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A numerical investigation of the evaporation process of a liquid droplet impinging onto a hot substrate

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A numerical investigation of the evaporation processof a liquid droplet impinging onto a hot substrate
N. Nikolopoulos
a
, A. Theodorakakos
b
, G. Bergeles
a,*
a
Department Mechanical Engineering, National Technical University of Athens, 5 Heroon Polytechniou, 15710 Athens, Greece
b
Fluid Research, Co, Greece
Received 20 December 2005Available online 22 August 2006
Abstract
A numerical investigation of the evaporation process of
n
-heptane and water liquid droplets impinging onto a hot substrate is pre-sented. Three diﬀerent temperatures are investigated, covering ﬂow regimes below and above Leidenfrost temperature. The Navier– Stokes equations expressing the ﬂow distribution of the liquid and gas phases, coupled with the Volume of Fluid Method (VOF) fortracking the liquid–gas interface, are solved numerically using the ﬁnite volume methodology. Both two-dimensional axisymmetricand fully three-dimensional domains are utilized. An evaporation model coupled with the VOF methodology predicts the vapor blanketheight between the evaporating droplet and the substrate, for cases with substrate temperature above the Leidenfrost point, and the for-mation of vapor bubbles in the region of nucleate boiling regime. The results are compared with available experimental data indicatingthe outcome of the impingement and the droplet shape during the impingement process, while additional information for the dropletevaporation rate and the temperature and vapor concentration ﬁelds is provided by the computational model.
2006 Elsevier Ltd. All rights reserved.
Keywords:
Droplet evaporation; Volume of Fluid Method; Kinetic theory; Leidenfrost temperature
1. Introduction
The liquid–vapor phase change process, plays a signiﬁ-cant role in a number of technological applications in com-bustion engines, cooling systems or refrigeration cycles. Inall the aforementioned applications, the dynamic behaviorof the impinging droplets and the heat transfer between theliquid droplets and the hot surfaces are important factors,which aﬀect the mass transfer associated with liquid–vaporphase change.The mechanism of the droplet spreading and theaccompanying heat transfer is governed not only by non-dimensional parameters as the droplet Weber (
We
), theReynolds (
Re
) number, Eckert (
E
c
) number, and Bond(
Bo
) number, but also by the temperature of the surface.As the droplet impacts upon the hot solid surface, heat istransferred from the solid to the liquid phase. This energytransfer to the droplet increases its mean temperature,while liquid vaporizes from the bottom of the droplet. If the heat transfer rate is large enough during the impact,liquid vaporized from the droplet forms a vapor layerbetween the solid and the liquid phase, which repels thedroplet from the solid surface. In this case the heat transferreaches a local minimum and the evaporation lifetime of the droplet becomes maximum. This phenomenon was ﬁrstobserved by Leidenfrost [1] in 1756 and hence the behavioris known as the Leidenfrost phenomenon. Based on theevaporation lifetime of a droplet, mainly four diﬀerentevaporation regimes can be identiﬁed depending on thewall temperature; ﬁlm evaporation, nucleate boiling, tran-sition boiling and ﬁlm boiling. This work contributes tothe study of transition and ﬁlm boiling impact regimesonly.
0017-9310/$ - see front matter
2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2006.06.012
*
Corresponding author. Tel.: +30 2107721058; fax: +30 2107723616.
E-mail addresses:
niknik@ﬂuid.mech.ntua.gr (N. Nikolopoulos),andreas@ﬂuid-research.com (A. Theodorakakos), bergeles@ﬂuid.mech.ntua.gr (G. Bergeles).
www.elsevier.com/locate/ijhmt
International Journal of Heat and Mass Transfer 50 (2007) 303–319
The collision dynamics of a liquid droplet impinging ona hot surface has been investigated mainly experimentally.Researchers have presented a sequence of photographsshowing the deformation process of liquid droplets impact-ing on a hot surface. Wachters and Westerling [2] wereamong the ﬁrst to investigate the impact of a saturatedwater droplet of about 2 mm in diameter impinging on apolished gold surface heated to 400
C, while Akao et al.[3] inspected the deformation behavior of various liquiddroplets of 2 mm diameter on a chromium-plate coppersurface heated to the same temperature. Xiong and Yuen[4] measured the time history of a
n
-heptane dropletimpinging on a stainless-steel surface heated to tempera-tures between 63
C and 605
C. Chandra and Avedisian[5] performed the same experiment with a temperaturerange from 24
C to 205
C keeping a constant Webernumber
We
= 43 while the same authors in [6] have pre-sented results for the deformation process of a dropletimpinging onto a porous ceramic surface. Naber and Far-rell [7] examined the deformation process of liquid dropletsof 0.1–0.3 mm in diameter impinging on a hot stainless-steel surface, while at the same time Anders et al. [8] inves-tigated the rebounding phenomenon of ethanol dropletsimpacting obliquely on a smooth chromium-plated coppersurface at 500
C.Ko and Chumg [9] investigated experimentally the eﬀectof wall temperature on the break-up process of
n
-decanefuel, in the Leidenfrost temperature range of 220–330
C,and demonstrated that wall temperature variation showsa peculiar nonlinear behavior in the droplet break-up prob-ability, especially near 250
C, which corresponds to thetemperature of local maximum droplet lifetime. Manzelloand Yang [10] examined the eﬀect of an additive in a waterdroplet on its collision dynamics on a stainless-steel surfacewith the wall temperature varying from ﬁlm evaporation toﬁlm boiling regime for three
Weber
number impacts.Bernardin et al. [11,12] realizing that the impact param-eters can alter the collision outcome, conducted a thoroughseries of experiments, concerning water droplets impingingon a polished aluminium surface, with the main controllingparameters of the phenomenon being droplet velocity,resulting in
We
number from 20 to 220 and surface temper-ature from 100
C to 280
C. They constructed dropletimpact regime maps, which distinguish between the variousboiling regimes for each of the three experimental
We
num-bers investigated. Moreover, the heat ﬂux from the surfacewas measured, for diﬀerent
We
numbers, drop impact fre-quency and surface temperature, determining the two veryimportant points in the regime map, the Leidenfrost point(LFP) and the critical heat ﬂux point (CHF). The ﬁrst
Nomenclature
Bo
Bond number,
ð¼
q
liq
gD
2o
=
r
Þ
C
p
non-dimensional pressure,
¼
D
P
=
12
q
liq
U
2o
c
p
heat capacity (J/kg K)
D
AB
diﬀusivity of gas
A
to gas
B
, (=
l
/(
Sc
q
))
E
c
Eckert number,
ð¼
U
2o
=
ð
c
p
ð
T
liq
T
w
ÞÞÞ
D
o
initial diameter of droplet
E
sur
surface energy
E
kin
kinetic energy
k
thermal conductivity (W/mK), (=
c
p
l
/
Pr
)
MB
molecular weight (kg/kmol)
~
n
vector normal to interface of the two phases
Oh
Ohnesorge number, (=
l
liq
/(
rq
liq
D
o
)
0.5
)
P
pressure
Pr
Prandtl number, (=
l
c
p
/
k
)
R
universal gas constant (J/kmol K)
R
computational radius
R
o
radius of initial droplet
Re
Reynolds number (=
q
liq
D
o
U
o
/
l
liq
)
Sc
Schmidt number (=
l
/(
q
D
AB
))
SYG
vapor concentration (mass of vapor (kg)/massof gas phase (kg))T temperaturet time
~
T
stress tensor
~
u
velocity
U
o
initial velocity of droplet
U
l
velocity of an equivalent droplet of the ring
V
volume
X X
-axis of computational ﬁeld
Y Y
-axis of computational ﬁeld
Z Z
-axis of computational ﬁeld
Z
h
height of spreading droplet
We
Weber number,
ð¼
q
liq
D
o
U
2o
=
r
Þ
Greek symbols
a
volume of ﬂuid (also noted as indicator func-tion)
d
vapor height
j
curvature (m
1
)
l
dynamic viscosity
q
density
r
surface tension
r
thermal accommodation coeﬃcient
Subscripts
gas gas phaseliq liquid phaseb basevap vaporcell computational cellsat saturation pointw substrate or wall
304
N. Nikolopoulos et al./International Journal of Heat and Mass Transfer 50 (2007) 303–319
corresponds to the minimum heat ﬂux point and the secondto the lower temperature boundary of the transitional boil-ing regime.Apart from the above-mentioned controlling parametersfor the description of such a phenomenon, secondaryparameters such as surface roughness, control the evolu-tion of this phenomenon. Most of the researchers ignoredthe eﬀects of surface roughness on droplet heat transfer.Cumo et al. [13], Baumeister et al. [14] and Nishio and Hir-
ata [15] observed that rough surfaces require a thickervapor layer between the droplet and the surface to sustainﬁlm boiling and, therefore, possess a higher LFP tempera-ture. Avedisian and Koplik [16] found that the LFP forwater droplets on porous ceramic surfaces increases withincreasing porosity. Engel [17] observed that surface rough-ness promotes droplet break-up, and Ganic and Rohsenow[18] reported surface roughness enhances liquid–solid con-tact in dispersed droplet ﬂow and hence increases ﬁlm boil-ing heat transfer. Fujimoto and Hatta [19] and Hatta et al.[20] conﬁrmed that the critical
We
number, above whichwhether or not the droplet is disintegrated during deforma-tion, depends on the kind of surface material. Wachtersand Westerling [2] observed experimentally that the critical
We
number, above which disintegration of a dropletimpinging on a hot wall once the droplet is transformedinto an expanding torus is around 80.Bernardin et al. [11,12] used three diﬀerent surface ﬁnishesand reported that although the temperature correspondingto the critical heat ﬂux (CHF) was fairly independent of sur-face roughness, the Leidenfrost point (LFP) temperature wasespecially sensitive to surface ﬁnish. They produced regimemaps illustrating not only the well-known boiling curveregimes of liquid ﬁlm, transition and nucleate boiling, butalso the complex liquid–solid interactions which occur dur-ing the lifetime of the impacting droplet.One more important secondary controlling parameter,essential not only for the description of physics of this phe-nomenon, but also for its numerical simulation, is the valueof contact angles. Bernardin et al. [11,12] using the sessiledrop technique measured the variation of contact anglesfor an aluminum surface, as a function of surface temper-ature, while Chandra et al. [21] studied the eﬀect of contactangles on droplet evaporation, adding varying amounts of a surfactant to water.A few studies have examined the eﬀect of reduced grav-ity. Siegel [22] has reviewed much of the work done on thistopic. The principal ﬁndings were that gravity has littleeﬀect on the nucleate pool boiling heat transfer coeﬃcients.For low wall heat ﬂux, vapor bubble diameters increase atlow gravity. Furthermore, the critical heat ﬂux decreases inthe absence of buoyancy forces while stable ﬁlm boiling canbe maintained at low gravity, but heat transfer is reduced.Qiao and Chandra [23] performed a series of experiments,using water and
n
-heptane, intending to isolate the eﬀect of buoyancy forces on droplet impact and boiling. Theirobjective was to study the eﬀect of gravity and liquid prop-erties on transition from nucleate to ﬁlm boiling.Due to the highly complex nature of these processes,development of methods to predict the associated heatand mass transfer has often proved to be a diﬃcult task.Nevertheless, research eﬀorts over several decades haveprovided an understanding of many aspects of vaporiza-tion or condensation. Important and interesting numericalsimulations of droplet collisions with a variety of methodshave also been published. The MAC-type solution methodto solve a ﬁnite-diﬀerencing approximation of the Navier– Stokes equations governing an axisymmetric incompress-ible ﬂuid ﬂow was used by Fujimoto and Hatta [19] andHatta et al. [20]. The simulation of the ﬂow ﬁeld insidethe liquid droplet has been performed assuming a simplethermal distribution such that temperature becomes lower(higher) on the upper (lower) side of the droplet and higherwith time. The unsteady thermal distribution inside thedroplet is not calculated, assuming the temperature of thedroplet’s bottom to be at the saturation temperature andthat a vapor layer exists between the droplet and solidsurface.A number of analytical studies by Gottfried et al. [24],Wachters et al. [25], Nguyen and Avedisian [26], and Zhang
and Gogos [27] are dealing with the Leidenfrost phenome-non and the steady-state droplet ﬁlm boiling. Indispensablecondition for these studies is that the droplet has a nearlysteady spherical shape, so that the heat transfer rates anddroplet evaporation times can be predicted successfully.Pasandideh et al. [28] used a complete numerical solu-tion of the Navier–Stokes and energy equations, basedon a modiﬁed SOLA-VOF method, to model dropletdeformation and solidiﬁcation, including heat transfer inthe substrate. The heat transfer coeﬃcient at the droplet-substrate interface was estimated by matching numericalpredictions of the variation of substrate temperature withmeasurements. Heat transfer in the droplet was modeledby solving the energy equation, neglecting viscous dissipa-tion, whilst the eﬀect of substrate’s cooling on the droplet’sevaporation was taken into account [29]. Following that,Pasandideh et al. [30], extended the model developed byBussmann et al. [31] and combined a ﬁxed-grid control vol-ume discretization scheme of the ﬂow and energy equationswith a volume tracking algorithm to track the droplet freesurface. Surface tension eﬀects were also taken intoaccount. The energy equation both in the liquid and solidportion of the droplet were solved using the Enthalpymethod in the case of solidiﬁcation. More recent three-dimensional codes have been used to model complex ﬂowssuch as impact on inclined surfaces resulting in dropletbreak-up, as shown by Zheng and Zhang [32] and splash-ing, according to Ghafouri-Azar et al. [33]. Zheng andZhang [32] developed an adaptive level set method formoving boundary problems in the case of droplet spread-ing and solidiﬁcation.Zhao and Poulikakos [34,35] studied numerically theﬂuid dynamics and heat transfer phenomena both in drop-let and the substrate, based on the Lagrangian formulationand utilizing the ﬁnite element method in a deforming
N. Nikolopoulos et al./International Journal of Heat and Mass Transfer 50 (2007) 303–319
305
mesh. The temperature ﬁelds developing in both the liquiddroplet and the substrate during the impingement processwere also determined. Waldvogel and Poulikakos [36]followed the Langrangian formulation including surfacetension and heat transfer with solidiﬁcation. They investi-gated the eﬀect of initial droplet temperature, impactvelocity, thermal contact resistance and initial substratetemperature on droplet spreading, on ﬁnal deposit shapesand on the times to initiate and complete freezing. Buttyet al. [37] solved the energy equation in both the dropletand substrate domain, implementing a time and space aver-aged thermal contact resistance between the two thermaldomains. During calculations a regeneration of mesh tech-nique is used, in order to enhance accuracy. Harvie andFletcher [38–40] coupled VOF methodology with a sepa-rate one-dimensional algorithm to model not only thehydrodynamic gross deformation of the droplet, impactingonto a hot wall surface, but also the ﬂuid ﬂow within theviscous vapor layer existing between the droplet and thesolid surface. The height of the vapor layer was assumedto be several orders of magnitude smaller than the dimen-sions of the droplet, resulting in a Knudsen numberapproaching values of the order of 0.1. It is important tonote that the height of the vapor layer does not result fromthe solution of the Navier–Stokes equations, but it wasassumed to be known. Furthermore, they used a kinetictheory treatment in order to calculate conditions existingat the non-equilibrium interface of the vapor layer, solvingthe heat transfer within the solid, liquid and vapor phases.This model was validated for a number of droplet impactconditions, covering a wide range of
We
numbers and ini-tial droplet and surface temperatures.The present investigation studies numerically theimpingement of
n
-heptane and water droplets on a hot sub-strate under various temperatures, covering regimes aboveand below the Leidenfrost temperature. Viscous dissipationand surface tension eﬀects are taken into account; the equa-tions are solved numerically with the ﬁnite volume method-ology, whilst the Volume of Fluid methodology of Hirt andNichols [41] is used for the tracking of the liquid–gas inter-face. The methodology is coupled with an adaptive localgrid reﬁnement technique, both in 2-D axisymmetric andfully three-dimensional cases, allowing the prediction of details of droplet’s levitation, above the Leidenfrost tem-perature, without any ‘a priori’ assumption for the vaporlayer height. Moreover in contrast to other methodologies,in the case of impact below the Leidenfrost point, theentrapment of vapor between the liquid droplet and thewall is predicted, The evaporation model coupled withVOF methodology is used in an in-house developed CFDcode, predicting not only the deformation of the liquiddroplet and the height of vapor blanket in the case of theabove Leidenfrost temperature, but also the correspondingtemperature and vapor ﬁelds. The used model is validatedfor a number of droplet impacts both for low and high
We
numbers and substrate temperatures. The heat transferinside the substrate is not solved, as the substrate tempera-ture is considered to be constant, the liquid–gas interface isassumed to be at saturation conditions, whilst the eﬀect of substrate roughness on the droplet spreading is not takeninto account.
2. The numerical solution procedure
2.1. Fluid ﬂow
The ﬂow induced by the impact of a droplet on a hotsurface, is considered as two-dimensional axisymmetricfor cases A, B, and C (
n
-heptane) and for case D (water)as three-dimensional; the details of the test conditionsinvestigated are summarized in Table 1. The volume frac-tion, denoted by
a
, is introduced following the Volume of Fluid Method (VOF) of Hirt and Nichols [41] in order todistinguish between the gas and the liquid phases. This isdeﬁned as:
a
¼
Volume of liquid phaseTotal volume of the control volume
ð
1
Þ
where the
a
-function is equal to:
a
ð
x
;
t
Þ ¼
1
;
forapoint
ð
x
;
t
Þ
insideliquidphase0
;
forapoint
ð
x
;
t
Þ
insidegasphase0
<
a
<
1
;
forapoint
ð
x
;
t
Þ
insidethetransitionalareabetweenthetwophases
8>>><>>>:
ð
2
Þ
For a single droplet splashing onto a wall ﬁlm, the VOFmethodology has been successfully applied and the methodis described in more detail in Nikolopoulos et al. [42].The momentum equation is written in the form:
o
ð
q
~
u
Þ
o
t
þ r ð
q
~
u
~
u
~
T
Þ ¼
q
~
g
þ
~
f
r
ð
3
Þ
where
~
T
is the stress tensor,
~
u
is the velocity,
q
is the densityof the mixture and
f
r
is the volumetric force due to surfacetension. The value of
f
r
is equal to
f
r
=
r
j
(
$
a
), where
r
is the numerical value of the surface tension (for immiscibleﬂuids the value is always positive) and
j
is the curvature of the interface region.The ﬂow ﬁeld is solved numerically on two or three-dimensional unstructured grids, using a recently developedadaptive local grid reﬁnement technique, following theﬁnite volume approximation, coupled with the VOF meth-odology; a detailed discussion of the ﬂuid ﬂow model ispresented by Nikolopoulos et al. [42], while the adaptivelocal grid reﬁnement technique is used in order to enhanceaccuracy of the predictions in the areas of interest (i.e. theliquid–gas interface), with minimum computational cost, asshown by Theodorakakos and Bergeles [43]. To accountfor the high ﬂow gradients near the free surface, the cellsare locally subdivided to successive resolution levels, onboth sides of the free surface. As a result, the interface is
306
N. Nikolopoulos et al./International Journal of Heat and Mass Transfer 50 (2007) 303–319
always enclosed by the densest grid region. A new locallyreﬁned mesh is created every 20 time steps for the cases thatwill be presented afterwards. The numerical cell at whichsubdivision is performed, is locally reﬁned by a factor of 3 for case D or 4 for cases A, B, and C (i.e. in two dimen-sions an initial cell is split into four cells). In that way a newgrid with 1 level of local reﬁnement is created. Obviously,computations are more time eﬃcient on the dynamicallyadaptive grid, than on the equivalent ﬁne resolution uni-form grid.The high-resolution diﬀerencing scheme CICSAM, pro-posed by Ubbink and Issa [44] in the transport equation for
a
(VOF-variable) is used. The discretization of the convec-tion terms of the velocity components is based on a highresolution convection-diﬀusion diﬀerencing scheme (HRscheme) proposed by Jasak [45]. The time derivative wasdiscretized using a second-order diﬀerencing scheme(Crank–Nicolson). Quadrangular (2D) or hexahedron(3D) computational cells are used. Finally, the contactangles at the advancing and receding contact lines areassigned as boundary conditions.
2.2. Heat transfer
Heat transfer in the droplet was predicted by solvingthe energy equation, calculating all physical propertiesas a function of the corresponding properties of the liquidand gas (air and vapor) phase. Such properties are den-sity, viscosity, heat capacity and Prandtl number. Allproperties were assumed to vary with temperature andpressure, including the diﬀusivity of vapor in air (
D
AB
).The surface tension coeﬃcient is assumed to vary alsowith temperature.Heat transfer within the liquid phase is described by thefollowing thermal energy transport equation (enthalpyequation) for incompressible ﬂuids:
q
D
h
0
D
t
¼ rð
k
r
T
Þ þ
D
P
D
t
þ
_
Q
;
ð
4
Þ
In this equation,
_
Q
is a source term due to evaporation andis equal to the amount of heat released, when liquid passesthrough the liquid–vapor interface and evaporates:
_
Q
¼
d
m
=
d
t V
cell
L
;
L
¼ ð
C
p
;
liq
C
p
;
vap
Þ
T
;
ð
5
Þ
where
L
is the latent heat of vaporization of liquid andd
m
/d
t
the evaporation rate of the liquid phase.The value of
~
Q
term is proportional to the mass ﬂux of liquid molecules which evaporate. Following Langmuir’s[46] approach, whereby the liquid and vapor phases areassumed to be separated by a discrete molecular layer,but including the Schrage’s correction [47] to account formolecular ﬂow towards or away from the liquid surface,the evaporated mass ﬂux is equal tod
m
=
d
t
¼
2
r
2
r
MB
vap
2
p
R
1
=
2
P
sat
;
liq
T
1
=
2liq
P
sat
;
vap
T
1
=
2vap
!
>
0
ð
6
Þ
where
r
is the thermal accommodation coeﬃcient.It seems that thermal accommodation coeﬃcient has notbeen measured with any real conﬁdence yet, but its value isin the range of 0–1. Here, a value of 0.5 has been chosenboth for
n
-heptane and water.Apart from the energy equation, an additional trans-port Eq. (7) for the concentration of vapor in the gas issolved
ð
1
a
Þ
q
air
D
C
D
t
¼ r½ð
1
a
Þ
q
air
D
AB
r
C
þ
d
m
=
d
t V
cell
ð
7
Þ
where
C
is the concentration of the vapor phase in the gasphase (kg
vapor
/kg
gas
). For the mixed phase of liquid andgas, physical and thermodynamic properties are calculated
Table 1Test cases examinedCase A B C DLiquid
n
-Heptane
n
-Heptane
n
-Heptane Water
R
o
0.00075 0.00075 0.00075 0.0015
U
o
0.8 0.8 0.8 2.34
We
34.52 34.52 34.52 222.10
Re
2156.09 2156.09 2156.09 7638.78
Oh
0.00273 0.00273 0.00273 0.00195
Bo
0 0 0 1.19
E
c
0.034 0.031 0.028 0.060
T
w
(
C) 178 190 210 180
T
liq
(
C) 25 25 25 27Computational domain(
X
tot
,
Y
tot
,
Z
tot
)13.33
R
o
6.67
R
o
13.33
R
o
6.67
R
o
13.33
R
o
6.67
R
o
10
R
o
10
R
o
6.67
R
o
Base grid 60
30 (4 levels localreﬁnement)60
30 (4 levels localreﬁnement)60
30 (4 levels localreﬁnement)45
45
30 (3 levels localreﬁnement)Maximum number of grid nodes 11314 16353 13183 499132
N. Nikolopoulos et al./International Journal of Heat and Mass Transfer 50 (2007) 303–319
307

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