droplet impact

1. Simulation of cooling of liquid Al–33 wt.% Cu droplet impinging
on a metallic substrate and its experimental validation
A. Kumar a
, S. Ghosh a,*, B.K. Dhindaw b
a
Department of Metallurgical and Materials Engineering, Indian Institute of Technology, Kharagpur 721302, India
b
Indian Institute of Technology Ropar (Punjab), Nangal Road, Rupnagar 140001, India
Received 20 June 2009; received in revised form 25 August 2009; accepted 28 August 2009
Available online 1 October 2009
Abstract
In the present work a model for heat transfer during collision of a falling liquid Al–33 wt.% Cu droplet on a 304 stainless steel sub-
strate has been developed on a FLUENT 6.3.16 platform. The model simultaneously takes into account the fluid flow and heat transfer
in the liquid droplet and the surrounding gas, and the heat transfer in the substrate. The liquid–gas interface was tracked using the vol-
ume of fluid method and the contact resistance between Al–33 wt.% Cu and the substrate was taken into account. The comprehensive
model correctly predicted the total spread in the droplet. As per the predicted transient thermal field, the solidification front speed
oscillated along the radius of the spread droplet. Based on the estimated front speeds at these locations and Jackson–Hunt plot for
Al–33 wt.% Cu, the variation of interlamellar spacing along the radial direction was found. It matched well with the variation of the
experimentally measured interlamellar spacing at different locations along the radius.
Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Eutectic; Alloy droplet; Substrate; Solidification; Modeling
1. Introduction
Spray forming or casting refers to the break-up of a
liquid metal stream into variously sized droplets, propelled
away from the atomization region by fast flowing atomiz-
ing gas to a substrate, where the droplets are deposited
[1–3]. The flight and deposition of the droplet are accompa-
nied by cooling and solidification, during which the cooling
rate determines the structural features at the mesoscale,
such as grain size, grain structure and eutectic spacing (in
the case of eutectic alloys). Thus, during the spray casting,
desired structural features can be obtained by controlling
the rate of cooling within the droplets.
The cooling rate within the droplets, in turn, depends on
the processing parameters such as superheat of liquid
metal, size of the droplet, velocity of impingement and
thermal diffusivity of the substrate material, and thus the
structure of solidified material can be controlled by con-
trolling these parameters. The processing condition for
achieving the desired structure can be identified by a vali-
dated comprehensive mathematical model that predicts
the flow and the thermal fields during the droplet deposi-
tion. However, due to interaction of several complex phe-
nomena, comprehensive modeling of the droplet
deposition is a challenging task. Further, the validation
of the model is complicated due to the small size of the
droplet and the high cooling rates involved.
In the past, mathematical models which simulate the
atomization and thermal history of gas atomized droplets
as a function of flight distance from the point of gas atom-
ization to the point of deposition [4–6] have been developed.
Gas atomization has been extensively modeled [7–13].
Zeoli and Gu [12] developed a numerical model to simulate
the critical droplet break-up during the atomization. In
this numerical model they coupled droplet break-up with
the flow field generated by high-pressure gas nozzle and
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2009.08.063
*
Corresponding author. Tel.: +91 3222283294; fax: +91 3222282280.
E-mail address: sudipto@metal.iitkgp.ernet.in (S. Ghosh).
www.elsevier.com/locate/actamat
Available online at www.sciencedirect.com
Acta Materialia 58 (2010) 122–133

2. reported that this numerical model can provide quantitative
assessment of the atomization process. Recently Zeoli and
Gu [13] developed an isentropic plug nozzle (IPN) to reduce
the shocks and maximize the kinetic energy being trans-
ferred from the gas in order to instablize the melt stream.
The performance of IPN was examined using a numerical
model which includes gas flow dynamics, droplet atomiza-
tion mechanism and particle tracking.
The in-flight thermal behavior during deposition over a
rotating substrate was modeled by Tinoco et al. [14]. The
calculations were performed by solving the momentum
and enthalpy equations for the gas. Ojha and coworkers
[15] studied solidification of undercooled droplets during
atomization process. The predicted cooling rate was com-
pared with the calculated cooling rate from the secondary
dendrite arm spacing measurement. Zeoli et al. [16]
described a numerical model which combines both cooling
and break-up in single computation (integrated model).
The dynamic history of droplets was solved as discrete
phase in Eulerian gas flow. The droplet solidification model
incorporated the detailed heat transfer mechanism of und-
ercooling, recalescence, peritectic and segregated solidifica-
tion during flight. The model establishes that the in-flight
distance is the major factor influencing the atomization
and solidification of droplets.
Several numerical models have been developed to simu-
late impact and solidification of molten droplets on a cold
substrate. Bennett and Poulikakos [17] and Kang et al. [18]
studied droplet deposition assuming that solidification
starts only after complete spreading of droplet in the form
of a disk. Theoretical and experimental studies done by
Bennett and Poulikakos [17] showed that the thermal con-
ductivity of the substrate significantly affects the cooling
rate of the splat. However, they did not incorporate the
convection.
Using numerical models and experiments, Zhao et al.
[19,20] studied heat transfer and fluid dynamics during
collision of a liquid droplet on a substrate. They extended
the earlier model of Fukai et al. [21] to account for the
relevant convection and conduction heat transfer phe-
nomena both in the droplet and in the substrate, in the
case when there is no solidification. Their results, there-
fore, are applicable to the pre-solidification stage of the
impact process. Liu et al. [22] used a one-dimensional
solidification model in conjunction with a two-phase flow
continuum technique to track the moving liquid–solid
boundary. The model, however, does not account for
the convection in the liquid and conduction in the
substrate.
Trapaga et al. [23] used a commercially available code,
FLOW-3D [24], to study the heat transfer and solidifica-
tion phenomena during droplet impact. They assumed
that the substrate was isothermal, and neglected any ther-
mal contact resistance at the liquid–solid interface. Ber-
tagnolii et al. [25] used a finite element approach with
an adaptive discretization technique to model the defor-
mation of the droplet and evolution of the thermal field
within the splat. Their model, however, does not take into
account the solidification and heat transfer to the sub-
strate. Waldvogel and Poulikakos [26] used a finite ele-
ment model to simulate spreading and solidification
during droplet impact.
A three-dimensional simulation of impingement of tin
droplet was carried out by Kamnis et al. [27]. This model
was based on a two-dimensional model of impact of tin
droplet on stainless steel using volume of fluid technique
(VOF) [28]. The VOF technique was employed along with
solidification to compute the tracking of dynamic impact,
spreading, solidification, splashing and air entrapment of
sequentially impinging droplets over the substrate. The
numerical model was validated with the results of high-
speed camera photography of the tin droplet, and excellent
agreement between simulation and experimental results
were reported. In particular the finger formation, which
cannot be predicted by an axisymmetric model, and air
entrapment could be predicted using the three-dimensional
model.
The heat transfer through the liquid metal substrate
contact will have significant influence on the cooling rate
of the impinging droplet. If the contact resistance is high
the cooling rate of the droplet will be low, even though
the substrate metal may have high conductivity and heat
capacity. Only a few investigators [27–32] have attempted
to estimate the contact resistance of the droplet–substrate
contact.
Simulation of the process of impingement and solidifica-
tion of a droplet by a comprehensive mathematical model
would incorporate the following:
1. cooling in the droplet during flight;
2. flow in the melt taking into account the surface tension;
3. coupled solidification, heat transfer to the surrounding
gaseous atmosphere and substrate;
4. solidification and undercooling;
5. variation of physical properties with temperature.
In the models of droplet deposition of liquid metals on a
solid substrate all these have not been simultaneously
incorporated. Most of the models have not taken into
account the contact thermal resistance between the liquid
droplet and substrate. Further, very few have attempted
validation of the model.
In the present study a comprehensive mathematical
model for impingement of liquid Al–33 wt.% Cu alloy on
a stainless grade 304 stainless steel substrate has been
developed. The model takes into account the coupled heat
transfer in the liquid droplet, surrounding gaseous med-
ium, solid substrate and the contact thermal resistance
between the droplet and substrate. It further takes into
account solidification, natural convection, flow in the drop-
let and the gaseous medium. The contact thermal resistance
between the liquid alloy droplet and a grade 304 stainless
steel substrate was estimated using an inverse approach.
The model was experimentally validated.
A. Kumar et al. / Acta Materialia 58 (2010) 122–133 123

3. 2. Mathematical modeling of droplet impingement
The present study aims at developing a mathematical
model of flow and heat transfer in a liquid metal which falls
freely through gaseous medium and finally impinges on a
cold substrate. The modeling of phenomena is complex
because it involves interactions among three phases, viz.,
liquid metal, solid substrate and gaseous medium. Further,
as the droplet falls under the action of gravity and impinges
on the substrate, its shape changes rapidly with time.
Fig. 1 is a schematic sketch of the phenomenon. A drop-
let, which is initially spherical, falls through gaseous med-
ium and impinges on a substrate. Since the initial shape
of the droplet is assumed to be spherical, the fluid flow
and the heat transfer are axisymmetric.
The following assumptions have been made while devel-
oping the model:
1. The droplet initially had spherical geometry.
2. The problem is axisymmetric about the axis shown in
Fig. 1.
3. Impinge velocity of the droplet is perpendicular to the
plane of the substrate and there is no rotation of drop-
let along the axis.
4. The flow of molten alloy and gas is laminar and incom-
pressible. The velocity of gaseous media at domain walls
is zero.
5. Walls of the domain (Fig. 1) are assumed to be at ambi-
ent temperature.
6. The heat transfer is dominated by convection and con-
duction modes. Therefore, radiation from the droplet
surface to the surroundings is ignored.
7. To estimate thermal contact resistance a special solid
material has been defined. The conductivity of the spe-
cial material is estimated from the measured value of
the contact resistance.
8. Initial velocity field is zero.
2.1. Governing equations
Numerical simulations of the axisymmetric droplet
deposition processes were conducted by solving the two-
dimensional (r–z) continuity, Navier–Stokes and energy
equations. The volume of fluid (VOF) approach [33] was
coupled with Navier–Stokes and energy equations to track
the surface of the impinging droplet on a fixed Eulerian
structured mesh.
The continuity or conservation of mass equation is given
by
1@
r@r
ðruÞ þ
@v
@z
¼ 0 ð1Þ
where u and v are velocity components in the r and z direc-
tions, respectively. The momentum conservation equations
in r and z directions are
@u
@t
þ u
@u
@r
þ v
@u
@z
¼ À
1
q
@P
@r
þ t
@2
u
@z2
þ
1
r
@
@r
r
@u
@r
À
u
r2
þ gr þ Sr ð2Þ
@v
@t
þu
@v
@r
þv
@v
@z
¼ À
1
q
@P
@z
þt
@2
v
@z2
þ
1
r
@
@r
r
@v
@r
þgz þSz
ð3Þ
where P, q, and t are pressure, density, and kinematic vis-
cosity of the fluid, respectively, g represents gravitational
force per unit mass and S represents any other source term
or body force term. The interfacial tension was incorpo-
rated in the Navier–Stokes equation as a body force term.
Enthalpy–porosity formulation [34] was incorporated in
the solution scheme in order to handle the effects of solid-
ification. The formulation is described below.
Solidification results in latent heat generation and a
modified form of the energy equation, incorporating latent
heat, was used. The modified energy equation is given by
@ðqHÞ
@t
þ r Á ðquHÞ ¼ r Á ðkrTÞ þ Sh ð4Þ
where H is the enthalpy per unit volume, k is thermal con-
ductivity and Sh is the rate of energy generation per unit
volume. The enthalpy of the material was computed as
the sum of the sensible enthalpy, h, and the latent heat,
DH. It is expressed as
H ¼ h þ DH ð5Þ
where
h ¼ href þ
Z T
Tref
cP dT ð6Þ
In Eq. (6) href is the sensible enthalpy at the reference tem-
perature and cP is the specific heat at constant pressure.
The liquid fraction, b, is defined as
b ¼ 0 if T Tsolidus
b ¼ 1 if T Tliquidus
b ¼ TÀTsolidus
TliquidusÀTsolidus
if Tsolidus T T liquidus
ð7Þ
Fig. 1. Schematic representation of the mathematical model of droplet
deposition.
124 A. Kumar et al. / Acta Materialia 58 (2010) 122–133

4. The latent heat content can be written in terms of the
latent heat of freezing, L:
DH ¼ bL ð8Þ
The source term appearing on the right-hand side of Eq.
(4) is given by
Sh ¼
@ðqDHÞ
@t
þ rðquDHÞ ð9Þ
Due to high temperature difference between the droplet
and substrate and the relatively low droplet velocity, vis-
cous dissipation of heat was negligible and thus not
included in the source term.
Solidification also results in phase change. Instead of
tracking the interface, Al–33 wt.% Cu was assumed to be
a single phase pseudo-porous medium whose porosity
was proportional to the liquid fraction. Thus the porosity
of this single phase pseudo-porous medium varied from
as low as 0 in the solidified region to maximum in the liquid
region. Since Al–33 wt.% Cu is a eutectic alloy a sudden
transition in the porosity is expected at the solidification
front. To avoid numerical difficulties associated with this
sudden transition, the solidification front was assumed to
be a narrow band and the porosity was varied from 0 to
the maximum value over this band. The effect of porosity
was incorporated in the momentum equations through a
momentum source terms (Eqs. (2) and (3)). The momen-
tum source term S is given by
S ¼
ð1 À bÞ
2
ðb3
þ eÞ
Amusht ð10Þ
where b denotes the liquid fraction, e is a small number
(0.001) to avoid divisionby zero, Amush is the mushy zone
constant, and t is the velocity term.
2.2. Boundary and initial conditions
2.2.1. Boundary conditions
Since the problem is axisymmetric, only half of the
domain shown in Fig. 1 was considered in the computation
domain. Axis of symmetry was like a free-slip wall. The
normal velocity was zero. The tangential velocity did not
have normal gradient. Thus along the axis of axisymmetry
u = 0 and ov/or = 0. At the other boundaries u = 0 and
v = 0. Since these boundaries are far away from the drop-
let, the temperature at these boundaries was set equal to the
ambient temperature, i.e., 300 K.
2.2.2. Initial conditions
All zones were initialized with a temperature of 300 K,
except the droplet. The droplet was given a pre-determined
higher temperature. The initial velocities at every point
inside the domain were zero, except within the droplet,
where it was set to ÀU0.
2.3. Discretization and the solution method
The above described model was simulated on a FLU-
ENT 6.3.16. FLUENT 6.3.16 solves the above described
equations using a control volume approach, which is fully
implicit in time and uses upwind differencing in space [35].
A segregated solution algorithm [36] with a control vol-
ume based technique was used in the numerical method.
The pressure and velocity were coupled with a semi-impli-
cit method for pressure linked equation (SIMPLE) algo-
rithm [37,38], which uses a guess-and-correct procedure
for the calculation of pressure on the staggered grid
arrangement.
3. Thermo-physical properties
Al–33 wt.% Cu alloy was selected as the alloy system for
two reasons. Firstly the thermo-physical properties of the
alloy are known, and secondly the Jackson–Hunt plot is
available for the comparison. The simulation was carried
out for Al–33 wt.% Cu liquid droplets sizes of 100–
1000 lm.
Table 1 gives the thermo-physical properties of the Al–
33 wt.% Cu alloy used in the present model [39].
The contact resistance between Al–33 wt.% Cu and the
grade 304 stainless steel substrate was determined to a first
approximation using an inverse approach, described in the
following section.
3.1. Determination of thermal contact resistance
In order to estimate the thermal contact resistance, two
chromel–alumel thermocouples were introduced in the
grade 304 stainless steel channel; one was placed at
the outer wall of the channel and the other was placed in
Table 1
Thermo-physical properties of Al–33 wt.% Cu alloy.
Properties of Al–33 wt.% Cu used in model Value
Thermal conductivity (W mÀ1
KÀ1
) Ks = 155
Kl = 71
Density (kg mÀ3
) qs = 3410
ql = 3240
Solidus temperature (K) 821
Liquidus temperature (K) 821
Melting heat (J kgÀ1
) 350,000
Specific heat (J kgÀ1
KÀ1
) Cs = 1070
Cl = 895
Viscosity*
(kg mÀ1
KÀ1
) At 943 K = 1.001 Â 10À3
[40]
At 973 K = 8.624 Â 10À4
[40]
At 1023 K = 5.65 Â 10À4
[40]
Surface tension (N mÀ1
) 0.868 [41]
s = solid; l = liquid.
*
Value of viscosity of Al–33 wt.% Cu has been calculated using Arrhenius
equation: g ¼ g0 exp À E
RT
À Á
up to the solidus temperature.
A. Kumar et al. / Acta Materialia 58 (2010) 122–133 125

5. the central position of the channel cavity as shown in
Fig. 2. Subsequently Al–33 wt.% Cu alloy was melted in
an electric resistance furnace and poured into the grade
304 stainless steel channel. Temperatures in the casting
were monitored using thermocouples connected to a data
acquisition system (DAS). The temperature data at the
two locations were used to estimate the thermal contact
resistance at the interface of the wall and the liquid metal
by matching them with the temperatures at the two posi-
tions obtained from simulation.
The computational domain for the simulation along
with the boundary conditions is shown in Fig. 3. The air
gap between Al–33 wt.% Cu and stainless grade 304 stain-
less steel channel has been assumed to be a solid layer with
equivalent thermal resistance.
3.2. Initial conditions
The input of initial conditions was taken from the
experiment:
Initia channel wall temperature = 398 K.
Liquid melt initial temperature = 956 K.
Ambient temperature = 300 K.
4. Experimental validation of the model
4.1. Uniform droplet deposition process
The master alloy was prepared from a 99.96% pure Al
and a 99.99% pure Cu by induction melting. The chemical
analysis was carried out to confirm the composition of the
master alloy. Fig. 4 shows the schematic diagram of droplet
deposition process. Al–33 wt.% Cu alloy was used for the
spray casting process. A metal charge of 50 g of master
alloy was heated in a crucible, under air atmosphere, to a
temperature above the liquidus temperature. The crucible
was made of quartz and was coated with alcohol soot to
avoid reaction between the melt and quartz crucible. The
diameter of the melt delivery nozzle was 3 mm. The cruci-
ble was fixed inside a resistance furnace. A graphite stopper
was placed on the delivery nozzle at the base of the cruci-
ble. A thermocouple in the center of the crucible and the
furnace insulation allowed continuous measurement of
temperature. When a pre-determined temperature was
reached, typically 150 °C above the alloy liquidus, the
graphite stopper rod was removed and the molten metal
flowed through the delivery nozzle. The molten metal
stream flowed into the deposition chamber in the form of
metal droplets which impinged upon the substrate and
were deposited.
The solidified droplets were prepared for metallographic
examination using standard metallographic procedure and
etched with Keller’s reagent. The microstructures were
studied using a JEOL JSM-6480LV scanning electron
microscope.
Fig. 2. Schematic two-dimensional view of the experimental set-up for determination of contact resistance.
Fig. 3. Computational domain along with the boundary conditions for
computing the thermal field in the experimental set-up shown in Fig. 2.
126 A. Kumar et al. / Acta Materialia 58 (2010) 122–133

6. 5. Results and discussion
5.1. Determination of contact thermal resistance
The value of the effective conductivity of the gap region
and the thermal contact resistance were determined by
matching the computed temperatures at (i) a selected loca-
tion in the Al–33 wt.% Cu liquid melt (Location 1 in Fig. 2)
after it was poured on a grade 304 stainless steel channel
(Fig. 2), and (ii) a selected location in the stainless steel
channel (Location 2 in Fig. 1), with the experimentally
measured ones. For the value of thermal contact resistance
7 Â 10À4
m2
K WÀ1
the simulated temperature profile was
similar to the experimental profile as shown in Fig. 5.
Therefore, 7 Â 10À4
m2
K WÀ1
was considered a value of
the thermal contact resistance in the droplet deposition
model.
Fig. 4. Schematic drawing of the uniform droplet forming process.
Fig. 5. Computed and experimentally measured temperature at Locations
1 and 2 (Fig. 2).
Fig. 6. Phase profile of an impinging Al–33 wt.% Cu droplet on the substrate.
A. Kumar et al. / Acta Materialia 58 (2010) 122–133 127

7. 5.2. Simulation of droplet deposition
In the present study simulations were carried out for the
droplets, with diameters varying between 3.82 mm and
8.18 mm falling through a height of 50 cm on a grade 304
stainless steel substrate. As stated earlier, the shape of the
droplet impinging upon the substrate was assumed to be
spherical and the cooling of droplet in the gaseous medium
during the free fall was ignored. The assumption of insig-
nificant temperature drop during flight was reasonable, as
the temperature drop during the period of flight is expected
to be negligible compared to that during the spreading.
This was substantiated by an additional simulation of a
droplet during its free fall through a height of 50 cm. In this
simulation the Weber number was $312. The shape of the
Al–33 wt.% Cu droplets does not change significantly dur-
ing flight. The substrate thickness was taken as low as
1 mm in order to reduce the computational time. This will
not affect the accuracy of the prediction of the thermal field
because during the short duration of droplet spreading and
solidification the temperature rise did not take place
beyond a distance of 0.3 mm.
The computed velocity, temperature, liquid fraction and
phase (Al–33 wt.% Cu and air) fields after different periods
of time were plotted. Fig. 6 shows the phase contour of Al–
33 wt.% Cu droplet having a diameter of 8.18 mm in air. In
this case the velocity of impingement was 3.13 m sÀ1
, and
therefore the Weber number was 302. The spread of the
droplet occurred within a short time frame of 8 Â 10À3
s
and no disintegration due to the surface tension force
was observed. It can also be seen that the solidification
starts 10À2
s after the spread has taken place. Fig. 7 also
shows the velocity profile of Al–33 wt.% Cu/air system. It
can be noted from the figure that air adjacent to Al–
33 wt.% Cu is dragged in the direction of spread and a
re-circulating flow in air can be observed.
The evolution of the shape of liquid droplet impinging a
solid substrate is a complex phenomenon. However, this
has been studied by many investigators [28,42–47]. An
important parameter which decides upon the evolution of
shape of the liquid droplet is the Weber number, i.e., den-
sity, viscosity, impingement velocity and surface tension.
Depending on the Weber number, different types of shape
evolution are possible.
Fig. 7. Velocity profile of the Al–33 wt.% Cu/air system.
128 A. Kumar et al. / Acta Materialia 58 (2010) 122–133

8. The simulated evolution of droplet shape (Fig. 6) has
been reported earlier [46]. The bump near the edge has been
observed even when there is no cooling [42]. The physical
reasons for the bump formation are not fully clear
although surface tension and the existence of the bump
because of the maximum cooling near the edge (Fig. 8)
[46] could be suggested as two of the reasons.
The thermal contour in Fig. 8 shows convective cooling
at the top surface, where Al–33 wt.% Cu is colder than the
inner region. Similarly conductive cooling of Al–33 wt.%
Cu near the substrate can be seen along with heating up
of the substrate as shown in the temperature profile plotted
(Fig. 8). Fig. 9 shows the liquid fraction within Al–33 wt.%
Cu system.
Solidification front speed was estimated based on the
liquid fraction profile at 20 different locations, 7 of which
are shown in Fig. 10. Fig. 11a shows the variation of front
speed with the radial distance. Interestingly the experimen-
tally observed interlamellar spacing (Fig. 11b), which is
related to the front speed by Jackson–Hunt relationship,
also correlated with the radial distance in a similar manner.
Fig. 11c and d shows the microstructure at the marked
radial distance (Fig. 11b).
The variation front speed with the radial distance can
be explained in terms of variation of splat thickness with
the radial distance. Fig. 10 shows the different locations
in the splat along the radial direction. Fig. 11a shows
the growth rates at those locations. It can be observed
that the front speed is significantly less at thicker
locations (L3, L6, L12, L18, L19) and highest front
speeds can be observed only at the thinnest locations
(L9,L15).
Fig. 8. Temperature profile of an impinging Al–33 wt.% Cu droplet on the substrate.
A. Kumar et al. / Acta Materialia 58 (2010) 122–133 129

9. 5.3. Sensitivity of font speed to variation in contact
resistance and natural convection
In order to study the sensitivity of front speed to varia-
tion in contact resistance, the contact resistance was varied
from 6.3 Â 10À4
m2
K WÀ1
(10% less than experimentally
determined value) to 7.7 Â 10À4
m2
K WÀ1
(10% less than
experimentally determined value). At the thicker locations
the front speed was found to be insensitive to the contact
resistance. However, at locations L9 and L15, which are
the thinnest locations, front speed is sensitive to the contact
resistance. Thus it can be concluded that the variation in
the front speed along the radial direction is due to variation
in the splat thickness.
Simulation was carried out where natural convection in
both Al–33 wt.% Cu and air was suppressed. The results
suggest that natural convection did not play a significant
role as compared to the conduction through the substrate.
5.4. Effect of the initial temperature
Simulations were carried out for the two values of initial
temperature; viz., 875 K and 973 K. The initial temperature
had a strong influence on the extent of spreading. Fig. 12
shows the spread in diameter vs. time for 8.18 mm droplets
having initial temperature of 875 K and 973 K. It can be
seen that the spread increases with the increase in initial
temperature or the superheat. This is expected, as due to
Fig. 9. Liquid fraction profile of the Al–33 wt.% Cu/air system.
Fig. 10. Profile of the deformed droplet location for determination of the cooling rate at different positions.
130 A. Kumar et al. / Acta Materialia 58 (2010) 122–133

10. the rise in the initial temperature, both the fluidity and time
required for the initiation of solidification increases.
Experimental validation of the simulated spreading was
carried out for 8.18 mm droplet having an initial tempera-
ture of 973 K. 3.3 cm spread was observed in the droplet,
which matches reasonably well with the predicted spread
%3.7 cm.
The initial temperature of the droplet had a strong influ-
ence on the front velocity. The front velocity significantly
decreased with increase in the initial temperature. The aver-
age front speed (averaged over 7 locations, as shown in
Fig. 10) of 8.33 mm sÀ1
for initial droplet temperature of
875 K dropped to 6.05 mm sÀ1
when the initial droplet
temperature was increased to 973 K. Front speed is related
to the microstructure and thus appropriate interlamellar
spacing can be obtained by controlling the initial tempera-
ture of the droplet.
5.5. Effect of the droplet diameter
Droplet diameter had a significant effect on the time to
spread and front velocity. The time to spread increased
with increase in the initial diameter of the droplet. This is
as expected because the time to spread is expected to
increase with increase in the size of the droplet. Fig. 13
shows the effect of the initial diameter of drops on the
average solidification front speed for the initial droplet
Fig. 11. Variation of (a) front speed with radial distance of droplet for different value of contact resistance, (b) measured interlamellar spacing (k) (initial
temperature = 973 K, droplet diameter = 8.18 mm), (c) and (d) microstructure of splat at marked radial distance.
Fig. 12. Effect of initial temperature on the spread.
A. Kumar et al. / Acta Materialia 58 (2010) 122–133 131

11. temperature of 973 K. The front speed initially increases
with the diameter and then decreases.
5.6. Validation of the model
The present model is a comprehensive one that takes
into account the convective flows within and outside the
melt, solidification effects, conductive heat transfer through
the contact interface and the surface tension effects. As
mentioned earlier, Al–33 wt.% Cu was selected due to
availability of physical properties and also due to the fact
that the simulated results can be validated using J–H plot.
The experimental validation was carried out only for
droplet with 8.18 mm diameter having a superheat of
150 K (initial temperature = 973 K). The solidified droplet
was sectioned at seven different locations (Fig. 10) and the
sectioned pieces were metallographically prepared to char-
acterize the microstructure.
For validation, solidification front velocities in a droplet
of size 8.18 mm of Al–33 wt.% Cu alloy were calculated
using the model developed in the present work. These val-
ues were plugged in the Jackson–Hunt relationship
k2
V = 88 lm3
sÀ1
[48] for calculation of k from those veloc-
ities. Fig. 14 shows the experimental values obtained for
various sections as shown in Fig. 10 super imposed on
those calculated from this model. A very good correlation
is seen in the plot between experimental and simulated val-
ues within standard errors of experimental measurements.
6. Concluding remarks
1. Comprehensive CFD based modeling of droplet deposi-
tion on a cold substrate was developed on a FLUENT
6.3.16 platform.
2. The model predicted a rapid spread of droplet much
before the initiation of solidification. The spread was
found to depend on superheat and droplet diameter.
3. Front speed was estimated accurately from the liquid
fraction profile predictions at different time steps. It
was sensitive to the droplet size and superheat. It ini-
tially increased with droplet size and then decreased.
The front speed decreased with increase in the
superheat.
4. The front speed oscillated along radius of the splat. This
is consistent with the variation of the interlamellar spac-
ing along the radial distance.
5. The prediction of the model on droplet spreading
matched well with the experimental findings and so
did the prediction of the interlamellar spacings at differ-
ent locations of the splat.
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