Modeling of Solution Droplet Evolution and Particle Morphologies in
Solution Plasma Spraying
Yuan Hu, Yanguang Shan
School of Energy and Power Engineering
University of Shanghai for Science and Technology, Shanghai, China
Abstract: A thermal physical model of an individual droplet containing ceramic precursor solution
is presented to study the heat and mass transfer within the droplet in Solution Plasma Spraying (SPS)
process. The temperature and concentration distributions within the droplet are predicted. The influence
of Stefan flow, the variable physical properties of the solution and environment gases, as well as the
internal circulation due to the relative velocity between the droplet and the plasma jet are considered.
Based on the distribution of the solute concentration within the droplet, the regions of droplets where the
solute might precipitate are predicted by employing the simple homogeneous nucleation hypothesis. The
different microcosmic morphologies of the particles with different injection parameters are compared.
Keywords: Solution Plasma Spraying, droplet evaporation, plasma, internal vortex, particle morphology
1. Introduction
Solution Plasma Spray (SPS) is developed to
attaining nano-structured coatings which is applied
to protect hardware against severe environment to
prolong engineering components service life. In this
process the precursor solution containing ceramic
salts is atomized into a spray by a pressure atomizer.
After injected into the plasma jet, the spray droplets
are rapid heated up and accelerated. As the droplets
moving towards the substrate location, the solvent of
the ceramic salts solution evaporates and the solute
concentration increases. Once reaching the critical
super saturation the precipitation is expected to
commence that might results in the formation of
particles with different microcosmic morphologies
such as solid or hollow shell. The particles arrive on
the substrate and generate the nano-structured
coatings[1]. The property of the coating is highly
depended on the particle morphologies and state on
the substrate.
In order to understand the SPS process and gain a
better performance of coating, a thermal physical
model is applied to simulate the trajectory and
transport phenomena of individual precursor droplet
in the high temperature plasma jet. This model
involves the motion and evaporation of the droplet
in the plasma environment and the heat and mass
transfer within the droplet. In this paper the model is
employed to simulate the performance of ZHC
(Zirconium hydroxychloride) solution droplets in the
argon plasma jet. The effect of Stefan flow, the
variable physical properties of the solution and
environment gases, as well as the internal circulation
due to the relative velocity between the droplet and
the plasma jet are considered. The temperature and
concentration distributions within the droplet are
predicted. The particle morphologies on different
operating parameters are predicted as well.
2. Mathematical models
The model is consisted of the droplet moving
model, gas-phase model and liquid phase model.
This physical model is based on the following
assumptions: (1) the droplet is assumed to be
spherical; (2) the influence of vaporizing droplets on
the plasma is neglected; (3) only single droplet is
considered, thus the effect of neighboring droplets
are neglected; (4) the vapor phase surrounding the
droplet is in quasi-steady state; (5) the gravitational
effect is neglected, the droplets motion is controlled
by aerodynamic drag.
In this paper the plasma temperature and velocity
fields predicted by the model developed in Ref 2 are
utilized as known conditions. The simulation is run
for an argon plasma jet issuing into air surroundings.
The argon flow rate is 35.4slpm and the plasma
power input is 13.68KW (the operational voltage is
15.4V, operational electric current is 900A). The
plasma temperature and velocity fields are shown in
Fig.1.
which influences the thicknesses of the gas film,
then the corrected Sh and Nu are given by [5]:

0.552 Re1/ 2 Sc1/3  ln(1 + BM )
(11)
Sh =  2 +


F ( BM )

 BM

0.552 Re1/ 2 Pr1/3  ln(1 + BT )
(12)
Nu =  2 +


F ( BT )

 BT
Figure 1. Temperature field and velocity field of the plasma jet
The droplet momentum and droplet size variation
equations are as follows[3]:
∂U 3CD ρ∞
=
U ∞ − U (U ∞ − U )
∂t
8rs ρ L
∂V
3C ρ
=− D ∞V2
8rs ρ L
∂t
(1)
evaluated as:
(4)
where the Reynolds Number is defined as:
Re = 2 ρ∞ U ∞ − U rs / µ g
(5)
The following expressions of the instantaneous
& for heat transfer and mass
vaporization rate m
transfer is from the film theory employed in this
study[3]:
m& = 2πρ g Drs Sh ln(1 + BM )
m& = 2π
BM
,
λg
C pv
rs Nu ln(1 + BT )
(6)
(7)
BT are Spalding mass and heat transfer
numbers which are calculated as:
m − mv∞
BM = vs
1 − mvs
C (T − T )
BT = pv ∞ s
L + Qg / m&
(13)
Where Qg is the heat transfer from ambient plasma
Qg = 2π r λ∞ (T∞ − Ts ) Nu
(14)
Qi is the heat conduction from the film at droplet
(2)
(3)
where CD is droplet drag coefficient which can be
24
Re(1 + BM )
& + Qi
Qg = mL
gas to drop surface and it is given by:
∂rs
m&
=−
∂t
4πρ L rs2
CD =
At the droplet’s surface the energy balance is
expressed as:
surface into its interior which is calculated as:
4
dT
Qi = π rs 3C p s
3
dt
(15)
The equations (6) and (7) are equated at the
droplet surface which yields the vaporization rate as
well as Nu . Once these quantities are determined,
the instantaneous droplet surface temperature at next
time step can be solved numerically using the
iteration method.
Due to the relative velocity between the droplet
and the plasma environment, there is an internal
circulation in the droplet caused by the surface
[3]
friction. It has been suggested by Sirignano that
the internal circulation could be represented by the
well-known Hill’s spherical vortex. The velocity
field in the spherical coordinate system inside the
moving droplet is given as:
Vr = U s (1 − r 2 / rs2 ) cos θ
(16)
Vθ = U s (1 − 2r 2 / rs2 )sin θ
(17)
Where U s is the maximum surface velocity which is
(8)
expressed as:
Us =
(9)
Sh and Nu are the Nusselt and Sherwood numbers,
respectively. In consideration of the Stefan flow
µ 
1
∆U ∞  g  Re∞ CF
32
 µL 
(18)
Where CF is the friction drag coefficient in the
following correlation considering the effect of Stefan
flow:
∂
λ
( ρT ) + div( ρ uT ) = div( gradT ) + ST (20)
∂t
CP
∂
( ρ ml ) + div ( ρ uml ) = div (Γgradml ) + Rl (21)
∂t
The trajectory, velocity and instantaneous
radius of the droplets are obtained from the motion
model. The gas-phase model is calculated to provide
the vaporization rate and the heat transfer into the
droplet as the boundary condition in solving the
liquid phase equations. Once this mass and energy
equations are solved, the temperature and
concentration distributions within the droplet can be
predicted. When the concentration in the droplet
reaches the critical super-saturation of the solution,
the computations will stop and the solute is assumed
to precipitation. If the concentration of the solution
everywhere within the droplet reaches equilibrium
saturation, the precipitation is volumetric. Otherwise
the solute precipitates at the point where the
concentration is greater than equilibrium saturation[7].
The final particle morphology can be predicted then.
3. Result and discussion
• The effect of injection velocity
In this paper, the simulation is performed for
30μm ZHC (Zirconium hydroxychloride) solution
droplets with injection velocity of 5m/s, 10m/s and
15m/s. The initial temperature and solute
concentration of the droplet is taken as 300K and 0.3.
The properties of ZHC are available in ref 8.
The trajectories and surface temperature of the
droplet with different velocities are shown in Fig.2.
The variations of droplet size and relative velocity at
the surface are shown in Fig.3. It is shown that the
droplet with higher injection velocity penetrates
deeper than the droplet at lower velocity. And hence
the variation of surface temperature and the
reduction of droplet size are faster. According to the
Hill’s spherical vortex model, the distribution of
temperature and solute concentration are determined
380
0.01
5m/s
10m/s
15m/s
5m/s
10m/s
15m/s
0.008
360
surface temperature(K)
The governing equation of the energy transfer
inside the droplet and the conservation of species are
[6]
expressed as :
by the distribution of velocity inside the droplet
which is decided by the relative velocity of the
droplet and the plasma environment. As shown in
fig.3, the relative velocity at the droplet surface
increases as the injection velocity increases.
0.006
340
0.004
320
0.002
300
0
0
0.01
0.02
0.03
0.04
0.05
0
0.06
0.0005
X(m)
0.001
0.0015
0.002
time(s)
Figure.2 Effect of velocity on droplet trajectories and surface
temperature
350
1
5m/s
10m/s
15m/s
300
250
relative velocity(m/s)
(19)
Y(m)
12.69
Re (1 + BM )
2/3
0.98
200
R/R o
CF =
0.96
150
5m/s
10m/s
15m/s
100
0.94
50
0.92
0
0
0.0005
0.001
time(s)
0.0015
0.002
0
0.0005
0.00 1
0.0015
0.00 2
time(s)
Figure.3 Effect of velocity on droplet radius variation and
relative velocity
Based on the distribution of temperature and
concentration inside the evaporating droplet, the
precipitated regions are estimated by employing the
simple homogeneous nucleation hypothesis.
According to the homogeneous nucleation
hypothesis, once the critical super-saturation
concentration is achieved, the precipitation takes
place in the droplet in those regions where the
equilibrium saturation concentration is exceeded.
The particles morphologies of droplets with different
injection velocities at the onset of precipitation are
shown in talble.1. For a 5m/s droplet, when the
critical super-saturation concentration is achieved,
equilibrium saturation concentrations are exceeded
everywhere since the vortex inside the droplet is
weak. This results in the formation of solid particle.
For a 10m/s droplet, the figure shows that at the
onset of precipitation, the minimum concentration
within the droplet is lower than the equilibrium
saturation, this results in the formation of a crust
around the droplet surface. Due to the effect of high
degree of internal convection, the solute
concentration is high around the droplet perimeter
350
1
20um
24um
30um
300
0.98
250
relative velocity(m/s)
Table.1 Particles morphology of droplet with different velocity
Concentration
Morphology
V
temperature the droplet exposes to, the surface
temperature increases faster and the droplet radius
decrease faster with increasing droplet size. Due to
the deeper penetration, the lager droplet experiences
higher velocity of plasma environment which cause
the higher relative velocity at the droplet surface.
And hence the effect of internal convection inside
the droplet will be more remarkable.
200
0.96
R/R o
which formed a thin shell trapping the interior liquid
for 15m/s droplet. As the droplet travels in the
plasma jet after precipitation, the internal liquid
might evaporate through the small opening of the
shell and form a solid particle finally reaching the
substrate, or could explode in flight and form a
series of small particles due to the built up of
internal pressure. The prediction of these possible
scenarios is another part of our project, which is not
included in this paper.
20um
24um
30um
0.94
150
100
0.92
50
0
5m/s
0.9
0
0.0002
0.0004
0.0006
0.0008
time(s)
0.001
0.0012
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
time(s)
15m/s
10m/s
Figure.5 Effect of droplet size on radius and relative velocity
• The effect of droplet size
The initial droplet size is another important factor
affecting the droplet vaporization and particle
morphology. In order to analyze the droplet size
effects, the initial size of 20, 24 and 30μm are
compared. The 300K droplets with initial solute
concentration of 0.3 are injected into the plasma jet
in 12m/s.
380
0.01
20um
24um
30um
20um
24um
30um
360
surface temperature(K)
0.008
0.006
Y(m)
340
0.004
The different precipitate zones inside the droplet
with different size are shown in table.2. As analyzed
earlier, the solute concentration increases uniformly
within the smaller droplet with weaker internal
vortex. So in the 20μm droplet, when the
precipitation occurs, the solute concentration within
the droplet exceeds the equilibrium saturation and
then forms a solid particle. For the 24μm droplet,
when the surface concentration reaches critical
super-saturation, the solute precipitates and forms a
zirconia shell which interconnects along the axis of
the droplet. This is cause by the high degree of the
internal vortex. In the case of 30μm droplet, the
precipitation takes place around the droplet surface
where the shell is thick at the region facing the wind.
This is due to the effect of internal convection and
experience of high environment temperature, which
results in faster evaporation and higher solute
concentrations in this region.
3. Conclusion
320
0.002
300
0
0
0.005
0.01
0.015
X(m)
0.02
0.025
0.03
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
time(s)
Figure.4 Effect of droplet size on trajectories and temperature
In Fig.4, the droplet trajectories and surface
temperature are presented. As expected, the smaller
the initial size, the less the droplet penetrates. A
comparison of the radius variation and relative
velocity of these droplets along their trajectories is
shown in fig.5. As a consequence of higher plasma
Solution Plasma Spray (SPS) process is widely
used in attaining nano-structured coatings. A
numerical model has been developed in this paper to
study the heat and mass transfer and precipitation of
solution droplets in SPS process. The effect of
droplet size and injection velocity on the droplet
motion, evaporation and particles morphology are
investigated. The results demonstrate that high
momentum droplets are able to penetrate deeper into
the plasma core and experience higher temperature
and velocity of plasma environment which in turn
affect the vaporization rate. The higher vaporization
rates result in the build up of high concentration at
the droplet surface where the precipitation is
expected to commence when the critical saturation
concentration is exceeded. Homogeneous nucleation
hypothesis and the distribution of solute
concentration are employed to estimate the particles
morphology when the precipitation takes place. It
has been shown that the lower injection velocity and
smaller initial size favor production of solid particles.
On the other hand, the droplet with higher injection
velocity and bigger initial size result in the formation
of a crust around the droplet surface trapping the
interior liquid due to the effect of high degree
internal convection.
Concentration
Morphology
30μm
24μm
20μm
Size
Table.2 Particles morphology of droplet with different size
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