Magnini 2013 International Journal of Thermal Sciences

8/12/2019 Magnini 2013 International Journal of Thermal Sciences

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Numerical investigation of the inuence of leading and sequentialbubbles on slug ow boiling within a microchannel

M. Magnini a,*, B. Pulvirenti b, J.R. Thome a

a Laboratory of Heat and Mass Transfer (LTCM), Ecole Polytechnique Fdrale de Lausanne (EPFL), EPFL-STI-IGM-LTCM, Station 9, CH-1015 Lausanne,

Switzerlandb Dipartimento di Ingegneria Industriale, Universit di Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy

a r t i c l e i n f o

Article history:

Received 21 December 2012

Received in revised form

5 April 2013

Accepted 7 April 2013

Available online 18 May 2013

Keywords:

Flow boiling

Microchannel

Evaporation

Bubbles

Slug ow

a b s t r a c t

Multiphase CFD simulations are presently employed to investigate the ow boiling of multiple sequential

elongated bubbles in a horizontal microchannel. Most of the computational studies published so far

explored the features of boiling ows within microchannels by simulating the uid-dynamics of a single

evaporating bubble, but the present work shows that multiple bubble simulations are necessary to

capture the essential features of the heat transfer process of a slug ow. In particular, it is shown that

leading and sequential bubbles interact thermally and hydrodynamically due to the evaporation process,

thus possessing different growth rates, velocities and thicknesses of the thin liquid lms trapped be-

tween the bubbles interfaces and the channel wall. The evaporation of this thin liquid lm is the

dominant heat transfer mechanism in the vapor bubble region and the transit of trailing bubbles strongly

enhances the time-averaged heat transfer coefcient of the bubble-liquid slug unit, by as much as 60%

higher relative to the leading bubble under the operating conditions presently set. Furthermore, the

presence of a recirculating vortex just after the tail of the bubble in the liquid slug trapped between the

bubbles was found in the simulations, signicantly improving the heat transfer between the wall and the

bulk liquid, thus maintaining the heat transfer coefcient much higher than otherwise expected in the

liquid slug region as well. Finally, a new multiple bubble heat transfer model is proposed to predict thelocal variation of the heat transfer coefcient, which might prove to be useful to improve the current

boiling heat transfer methods, such as the three-zone model of Thome et al. [1,2]. The numerical

framework employed to perform this study was the commercial CFD solver ANSYS Fluent 12 with a

Volume Of Fluid interface capturing method, which was improved here by implementing external

functions, in particular a Height Function method to better estimate the surface tension force and an

evaporation model to compute the phase change.

2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

Flow boiling in microchannels is nowadays one of the most

attractive cooling technologies to dissipate high heat uxes

through small areas. Compared with conventional channels,evaporation in narrow channels provides higher heat transfer

performance due to the large interfacial area per unit volume of the

uid in close proximity to the higher temperature wall. The physics

of the two-phase ow and the heat transfer mechanisms in

microchannels are substantially different from those in macro-

channels due to the connement effect of the channels walls, and

hence several studies have been conducted in the last decade to

investigate the characteristics of suchow. Comprehensive reviews

on microchannel ow boiling are available in Garimella and Sobhan

[3], Bertsch et al.[4], Thome[5]and Baldassari and Marengo[6].

Within microchannels, once that nucleation begins the vapor

bubbles grow rapidly and ll the entire cross-section of the chan-nel, such that slug ows appear already at low values of the vapor

quality[7]. The slug ow regime shows numerous ow structures

which make it a favorable pattern to achieve efcient heat transfer.

The presence of a recirculation pattern in the liquid slugs which

separate the bubbles enhances the convective heat transfer be-

tween this liquidand the wall [8]. The evaporation of the thin liquid

lm trapped between the bubble and the wall strongly increases

the local heat transfer coefcient [9]. For what concerns the reliable

prediction of the boiling heat transfer coefcient, Thome pointed

out in Ref. [5] that physics-based boiling heat transfer models,

which attempt to reconstruct the actual ow conguration, are* Corresponding author. Tel.: 41 021 6937343.

E-mail address: [email protected](M. Magnini).

Contents lists available at SciVerse ScienceDirect

International Journal of Thermal Sciences

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c om / l o c a t e / i j t s

1290-0729/$ e see front matter 2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.ijthermalsci.2013.04.018

International Journal of Thermal Sciences 71 (2013) 36e52
mailto:[email protected]:[email protected]:[email protected]://www.sciencedirect.com/science/journal/12900729http://www.elsevier.com/locate/ijtshttp://dx.doi.org/10.1016/j.ijthermalsci.2013.04.018http://dx.doi.org/10.1016/j.ijthermalsci.2013.04.018http://dx.doi.org/10.1016/j.ijthermalsci.2013.04.018http://dx.doi.org/10.1016/j.ijthermalsci.2013.04.018http://dx.doi.org/10.1016/j.ijthermalsci.2013.04.018http://dx.doi.org/10.1016/j.ijthermalsci.2013.04.018http://www.elsevier.com/locate/ijtshttp://www.sciencedirect.com/science/journal/12900729http://crossmark.dyndns.org/dialog/?doi=10.1016/j.ijthermalsci.2013.04.018&domain=pdfmailto:[email protected]


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preferable to completely empirical ts of experimental data. To this

end, Thome et al.[1] developed a three-zone model for the evap-

oration of elongated bubbles in microchannels, in which the liquid

sluge elongated bubble unit is split into three regions. The evap-

oration of the thin liquid layer surrounding the bubble is consid-

ered to be the dominant heat transfer mechanism in the liquid lm

region and the local heat transfer is estimated by assuming that

heat is transferred by steady one-dimensional heat conduction

across the stagnant liquid lm. Instead, single-phase methods are

used for the liquid slug and dried out vapor region. Different pub-

lications[2,10e13]assessed the capability of the model to predict

the time-averaged boiling heat transfer coefcient for a wide range

ofuids, channel sizes, and operating conditions. The tight linkage

between liquid lm thickness and heat transfer performance was

supported also by Han et al. [14], who performed liquid lm

thickness and wall temperature measurements under ow boiling

conditions for water and ethanol in a tube of diameter 0.5 mm,

covering mass ow rates from 169 to 381 kg/m2s and heat uxes

from 77 to 735 kW/m2. They found a good agreement between the

heat transfer coefcient calculated from the measured liquid lm

thickness and that obtained directly from wall temperature

measurements.

The pursuit of more accurate boiling heat transfer predictionmethods for the slug ow regime in microchannels will benet

from a more detailed knowledge of the actual local uid and

thermal dynamics. With this aim, Han and Shikazono [15,16]per-

formed experimental measurements on the thickness of the liquid

lm surrounding elongated bubbles owing at constant velocity

and under accelerated conditions. Air, ethanol, FC-40 and water

were used as working uids, covering tube diameters from 0.3 to

1.3 mm in Ref. [15]and from 0.5 to 1 mm in Ref. [16]. In Ref.[16]

they proposed a comprehensive correlation to predict the liquid

lm thickness as a function of the operating conditions in terms of

Capillary number Ca mU/s, tubular Reynolds number Re rUD/m

and acceleration Bond number Boa raD2/s.

Due to the limitations of the current experimental techniques

when applied to the microscale, CFD studies aim to better under-stand the local features of the ow. This is possible thanks to the

recent improvements on robustness and accuracy of multiphase

methods, such as the Level Set (LS)[17]and Volume Of Fluid (VOF)

[18]algorithms. Mukherjee and Kandlikar [19]simulated the ow

boiling of a single water vapor bubble at atmospheric pressure

within a 200mm square microchannel using a LS framework. They

reported that when the bubble began to elongate, its growth rate

became exponential and the velocity of the liquid ahead of the

bubble increased by one order of magnitude with respect to that

before evaporation occurred. The formation of a thin liquid layer

between the elongated bubble and the wall enhanced the heat

transfer. Mukherjee [20] simulated the ow boiling of a single

bubble in contact with the heated surface of a microchannel and

observed that, when dryout occurred, a smaller contact angleincreased the heat transfer as it promoted the formation of a liquid

layer at the wall.

Yan and Zu [21] employed a VOF method to simulate the

nucleate and ow boiling of water within a rectangular micro-

channel and reported that vortices were generated at the front and

rear of the vapor bubbles as well as in the thin liquid layer, thus

suggesting that the heat transfer rates might be enhanced locally.

Lee et al.[22]performed numerical simulations ofow boiling of

water at 1 atm in a nned microchannel of cross-section

(0.24 0.32) mm by means of a LS method. They showed that an

optimal sizing of the ns augmented the heat transfer as they

increased the liquidevaporesolid interface contact region. Dong

et al. [23] employed a lattice Boltzmann method to analyze the

effect of single and multiple bubbles in flow boiling conditions on

the ow and heat transfer in a microchannel of 0.2 mm height,

which was modeled with a two-dimensional geometry. Carbinol

was adopted as working uid. They reported that the heat transfer

was enhanced by the superposition of the effects of multiple bub-

bles on the temperature eld in the microchannel.

Magnini et al. [24] studied the hydrodynamics and heat

transfer given by the ow boiling of single elongated bubbles in a

circular microchannel by means of a VOF method. A channel of

diameter 0.5 mm was modeled with an axisymmetrical geometry,

involving three different refrigerant uids, mass uxes ranging

from 500 kg/m2s to 600 kg/m2s, heat uxes from 5 kW/m2 to

20 kW/m2 and saturation temperatures of 31 C and 50 C. For a

thickness of the liquid lm on the order of 105 m, they argued that

the thermal inertia of the liquid within the lm could not be

neglected in the estimation of the local heat transfer trends and

obtained good predictions of the heat transfer coefcient by means

of a model based on the transient heat conduction across the lm.

The enhancement of the heat transfer was observed to reach a

maximum in the bubble wake, due to the disturbance on the ow

and thermal eld induced by the bubble transit.

Since most of the above cited computational studies dealt with

the ow and evaporation of single bubbles, the objective of this

paper is to explore computationally the inuence on the bubbledynamics and the wall heat transfer by two successive evaporating

bubbles within a circular horizontal microchannel. The conse-

quence of the transit of two consecutive bubbles is the overlapping

of their effect on theuid ow and heat transfer, which is presently

compared with the results of single bubble cases. This study con-

tributes to a better understanding of the thermal behavior of the

liquid lm and the liquid slug when multiple bubbles ow within a

microchannel, as is peculiar to the slug ow regime. A boiling heat

transfer model, which incorporates the major ndings obtained in

this study, is nally proposed. Simulations are performed by means

of the nite-volume commercial CFD solver ANSYS Fluent release

12.1 and the VOF method is adopted to numerically deal with the

liquidevapor interface. The default solver is improved by a self-

implementation of a Height Function algorithm[25]to better es-timate the interface curvature involved in the surface tension force

calculation, and an evaporation model to compute the rates of

mass and energy exchange at the interface due to evaporation.

Both the algorithms are introduced in the solver as User-Dened

Functions (UDF) developed here and are capable of parallel

computing.

2. Numerical model

2.1. Governing equations

The VOF algorithm belongs to the class of the single-uid

multiphase methods, as the phases are treated as a single uid

whose properties change abruptly across the interface. A uniquevelocity, pressure and temperatureeld is shared among the pha-

ses, such that a single set ofow equations is written and solved

throughout the ow domain. Among the single-uid approaches,

the VOF method is a so-called interface capturing scheme, as the

interface is not tracked explicitly on the ow domain, but it is

captured by means of a color function eld, namely the volume

fraction. On a discretized domain, the volume fractionarepresents

the ratio of the cell volume occupied by the primary phase: it is 1 if

the cell is lled with the primary phase, 0 if lled with the sec-

ondary phase and 0< a


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phase of a vaporeliquid two-phaseow, the value of the density on

the generic computational cell is given by:

r rl rvrla (1)

wherea is the vapor volume fraction value in the cell and rv,rlare

the vapor and liquid phasesspecic densities.

Since the volume fractioneld is transported as a passive scalar

by the ow eld, the interface location is evolved by solving aconservation equation. Therefore, the set ofow equations includes

mass, volume fraction, momentum and energy equations. Theow

problem is treated as incompressible, as the variation of the vapor

density due to the pressure drop along the channel is estimated to

be less than 1% under the working conditions analyzed here. As a

consequence, the mass conservation equation for incompressible

ow with phase change takes the following form:

V$u

1

rv

1

rl

_mjVaj (2)

where the r.h.s. of Eq. (2) accounts for the uid expansion due to

phase change by introducing the interphase mass ux _m(see the

Section2.3for its computation algorithm).The conservation equation for the volume fraction of the vapor

phase is expressed as follows:

va

vt V$au

1

rv_mjVaj (3)

Only the vapor volume fraction Eq.(3)is solved, while the liquid

volume fraction eld is obtained as 1 a at the end of the

calculation.

The momentum equation is written for a Newtonian uid as:

vru

vt V$ru$u Vp V$

hmVuVuT

irgFs (4)

whereFsis the surface tension force. By means of the ContinuumSurface Force (CSF) method proposed by Brackbill et al. [26], the

capillary force is expressed as a body force:

Fs skVa (5)

where s is the surface tension coefcient, which is considered

constant in this work, and k is the local interface curvature. ANSYS

Fluent release 12.1 and earlier versions computes the local curva-

ture by differencing the volume fractions. In this work, the ANSYS

Fluent default interface reconstruction algorithm was replaced by a

User-Dened Function (UDF) self-implementation of a Height

Function algorithm to improve the estimation of the local interface

topology, namely its interface normal vector and curvature (see

later in Section2.4a brief introduction to the algorithm).Finally, the energy conservation equation is solved as:

vrcpT

vt

V$rcpuT

V$lVT _hjVaj (6)

which at the r.h.s. shows the energy source term given by the

evaporation, expressed by means of the interfacial enthalpy ux _h.

This parameter accounts not only for the enthalpy sink due to the

evaporation, but also the enthalpy of the vapor created and that of

the liquid removed:

_h _mh

hlv

cp;vcp;l

Ti

(7)

withhlvbeing the latent heat of vaporization.

2.2. Basic assumptions

Besides the already mentioned general assumptions, which

were made to derive the governing equations presented in the

Section2.1, the specic ow conguration and the operating con-

ditions simulated allowed the following additional simplications

on the mathematical and numerical treatment of theow problem:

The variation of the saturation temperature due to the pressure

drop along the channel is only a few tenths of degree Kelvin,

and hence the saturation temperature is considered constant

throughout the microchannel.

The variation of the uid temperature is assumed to be suf-

ciently small such that the liquid and vapor specic properties

are considered constant throughout the ow domain.

Ong and Thome [27] observed that gravitational forces are fully

suppressed in slug ow within horizontal microchannels when

the Connement number Co s=gDrD21=2 is above 1. Only

operating conditions which respect this condition are chosen

for the simulations discussed in this work. Therefore, the

gravitational force appearing within Eq. (4) is dropped and a

two-dimensional axisymmetrical formulation of the ow

problem is possible, greatly reducing computational time. Only working conditions which prevent wall dryout are cho-

sen, i.e. the rst two zones of the three-zones model of Thome

et al. [1,2], such that wall adhesion does not need to be

modeled.

Equation (6) does not include the viscous heating term

because, following the analysis proposed by Morini[28], it was

estimated to be negligible for the operating conditions simu-

lated in this work.

2.3. Evaporation model

The evaporation model implemented here via UDF code corre-

sponds to the framework originally proposed by Hardt and Wondra[29]. This model consists of a physical relationship to compute the

local interphase mass ux _mand a smoothing procedure to smear

the mass and energy source terms over a few computational cells

across the liquidevapor interface.

A common approach, which is widely used to evaluate the local

mass transfer in macroscale problems, is to assume that the inter-

face is at the equilibrium saturation temperature corresponding to

the system pressure, and then to compute the mass ux propor-

tionally to the component of the temperature gradient normal to

the interface[19,20,22]. Such an assumption may be untrue in the

microscale, where interfacial resistance, disjoining and capillary

pressures tend to create an interfacial superheating above the

saturation temperature. A more suitable physical relationship for

microscale phase change problems was derived by Schrage [30],who assumed that the vapor and liquid temperatures are at their

thermodynamic equilibrium saturation values at the interface, but

he supposed an interfacial jump in the temperature to exist, such

that at the interface Tsatpl TlsTv Tsatpv. By assuming a

Maxwellian distribution for the velocity of the gas molecules in

proximity of the interface, Schrage applied the kinetic theory of

gases to analyze the mass transfer process at the liquidevapor

interface and obtained an expression relating the local massux to

the local temperature and pressure at the interface. Subsequently,

Tanasawa[31] simplied Schrages expression by suggesting that

for a low interface superheating over the local vapor equilibrium

saturation temperature (such that TiTv=Tv 1), the following

linear relationship between mass ux and interface superheating

yields:

M. Magnini et al. / International Journal of Thermal Sciences 71 (2013) 36e5238


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_m fTiTv (8)

with Ti being the interface temperature, Tv Tsat(pv) is the vapor

temperature at the interface, and f is the kinetic mobility of the

interface given by:

f

2g

2 g M

2pRg1=2rvhlv

T3=2v (9)

where gis the evaporation coefcient,Mis the molecular weight of

the uid andRg is the universal gas constant. The evaporation co-

efcientgrepresents the fraction of molecules that evaporate from

the bulk phase and then strike and cross the interface, that is

evaporate or condense, while the fraction 1 g is reected.

In the last two decades there has been much debate on the

values assumed for the evaporation (or condensation) coefcientgas phase change occurs. According to published experimental re-

sults [32,33] and preliminary validation benchmarks, the numerical

results discussed in the present work were obtained with an

evaporation coefcient of 1.

With respect to an evaporation model which considers the

interface to be at the saturation temperature, the model discussedabove allows us to easily incorporate many microscale effects

which come into play as the scale of the problem is reduced, by

introducing more sophisticated versions of Eq. (8) as reported in

the detailed analysis of Juric and Tryggvason [34]. In the evapo-

ration model implemented here, the local interphase mass

transfer is estimated by means of the Tanasawa expression (8).

The local interfacial temperature Ti is computed as the cell-

centroid temperature of the computational cell, as it is given by

the solution of Eq.(6). Since the operating conditions simulated in

this work involve small Laplacian pressure jumps across the

interface (on the order of 102e103 Pa), the vapor temperature at

the interface is considered equal to the saturation temperature

referred to the system pressure pN, and hence the termTvwithin

Eqs.(8) and (9)is estimated as Tsat(pN). Due to the high values ofthe kinetic mobility of the interface for the working uids and

operating conditions simulated in the present study, the tem-

perature of the interface is thus always very close to the satura-

tion value.

An evaporation model which computes the interphase massux

according to Eq.(8)was already used by Kunkelmann and Stephan

[35] to simulate the growth of a bubble from a heatedsteelfoil.Juric

and Tryggvason [34] adopted a more complete formulation, also

accounting for the jump in the Gibbs function across the interface,

the irreversible production of entropy at the interface due to phase

change, and the capillary effect induced by a curved interface, to

simulate lm boiling on a heated surface. Nebuloni and Thome[36]

modeled condensation in microchannel annular ows by adopting

Eq.(8) with an additional pressure jump term on the r.h.s. to ac-count for disjoining pressure effects, which may become dominant

when very thin liquid lms occur. The liquid lms observed in the

present simulations are sufciently thick (w105 m) such that the

disjoining pressure jump (estimated to be on the order of 10 5 Pa)

term is negligible here.

The source terms implemented in the numerical model actually

differ slightly from the r.h.s. terms of Eqs. (3) and (6), because the

model also includes the smoothing procedure for the evaporation

source terms described in Ref. [29]. This smoothing procedure

smears the mass and energy source terms over few mesh cells

across the interface, in order to avoid numerical instabilities when

the rate of evaporation is high. A brief description of the smoothing

procedure is provided below, and the reader is referred to Ref.[29]

for details:

1. An initial mass transfer rate f0is estimated as follows:

40 N1 a _mjVaj (10)

where Nis a normalization factor based on the integration of

the volume fractioneld over the whole computational domain

[29].

2. This initial mass transfer rate, which is concentrated only on a

couple of computational cells across the interface, is smeared

by solving a diffusion equation in which 40 represents the

known term, and a Neumann boundary condition applied at

theow domain boundary ensures that the global rate of mass

transfer is conserved for the new smeared mass transfer rate 4.

3. The smeared mass transfer rate is nally used to compute the

new source terms for the mass, volume fraction and energy

equations. In order to concentrate vapor creation on the vapor

side of the interface and liquid disappearance on the liquid

side, the r.h.s. of Eq. (2) is rewritten as:

1rv

Nva1rl

Nl1 a4 (11)

whereNvand Nlare normalization factors which ensure global

mass conservation, i.e. the global amount of liquid disappeared

actually reappears as vapor on the vapor side of the interface.

According to this formulation, the r.h.s. of the energy Eq. (6) is

modied as follows:

40hlvh

Nvacp;vNl1 acp;l

i4T (12)

Hardt and Wondra [29] validated their evaporation model byimplementing it in ANSYS Fluent by means of User-Dened Func-

tions. They ran numerous benchmarks, i.e. one-dimensional Stefan

problems, droplet evaporation and lm boiling tests, and obtained

verygoodagreement withthe analogous analytical solutions.Tests of

the present implementation of the model against the same bench-

marks yielded identical results, and therefore are not depicted here.

Asa further validation case,the growth of a sphericalvaporbubble in

a uniformly superheated liquid was simulated, and it was shown in

Ref. [24] that the numerical bubble growth rateobtained was in good

agreement with analytical solutions for three working uids.

2.4. Height Function algorithm

It is well-known that the accuracy of the interface reconstruc-tion is fundamental when employing the CSF method to estimate

the surface tension force, in particular when simulating capillary

driven ows such as the conned motion of bubbles in micro-

channels. It was already showed in Ref. [24]that, by replacing the

ANSYS Fluent (release 12.1 and earlier) standard interface recon-

struction algorithm with our UDF Height Function method, the

magnitude of the spurious velocity eld generated by the errors in

the curvature estimation decreased by several orders of magnitude.

The Height Function algorithm is based on the local integration

of the volume fractioneld to obtain a discreteeld of local heights

of the interface above a reference axis. This is accomplished by

summing the volume fractions columnwise (or rowwise) within a

local block of cells surrounding the i,j cell for which the interface

curvature is being computed:

M. Magnini et al. / International Journal of Thermal Sciences 71 (2013) 36e52 39


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Hj Xt tup

t tlow

ait;jD (13)

where iandjare the indexes respectively for rows and columns,tupand tlow represent the vertical extension of the block of cells

respectively above and below the i-th row and D is the computa-

tional grid spacing, which is considered constant here. Once three

consecutive values of the height of the interface are reconstructed,the rst and second order derivatives of the height for the central

column can be computed by means of a central nite-difference

scheme. Finally, the interface unit normal vector and curvature are

estimated by resorting to geometrical considerations. For instance,

for an axisymmetrical domain with revolution around thez-axis:

n 1h

1 Hz2i1=2Hz; 1 (14)

k V$n Hzzh

1 Hz2i3=2

HzzjHzzj

1

fzh

1 Hz2i1=2 (15)

where Hzand Hzzdenote the rst and second order derivatives with

respect tozand f(z) is the local elevation of the interface over the

revolution axis. The accuracy of our UDF implementation of the HF

method was tested by means of several validation benchmarks, see

Magnini[37]and Magnini et al. [24]for the results.

2.5. Theow solver

The ow equations reported in Section 2.1 were solved by

means of the nite-volume method which is employed in the

commercial CFD solver ANSYS Fluent release 12.1. The evaporation

and Height Function models are implemented within the solver by

means of User-Dened Functions, which are written inCcode and

are capable of parallel computing. The double precision version ofthe solver was preferred to improve the accuracy of the solution.

In the following, a list of the chosen solver options for the dis-

cretization of the various terms appearing in the ow equations is

presented, while detailed descriptions are given in Ref.[38].

1. Time discretization of the volume fraction equation: rst order

explicit.

2. Time-step for the volume fraction equation: variable time-step

calculated by the solver according to a maximum Courant

number of 0.25 allowed for interface and near-interface cells.

The value chosen is the default value within the solver and it

was adopted by many different authors (see e.g. Refs. [39e41])

to set up the VOF algorithm within ANSYS Fluent.

3. Discretization of the convective term within the volume frac-tion equation: geometrical reconstruction of the uxes across

the boundary faces of the cells as given by the PLIC (Piecewise

Linear Interface Calculation) formulation, originally proposed

by Youngs [42]. The PLIC algorithm avoids the numerical

diffusion of the interface and the oscillation of the volume

fraction values across the interface, which may occur when

standard interpolation schemes are used to compute face-

centered values of the volume fraction eld.

4. Time discretization of momentum and energy equations: rst

order implicit.

5. Time-step for momentum and energy equations: variable time-

step calculated by the solver according to a maximum Courant

number of 0.5. The largest velocity associated with the ow

boiling may vary by one order of magnitude as time elapses,

and hence a variable time-step ensures a good compromise

between accuracy and computational cost of the simulation

compared to a xed time-step.

6. Discretization of convective terms within momentum and en-

ergy equations: third order MUSCL (Monotonic Upstream-

centered Scheme for Conservation Laws)[43]scheme.

7. Discretization of diffusive terms within momentum and energy

equations: central nite-difference scheme.

8. Reconstruction of cell-centered gradients: Green-Gauss node-

based formulation. It was proved to be the best option to

minimize the spurious velocities arising from the unbalance of

pressure and capillary terms within the momentum equation.

9. Pressureevelocity coupling: segregated pressure-based PISO

(Pressure Implicit Splitting of Operators) [44] algorithm. The

mass conservation equation is turned into a pressure correction

equation which is solved iteratively together with the mo-

mentum equation. The PISO algorithm was found to converge

more quickly than the other options available in the solver.

10. Evaluation of face-centered values of the pressure: PRESTO

(PRessure STaggering Option) option. In principle, ANSYS

Fluent adopts a collocated technique, in which the ow equa-

tions are solved for cell-centered variables. However, with the

chosen option, the pressure correction equation is solved for astaggered control volume, thus providing face-centered pres-

sures without the need of interpolations. This option led to a

lower magnitude of the spurious velocity elds compared with

the other available.

11. Convergence criterion: absolute residuals below 106 for all the

equations solved. However, no appreciable differences were

observed in the results obtained with the threshold increased

to 103.

This implementation is similar to the adiabatic scheme of

Nichita and Thome [45], which was well benchmarked versus

numerous standard cases.

3. Validation of the numerical framework

In the absence of a heat load, the balance of the forces acting on

an elongated bubble owing within a microchannel determines the

shape and the velocity of the bubble and the thickness of the liquid

lm trapped between the bubble interface and the channel wall. Inow boiling conditions, the thickness of this liquidlm is known to

play a primary role in determining the heat transfer magnitude and

trend[1,9,14], which in turn inuences the evaporation rate and

hence the bubble dynamics itself. Therefore, a CFD solver whose

aim is to simulate ow boiling of elongated bubbles within

microchannels has rstto be proven to be accuratein predicting the

actual liquid lm thickness. In order to test the present numerical

framework, the adiabatic ow of an elongated bubble within a

horizontal circular microchannel (with negligible gravitational ef-fects) was simulated for 8 different operating conditions, involving

Capillary numbers of 0.025 and 0.0125 and Reynolds numbers in

the range from 15.625 to 625. The bubble was pushed downstream

to the channel by a constant ow rate of liquid set at the channels

inlet, and the terminal liquid lm thickness in the simulations was

compared to the very well documented experimental correlation

proposed by Han and Shikazono[15]. It was found that, using the

actual terminal bubble velocity as recommended by the authors,

the numerical results deviated from the predicted values at

maximum by 5% for 6 out of 8 runs and by 16% for the remaining 2

cases, which is close to the accuracy of their method.

Then, in order to test the bubble dynamics given by the simu-

lations in ow boiling conditions, 5 test cases were run involving

three different refrigerant

uids for a channel diameter of 0.5 mm



6/17

for saturation temperatures in the range of 31e50 C, mass uxes of

500e600 kg/m2s and wall heat uxes of 5e20 kW/m2. The time-

law of the bubble nose position for each benchmark was

compared successfully with a theoretical model developed by

Consolini and Thome[46].

Experimental measurements of the wall temperature variation

during the transit of an elongated bubble in ow boiling conditions

are still difcult to achieve, and hence no benchmarks are available

for the local heat transfer coefcients in simulations. However, we

compared the simulation results for 4 of the mentioned 5 test runs

with an analytical boiling heat transfer model based on transient

heat conduction across the liquid lm [24]. The heat transfer co-

efcientsmagnitudes and time-trends for the analytical solutions

and the simulations were observed to match closely.

4. Results and discussion

4.1. Simulations matrix

Four different case studies were performed involving two

different low pressure refrigerants and their operating conditions

are summarized in Table 1.The Cases 1e3 involve the simulation of

a single vapor bubble and they serve as a preliminary study. Case 4

involves the simulation of two successive bubbles and allows us to

investigate their mutual effects on the dynamics of theowand the

wall heat transfer performance.

The circular microchannel is modeled as a two-dimensional

axisymmetrical channel with a diameter D 0.5 mm and a

lengthL that varies depending on the simulation run. The channel

is always split into an initial adiabatic region of length La, followed

by a heated region of lengthLh, such thatL La Lh. The length of

the adiabatic region is chosen in such a way that the bubble enters

in the heated region of the channel in a steady-state ow condition.

The length of the heated region is limited to avoid an excessive

computational expense of the numerical simulation. Elongated

vapor bubbles of length 3Dare initialized as cylinders with spher-

ical rounded ends and placed at the upstream of the microchannel.This starting conguration, rather than the initialization of small

spherical bubbles, guarantees that the ow is axisymmetrical at the

beginning of the simulation, and in ow boiling experiments vapor

may be present upstream to the heated section of the channel

when bubbles are created by ashing rather than nucleate boiling

[12]. Such a technique avoids the typical temperature overshoots

necessary to initiate nucleation in microchannels. As boundary

conditions, a saturated liquid inow of constant mass ux G issetat

the channel inlet. The value ofG is achieved with a at velocity

prole of magnitude Ul G/rl. Atthe outlet section of the channel, a

zero gradient condition (as implemented in the ANSYS Fluents

outow boundary treatment[47]) is set for the velocity and tem-

perature elds. A constant and uniform heat uxqis applied at the

wall of the heated region of the channel. For each run, the initialvelocity and temperature elds are taken from a preliminary

steady-state simulation for a single phase liquid ow run under the

same operating conditions. For reference,Fig. 1reports for Case 1

the initial bubble prole, temperatureeld within the channel, wall

temperature prole and local heat transfer coefcient computed as:

hz q

Twz Tsat(16)

whereTw is the local wall temperature andzis the axial coordinate.

The saturation temperatureTsatis considered constant throughout

the ow domain and the value set foreach simulation is reported in

Table 1. The heat load applied at the wall of the microchannel

generates a thermal developing region characterized by a super-

heated thermal boundary layer at the wall that thickens in the

streamwise direction, while the heat transfer coefcient decreases

accordingly as shown inFig. 1. The liquid Reynolds number is al-

ways below 1000 under the operating conditions set and thus the

ow is laminar. The h(z) curve reported in Fig.1 compares well with

the London and Shah correlation given by the VDI [48]for laminar

developing ow.

The ow domain is discretized by a uniform computational grid

made by square cells. The mesh element size adopted for all the

simulated cases is D D/300, which ensures that at least 8

computational cells always discretize in the radial direction the

liquid lm trapped between the bubble and the channel wall, ac-

cording to the results of a preliminary grid convergence analysis.

Due to the ne computational mesh necessary to obtain reliable

results, the computational cost of these simulations is typicallyhigh. For instance, Case 4, which has the longest computational

domain (72D), involved more than 3 million mesh cells and 75,000

time steps to run about 60 ms of simulation time. The simulations

were performed on the EPFLPleiades cluster, having computing

nodes of 2 dual-core Intel Xeon 5150 processors at 2.67 GHz and

8 GB of RAM, and Giga-Ethernet network interconnection among

the nodes. The simulation for Case 4, run with 96 parallel cores and

a Message-Passing-Interface protocol, took approximately 3 weeks

to run, while those involving shorter channels (30D) took about 1

week with 64 cores.

4.2. Flow boiling of a single elongated bubble

In Ref.[24]it was reported that, as a consequence of the evap-oration, the nose of the bubble accelerated downstream to the

channel while the velocity of the rear of the bubble remained equal

to the adiabatic velocity of the bubble. The analysis of the ow and

temperature eld in the wake region behind the bubble showed

that the bubble passage generated a thermal time-developing re-

gion along the heated wall because the bubble partially erased the

thermal boundary layer, thus enhancing locally the heat transfer.

The liquid in the region ahead of the bubble accelerated strongly

due to the evaporation, showing values of the velocity much higher

than that set as the boundary condition at the channels inlet

section.

Below, the heat transfer performances for the simulations for

Cases 1e3 are presented and discussed. Fig. 2(a)e(c) report, for

each case run, the two-phase heat transfer coefcient htp as afunction of time during the bubble passage at a given axial location.

The heat transfer coefcient is made dimensionless by the value of

the single phase heat transfer coefcienthspcomputed at the same

location for the single phase preliminary simulation and reported

in the gures captions. For both two-phase and single phase ows,

the heat transfer coefcient is computed as expressed in the Eq.

(16). The axial location analyzed for each case studied is reported as

dimensionless axial distance zh/Dfrom the entrance in the heated

region of the channel.

The trends of the heat transfer coefcients plotted in Fig. 2

conrm what was already observed in Ref. [24]. As the bubble

nose is approaching the axial location under analysis, the acceler-

ation of the liquid ahead of the bubble enhances the liquid-wall

heat convection and the heat transfer coef

cient increases

Table 1

Operating conditions for ow boiling simulation runs.Lastands for adiabatic length

of the channel, Lhfor the heated length.

Case Bubbles Fluid G[kg/m2s] Tsat [ C] q[kW/m2] La Lh

1 1 R113 600 50 9 8D 12D

2 1 R113 600 50 20 8D 22D

3 1 R245fa 600 50 20 8D 22D

4 2 R245fa 550 31 5 (16 34)D 22D



7/17

accordingly by a few percent. As the bubble nose crosses zh, which

happens attz 9 ms for the Cases 1 and 2 and attz 11 ms for Case

3, the liquidlm trapped between the bubble and the channel wall

gets thinner from the bubble nose to the rear and the lm evapo-

ration becomes the governing heat transfer mechanism. As a

consequence, the heat transfer performance improves mono-tonically while the bubble is crossingzh. The maximumvalue of the

heat transfer coefcient for the liquid lm region, which in the

present simulations is 20e30% more than the local single phase

value and it is detected at the transit of the rear of the bubble, is

strictly related to the minimum value of thelm thickness, which is

on the order of 105 m here. Note that much larger multipliers

typical of experimental data would be found for longer bubbles

with the lm thickness approaching its dryout condition at about

0.3$

10

6

m used in Refs. [1,2]or at much higher heat

uxes.Even after the passage of the bubble rear, the heat transfer co-

efcient still grows for a few milliseconds in all the simulations

performed. In order to clarify this behavior,Figs. 3and4report the

8 10 12 14 16 181

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Time [ms]

htp

/hsp

liquid bubble liquid

8 10 12 14 16 18 20 221

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Time [ms]

htp

/hsp


10 12 14 16 18 20 22 24 261

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Time [ms]

htp

/hsp


Fig. 2. Two-phase (subscript tp) to single phase (sp) heat transfer coefcient ratio plotted versus time at a given axial location for the simulations run with a single bubble. zhrefers

to the axial distance from the entrance in the heated region of the channel. The vertical dashed lines identify the transit of bubbles nose and rear.

2

4

6

8

Tw

T

sat

[K]

1000

2000

3000

4000

h[W/m2K]

z/D

0 2 4 6 8 10 12 14 16 18 20

0 1 2 3 4 5 6 7

TTsat

[K]

R113 liquid

G=600 kg/m2s,

T=323.15 K

q=0 q=9 kW/m2

Fig.1. Initial temperature eld within the channel, wall temperature (dashed line) and heat transfer coefcient (solid line) for simulation Case 1. The bubble interface is represented

by the white line prole at the upstream of the channel. The channel image is stretched vertically to enlarge the thermal boundary layer at the heated wall.



8/17

radial prolesof the velocity and temperature within the channel at

a xedaxial location (zh/D 5) for Case 1, for different time instants

after that the bubble rear (t 13.1 ms) has passed. Note that the

centerline velocity of the liquid-only ow is less than 2 times the

average value Ul since the ow is not yet fully developed. The

bubble transit generates a hydrodynamically developing region

characterized by a plug-like velocity prole at the bubble rear (see

the curve t 13.1 ms in Fig. 3), which restoresitself to the reference

liquid-only undisturbed prole within around 2 ms. The tempera-

ture of the bulk liquid drops more rapidly than that of the liquid

nearby the channel wall, as it can be argued by comparing the

temperature proles at t 13.1 ms and t 13.7 ms in Fig. 4, becausethe ow recirculation is more effective in the proximity of the

channel axis. Hence, the wall temperature responds with a little

delay to the ow dynamics in the bulk liquid. For this reason, even

though the velocity prole aftert 15.2 ms corresponds to that of

the undisturbed ow and thus the temperature in the bulk liquid is

increasing as the time elapses (seeFig. 4), the wall temperature is

still decreasing until aroundt 16 ms as proven by the prole of

the heat transfer coefcient reported inFig. 2(a).

It is worth to note that the hydrodynamically developing region

generated by the bubble is much shorter than the developing

length of the liquidow eld atzh/D5, which is 13D, because the

velocity prole just behind the bubble (t 13.1 ms inFig. 3) is not

entirely at. Hence, the liquid ow behind an elongated bubble

develops more rapidly with respect to a hydrodynamically devel-

oping region within a channel and this is an important outcome for

the modeling of the heat transfer in the liquid slug region trapped

between two bubbles. In particular, this suggests that the length

between sequential bubbles plays an important role in the hydro-

dynamics and heat transfer in the liquid slug.

The heat transfer coefcients for Cases 1e3 show similar trends

and values, as the operating conditions set in the simulations

are not very different from each other. However, the following

differences can be detected. Given the same uid and operating

conditions (Cases 1 and 2), the increase of the heat load (from 9 to

20 kW/m2) shifts the prole of the heat transfer coefcient to

higher values. The maximum of the heat transfer coefcient in thebubble region, measured at the time instant of the transit of the

bubble rear, grows from 24% to 30% of the single phase case value,

see Fig. 2(a) and (b).Cases 2 and 3 are run under the same operating

conditions but for different working uids.Fig. 2(b) and (c) shows

that the uid R113 leads to better heat transfer performance than

R245fa for the given operating conditions. This is likely due to the

thinner liquid lm surrounding the bubble for the R113 uid case

(24mm) than that for R245fa (30mm).

According to the thermal and uid dynamics discussed here for

the single bubble case, when multiple bubbles ow in sequence

within a microchannel and evaporate, they may inuence each

other in several ways. First of all, the uid accelerated by a trailing

bubble pushes the leading one, and hence the rear of the leading

bubble will not ow with a constant velocity anymore. If the bub-bles are sufciently close, the thermal developing regions gener-

ated by their passage may partially overlap, thus leading to better

heat transfer performance than for the single bubble case. The

investigation of the ow boiling of consecutive bubbles is

addressed in the next section.

4.3. Flow boiling of multiple bubbles

The bubble dynamics, ow and thermal eld, and heat transfer

for the simulation Case 4 arediscussed separately below, andnally

a theoretical model for the resulting heat transfer is proposed. Two

elongated vapor bubbles are initialized at the upstream of the

microchannel. The bubbles are separated by a trapped liquid slug of

6Dlength, which was chosen arbitrarily. The adiabatic region of thechannel is doubled with respect to the single bubble simulations to

allow both the bubbles to reach a steady ow. Numerical errors

occurred when the vaporeliquid interface crossed the outlet sec-

tion of the channel. In order to avoid such errors while the bubbles

are still within the heated region of the channel, the computational

domain ends with a terminal adiabaticregion of length 34D, chosen

to store both the bubbles after the evaporation stage.

4.3.1. Dynamics of the bubbles

The bubbles quickly achieve a steady-stateow in the adiabatic

region of the channel. The adiabatic velocity of the bubbles is

0.485 m/s, which exceeds by around 17% the average velocity of the

liquid inow, i.e. equivalent to a void fraction 3of 0.28 based on the

de

nition of

rvUb/(Gx) where the vapor quality x is 0.02. The

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

uz/U

l

r/D

Liquid only

Twophase

time

Fig. 3. Dimensionless axial velocity proles for Case 1 at zh/D 5. The black dashed

line refers to the preliminary liquid-only simulation. The two-phase proles refer tot 13.1,13.4,14,14.6,15.2 ms.

0 1 2 3 4 5

0.38

0.4

0.42

0.44

0.46

0.48

0.5

TTsat

[K]

r/D Liquid onlyt=13.1 ms

t=13.7 ms

t=15.2 ms

t=16.1 ms

t=17.3 ms

Fig. 4. Temperature proles along the radial coordinate for Case 1 at zh/D 5. The

black dashed line refers to the preliminary liquid-only simulation while the colored

lines refer to the two-phase

ow simulation.



9/17

thicknessd of the liquid lm surrounding the bubbles is the same

for both the bubbles,d/D 0.04, which is in good accord with the

value of 0.0373 predicted by the Han and Shikazonocorrelation [15]

for the lm thickness of elongated bubbles owing within circular

microchannels with constant velocity, under the operating condi-

tions presently set. It is reasonable to consider that the hydrody-

namics of the trailing bubble is inuenced by the leading one in an

adiabatic slugow, such that the thickness of the liquidlm (which

depends mainly on the bubble velocity and the velocity prole of

the liquid ahead of the bubble[15]) may be different for consecu-

tive bubbles. However, the length of the hydrodynamic disturbance

by the leading bubble is very short in the present case (around 1.5

diameters) and since the liquid slug trapped between the bubbles is

much longer, the bubbles do not hydrodynamically inuence each

other and show the same value of the liquid lm thickness.

Fig. 5depicts the evolution of the bubbles during their growth

occurring within the heated region of the channel, which is

included between the sectionsz/D 16 and 38. It is evident that the

trailing bubble grows less rapidly than the one ahead (viz. the

length of the bubble at the same z/D locations). This happens

because the transit of the leading bubble has cooled down the su-

perheated liquid near the wall and the thermal boundary layer has

not hadenough time to rearrangeto the steadysituation. Therefore,the trailing bubble seesless superheated liquid than the leading

one.

Figs. 6and7show respectively the position and the velocity of

bubbles nose and rear versus time. The velocity is computed as

Ub dz/dt, wherezmay refer to the position of the leading bubble

(b1 in Figs. 6 and 7) or trailing bubble (b2)nose (N)orrear(R),andit

is made dimensionless by the average velocity of the liquid inletUl.

The leading bubble enters the heated region of the channel after

4 ms and the nose accelerates due to the evaporation around 1 ms

later, when the bubble interface comes in contact with the super-

heated thermal boundary layer developing at the wall. The oscil-

lations of the bubble rear, which diminish as the bubble starts to

evaporate, are consistent with the observations of Polonsky et al.

[49] whilst Liberzon et al. [50] has explained and characterized

these as capillary waves. As long as the trailing bubble is still

owing within the adiabatic region, the dynamics of the rst

bubble during the evaporation proceeds as though the bubble was

owing alone in the channel. At t 14 ms the trailing bubble enters

within the heated region and starts to grow. Its evaporation rate is

lower than that of the leading bubbledue to the cooler liquid region

crossed, and hence the acceleration of the nose is lower too. The

dynamics of the leading bubble is signicantly affected by the

presence of the trailing evaporating bubble because the liquid

accelerated by the nose of the second bubble pushes the rear of the

rst one. As a consequence,Fig. 7suggests that the velocity of therear of the leading bubble is no longer constant but it increases

accordingly, such that the leading bubble as a whole accelerates

further.

Fig. 5. Evolution of the bubbles while

owing across the heated region of the channel.

0 10 20 30 40 500

10

20

30

40

50

60

Time [ms]

z/D

b1,N

b1,R

b2,N

b2,R

Fig. 6. Position of bubbles nose and rear versus time. Nand R stand respectively for

the nose and rear, with b1 and b2 for the leading and trailing bubbles.

0 10 20 30 40 50

1

1.5

2

2.5

Time [ms]

U/Ul

b1,N

b1,R

b2,N

b2,R

Fig. 7. Velocity of bubbles nose and rear versus time. Nand R stand respectively for

the nose and rear, with b1 and b2 for the leading and trailing bubbles.



10/17

After 19 ms, the nose of the leading bubble exits the heated

region, but it continues to accelerate and grow until t 22 ms

because of the superheated liquid transported by the ow within

the terminal adiabatic zone and by the evaporation of its liquidlm

still within the adiabatic zone. Fig. 8 depicts the prole of the

leading bubble att 19 ms. It can be observed that the bubble has

grown to a length of 9Dfrom 3D, the bubble nose is less blunt than

the adiabatic prole due to the augmented effect of the inertial

forces, which are also responsible for the increase of the liquid lm

thickness tod/D 0.053, measured downstream to the interfacial

wave occurring at the bubble rear. As the rise velocity of buoyant

bubbles in a pool are increased by such sharper proles according

to Tomiyama et al.[51], this also is seen to aid the vapor ow here.

Despite that the amplitude of the oscillations of the bubble rear

decrease as the bubble grows (see the red plot in Fig. 7for times

from 10 to 25 ms), the portion of the liquid lm which is disturbed

by the capillary wave has grown in comparison to that of the

adiabatic situation. This may appear to be in contrast with the

Liberzon et al.[50]interpretation about the nature of these capil-

lary waves, which they considered to be excited by the oscillation of

the rear of the bubble. However, one has to consider that the

experimental observations reported in Ref. [50] concerned short

Taylor bubbles (1.75D long at maximum) of air rising in stagnantwater due to gravity, and thus their working conditions are very

different from those investigated here. In the present case, the

evaporation phenomenon accelerates the bubble with respect to

the adiabatic case (but, notably, not the liquid within the lm in

proximity of the bubble rear, which remains almost stagnant, see

Magnini et al. [24] as reference), thus increasing the velocity dif-

ference between the liquid phase in the lm and the vapor phase

within the bubble and thus promoting the local instability of the

bubble interface, which is also triggered by the increased length of

the bubble.

The nose of the trailing bubble reaches the end of the heated

section after 31 ms. Due to the lower growth rate,Fig. 8shows that

the bubble is noticeably shorter (7D) than the leading one. This in

turn gives rise to less acceleration of the bubble, and hence a morerounded prole of the bubble nose and a liquidlm slightly thinner

(d/D 0.047) than the leading bubble. Notably, if one considers

simple one-dimensional steady-state heat conduction across the

liquid lm as in Refs. [1,2], this would locally result in an increase of

the heat transfer coefcient by 13% in the second bubble with

respect to the rst bubble.

4.3.2. Flow dynamics within the liquid slug

The velocity and temperature eld within the liquid slug trap-

ped between the evaporating bubbles is analyzed here. The ow is

captured at t 22.4 ms, when the leading bubble is partially

downstream to the heated section of the channel while the trailing

bubble is still entirely inside. The velocity of the nose of the trailing

bubble isUb2,N 0.624 m/s and it is equal to that of the tail of the

bubble ahead.

Fig. 9(a) and (b) depicts respectively the streamlines of the ow

eld (ur,uz) and the streamlines of the ow eld observed from a

reference frame moving at the velocity of the trailing bubble nose,

obtained as (ur,uz Ub2,N). The streamlines are computed as iso-

level curves of the streamfunction j, dened as:

1

r

vj

vr uz;

1

r

vj

vz ur (17)

while the velocity of the bubble noseUb2,Nis subtracted fromuzin

Eq.(17) to calculate the streamfunction of the relative ow eld.

Fig. 9(a) shows that the streamlines of the velocity eld are parallel

to the channel axis in the liquid slug region trapped between the

growing bubbles, and this indicates that the ow is moving

downstream across the entire cross-section of the channel. Some

small recirculation patterns appear upon each crest of the capillary

waves occurring in the liquid lm at the rear of the leading bubble,

due to the local oscillations in the pressure eld induced by the

change in sign of the liquidevapor interface curvature along the

bubble prole. The plot of the streamlines of the relative velocityeld reported inFig. 9(b) shows that the liquid ow eld in the slug

can be split along the radial direction into a reversed ow occurring

in the proximity of the channel wall and a recirculating ow pre-

sent on the core region of the channel. The reversed ow is

constituted by the liquid which bypasses the bubble through the

liquidlm region, as the streamlines depicted inFig. 9(b) along the

channel wall maintain a constant backward direction. The recir-

culatingow at the core of the liquid slug indicates that the liquid

velocity near the centerline of the channel exceeds the bubble ve-

locity, and hence the liquid impinges on the leading bubbles tail

and then moves radially toward the channel wall, as suggested by

the plot of the streamlines. This anti-clockwise rotating toroidal

vortex feeds the wall with fresh liquid, thus enhancing the heat and

mass transfer within the slug. The bypass liquid acts as a thermalresistance to the heat transfer between the recirculating bulk liquid

and the channel wall and is responsible for the delay of the wall

temperature feedback to the variation of the temperature eld of

the bulk liquid, which had already been observed inFig. 4. At short

distance from the bubbles interfaces, the streamlines within the

recirculating region are straight, indicating that the ow in the

liquid slug is a fully developed Poiseuille ow, and hence the

bubbles do not inuence each other from a hydrodynamical point

of view. This outcome agrees with the experimental ndings of

Thulasidas et al.[52]which, for adiabatic slug ows, observed that

only liquid slugs shorter than 1.5 times the channel diameter pre-

vented the streamlines from becoming straight in the region be-

tween the bubbles. An important parameter, which may be

useful to model theow within the liquid slug, is the radialpositionof the streamline which divides bypassing and recirculatingows, for which an analytical expression is provided in Ref. [52].

For the operating conditions in the present simulation, the rela-

tionship given in Ref. [52] suggests the dividing streamline to be

located at r/D 0.457, which is in excellent agreement with the

value 0.45 given by the numerical simulation here.

Interestingly, Fig. 9(b) also depicts a large recirculation zone

inside the trailing bubble near its nose, which matches the liquid

recirculation in the liquid slug, while a much smaller recirculation

zone is instead observed inside the leading bubble near its rear.

Similarly, Lakehal et al.[53]and Fukagata et al. [54]observed that,

for elongated bubbles, the vortices appearing near the nose are

larger than those occurring near the rear of the bubbles. This is a

direct consequence of the different shape of the nose and rear of

9 8 7 6 5 4 3 2 1 00

0.1

0.2

0.3

0.4

(zzN

)/D

r/D

Adiabatic

Bubble ahead

Bubble behind

Fig. 8. Proles of the bubbles. The adiabatic prole refers to that of the leading bubble

before it enters in the heated region of the channel. The prole of the leading bubble is

captured after 19 ms, that of the trailing bubble refers to t 31 ms. The proles are

shifted in order to match the nose positions zNfor a comparison.



11/17

elongated bubbles, the former being spherical and the latter nearly

at.

Fig.10shows the isotherms in the liquid slug region, along with

the reference isotherm T Tsat 0.3 K for the liquid-only single

phase simulation case. The crowding of the isotherms next to the

leading bubble rear indicates that the bubble transit has squeezed

the thermal boundary layer against the wall, while the evaporation

of the liquid lm has cooled it down. The lower thickness of the

thermal layer with respect to the liquid-only case suggests that theheat transfer performance is enhanced. The superheated thermal

layer thickness increases at axial locations close to the nose of the

trailing bubble since the heat ux applied at the channel wall tends

to reform it. This phenomenon is slowed down by the presence of

the recirculation pattern in the liquid slug and by the augmented

velocity of the liquid due to the growth of the trailing bubble. In the

proximity of the nose of the trailing bubble, the thermal layer is still

thinner than the single phase case, such that it can be argued that

the disturbance generated by the bubbles on the thermal elds

overlap. The convective motion generated by the recirculating

vortex within the liquid slug moves a portion of superheated liquid

from the thermal boundary layer to the bulk region of the ow,

such that also the bubble nose contributes to evaporation, even if

only to a minor extent. Such an effect is increased by a thicker

thermal layer, and hence it is more effective for the leading bubble

whichows through a thermally undisturbed region.

Note that the vapor temperature always stays very close to the

saturation value, actually it is only few hundredths of degree Kelvin

above. Due to the high value of the kinetic mobility of the interface

for the uids and the working conditions simulated, almost all theheat ux which crosses the liquidevapor interface is used to

evaporate the liquid and the interfacial temperature stays close to

the saturation value. Hence, the heat ux transferred from the

interface to the vapor phase is minimum and the increase of the

vapor temperature is unperceived.

4.3.3. Heat transfer performance

The heat transfer performance is analyzed by means of the

instantaneous and time-averaged values of the heat transfer coef-

cient, associated with the ow of the bubbles at different axial

locations within the heated region of the channel.Fig. 11shows the

Fig.10. Isotherm lines with DT 0.3 K, att 22.4 ms. The dashed line identies the isothermT Tsat 0.3 K for the liquid-only single phase case. The thick black lines identify the

bubbles pro

les. (For interpretation of the references to color in this

gure legend, the reader is referred to the web version of this article.)

Fig. 9. Streamlines of the (a) velocity eld (ur,uz) and (b) relative velocity eld (ur,uz Ub2,N) whereUb2,N 0.624 m/s is the velocity of the nose of the trailing bubble, att 22.4 ms.

The blue lines identify the bubbles proles which are superimposed to the streamlines plots.



12/17

heat transfer coefcient as a function of time at four chosen axial

locations, which are identied by their non-dimensional axial

distance zh/D from the entrance in the heated region. Each plot

reports a black horizontal dashed line identifying the value of the

heat transfer coefcient in the preliminary liquid-only simulation

at the given location. The time instants at which the bubblesnoses

and rears cross the axial location observed are identied by vertical

black lines. Time-averaged heat transfer coefcients for each bub-

ble cycle are evaluated by integrating h within a time interval thatincludes the passage of the bubble and one liquid slug length. The

integration begins when the position of the bubble nose zN is

located at one-half of the liquid slug lengthLs upstream tozhand

ends when the position of the bubble rear zRis one-half liquid slug

length downstream tozh:

hz 1

Dt

ZtzRzhLs=2

tzNzhLs=2

hz; tdt (18)

and hence one-half liquid slug length is considered ahead of the

bubble and one-half behind it. The averaged heat transfer co-

ef

cients are reported as horizontal dashed blue lines inFig. 11and

each lines length is that within the time window considered to

compute the average value. The limits of each window are plotted

as vertical red lines.

The rst axial location explored inFig. 11is placed 4 diameters

downstream to the entry in the heated region. The local heat

transfer coefcient for the preliminary liquid-only single phase

simulation is 2234 W/m2K, which is about three times the value for

thermally fully developed laminar ow with constant heat ux

(4.36$

(ll/D) 769 W/m2

K). Similar to what was observed inFig. 2,the heat transfer coefcient for the two-phase ow grows slowly as

the bubble nose of the leading bubble is approaching and crossing

zh, then it rises sharply after about half of the residence time of the

bubble atzh. The peak of the heat transfer coefcient is achieved in

the wake region behind the bubble rear and for the leading bubble

cycle it is 3331 W/m2K, which is 49% higher than the local value for

the preliminary liquid-only simulation. The average heat transfer

coefcient for the rst bubble cycle, computed by means of the Eq.

(18), is 2626 W/m2K which is the 18% higher than the single phase

value.

After the peak detected next to the bubble wake region, the heat

transfer coefcient drops because the thermal boundary layer at

the wall is being restoredwhile the liquid slug trapped between the

bubbles is passing. During this stage, the heat transfer coef

cient

10 20 30 401

1.5

2

2.5

3

3.5

4

zh/D=4

Time [ms]

h[kW/m

2K]

10 20 30 40

zh/D=10

Time [ms]

10 20 30 40 50 601

1.5

2

2.5

3

3.5

4

zh/D=16

Time [ms]

h[kW/m2K]

20 30 40 50 60

zh/D=21

Time [ms]

Fig. 11. Heat transfer coefcient at various axial locations. The black vertical lines identify the transit of the bubblesnose and rear, while the red lines identify the limits of the time

intervals which the coefcients are averaged within. The black dashed lines identify the value of the heat transfer coefcient for the liquid-only simulation and the dashed blue lines

the average coefcient for the two-phase ow. (For interpretation of the references to color in this gure legend, the reader is referred to the web version of this article.)



13/17

dropsto 84% of the maximum value measured in the leading bubble

cycle. Then, the transit of the trailing bubble increases the value ofh

to a maximum of 3571 W/m2K (60% higher than the local single

phase value), leading to a time-averaged value for the second

bubble of 3195 W/m2K (43%). The better heat transfer performance

measured for the trailing bubble cycle with respect to the leading

bubble (w20%) is a direct consequence of the overlap of the per-

turbations generated by the bubbles on the local thermal eld. Note

that, at the operating conditions simulated, the temperature eld

takes around 20 ms to achieve a steady-state situation, which is

much more than the residence time of the bubbles and the trapped

liquid slug atzh/D 4 (12.5 ms). By analyzing axial locationsfurther

downstream to the microchannel, the plots inFig. 11suggest that

the heat transfer coefcient decreases as the ow develops ther-

mally. The drop of the heat transfer in the trapped liquid slug region

becomes less evident, rstly because the residence time of the

liquid slug is decreasing due to the higher velocity of the bubbles,

such that the thermal boundary layer has less time to restore itself.

Secondly, the liquid within the trapped slug is accelerating due to

the growth of the trailing bubble, so the Peclet number of the liquid

slug is increasing and this turns into a higher Nusselt number due

to the thermally developing conditions. Since the heat transfer

coefcient stays high even during the transit of the liquid slug, thepeak and the average of the heat transfer performance for the

trailing bubble cycle increases with respect to the leading bubble

cycle and to the liquid-only case. After 21 heated diameters, the

time-averaged heat transfer coefcient for the bubble ahead has

grown to 24% over the local single phase value (1447 W/m2K

against 1171 W/m2K) while for the trailing bubble cycle the average

coefcient (2343 W/m2K) has become twice the liquid-only value

(and w60% higher than the leading bubble). Thus, for a real slug

ow in microchannels, multiple bubbles must be simulated to

emulate the heat transfer process, not single bubbles.

To better illustrate the heat transfer coefcient trends as theow develops thermally along the channel, Fig. 12 depicts the

time-averaged heat transfer coefcient versus the axial coordinate

for the two-phase and single phase cases. This plot shows that,within a thermal entry region, the heat transfer performance for

both the bubble cycles varies along the streamwise direction as an

inverse function of the axial coordinate, similarly to the liquid-only

case. The prole of the heat transfer coefcient for the trailing

bubble cycle reported in Fig. 12 is shifted to much higher values

than that of the leading bubble, thus conrming that due to the

overlap of the effects of the bubbles transits on the thermal eld,

the trailing bubble cycle enjoys a signicantly better heat transfer

performance.

Fig. 13 displays the time-averaged two-phase to single phase

heat transfer coefcient along the axial coordinate for both the

bubble cycles. The enhancement of the heat transfer performance

by the leading bubble grows in the streamwise direction due to a

more efcient liquid-wall heat convection in the liquid region

ahead of the bubble, which is improved by the higher velocity of

the liquid accelerated by the evaporation phenomenon. However,

this heat transfer mechanism has little effect on the average per-

formance of the bubble cycle, and as a consequence the average

heat transfer coefcient for the leading bubble increases only from

18% to 24%. In contrast, due to the already mentioned uid dy-

namics occurring within the trapped liquid slug, the trailing bubble

cycle exhibits a two-phase to single phase heat transfer coefcient

which rises steeply along the axial direction andFig. 13suggests it

exceeds the value 2 (100%) forzh/D> 21.

Consolini and Thome[11]measured the boiling heat transfer

coefcient for R245fa and similar channel diameter (0.51 mm)and saturation temperature (31 C). For a vapor quality of 0.02,

they reported a heat transfer coefcient of about h 800 W/m2K

forG 305 kg/m2s andq 3 kW/m2 and abouth 2100 W/m2K

forG812 kg/m2s andq14 kW/m2. The values of the mass ux

and heat ux set in the present simulation are between the

range of values set in this experiment. By considering the time-

averaged coefcient computed for the trailing bubble, this value

is 2343 W/m2K, to be representative for the computational case,

it is slightly above the experimentally measured range. However,

in the experimental study the measurements are performed in

thermally fully developed conditions (after 130 heated di-

ameters), while in the present simulation the ow is still ther-

mally developing (after 21 heated diameters). By extrapolating

the plot ofFig. 12to higher values ofzh/D, it is realistic to assumethat the averaged heat transfer coefcient for the trailing bubble

in fully developed conditions will be well-within the range

measured in Ref. [11].

5 10 15 201000

1500

2000

2500

3000

3500

zh/D

h[W/m

2K]

Liquidonly

Leading bubble

Trailing bubble

Fig. 12. Time-averaged heat transfer coefcient along the axial coordinate. The heat

transfer coef

cient for the liquid-only fully developed

ow is 769 W/m

2

K.

5 10 15 201

1.2

1.4

1.6

1.8

2

2.2

zh/D

htp

/hsp

Leading bubble

Trailing bubble

Fig. 13. Enhancement of the two-phase heat transfer coefcient with respect to the

local single phase value.



14/17

4.3.4. Heat transfer modeling

A theoretical heat transfer model is proposed here to emulate

the numerical results and is schematically represented inFig.14. At

a given axial location, the two-phase ow shows the alternate

transit of a vapor bubble and a liquid slug. In Ref. [24] the heat

transfer in the vapor bubble region was modeled by assuming one-

dimensional unsteady heat conduction across the liquid lm sur-

rounding the bubble. Since dR, the curvature of the channel and

bubble interface was neglected and the one-dimensional Fourier

equation for heat conduction was solved for a vertical coordinate y

ranging from the channel wall (y 0) to the bubble interface

(y d). With constant heat ux at the channel wall and tempera-

ture xed at the saturation value at the bubble interface as

boundary conditions, the solution of the thermal problem with a

constant liquid lm thickness gave the following analytical

expression for the heat transfer coefcient:

ht lld

1

1 llqd

XNm 1

cmYmdexp

at;lb2mtt0

(19)

where at,l is the liquid thermal diffusivity, bm and Ym represent

respectively the m-th eigenvalue and eigenfunction of the spatial

solution of the thermal problem and cm is a constant which ac-

counts for the initial temperature prole within the lm. The heat-

conduction controlled stage for the heat transfer is considered tobegin att0.

In Section4.3.2, it was shown that the liquid slug can be split

into an adherent lm and a bulk recirculating ow, as sketched in

Fig.14. Due to the no-slip condition at the channel wall, the velocity

inside the adherent lm is assumed to be negligible such that it is

considered stagnant. Hence, the heat transfer can be modeled by

assuming one-dimensional unsteady heat conduction across the

adherent lm. In this case, the boundary condition which applies at

the interface between the adherent lm and the recirculating ow

region is:

llvT

vy hsTTs (20)

wherehsis the heat transfer coefcient between adherent lm andrecirculating ow and Ts is a reference temperature for the liquid

slug. By considering a constant depth of the adherent lmds, the

heat transfer coefcient along the liquid slug region can be esti-

mated as:

wherecm,bmandYmdiffer from those in Eq.(19)due to the change

in the boundary conditions.

In the numerical simulation discussed earlier, there are two

vapor bubbles and two liquid slugs, one trapped between the

bubbles and onewhich follows the trailing bubble. An estimation of

the heat transfer coefcient at a given axial location during the

whole simulation is obtained by applying respectively Eq. (19)or

Eq.(21)when a vapor bubble or a liquid slug is passing. For each

bubble or liquid slug region, t0, d and ds are taken from the simu-

lation. The initial time instantt0for Eq.(21)is the instant at which

the bubble rear crosses the axial location under analysis zh, i.e.

t0 tzb;R zh. In the simplest implementation of the model, t0in Eq.(19)is the time instant at which the bubble nose crosseszh,

i.e. t0 tzb;N zh, and the slug temperature is taken equal to

the saturation temperature. The temperature prole at t0 for the

liquid lm region of the leading bubble is exported from the

simulation results. The initial temperature proles for the succes-

sive regions are obtained by the model itself. The heat transfer

coefcienthsfor the liquid slug region is obtained by means of the

following correlation proposed by He et al. [55]:

hs ll

D24:7 0:54Pe0:45Ls=D1:34 (22)

where the liquid Peclet number (Pe rcpUD/l) is computed by

referring to the average velocity of the liquid within the slug. It is

assumed that Ls/D/N for the terminal liquid slug as it is innitely

extended.

Fig. 15 shows the comparison of the heat transfer coefcient

given by the simulation and that predicted by the model discussed

so far (Model1 curve) at zh/D 21. The analytical model is able to

represent very well the heat transfer coefcient trend, which yields

the rise in the heat transfer in the vapor bubble region, the plateau

in the trapped liquid slug region, the peak and the subsequent

decrease in the liquid region behind the trailing bubble. As the

vapor bubbles are crossing the axial location under analysis, the

model, which does not account for the actual decreasing of the

liquid lm thickness, suggests that the heat transfer performanceincreases because the temperature prole within the lm evolves

T=Tsat

s

liquid filmregion

liquid slugregion

Ty

l s=h (TT )s

z

y

liquid filmadherent film

vapor bubblerecirculating zone

R

Fig. 14. Scheme of the decomposition of the

ow

eld within the microchannel. Sinced

R, the radial coordinate is here replaced by the vertical coordinate y.

ht

ll

ds

1

1 llqds

XNm 1

cmYmdsexp

at;lb2mtt0

llhsds

llqds

TsTsat(21)



15/17

toward an asymptotic steady-state situation. As time elapses, the

sum at the denominator of Eq. (19)tends to zero, thus identifying

the largest heat transfer coefcient achievable in the vapor bubble

region as h l l/d, i.e. that given by one-dimensional steady-state

heat conduction across the trapped liquid lm. This suggests that,

not only the thinning of the lm itself, but also the time-evolution

of the transient thermal phenomenon occurring within the liquid

lm contributes to the increase of the heat transfer coefcient as

the bubble is crossing the axial location under analysis.

The positive deviation observed between the numerical results

and the prediction (Model1) arises from the assumptions that the

heat-conduction-controlled stage for the heat transfer in the bub-ble region begins whenzb,Nzhand that the temperatureTsof the

liquid within the recirculating region of the slug is constant. Since

the heat-conduction-controlled stage starts when a liquid lm has

been formed between the bubble and the wall, the former

assumption speeds up the growth of the heat transfer coefcient

compared to the numerical results. The latter assumption leads to

the underestimation of the drop of the heat transfer in the liquid

slug as Ts is actually increasing as time elapses. Nevertheless, the

prediction of the magnitude of the heat transfer coefcient within

each region given by the theoretical model is satisfactory, yielding

average errors respectively of 8% and 18% for the leading and

trailing bubble regions and 11% for the trapped liquid slug zone.

The prediction of the heat transfer given by the theoretical

model can be improved by delaying the beginning of the heat-conduction-controlled stage for each vapor bubble region and by

introducing an appropriate estimation for the time-law of the

temperature within the liquid slugs. The magnitude of the delay

depends on the shape of the nose of the bubble and it is presently

set as the time that it takes for the nose of the bubble to ow two

diameters downstream to zh:

t0 tzb;N zh

2D

Ub;N(23)

whereUb,Nis the velocity of the nose of the bubble. The tempera-

ture within the slug is estimated by means of an energy balance

between the rear of the bubble at the downstream end of the slug

and thezhlocation:

Ts TsatqUb;Rtt0

Gcp;lD (24)

whereUb,Ris the velocity of the rear of the bubble. The improved

implementation Model2still uses Eqs.(19) and (21)to estimate the

heat transfer coefcient, but it includes Eqs. (23) and (24) to

compute respectively the initial time instant for each bubble region

and the temperature within the slug. The curve Model2 in Fig. 15shows that the improved model matches closely the simulation,

providing an effective theoretically-based explanation of the ther-

mal mechanisms governingow boiling in the slug ow regime in

microchannels.

5. Conclusions

A numerical framework based on the commercial CFD solver

ANSYS Fluent and the multiphase VOF method, along with a self-

implementation of a Height Function algorithm and an evapora-

tion model, were employed to investigate the thermal-hydraulic

details of ow boiling of single and multiple elongated bubbles

within a horizontal circular microchannel. The dynamics of the

evaporating bubbles, the wall heat transfer performance induced

by the transit of the bubbles, the uid and thermal dynamicsoccurring within the liquid slug separating the bubbles, and that

within the liquid lm trapped between the bubble interface and

the channel wall, were the object of this work. Since the liquid

slug and trapped liquid lm regions are two of the three zones in

which a liquid sluge bubble unit can be split according to Thome

et al.[1], the numerical results obtained offer new insight into the

governing heat transfer mechanisms of evaporation in micro-

channels, which can potentia

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Magnini 2013 International Journal of Thermal Sciences