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:
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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:
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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
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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
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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
liquid bubble liquid
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
liquid bubble liquid
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.
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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.
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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.
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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.
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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.
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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.)
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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.
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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)
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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