a separated-flow model for predicting flow boiling

A Separated-Flow Model for Predicting Flow Boiling CriticalHeat Flux and Pressure Drop Characteristics in Microchannels

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Citation Zhang, Tie Jun, Siyu Chen, and Evelyn N. Wang. “A Separated-Flow Model for Predicting Flow Boiling Critical Heat Flux andPressure Drop Characteristics in Microchannels.” ASME 2012 10thInternational Conference on Nanochannels, Microchannels, andMinichannels, 8-12 July, 2012, Rio Grande, Puerto Rico, USA, ASME,2012. © 2012 by ASME

As Published http://dx.doi.org/10.1115/ICNMM2012-73046

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http://hdl.handle.net/1721.1/120014

A SEPARATED-FLOW MODEL FOR PREDICTING FLOW BOILING CRITICAL HEATFLUX AND PRESSURE DROP CHARACTERISTICS IN MICROCHANNELS

TieJun Zhang∗Masdar Institute of Science and Technology,

P. O. Box 54224, Abu Dhabi, UAEMassachusetts Institute of Technology,

Cambridge, MA 02139, USAEmail: [email protected], [email protected]

Siyu Chen, Evelyn N. WangMechanical Engineering Department,

Massachusetts Institute of Technology,Cambridge, MA 02139, USA

Email: [email protected], [email protected]

ABSTRACTTwo-phase microchannel cooling promises high heat flux re-

moval for high-performance electronics and photonics. How-ever, the heat transfer performance of flow boiling microchan-nels is limited by the critical heat flux (CHF) conditions. Forvariable heat inputs and variable fluid flows, it is essential topredict CHFs accurately for effective and efficient two-phase mi-crochannel cooling. To characterize the CHF and pressure dropin flow boiling microchannels, a separated-flow model is pro-posed in this paper based on fundamental two-phase flow mass,energy, momentum conservation and wall energy conservationlaws. With this theoretical framework, the relationship amongliquid/vapor interfacial instability, two-phase flow characteris-tics and CHF is further studied. This mechanistic model alsoprovides insight into the design and operational guidelines foradvanced electronics and photonics cooling technologies.

NOMENCLATUREA cross-sectional area (m2)D hydraulic diameter (m)G mass flux (kg/m2-s)h specific enthalpy (J/kg)L channel length (m)m mass flowrate (kg/s)p channel perimeter (m)

∗Address all correspondence to this author.

P absolute pressure (Pa)q′′ heat flux (W/m2)r vapor core radius (m)t time (s)T temperature (◦C)u velocity (m/s)z channel location (m)

Greek letters

α heat transfer coefficient (W/m2-K)δ liquid film thickness (µm)ρ density (kg/m3)σ surface tension (N/m)

Subscripts

b nucleating bubblef single-phase fluidh heatedi interfacein inletl liquid phaseout outlett thermalv vapor phasew wall (wetted)

Proceedings of the ASME 2012 10th International Conference on Nanochannels, Microchannels, and Minichannels ICNMM2012

July 8-12, 2012, Rio Grande, Puerto Rico

1 Copyright © 2012 by ASME

ICNMM2012-73046

Downloaded From: https://proceedings.asmedigitalcollection.asme.org on 01/09/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use

INTRODUCTIONThermal management is a critical issue for safe and energy-

efficient operation of high-power-density electronic, avionic andphotonic systems [1, 2], such as high-concentration photovoltaic(HCPV) modules [3, 4], light emitting diodes [5], laser diodearrays and high-performance data centers [6]. Existing thermalmanagement technologies are not adequate for high heat flux re-moval while maintaining the operating temperature, i.e., silicon-based electronic components need to be below 125 ◦C [1, 7]. Inaddition, low temperatures can improve both the computing per-formance and the reliability of the data center system [8] as wellas increase the solar-to-electric conversion efficiency of photo-voltaic cells [4].

Microchannel and microjet impingement technologies haveshown their great capability of high heat flux removal for micro-electronics and microphotonics [1,7,9]. It is well-known that mi-crochannel heat sinks can increase the heat transfer surface areasignificantly. However, single-phase microchannel heat sinksyield high pressure drops because they require high flowratesand rely on the sensible heat rise to achieve cooling, there-fore, a large temperature gradient exists in the direction of fluidflow [1]. Two-phase microchannel cooling has been identifiedas a promising strategy for efficient thermal management of ul-tra high-power applications [7] by taking advantage of the latentheat associated with phase change.

The critical heat flux (CHF) condition, however, dictates thepractical thermal limit in flow boiling systems, at which pointthe heat transfer performance degrades significantly. In high fluxremoval applications, knowledge of the CHF condition is ex-tremely critical to prevent device burnout. There have thereforebeen lots of microchannel CHF studies reported in open liter-ature. Qu and Mudawar [10] conducted CHF tests in a water-cooled micro-channel heat sink containing 21 parallel micro-channels with dimension of 215×821µm. Deionized water wasused as working fluid. The experiments were performed at var-ious mass velocities from 86-368 kg/(m2s), an outlet pressureof 1.13 bar and two different inlet temperatures: 30 ◦C and 60◦C. Based on their CHF data and the Bowers and Mudawar’sCHF data for R-113, Qu and Mudawar developed a new CHFcorrelation for multiple channel mini/micro-channel heat sinks.Compared to the well-known Katto correlation [11], which wasdeveloped from CHF data of single circular channels, it can bet-ter predict the CHF data in multiple channels. While many othercorrelations have been developed for the prediction of CHF, veryfew studies have focused on exploring the physical insight of dry-out in flow boiling microchannels except that in [12]. In thatwork, Revellin and Thome proposed an interesting separated-flow model to predict flow boiling CHF in uniformly heated mi-crochannels. It is supposed that the annular liquid film becomesthiner and thiner along the microchannel until it is destablizedby the so-called “Kelvin-Helmholtz interfacial instability”. Themodel was applied to fit various CHF database including three

different refrigerants from two different laboratories. Recently,Park and Thome successfully applied the same model to predictthe CHF of three refrigerant flows (R134a, R236fa, R245fa) intwo different microchannel test sections [13].

The liquid/vapor separated-flow model used in the past [12]is illustrated in Figure 1(a), where the following three assump-tions were made:

1. Constant streamwise (axial) liquid and vapor densitiesHowever, in the case of fluid flow in microchannels withhigh viscosity (i.e., water), the axial flow pressure drop mayno longer be neglected [20], and could cause significantvariations in the properties of the vapor and liquid flows.

2. No inlet subcooling (inlet two-phase mixture, rb = 0.22R)An initial vapor core radius, rb, was chosen as a fittingparameter even for microchannel fluid flow with highinlet subcooling; In fact, rb has the physical meaning ofrepresenting the nucleating bubble size.

3. Equal axial and radial vapor expansions (dz = dr)This is a demanding assumption as seen in Figure 1(a). In a“micro” channel, the radial vapor bubble expansion is mostlikely confined; instead, the axial vapor expansion is free togo. However, the radius r is a function of z.

These assumptions have limited the model prediction per-formance especially for the purpose of predicting two-phase flowpressure drop. In fact, no overall pressure drop predictions werereported in [12] for uniformly heated flow boiling microchan-nels; only adiabatic two-phase pressure drop predictions werediscussed there. Pressure drop is actually very important formicrochannel cooling systems because it determines the sup-ply pumping power consumption and even triggers flow boil-ing instabilities. For flow boiling microchannels with high inletsubcooling, it makes more sense that a complete pressure dropmodel represents all the flow characteristics of subcooled liquid,two-phase mixture and superheated vapor, as illustrated in Figure1(b). Poor prediction in single-phase flow region can render two-phase pressure drop prediction large errors, which compromisesthe accuracy of the overall microchannel pressure drop predic-tion regardless of the two-phase model prediction accuracy [1].

Moreover, two-phase flow characteristics and CHF wereusually studied in a separate way [10, 12, 14, 15]. In a practi-cal flow boiling system, the pressure drop characteristics, two-phase flow instabilities and CHF are highly coupled. Due to steepnegative pressure drop-mass flux slope and high length-diameterratio of microchannels, microchannel flow boiling systems areprone to various flow boiling instabilities, which could easilycause a premature initiation of CHF conditions [14–17, 21]. Asan extended effort, this paper presents a modified liquid/vaporseparated-flow model to predict both the pressure drop flow char-acteristics and critical heat flux conditions in flow boiling mi-



FIGURE 1. Separated-flow control volume and liquid/vapor interfa-cial wave in a flow boiling microchannel: (a) inlet two-phase flow [12];(b) inlet subcooled flow (this paper)

crochannels. This generalized model captures the stream-wisesubcooled/superheated flow characteristics, back flow and liq-uid film dryout phenomena in microchannels. Both the predictedpressure drop and critical heat flux agree well with available ex-perimental data.

GENERALIZED SEPARATED-FLOW MODELIn this section, a generalized separated-flow model is devel-

oped to characterize both the overall pressure drop and criticalheat flux in uniformly heated microchannels. It will be shownthat the separated-flow model developed by Revellin and Thome[12] is a special case of the proposed model.

Before presenting more modeling details, 1-D mass, energyand momentum conservation laws for the steady flows in heatedmicrochannels are summarized below

dmdz

= 0 (1)

d(mh)dz

+αw pw(Tf −Tw) = 0 (2)

ddz

(m2

ρA

)+

d(PA)dz

+Fvisc = 0 (3)

which are a set of nonlinear differential-algebraic equations. Twand Tf are the wall and fluid temperature, respectively. Fvisc isthe friction force due to the fluid viscosity.

In addition, there is an energy balance between the heat ex-changer wall and the working fluid

kwAwd2Tw

dz2 +q′′ph−αw pw(Tw−Tf ) = 0 (4)

where αw is the heat transfer coefficient, the Boelter and Kan-dlikar correlations are used to predict the single-phase and two-phase boiling heat transfer coefficients [19]. q′′ is the base heatflux imposed on a microchannel heat sink and to be removed.

Phase-Change Fluid FlowInside a microchannel evaporator, the inlet subcooled liquid

is heated until the vapor/liquid mixture emerges. If the fluid flowis overheated, superheated flow exit from the microchannels. Toquantify the fluid flow and heat transfer in uniformly heated mi-crochannels, one-dimensional mass, energy and momentum bal-ances (1)-(3) are applied to subcooled liquid or superheated va-por single-phase flow,

ddz

(ρ f A f u f

)= 0 (5)

ddz

(ρ f A f u f h f

)= q′′ph (6)

ddz

(ρ f A f u2

f +Pf A f)=−pw|τw| (7)

where the flow cross-sectional area is, A f = πR2, R = D/2. Forsingle-phase fluid flow, the velocity u f , pressure Pf and specificenthalpy h f can be chosen as independent states, leading to

ρ f A fdu f

dz+A f u f

dρ f

dPf

dPf

dz+A f u f

dρ f

dh f

dh f

dz= 0 (8)

ρ f A f u fdh f

dz= q′′ph (9)

ρ f A f u fdu f

dz+A f

dPf

dz=−pw|τw| (10)

where the the shear stress between the liquid flow and the chan-nel wall, τw = 0.5C f ρ f u2

f when Re f = 2ρ f ulR/µ f < Retrs, C f =

PoRe−1f ; when Re f >= Retrs, C f = 0.078Re−0.25

f , Retrs = 2300,Po is the Poiseuille number dependent on the microchannel as-pect ratio [19].

These principles are applicable to the separated liquid andvapor flows, respectively

ddz

(ρlAlul +ρvAvuv) = 0 (11)

ddz

(ρlAlulhl +ρvAvuvhv) = q′′ph (12)

ddz

(ρlAlu2

l +PlAl)= pi cosθ |τlv|− pw|τw| (13)

ddz

(ρvAvu2

v +PvAv)=−pi cosθ |τlv| (14)

Pv−Pl =2σ

r(15)



where the last equation is the Laplace-Young equation [18],r is the equivalent vapor core radius. For the separated liq-uid/vapor flow in Figure 1(b), dr is not necessarily equal todz, the vapor cone perimeter can be represented by, pi =π(2r + dr)

√(dr/dz)2 +1, cosθ = dz/

√(dr)2 +(dz)2, there-

fore, pi cosθ = π(2r + dr). Note that the shear stress be-tween the liquid and vapor flows, τi = 0.5C f ρvu2

v , when Rev =2ρvuvr/µv < Retrs, C f = Po ·Re−1

v ; when Rev >= Retrs, C f =0.078Re−0.25

v . Correspondingly, the shear stress between the liq-uid flow and the channel wall is, τw = 0.5C f ρlu2

l , when Rel =

2ρlul(R− r)/µl < Retrs, C f = PoRe−1l ; when Rel >= Retrs,

C f = 0.078Re−0.25l , Retrs = 2300, Po is the Poiseuille number.

With the independent states, X = [r,uv,ul ,Pv,Pl ]T , the above

separated-flow conservation equations become

Z(X) · dXdz

= Y (X) (16)

where the elements of matrices Z and Y are:

Z11 = 2πr(ρvuv−ρlul),

Z12 = ρvAv, Z13 = ρlAl ,

Z14 = Avuv∂ρv

∂Pv, Z15 = Alul

∂ρl

∂Pl,

Z21 = 2πr(ρvuvhv−ρlulhl),

Z22 = ρvAvhv, Z23 = ρlAlhl ,

Z24 = Avuvhv∂ρv

∂Pv+ρvAvuv

∂hv

∂Pv,

Z25 = Alulhl∂ρl

∂Pl+ρlAlul

∂hl

∂Pl,

Z31 = (−2πr)(ρlu2l )−π(dz)|τi|,

Z33 = 2ρlAlul , Z35 = Al

(∂ρl

∂Plu2

l +1),

Z41 = 2πr(ρvu2v)+π(dz)|τi|,

Z42 = 2ρvAvuv, Z44 = Av

(∂ρv

∂Pvu2

v +1),

Z51 =2σ

r2 , Z54 = 1, Z55 =−1;

Y11 = 0, Y21 = q′′ph,

Y31 = 2πr|τi|−2πR|τw|,Y41 =−2πr|τi|, Y51 = 0.

and X = [r,uv,ul ,Pv,Pl ]T , Av = π ·r(z)2, Al = π(R2−r(z)2), ρv =

ρsat(Pv(z)), ρl = ρsat(Pl(z)), σ = σsat(Pl(z)). The local vaporquality can be subsequently obtained

xv =mv

mv + ml, mv = ρvAvuv(z), ml = ρlAlul(z) (17)

and the local liquid film thickness, δ (z) = R− r(z). Note thatthe saturated property derivatives in Eq.(16), ∂ρv/∂Pv, ∂ρl/∂Pl ,∂hv/∂Pv, ∂hl/∂Pl , are also functions of the state variables Pl(z)and Pv(z). The microchannel flow boiling heat transfer coeffi-cient also depends on the vapor flow quality xv in Eq.(17).

Onset of Nucleate BoilingTo characterize the phase transition from the subcooled

flow to two-phase liquid/vapor mixture, bubble nucleation the-ory [18, 19] are applied here. For the liquid film near the wallin Figure 2(a), there exists a radial energy balance between themicrochannel wall and subcooled fluid to obtain the followingconjugated heat conduction and convection equations

q′′ = αw(Tw−Tf

)= kl

(Tw−Tf

)/δt (18)

where Tf is the subcooled fluid temperature, Tf < Tsat(Pl).With the Clausius-Clapeyron equation, dP/dT =

hlv/(T (vv − vl)), ρv � ρl , vv � vl , and the Laplace-Youngequation, Pv−Pl = 2σ/rb,

Tsat(Pv) = Tsat(Pl)+2σ

rb

Tsat(Pl)

ρvhlv(19)

On the other hand, from the temperature/pressure distribu-tions around a nucleate bubble in Figure 2(a),

Tw−Tsat(Pv)

Tw−Tf=

rb

δt(20)

where δt the thickness of thermal boundary layer around theheated channel wall. Subtracting Tw from both sides of Eq.(19)and substituting Eqs.(20) and (18) yield

q′′

klr2

b−∆Twshrb +2σTsat(Pl)

ρvhlv= 0 (21)

where ∆Twsh is the degree of wall superheat, ∆Twsh = Tw −Tsat(Pl). By solving the quadratic equation (21), the initial ra-dius of a bubble, or equivalently, the initial vapor core in Figure2(b), can be determined

rb =kl∆Twsh

2q′′+

√(kl∆Twsh

2q′′

)2

− kl2σTsat

q′′ρvhlv(22)

where Tsat and hlv are the saturated temperature and latent heatof evaporation at the local liquid pressure, Pl , respectively.



FIGURE 2. Schematic diagram of temperature and pressure around anucleate bubble in microchannels

Critical Heat Flux

Critical heat flux is a key parameter for better design andsafe operation of high-heat-flux removal systems. Katto pro-posed a widely-used correlation to predict the CHF in macroscaleboiling tubes [11]. However, this lumped correlation did notreveal how the local fluid flow causes the occurrence of CHF.One physical approach is to investigate the interaction betweenthe annular liquid film thickness and the vapor/liquid velocityratio. Recent studies [12] indicate that the saturated criticalheat flux condition occurs when the local liquid film thickness,δ (z) = R− r(z), as in Eq.(16) and Figure 1(a), is thinner than thelocal height of the interfacial waves, ∆δi, which was originallyderived from the Kelvin-Helmholtz interfacial instability. In thispaper, a modified critical wave height model (23) is presented totake into account the effect of inlet subcooling

∆δi =

(K1 +K2

hl−hin

hv−hl

)R(

uv

ul

)−3/7( (ρl−ρv)gR2

σ

)−1/7

(23)where the gravity, g=9.8 m/s2, under the standard earth condi-tion, hin is the enthalpy of subcooled inlet flow, and hl/hv arethe specific enthalpies of saturated liquid/vapor corresponding toPl(z)/Pv(z) in Eq.(16), respectively. Compared with the modelin [12], the new parameter K2 is introduced to represent the ef-fect of inlet subcooling. Hence, the proposed critical heat fluxcondition is a generalization of the one in [12]. The result thusagrees with Katto’s correlation [11]: CHF increases with massflow rate and inlet subcooling. In this paper, we use this theoreti-cal model to determine CHF of the separated-flow boiling systemin microchannnels.

MICROCHANNEL FLOW BOILING CHARACTERISTICSAs discussed in the previous section, a complete theoretical

model is presented to characterize the microchannel flow boil-ing system with inlet subcooling. As illustrated in Figure 1,the proposed model, including Eqs. (8)-(10) and (16), describesthe physical behavior in subsequent flow regions: subcooled liq-uid, liquid/vapor two-phase mixture and superheated vapor. Forsimplicity, the two-phase region (incipient boiling) starts at thethermodynamic equilibrium of saturated liquid, which is also re-garded as the onset of nucleate boiling with the initial bubblesize represented in Eq. (22). The critical heat flux condition oc-curs when the local liquid film thickness, δ = R− r(z), is lessthan the critical wave length ∆δi(z) in Eq.(23); consequently, thesuperheated vapor would flow out of the microchannel.

Flow Boiling Model PredictionThe above flow boiling model is used to predict pressure

drop and CHF in microchannels. As shown in Figure 3 and Fig-ure 5, both pressure drop and CHF increases with mass flux,where the model predictions and experimental data are markedas blue circles and red stars, respectively. Notice that experi-mental data were collected from [10], where Qu and Mudawarreported their experimental critical heat flux data and the corre-sponding inlet/outlet pressure data under different mass flux andinlet subcooling conditions in 21 water microchannels. The inletand outlet pressures were measured using two absolute pressuretransducers, respectively. The uncertainty in these pressures was3.5%. The error in heating power input was 0.5%, and CHFshould carry a similar level of error (Qu, private communica-tion). This data set is used in this paper to evaluate the predictionperformance of the proposed separated-flow model. Water prop-erty data are provided by the NIST REFPROP universal function.Note that in subsequent studies, the agreement between the ex-periments and the model predictions can be assessed by meanabsolute error (MAE):

MAE =1M ∑

|experiment−prediction|experiment

×100% (24)

Under the same experimental conditions for imposed (crit-ical) heat flux q′′, inlet fluid temperature ∆Tin, mass flux Gand inlet pressure Pin, the model predicts the exit pressure ofa flow boiling microchannel. As evidenced by the small MAE(6.5918%) and Figure 3(a), the microchannel pressure drop pre-dictions agree well with the experiments, where 12 of 18 data arepredicted by the model within a ±12% error band (Figure 3(b)).Moreover, the relative bubble nucleating size, rb/R, is also pre-dicted by Eq.(22), which is approximately 0.458 with the stan-dard deviation of 0.0002. Note that in these circumstances, thehydrodynamic entrance length of the developing laminar flow,Lh = 0.05 ·Re ·D, fluctuates from 0.05L to 0.23L with varying



50 100 150 200 250 300 350 4000

5

10

15

20

25

30

35

40

Mass flux G (kg/m2−s)

Pre

ssur

e dr

op Δ

P (

kPa)

MAEΔP

= 6.5918 %

predictionexperiment

(a)

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Experimental pressure drop (kPa)

Pre

dict

ive

pres

sure

dro

p (k

Pa)

MAEDP

= 6.5918 %

16 of 18 data in ± 12%All 18 data in ± 28%

28% 12%

(b)

FIGURE 3. Prediction of flow boiling pressure drop in microchannelsunder imposed CHF conditions (experimental data collected from [10]):(a) pressure drop versus mass flux; (b) comparison of pressure drop datawith prediction

mass flux and inlet pressure conditions, which is a small portionof the total microchannel length L and before the occurrence ofphase change. Therefore, only the fully developed flow and theassociated subcooled liquid model (8)-(10) are considered in thisstudy. Since all the critical heat flux conditions are taken as theimposed heat fluxes, the liquid film thickness at the microchan-nel exit corresponds to the critical wave height (23) due to theKelvin-Helmholtz liquid/vapor interfacial instability; therefore,one can estimate the coefficients in (23), K1=0.1923, K2=1.3178,with an MAE of 5.3452 %.

50 100 150 200 250 300 350 4000

5

10

15

20

25

30


Liqu

id fi

lm th

ickn

ess

δ / w

ave

heig

ht Δ

δ i (μm

)

MAEδ = 5.3452 %

film thicknesswave height

FIGURE 4. Critical liquid film thickness and liquid/vapor intefacialwave height in microchannels

Critical Heat Flux PredictionOnce the coefficients of the critical liquid/vapor interfacial

wave height model (23) are determined, the critical heat flux canbe predicted under given conditions of mass flux G, inlet pressurePin and temperature Tin. As seen from Figure 5(b), the CHF pre-dictions match the experiments well, with an MAE of 5.8889%for all data and a ±12% error band for 16 of 18 data. Figure 5(a)shows that CHF increases as mass fluxes and inlet subcoolingincrease, which is consistent with the trend predicted by Katto’sCHF correlation [11].

Distributed Thermal-Fluid CharacteristicsIn fact, the proposed theoretical model can provide more

physical insights of a flow boiling microchannel under variousoperating conditions, such as the stream-wise (axial) distribu-tions of vapor core radius, pressure, velocity, heat transfer coef-ficient, wall and fluid temperatures as well as vapor flow quality.Here, axial microchannel surface temperature, Tw(z), can be pre-dicted by the wall heat conduction equation (4). For instance, atthe imposed base heat flux of q′′=164.93 W/cm2, correspondingto the wall critical heat flux q′′w=41.24 W/cm2, when the massflux G=194.5 kg/(m2-s), the axial velocity and pressure distribu-tion of separated liquid and vapor flows are shown in Figure 6.At the microchannel exit, the vapor velocity reaches 108.2 m/swhile the liquid velocity is only 0.63 m/s. No significant wall(surface) temperature increase is observed in Figure 7 from thesubcooled flow inlet to two-phase flow exit along the microchan-nel evaporator, where the liquid film thickness just reaches theboundary of critical wave height, δ = R− r = ∆δi = 16.5 µm.Even a slight flow or thermal condition deviation could triggerthe critical heat flux condition, as seen in Figure 8 and Figure 9,



50 100 150 200 250 300 350 40010

20

30

40

50

60

70


Wal

l crit

ical

hea

t flu

x (W

/cm

2 )

MAECHF

= 5.8889 %

predictionexperiment

(a)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

Experimental critical heat flux (W/cm2)

Pre

dict

ive

criti

cal h

eat f

lux

(W/c

m2 ) MAE

CHF = 5.8889 %

16 of 18 data in ± 12%All 18 data in ± 24%

12%24%

(b)

FIGURE 5. Prediction of flow boiling critical heat flux in microchan-nels with modified Kelvin-Helmholtz instability conditions (experimen-tal data collected from [10]): (a) CHF versus mass flux; (b) comparisonof CHF data with prediction

where a negative perturbation is applied to the normal mass flux.At the same heat flux condition, q′′=164.93 W/cm2, as in

Figures 6-7, when the mass flux G is reduced from 194.5 to 180kg/(m2-s), the axial velocity and pressure distribution of sepa-rated liquid and vapor flow are shown in Figure 8. Because thecritical heat flux condition occurs, significant wall temperaturerise and vapor back flow are observed in Figure 8 and Figure 9 atthe exit of the microchannel evaporator.

On the other hand, for the same heat flux condition,q′′=164.93 W/cm2, when the mass flux G increases from 194.5to 220 kg/(m2-s), the axial velocity and pressure distribution ofseparated liquid and vapor flow are shown in Figure 10. Liq-

0 5 10 15 20 25 30 35 40 450

100

200

Rad

ius

(μm

)

total Rvapor r

0 5 10 15 20 25 30 35 40 450

100

200

Vel

ocity

(m

/s)

liquid u

l

vapor uv

0 5 10 15 20 25 30 35 40 45110

120

130

Location z (mm)

Pre

ssur

e (k

Pa)

liquid Pl

vapor Pv

FIGURE 6. Axial distribution of boiling fluid flow in microchannelswhen q′′=164.93 W/cm2, G=194.5 kg/m2-s (δ = ∆δi = 16.5088 µm)

0 5 10 15 20 25 30 35 40 450

100

200α w

(kW

/m2 °C

)

0 5 10 15 20 25 30 35 40 450

100

200

Tw /

T f (°C

)

0 5 10 15 20 25 30 35 40 450

0.2

0.4

Vap

or q

ualit

y x

Location z (mm)

wall Tfluid T

ONB

FIGURE 7. Axial distribution of boiling heat transfer in mi-crochannels when q′′=164.93 W/cm2, G=194.5 kg/m2-s (δ = ∆δi =

16.5088 µm)

uid/vapor mixture exits from the microchannel evaporator with-out significant wall temperature increase in Figure 11.

MICROCHANNEL TWO-PHASE FLOW INSTABILITIESAs demonstrated in Figure 3 and Figure 5, the generalized

separated-flow model proposed in this paper can capture theflow boiling pressure drop, critical heat flux and other importantphase-change phenomena in inlet subcooled microchannels. Itprovides a basis to investigate two-phase flow characteristics inmicrochannels, i.e., the demand pressure drop versus mass flux



0 5 10 15 20 25 30 35 40 450

100

200

Rad

ius

(μm

)

total Rvapor r

0 5 10 15 20 25 30 35 40 450

50

100

Vel

ocity

(m

/s)

liquid u

l

vapor uv

0 5 10 15 20 25 30 35 40 45110

120

130

Location z (mm)

Pre

ssur

e (k

Pa)

liquid Pl

vapor Pv

Back Flow

FIGURE 8. Axial distribution of boiling flow in microchannels witha negative flow perturbation when q′′=164.93 W/cm2, G=180 kg/m2-s(δ < ∆δi = 16.5442 µm) (back flow arrow: downstream Pv larger thanupstream Pv)

0 5 10 15 20 25 30 35 40 450

100

200

α w (

kW/m

2 °C)

0 5 10 15 20 25 30 35 40 450

1000

2000

Tw /

T f (°C

)

wall Tfluid T

0 5 10 15 20 25 30 35 40 450

0.5

1

Vap

or q

ualit

y x

Location z (mm)

Burnout

ONB

FIGURE 9. Axial distribution of boiling heat transfer in microchan-nels with a negative flow perturbation when q′′=164.93 W/cm2, G=180kg/m2-s (δ < ∆δi = 16.5442 µm)

under constant heat flux (Figure 12). When the flow rate is suf-ficiently large, the heat load is not enough to boil the subcooledliquid, so the single-phase liquid flow remains at the microchan-nel exit (the right portion of the curve in Figure 12). As the massflowrate is continuously reduced, for fixed heat load and inletsubcooling, flow boiling will start, i.e., onset of nucleate boiling(ONB); Further reduction in the mass flux will gradually enhanceflow boiling and bubble generation. Frictional and accelerationalpressure drops significantly increase with the void fraction and

0 5 10 15 20 25 30 35 40 450

100

200

Rad

ius

(μm

)

total Rvapor r

0 5 10 15 20 25 30 35 40 450

100

200

Vel

ocity

(m

/s)

liquid u

l

vapor uv

0 5 10 15 20 25 30 35 40 45110

120

130

Location z (mm)

Pre

ssur

e (k

Pa)

liquid Pl

vapor Pv

FIGURE 10. Axial distribution of boiling flow in microchannels witha positive flow perturbation when q′′=164.93 W/cm2, G=220 kg/m2-s(δ > ∆δi = 17.6644 µm)

0 5 10 15 20 25 30 35 40 450

100

200

α w (

kW/m

2 °C)

0 5 10 15 20 25 30 35 40 450

100

200

Tw /

T f (°C

)

wall Tfluid T

0 5 10 15 20 25 30 35 40 450

0.2

0.4

Vap

or q

ualit

y x

Location z (mm)

ONB

FIGURE 11. Axial distribution of boiling heat transfer in microchan-nels with a positive flow perturbation when q′′=164.93 W/cm2, G=220kg/m2-s (δ > ∆δi = 17.6644 µm)

vapor quality. The point of minimum pressure drop betweensubcooled liquid and liquid/vapor mixture is widely called as theonset of flow instability (OFI) [16, 17, 20]. Furthermore, whenthe mass flux decreases further, the saturated flow boiling criti-cal heat flux condition is triggered, superheated vapor flow exitsfrom the microchannel, which corresponds to the left portion ofthe pressure drop - mass flow characteristic curve in Figure 12.



100 200 300 400 500 600 700 800 900 10000

2

4

6

8

10

12

14

16


Pre

ssur

e dr

op Δ

P (

kPa)

Onset of Flow Instability

Onset of Nucleate Boiling

Critical Heat Flux

FIGURE 12. Two-phase flow characteristics of flow boiling mi-crochannel (demand pressure drop ∆Pd vs. mass flux G under fixedheat flux q′′=164.93 W/cm2)

By examining the momentum balance (3) or equivalently

dGdt

=1L

[Pin−Pout −∆Pd(G,q′′)

](25)

the balance between the pump supply pressure drop Pin − Poutand the demand pressure drop ∆Pd (pressure drop in Figure 12)should be considered to maintain stable microchannel flow boil-ing. If the pump supply curve has the steeper negative slopein two-phase region than the demand flow characteristic curve,the flow boiling system is stable, ∆Pd ≡ Pin − Pout ; otherwise,flow excursions occur in microchannels because the pump can-not counteract even a small perturbation in the mass flow fromthe two-phase equilibrium condition, i.e., a spontaneous shift toa more stable superheated (left portion of Figure 12) or subcooledflow condition (right portion of Figure 12), which deteriorates theheat dissipating performance. Recent studies have shown that in-creasing system pressure and channel diameter, reducing parallelchannel number and channel length, and including an inlet re-strictor can enhance flow stability in microchannels [20], whichbasically means less negative pressure-drop slope.

When the flow is close to the local minimum of flow char-acteristic curve in Figure 12, pure pressure-drop oscillatory be-havior is usually exhibited; when the flow is close to the localmaximum of the flow characteristics curve, pressure-drop oscil-lations superimposed with density-wave oscillations and parallelflow instabilities may occur [21, 22].

CONCLUSIONSThis paper presents a generalized separated-flow model for

the prediction of both two-phase flow characteristics and criticalheat flux conditions in inlet subcooled flow boiling microchan-nels. The previous model in [12] is a special case of the pro-posed model since three unreasonable assumptions made therehave been removed in this paper, and this proposed theoreticalmodel can even predict the overall microchannel pressure dropwhile the previous model is not able to do that. The critical waveheight condition from the Kelvin-Helmholtz interfacial instabil-ity is modified to accommodate the effect of inlet subcooling. Ithas been shown that the model predictions in this paper agreewell with experimental pressure drop and CHF data from a wa-ter microchannel heat sink. Nevertheless, more model validationwork needs to be done to demonstrate the predictive capabilityof this theoretical model for refrigerant flow boiling microchan-nels. The limitation of this paper comes from the onset of nucle-ate boiling part (incipient boiling), where bubbles are assumedto suddenly emerge at the thermodynamic equilibrium quality ofzero (saturated liquid). In practice, subcooled flow boiling hap-pens before the bulk flow reaches the saturated liquid flow con-dition. Furthermore, it is also interesting to predict experimentalpressure-drop flow oscillations observed in microchannels.

ACKNOWLEDGMENTSThis work was funded by the Cooperative Agreement Be-

tween the Masdar Institute of Science and Technology (Mas-dar Institute), Abu Dhabi, UAE and the Massachusetts Insti-tute of Technology (MIT), Cambridge, MA, USA, ReferenceNo. 196F/002/707/102f/70/9374. The authors would like tothank Profs. Remi Revellin and John R. Thome for instruc-tive responses to inquiries on their theoretical critical heat fluxmodel [12]. The authors would also like to thank Prof. WeilinQu for providing experimental uncertainties and measurementerrors on his CHF and pressure drop data.

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a separated-flow model for predicting flow boiling

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