Magnini Et Al. 2013 Numerical Investigation of Hydrodynamics and Heat Transfer of Elongated Bubbles...

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Numerical investigation of hydrodynamics and heat transfer of elongated

bubbles during flow boiling in a microchannel

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

a Dipartimento di Ingegneria Energetica, Nucleare e del Controllo Ambientale, Universit di Bologna, Bologna, Italyb Laboratory of Heat and Mass Transfer (LTCM), Ecole Polytechnique Fdrale de Lausanne (EPFL), Lausanne CH-1015, Switzerland

a r t i c l e i n f o

Article history:

Received 29 June 2012

Received in revised form 21 November 2012

Accepted 2 December 2012

Available online 23 January 2013

Keywords:

Flow boiling

Microchannel

Volume Of Fluid

Evaporation

Heat transfer

a b s t r a c t

Flow boiling within microchannels has been explored intensively in the last decade due to their capabil-

ity to remove high heat fluxes from microelectronic devices. However, the contribution of experiments to

the understanding of the local features of the flow is still severely limited by the small scales involved.

Instead, multiphase CFD simulations with appropriate modeling of interfacial effects overcome the cur-

rent limitations in experimental techniques. Presently, numerical simulations of single elongated bubbles

in flow boiling conditions within circular microchannels were performed. The numerical framework is

the commercial CFD code ANSYS Fluent 12 with a Volume Of Fluid interface capturing method, which

was improved here by implementing, as external functions, a Height Function method to better estimate

the local capillary effects and an evaporation model to compute the local rates of mass and energy

exchange at the interface. A detailed insight on bubble dynamics and local patterns enhancing the wall

heat transfer is achievable utilizing this improved solver. The numerical results show that, under operat-

ing conditions typical for flow boiling experiments in microchannels, the bubble accelerates downstream

following an exponential time-law, in good agreement with theoretical models. Thin-film evaporation is

proved to be the dominant heat transfer mechanism in the liquid film region between the wall and the

elongated bubble, while transient heat convection is found to strongly enhance the heat transfer perfor-

mance in the bubble wake in the liquid slug between two bubbles. A transient-heat-conduction-basedboiling heat transfer model for the liquid film region, which is an extension of a widely quoted mecha-

nistic model, is proposed here. It provides estimations of the local heat transfer coefficient that are in

excellent agreement with simulations and it might be included in next-generation predictive methods.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

Microscale flow boiling is one of the most promising cooling

technologies to dissipate high heat fluxes from microprocessors.

The two-phase cooling, applied directly on the chip through micro-

channels evaporators, is nowadays succeeding in removing more

than 300 W/cm2 from the electronic chip itself. Besides the capa-

bility of removing high heat power densities, Agostini et al.[1]ar-gued that the main advantages of two-phase flow boiling heat

transfer compared to other high heat flux cooling methods are:

lower mass flow rate of the coolant due to the high energy absorp-

tion by the latent heat of vaporization, lower pressure drop due to

this lower mass flow rate, lower temperature gradients due to sat-

urated flow conditions and the heat transfer coefficient increases

with heat flux. Within microchannels, once nucleation begins at

one location, the vapor bubble grows rapidly and fills the entire

cross-section of the channel. Hence, bubbly flows are suppressed

already at low values of vapor quality, while slug (elongated bub-

ble) and annular flow regimes occupy a large area on the flow map

[2]. In particular, slug flows lead to very efficient heat transfer

mechanisms due to the following local flow structures: the recircu-

lating flows within the liquid slugs enhance heat and mass transfer

from the liquid to the wall; the large interfacial area promotes li-

quidvapor mass transfer; the presence of the bubbles separatingthe liquid slugs prevents the flow to become thermally fully devel-

oped and shorter liquid slugs lead to higher local Nusselt number

[3]; the evaporation of the thin liquid film surrounding the bubble

increases strongly the local heat transfer coefficient[4].

Due to the importance of the slug flow regime in microchannels,

the availability of reliable predicting methods for boiling heat

transfer and pressure drop is fundamental for industrial manufac-

turing of cooling systems. Models and correlations developed for

the macroscale do not apply well when extrapolated to channel

sizes below 1 or 2 mm, thus highlighting the presence of a macro-

to microscale transition. Therefore, in the last decade, researchers

have focused on the development of new physically-based models

0017-9310/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12.010

Corresponding author. Address: via Terracini, 34, 40128 Bologna, Italy. Tel.: +39

051 2090541.

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

International Journal of Heat and Mass Transfer 59 (2013) 451471

Contents lists available atSciVerse ScienceDirect

International Journal of Heat and Mass Transfer

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 o m / l o c a t e / i j h m t
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12.010mailto:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12.010http://www.sciencedirect.com/science/journal/00179310http://www.elsevier.com/locate/ijhmthttp://www.elsevier.com/locate/ijhmthttp://www.sciencedirect.com/science/journal/00179310http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12.010mailto:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12.010


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accounting for the actual two-phase flow structure. However, the

contribution of experiments to the understanding of the local fea-tures of the two-phase flow is still limited, due to the small scales

involved, while multiphase CFD techniques are emerging as pow-

erful tools to provide detailed and interesting insights into the local

hydrodynamics and wall heat transfer.

2. Literature review

2.1. Experiments on flow boiling in microchannels

Excellent reviews on microchannel flow boiling are available in

Garimella and Sobhan[5], Bertsch et al. [6]and in the Wolverine

Engineering Data Book III[7]written by Thome.

Agostini and Thome [8] analyzed 13 published experimental

studies on flow boiling heat transfer in microchannels. These stud-ies reported a broad agreement on the increasing of the heat trans-

fer coefficient with heat flux, weak effect of the mass flux, but

conflicting trends with the vapor quality. This suggested to theauthors that additional phenomena, negligible in the macroscale,

must come into play in microchannels.

Due to the substantial heat flux dependency of the heat transfer

coefficient which is typical of a nucleate boiling controlled regime

in the macroscale, many authors concluded that nucleate boiling is

the governing heat transfer mechanism in the microscale as well.

However, Thome observed in [7] that there is not any experimental

proof to conclude that nucleate boiling is the prevalent regime in

microchannels, hence he advised against the application of macro-

scale ideas to derive microscale flow boiling methods.

Bertsch et al.[6]compared predictions of 25 published correla-

tions for flow boiling heat transfer against 10 independent data

sets from the published literature. They reported that models

developed specifically for the microscale gave no better resultsthan those for conventional channels, and that Coopers pool

Nomenclature

Roman LettersA areaa accelerationB generic interfacial effectBoa bubble acceleration Bond number

qaD2

r b generic fluid property

Ca capillary number lUr

Co confinement number r

gDqD2

h i1=2

Cr Courant number DtV=PNf

f ufnfAf

!

cp constant pressure specific heatD diameterFr surface tension force vectorf interface lineG mass fluxg gravity acceleration vectorH height functionh heat transfer coefficient

hlv latent heatI VOF indicator functionL lengthLa adiabatic lengthLh heated lengthLs slug lengthL2, L1 error normM molecular weight_m interphase mass transfer_mg global rate of vapor creation_mi mass flow rate across the inlet section_mo mass flow rate across the outlet section

N number of computational cellsn interface unit norm vectorp pressure

p dimensionless pressureq heat fluxqe evaporation equivalent heat fluxR radiusRe Reynolds number qUDl

Rg universal gas constantr,z cylindrical reference frameT temperaturet timeU velocity

u velocity vectoru dimensionless velocity vectorV volumeWe Weber number qU

2Dr

x,y Cartesian reference framex position vectorxS interface position vectorY eigenfunctionzG axial position of the center of gravityzh axial distance from the entrance in the heated region

Greek Lettersa volume fractionat thermal diffusivityb eigenvaluec accommodation coefficientD mesh element sized liquid film thicknessdS delta-function

dT thermal boundary layer thicknessj interface curvaturek thermal conductivityl dynamic viscosityn Scriven model growth constantq densityr surface tension coefficient/ interface kinetic mobility

Subscripts0 initial conditions1, 2 primary, secondary phaseb bubblec computational cell centroid valueex exact value

f computational cell face centroid valueif interfaciall liquidN bubble nosesat saturationsp single phasetp two-phasev vaporw wallz, zz first, second order derivatives with respect to z1 far system conditions

452 M. Magnini et al. / International Journal of Heat and Mass Transfer 59 (2013) 451471


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boiling correlation[9]provided the overall best prediction. How-

ever, Coopers correlation was able to predict only the 48% of the

data set with a deviation within 30%. Thus, Bertsch and co-

authors remarked the clear need for additional research into the

mechanism of flow boiling in small channels.

Thome et al.[4]developed a three-zone heat transfer model for

the evaporation of elongated bubbles in microchannels. The model

assumes that the evaporation of the thin liquid film trapped

between the bubble and the channel wall is the dominant heat

transfer mechanism rather than nucleate boiling and estimates a

time-averaged heat transfer coefficient in the liquid film zone by

one-dimensional steady-state heat conduction across the film.

The bubble is modeled as a trapezoid and empirical correlations

are used to evaluate its size, which in turn lead to five adjustable

empirical parameters. Recent papers [1012] have eliminated

one of these, finding that the film dryout thickness can be set to

the measured wall roughness. The three-zone model is very sensi-

tive to the set of parameters chosen, this has led to very good pre-

dictions [1315] as well as poor ones [6,12] when employed to

predict independent flow boiling heat transfer databases.

Recently, Harirchian and Garimella [12] employed the Thome

et al. three-zone model [4] to estimate boiling heat transfer in

the slug flow regime, as part of a comprehensive flow regime-

based heat transfer model. They modified the three-zone model

original correlations for initial and minimum film thickness. With

a new set of empirical parameters, the percentage of data predicted

within 30% increased from 36% obtained with the original

three-zone model to 82% of the new one with the proposed

modifications when fit to their data set for FC-72. Han et al. [16]

performed liquid film thickness and wall temperature measure-

ments under flow boiling conditions for water and ethanol and

they found a good agreement between the heat transfer coeffi-

cient calculated from measured liquid film thickness and that

obtained directly from wall temperature measurements. This

confirmed that thin-film evaporation played a dominant role on

the heat transfer within microchannels.

The three-zone model approach, based on thin film evaporationas the prevalent heat transfer mechanism and attempting to recon-

struct the actual flow configuration, laid the foundation for a more

reliable physics-based modeling of microscale flow boiling. How-

ever, new generation methods cannot prescind from accurate mod-

els for the macro-to-micro transition, flow patterns and geometry

of the liquid-vapor interface. Kew and Cornwell [17]for example

recognized the unimportance of gravitational forces as a peculiar

effect of microscale conditions. They observed that heat transfer

and flow characteristics deviate considerably from macroscale

trends when the Confinement number Co rgDqD2

1=2had values

above 0.5. Recently, Ong and Thome[18]measured the film thick-

ness above and below elongated bubbles in flow boiling conditions

and observed that gravity forces are fully suppressed, such that theflow is symmetric, when Co > 1. Revellin and Thome [19] and

Harirchian and Garimella [20] proposed diabatic vapor quality ver-

sus mass flux flow pattern maps, with transition lines captured

through mechanistic models in order to quantitatively distinguish

the different flow regimes. The accurate estimation of the liquid

film thickness d surrounding the bubble in the slug flow regime

is fundamental for boiling heat transfer models based on thin film

evaporation[4,12], as the local heat transfer coefficient in the film

region is computed as h= k/d. Han and Shikazono performed a

large experimental study of liquid film thickness measurements

for bubbles in steady motion [21] and under acceleration [22].

By applying a scaling analysis to the forces acting on the bubble

to fit their experimental data, they proposed the following

relationship to estimate the film thickness in laminar flow

conditions:

d

D

min d

D

steady

; d

D

accel

" # 1

whered is the liquid film thickness at the beginning of the flat film

region which follows the bubble nose region. Values of the thick-

ness under steady and accelerated conditions are obtained by the

following empirical expressions:

d

D

steady

0:67Ca2=31 3:13Ca2=3 0:504Ca0:672Re0:589 0:352We0:629

2

d

D

accel

0:968Ca2=3Bo0:414a

1 4:838Ca2=3Bo0:414a3

where the Capillary number Ca lUr , the Reynolds number Re qUDl ,the Weber number We = Ca Re and the bubble acceleration Bondnumber Boa qaD

2

r have to be evaluated at the actual bubble

velocity.

2.2. Numerical simulations of flow boiling within microchannels

Recent advances on multiphase computational fluid dynamics

allow numerical solution of boiling flows within microchannels

to be performed, thus providing essential information on the local

structure of the flow. Interface capturing techniques for fixed com-

putational grids, such as Level Set (LS) [23] or Volume Of Fluid

(VOF)[24]methods, are emerging as one of the best mathematical

and numerical treatments of multiphase flow physics due to their

easiness of implementation, accuracy and robustness of the

algorithms.

Talimi et al. [25] provided a comprehensive review of numerical

studies concerning adiabatic and diabatic slug flow in microchan-

nels without phase change. The simulations of slug flows with heat

transfer reported impressive enhancement of the wall heat transfer

performance along the liquid film region, as well as remarkable in-

crease of heat transfer coefficients in the wake behind the bubble,

due to local recirculation patterns forced by the bubble motion.

Mukherjee and Kandlikar [26]simulated the flow boiling of awater vapor bubble within a square microchannel, by use of a LS

method to track the interface. They studied the bubble growth rate

for different liquid superheats and flow velocities and observed

that the vapor bubble grew spherically with a linear timelaw

for the growth rate until it approached the channels walls. Subse-

quently, the bubble stretched and generated a thin liquid film,

eventually forming some dry patches, while the growth rate

time-law became exponential.

Suh et al. [27] studied, by means of a LS method, the bubble

dynamics and the associated flow and heat transfer in parallel

microchannels, in order to investigate the conditions leading to

flow reversal. They showed that backflows may occur in parallel

microchannels when the bubble formation is not simultaneous in

adjacent channels. This leads to a drop in the heat transfer perfor-mance at the wall of the channel where reversed flow occurs and

such an instability is boosted by higher wall superheats and smal-

ler contact angles.

Mukherjee[28], simulating the flow boiling of a bubble in con-

tact with the heated surface of a microchannel, investigated the

role of advancing and receding contact angles between the bubble

interface and channel wall. He reported that the wall heat transfer

is improved by a smaller contact angle, as it promotes the forma-

tion of a thin liquid layer trapped between the bubble interface

and the channel wall, thus indicating that thin film evaporation

is the primary wall heat transfer mechanism in microscale flow

boiling as proposed earlier in[29].

Mukherjee et al. [30] performed a parametric study to assess

the influence of wall superheat, Reynolds number, surface tensionand contact angles on bubble growth rate and wall heat transfer.

M. Magnini et al. / International Journal of Heat and Mass Transfer 59 (2013) 451471 453


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They observed that an increase of the wall superheat favored the

bubble nose to move downstream to the channel, allowing a longer

liquid film to exist thus increasing the wall heat transfer. The Rey-

nolds number was observed to have little effect on the flow, be-

cause the velocities associated to the evaporation phenomena

were much higher than that of the liquid inflow.

Zuet al.[31] carried out numerical simulations of bubble nucle-

ation, detachment and then flow boiling within a rectangular

microchannel by employing the VOF method included in the com-

mercial CFD software ANSYS Fluent. The phase change at the li-

quidvapor interface due to evaporation was modeled by

implementing an approximate method based on the concept of

pseudo-boiling. Bubble deformation while growing at the wall

and its trajectory during partially confined growth were success-

fully compared with experimental results and 1-D theoretical

models.

Zhuan and Wang[32]studied flow patterns and related transi-

tions within circular microchannels in flow boiling conditions by

means of a VOF algorithm. They explored the effect of bubble

lift-off size, heat flux, mass velocity, frequency of bubble genera-

tion and fluid properties on the location of transition lines from

bubbly flow to semi-annular flow and obtained results in good

agreement with the Revellin et al. [2] experimental findings.

The direct numerical simulation of the interface has intrinsic

limitations related to the specific interface capturing scheme

adopted. Level Set methods have issues with mass conservation

while the Volume Of Fluid approach tends to suffer from poor esti-

mation of the interface topology, which is involved in the surface

tension force calculation.

In the microscale, the interface temperature condition has to ac-

count for the interfacial resistance to mass transfer and the Lapla-

cian jump in pressure across the interface [33]. Microscale effects

such as disjoining pressure and microlayer evaporation, which

act at length scales several orders of magnitude smaller than typ-

ical mesh element sizes, should be modeled and coupledwith mac-

roscopic calculations when dealing with very thin liquid films or

wall adhesion [34,35] and dynamic contact angles at the solidliquidvapor three-phase contact line should be assigned.

Boiling flows require typically very fine computational grids,

such that computations become extremely time-consuming, espe-

cially when performing three-dimensional simulations.

3. Objectives and methodology

The objective of this study is to analyze the hydrodynamics and

heat transfer of slug flow in microchannels in flow boiling condi-

tions. This is accomplished by performing numerical simulations

of a single vapor bubble flowing within a heated channel and

growing as a consequence of evaporation of liquid at the interface.

An elongated bubble at saturation conditions is patched at the up-stream of a horizontal circular microchannel. The dynamics of the

bubble during evaporation, the thermal and flow fields within the

channel and the local variations of the heat transfer coefficient at

the heated wall are investigated, for different refrigerant fluids

and operating conditions. The effect of the fluid properties on the

bubble growth rate and the governing heat transfer mechanisms

along bubble and bulk liquid regions are explored. The results of

the computations suggest modifications to the above mentioned

three-zone model for boiling heat transfer in microchannels.

Simulations are performed by means of the finite-volume com-

mercial CFD solver ANSYS Fluent version 12 where the solvers de-

fault VOF algorithm is adopted to capture the interface. In order to

overcome VOF issues on poor interface reconstruction, a Height

Function algorithm [3639] is implemented by self-developed sub-routines to replace ANSYS Fluent default estimation of the surface

tension force. An evaporation model is introduced within the sol-

ver through additional subroutines to estimate the rates of mass

and energy exchange at the interface due to evaporation. The evap-

oration model allows the interface temperature to deviate fromthe

saturation condition, according to a physical model developed by

Schrage[40]for interphase mass transfer. Dryout and microlayer

effects are avoided in our simulations by appropriate choice of

the operating conditions for each case. Gravitational effects are

made negligible by choosing operating conditions which lead to

Co> 1[18]. This allows a two-dimensional axisymmetrical formu-

lation of the flow problem, such that the entire computational ef-

fort is aimed to very fine mesh grids and long channels, up to 72

diameters. Abundant use of parallel computations were imple-

mented to decrease the computational time, using up to 128 pro-

cessors for the simulation run with the longest channel.

4. Numerical framework

4.1. The VOF method

The two-phase flow problem is formulated through a single-

fluid approach, such that a unique velocity, pressure and tempera-ture field is shared among the phases. A single set of flow equations

is written and solved throughout the domain and the phases are

treated as a single fluid, whose properties change abruptly across

the interface. The flow problem, along with boundary conditions,

is similar to that of a single phase flow; however additional

arrangements are necessary: definition of a marker function to

identify each fluid, a method to update the marker function as

the interface evolves, mathematical modeling of interfacial effects

and discretization on the computational grid.

The Volume Of Fluid method defines a marker function I(x,t) as

a multidimensional Heavyside step function with the value 1 in the

primary phase and 0 in the secondary phase. The discrete version

of the indicator function is the volume fractiona, obtained by inte-gration ofI(x,t) over the computational cell of volume V:

a1V

ZV

Ix; tdV 4

The so-defined volume fraction represents the ratio of the cell vol-

ume occupied by the primary phase. It is 1 if the cell is filled with

the primary phase, 0 if filled with the secondary phase and

0 < a < 1 for an interfacial cell with both phases inside. The genericfluid property b for every domain cell can be expressed in terms ofaas follows:

bb2 b1b2a 5where b1andb2are primary and secondary phases specific proper-

ties. Since the volume fraction is transported as a passive scalar by

the flow field, its values can be updated by solving a transport equa-tion. The interfacial effects are modeled as delta functions concen-

trated at the phases interface. By referring to dS= d(x xS) as amultidimensional delta function which is non-zero only on inter-

face points xS, the generic interfacial effect B(x) is introduced within

the flow equation as the source term B(x)dS. According to the VOF

approach, the delta function is represented in the computational

grid as dS= jraj. Sincera 0 on the few layers of cells layingacross the interface, the interface is meant as a transition region

with finite thickness, where interfacial effects are concentrated

and the fluid properties vary according to Eq. (5).

4.2. Governing equations

In this work both phases are always assumed incompressible. Itis worth to note that the density of the vapor phase may decrease



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considerably in the streamwise direction due to the pressure drop

which occurs within the microchannel, but the operating condi-

tions simulated in the present work made such effect negligible.

Hence, the mass conservation equation for an incompressible flow

is expressed as:

ru 1q

v

1ql _mdS

1

qv

1ql _mjraj 6

where the r.h.s. of Eq. (6) accounts for the fluid expansion due to

phase change by introducing the interphase mass flux _m. The for-

mulation and computation of _mis the task of the evaporation mod-

el, which is presented in Section 4.3. Note that, within the bulk

phases, the Eq.(6)reduces to the well-known divergence-free con-

dition for the velocity field r u= 0.To evolve the interface location, the following volume fraction

conservation equation for flow with phase change is solved:

@a@t

r au 1q

v

_mdS 1qv

_mjraj 7

where a represents the vapor volume fraction. Only the primary

phase volume fraction Eq.(7) is solved, while the secondary volumefraction field (the liquid phase in this case) is obtained as 1 a atthe end of the calculation.

The single-fluid momentum conservation equation for Newto-

nian fluids in laminar flow takes the following form:

@qu@t

r quu rpr lruruT qgFr 8where Fr is the surface tension force. By means of the Continuum

Surface Force (CSF) method proposed by Brackbill et al. [41], the

capillary force is converted into a volume force:

FrrjndSrjnjraj 9which is concentrated only at interfacial cells, where

ra 0. n

identifies the interface unit norm vector. The surface tension coeffi-

cient ris considered constant. The local interface curvature j is notavailable explicitly in the VOF method, but it is implied in the vol-

ume fraction field. Various approaches exist to estimate the curva-

tures from the volume fractions, see Cummins et al. [36] for

reference. In this work ANSYS Fluents default scheme is replaced

by a self-implementation of the Height Function interface recon-

struction algorithm, which is briefly introduced in Section 4.4.

The set of flow equations is completed by the energy conserva-

tion equation:

@qcpT@t

r qcpuT r krT _mhlv cp;vcp;lTjraj

10

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

evaporation _mhlv, with h lv being the latent heat of vaporization,the enthalpy of the vapor created _mcp;vTand that of the liquid re-

moved _mcp;lT, withcp being the constant pressure specific heat.The energy equation(10)does not include the viscous heating

term. Following the dimensional analysis proposed by Morini

[42], the viscous heating contribution was estimated here on the

fluid bulk temperature for a single phase flow under operating con-

ditions representative of the cases simulated. It was found to be of

the order of 104 compared with the rise in temperature generatedby the wall heat flux, and therefore the viscous heating effect is

negligible in our simulations. The variation of the fluid tempera-

ture in the simulations performed is sufficiently small such that

the fluid specific properties are considered constant throughoutthe flow domain.

4.3. Evaporation model

The task of the evaporation model is to provide an estimation of

the local interphase mass transfer _mdue to the evaporation of the

liquid at the interface, depending on the local temperature field.

The foundation of the evaporation model is the interfacial condi-

tion assigned to the temperature. Schrage [40] assumed that, at

the interface, vapor and liquid temperatures are at their thermody-

namic equilibrium saturation values, but he supposed an interfa-

cial jump in the temperature to exist, such that at the interface

Tsat(pl) = Tl Tv = Tsat(pv). Schrage applied the kinetic theory of

gases to express the net flux of molecules crossing the interface

due to the phase change, as a function of the temperature and pres-

sure jumps. When phase change occurs, a fraction c of the mole-cules from the bulk phase strike and cross the interface, that is

evaporate or condense, while the fraction 1 c is reflected. Theevaporation and condensation fractions ceandccare often consid-ered equal and are referred to as the accommodation coefficient.

The net mass flux across the interface _mis given by the difference

in the liquid-to-vapor and vapor-to-liquid mass fluxes and accord-

ing to[43]is given by the following expression:

_m 2c2 c

M2pRg

1=2 pvffiffiffiffiffiffi

Tvp plffiffiffiffi

Tlp

11

where Mis the molecular weight and Rg= 8.314 J/mol K is the uni-versal gas constant,pvandTvare the vapor pressure and tempera-

ture at the interface, pl and Tl are the liquid pressure and

temperature at the interface.

The accommodation coefficient is difficult to be measured

experimentally and it is known only for a few liquids, with a large

degree of uncertainty. Marek and Straub [44] analyzed the pub-

lished data for water and reported values in the range from 103

to 1. Rose [45] performed a review of experimental results on drop-

wise condensation and concluded that the most reliable values for

c were close to unity. Wang et al. [46] showed that non polar-

liquids have an experimentally determined accommodationcoefficient of unity. As it will be discussed in the Section 5.3, we

found the best agreement with analytical solutions by settingc = 1.Tanasawa [47] assumed that for small interface temperature

jumps, such that (Tv Tif) Tv, the interphase mass flux dependslinearly on the temperature jump between the interface and the

vapor phase:

_m 2c2 c

M

2pRg

1=2 qvhlvTifTv

T3=2v

12

The operating conditions simulated in our work involve small

Laplacian pressure jumps across the interface, such that at the inter-

face Tv Tsat(p1) withp1being the system pressure. Therefore, theevaporation model implemented in the numerical framework com-

putes the interphase mass transfer through the following modifiedversion of the Tanasawa expression(12):

_m/TTsatp1 13where / is the so-called kinetic mobility:

/ 2c2 c

M

2pRg

1=2 qvhlv

T3=2satp114

and 1//can be meant as the interfacial resistance to mass transfer.

In Eq.(13), the kinetic mobility and the saturation temperature are

constant throughout the domain, and therefore the evaporation

model computes the rate of mass transfer at the interface propor-

tional to the local interface superheating. For each superheated

interface cell, an amount _

mjraj of vapor is created and the samemass of liquid disappears. The latent heat of the evaporating liquid



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is subtracted from the energy stored within the cell, such that the

temperature drops locally to a value close to the saturation condi-

tion. Therefore, the interface temperature is always equal or little

above the saturation value.

The mass and energy source terms at the r.h.s. of Eqs. (7) and

(10)are concentrated at the 23 cells laying across the interface.

Such source terms localized in a narrow region may give rise to

numerical instabilities when the rate of mass and energy produc-

tion is high. Hence, a mathematical procedure was implemented

to smear the source terms over a few cells across the interface,

as proposed by Hardt and Wondra[48]. The solution of a diffusion

equation for the original evaporation rate provides a smooth evap-

oration rate field. Then, the use of volume fractions and normaliza-

tion factors allows the newly created vapor to concentrate on the

vapor side of the interface and liquid to disappear on the liquid

side, always ensuring global mass conservation. The reader is re-

ferred to [48] for a detailed description and validation of the proce-

dure. In order to test the entire evaporation model, a vapor bubble

growing in superheated liquid was simulated and the results are

reported in Section5.3.

4.4. Height Function algorithm

Generally speaking, once an approximation of the interface unit

normal vector n is built, the local curvature can be derived as

j =r n [41]. ANSYS Fluent (version 12 and earlier) computesthe interface unit norm vector as n=ra/jraj according to earlyYoungs PLIC (Piecewise Linear Interface Calculation) formulation

[49], and it estimates the curvature by differencing volume frac-

tions. However, such an approach is known to have poor accuracy

as the volume fraction changes abruptly across the interface and

standard derivation schemes do not converge when applied to

strongly discontinuous functions. The consequence is the creation

and growth of unphysical velocities, known as spurious velocities

or parasitic currents[50], which may lead to unreal deformation

of the interface, up to its breaking-off. Furthermore, these numer-

ical artifacts artificially increase heat convection at interfaces,speeding up evaporation or condensation phenomena through a

purely numerical process.

To overcome this limitation, a Height Function algorithm was

implemented to replace the ANSYS Fluent default method to esti-

mate curvatures. Lety=f(x) be the mathematical function identify-

ing the interface line in a Cartesian (x,y) reference frame, as shown

inFig. 1. The height functionH(x;D) represents the height of the

interface line f(x), averaged within a local stencil of width D and

centered onx:

Hx; D 1D

Z xD=2xD=2

ftdt 15

For axisymmetrical domains with revolution around the z axis,the interface unit normal vector and curvature can be represented

by geometrical considerations as:

n 11 Hz21=2

Hz;1 16

j r

n Hzz

1 Hz2

3=2

Hzz

jHzz

j 1

fz1 Hz2

1=2

17

where Hzand Hzzdenote the first and second order derivatives with

respect tozandf(z) is the local elevation of the interface over the

revolution axis.

The HF algorithm implemented here is a combination of the

Malik et al.[51]and Hernandez et al.[52]versions, with the addi-

tion of a self-developed routine to estimate the local interface ele-

vation f(z). The algorithm is written for two-dimensional and

axisymmetrical geometries, with constant grid spacings. In Sec-

tion 5, the performance of our implementation of the HF algorithm

is assessed by several validation benchmarks.

4.5. The flow solver

ANSYS Fluent discretizes and solves the flow equations bymeans of a finite-volume scheme. The volume fraction Eq. (7) is

discretized in time with a first order explicit scheme and the con-

vective term is computed through a geometrical PLIC [49] recon-

struction of the fluxes across the faces of each interfacial cell.

The numerical stability of the explicit PLIC scheme poses a limita-

tion on the maximum time step allowed to solve the volume frac-

tion equation, since the interface must travel less than one grid cell

at each time interval. The time step for the volume fraction equa-

tion is calculated by the solver according to the maximum Courant

number (Cr) allowed for interface and near-interface cells. The

Courant number is a dimensionless number that compares the

simulation time step Dtand the time it would take for the fluid

to empty out of the cell:

Cr DtV=PNf

f ufnfAf

18

where V is the volume of the cell and the sum loops on the Nfboundary faces of the cell. The Fluent default value of Cr = 0.25

was used here.

The momentum and energy equations are discretized in time

with a first order implicit formulation, which allows a coarser time

step than the volume fraction one. A variable time step technique

was adopted with a maximum Courant number of 0.5 and there-

fore the volume fraction field is updated more frequently than

the velocity and temperature fields. The convective terms within

momentum and energy equations are discretized with a third or-

der MUSCL (Monotonic Upstream-centered Scheme for Conserva-

tion Laws) [53]scheme, while the diffusive terms are discretizedwith a central finite-difference scheme. The cell-centered gradients

of each scalar field are computed from the scalar values at the cell

face centroids, by means of the Green-Gauss theorem. The Fluent

GreenGauss node-based formulation [54] was proved to be the

best option to enforce the balance among pressure and surface ten-

sion within the momentum equation when the HF algorithm is

employed, leading to spurious velocities of two orders of magni-

tude smaller than other options available in the solver. Further-

more, Gupta et al. [55] reported smaller unphysical pressure

oscillations at the interface of a bubble when the node-based for-

mulation was employed rather than a cell-based one.

The mass conservation equation (6)is turned into a pressure

correction equation which is coupled to the momentum equation

(8). The pressurevelocity coupling is handled by a PISO (PressureImplicit Splitting of Operators)[56]algorithm, which was provedFig. 1. Illustration of the height function on the continuous domain.



7/21

to converge more quickly than the other options available in the

solver. The control-volume integration of the pressure gradient

within the momentum equation turns the pressure gradient into

differences among face-centered pressures. Fluent uses a collo-

cated technique, in which the flow equations are solved for cell-

centered variables. Fluent offers a body-force-weighted scheme

to interpolate cell-centered pressures at the cell faces and a PRE-

STO (PRessure STaggering Option) algorithm that solves the pres-

sure correction equation for a staggered control volume, thus

leading to face-centered pressures without the need of interpola-

tions. An evaluation here showed that the PRESTO algorithm gen-

erated spurious velocities of three orders of magnitude lower than

the body-force-weighted scheme.

The Height Function and evaporation models are implemented

in the ANSYS Fluent software by means of user-defined subrou-

tines. Fluent default surface tension computation is disabled by

setting a zero value of the surface tension in the solver. For each

interfacial cell, the HF-computed local curvature is used to esti-

mate the local surface tension force by means of Eq. (9), where

the actual value of the surface tension coefficient is set. Finally,

the estimated local surface tension force is introduced within the

momentum equation as a source term. ANSYS Fluent does not han-

dle any default evaporation model. Therefore, the evaporation

model discussed in Section 4.3 computes the r.h.s. terms of Eqs.

(7) and (10), and hence they are introduced in the related equa-

tions as source terms.

5. Validation benchmarks

In Sections 5.1 and 5.2 the HF performance in the reconstruction

of a circular interface and simulation of an inviscid static droplet

are compared with the Fluent-default PLIC-based algorithm, in

the following referred to as Youngs method. In Section 5.3, the

HF and evaporation models are tested by the simulation of a vapor

bubble growing in superheated liquid and compared with analyti-

cal solutions. Finally, in Section5.4the adiabatic flow of an elon-gated bubble within a circular horizontal microchannel is

simulated and results are validated through comparison with cor-

relations available in the literature. Further benchmarks for our

implementation were discussed in[57].

5.1. Reconstruction of a circular interface

This test case involves the curvature calculation algorithm

alone, without solving the flow equations. A two-dimensional cir-

cular droplet of radius R = 5 mm is placed within a L= 4R side

square domain and the HF and Youngs algorithm performances

on local interface curvature estimation under mesh refinement

are compared. The coarsest computational mesh has 10 10 ele-

ments, withR/D

= 2.5. The most refined mesh has 160 160 ele-ments, with R/D = 40. The circle center is placed randomlyaround the domain center (0, 0) in the interval ([0, D], [0, D]). For

each test case, the analytical surface representing the droplet is

intersected with the domain mesh, then the volume fraction field

is mapped through the numerical computation of the areas. For

each mesh element size, 50 runs are performed to span the range

of possible positions for the circle interface, then the results are

averaged. The comparison is performed by computing the follow-

ing curvature error norm for each test run:

L1j maxjjijexjjex ; for i1;. . .;Ni 19

where Ni is the number of interfacial cells, j i is the ith cell com-

puted curvature andjex= 1/R is the exact curvature of the droplet.Fig. 2 reports the values ofL1(j) under mesh refinement for HF and

Youngs algorithms. Youngs curvature estimation worsens as the

mesh is refined, because the standard derivation schemes do not

converge when differencing volume fractions. On the other hand,

the HF approach gives estimations that converge with the second

order of the mesh size, which is consistent with the second order

accurate finite-difference schemes used to differentiate the local

heights. A second order convergence rate was detected for the

Height Function algorithm by various authors as well [36,38,39],

thus proving the accuracy of our implementation. Note that at the

highest mesh resolution tested, the HF evaluates curvatures with

a maximum error of four orders of magnitude lower than Youngs

one, which is a considerable improvement.

5.2. Inviscid static droplet

Height Function and default Youngs schemes are separately em-

ployed to compute curvatures in the numerical simulation of aninviscid static droplet in equilibrium without gravity. The absence

of gravitational and viscosity effects tests exclusively the accuracy

of the implementation of the surface tension term within the

momentum equation and the solution algorithm. A circular droplet

is centered on a square domain and the mesh size range R/D = [5,40]

is investigated. The geometrical configuration is the same as de-

scribed in Section5.1. Surface tension and phase densities are set

to unity. Viscous and gravity effects are neglected. A constant value

of the pressure is set on all domain boundaries. The time step for the

solution of the momentum equation is fixed to Dt= 5 107 s. Theinitial velocity field is null throughout the domain. With such oper-

ating conditions, the exact solution of the momentum Eq. (8)is a

null velocity field, a constant pressure within and outside the drop-

let and a pressure jump at the interface given by the Laplace lawDpex= r/R. Errors in surface tension estimation generates unphysicalflows and an erratic pressure field, hence the magnitude of the

velocities and the errors in the pressure are optimal parameters to

compare the efficiency of HF and Youngs schemes within the

numerical solver. The following dimensionless velocity and interface

pressure jump error norms are observed:

L1juj 2qRr 1=2

maxjuij for i1;. . .;N 20

whereNis the number of domain cells and (r/2qR)1/2 is a chosenvelocity scale,

L2Dp

1

DpexffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

Nint

i1

DpiDpex2

Nintvuut 21

Fig. 2. L1(j) error norm convergence rate. White circles are HF errors and blackdiamonds arethoseof Youngs. The solid line is thesecond order convergence curve.



8/21

where Nintis the number of interior droplet cells. Fig. 3(a) reports

the velocity error norms after one simulation time step. The

Youngs-based algorithm generates spurious vortices whose magni-

tudes increase with the mesh refinement, as the curvature estima-

tion gets worse by refining the numerical grid. The Height Function

method exhibits velocities which scale with a convergence rate be-

tween the first and second order with respect to the mesh element

size, in good agreement with Francois at al. [38] observations. When

the finest R/D = 40 computational grid is employed, the parasitic

currents induced by the HF algorithmare three orders of magnitude

lower than Youngs generated ones. Such an improvement of the

standard commercial code is favorable to simulate elongated bub-

bles within microchannels, because the domain mesh is typically fi-

ner than R/D = 40. When the Fluent default estimation of curvatures

is employed, the magnitude of the spurious flows may easily ap-

proach that of the physical phenomenon being simulated, thus ren-

dering the results unusable. Fig. 3(b) plots the errors in the

computed pressure field after one simulation time step. The Height

Function computation exhibits a convergence rate between the first

and second order as the mesh is refined. The Youngs algorithmgives

a converging pressure field only when R/D 6 10; however at higher

mesh resolutions the average pressure within the droplet deviates

only by 1% from the exact value, which is a reasonable error.

5.3. Vapor bubble growing in superheated liquid

The heat-transfer-controlled growth of a spherical vapor bubble

in an infinitely extended superheated liquid was simulated.

According to the analysis of Plesset and Zwick[58]on the bubble

growth process, the heat-transfer-controlled growth reaches an

asymptotic stage in which the growth of the bubble is limited by

heat transport to the interface. The pressure within the bubble is

equal to the liquid pressure increased by the pressure jump at

the interface and the temperature of the vapor is equal to the sat-

uration temperature for that pressure. Scriven [59]has derived an

analytical solution for this stage neglecting viscous and surface

tension effects and considering the interface to be at the saturationtemperature. He obtained the following time-law for the bubble

radius:

Rt 2n ffiffiffiffiffiffiffiattp 22where n is a growth constant whose detailscan be found in [59] and

at is liquid thermal diffusivity. This solution is used to validate thesimulations here.

A spherical vapor bubble of radius R0= 0.1 mm is initialized at

the center of the axis of an axisymmetrical domain of radius 4 R0

and length 8R0. A uniform mesh size is chosen, with 1lm elementsize. Such a fine grid is necessary in order to solve the thin thermal

boundary layer surrounding the bubble interface. A constant pres-

sure is set at all boundaries except for the axis. Gravity effects are

neglected. The initial bubble size is large enough to neglect vapor

saturation temperature rise due to pressure jump across the inter-

face, so that the saturation temperature is equal in both phases. As

initial conditions, the velocity field is zero, the vapor phase is at the

saturation temperature while the liquid is superheated at T1=Tsat+ 5 K. Since the simulation starts at t= t0, when R(t0) = R0, a

thermal boundary layer has already been developing on the liquid

side around the bubble since the beginning of the heat-transfer-

controlled growth stage. The temperature field within the layer

at t= t0, and thus its thickness dT, canbe extrapolated from the Scri-

ven solution[59]. The temperature profile within the layer serves

as the initial condition for the temperature field. In order to avoid

that the thermal layer overlaps the vaporliquid interface on the

computational grid, it is initialized with a bit of misplacement,

about 12 cells, outside the bubble interface. Three different fluids

were tested. Water at atmospheric pressure and HFE-7100 at

0.52 bar, both withn = 15.1 anddT= 7 lm, and R134a at 0.84 bar,with n = 9.34 anddT= 11lm. The choice of each system pressurewas done in order to have similar growth constants for the fluids.

All vapor and liquid properties for the fluids are considered con-

stant at the saturation temperature. The accommodation coeffi-

cient within Eq.(14)for the evaporation model is set as unity.

Fig. 4shows the bubble radius evolution compared to the ana-

lytical solutions for all three fluids. Numerical data show very good

agreement with the analytical results. For each fluid, the bubble

numerical growth rate follows affiffit

p proportional law, as it should

be from Eq. (22). This does not happen during the initial growth

phase, when the initially misplaced thermal boundary layer rear-

ranges itself to fit the interface position. This settlement phase is

reflected on the numerical growth rates being lower than analyti-

cal ones at the beginning of the simulations. As detected by Kun-

kelmann and Stephan[35], the liquid thermal conductivity is the

parameter that rules the length of this thermal layer settlementphase. The higher the liquid thermal conductivity is, the faster is

thermal layer arrangement. HFE-7100 has the lowest thermal con-

ductivity among the fluids employed, for this reason its numerical

bubble growth rate deviates slightly from the analytical curve in

Fig. 4. The deviation for water at its higher time steps is due to

the presence of weak parasitic vortices across the interface, which

neither the use of the Height Function algorithm can overcome at

all. Their effect is more evident for water because it has the highest

surface tension coefficient (around four times higher than the

Fig. 3. (a)L1(ju

j) and (b)L2(Dp

) error norms after one simulation time step. White circles are HF errors and black diamonds are Youngs ones. The dashed line is the firstorder convergence curve and the solid line is the second order curve.



9/21

refrigerant fluids considered here) and, as observed by various

authors [50,60,61], the spurious velocities magnitude is propor-

tional tor. However, the maximum deviation between numericaland analytical bubble radius for water stays under the 10%

throughout the simulation.

5.4. Simulation of the adiabatic flow of an elongated bubble inside a

horizontal microchannel

The isothermal flow of an elongated bubble within a horizontal

microchannel, pushed by a liquid flow rate, is simulated under sev-

eral operating conditions. Such a flow configuration corresponds to

the adiabatic version of the evaporating flow which this work deals

with, and therefore this test is an optimal benchmark to evaluatethe accuracy of the numerical framework by comparison of simu-

lation results with published correlations. An elongated gas bubble

is initialized at the upstream of a horizontal circular channel and a

constant liquid flow rate feeds the channel. The bubble, pushed by

the liquid flow, accelerates and deforms until a steady motion is at-

tained. The flow domain is modeled as a two-dimensional axisym-

metrical channel with diameter D = 1 mm and length L = 8D. The

bubble is initialized as a cylinder with spherical rounded ends.

The initial length of the bubble is 3D and the film thickness is d/

D= 0.045. However preliminary tests showed that the steady flow

achieved is independent of the bubble initial shape and size. The

liquid flow into the channel is modeled as a fully developed lami-

nar velocity profile set as the boundary condition at the channel in-

let and outlet. The average velocity of the liquid at the inlet andoutlet boundaries is fixed at Ul= 0.25 m/s for all the simulations.

The specific properties of gas and liquid phases are different for

each simulation run. The values chosen give the Capillary numbers

of Ca = 0.025 and 0.0125 and Reynolds numbers within the range

of 15.625625, with both the groups computed by referring to

the liquid average velocity. The liquid to gas density ratio is set

to 1000 and the viscosity ratio to 50. The computational grid is a

uniformD/D = 100 mesh for the simulations with Ca = 0.025 andD/D = 200 for those with Ca = 0.0125, in order to always have at

least five cells discretizing the predicted liquid film thickness in

accordance with the Gupta et al. [55]recommendation. Each sim-

ulation is run until a steady state condition for the bubble velocity

is reached. The bubble velocity is estimated at every time step by

differencing in time the position of the center of gravity zG com-puted as:

zGPN

c1zcacVcPNc1acVc

23

whereNis the number of computational cells, zcthe axial location

of thecth cell centroid andVthe volume of the cell.

The width of the liquid film surrounding the bubble, the bubble

terminal velocity and the pressure drop across the channel at the

steady state of the flow are compared with correlations availablein the literature. The Han and Shikazono correlation for the liquid

film thickness at steady conditions reported in Eq. (2) is consid-

ered. It involves dimensionless groups computed by referring to

the bubble velocity. A prediction for the steady velocity of the bub-

bleUb can be obtained by applying a mass balance to part of the

channel which includes the flowing bubble, which for axisymmet-

rical flows leads to the following relationship:

Ub Ul1 4 d

D 1 d

D

24where d is the thickness of the liquid film. Predictions of terminal

liquid film thickness and bubble velocity are obtained by solving

iteratively Eqs.(2) and (24), introducing the average liquid velocity

Ul as the initial guess.Kreutzer et al. [62]suggested the following correlation for the

pressure drop in a channel with flow of an elongated bubble:

Dp1:08 rD

3Ca2=3Re1=3 64

Re

1

2qU2

LsD

25

whereLsis the length of the liquid slug,Uis the superficial velocity

of the flow, Ca and Re have to be evaluated by referring to U. Eq.

(25)is used as the benchmark for pressure drop in the simulations.

Fig. 5 shows the bubble terminal shapes obtained by the numer-

ical simulations. At a fixed Ca, an increase of the Reynolds number

tends to sharpen the bubble nose, to flatten the rear and to thicken

the liquid film due to the effect of inertia, as already observed by

Aussillous and Qur[63]and Kreutzer et al.[62]. At low Reynolds

numbers the liquid film is flat but, as Re is increased, the bubblerear enlarges thus squeezing locally the film. At the highest Rey-

nolds number simulated, the bubble rear inFig. 5(a) and (b) shows

some capillary waves whose wavelength seems to be a function of

the surface tension. Liberzon et al.[64]observed similar waves on

the surface of short Taylor bubbles rising in vertical pipes and con-

cluded that their length is a function of liquid density, surface ten-

sion and distance from the bubble nose. Due to these waves, the

liquid film does not reach a constant thickness because from the

bubble nose toward the rear the film becomes thinner and then

it also becomes wavy. By comparing the shapes obtained with

Ca = 0.0125 and Ca = 0.025, the thinning of the liquid film is evi-

dent, as the influence of the higher surface tension tends to round

off and thus shorten the bubble.

Table 1summarizes the numerical results on pressure drop, li-quid film thickness and bubble terminal velocity compared with

the considered correlations. The simulations reproduce quite well

the rise in the pressure drops with the increase of the Capillary

and Reynolds numbers, where the maximum deviation from Eq.

(25)is 17.6% and the average error is around 10%. The thickness

of the liquid film in each simulation is not constant along the bub-

ble. At low Re the value reported refers to the thickness at the cen-

tral region of the film, where it is almost constant. At high Re, a

zone with constant thickness is not observed and the value re-

ported refers to the location where the smooth profile of the bub-

ble matches the wavy profile at the rear. The thickening effect of

the inertia on the liquid film is well captured by the simulations

as well as the thinning effect of the capillary forces, such that the

film thickness increases as Ca and Re grow. The agreement withHan and Shikazono correlation is very good and the errors are

Fig. 4. Vapor bubble radius over time for analytical (lines) and numerical (symbols)

solutions.



10/21

within 5% with the exception of the cases with Ca = 0.0125 and

Re = 15.625 and 62.5, whose errors are about 15%. However, the lo-

cal film thicknesses in the simulations range within 0.0220.034for Re = 15.625 and from 0.02 to 0.038 for Re = 62.5, thus including

the predicted values. Eq. (24) estimates the terminal velocity of the

bubble in excellent agreement with the numerical simulations for

all the simulation runs, with errors within 3%.

6. Flow boiling of an elongated bubble within a horizontal

microchannel: simulation set-up

6.1. Flow conditions

Five different simulation runs are performed and their operat-

ing conditions are summarized inTable 2. The circular microchan-

nel is modeled as a two-dimensional axisymmetrical channel with

a diameterD = 0.5 mm and lengthL that varies depending on thesimulation run. The channel is always split into an adiabatic region

of lengthLafollowed by a heated region of length Lh. An elongated

vapor bubble of length 3Dat saturation conditions is initialized as

a cylinder with spherical rounded ends, placed at the upstream ofthe channel as depicted inFig. 6, which reports an illustration of

the initial configuration for the Case 1. The bubble is pushed by a

saturated liquid inflow of mass fluxG, introduced within the chan-

nel through the inlet section upstream to the channel. A constant

heat flux q is applied at the wall of the heated region of the

Fig. 5. Bubble terminal shapes for different test conditions.

Table 1

Comparison of numerical results and correlations. The errors between parenthesis are computed as jcorrnumjcorr

100.

Ca 0.025

Re 15.625 62.5 312.5 625

Dp [Pa] Eq.(25) 60 65 69 73

num (err) 52 (13.3) 54 (16.9) 71 (2.9) 80 (9.6)

d/D Eq.(2) 0.0494 0.0495 0.0526 0.0578

num (err) 0.05 (1.2) 0.047 (5.1) 0.05 (4.9) 0.06 (3.8)Ub [m/s] Eq.(24) 0.308 0.308 0.312 0.32

num (err) 0.308 (0) 0.305 (1) 0.305 (2.2) 0.315 (1.6)

Ca 0.0125

Re 15.625 62.5 312.5 625

Dp [Pa] Eq.(25) 62 67 74 80

num (err) 57 (8.1) 58 (13.4) 61 (17.6) 82 (2.5)

d/D Eq.(2) 0.0331 0.0333 0.0352 0.0379

num (err) 0.0279 (15.7) 0.028 (15.9) 0.035 (0.6) 0.038 (0.3)

Ub [m/s] Eq.(24) 0.287 0.287 0.289 0.293

num (err) 0.279 (2.8) 0.278 (3.1) 0.285 (1.4) 0.291 (0.7)

Table 2

Operating conditions for flow boiling simulation runs.Lastands for adiabatic length of

the channel, Lh for the heated length.

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

1 R113 600 50 9 8D 12D

2 R113 600 50 20 8D 22D

3 R245fa 600 50 20 8D 22D

4 R134a 500 31 20 8D 22D

5 R245fa 550 31 5 16D 56D



11/21

channel. The gravitational force is neglected. A constant velocity of

the liquid Ul= G/ql is imposed at the inlet section of the channel. Atthe channels outlet, the Patankars outflow boundary treatment

[65] is set for the velocity and temperature fields. This mimics azero gradient boundary condition for velocity and temperature,

which is a valid assumption provided that the Peclet number of

the flow exceeds 1 and that no flow reversal occurs at the outlet

section. The initial velocity and temperature fields are obtained

as results of a preliminary liquid-only steady state simulation

run under the same flow conditions. For reference, Fig. 6reports

for Case 1 the initial temperature field within the channel, the wall

temperature profile and the heat transfer coefficient computed as:

h qTwTsat 26

whereTw is the local wall temperature.

Due to the implemented version of the Height Function algo-

rithm, only uniform grids with square elements can be adopted.The choice of the computational grid for the simulations is ad-

dressed in the next section.

6.2. Grid convergence analysis

The minimum number of mesh elements to be fitted within the

liquid filmto obtain converged results is the object of this grid con-

vergence analysis.Fig. 6shows that the heat load applied to the

diabatic wall of the channel generates a superheated thermal

boundary layer. The bubble flows downstream and when the va-

porliquid interface reaches the superheated thermal layer the

evaporation process starts. Due to the elongated profile and the

sharp nose of the bubble, such contact likely occurs in the liquid

film region, where the thickness of the thermal layer approachesthe height of the film. The evaporation model cools down locally

the liquid temperature to maintain the bubble interface close to

the saturation temperature, and hence the liquid temperature

within the film drops sharply from the highest value at the channel

wall to the lowest value at the bubble interface. For this reason, in

the filmregion high temperature gradients take place and the com-

putational grid needs to be fine enough in order to solve the gradi-

ents properly. The accurate discretization becomes fundamental in

conjunction with the evaporation model, which computes the rate

of evaporation proportional to the local interface superheating. The

Gupta et al. [55] criterion is valid for adiabatic or diabatic flows

without phase change, but it cannot be assumed a priori in the

present configuration.

Four computational grids are employed to perform the Case 1run, whose operating conditions are listed in Table 2. For such

operating conditions, Eq. (2) predicts a liquid film thickness of

d/D= 0.04. The four grids used have channel diameter to mesh ele-

ment size ratio ofD/D = 100, 200, 300, 400, such that the predicted

film thickness to mesh element ratio ranges from 4 to 16. For eachcomputational grid the simulation is run until the nose of the bub-

ble reaches the end of the channel. The grid convergence analysis is

performed by examining the differences in the bubble dynamics

obtained with the computational grids employed. The bubble

dynamics are reconstructed by computing at each time instant

the bubble nose velocity as dzN/dtand the bubble growth rate as

dVb/dt, wherezNis the bubble nose position and Vb is the volume

of the vapor bubble.

Fig. 7(a) reports the bubble nose velocity as a function of the

time. There is a short stage that lasts less than one millisecond in

which the bubble shape modifies from the initialized shape to at-

tain a steady motion. After this settlement period the bubble flows

steadily up to the heated region with a constant velocity of

Ub= 0.47 m/s and liquid film thicknessd/D= 0.045 which are thesame for all the computational grids employed. Hence, even the

coarsest mesh is sufficient to accurately capture the dynamics of

the adiabatic stage of the flow. Fig. 7(a) suggests evaporation to

start at around t= 6 ms, when the bubble nose begins to accelerate.

As well, the beginning of the evaporation phenomenon is clearly

illustrated by the plot of the bubble volume growth rate in

Fig. 7(b). Henceforth the different performances of the computa-

tional grids become evident. The computed evaporation rate de-

creases as the mesh is refined, converging to similar profiles for

the grids with D/D = 300 and 400. Instead, an inadequate mesh ele-

ment size within the liquid film leads to a faster growth of the bub-

ble, as a consequence of the inaccuracy on the discretization of the

local temperature gradients. The computational grid with D/

D = 300 is a good compromise between accuracy and computa-tional cost of the simulations. The minimum film thickness, mea-

sured at the highest crest of the interfacial wave occurring at the

bubble rear, to mesh element size ratio is dmin/D 7, thus suggest-ing that 7 cells is the minimum resolution of the liquid film able to

solve the local temperature field in flow boiling conditions. All the

test cases reported inTable 2are characterized by a similar liquid

film thickness, and hence the mesh grid employed is always the

one withD/D = 300.

6.3. Error evaluation

The error on the mass balance throughout the simulation is

observed for Case 1 run with the chosen D/D = 300 mesh grid.

The mass balance between the terminal sections of the channelyields:

Fig. 6. Initial temperature field within the channel, wall temperature (dashed line) and heat transfer coefficient (solid line) for simulation Case 1. The bubble interface is

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


UP


12/21

_mot _mi _mgt qlqvqv

27

where _miand _mo are the mass flow rates across the inlet and outlet

sections and _mg is the global rate of vapor creation, obtained by

integration of _mjraj throughout the domain. Fig. 8depicts the er-rors on the mass balance expressed in Eq. (27). The relative error,

intended as absolute error normalized by the l.h.s. of Eq. (27), al-

ways remains under 106 as time elapses and vapor is being cre-ated. This means that mass conservation is ensured during bubble

growth. In addition, according to tests, at each time step the unbal-ance between vapor created and liquid disappeared is of the order

of machine accuracy, and hence all the liquid removed due to evap-

oration is actually converted into vapor.

7. Flow boiling of an elongated bubble within a horizontal

microchannel: results and discussion

7.1. Bubble dynamics and comparison with a theoretical model

The bubble dynamics during flow boiling for Case 1 is analyzed.

Fig. 9reports some snapshots of the bubble evolution while flow-

ingwithin the microchannel. The bubble nose enters the heated re-

gionof the channel a little after 4.5 ms, while the plot of the bubble

growth rate shown inFig. 7(b) (blue color) suggests evaporation tostart at 6 ms, when the bubble interface gets in contact with the

superheated thermal boundary layer developing at the heated

wall. Before evaporation starts, we observed that the bubble rear

oscillates with a constant time-period. Liberzon et al. suggestedin[64]that these oscillations generate capillary waves on the bub-

ble surface, which are clearly visible here in Fig. 5(g) and (h). While

evaporation occurs, the rear of the bubble stabilizes and proceeds

downstream with a constant velocity, equal to the adiabatic veloc-

ity of the bubble, while the bubble nose accelerates making the

bubble longer. Att= 12.5 ms the bubble volume is twice the value

before evaporation began. As an effect of the acceleration, the

thickness of the liquid film is increased from the adiabatic value

of d/D= 0.045 to d/D = 0.05, in good agreement with both Han

and Shikazono correlations for adiabatic [21] and accelerated

[22]flows, which predict respectively d/D= 0.056 and d/D= 0.054

if the actual bubble velocity and acceleration at 12.5 ms are used.

InFig. 7(a) the velocity of the bubble nose appears to increase

linearly with time. Actually, it has an exponential behavior whichis little perceptible as the simulation ends after few milliseconds.

Such hypothesis is proved below by comparing the simulation re-

sults with an exponential law for the bubble nose position derived

theoretically by Consolini and Thome[66]. Under the assumptions

of thermodynamic equilibrium between the phases, thus no liquid

superheating, axisymmetrical flow configuration and applying a

constant heat flux only to evaporate the liquid, they obtained the

following time-law governing the position of the bubble nose:

zNt z0Gqlq

vhlvD

4q exp

4q

qvhlvD

tt0

1

28

wherez0andt0are the axial location and time instant at which the

bubble nucleates. In the present simulations, the bubble is gener-

ated before entering within the heated region, hence z0 locatesthe entrance in the heated region and t0 the time instant at which

the nose of the bubble crossesz0. In the theoretical model the veloc-

ity of the bubble when t= t0equals the velocity of the liquid inflow,

while in the simulations the bubble velocity before evaporation be-

gins exceeds that of the liquid as expressed by Eq.(24). Thus, in or-

der to have the sameinitial velocity of the bubble for the simulation

and the model, the second term on the r.h.s. of Eq.(28) is multiplied

by the reciprocal of the denominator of the r.h.s. of Eq. (24).

Fig. 10(a) shows the comparison of simulation and theoretical

bubble nose positions for Case 1. The exponential time-law is well

captured by the simulation. The curves overlap at the initial stage

of the bubble evaporation, but afterwords the model underesti-

mates the bubble velocity and, as time elapses, this deviation

slowly grows. The origin of the gap is within the mentionedassumptions of the theoretical model. In the model the bubble

Fig. 7. (a) Velocity of the bubble nose, (b) bubble growth rate.

Fig. 8. Errors in mass conservation throughout the simulation for Case 1 run with

theD/D = 300 computational grid. The absolute error is reported in kg/s units.



13/21

grows only due to the wall heat flux, while in the simulation the

bubble grows also absorbing the sensible heat of the superheated

liquid, which is able to store energy before the bubble transits.

Thus, the heat used to evaporate the liquid may be smaller than

the wall heat flux or exceed it. In order to obtain an estimation

of the time-varying equivalent heat fluxqeabsorbed by the evapo-

ration, the heat absorbed to generate vapor is divided by the

heated surface actually traveled by the bubble at the time instant t:

qedVb=dtqvhlvpDzNt z0 29

Fig. 10(b) shows that qe> qfrom t= 8 ms on, where 8 ms is thetimeinstant at which the bubble has entered completely within the

heated region. As time elapses and the bubble nose flows down-

stream to the heated channel, it comes across regions more and

more superheated, and thereforeqerises monotonically as a conse-

quence of the increasing evaporation rate. Within the hypothesis of

wall heat flux used only to evaporate liquid,qe represents the wall

heat flux actually felt by the bubble, and therefore when qe> q Eq.

(28)leads to the underestimation observed inFig. 10(a).

Numerical and theoretical results are compared in Fig. 11 for

simulation cases 2, 3, 4 and 5, and despite the systematical under-

prediction of the model at the later stage of the growth as noted

above, the agreement is good. Note that among the simulations

run under the same heat flux (cases 2, 3, 4), the operating fluid

R113 gives the fastest growing bubble, while the vapor bubble ofR134a is the last to reach the end of the channel. Due to similar

Fig. 9. Bubble evolution during evaporation for Case 1. The black dashed line indicates the entrance in the heated region.

Fig. 10. (a) Time evolution of bubble nose position for Case 1 obtained with simulation compared with Consolini and Thome theoretical model[66].(b) Equivalent heat flux

absorbed by evaporation.



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heat fluxes and mass flow rates, the bubble growth rate depends

mainly on liquid-to-vapor density ratio, kinetic mobility and liquidspecific heat of the operating fluids. The specific heat represents

the capacity of the fluid to store thermal energy with low temper-

ature gradients, hence it is responsible for the rate of liquid super-

heating at the wall which determines the local interphase mass

flux, see Eq. (13). Furthermore, from Eq. (13) it follows that the

mass of vapor created is proportional to the kinetic mobility. The

density ratio has a strong effect on bubble growth in volume, as

the lighter is the gas phase than the liquid which it replaces, the

more the liquid evaporated expands when it is converted into va-

por. The R113 has the lowest kinetic mobility, 24 kg/m2s K against

58 kg/m2s K of R245fa and 111 kg/m2s K of R134a at the given

operating conditions. Thus its rate of vapor creation would be the

lowest one if all the fluids share the same temperature field. How-

ever, R113 has also the lowest liquid specific heat (943 J/kg Kagainst about 1400 J/kg K of the others), and thus it leads to the

highest liquid superheating and this partially balances the lower

kinetic mobility. What makes the R113 the fastest growing vapor

bubble is its higher liquid-to-vapor density ratio, which is three

times that of R245fa and six times R134a. Therefore, even though

at the end of the simulations R134a has the highest rate of mass

of vapor created, the evaporation of R113 generates the largest

bubble and it is the first to reach the end of the channel.

7.2. Analysis of flow and temperature field around the bubble

The temperature and flow fields obtained by the simulation of

Case 1 (fluid R113, see Table 2) are analyzed, in order to investigate

the local patterns of the flow and the dominating heat transfermechanisms. The flow is captured at the time instant t= 12.5 ms,

before the bubble exits the flow domain. At this time instant the

velocity of the nose of the bubble is 1.07 m/s, the center of gravitymoves at 0.76 m/s and the bubble rear at 0.47 m/s. The axial loca-

tion of the bubble nose at the center of the channel is z/D= 18.91

and the location of the rear is z/D= 12.43.

Fig. 12 depicts the contours and isolines of the velocity field

within the heated region of the channel, together with the profiles

of the axial and radial velocities averaged within the cross-

sectional area occupied by the liquid. Average values are evaluated

as:

umeanz 2R2 Rdz2

Z RRdz

ur;zrdr 30

whered(z) = Rin the absence of the bubble. Isotherms and temper-

ature contours are reported in Fig. 13 together with the wall heattransfer coefficient estimated as expressed in Eq. (26). The single

phase heat transfer coefficient refers to the preliminary steady

state simulation run with only liquid. A black dashed line repre-

sents the thickness of the thermal boundary layer dTfor the single

phase case computed as dT(z) = R r(T= Tsat+ 0.01(Tw(z) Tsat)),where the value 0.01 was arbitrarily chosen to identify the thermal

boundary layer as that region whose temperature exceeds the

saturation value by at least 1% of the local wall superheating

(Tw(z) Tsat). Fig. 14reports the two-phase boiling heat transfercoefficient htp along the heated wall, relative to the local single

phase value hsp obtained as result of the preliminary single phase

simulation.

Four separate regions can be identified by observation of the

flow and temperature fields across the bubble and each region isruled by specific wall heat transfer mechanisms:

Fig. 11. Time evolution of bubble nose position obtained with simulations compared with Consolini and Thome theoretical model [66].



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wake region far away from the bubble, located approximately atz/D= [8,10]. The isolines of velocity reported inFig. 12are hor-

izontal within this region and the mean radial velocity of the

liquid is null, thus suggesting that the bubble disturbance on

the liquid flow is negligible. This hypothesis is confirmed by

the profiles of axial velocity plotted inFig. 15(a), which suggest

that deviations from a single phase flow become noticeable only

forz/D> 11, thus from one and a half diameters behind the bub-

ble. Differently, the bubble influence on the thermal field is

noticeable up to the entrance in the heated region (z/D= 8).

The temperature contours reported in Fig. 13 show that the wall

thermal boundary layer is still restoring to the steady situationthat was holding before the bubble transited, thus during this

transient stage the thickness of the layer is smaller and the

liquid is cooler.Fig. 16proves that atz/D= 10 the wall temper-

ature is decreased by 1 K compared with the single phase case.

In this region the wall heat transfer is enhanced by the transient

heat convection mechanism, such that the improvement with

respect to the single phase case increases monotonically in

the streamwise direction, as shown inFig. 14.

wake region nearby the bubble, located within z/D= [10,12.5].The velocity isolines located atr/D< 0.35 converge on the chan-

nel axis, thus indicating that the liquid is slowing down along

the centerline of the channel. The undisturbed liquid axial

velocity around the channels axis is higher than that of thebubble rear, and hence the bubbles presence slows down

Fig. 12. Average liquid axial and radial velocity (above) and contours of the velocity field (below). The thick black solid line identifies the bubble interface.

Fig. 13. Heat transfer coefficient(above) and contours of temperature field (below). The thick blacksolid line identifies the bubble interface. The black dashed line represents

the width of the thermal boundary layer for the single phase case.



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locally the liquid. As a consequence, the axial velocity nearby

the wall has to increase in order for the flow to maintain a con-

stant liquid volumetric flow rate behind the bubble, and a flat

velocity profile is generated. This is demonstrated by the pro-

files of axial velocity in the wake reported in Fig. 15(a). At sec-

tions closer to the bubble rear, the liquid velocity profile

deviates from an expected parabolic to flat profile and at

z/D = 12.4 the liquid velocity at the centerline of the channel

matches the bubble rear velocity (0.47 m/s). Note that the inter-

sections between flat and undisturbed velocity profiles behind

the bubble occur atr/D 0.35 at all the axial locations reportedinFig. 15(a), in agreement with the observations of Gupta et al.

[67] for flow without phase change. The higher liquid velocity at

the wall delays the time-development of the thermal layer,

which appears thinner than upstream locations inFig. 13. The

temperature profiles depicted in Fig. 16 show that the wall tem-

perature drops by more than 1 K. In this region the combined

effect of transient heat convection and flat velocity profile aug-

ments the heat transfer coefficient up to 30% with respect to the

liquid-only case.

liquid film region,z/D= [12.5,19]. In this region the bubble inter-face is in contact with superheated liquid and evaporation

occurs. The interfacial resistance to mass transfer is very low,

such that the interface always stays at the saturation tempera-

ture. The bubble interface squeezes the thermal boundary layer

against the channel wall and film evaporation cools down

locally the superheated liquid to the saturation condition, and

thus the isotherms inFig. 13are more dense. The undulations

of the interface profile near the rear of the bubble create local

recirculation patterns, which make the liquid average velocities

oscillate around zero, but with values low in magnitude. Clock-

wise vortices are detected upon each valley while anticlockwise

vortices are evident at each crest of the interfacial wave.

Fig. 15(b) reports the profiles of axial velocity within the liquid

film and at z/D= 13, where an anticlockwise vortex occurs, the

axial velocity is negative. Moving downstream along the bubble,

the amplitude of the interfacial wave decreases and the shape of

Fig. 14. Enhancement of the heat transfer coefficient given by the boiling two-

phase flow (subscript tp) relative to the single phase case (sp). The black dashed

lines identify the bubble rear and nose positions.

Fig. 15. Profiles of liquid axial velocity in (a) the bubble wake and (b) liquid film.

Fig. 16. Profiles of liquid temperature in the bubble wake. Solid line: two-phase flow, dashed line: single phase flow.

466 M. Magnini et al. / International Journal of Heat and Mass

Magnini Et Al. 2013 Numerical Investigation of Hydrodynamics and Heat Transfer of Elongated Bubbles...

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