Flattening of thermally sprayed particles

February 1995

MaterialsLetters22 (1995) 209-213

Flattening of thermally sprayed particles V.V. Sobolev, J.M. Guilemany Metal&a

Fisica, Ciencia a’e Materiales. Departamento de lngenieria Quimica y Metalurgia, Vniversidad de Barcelona, Marti i Franques 1, 08028 Barcelona. Spain

Received2 August 1994;in final form 30 October 1994;accepted2 November 1994

Abstract The analytical formulae describing the time evolution as well as the final values of the splat thickness and radius in the process of flattening of droplets at a substrate surface during thermal spraying are obtained. The theoretical results agree well with the experimental data and the results of the other authors. They can be used for the predictionof the splat flatteningparameters.

1. Introduction The coating quality during thermal spraying depends very much on the flattening of the powder particle droplets [ 1,2]. The engineering practice needs rather simple formulae which permit one to estimate this process. Important theoretical results concerning the process of droplet flattening were obtained in Refs. [ 3-61. In Ref. [ 31 some analytical formulae were found for the ratio &= R/R, of a flattened splat (disk) radius R to that of a droplet R, which in general is a solution of the integro-differential equation with respect to time. In Refs. [4-6] the mathematical simulation of the splat flattening was carried out and in Ref. [ 41 some important analytical results were obtained. Information about the time evolution of droplet flattening depending significantly on a powder particle inflight behaviour plays an essential role in the investigation of the coating formation [ 2,7-121. Besides the experimental way which is both difficult and expensive, this information can be obtained by means of the mathematical simulation or/and the analytical formulae describing not only the final values of the splat radius R but also those of its thickness b and the transient characteristics of the flattening process. 0167-577x/95/$09.50 Q11995 Elsevier Science B.V. All rights reserved SSDIOl67-577x(94)00245-2

Although these formulae are approximate they reflect all the main features of the process and permit one to obtain the necessary estimates which do not differ markedly from the results of mathematical simulation which demand significantly more efforts to be established. This paper is devoted to obtaining such analytical formulae which agree well both with the results of the other authors [3-51 and with the experimental data [ 43Sl.

2. Main equations We shall consider a droplet of radius Rp impinging normally onto the substrate surface and forming a cylindrical splat (disk) of radius R and thickness b which vary with time during flattening (Fig. 1). The experimental data presented in Ref. [ 1 ] show that the characteristic times of droplet flattening are considerably shorter than the time intervals of subsequent droplet solidification and hence the latter can be neglected. The experimental results established in Ref. [ 33 also give evidence of this fact. There it is noted that the thermal properties of the cold substrate surface, on

V. V. Sobolev, J.M. Guilemany /Materials Letters 22 (1995) 209-213

210

I’

where U is the droplet impinging velocity. We shall suggest that the major contribution to the energy balance is due to the splattening when R I r I R,, b I z I 0 [ 131. The kinetic energy can be determined from the formula R

W,=wp

b

a

Fig. 1. Initial (a) and final (b) stages of a droplet impingement at the substrate surface.

which the impinged droplets solidify, have essentially no influence on the value of 6, and this fact has been observed for isolators as well as for good conductors as aluminium or copper. Taking into account that the substrate thermal properties influence very much the droplet solidification [ 7,8], from the experimental results of Ref. [ 31 it also follows that the characteristic time of droplet solidification significantly exceeds that of droplet flattening. Therefore, droplet solidification need not be accounted for when droplet flattening during thermal spraying is considered. The same refers to the surface tension effects which can be neglected as the Weber numbers during thermal spraying exceed significantly theunity [3,4,13]. During the splat liquid motion in the process of flattening, energy conservation takes place:

where W,, W, are the kinetic and potential energies respectively and W, is the energy of friction forces. Determining W,, W,, and W, we shall use the corresponding formulae from Ref. [ 31. To find W, and W, we need to know the velocity field in the splat. We shall use the cylindrical coordinate system (r, rp,z) in which the flow velocity Vin general has three components: V,., V,, V,. Since the molten particle flow is symmetrical with respect to the z axis, V, = 0 and V, and V, do not depend on Q. We present V, and V, in the following form which satisfies the continuity equation and is similar to that presented in Ref. [9]: v,=F(t)z2r-2,

(2)

V,= F(t)z3/3r3,

(3)

c=exp(at),

rdr

RO

1

(VF+V:)

a= UR;‘,

dz,

(4)

0

where p is the droplet material density, b is the splat height and R. is the splat initial radius. We shall consider the most interesting case when RB R@ Substituting (2), (3) in (4) and integrating, we obtain the following approximate expression for W, with an accuracy up to the terms of the order O(R;R-*): W,=0.l~R,~pF~b’.

(5)

The friction power can be presented in the form:

(6) RO

Here the shear stress r depends on the dynamic viscosity p of the molten droplet and it is determined with the formula:

dvr

r=pxdz

$ tw,+w,+wf)=o,

F(t) =cU,

I

b

(7)

also 6

UsingEqs. (2), (7) and (8) wehavefiom (6) with the accuracy up to the terms of the order O(RgR-*) that dW, -=dt

2 n-pF*b*z. 3R;

(9)

The potential energy W, created by the surface tension acting on the splat free surface in our case can be neglected in comparison with W, and Wf because the Weber numbers are extremely high [ 141. For our purposes it is more convenient to use the mean values of the time derivatives in ( 1) with respect

V. V. Sobolev, .I.M. Guihnany /MateriaLs Letters 22 (1995) 20%213

to z which are determined with formula (8). Substituting (5) and (9) in (I ) and carrying out an averaging with respect to z we obtain p 5b4F2 $

+ $! pF2b3 = 0.

+ 2Fb5 $ )

(

(10)

From the mass conservation condition it follows that R2b =

4R3 3 P’

(11)

3. Resultsand discussion Rutting the initial value b. of b at t= 0 equal to R, we have from ( 10) that

1, 112

exp(-0.8at)-

b=Rp

&

Re = 2R,UpIp.

( 12)

It follows from ( 11) that R=Rp

(13)

R = R,fi

exp( 0.2ut).

2

In( 1+0.3Re).

Let us estimate this value for example for the plasma sprayed molybdenum particles. Rutting U= 100 m SC’, R,=2X 10e5 m, p= lo4 kg me3, ~=0.004 N s mP2 we obtain from (20) that t, = 2 ps. This value exceeds the characteristic time of flattening of these particles [ 11. This means that the obtained formulae ( 12)-( 15) are valid in the whole interval of the particle flattening while Eqs. (16)-(19) can be used when r-~ r, = R,iT ‘. Under the given parameters of the plasma sprayed molybdenum particles tC= 0.2 /..K+. Let us find the values of b and R at t= c*. Using ( 12)) ( 13) and (20) we have the following b = 10Rp13Re,

(21)

R = 0.632Rp&.

(22)

When ReB 1, from (12), (13) and (20) we obtain b=Rp&%,

(23)

R = 0.8546R$e1’4.

(24)

(14)

dbldt = - 0.4U exp( - 0.4ut).

(15)

At t=t.,

Formula ( 14) shows that the flattening is a continuous process if the viscous effects are negligible. At the initial moments of flattening when at -K 1 we obtain from ( 12) and (13) the following:

R=Rpfi[l+

$I+

(16)

&)I,

:(I+

g)].

(17)

When Re >> 1 and at cz 1 we have

(18) (19) The analytical results obtained are valid up to f < t, where

(25)

from (25) we obtain

dbldt= -0.73URe-“2.

(26)

At the initial moments of time when t-~ f, and Re =EZ1, from(18) wehave dbldt= -0.4U.

b=Rp[ly

(20)

We can also obtain the rate characteristics of the flattening process. When ReB 1, from ( 14) we have

When ReB 1 we have from ( 12) and ( 13) that b = Rp exp( - 0.4at),

t. =

211

(27)

It is seen that at the initial stage of flattening the rates of changes of the splat thickness and length are determined only with the particle impinging velocity and do not depend on the Reynolds number. The similar situation was observed in Refs. [ 4,6]. From (23) it follows that the final thickness of flattened droplet for the plasma sprayed molybdenum particles under the given Reynolds numbers is about 2% of the droplet initial radius which agrees with the results presented in Ref. [ 41. This value should be somewhat higher when taking into account the surface tension effects characterized with the Weber number which influences mainly the final stage of flattening [ 3,4].

212

V. V. Sobolev, J.M. Guilemany /Materials

Letters 22 (1995) 209-213

l-Formula~24) Z-Modejski's formula

200

400

600

800

1000

Reynolds number

Fig. 2. Comparisonof analytical and experimentalresults describingfinal splat radius (flatteningdegree); ( ...) - experimentalresults from Ref. [El.

IO3

2.104

lo4

Reynolds

l-formula

1281

Z-formula

(29)

3.104

4404

number

Fig.3. Comparisonof analyticalformulaedescribingtime of splat flattening.

In Fig. 2 the comparison of the flattening degree (dimensionless splat radius) r= R/R, calculated with formula (24) with that found by Madejski [ 31 (l= 1.2941Re1’5) as well as withtheexperimentaldata is presented. A good agreement between two analytical trends can be observed. It also follows that the results obtained agree well with the experimental data reported inRef. [15]. From (20) we can obtain a parameter tf which approximately describes a characteristic flattening time which is defined as a time required to reach 90% completion of flattening [ 43. When Re B 1 from (20) we have

From Fig. 3 it follows that the values of tf from (28) and (29) are in close agreement. The predicted evolution of the splat thickness calculated by means of ( 14) was compared with the experimental results for a copper droplet impinging on a

(28) In Ref. [4] for the practical purposes of plasma spraying the similar value was introduced: tr=

2

Re”5.

(29)

Flattening time 0.5U RG t

Fig. 4. Comparisonof theoreticaland experimentaldatafor evolution of splat thickness.

V.V. Sobolev, J.M. Guilemuny /MateriaLF Letters 22 (1995) 209-213

copper substrate (U:=2.52 m s-i, D,,=2Rp=4.67 mm) given in Ref. [5] (Fig. 4). Formula ( 14) for b is used as in this case Re = 23500. The theoretical predictions are seen to be in good agreement with the experimental measurements. 4. Conclusions ( 1) The approximate formulae describing the time evolution both of the thermally sprayed splat thickness and radius during the flattening process are established. (2) The realistic correlations between the final thickness and final radius of splat as well as its flattening time and the Reynolds number are introduced. (3) The theoretical results obtained agree well with the experimental data and those of the other authors. They can be used for the prediction purposes.

Acknowledgements The authors are grateful to CICYT for the financial support received with the project MAT94-0013. WS also thanks CICYT for the concession of the Sabbatical Year SAB 94-0066 at the University of Barcelona.

213

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