## Abstract

An analytic model of the flattening process based on assumptions considering both the particle deformation shape and flow field is presented.The deformation shape assumes that the thin disk begins to spread from the point of the particle impinging on the surface,and that the disk thickness is constant during the flattening process.The particle is divided into three regions,each with an assumed that depends on time exponentially.The model describes the flattening ratios of molten particles as a function of time.The model was compared with experimental results of molten droplets allowing to fall freely onto flat surfaces.The flattening behavior of impinging molten tin and zinc droplets was photographed by high-speed camera and the spreading diameters of molten droplets were measured against time after impingement.

THE IMPINGING AND DEFORMATION PROCESS of a heated particle can be divided into four categories,as shown in Fig.1.i)the molten viscous particle impinges and spreads as a thin disk,ii)the molten viscous particle impinges and splashes with some part of the particle remaining on the substrate,iii)the plastic particle impinges and is deformed plastically,iv)the elastic particle impinges and bounces off the substrate.

The impingement of viscous and plastic particles is relevant for thermal spray coatings.Elastic particle impingement can be neglected.In general,particle velocity is lower in plasma spray,so only molten particles may deposit on the substrate or previously deposited layers.When these viscous particles impinge on the substrate the particles spread radially in contact with the substrate or they splash in the process of spreading.The particular conditions that cause splashing are not readily apparent,and they have not been extensively studied.The particle velocity is considerably faster In HVOF splaying that both plastic and viscous particles may deposit.Plastic deformation becomes particularly important with high velocity particles where the impinging force causes the softened particles to mechanically bond to the substrate

surface.The elasticity of the softened particles is small compared to the kinetic energy on contact with the substrate and so the particles do not bounce off.

This paper describes a model for radial flow behavior of the impinging and spreading processes of molten tin and zinc particles were observed by a high-speed camera and the model was compared with the experimental values.

## Modeling of The Flattening Process

The deformation of molten particles on contact with the substrate changes with time and the flow field in the particle and so these must be determined in order to model the flattening of an impinged molten particle on a flat surface.When the deformed particle shape and flow field are known,the dissipation energy by viscous friction in the fluid can be determined. It is assumed that the flattening process is isothermal.The particle deformation assumes that a thin radial disk begins to spread from the contact point of the particle on the substrate surface immediately after impingement.The disk

thickness,*h*,is constant during the flattening process as shown in Fig.2.The disk thickness *h*is influenced by the impinging velocity and the particle diameter and viscosity.The particle is divided into three regions in the flattening process as shown Fig.3,Region I is the part of the sphere on the disk,Region II is the disk of radius *R0 = d0 / 2* where *d0* is the particle diameter before impingement and Region III is outside of Region II in the disk.

If *φI*,*φII* and *φIII* are the dissipated energies by viscous friction in Region I,II and III during particle spreading,respectively,the kinetic energy of particle,*Ekin*,on impingement is,

where *Esuf* and *Eint* is the surface and interfacial work for the spreading process,respectively.Assuming that the flow is laminar in Region I and II,the flow field *** expressed in cylindrical coordinates is,

where ***,*** and *** are coefficients which are determined

from boundary conditions,and *** is time.It is reasonable to assume that the flow field exponentially decreases with time,because it is considered the flow field rapidly decreases during the flattening process.Eq.2 and 3 mean that the radial flow velocity is proportional to *** in Region II and is inversely proportional to *** in Region III.The flow field *** satisfies the equation of continuity,namely div ***.If the dissipated energy,***,is small in Region I,due to the small velocity gradient,then,

and,*** and *** are expressed as follows,

where *** and *** are the fluid volumes of Region II and III,and *** is particle viscosity(1).If"***"is taken as the particle surface tension,then surface work can be written as the product of surface tension and increased surface area,

where *** is the disk diameter after the spreading is complete.Taking *** as the impinging velocity and *** as the particle density,the expression becomes,

By assuming that interfacial work *** is negligible,and substituting Eq.4,5,6,7 and 8 into Eq.1,the expression becomes,

The next boundary condition and assumption allow

coefficients ***,*** and *** to be determined.i)the fluid volume flowing out of Region II at time *** through the boundary between Region II and III equals the fluid volume flowing into Region III.ii)The assumption is that the particle moves at the impinging velocity *** for a very short time after impinging,and the particle bottom deforms into the thin disk.From the condition i,the next equation holds,

Eq.10 can be simplified taking.

If the particle forms a disk with radius *** at *** and *** at *** impinging at time ***,then the volume of Region III at time *** can be expressed as,

Since the disk volume of radius *** equals the sum of the volumes of Region II and III,then,

and Eq.13 can be simplified to,

After differentiating Eq.14 by ***,and substituting *** and ***,the expression becomes,

While it is assumed that the portion OAB in the sphere deforms into a disk of radius *** and thickness ***,as shown in Fig.2,then

When ***,it is assumed that *** moves with particle velocity ***,and so,

Differentiating Eq.16 by ***,the expression becomes,

At time,***,when the particle contacts the surface,Eq.18 becomes,

Substituting ***,*** and *** into Eq.16,when *** is negligible compared to ***,*** is solved to,

Substituting Eq.20 into *** in Eq.18,and assuming ***,the expression becomes,

When the second term in the parenthesis in Eq.21 is negligible compared to 1,Eq.21 is simplified as,

That is,*** can be considered as *** from Eq.19 and 22.Combining Eq.22 and the equation obtained by substituting *** into Eq.15 yields,

Substituting *** and *** into Eq.14 and *** is the disk radius at the time ***,the expression becomes,

***,*** and *** are determined from Eq.11,23 and 24 as,

Substituting ***,***,***,***,Eq.26 and 27 into Eq.14,*** is solved as,

When the disk diameter at time *** is ***,then disk

diameter *** at time *** is solved from Eq.28.When ***,Eq.28 is simplified as,

Since coefficients ***,*** and *** are determined,the flow field is determined,therefore,Eq.9 can be solved as,

Since *** and ***,Eq.30 is simplified as,

As a result *** is solved.When Reynolds number *** and Weber number *** is defined as,

and approximating Eq.31,the expression is simplified as,

When ***,so that surface work is negligible,Eq.34 is simplified as,

Substituting Eq.35 into Eq.29,the expression becomes

The flattening ratio as a function of time is thus obtained.

## Experimental procedure

Molten tin and zinc droplets were allowed to fall freely and impinge on flat,smooth polished substrates,and their spreading behavior was photographed by a high-speed camera.The schematic of the experimental equipment is shown in Fig.4.The glass tube filled with metallic tin or zinc was set in the electric furnace.The tube temperature was controlled by the thermocouple contacting the tube surface.The tube had an inner diameter of 10mm and which was reduced to

approximately 0.35mm inner diameter at the tip.Experimental conditions are shown in Table 1.Molten droplets fell from the glass tube tip.Impinging velocity was determined by applying the equation *** for molten particles falling in gravity,where *** and *** are gravity and the height of the tube,respectively.Molten particle diameter was determined by measuring the weight of splats.Solidified splat diameters were measured by a micrometer.The disk diameters in the photographs were measured as the reference length of the solidified splat diameters.

## Experimental Results and Discussions

A flattening behavior is shown in Fig.5.The picture shows that a radial thin disk begins to spread from the particle bottom,the particle above the disk surface retains its shape for a short time after impinging,then the disk rapidly increases and the spreading velocity then slows down.It agrees well with the assumed deformation shape.

Physical properties for the theoretical calculation and the experimental results of the flattening ratios are shown in Table 2.The experimental results of the flattening ratios are plotted against time in Fig.6,7 and 8.

The theoretical curves which consider the contribution of surface work are shown in the figures.Fig.6-(a)shows that the model agrees well with the experimental results,in particular,the curve considering surface work.Surface work can not be negligible for the slow impinging velocity of the experiments.Figures 6-(a),(b)and(c)show that the deviation of experimental results from the theory increases with decreasing particle diameter.Since the particle is smaller,the particle decreases temperature more rapidly,and the viscosity of the smaller particle increases more quickly compared to larger particles.

The experimental results illustrated in Fig.6,7 and 8 show that the disk diameters rapidly increase from the point of impingement,then slow down and eventually the disk diameters slightly decrease.This is evidence that the disks do

not solidify at the time of contact.Solidification occurs after flattening and the surface tension causes a slight decrease in disk diameter.Figures 6-(a),(b)and(c)show that as the particle diameter decrease,the time at which the disk diameter finally decreases becomes shorter.The temperature drop is thus faster for smaller particles.Viscosity increases more rapidly with faster drop in temperature,and the time for reaching the naximum diameter becomes shorter.

Fig.6-(b)and Fig.7 show that,although the particle

diameters are the same at impingement,the deviation of the experimental disk diameter from the theory is larger on the stainless steel substrate compared to ***.This is due to the particle temperatures are different.

The zinc particle on stainless steel appears to expand more than the theory as shown in Fig.7.This can not be explained.

Fig.9 shows the flattening ratios of experimental results related to Reynolds number.This model theory,and the Madejski and Jones theories are compared with the experimental results(2.3).Madejski and Jones equations are expressed as,

The Madejski theory is too large compared to the experimental results even if surface work is considered and the experimental condition is not isothermal.On the other hand,the Jones theory is too small,and if the contribution of surface work and temperature drop from heat transfer is considered,the Jones values becomes much smaller.This model agrees quite well with the experimental values;if the particle temperature decrease by heat transfer was considered,the results would agree with our suggested model.

## References

- Landau and Lifshitz,Fluid Mechanics 2nd edition,Chapter I,Section 16,Pergamon Press,(1987).
- J.Madejski,Solidification of Droplets on A Cold Surface,Int.J.Heat Transfer,19,1009-1013(1976).
- H.Jones,Cooling,freezing and substrate impact of droplets formed by rotary atomization,Journal of Applied Physics,4,1657-1660(1971).