Modelling of particle movement and thermal behaviour during high velocity oxy-fuel spraying

Surface and Coatings Technology, 63 (1994) 181-187

181

Modelling of particle movement and thermal behaviour during high velocity oxy-fuel spraying V. V. S o b o l e v a, J. M. G u i l e m a n y a, J. C. G a r m i e r b a n d J. A. C a l e r o a aPhysical Metallurgy Materials Science, Department of Chemical Engineering and Metallurgy, University of Barcelona, E-08028 Barcelona (Spain) bPlasma Technik SA, F-03809 Villefontaine (France) (Received May 14, 1993; accepted in final form August 4, 1993)

Abstract A theoretical model is presented to investigate the particle mechanical and thermal behaviour in the process of high velocity oxy-fuel (HVOF) spraying. The model accounts for the various phenomena which influence the momentum and heat transfer during particle movement and heating. The mathematical formulation takes into account the fluid velocity dependence on the particle density, internal heat conduction in particles and their composite structure, steep temperature gradients near the particle surface and fluid parameter variations. Comparison with experimental data shows that the present model enables accurate predictions to be made and optimal conditions of HVOF spraying to be established.

1. Introduction High velocity oxy-fuel ( H V O F ) spraying has been one of the most rapidly developing areas of thermal spraying in the past few years, gaining wide industrial acceptance. In applications such as the deposition of carbide coatings, H V O F spraying produces coatings comparable with those obtained by the detonation flame spray process and possibly superior to those obtained by air and vacuum plasma spraying [ 1 ]. The mathematical simulation of gas-solid transport processes offers great possibilities of improving the technology of H V O F spraying in particular, taking into account that in situ measurements of particle velocity and temperature are more difficult and expensive. The purpose of this paper is to introduce a realistic prediction model which includes the main characteristic features of H V O F spraying and to investigate by means of this model the mechanical and thermal behaviour of various powder particles during H V O F spraying.

2. Model description In the last decade several mathematical models for analysis of the movement and heating of particles injected into a plasma torch during plasma spraying have been developed [2, 3]. With respect to H V O F spraying, similar problems have been considered in ref. 4, but that model does not take into account some very important characteristic features of H V O F spraying such

0257-8972/94/$7.00 SSDI 0 2 5 7 - 8 9 7 2 ( 9 3 ) 0 2 2 0 5 - Q

as the strong dependence of the fluid velocity on the particle density and its considerable variation with the spraying distance, the composite structure of particles consisting of carbides and binding elements, the comparatively small influence of Knudsen discontinuum effects on heat and momentum transfer, the thermal behaviour of particles inside the spray gun of the H V O F system and the dependence of the initial values of the particle temperature and velocity on the particle prehistory, particularly in the combustion chamber of the gun. The mathematical model of particle behaviour developed in this paper takes into account the above-mentioned characteristic features of H V O F spraying as well as internal heat conduction in particles, fluid temperature changes both in the gun and in the spraying distance, the possible creation of relatively steep temperature gradients near the particle surface and variable properties of the fluid. One of the most important features of the process of H V O F spraying is the strong dependence of the fluid velocity on the particle density. A density increase leads to a decrease in the pressure gradients in the spray gun as a result of the additional energy dissipation caused by the solid powder particles [5]. Consequently the fluid velocity is diminished. A particle density enhancement also leads to a decrease in particle velocity, causing an increase in particle residence time as high temperatures and thus an increase in particle temperature. It is also important to take into account that in the case of particles consisting of carbides with binding

© 1994

Elsevier Sequoia. All rights r~served

V.V. Sobolev et al. / Modelling of particle behaviour during HVOF spraying

182

metals, only the fusion of the latter needs to be considered in the process of H V O F spraying, because the carbide melting points are usually higher than the conventional fluid temperatures in this process. By taking into account the above-mentioned factors, it is possible to predict correctly the particle behaviour during H V O F spraying.

l <~z <~L : Vr= Az2 + Bz + C,

where A=-[(L

l)(L-zm)(zm-l)]-i

X [(Vfp -- Vfs)(Z m -- l) -I- (Vfm

B = ( z m _ _ l ) - l [ g f m _ _ gfp_ A(z21 2.1. Fluid parameters

The coordinate system is chosen in such a way that the origin of the longitudinal coordinate z coincides with the point of powder particle acceleration and its temperature increase due to the combustion resulting from chemical reaction of the combustion gas (in our case propane) with oxygen. According to experimental data [6 8], the fluid velocity Vr increases and the fluid temperature Tf decreases with the distance from the gun throat (z = 0) to its exit (z = l) from Vr0 to Vfp and from Tro to Tfv respectively. As a result of the fluid expansion, the fluid velocity grows further and its temperature continues to decrease in the supersonic flame of length lc = Zm- l arising at the gun exit in the form of shock waves. The increasing fluid velocity attains its maximum value Vrmat the point z = Zm and then diminishes towards the substrate, taking the value Vrs at the substrate surface (z = L). The fluid temperature decreases from the gun exit, takes the value Tfm at the flame end (z = Zm), then continues to diminish and achieves the value Trs at the substrate surface (z = L). By means of linear interpolation, the fluid velocity and temperature can be presented as the following functions of z: Vf = Vfo @ (Vfp - Vfo)zl -1, Vf = (Zm -- l ) - 1 [(Vfm

(2)

Zm)-l[--(Vrm

Vrs)Z+ VrmL Vfszm],

Zm<
0~
(4)

(rfp__Tfm)z+rfpzm__rfml],

I~Z~Zm Tr=(L

(5)

Zm) l [ - ( r f m - r f s ) Z - t - r f r n L - r f s Z m ] ,

Zm~Z~L

Al 2

2.2. Momentum transfer

The equation of motion of a spherical particle in the process of H V O F spraying describing the gas-particle momentum transfer can be written as [2] dVp _ 3 CD pf (Vf- Vp)l Vf- Vp[ dt 4 dp pp

(8)

where CD --

23.707 Re

( 1 + 0.165Re 2/3 -- 0.5Re-°l ),

0.15 < Re < 500 Re = dp]

Vp - -

VrltOf.tt( 1

(9)

The initial condition for solving the governing equation of motion (8), corresponding to t = 0, is Vp(O) = Vpo

(10)

Since the initial position (z = O) of the particle is known, its axial location z at any given moment of time is given by (11)

2.3. Heat transfer

(3)

T~= T~0- (T~o- T~p)zl 1, rf = ( Z m - / ) 1[

12)]

The thermal behaviour of a spherical particle in the process of H V O F spraying is described by the heat conduction equation

l<~z<~z m

Vf=(L

C = V f p - Bl

Vfp)(L- l)]

z ( t ) = i ' Vp(t) dt 3o

O~ z ~ l

Vfp)Z + Vfpz m -- Vrml ],

(7)

l <~z <~L

(6)

The approximation of the fluid velocity out of the gun is very important for the particle behaviour in the process of H V O F spraying. In accordance with the experimental data, the following formula for Vr obtained by square interpolation was also used in the interval

[)pep 8t - r 2 81" r2)~P 8 r J

O<~r<~RP

(12)

The powder particle temperature T is a function of time t and radial coordinate r. At the particle centre the boundary condition of the thermal field symmetry is introduced and at the particle surface the boundary condition describing gas particle heat exchange is applied. These boundary conditions are 8T

8rr (0, t ) = 0

(13)

8T 2p ~ (Rp, t) = c~[Tf-- T(Rp, t)]

(14)

V. V. Sobolev et al. / Modelling of particle behaviour during HVOF spraying At the initial m o m e n t of time ( t = 0 ) temperature is T(r, 0 ) = Tpo

the particle (15)

The heat transfer coefficient c~in (14) can be found by means of the semiempirical Ranz-Marshall equation

[2]

(16)

Pr = Cffif)~f 1 For the calculation of the Reynolds number Re, the Nusselt number N u and the Prandtl number Pr, the integral mean values of pf, /~f, cf and 2f are used in the temperature interval Tps ~< T ~< Tf: fif=b 1

pfdT ps

fif=b I

;?

2- a ? )~fg ~0= 1 + 4 - -a 1 q- ? gfg VmoIPfs Rp

ps

ff<

(17) ]~f d T

ps

2~fg Prs Pfs VinciCfg

Kn, -

During H V O F spraying a particle is gradually heated and its surface can attain the melting temperature. In this case the Stefan problem must be solved taking into account a heat balance at the moving liquid-solid interface r = r i which states that the difference between the heat fluxes associated with the liquid and solid phases is equal to the rate of heat absorption due to fusion [9]: ~ (0T2~ dFi -- Ap2 k ar)r=r = Z P p l ~ i

2-a a

7

4

1 + 7 Prs

Kn,

(23)

/'],

b I = ( T f - Tps) 1

(~Zl~ "~pl k a r ) r : r

(22)

where Pr = c f # f 2 - 1 is the Prandtl number at the particle surface temperature Tp~. Taking into account (22), the correction factor ~0 can be rewritten as [ 10] q) = 1 +

where

(21)

where Pfs is the value of Pf at the temperature Tps and 2rg and Gg are the integral mean values of 2rg and Crg respectively in the temperature j u m p interval Tps ~< T ~< Tpg calculated using formulae similar to (17). The effective gas mean free path q, can be introduced [10]:

11,

cfdT

(f=b I

(20)

In ref. 10 a correlation factor q~ was introduced to take into account the influence of the Knudsen effect:

flfdT

ps

'~f = bl

surface is higher than the particle surface temperature Tps. This leads to a reduction in the rates of heat and m o m e n t u m transfer [10]. The Knudsen effect is characterized by the Knudsen number Kn = - dp

Nu - c~dv - 2 + 0.6 Re 1/2 Pr 1/3 ~f

183

(18)

i

There is also another boundary condition which is applied in this case at the liquid-solid interface: Tl(r i, t ) = T2(ri, t ) = Tc

(19)

2.4. Influence of the Knudsen effect Since the characteristic dimensions of the powder particles used in H V O F spraying can have the same order of magnitude as the length of the gas mean free path, it is necessary in the general case to take into consideration the Knudsen effect, which manifests itself in the form of a temperature j u m p at the particle surface [10]. In this case the temperature Tpg near the particle

dp

(24)

Expression (24) determines the effective Knudsen number. In ref. 10 the values of K n , for particles in a nitrogen plasma were calculated starting from a temperature of 3000 K. The main gases in the process of H V O F spraying under consideration are oxygen and propane (with an o x y g e n - p r o p a n e ratio of 7 : 1) and the combustion gases (steam, carbon dioxide, unreacted oxygen). Outside the gun the air should also be taken into account. Since the thermal properties of these gases are similar to those of nitrogen, the results of the calculations presented in ref. 10 can be used to estimate the value of ~0, taking into account that the value of the accommodation coefficient a is about unity [ 10]. In the case of nitrogen 7 = 1.4, Pr s = 2.287 and, according to ref. 10, the effective Knudsen number K n , 0.02250 for d p = 2 0 ~ t m and K n , ~ 0 . 0 1 1 2 5 for d p = 40 gin. The corresponding values of q) are 1.0450 and 1.02250. The heat transfer reduction is determined by the value ~o-1. It follows that for particles with dp = 20 lam this reduction is about 4% and for particles with dp = 40 gin it is about 2%. The Knudsen effect also causes a reduction in the drag coefficient CD in the equation of motion (8). This

K K Sobolev et al. / Modelling q['particle behaviour during HVOF spraying

184

reduction is defined by the value ~0 0.45. From the calculations it follows that in the case of particles with dp = 20 lam it is equal to 1%. Even in the case of smaller particles with d p = 10 gm the reductions in heat and m o m e n t u m transfer associated with the Knudsen effect are about 8% and 4% respectively. The present results show that in the process of H V O F spraying, where the temperatures are much lower than those in plasma spraying, the influence of the Knudsen effect on both heat and m o m e n t u m transfer is rather small. The calculated values of the corresponding reductions are in fact even smaller owing to the pressure, which is higher in the process of H V O F spraying than in plasma spraying [1, 8]. Thus in the case of H V O F spraying it is possible to ignore the Knudsen effect, in particular for powder particles with dp = 10 4 0 g m . in this paper, to increase the mathematical simulation accuracy, the formation of relatively steep temperature gradients near the particle surface is taken into consideration by means of the introduction of a correction factor [2]. In this case

CD= Coo

(25)

\vr/

where the value of CDo is determined by eqn. (9). The integral mean value of the kinematic fluid viscosity vf is calculated in the same way as in (17).

3. Mathematical simulation The mechanical problem (8)-(10), (25) was solved numerically by the R u n g e - K u t t a method. The solution of the thermal problem (12)-(19) was obtained by the method of finite differences in the implicit form with absolute stability. The numerical algorithm is similar to that described in refs. 11 and 12. The behaviour of the particles of AI20 3 (dp = 10-15 p.m), Ni (dp = 15 40 gm) and WC 12%Co ( d p = 20-40 gm) was investigated. The parameters of the process of H V O F spraying corresponding to the H V O F system installed at the University of Barcelona (Plasma Technik

PT 100) were used: 1 = 0.1 m, zm = 0.22 m, L = 0.4 m, Tro = 2700 °C, Tfp = 2600 °C, Tfm = 2350 °C, Tf~ = 540 '~C. The parameters of the fluid velocity variation depending on the particle material were chosen according to the experimental data and the characteristics of the H V O F system used. The properties of the powder materials considered are given in Table 1 [4]. In the case of tungsten carbide with cobalt the melting point of cobalt is indicated, because only this component of the mentioned composite can be melted under the conditions of H V O F spraying considered.

4. Results and discussion 4.1. Particle mechanical behaviour In Fig. 1 the particle velocity variations outside the gun are presented along the spraying distance. After the gun exit the particle velocity increases, achieves its maximum value and then diminishes on moving towards the substrate surface. A particle density increase leads to its velocity reduction and a velocity maximum displacement towards the substrate. The particle velocity diminishes with its size enhancement and, as can be seen from a comparison of curves 1 and 2 in Fig. 1, the velocity of a large particle exceeds that of a small particle during some period of time only when the former has a higher initial velocity. As has already been mentioned, the form of the fluid velocity variation has a strong influence on the particle velocity behaviour. This influence increases with particle 11 O 0 I 1000

"E 800 ~-~>-

~~

\\\\\ 4

1

\\\

6o0 (.9

4oo

i

TABLE l. Thermophysical properties of powder materials Property

AlzO 3

Ni

WC 12%Co

Density (kg m a) Melting point (K) Heat of fusion ( l06 J k g 1) Specific heat (J kg ~ K 1) Thermal conductivity ( W m ~K 1)

3900 2326 1.065 1242

8900 1727 0.30 541

14320 1768 ~ 0.42 295

6.3

73

~This is the melting point of cobalt.

45

200

100

200

300

400

AXIAL DISTANCE (rnm)

Fig. 1. Variation in particle velocity with spraying distance and its comparison with experimental data: 1, A1203, dp = | 0 p.m, Vfparabolic; 2, AlzO 3, 20 p.m, Vf parabolic; 3, A1203, 5 15 p_m, experimental data; 4, Ni, 30gm, Vf linear; 5, W C 12%Co, 30~m, Vr parabolic; 6, WC 12%Co, 30 p.m, Vf linear; 7, WC 12%Co, 22 44gm, experimental data.

V. V. Sobolev et al. / Modelling of particle behaviour during HVOF spraying

density reduction and is more important for alumina particles than for particles of tungsten carbide with cobalt. In general the fluid velocity variation in the form obtained by square interpolation of the HVOF spraying parameters [8] corresponds better to the real situation than the similar variation obtained by means of linear interpolation. From Fig. 1 it is seen that the mathematical simulation results agree well with the particle velocity experimental data for particles of alumina and of tungsten carbide with cobalt supplied by Plasma Technik AG. [8]. In terms of the technology of HVOF spraying, the variations in particle velocity maximum and its position with particle diameter are interesting (Fig. 2). It is seen that the maximum velocity becomes smaller and its position is displaced towards the substrate as the particle diameter grows. 4.2. Particle thermal behaviour

185

3000

dp=2Opm Tf

2500

w

2000

w u W

d

1500

1oo0

50O

50

100

150

200

AXIAL DISTANCE (ram) Fig. 3. Variation in core temperature of particles with their axial position for a given fluid temperature distribution: 1, Ni, Tpo= 20 °C; 2, Ni, I00°C; 3, Ni, 300°C; 4, Ni, 500°C; 5, Ni, IO00°C; 6, W C - 1 2 % C o , 300 °C; 7, W C - 1 2 % C o , 500 °C,

The mathematical simulation of the powder particle thermal behaviour was carried out up to its full melting or in the case of particles of tungsten carbide with cobalt up to the full melting of cobalt. The curves depicted in Figs. 3 and 4 show that the axial distance corresponding to full melting is enhanced with decreasing particle initial temperature and thermal diffusivity as well as with increasing particle size. The particle diameter influence is shown in more detail in Fig. 5. The displacement of the particle melting

i

1400

Tpo:I000°C

1

s

~ 12oo

g 1100 -

-

A[70~

0"2t'5

10000 1095

I00

200

300

AXIAL DISTANCE (mm)

0.235

Fig. 4. Variation in core temperature of particles with their axial distance for the same particle initial temperature: 1, Ni, dp = 15 lam; ~" I090

00 ZZ5

PARTICLE D I A M E T E R (pro) 0.30

~95

W[-12% Co

rr f~

olii;

2 30 PARTICLE DIAMETER (pro)

--Nl

6z"O

O

2, Ni, 30 lam; 3, Ni, 40 ~tm; 4, WC-12%Co, 30p.m; 5, WC-12%Co, 40 t~m.

o

PARTICLE DIAMETER (pm) 12,5

0.310

E tu

025

15

Tpo:I000~C

g 0

WC-12°/oCo

~D 0.2 z

630 ~

0.2t*

2~

13- 0.120

6202

$

2 30 0

00

30

40

PARTICLE DIAMETER (pro)

PARTICLE DIAMETER (pro)

Fig. 2. Variations in particle maximum velocity and its axial position with particle diameter.

Fig. 5. Variation in particle melting axial position with particle diameter.

186

l~5 1~: Sobolev et al. / Modelling of particle behaviour during H VOF spraying

axial position in the direction of the substrate with increasing particle diameter is more pronounced in the case of nickel and tungsten carbide- cobalt particles than for alumina particles, since the former have a higher thermal diffusivity. The particle initial temperature increases with increasing particle density. This is why heavy particles have better conditions for melting than light particles, because their initial temperature and residence time under the combustion flame conditions are higher than those for light particles. From Figs. 3 and 4 it follows that under the conditions considered for the process of H V O F spraying, all the particles are melted within the spraying distance, taking into account that in the case of particles of tungsten carbide with cobalt only the latter is melted.

initial temperature and thermal diffusivity as well as by an increase in particle size. (9) In general heavy powder particles have better conditions for melting than light particles, since their initial temperature and residence time under the combustion flame conditions are higher than those for light particles. (10) The presented mathematical model reflects all the main characteristic features of the mechanical and thermal behaviour of various powder particles in the process of H V O F spraying. The mathematical simulation results obtained on the basis of this model agree well with the experimental data. This means that the presented model can be used to predict the parameters of powder particle behaviour in the process of H V O F spraying in order to obtain the optimal coating structure [13, 14].

5. Conclusions (1) A mathematical model describing the mechanical and thermal behaviour of powder particles in the process of H V O F thermal spraying is presented. (2) The Knudsen effect is shown to have only a small influence on the mechanical and thermal behaviour of powder particles during H V O F spraying, in particular when the particle diameter is 10 40 lain. To increase the mathematical simulation accuracy, a correction factor is used to take into account the formation of relatively steep temperature gradients near the particle surface. (3) The dependences of the fluid velocity and particle initial temperature on the particle density are among the most important factors of the presented mathematical model. A particle density increase causes a fluid velocity decrease and a particle initial temperature enhancement. (4) Within the spraying distance the various powder particle velocities increase, attain their maximum values in the region of the combustion flame end and then decrease in the direction of the substrate surface. (5) A particle density increase leads to a particle velocity reduction and a displacement of the particle velocity maximum value in the direction of the substrate. (6) The particle maximum velocity becomes smaller and its axial position is displaced in the substrate direction as the particle diameter increases. (7) The form of the fluid velocity variation has a strong influence on the particle velocity behaviour and this influence is amplified with decreasing particle density. The fluid velocity parabolic variation obtained by square interpolation of the H V O F spraying parameters agrees better with the real situation than the similar variation obtained by linear interpolation. (8) The axial distance corresponding to the full particle melting is enhanced by a reduction in the particle

Acknowledgments The authors would like to thank CICYT for the financial support received with the project MAT 91-0044 and Plasma Technik AG. for cooperation and useful discussions. V. V. Sobolev also expresses his gratitude to CICYT for the concession of the sabbatical year SAB 92-0207 at the University of Barcelona.

References l A. J. Sturgeon, High velocity oxyfuel spraying promises better coatings, Met. Mater., (1992) 547 548. 2 D. Apalian, R. Paliwal, R. W. Smith al:d W. F. Schilling, Melting and solidification in plasma spray deposition--phenomenological review, Int. Met. Rev., 28(5) (1983) 271 294. 3 D. K. Das and R. Sivakumar, Modelling of the temperature and the velocity of ceramic powder particles in a plasma flame 1. Alumina, Acta Metall. Mater., 38(11) (1990) 2187 2192. 4 S. V. Joshi and R. Sivakmnar, Particle behaviour during high velocity oxy-fuel spraying, Surf. Coat. Technol., 50 (1991) 67 74. 5 J. H. Perry (ed.), Chemical Engineers Handbook, McGraw-Hill, New York, 1985. 6 K. V. Rao, D. A. Somerville and D. Lee, Properties and characterization of coatings made using Jet-Kote thermal spray technique, Proc. 1 lth Int. Thermal Spray Conf., Montreal, Pergamon, Toronto, 1986, pp. 873 882. 7 M. R. Dorfman, B. A. Kushner, J. Nerz and A. J. Rotolico, A technical assessment of H V O F versus high energy plasma tungsten carbide for wear resistance, Proc. 12th Int. Thermal Spray Con[:, London, 1989, Weld. Institute, Cambridge, 1989, pp. 1 12. 8 Introduction to the CDS Technology, Plasma Technik AG., Wohlen, 1990. 9 J. Szekely and N. J. Themelis, Rate Phenomena in Process Metallurgy, Wiley Interscience, New York, 1971. 10 X. Chen and E. Pfender, Effect of Knudsen number on heat transfer to a particle immersed into a thermal plasma, Plasma Chem. Plasma Process., 3(1) (1983) 97 113.

V. V. Sobolev et al. / Modelling of particle behaviour during HVOF spraying 11 V. V. Sobolev and P. M. Trefilov, Thermophysics of Metal Solidification During Continuous Casting, Metallurgia, Moscow, 1988. 12 V. V. Sobolev and P. M. Trefilov, Optimization of Thermal Regimes of Liquid Metals Solidification, Krasnoyarsk University Press, Krasnoyarsk, 1986. 13 J. M. Guilemany, N. Llorca and J. Nutting, Ceramic coatings obtained by HVOF thermal spraying, Powder Metall. Int., 25(4) (1993) 176 179. 14 J. M. Guilemany, N. Llorca, M. D. Nufiez and J. Garcia, Characterization of (W,Ti)C/Ni powder for thermal spraying, Surf Coat. Technol., 58(1993) 173-177.

T V Vmol z

Greek letters 7 q 2 P y

Appendix: Nomenclature a A B c C CD d Kn l L Nu Pr r R Re t

accommodation coefficient coefficient of eqn. (7) (m s) 1 coefficient of eqn. (7) (s -1) specific heat (J kg ~ K -1) coefficient of eqn. (7) (ms - 1) drag coefficient diameter (m) Knudsen number gun length (m) gun length plus spraying distance (m) Nusselt number Prandtl number radial coordinate (m) radius (m) Reynolds number time (s)

temperature (°C) velocity (ms =l) mean molecular velocity (m s 1) longitudinal coordinate (m)

P X

heat transfer coefficient (W m -2 K -I) specific heat ratio gas mean free path (m) thermal conductivity (W m 1 K=a) viscosity (kg m -1 s 1) kinematic viscosity (m 2 s 1) density (kg m -3) correlation factor latent heat of fusion (J kg 1)

Subscripts c f i m p s 0 1 2

crystallization fluid interface maximum particle particle surface initial solid liquid

Superscript (-)

mean value

187