adiabatic two-phase frictional pressure drops in microchannels

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Experimental Thermal and Fluid Science 31 (2007) 673–685

Adiabatic two-phase frictional pressure drops in microchannels

Remi Revellin, John R. Thome *

EPFL, STI ISE LTCM, ME Gl 464, Station 9, CH-1015 Lausanne, Switzerland

Received 21 March 2006; received in revised form 11 July 2006; accepted 11 July 2006

Abstract

Two-phase pressure drops were measured over a wide range of experimental test conditions in two sizes of microchannels (sight glasstubes 0.509 and 0.790 mm) for two refrigerants (R-134a and R-245fa). Similar to the classic Moody diagram in single-phase flow, threezones were distinguishable when plotting the variation of the two-phase friction factor versus the two-phase Reynolds number: a laminarregime for ReTP < 2000, a transition regime for 2000 6 ReTP < 8000 and a turbulent regime for ReTP P 8000. The laminar zone yields amuch sharper gradient than in single-phase flow. The transition regime is not predicted well by any of the prediction methods for two-phase frictional pressure drops available in the literature. This is not unexpected since only a few data are available for this region in theliterature and most methods ignore this regime, jumping directly from laminar to turbulent flow at ReTP = 2000. The turbulent zone isbest predicted by the Muller-Steinhagen and Heck correlation. Also, a new homogeneous two-phase frictional pressure drop has beenproposed here with a limited range of application.� 2006 Elsevier Inc. All rights reserved.

Keywords: Microchannels; Frictional; Pressure drop; Two-phase flow

1. Introduction

Micro- or mini-heat spreaders are used in the interest ofproviding higher cooling capability for microtechnologies.They are characterized by a high heat flux dissipationand a better heat transfer coefficient compared to conven-tional processes. Higher effectiveness means, for an identi-cal power, a reduction of size and cost. Compactness alsoreduces the amount of charge of the fluid, which has alsoa direct positive impact on safety and environment. How-ever, the negative point is possibly a higher pressure droprelated to the micro- or mini-flow channels.

The total pressure drop of a fluid is due to the variationof kinetic and potential energy and that due to friction, sothat the pressure drop is the sum of the static pressure drop(elevation head), the momentum pressure drop (accelera-tion) and frictional pressure drop:

0894-1777/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.expthermflusci.2006.07.001

* Corresponding author.E-mail addresses: [email protected] (R. Revellin), john.thome@

epfl.ch (J.R. Thome).

dPdz

� �t

¼ dPdz

� �s

þ dPdz

� �m

þ dPdz

� �f

ð1Þ

The static pressure drop in a horizontal microchannel is 0

dPdz

� �s

¼ 0 ð2Þ

The momentum pressure drop takes into account the accel-eration of the flow due to the flashing or diabatic effect andis defined as follows:

dPdz

� �m

¼ G2 DxðqL � qVÞðLqVqLÞ

ð3Þ

The frictional pressure drop dPdz

� �f

can be determined bydifferent models or correlations for macrochannels as thosereported in Table 1. The simplest one is the homogeneousmodel that makes analysis of two-phase flows easier: thisideal-fluid obeys the usual equation of a single-phase fluidand is characterized by suitably averaged properties. Threepossible forms of the two-phase viscosity models arereported in Table 1.

mailto:[email protected]

mailto:john.thome@ epfl.ch

mailto:john.thome@ epfl.ch

Nomenclature

B Chisholm parameter, –C constant of Lockhart and Martinelli or Chis-

holm parameter, mC0 confinement number, –D diameter, me surface roughness, lmE Friedel parameter, –F Friedel parameter, –F Muller-Steinhagen parameter, bar/mFr Froude number, –f friction factor, –G mass flux, kg/m2 sg acceleration of gravity, m/s2

H Friedel parameter, –L length, mP pressure, barq heat flux, W/m2

Re Reynolds number, –T temperature, �CWe Weber number, –x vapor quality, –X Lockhart–Martinelli parameter, –Y Chisholm parameter, –z longitudinal abscissa, m

Greek lettersa mean absolute error, a ¼ 1

N

�PN

1predicted value�experimental value

experimental value

�� , %

b fraction of data predicted to within ±20%, %e void fraction, –k Lee and Lee parameter, –

/ two-phase multiplier, –w Lee and Lee parameter, –q density, kg/m3

l dynamic viscosity, Pa sr surface tension, N/m

Subscripts

crit criticalf frictionalG gash heatedh hydraulicHom homogeneous modelint internalin inletL liquidm momentumMEV microevaporatorMPH micropreheaterO onlyout outlets staticsat saturationsub subcoolingt totalt turbulentTP two-phaseTS test sectionv viscousV vapor

674 R. Revellin, J.R. Thome / Experimental Thermal and Fluid Science 31 (2007) 673–685

The separated flow model considers the two-phases tobe artificially separated into two streams, each flowing inits own pipe. The cross-sectional flow areas of the twopipes are proportional to the void fraction. The basicequations for the separated flow model are not dependenton the particular flow configuration adopted. It is assumedthat the velocities of each phase are uniform, in any givencross-section, within the zone occupied by the phase. Thefirst of these analyses was performed by Lockhart andMartinelli [1].

The Friedel [2] correlation for the two-phase frictionalpressure drop was specially developed for conventionalchannels as well as that of Chisholm [3].

A new correlation by Muller-Steinhagen and Heck [4]for the prediction of frictional pressure drop for two-phaseflow in pipes was suggested which is simple and more con-venient to use than other prior methods. The correlationwas developed using a data bank containing 9300 measure-ments of frictional pressure drop for a variety of fluids andconditions, including channel diameters from 4 to 392 mm.

In Table 2, are reported the modified correlations ormodels for microchannels.

Mishima and Hibiki [5] measured the frictional pressureloss for air–water flows in vertical capillary tubes withinner diameters in the range from 1 to 4 mm. The resultswere compared with the Lockhart and Martinelli model.The frictional pressure loss was reproduced well byChisholm’s equation with a new equation for Chisholm’sparameter C as a function of inner diameter. Lee and Lee[6] proposed new correlations for the two-phase pressuredrop through horizontal rectangular channels with smallgaps (heights) based on 305 data points. The gap betweenthe upper and the lower plates of each channel ranges from0.4 to 4 mm while the channel width was fixed to 20 mm.Water and air were used as the test fluids. The authorsexpressed the two-phase frictional multiplier using theLockhart–Martinelli type correlation but with the modifi-cation on parameter C (see Table 3).

Lee and Mudawar [7] measured the two-phase pressuredrop across a microchannel heat sink that served as an

Table 1Pressure drop models and correlations for macrochannels

Authors Equation

Homogeneousmodel

dPdz

� �f¼ 2f TPG2

DqTP

withqTP ¼ x

qVþ 1�x

qL

� ��1

fTP ¼ 16ReTP

for ReTP < 2000 or

f TP ¼ 0:079Re�0:25TP for ReTP > 2000

and ReTP ¼ GDlTP

Three possible forms of the two-phaseviscosity models are:McAdams et al. [13]: lTP ¼ x

lVþ 1�x

lL

� ��1

Cicchitti et al. [16]: lTP = (xlV + (1 � x)lL)

Dukler et al. [17]: lTP ¼ qTP x lV

qVþ ð1� xÞ lL

qL

� �Lockhart and

Martinelli [1]

dPdz

� �f¼ dP

dz

� �LU2

L

withU2

L ¼ 1þ CX þ 1

X 2 with C given by:

Ctt = 20 (liquid turbulent, gas turbulent),Cvt = 12, Ctv = 10, Cvv = 5

X ¼dPdzð ÞLdPdzð ÞV

0:5

with dPdz

� �L¼ fL

2G2

DqLð1� xÞ2

and dPdz

� �V¼ fV

2G2

DqVx2

fL ¼ 16ReL

for ReL < 2000 or fL ¼ 0:079Re�0:25L

for ReL > 2000

fV ¼ 16ReV

for ReV < 2000 or fV ¼ 0:079Re�0:25V

for ReV > 2000

ReL ¼ Gð1�xÞDlL

and ReV ¼ GxDlV

Friedel [2] dPdz

� �f¼ dP

dz

� �LO

U2LO

with

U2LO ¼ E þ 3:24FH

Fr0:045We0:035

Fr ¼ G2

gDq2H, F = x0.78(1 � x)0.224,

H ¼ qL

qV

� �0:91lV

lL

� �0:191� lV

lL

� �0:7

We ¼ G2DrqH

with qH ¼ xqVþ 1�x

qL

� ��1

E ¼ ð1� xÞ2 þ x2 qLfVO

qVfLO

fLO ¼ 16ReLO

for ReLO < 2000 or

fLO ¼ 0:079Re�0:25LO for ReLO > 2000

fVO ¼ 16ReVO

for ReVO < 2000 or

fVO ¼ 0:079Re�0:25VO for ReVO > 2000

ReLO ¼ GDlL

and ReVO ¼ GDlV

and dPdz

� �LO¼ fLO

2G2

DqL

Chisholm [3] dPdz

� �f¼ dP

dz

� �LO

U2LO

with

U2LO ¼ 1þ ðY 2 � 1Þ Bx

2�n2 ð1� xÞ

2�n2 þ x2�n

h iwhere the exponent n = 0.25 and

Y 2 ¼ ðdP=dzÞVO

ðdP=dzÞLOand dP

dz

� �VO¼ fVO

2G2

DqV;

if 0 < Y < 9.5:B ¼ 55

G0:5 for G P 1900 kg/m2 s

B ¼ 2400G for 500 < G < 1900 kg/m2 s

B = 4.8 for G < 500 kg/m2 sIf 9.5 < Y < 28:B ¼ 520

YG0:5 for G6 600 kg/m2 s

B ¼ 21Y for G > 600 kg/m2 s

For Y > 28:B ¼ 15000

Y 2G0:5

Muller-Steinhagenand Heck [4]

dPdz

� �f¼ F ð1� xÞ1=3 þ dP

dz

� �VO

x3

with F ¼ dPdz

� �LOþ 2 dP

dz

� �VO� dP

dz

� �LO

h ix

Table 2Pressure drop models and correlations for microchannels

Authors Equation

Mishima and Hibiki [5] C = 21(1 � e�319D)using the Lockhart–Martinelli model

Lee and Lee [6] C ¼ AkqwrResLO

using the Lockhart–Martinelli model

k ¼ l2L

qLrD and w ¼ lLjr

The exponents A, q, r and s aregiven in Table 3

Lee and Mudawar [7] For laminar liquid and laminar vapor:Cvv ¼ 2:16Re0:047

LO We0:6LO

For laminar liquid and turbulent vapor:

Cvt ¼ 1:45Re0:25LO We0:23

LO

with WeLO ¼ G2DqLr

using the Lockhart–Martinelli model

Zhang and Webb [8] dPdz

� �f¼ dP

dz

� �LO

U2LO

with U2LO ¼ ð1� xÞ2 þ 2:87x2 P

P crit

� ��1þ

1:68x0:8ð1� xÞ0:25 PP crit

� ��1:64

Tran et al. [9] dPdz

� �f¼ dP

dz

� �LO

U2LO

with U2LO ¼ 1þ ð4:3Y 2 � 1Þ

Cox0:875ð1� xÞ0:875 þ x1:75h iCo ¼ r

D2gðqL�qVÞ

� �0:5and Y 2 ¼ ðdP=dzÞVO

ðdP=dzÞLO

R. Revellin, J.R. Thome / Experimental Thermal and Fluid Science 31 (2007) 673–685 675

evaporator in a refrigeration cycle. The microchannels wereformed by machining 231 lm wide · 713 lm deep groovesinto the surface of a copper block. Experiments were per-formed with refrigerant R-134a.

Zhang and Webb [8] measured adiabatic two-phase flowpressure drops for R-134a, R-22 and R-404a flowing in amulti-port extruded aluminum tube with a hydraulic diam-eter of 2.13 mm, and in two copper tubes having insidediameters of 6.25 and 3.25 mm, respectively. They foundthat the Friedel correlation did not predict the two-phasedata accurately, especially for high reduced pressure. Usingthe data taken in their present and in a previous study, anew correlation for two-phase friction pressure drop insmall tubes was developed by modifying the Friedel corre-lation. The new correlation predicts 119 data points with amean deviation of 11.5%.

Two-phase flow pressure drop measurements were madeby Tran et al. [9] during a phase-change heat transfer pro-cess with three refrigerants (R-134a, R-12 and R-113) at sixdifferent pressures ranging from 138 to 856 kPa, and in twosizes of round tubes (2.46 and 2.92 mm inside diameters)and one rectangular channel (4.06 · 1.7 mm). The datawere compared with those from large tubes under similarconditions, and state-of- the-art correlations were evalu-ated using the R-134a data. The state-of-the-art large-tubecorrelations failed to satisfactorily predict the experimentaldata.

Garimella et al. [10] proposed the first mechanisticmodel for two-phase pressure drop during intermittentflow of refrigerant in circular microchannel. The modelwas developed for a unit cells in the channel based on the

Fig. 2. Schematic of the test section.

Table 3Constant and exponents for parameter C of Lee and Lee [6]

Flow regime liquid Gas A q R S Range of X Range of ReLO Number of data

Laminar Laminar 6.833 · 10�8 �1.317 0.719 0.557 0.776–14.176 175–1480 106Laminar Turbulent 6.185 · 10�2 0 0 0.726 0.303–1.426 293–1506 2Turbulent Laminar 3.627 0 0 0.174 3.276–79.415 2606–17,642 85Turbulent Turbulent 0.408 0 0 0.451 1.309–14.781 2675–17,757 62

Laminar: ReL,ReG < 2000; turbulent: ReL,ReG > 2000.


observed slug/bubble flow pattern for these conditions. Theunit cell comprises a liquid slug followed by a vapor bubblethat is surrounded by a thin, annular liquid film. Authorstake into account different phenomena. They first of allaccounted for the continual uptake of liquid from film intothe front of the slug (DPfilm�slug transitions) as the bubbletravels faster than the liquid slug. They also accountedfor the pressure drop in bubble/film interface (DPf/b) andthe pressure drop in the liquid slug (DPslug). The total pres-sure drop is given by the following equation:

DP ¼ DP slug þ DP f=b þ DP film�slug transitions ð4Þ

DPfilm�slug transitions is directly related to the number of unitcell per meter. An experimental correlation is proposed forpredicting the slug frequency. It was assumed in the modelthat the length/frequency/speed of bubble/slugs is con-stant, with no bubble coalescence and a smooth bubble/film interface. The model has been experimentally validatedfor condensing R-134a.

2. Description of the test facility

The microchannel test facility is described in detail inRevellin [11] and available at the university website. Thetest facility was designed to operate using either a speedcontrolled micropump or the pressure difference betweenits two temperature-controlled reservoirs (the latter modewas used for all the present tests and is presented inFig. 1). A valve installed between the upstream reservoirand the test section is used to avoid flow oscillations in

Fig. 1. Schematic of reservoir loop.

the loop and a wide range of stable operating conditionsis thus achieved.

2.1. Test section

The test section consisted of four subsections: (i) an80 mm long, thin wall stainless steel tube used as a pre-heater, (ii) a 20 mm long glass tube for electrical insulation,(iii) a 110 mm long stainless steel tube microevaporator and(iv) a 100 mm long glass tube for flow pattern visualizationand pressure drop measurements, as shown in Fig. 2. Twocopper clamps were attached to both the preheater andthe evaporator and heating was provided by two SorensenDC power supplies. Two pressure transducers were installedat the inlet and outlet of the test section and two 0.25 mmthermocouples were placed in the fluid at the same loca-tions. A good agreement between the two different measure-ments (pressure and temperature) was observed at the outletof the test section (saturation conditions). Four 0.25 mmthermocouples were also attached on the external surfaceof the sight glass tubes (before the inlet and after the outletof both the preheater and the evaporator) to measure localfluid saturation temperatures. After calculation it has beenfound that the heat conduction through the thermocoupleswas negligible (2–3%). Furthermore, the internal thermalresistance of the flow is much lower than the wall thermalresistance thus, the longitudinal heat conduction can alsobe neglected. Two more 0.25 mm thermocouples wereinstalled on the two heated tubes to avoid dryout and over-heating. All the test section was thermally insulated. Carefulattention was made to match up the ends of the glass andstainless steel tubes.

3. Measurements and accuracy

A Coriolis mass flow meter was used to measure theflow rate of the subcooled refrigerant. Joule heating was

100

101

]


determined by measuring the DC voltage (±0.02%) and thecurrent by a DC current transformer (±3.5% for low cur-rents and ±1% for high ones). The absolute pressure trans-ducers for monitoring the local pressures were accurate to±5 mbar and the thermocouples to ±0.1 �C, according totheir calibrations. The vapor quality entering the flow visu-alization tube was estimated to be accurate to ±2% formost test conditions.

The pressure drops in the visualization tube reached upto 0.174 MPa, which were obtained indirectly from the mea-sured differences in saturation temperature between theinlet (0.25 mm thermocouple attached on the adiabatic sur-face of the microtube after the microevaporator) and outlet(0.25 mm thermocouple installed in the fluid after the glasstube) of the visualization tube. This temperature measure-ment is preferred to pressure measurement as no openingis made in the microtube to install a pressure tap. The flowis thus not deformed or disturbed. Moreover, this techniqueavoids also the capillary pressure due to the meniscusliquid/vapor before the pressure transducer that could influ-ence the measurement. However, for low pressure drops,the inaccuracy is higher with this method than with the pres-sure measurement. But as the pressure drop is high inmicrotubes, the accuracy of the measurement remains good.

The total database from this study and our prior publi-cations covers two different refrigerants (R-134a andR-245fa) and two diameter tubes (0.509 and 0.790 mm).The microevaporator heated length varied from 20 to70 mm and the inlet subcooling from 2 to 15 �C. The massfluxes ranged from 200 to 2000 kg/m2 s and the maximumheat flux was 415 kW/m2. Three different saturation tem-peratures were tested: 26, 30 and 35 �C. Experimental con-ditions and uncertainties are summarized in Table 4.

Notably, the present setup involves actual two-phaseflows exiting a microevaporator channel. The bubblesand subsequent flow regimes observed here originated fromnucleation in the evaporator, just like in a microchannelcooling element attached to a computer chip, for instance.Thus, here the resulting flow pattern and bubble character-istics are determined by the boiling process itself, notthe hydrodynamics of an injector, mixer or header usedin adiabatic tests.

Table 4Experimental conditions and uncertainties

Parameter Range Uncertainties Units

Fluid R-134a, R-245fa – –D 0.509, 0.790 ±1% mmeD <0.002% – –LMEV 30–70 <2.5% mmG 210–2094 ±2% kg/m2 sq 3.1–415 <5.7% kW/m2

Tsat 26,30,35 ±0.1 �CPsat 6.9,7.7,8.9 (for R-134a) <0.07% bar

2.1 (for R-245fa) <0.23%DTsub 2–6 ±0.1 �CxMEV,out 0–0.95 >1.3% –

dPdz

� �t

0–14.5 >1.1% bar/m

4. Experimental results

There were 2210 experimental two-phase pressure dropsdata points measured in this study. Thirty data points arenot taken into account as their values are below zero dueto small instabilities and error measurements. The totalpressure drop is calculated using pressure difference.According to Eqs. (1) and (3) the momentum pressure dropis subtracted from the total pressure drop to obtain thetwo-phase frictional pressure drop dP

dz

� �f. The momentum

pressure drop is caused by the slight flashing effect due tothe pressure drop but it is negligible (less than 0.1%) asshown in Fig. 3, but is taken into account anyway.

The general trend for the two-phase frictional pressuredrop is illustrated in Fig. 4. The data are plotted accordingto the vapor quality calculated at 20 mm from outlet of themicroevaporator. In general, the higher the mass flux, thehigher the vapor quality, the higher the two-phasefrictional pressure drop is (the expected trend). However,a change in the trend can be observed for G = 1000 kg/m2 s and G = 1200 kg/m2 s. Three arrows show thischange. It corresponds to a change in flow pattern. Theflow is annular for case (a) and (b) according to the flowpattern map of Revellin et al. [12]. But the annular flowis wavy annular for case (a) and tends to be separated intosmooth annular and wavy annular flow for case (b) asshown in Fig. 5(a) and (b). (This can happen also forsemi-annular flow which exhibits both smooth annularand churn flow zones.) The friction factor is definitely lessfor smooth than for wavy annular flow so the two-phasepressure drop is less for case (b) than for case (a).

For very low vapor quality one can surprisingly observethe two-phase frictional pressure drops decreasing withincreasing vapor quality. Afterwards the two-phase fric-tional pressure drop increases with the vapor quality with

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410

–3

10–2

10–1

Vapor quality [–]

Pre

ssu

re d

rop

[b

ar/m

Total pressure dropFrictional pressure dropMomentum pressure drop

Fig. 3. Comparison between the total, the frictional and the momentumpressure drop as a function of vapor quality from laser 1 for R-134a,D = 0.509 mm, LMEV = 70.70 mm, Tsat = 30 �C, G = 1800 kg/m2 s andDTsub = 3�C.

0 0.2 0.4 0.6 0.8 10

5

10

15

Vapor quality [–]

Tw

o–P

has

e fr

icti

on

al p

ress

ure

dro

p [

bar

/m]

G=1000 kg/m2sG=700 kg/m2sG=500 kg/m2sG=350 kg/m2s

0 0.2 0.4 0.6 0.8 10

5

10

15

Vapor quality [–]

Tw

o–P

has

e fr

icti

on

al p

ress

ure

dro

p [

bar

/m]

G=2000 kg/m2sG=1800 kg/m2sG=1500 kg/m2sG=1200 kg/m2s

a

b

Fig. 4. Two-phase frictional pressure drop as a function of the vaporquality from laser 1 for R-134a, D = 0.509 mm, LMEV = 70.70 mm,Tsat = 30 �C and DTsub = 3 �C.

Fig. 5. Flow patterns for R-134a and D = 0.509 mm. (a) Wavy annularflow and (b) smooth annular flow.

0 0.2 0.4 0.6 0.8 10

5

10

15

Vapor quality [–]

Tw

o–P

has

e fr

icti

on

al p

ress

ure

dro

p [

bar

/m]

Δ Tsub =2ºCΔ Tsub =3ºCΔ Tsub =5ºC

Fig. 6. Influence of the inlet subcooling at entrance to the microevapora-tor on the two-phase frictional pressure drop for R-134a, D = 0.509 mm,LMEV = 70.70 mm, Tsat = 35 �C and G = 1500 kg/m2 s.

0 0.2 0.4 0.6 0.8 10

5

10

15

Vapor quality [–]

Tw

o–P

has

e fr

icti

on

al p

ress

ure

dro

p [

bar

/m]

Tsat =26ºCTsat =30ºCTsat =35ºC

Fig. 7. Influence of the saturation temperature on the two-phase frictionalpressure drop for R-134a, D = 0.509 mm, LMEV = 70.70 mm, DTsub = 3 �Cand G = 1800 kg/m2 s.


two different gradients, where the higher gradient is forlower vapor qualities. The change in the gradient of thetwo-phase frictional pressure drop will be explained later.

4.1. Effect of the inlet subcooling

As expected, the inlet subcooling at the entrance to themicroevaporator had no effect on the saturated two-phase

frictional pressure drops as shown in Fig. 6. The small inletsubcoolings tested here had no influence on the flow pat-terns and thus the two-phase mechanisms are not affected.

4.2. Effect of the saturation temperature

As presented in Fig. 7, the saturation temperature influ-ences the two-phase frictional pressure drop. The higherthe saturation temperature, the lower the two-phase fric-tional pressure drop is. When increasing the saturatizontemperature, the vapor density increases, the vapor velocitydecreases and by consequence the two-phase frictionalpressure drop decreases. The results here show the expectedtendency from macroscale theory.

0 0.2 0.4 0.6 0.8 10

5

10

15

Vapor quality [–]

Tw

o–P

has

e fr

icti

on

al p

ress

ure

dro

p [

bar

/m]

L MEV =30 mmL

MEV =50 mm

LMEV =70 mm

Fig. 8. Influence of the microevaporator length on the two-phase frictionalpressure drop for R-134a, D = 0.509 mm, Tsat = 30 �C, DTsub = 3 �C andG = 1800 kg/m2 s.

0 0.2 0.4 0.6 0.8 10

5

10

15

Vapor quality [–]

Tw

o–P

has

e fr

icti

on

al p

ress

ure

dro

p [

bar

/m]

R–134aR–245fa

Fig. 10. Influence of the fluid on the two-phase frictional pressure dropfor D = 0.509 mm, Tsat = 35 �C, LMEV = 70.70 mm, DTsub = 5–6 �C andG = 700 kg/m2 s.


4.3. Effect of the microevaporator length

As the microevaporator length has no effect on the flowpatterns, it was expected that the two-phase frictional pres-sure drops measured in the sight glass tube would not beaffected by this variation. Fig. 8 shows this to be the case.

4.4. Effect of the microevaporator diameter

As can be seen in Fig. 9, the diameter of the sight glasstube has a strong influence on the two-phase frictionalpressure drop. As expected, the larger the diameter, thelower the two-phase pressure drop is. The same trendsare observed for both diameters.

0 0.2 0.4 0.6 0.8 10

5

10

15

Vapor quality [–]

Tw

o–P

has

e fr

icti

on

al p

ress

ure

dro

p [

bar

/m]

D=0.509 mmD=0.790 mm

Fig. 9. Influence of the microevaporator diameter on the two-phasefrictional pressure drop for R-134a, Tsat = 30 �C, LMEV = 70.70 mm,DTsub = 3 �C and G = 1200 kg/m2 s.

4.5. Effect of the fluid

The fluid has an important influence on the two-phasefrictional pressure drop. The values for R-245fa arestrongly higher than those for R-134a. ActuallyqV(R-134a) ’ 3qV(R-245fa) so the vapor velocity forR-245fa is higher than for R-134a and the result is theincrease of the pressure drop for R-245fa (see Fig. 10).

4.6. The two-phase friction factor

As observed in the previous figures, the variation of thetwo-phase frictional pressure drop can be divided intothree different zones according to the two-phase Reynoldsnumber. Firstly, the two-phase frictional pressure dropdecreases, secondly the two-phase frictional pressure dropincreases with a certain gradient and thirdly the two-phasefrictional pressure drop increases with a lower gradient ascan be seen in Fig. 4.

It is possible to plot the pressure drop data as the two-phase friction factor versus the two-phase Reynolds num-ber as shown in Fig. 11. lTP [13] and qTP are calculatedusing the definitions of Table 1 for the homogeneous model.The two-phase friction factors are determined from themeasured pressure gradients using Eq. (5). The values ofReTP are obtained using the definition of Table 1 for thehomogeneous model. Three different zones are detected.An analogy with the behavior of the single-phase frictionfactor is made as follows:

dPdz

� �f

¼ dPdz

� �TP

¼ 2f TPG2

DqTP

ð5Þ

• Re < 2000: The first zone is called the laminar zone.
TP
The two-phase friction factor diminishes when increas-ing the two-phase Reynolds number.

103

104

10–3

10–2

10–1

100

Two–phase Reynolds number [–]

Tw

o–p

has

e fr

icti

on

fac

tor

[–]

103

104

10–3

10–2

10–1

100


Tw

o–p

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G=1200 kg/m2 sG=1000 kg/m2 sG=700 kg/m2 sG=500 kg/m2 s

Laminar Transition Turbulent

G=1000 kg/m2 sG=700 kg/m2 sG=500 kg/m2 sG=350 kg/m2 s


103

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G=1000 kg/m2s

G=200 kg/m2s

G=500 kg/m2 sG=350 kg/m2 sG=200 kg/m2 s


Fig. 11. Two-phase friction factor versus two-phase Reynolds number. (a) R-134a, D = 0.509 mm, Tsat = 30 �C, LMEV = 70.70 mm, DTsub = 3 �C.(b) R-134a, D = 0.790 mm, Tsat = 30 �C, LMEV = 70.70 mm, DTsub = 3 �C. (c) R-245fa, D = 0.509 mm, Tsat = 35 �C, LMEV = 70.70 mm, DTsub = 6 �C.


• 2000 6 ReTP < 8000: The second zone is the transitionzone. The two-phase friction factor increases with theReynolds number or decreases, depending to the condi-tions. A case could also be made here to set the upperboundary at 4500 rather than 8000 in viewing Fig. 11but in (b) and (c) some transition effects still appear tofall in this range.

• ReTP P 8000: The third zone corresponds to the turbu-lent zone. The data are grouped and decrease withincreasing two-phase Reynolds number.

The two-phase friction factor behavior is the same as forthe single phase friction factor. It is interesting to empha-size here that each zone corresponds to each trend observedearlier when plotting the two-phase frictional pressure dropversus the vapor quality.

The laminar zone corresponds most often to the bubblyor bubbly/slug regime [12]. Usually, the pressure dropshould not decrease and then increase afterwards as thereshould be a continuous increase of the pressure drop withthe two-phase Reynolds number. However, macroscale

two-phase pressure drop data nearly always fall only inthe turbulent zone and this trend is not thus observed. Fur-thermore, it is interesting to emphasize that the transitionzone occurs in the same range of Reynolds number as forsingle-phase flow. The laminar zone will be ignored forthis study as it is difficult to make a detailed analysissince it corresponds to only 74 data points, or 3.3% ofthe database.

The rest of the database will be divided into two sets ofdata (936 data points for the transition zone and 1200 datapoints for the turbulent region) and compared to availablecorrelations or models. As the transition zone cannot bepredicted by any of the existing methods, only the turbu-lent zone will be presented here.

5. Comparison with existing methods

The comparison between the present experimental data(for R-134a, R-245fa and both sight glass tube diameters)and twelve models and correlations available in the litera-ture (described in Section 1) is shown in Table 5 for the

Table 5Statistical analysis of pressure drop (the fraction of data predicted to within ±20%, b, and the mean absolute error, a)

Frictional pressure drop prediction method Transition regime Turbulent regime2000 6 ReTP < 8000 ReTP P 8000

b (%) a (%) b (%) a (%)

McAdams et al. [13] 28.1 45.9 5.75 39.8Dukler et al. [18] 23.8 46.2 3.58 43.5Cicchitti et al. [17] 34.7 45.2 52.3 25.9Lockhart and Martinelli [1] 26.6 69.3 22.8 51.4Friedel [2] 33.3 55.9 44.7 28.1Chisholm [3] 22.8 162 28.7 65.2Mishima and Hibiki [5] 10.5 56 1.58 51.7Lee and Lee [6] 23.7 69 18.9 51.4Lee and Mudawar [7] 16.9 109 18.1 70Zhang and Webb [8] 30.2 81.9 46.8 34.7Tran et al. [9] 11.2 474 1.75 355Muller-Steinhagen and Heck [4] 30.1 68 62.5 18.3

0 1 2 3 4 50

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Experimental pressure drop [bar/m]

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Garimella model

Experimental data± 20 % error band

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

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Fig. 12. (a) Comparison between the experimental data of two-phasefrictional pressure drop in the slug flow regime and the Garimella et al.[10] model and (b) repartition of the experimental data of two-phasefrictional pressure drop as a function of the two-phase Reynolds number.


transition and turbulent region. No prediction methodworks very well to predict the present experimental data,except for the Muller-Steinhagen and Heck [4] correlationthat predicts more than 62% of the data within a ±20%error band for the turbulent data with a mean absoluteerror of 18.3%. Furthermore, it predicts 83% of the datafor R-134a in the 0.509 mm tube within this ±20% errorband. For R-134a in the 0.790 mm tube, the results arenot well predicted, as only 29% of the data fall within a±20% error band while 62% of the data are within ±20%for R-245fa. Ribatski et al. [14] found also that Muller-Steinhagen and Heck [4] was the best correlation to predicttwo-phase frictional pressure drop in microchannels. Theyused a huge database from the literature covering eight flu-ids and compared with twelve existing predictions methods.

The homogeneous models under predict the data. Infact, some methods over predict the data while othersunder predict them. A new prediction method has beendeveloped and is proposed below.

6. Comparison to a phenomenological model

Fig. 12(a) shows a comparison between experimentaldata of two-phase frictional pressure drop and the Garim-ella et al. [10] model for intermittent flows of R-134a andR-245fa in a 0.509 and 0.790 mm tube (874 data points).As can be seen, only 20.4% of the data fall within a±20% error band. The mean deviation is 51.7%. The repar-tition of the data as a function of the two-phase Reynoldsnumber is represented in Fig. 12(b). Most of the data are inthe transition zone and only a few of them in the turbulentzone. The comparison has also been done inputting ouroriginal experimental bubble frequencies directly into themodel. The results are then in even poorer agreement thanthe model.

Some explanations may explain the difference betweenthe experimental data and the Garimella et al. model. Firstof all, the model has been experimentally validated for con-densing R-134a whereas the data in the present study arefor evaporating R-134a and R-245fa. The break down of

annular to intermittent flow in condensation has differentmechanisms than coalescence in evaporation. Secondly,


in the model, a constant frequency has been assumed butactually the bubble frequency is not constant due to thecoalescence or break down phenomena. Furthermore, thedifference between the model and the experimental datacould be explained by the wide range of ReTP. AS shownin Fig. 12(b), most of the data in this work are in the tran-sition zone. As expected, this transition is difficult to pre-dict accurately with any of the existing methods. Themodel of Garimella et al. has been validated for parallelcircular multichannels whereas here a single circular chan-nel is used. This particularity could also explain the differ-ence between the model and the present study.

7. New prediction method

Two possibilities are offered to develop a new predictionmethod: In first, modification of the homogeneous modelusing the two-phase friction factor. This assumption is

104

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Experimental dataFit curve0.079Re

–1/4TP

Experimental dataFit curve0.079Re –1/4

TP

Fig. 13. Two-phase friction factor versus two-phase Reynolds numberfor ReTP P 8000. (a) 768 data points for R-134a, R-245fa, D = 0.509 mm.(b) 432 data points for R-134a, D = 0.790 mm.

not far away from reality as it has been shown in Revellinet al. [12] that the homogeneous model predicted the voidfraction rather well. Moreover, Agostini and Bontemps[15] found that homogeneous model predicted their R-134a two-phase pressure drop data very well. Secondly,the classic Lockhart–Martinelli method can be modifiedwith a new C parameter. Both approaches are widely usedin the literature to predict two-phase pressure drops. Onlythe results for the homogeneous model will be shown hereas this approach gives the best prediction.

Based on Section 4.6, it is possible to determine a newtwo-phase friction factor for the turbulent data where8000 6 ReTP. Fig. 13 shows the new predicted two-phasefriction factor determined with a least square method andgiven by the following equations:

fTP ¼ 0:08Re�1=5TP for D ¼ 0:509 mm ð6Þ

fTP ¼ 6Re�3=5TP for D ¼ 0:790 mm ð7Þ

The friction factor for smooth tubes used in the homoge-neous model (see Table 1) is also plotted for comparison.The experimental two-phase friction factor is higher thanthat for the single-phase smooth tubes. Actually, the two-phase friction takes into account the friction between theliquid and the wall as well as the friction between the liquidand the vapor, the latter being much higher in case ofwavy-annular regime (see Fig. 5(a)). This difference is thusexpected as the annular regime is the flow pattern presentin the turbulent zone.

There are two different relations for fTP, one for eachtube. This could be explained by the fact that even if thesight glass tubes come from the same manufacturer, therecould be a difference between them; however, they are verysmooth (see Table 4) as can be expected with good opticalquality glass and hence surface roughness can be excludedas influencing the results. More likely, it is the buoyancyeffect that is responsible, which is greater for the 0.790 mmtube than for the 0.509 mm tube as shown in Fig. 14(a)–(c) and this effect explains the difference in the results. Buoy-ancy effects for R-245fa are the same as for R-134a in the0.509 mm glass tube as shown in Fig. 14(d). Supporting thisconclusion, the data for R-134a and R-245fa in the0.509 mm glass tube are well grouped together and are wellpredicted by Eq. (6) whereas those for the 0.790 mm tube arewell predicted by Eq. (7).

Fig. 15 presents the comparison between the 1200 exper-imental data points for the turbulent zone with the homo-geneous model using a new two-phase friction factorexpressions. It can be seen that the data are very well pre-dicted. The trends are captured and the data are wellgrouped with 85.7% of the data falling in a ±20% errorband and more than 96% within ±30%. The transition zonedata cannot however currently be correctly predicted byany of the existing methods and no method is proposedhere for this regime. In fact none of the literature methodswere specifically developed to handle the transition regime,so the lack of accuracy is not surprising. On the other

Fig. 14. Elongated bubble in 2.0, 0.8 and 0.5 mm tubes. (a) 2.0 mm tube for R-134a. (b) 0.8 mm tube for R-134a. (c) 0.5 mm tube for R-134a. (d) 0.5 mmtube for R-245fa.

0 5 10 150

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0 5 10 150

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0 5 10 150

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Fig. 15. Comparison between the experimental data of two-phase frictional pressure drop and the homogeneous model using a new two-phase frictionfactors for ReTP P 8000. (a) R-134a and D = 0.509 mm. (b) R-134a and D = 0.790 mm. (c) R-245fa and D = 0.509 mm. (d) R-134a, R-245fa,D = 0.509 mm and D = 0.790 mm.


hand, most literature methods eliminate the transitionregime in the two-phase analysis, jumping directly fromlaminar to turbulent (see for example Friedel [2]). Fig. 11provides a very good example of why this is not at allappropriate.

The new prediction method has also been comparedwith data from the literature. Cavallini et al. [16] haveperformed two-phase frictional pressure gradient tests forR-134a inside multi-port minichannels. The test sectionconsisted of eleven parallel rectangular cross-section

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Cavallini et al.Zhang & WebbTran et al.± 30 % error band

Fig. 16. Comparison between the new prediction method (Eq. (7)) anddata from the literature for ReTP P 8000.


channels with a hydraulic diameter of 1.4 mm. Zhang andWebb [8] measured adiabatic two-phase pressure drop forR-134a flowing in a multi-port extruded aluminum tubewith hydraulic diameter of 2.13 mm. Tran et al. [9] carriedout two-phase flow pressure drop measurements forR-134a in a 2.46 mm diameter tube. The test sections usedin these three studies have a hydraulic diameter in a so-called transition zone, i.e. minichannels and not microchan-nels. As a result, these data have been compared with theprediction method detailed in Eq. (7) as it has been devel-oped for a transition diameter. Eq. (6) cannot be testeddue to the lack of data in the literature for refrigerants flow-ing in microchannels. Fig. 16 shows the comparisonbetween the prediction model and the data for ReTP P 8000(lTP calculated with the McAdams relation). The data arewell grouped with 70.3% of the data falling in a ±30% errorband and an absolute error of 26%. The same comparisonwith the conventional homogeneous model would give16.5% of the data falling in a ±30% error band and an abso-lute error of 47%. As a result, the new prediction methodsare validated.

8. Conclusions

In this study, 2210 experimental two-phase frictionalpressure drop data points were taken in 0.790 and0.509 mm adiabatic glass tubes for R-134a and R-245fafor a wide range of test conditions. The following conclu-sions can be made here:

• Similar to the classic Moody diagram in single-phaseflow, three zones were distinguishable when plottingthe variation of the two-phase friction factor versusthe two-phase Reynolds number: a laminar zone forReTP < 2000 (74 data points), a transition zone for2000 6 ReTP < 8000 (936 data points) and a turbulentregion for ReTP P 8000 (1200 data points).

• The laminar zone is actually not taken into account asthe gradient for the pressure drop versus vapor qualityis negative. This zone corresponds to small instabilities.

• The transition region is not predicted well by any of theprediction methods. In the literature, only a few data areavailable for this region and most methods ignore thisregion, jumping directly from laminar to turbulent flowat ReTP = 2000.

• The turbulent zone can be reasonably well predicted bythe Muller-Steinhagen correlation.

• New prediction methods have been proposed here forboth size of tubes using the homogeneous model anddetermining a new two-phase friction factor fTP for thetwo different tubes. More than 85.7% of the data fallin a ±20% error band and more than 96% within±30%. The trend is captured for both diameters andboth fluids. The new prediction methods have also beensuccessfully validated with data from literature.

In summary, none of the methods are able to capturethe laminar, transition and turbulent trends in the presenttwo-phase friction factors shown in Fig. 11, not even theLockhart–Martinelli method with its different C valuesfor laminar–laminar, turbulent–laminar, laminar–turbu-lent and turbulent–turbulent combinations in the liquidand vapor phases. Since most methods either ignore thetransition regime and laminar regime all together, or jumpdirectly from a laminar to turbulent formulation, it is notsurprising that databases containing transition regime dataare poorly predicted (as these data are not usually segre-gated from the database as was done here). The Garimellaet al. model is compared with experimental pressure dropfor intermittent flow. Most of the data fall in the trouble-some transition region and are poorly predicted. Proposinga new prediction method that captures all these threeregimes requires more laminar data and more accuratemeasurements at low pressure drops to do this.

Acknowledgement

R. Revellin is supported by the European Community’sHuman Potential Programme under contract HPRN-CT-2002-00204 [HMTMIC] funded by OFES, Bern,Switzerland.

References

[1] R.C. Lockhart, R.W. Martinelli, Proposed correlation of data forisothermal two-phase, two component flow in pipes, Chem. Eng.Progr. 45 (1949) 39–48.

[2] L. Friedel, Improved friction pressure drop correlations for horizon-tal and vertical two-phase pipe flow, in: European Two-Phase FlowGroup Meeting, Ispra, Italy, 1979, paper E2.

[3] D. Chisholm, Pressure gradients due to friction during the flow ofevaporating two-phase mixtures in smooth tubes and channels, Int. J.Heat Mass Transfer 16 (1973) 347–348.

[4] H. Muller-Steinhagen, K. Heck, A simple friction pressure dropcorrelation for two-phase flow pipes, Cem. Eng. Process. 20 (1986)297–308.


[5] K. Mishima, T. Hibiki, Some characteristics of air–water flow insmall diameter vertical tubes, Int. J. Multiphase Flow 22 (1996) 703–712.

[6] H.J. Lee, S.Y. Lee, Pressure drop correlations for two-phase flowwithin horizontal rectangular channels with small heights, Int. J.Multiphase Flow 27 (2001) 783–796.

[7] J. Lee, I. Mudawar, Two-phase flow in high heat flux microchannelheat sink for refrigeration cooling applications: Part I—pressure dropcharacteristics, Int. J. Heat Mass Transfer 48 (2005) 928–940.

[8] M. Zhang, R.L. Webb, Correlation of two-phase friction for refriger-ants in small-diameter tubes, Exp. Therm. Fluid Sci. 25 (2001) 131–139.

[9] T.N. Tran, M.C. Chyu, M.W. Wambsganss, D.M. France, Two-phase pressure drop of refrigerants during flow boiling in smallchannels: an experimental investigation and correlation development,Int. J. Multiphase Flow 26 (2000) 1739–1754.

[10] S. Garimella, J.D. Killion, J.W. Coleman, An experimental validatedmodel for two-phase pressure drop in the intermittent flow regime forcircular channel, J. Fluid Eng. 124 (2002) 205–214.

[11] R. Revellin, Experimental two-phase fluid flow in microchannels,Ph.D. thesis, Ecole polytechnique Federale de Lausanne, 2005.Available from: <http://library.epn.ch/en/theses/?nr=3437>.

[12] R. Revellin, V. Dupont, T. Ursenbacher, J.R. Thome, I. Zun,Characterization of diabatic two-phase flows in microchannels: Flow

parameter results for R-134a in a 0.5 mm channel, Int. J. MultiphaseFlow, 32, 755–774, in press.

[13] W.H. McAdams, W.K. Woods, R.L. Bryan, Vaporization insidehorizontal tubes—II—benzene–oil mixtures, Trans. ASME 64 (1942)193.

[14] G. Ribatski, L. Wojtan, J. Thome, An analysis of experimental dataand prediction methods for two-phase frictional pressure drop andflow boiling heat transfer in microscale channels, Exp. Therm. FluidSci., in press, doi:10.1016/j.expthermflusci.2006.01.006.

[15] B. Agostini, A. Bontemps, Vertical flow boiling of refrigerant R-134ain small channels, Int. J. Heat Fluid Flow 26 (2005) 296–306.

[16] A. Cavallini, D.D. Col, L. Doretti, M. Matkovic, L. Rossetto, C.Zilio, Two-phase frictional pressure gradient of R-236ea, R-134a andR-410a inside multi-port minichannels, Exp. Therm. Fluid Sci. 29(2005) 861–870.

[17] A. Chicchitti, C. Lombardi, M. Silvestri, G. Soldaini, R. Zavattarelli,Two-phase cooling experiments-pressure drop, heat transfer andburnout measurements, Energ. Nucl. 7 (6) (1960) 407–425.

[18] A.E. Dukler, M. Wicks, R.G. Cleveland, Pressure drop and hold-upin two-phase flow part A—a comparison of existing correlations andpart B—an approach through similarity analysis, AIChE J. 10 (1)(1964) 38–51.

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adiabatic two-phase frictional pressure drops in microchannels

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