinelli

C ¼ 21 1� e�0:319Dh� �

�ð0:00418Gþ 0:0613ÞThis correlation is based on the combination of laminarliquid and laminar vapor flow. Pressure drop increasesappreciably upon commencement of boiling inmicrochannels

Yue et al. [214] 2004 Water, N2 T 333–528 10–1000 Lockhart–Martinelli

C ¼ aXbRecLO

a ¼ 0:411822;b ¼ �0:0305; c ¼ 0:600428

The Lockhart–Martinelli method generallyunderpredicts the frictional pressure drop. Among thehomogenous flow models, the viscosity correlation ofMcAdams indicates the best performance in correlatingthe frictional pressure drop data

Coleman and Krause [152] 2004 R134a R 830 400–40,000 – – The experimentally determined pressure drops in themicrochannel headers were significantly higher thanany of the existing models including the homogeneousmodel and the separated model

Chung and Kawaji [151] 2004 Water, N2 R 50–100 0.0014–0.012

Mechanistic model dPdz

� �F=B ¼ f B

qG UB�U1ð Þ22DB

The modified model was applied to all flow conditionsand ensuing flow patterns in the microchannels, sincethe flow patterns in a microchannel of 100 lm diameterhave been observed to be entirely intermittent

Lee and Mudawar [134] 2005 R134a R 349 – Lockhart–Martinelli

C ¼ 2:16Re0:047fo We0:6

fo : L–L L–L: laminar–liquid–laminar–vapor

C ¼ 1:45Re0:25fo We0:23

fo : L–T L–T: laminar–liquid–turbulent–vapor

Pamitran et al. [217] 2008 CO2, R–22 C 1500–3000

– Lockhart–Martinelli

C ¼ 1:2897� 106Re0:5674tp We�3:3271

tpThe homogeneous model with two-phase viscositysuggested by Dukler predicted the experimental databetter than the other models

Yue et al. [215] 2008 CO2, water C 200–667 0.4–2300 SFM /2L ¼ 0:217b�1=2

L Re0:3LS

Two-phase frictional pressure drop in microchannelsshould be described by different models depending onthe flow pattern investigated. For flow patterns such asslug-annular flow, annular flow and churn flow, thetraditional separated flow model is more realistic

Lee and Garimella [92] 2008 Water R 160–538 – Lockhart–Martinelli

C ¼ 2566G0:5466D0:8819h ð1� e�0:319Dh Þ The pressure drop across the microchannels increases

rapidly with heat flux when the incipience heat flux isexceeded

Kawahara et al. [136] 2009 Water,ethanol

C 250–500 – Lockhart–Martinelli

C ¼ 1:38Bo0:04Re0:52L We�0:12

G : withoutcontraction

The correlation was developed by using data for two-phase flows in 50–250 lm diameter microchannels.

C ¼ 0:55Bo0:04Re0:52L We�0:12

G : with contractionMegahed and Hassan

[135]2009 FC-72 R 70–304 19–4443 Lockhart–

MartinelliC ¼ 0:0053Re0:934

fo

Co0:73 X2ð Þ0:175; laminar liquid–laminar vapor The Qu and Mudawar correlation gives the bestprediction among the all methods. The two-phasepressure drop depends strongly on the mass flux, andincreases almost linearly with increasing exit quality at aconstant mass flux

C ¼ 0:0002Re1:7fo

Co0:7 X2ð Þ1:24; laminar liquid–laminar vapor

Choi et al. [117] 2009 Propane R 1500–3000

– Lockhart–Martinelli

C ¼ 1732:953Re�0:323tp We�0:24

tpThe pressure drop models of Mishima and Hibiki, Friedeland Chang gave a better prediction, among the othermethods, with a mean deviation less than 40%

Venkatesan et al. [162] 2011 N2 C 600 – Lockhart–Martinelli

C ¼ 4 WeLð Þ0:3 ReGReL

� �0:5; Bo 1

Friedel’s correlation over predicts the entire data for alltube diameters with mean deviation greater than 100%

C ¼ 2 WeLð Þ0:5 ReGReL

� �0:5; Bo < 1

46M

.Asadi

etal./International

Journalof

Heat

andM

assTransfer

79(2014)

34–53

Table 7Mean deviation between experimental data and SFM models.

L–M Chisholm Mishima andHibiki

Lee andLee

Qu andMudawar

Kawahara Zhang andWeb

Trant Lee andMudawar

Li andWu

Kawahara et al. [136], distilledwater

– 21.6 25.8 36.0 25.8 22.8 – – – –

Kawahara et al. [136], ethanol4.8 wt%

– 23.9 17.0 27.6 17.8 15.4 – – – –

Kawahara et al. [136], ethanol100 wt%

– 45.4 12.5 8.3 24.4 18.8 – – – –

Choi and Kim [212], Dh = 490 -lm

27.50 71.13 – 51.8 26.65 – 45.32 73.00 43.3 45.0

Choi and Kim [212], Dh = 322 -lm

29.24 60.18 – 41.07 19.75 – 30.52 92.99 23.7 29.37

Choi and Kim [212], Dh = 143 -lm

100.84 42.91 – 18.88 23.7 – 12.40 283.17 19.42 11.12

Qu and Mudawar [218] 28.6 378.4 13.9 19.1 12.7 – – 828.3 – –Lee and Mudawar [134] – 34.37 16.04 – – 50.07 – 5.62 –Megahed and Hassan [135] 30.3 – 11.7 14.7 10.4 – – – – –Lee and Garimella [92] 27.3 – 21.7 34.5 16.4 – – – – –Cioncolini et al. [186] 41.3 – 47.2 – – – – 185.0 – –Average 37.49 91.93 23.02 26.79 19.73 19.00 34.58 292.4 23.00 28.47

L–M: Lockhart–Martinelli.

Table 8Values of the fluid-surface parameter in the correlation of Kandlikar et al. [158].

Fluid FfL Fluid FfL

Water 1.00 R-1132 3.30R-11 1.30 R-124 1.00R-12 1.50 R-141b 1.80R-13BI 1.31 R-134a 1.63R-22 2.20 R-152a 1.10R-113 1.30 Kerosene 0.488R-114 1.24 Nitrogen 4.70R-32 3.30 Neon 3.50

M. Asadi et al. / International Journal of Heat and Mass Transfer 79 (2014) 34–53 47

Chisholm’ correlation within 14% mean deviation compared to 43%with the HFM. For a channel with D = 1.7 mm, the bubbly flowregime is well predicted by the HFM with 7% mean deviation, whileChisholm’s correlation predicts the slug and Taylor flow regimeswith 22% and 28% mean deviation, respectively. These results tellthat the SFM method is more realistic for the slug and churn flowregimes, while the HFM model can predict the data for the bubblyand Taylor flow regimes well. However, Venkatesan et al. [162]found that the annular flow regime is not observed in the0.6 mm tube even at high gas velocities, similar to with Chungand Kawaji’s results. May be this is due to the stronger surfacetension effects in a microchannel that allows the liquid film tobridge the gas core more easily than in a minichannel so that theformation of annular flow would be less likely.

In microchannels in two-phase flow, the capillary force is typi-cally negligible compared to the viscous and inertia forces [132].However, by decreasing the hydraulic diameter the capillary forcestarts to play a key role in determining the behavior of thetwo-phase flow patterns. Li and Wu [140] expressed that thereare theoretically four forces related to two-phase flow in channels;gravitational, viscous, inertia, and surface-tension forces. The com-parison of the channel dimension and the nominal bubble size canbe expressed in terms of the Bond number, which is a measure ofthe importance of body forces compared to the surface-tensionforces. The significance of the inertia force to viscous force ratiois determined by the liquid and gas Reynolds numbers, while thatof the inertia force to surface-tension force is known from theWeber number [204]. Also, the ratio of the viscous force to thesurface-tension force is called the Capillary number. Lee and Lee[128], Yue et al. [214,215], Lee and Mudawar [134], Pamitranet al. [217], Kawahara et al. [136], Megahed and Hassan [135], Choiet al. [117], and Venkatesan et al. [162] used surface-tension andinertial forces to determine the two-phase pressure drop.

It is observed from Table 7 that Qu and Mudawar [218]predicted the two-phase pressure drop with a reasonable meandeviation among all models for microchannels. After that, Mishimaand Hibiki [127] and Lee and Lee [128] models have the best per-formance with the average mean deviation of 23.03% and 26.79%,respectively. Megahed and Hassan [135] also reported that theQu and Mudawar’s model gives the best prediction. Note that thiscorrelation has been developed for microchannel sizes, comparedto Mishima and Hibiki [127] and Lee and Lee [128] models thatcover a wide range of conventional and mini size channels. There-fore, in order to select a proper model for predicting the two-phase

pressure drop, it is important to notice the hydraulic diameter ofmodel and experimental apparatus.

3.2. Heat transfer

The correlation of Chen [137] is among the oldest, most success-ful and widely used correlations for saturated boiling. It works wellfor water at relatively low pressure and has been applied to varietyof fluids. The correlation can be used for dh P 1 mm,P = 0.09–3.45 MPa, x = 0–0.7 and

q00 ¼ 0� 2:4MW=m2 ð49Þh ¼ hNB þ hFC ð50Þ

The forced convection component can be found from,

hFC ¼ 0:023Re0:8f Pr0:4

f FKdh

� �ð51Þ

where,

Ref ¼ Gð1� xÞdh=lf ð52Þ

Prf ¼ ðlCp=KÞf ð53Þ

The enhancement factor F is represented by (ReTP/Ref)0.8 and thiswas correlated by Chen [137] empirically in a graphical form.

The nucleate boiling is based on the correlation by Forster andZuber [138], modified to account for the reduced averagesuperheat in the thermal boundary layer for bubble nucleationon wall cavities.

hNB ¼ 0:00122K0:79

f C0:45pf q0:49

f g0:43c

r0:5l0:29f h0:24

fg q0:24g

( )DT0:24

sat DP0:75sat S ð54Þ

Table 9Selected literature for two-phase heat transfer.

Author Year Fluid Channel Nature of work General outcome

Lazarek and Black [141] 1982 R113 Circular Experimental, Analytical Plot, CorrelationCornwell and Kew [142] 1992 R113 Rectangular Analytical PlotMoriyama and Inoue [143] 1992 R113 Rectangular Analytical Plot, Numerical dataBowers and Mudawar [144] 1993 R113 Minichannels Analytical PlotWambsganss et al. [145] 1993 R113 Circular Experimental PlotMertz et al. [146] 1996 Water, R141b Rectangular Experimental PlotKew and Cornwell [147] 1997 R141b Circular Experimental Plot, CorrelationRavigururajan et al. [148] 1998 R124 Rectangular Experimental, Analytical PlotMehendale and Jacobi [149] 2000 R134a Rectangular Experimental PlotKawahara and Chung[150] 2002 Water, Nitrogen Circular Experimental, Analytical Plot, Numerical dataLee and Mudawar [134] 2005 R134a Rectangular Experimental PlotZhao et al. [153] 2006 Water, Kerosene Rectangular Experimental, Analytical Plot, CorrelationYue and Luo [103] 2008 Water, CO2 Rectangular Experimental, Analytical Plot, CorrelationAgostini et al. [155] 2008 R236fa Rectangular Experimental PlotMegahed and Hassan [135] 2009 FC-72 Rectangular Experimental, Analytical Plot, CorrelationErgu and Sara [156] 2009 Water Rectangular Experimental PlotAlapati and Kang [157] 2009 NA Rectangular Numerical Numerical dataNa and Chung [163] 2011 ID Circular Numerical Plot, Numerical dataMegahed et al. [164] 2012 FC-72 Rectangular Experimental PlotLiu and Fu [33] 2012 Methanol, Helium Rectangular Experimental Plot, Numerical dataAutee and Rao [165] 2012 Water, Air Circular Experimental, Analytical Plot, Numerical data, CorrelationIde and Kimura [166] 2012 Water, Nitrogen, Circular Experimental PlotCosta-Patry et al. [167] 2012 R245fa Rectangular Experimental PlotSzczukiewicz and Magnini [168] 2013 R236fa, R245fa Circular Experimental, Numerical Plot, Numerical dataNascimento et al. [169] 2013 R134a Rectangular Experimental PlotGoss and Passos [170] 2013 R134a Circular Experimental Plot

NA = not applicable.

48 M. Asadi et al. / International Journal of Heat and Mass Transfer 79 (2014) 34–53

where DTsat = TW � Tsat, and DPsat = Psat(TW) � P. The parameter S is

Chen’s suppression factor and is represented as S ¼ DTeffDTsat

� �0:99, where

DTeff is the effective liquid superheat in the thermal boundary layer.The correlation of Kandlikar et al. [158] is based on 10,000 data

points for water, refrigerants and cryogenic fluids. The correlationcan be used for dh P 1 mm.

h¼max hNBO;hCBCð Þ ð55Þ

hNBO ¼ 0:6683Co�0:2ð1�xÞ0:8f 2ðFrf 0Þþ1058BL0:7ð1�xÞ0:8FfL

h ihf 0 ð56Þ

hCBC¼ 1:136Co�0:9ð1�xÞ0:8f 2ðFrf 0Þþ667:2BL0:7ð1�xÞ0:8FfL

h ihf 0 ð57Þ

where,

hf 0 ¼Kf

dh

Ref � 1000� �

ðf=2ÞPrf

1þ 12:7 Pr2=3f � 1

� �ðf=2Þ0:5

h i ð58Þ

The parameters in Kandlikar’s correlation are the convectionnumber Co, the boiling number BL, and the forced number whenall mixture is saturated liquid, Frf0:

Co ¼ qg=qf

� �0:5ð1� xÞ=x½ �0:8 ð59Þ

BL ¼ q00w=Ghfg ð60ÞFrf 0 ¼ G2=ðq2

f gdhÞ ð61Þ

FfL is the fluid-surface parameter and is shown in Table 8.

f 2ðFrf 0Þ ¼1 : Frf 0 P 0:4

ð25Frf 0Þ0:3 : Frf 0 < 0:4

(ð62Þ

Gungor and Winterton [139] presented a correlation based on the3700 data points for water, refrigerants, and ethylene glycol. Thecorrelation is used for dh P 1 mm.

h ¼ hf 1þ 3000BL0:86 þ 1:12 ð1� xÞ=x½ �0:75ðqg=qf Þ0:41

n oE2 ð63Þ

E2 ¼1 : Frf 0 P 0:05

Frð1�2Frf 0Þf 0 : Frf 0 < 0:05

(ð64Þ

Shah and London [105] presented some correlation, to be used inmini and microchannels. They considered nucleate and convectiveboiling both to be important for the two-phase flow evaporativeheat transfer. Their method is applicable for both horizontal andvertical tubes. When N > 1.0 and BL > 0.0003, then h is calculated as,

h ¼ 230BL0:5hf ð65Þ

N ¼ 1� xx

� �0:8 qg

qf

!0:5

ð66Þ

when N > 1.0 and BL < 0.0003, then h is calculated from,

h ¼ ð1þ 46BL0:5Þhf ð67Þ

when 0.1 < N < 1.0,

h ¼ FsBL0:5 expð2:74N � 0:1Þhf ð68Þ

and when N < 0.1,

h ¼ FsBL0:5 expð2:74N � 0:15Þhf ð69Þ

here, Fs is Shah’s constant:

Fs ¼15:43 : BL < 0:001114:7 : BL > 0:0011

ð70Þ

Recently, Li and Wu [140] obtained a correlation using the boilingnumber, Bond number, and Reynolds number. The correlation ispresented based on the 3744 data points, covering a wide rangeof working fluids, operating conditions, and different microchanneldimensions. They showed that the Bond number in predicting heattransfer coefficients can be used as a criterion to classify a flow pathas a microchannel or as a conventional macrochannel. Thecorrelation can be used for 0:19 mm 6 dh 6 2:01 mm:

M. Asadi et al. / International Journal of Heat and Mass Transfer 79 (2014) 34–53 49

h ¼ 334BL0:3ðBoRe0:36f Þ0:4ðKf =dhÞ ð71Þ

Bo ¼gðqL � qgÞd

2h

rð72Þ

Table 9 summarizes some works related to two-phase heat transferin microchannels.

From much results published on the pressure drop and heattransfer characteristics of single- and two-phase flows, it is evidentthat if the experimental results are used to validate the conven-tional theories for microchannels, the answer obtained is not univ-ocal (especially for heat transfer characteristics). In fact, someauthors found their experimental data in agreement with the con-ventional theory. On the other hand, for the same range of hydrau-lic diameter and working fluid, some found the opposite results.Those authors who reached the opposite results proposed newcorrelations to predict the Nusselt number and friction factor.The proposed correlations are in general based on a limited num-ber of data points and no theoretical analysis originated them,hence, the reliability of the new correlation is yet questionable.

4. Summary and conclusion

In this paper a comprehensive review has been performed onsingle and two-phase microchannels. The investigation has beencarried out on heat transfer and pressure drop characteristics. Forsingle-phase channels, in the fully developed, incompressible,and isothermal laminar flow regime, the fluid behavior in micro-channels with diameter of several hundred micro meters obeysthe classical theory. At low Reynolds numbers the experimentalresults matched the predictions for thermally developing flowsas the Reynolds number increases the experimental results departfrom the laminar predictions. It is noteworthy that when thehydraulic diameter increases the experimental measurements fol-low the predicted thermal developing behavior. However, for thepressure drop there is a little agreement between the results fromdifferent investigations in both the laminar and turbulent regimes.The measurements lie above or below the predictions, and showdifferent trends relative to conventional predictions in the laminarand turbulent regimes, mainly due to this fact that many research-ers assumed macroscale behavior for deriving correlations.Another reason is that, just 57% of the researchers consideredentrance and exit losses in their investigations for calculating thepressure drops, and investigators who considered these effectsreported a better C⁄ parameter values. About 76% of the investiga-tors studied heat transfer and pressure drop of single-phasemicrochannels in the laminar region, and this shows that thebehavior of fluids in the turbulent region remained unexploredand this is the main reason why the experimental results of manyresearchers were over-predicted or under-predicted.

For two-phase microchannels, we found that choosing a reason-able and proper mixture viscosity correlation is crucial to a suc-cessful interpretation of the two-phase frictional pressure dropdata. We found that all relevant correlations except that of McAd-ams et al. [122] over predict the two-phase pressure drop. For heattransfer characteristics, the correlations by Chen [137], Kandlikaret al. [158], and Gungor and Winterton [139] predict experimentaldata well for cases where the channel hydraulic diameter is largerthan 500 lm, but they are not accurate for microchannels.Nevertheless, our study indicated that the correlation proposedby Li and Wu [140] can predict the experimental data well forthe range of hydraulic diameters from 190 to 2000 lm.

Conflict of interest

None declared.

Acknowledgements

This work was supported by National Natural Science Founda-tion of China (11202164) and NPU Foundation for FundamentalResearch (NPU-FFR-JC20130115). The authors are grateful to theauthors whose articles were reviewed in this manuscript for theirvaluable research and findings.

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