- Email: [email protected]
Mathematical simulation of particle velocities in a low pressure plasma jet and its experimental verification
Surface and Coatings Technology, 41(1990) 117
-
126
117
MATHEMATICAL SIMULATION OF PARTICLE VELOCITIES IN A LOW PRESSURE PLASMA JET AND ITS EXPERIMENTAL VERIFICATION H.-D. STEFFENS, M MACK, B. ECKHARDT and R. LAUTERBACH Institute of Materials Technology, University of Dortmund (F.R.G.) (Received July 24, 1989)
Summary A laser two-focus (L2F) velocimeter is used to evaluate particle velocities in a low pressure plasma jet with fixed parameters. The particle velocities are simulated by numerical computation and are related to particle diameter and density by using the drag equation. The solutions are checked and found to be in good agreement with experimentally observed velocity fields.
1. Introduction Vacuum plasma spraying is an example of industrial plasma applications [1] which still lack a detailed understanding of the process. Recently, there have been various efforts to model heat and momentum transfer from the low pressure plasma jet to the injected powder particles [2]. Apart from plasma temperatures, the velocity of the powder particles affects the heating of the powder, because it determines the particle dwell time inside the plasma jet. The state of heating of the particle upon impact determines coating parameters such as density, adhesive strength and surface roughness. Little detailed experimental work has been published over the last few years to understand the different effects of variation in process parameters [3 6]. Some analytical work has been done by Smith and Dykhuizen [4]. They developed a simple model for the calculation of particle velocities at different chamber pressures that was based on the assumption of a constant plasma gas velocity. The model shows good correlation between measured and calculated particle velocities only for very short distances (30 40 mm from the nozzle exit for a chamber pressure of approximately 400 mbar and 60 80 mm for 65 mbar). Thus this model does not allow the calculation of particle velocities on impact, which is most important for the structure and adhesion of the sprayed layer [7]. This paper presents an analytical model for particle velocities that takes into account a decrease in the plasma gas velocity along the jet axis. Experi-
-
-
0257-8972/90/$3.50
© Elsevier Sequoia/Printed in The Netherlands
118
mental investigations with laser optical methods have been used to check this model.
2. Experimental procedure The plasma was produced using a commercial low pressure plasma spraying (LPPS/VPS) system equipped with a standard nozzle [8] and operated at a chamber pressure of 4 kPa, net input power of 22 27 kW and an argon flow of 130 standard 1 min’. This set of parameters was taken as a standard configuration for all measurements. The powder particles were injected via a single injection port situated 25 mm upstream of the nozzle exit at a radial distance of 6 mm from the jet axis. The injection was nearly horizontal with an inclination of 15~towards the nozzle exit. The velocities of the particles were determined using a laser two-focus (L2F) system [9]. The L2F system consists of an optical head, including a 2 W Ar~laser and a transceiver system, a signal processing unit and a personal computer. The L2F technique operates by detecting scattered light from a small particle that passes through two focal volumes generated by two separate and focused laser beams. The distance between the two beams is known (425 ~tm) and by measuring the time of flight, the velocity of the crossing particles can be calculated. A calibration of the system is not necessary as the L2F technique is an absolute method. By rotating the off-axis laser beam around the beam which lies on the optical axis it is possible to obtain information about the turbulence intensity and the mean flow angle of the powder particle investigated. The measurements were taken all along the visible plasma jet. The vacuum chamber was therefore equipped with a large observation window. Different powder materials were used to cover a broad range of specific weights (from 3.6 to 19.3 g cm3). The powders were of high purity, consisting of fully dense spherical fused particles (except the alumina powder, which had a blocky and angular shape). The diameter varied from 26 to 68 pm with full width at half-maximum values lower than 10 pm. -
3. Calculation of particle velocities The drag equation for spherical particles in a flowing stream is mz=cdA-(vPl—z)
(1)
The connection between the drag coefficient Cd and the Reynolds number Re can be taken from the work of Groma and Veto [10]. Here, g(Re) is a correction factor depending on the value of the Reynolds number
119
~-g(Re)
Cd
(2)
Re
The numerical value of the Reynolds number can be calculated using the formula Re
2rp(vp1—~)
(3)
These considerations lead to the reduced drag equation z
—~-g(Re)(vp1-—z) pD
(4)
The general solution of this differential equation can be found by solving the homogeneous equation and adding an inhomogeneous solution. The following abbreviation is used k
18n
=
_~~~~(Re)
(5)
The required viscosities at various temperatures were taken from Nuss [11] and the equations to determine the position, velocity and acceleration of powder particles in time space are as follows: position of particle ~
VP1_VO((k)l)
(6)
velocity of particle Z
=
Up~— (Vp 1
—
v0) exp(—kt)
(7)
acceleration of particle =
k(u~
—
v0) exp(—kt)
(8)
These mathematical solutions to the drag equation are valid only for a constant plasma gas velocity. However, the velocity of a low pressure plasma jet falls with increasing distance from the nozzle exit, especially if the turbulence of the plasma jet is taken into account. Neglecting turbulence effects leads to a non-realistic behaviour of the injected particles. An iterative procedure therefore was derived, in which the solutions for constant plasma gas velocities were maintained only for small distance intervals. A particle velocity measured just below the nozzle exit gives the value of the initial velocity v0. Then the particle velocity is calculated for the next small interval. This calculated velocity serves as a new starting value for the next interval. This method leads to the recursive formula shown in eqn. (9). The axial particle velocity distribution can be evaluated by computing the following
120 Vp~--
=
(Up~
0—
~)
exp(—k ~11/~~)
n Z0
Vp11 ~Vp11
Z2
= =
1,
...,
i
V0
vp10)exp(kL~11/~1)
(Vp10 —~)
exp{—k~1(1/~1+ 1/~~)}
:
-
=
Vp~~~
(Vp1~~1 —
(9)
Vp11~_2) exp(—k~l1/~~1)
(v~1~_2 — v~1~_3) exp~—k~l(1/~~~
—(vp1~_0—~0)exp{—k~1(1/~_1 +
1
--.
+ 1/z,~2)}
+1/h +1/~~)}
A convergence of this calculation procedure is guaranteed only for a large number of grid points downstream of the plasma jet. To take the sensitivity of the solution into account the steps are kept very small (~z< 0.1 mm). The plasma gas velocity is taken from calculations by Chang and Pfender [12] for a turbulent plasma jet (turbulence intensity of about 10% at the nozzle exit). These calculations are based on the popular k—c model [131. For a fast estimation of particle velocities the analytical solution of the drag equation in position space can be used: ~(z)=
(2k)h/2(fu~idz_f~dz)
(10)
It turns out that, almost independent of the assumed decrease in plasma gas 2 in the following velocity, the particle velocities depend on the product p0D way (11) z
\mD1~!
2
This relation is valid in the z regions, where the integral of the particle velocity is much smaller than the integral of the plasma velocity. Details of a calculation for an assumed plasma gas velocity of the form of, for example Up1
=
Up10
exp(—Gz)
(12)
are given in Appendix B. However, if the velocity of one powder particle fraction with a defined mean diameter is known, velocities of other particle fractions can be calculated with a good accuracy, as will be shown. 4. Results and discussion The simple model described here and the formula for estimating particle velocities in a very straightforward way are tested by comparison with
121 Pressure: 4000 Pa Power: 22.4 kW (Mo 28.6 kW) 1 Gas flow: 130 standard I min
400 0
I: : t0o.o.~/tg. 3
0.0
F 0.0
40.0
I
go.o
120.0
T
tSO.O
Distance to Nozzle Exit (mm) Fig. 1. Centre-line velocity distribution.
measurements carried out for different powder materials with different particle diameters. The injection of the powder particles can be influenced by the carrier gas flow. This process parameter was adjusted such that the powder trajectories follow the jet axis exactly. This can be checked by measuring the mean flow angle of the powder particles downstream of the plasma jet. Measured particle velocities are plotted as a function of the distance to the nozzle exit in Fig. 1. The particle velocities for the different types of powder vary from 50 to 450 m s~. The particles are accelerated up to a distance of 200 mm from the nozzle exit. Here the visible part of the plasma flame (described by an isotherm of approximately 9000 K for an argon plasma jet) ends. The results of the model are shown in Fig. 2. The calculated velocities are plotted vs. distance from the nozzle exit. The computations were carried out for the same values of density and particle diameter as used for the measurements. The initial particle velocity has been taken from L2F measurements close to the nozzle exit. Measured and calculated particle velocities are in excellent agreement for high density, large and spherical particles. The model presented fails for broken, irregularly-shaped (crushed) low density materials such as small-sized alumina. A major reason for this is the rather arbitrary choice of the plasma
122 Pressure: 8000 Pa Power: 22.4 1 kW Argon Gas flow: 130 standard I min Pl.-Vel.: Splines Grid points: >2000
400.0
~4
A1203r 35j~m/3.6
300.0
~0
U
NiCr~: 26~m/8.O
200 . 0 100
W:
0
-
0.0
-
0.0 Fig. 2. Simulation
TABLE
F
I
I
40.0
260rn/19.3
W: 6.8~im/19,3
7~~~4O~m/1O.2
I
I
F
I
I
80.0 120.0 160.0 Distance to Nozzle Exit (mm)
of centre-line
velocity
distribution.
I
Calculated
and measured
Powder (diameter)
particle velocities
Velocity
(m
Distance to nozzle, 100 mm Experiment
tungsten (68 pm)
s’)
Theory
—.______________________________
Distance to nozzle, 160 mm Experiment
Theory
94
70
~09
80
molybdenum (40 pm)
163
163
187
187
tungsten (26 pm)
193
182
215
209
NiCrAl (26 pm)
283
283
333
324
Al203 (35 pm)
378
314
450
360
123 TABLE 2 Calculated values for the simulated and measured velocity relations (eqn. 11)
Experiment
Theory
4.6
4.5
1.8
1.75
1.1
1.1
v(A1
203, 35 pm)
v(W, 68 pm) u(NiCrA1, 26 pm) v(Mo, 40 pm) v(W, 26 pm) v(Mo, 40 pm)
gas velocity. Small and light particles are more influenced by variations in the plasma gas velocity than are particles with a high specific weight. Table 1 gives some values for calculated and measured velocities. Table 2 shows a few results for calculated and measured velocity relations (eqn. (11)) which are also in excellent agreement. In the case of small and light particles, e.g. alumina, it turns out that the assumed plasma gas velocity has to be corrected. The results indicate how the decrease in plasma gas velocity along the jet axis should be modified. The calculated particles velocities become closer to measured values if the following assumptions are made. (1) Lower initial value Up~0. (2) Faster decrease in the plasma gas velocity for distances z <100 mm and a slower decrease for z> 100 mm. (3) Correction of the parameters depending on pressure (e.g. the viscosity has a lower value for low pressure). Furthermore, the differences between measured and calculated particle velocities can be partly explained by different deviations of the sphericity for the different powder types.
5. Conclusions For a mathematical simulation of particle velocities the plasma gas velocity field, the dynamic viscosity and the injection velocity of the particles have to be known. The model presented neglects the influence of thermophoresis and evaporation of particles. Nevertheless, it is in good agreement with measurements for large, spherical particles with high specific weights. The differences in velocity between calculated and measured values for the very light particles (e.g. alumina, 26 pm) may be used to recalculate the assumed plasma gas velocity. Particle velocities in a low pressure plasma can be related to the density and diameter of powder particles. It is therefore easy to estimate dwell times
124
of powder particles in a plasma jet, if the velocity of a certain powder fraction is known. Acknowledgment This work has been supported by the Stiftung Volkswagenwerk, Hannover, F.R.G. References 1 H.-D. Steffens and M. Dvorak, Arc and Plasma Spraying Today and in the 90th SIMAP ‘88, Trans. JWRI, 17(1988) 57 - 70. 2 Y. C. Lee and E. Pfender, Plasma Chem. Plasma Proc., 7 (1987) 1ff. 3 E. Fleck, Y. C. Lee and E. Pfender, Experimental and numerical studies of particle velocities in thermal plasmas. In Proc. 8th mt. Symp. on Plasma Chemistry. Vol. 1, Tokyo, Japan, 1982, p. 392. 4 M. F. Smith and R. C. Dykhuisen, Surf. Coat. Technol., 34 (1988) 25 - 31. 5 A. Vardelle, M. Vardelle and P. Fauchais, Plasma Chem. Plasma Proc., 2 (1982) 255 291. 6 M. F. Smith, Laser measurement of particle velocities in vacuum plasma spray deposition. In Proc. 1st Plasma-Technik-Symp., Vol. 1, Plasma-Technik, Wohlen, Switzerland, 1988, pp. 77 - 84. 7 J. M. Houben, Relation of the adhesion of plasma sprayed coatings of the process parameters size, velocity and heat content of spray particles, Dissertation, Technische Hoogeschool Eindhoven, 1988. 8 H.-D. Steffens, M. Mack and R. Lauterbach, Measurement of particle velocities for an analytical model of low pressure plasma jets. In Proc. 1st Plasma-Technik-Symp.. Vol. 1, Lucerne, Switzerland, 1988, pp. 67 - 76. 9 R. Schodl, Laser-two-focus velocimetry (L2F) for use in aero engines, DFVLRInstitut für Luftstrahlantriebe, Linder Hbhe, Köln, F.R.G., July, 1977. 10 I. Groma and Veto, ml. J. Heat Mass Transf., 29(1988)549-554. 11 H. Nuss, Bestimmung der Plasmaparameter des Niedervoltbogens in Edelgasen, D. Sc. Thesis, University of Stuttgart, 1970. 12 C. H. Chang and E. Pfender, Non-equilibrium behaviour of low pressure plasma jets. In Proc. Conf. on the High Temperature Dust Laden Jets in the Processes of Treatrnent of Powder, Novosibirsk, USSR, 1988. 13 N. El-Kaddah, J. McKelliget and J. Szekely, Metall. Trans. B. 15 (1984) 59.
Appendix A: nomenclature Cd
g(Re) m r Re tdw
drag coefficient correction factor for slip flow correction factor for variable boundary layer properties correction factor of Reynolds number mass of particles radius of particle Reynolds number dwell time
125
velocity of plasma gas position, velocity and acceleration of particles step size viscosity of plasma gas density of plasma gas
Up~
z,
Pp,
Appendix B For fast estimates of the particle velocity an analytical approximation may be used. The starting point is the reduced drag equation (4) (Bi) In this, it is assumed that the relation between drag coefficient Cd and the Reynolds number Re is of the form Cd = 24/Re. Then k is given by 1 8i~ 2g(Re) p0D0
k=f1f2
(B2)
with correction factors f and f2 for slip flow and variable boundary layer properties which may be taken from the literature [Bi]. The viscosity is taken at the film temperature (average of the particle surface temperature and the temperature of the plasma adjacent to the boundary layer). The solution of eqn. (Bi) in position space, given by ~,
2
/‘vP1~
dz
—
/‘~~
(B3)
= (2k)”
is evaluated under the assumption f~dz~Jvpith
(B4)
which holds for a wide range of distances from the injection point. Then the second integral in eqn. (B3) may be replaced by
(2k)1~’2~f(
f
vP 1dz)dz’~
(B5)
This estimate leads to a particle velocity which tends to be slightly too low at larger distances. Assuming that the decrease in plasma gas velocity is exponential, e.g. of the form Vp1(Z)
=
v,,,0 exp(—Gz)
(B6)
126
the integrals may he evaluated analytically. Then it turns out that the approximate particle velocity depends only on two parameters, namely a parameter ct, independent of z and defined by 2k = (B7) G
Vp 10
and the integral of the plasma gas velocity decrease, here introduced as = 1 exp(—Gz) —
(B8)
The particle velocity, relative to the initial plasma gas velocity at the starting point, is then 2—~
-2~<1
(BlO)
Under normal conditions, eqn. (BlO) will hold for almost the whole particle acceleration region. Then for this region, the particle velocity is approximately proportional to the square root of the integral of the plasma gas velocity and the square root of the constant k (see eqn. (B2)). The maximum particle velocity ~max is reached at the distance z,, where the plasma gas velocity equals the particle velocity. Usually Zn5ax ~ ~ holds; therefore, series expansion renders a crude but rather reliable estimate for ~ given by 1 —In 1+—I—lnct (Bli) * G 21 Finally, the dwell time of particles in the plasma jet is estimated. The dwell time is most critical at the start of the acceleration phase, where the particle traverses the hottest regions of the jet. Here, tdv., may be estimated by ln(1 +~3/1--—j3) tdw
(2kGv~10)
(B12)
,>
Equation (B12) will underestimate tdw for larger distances. The approximations presented in eqns. (Bi) -(B12) are only intended to demonstrate how simple, fast and reliable scaling of parameter dependencies may be obtained when using different powder materials, different powder fractions or different plasma gas velocities (e.g. by changing the chamber pressure). More elaborate calculations are certainly needed, e.g. to predict the decrease of the plasma gas velocity.
Reference for Appendix B BI
a
Pfender, Pure ,4pp/. (‘hem., .57 (1985) 1179
-
1195.
-
126
117
MATHEMATICAL SIMULATION OF PARTICLE VELOCITIES IN A LOW PRESSURE PLASMA JET AND ITS EXPERIMENTAL VERIFICATION H.-D. STEFFENS, M MACK, B. ECKHARDT and R. LAUTERBACH Institute of Materials Technology, University of Dortmund (F.R.G.) (Received July 24, 1989)
Summary A laser two-focus (L2F) velocimeter is used to evaluate particle velocities in a low pressure plasma jet with fixed parameters. The particle velocities are simulated by numerical computation and are related to particle diameter and density by using the drag equation. The solutions are checked and found to be in good agreement with experimentally observed velocity fields.
1. Introduction Vacuum plasma spraying is an example of industrial plasma applications [1] which still lack a detailed understanding of the process. Recently, there have been various efforts to model heat and momentum transfer from the low pressure plasma jet to the injected powder particles [2]. Apart from plasma temperatures, the velocity of the powder particles affects the heating of the powder, because it determines the particle dwell time inside the plasma jet. The state of heating of the particle upon impact determines coating parameters such as density, adhesive strength and surface roughness. Little detailed experimental work has been published over the last few years to understand the different effects of variation in process parameters [3 6]. Some analytical work has been done by Smith and Dykhuizen [4]. They developed a simple model for the calculation of particle velocities at different chamber pressures that was based on the assumption of a constant plasma gas velocity. The model shows good correlation between measured and calculated particle velocities only for very short distances (30 40 mm from the nozzle exit for a chamber pressure of approximately 400 mbar and 60 80 mm for 65 mbar). Thus this model does not allow the calculation of particle velocities on impact, which is most important for the structure and adhesion of the sprayed layer [7]. This paper presents an analytical model for particle velocities that takes into account a decrease in the plasma gas velocity along the jet axis. Experi-
-
-
0257-8972/90/$3.50
© Elsevier Sequoia/Printed in The Netherlands
118
mental investigations with laser optical methods have been used to check this model.
2. Experimental procedure The plasma was produced using a commercial low pressure plasma spraying (LPPS/VPS) system equipped with a standard nozzle [8] and operated at a chamber pressure of 4 kPa, net input power of 22 27 kW and an argon flow of 130 standard 1 min’. This set of parameters was taken as a standard configuration for all measurements. The powder particles were injected via a single injection port situated 25 mm upstream of the nozzle exit at a radial distance of 6 mm from the jet axis. The injection was nearly horizontal with an inclination of 15~towards the nozzle exit. The velocities of the particles were determined using a laser two-focus (L2F) system [9]. The L2F system consists of an optical head, including a 2 W Ar~laser and a transceiver system, a signal processing unit and a personal computer. The L2F technique operates by detecting scattered light from a small particle that passes through two focal volumes generated by two separate and focused laser beams. The distance between the two beams is known (425 ~tm) and by measuring the time of flight, the velocity of the crossing particles can be calculated. A calibration of the system is not necessary as the L2F technique is an absolute method. By rotating the off-axis laser beam around the beam which lies on the optical axis it is possible to obtain information about the turbulence intensity and the mean flow angle of the powder particle investigated. The measurements were taken all along the visible plasma jet. The vacuum chamber was therefore equipped with a large observation window. Different powder materials were used to cover a broad range of specific weights (from 3.6 to 19.3 g cm3). The powders were of high purity, consisting of fully dense spherical fused particles (except the alumina powder, which had a blocky and angular shape). The diameter varied from 26 to 68 pm with full width at half-maximum values lower than 10 pm. -
3. Calculation of particle velocities The drag equation for spherical particles in a flowing stream is mz=cdA-(vPl—z)
(1)
The connection between the drag coefficient Cd and the Reynolds number Re can be taken from the work of Groma and Veto [10]. Here, g(Re) is a correction factor depending on the value of the Reynolds number
119
~-g(Re)
Cd
(2)
Re
The numerical value of the Reynolds number can be calculated using the formula Re
2rp(vp1—~)
(3)
These considerations lead to the reduced drag equation z
—~-g(Re)(vp1-—z) pD
(4)
The general solution of this differential equation can be found by solving the homogeneous equation and adding an inhomogeneous solution. The following abbreviation is used k
18n
=
_~~~~(Re)
(5)
The required viscosities at various temperatures were taken from Nuss [11] and the equations to determine the position, velocity and acceleration of powder particles in time space are as follows: position of particle ~
VP1_VO((k)l)
(6)
velocity of particle Z
=
Up~— (Vp 1
—
v0) exp(—kt)
(7)
acceleration of particle =
k(u~
—
v0) exp(—kt)
(8)
These mathematical solutions to the drag equation are valid only for a constant plasma gas velocity. However, the velocity of a low pressure plasma jet falls with increasing distance from the nozzle exit, especially if the turbulence of the plasma jet is taken into account. Neglecting turbulence effects leads to a non-realistic behaviour of the injected particles. An iterative procedure therefore was derived, in which the solutions for constant plasma gas velocities were maintained only for small distance intervals. A particle velocity measured just below the nozzle exit gives the value of the initial velocity v0. Then the particle velocity is calculated for the next small interval. This calculated velocity serves as a new starting value for the next interval. This method leads to the recursive formula shown in eqn. (9). The axial particle velocity distribution can be evaluated by computing the following
120 Vp~--
=
(Up~
0—
~)
exp(—k ~11/~~)
n Z0
Vp11 ~Vp11
Z2
= =
1,
...,
i
V0
vp10)exp(kL~11/~1)
(Vp10 —~)
exp{—k~1(1/~1+ 1/~~)}
:
-
=
Vp~~~
(Vp1~~1 —
(9)
Vp11~_2) exp(—k~l1/~~1)
(v~1~_2 — v~1~_3) exp~—k~l(1/~~~
—(vp1~_0—~0)exp{—k~1(1/~_1 +
1
--.
+ 1/z,~2)}
+1/h +1/~~)}
A convergence of this calculation procedure is guaranteed only for a large number of grid points downstream of the plasma jet. To take the sensitivity of the solution into account the steps are kept very small (~z< 0.1 mm). The plasma gas velocity is taken from calculations by Chang and Pfender [12] for a turbulent plasma jet (turbulence intensity of about 10% at the nozzle exit). These calculations are based on the popular k—c model [131. For a fast estimation of particle velocities the analytical solution of the drag equation in position space can be used: ~(z)=
(2k)h/2(fu~idz_f~dz)
(10)
It turns out that, almost independent of the assumed decrease in plasma gas 2 in the following velocity, the particle velocities depend on the product p0D way (11) z
\mD1~!
2
This relation is valid in the z regions, where the integral of the particle velocity is much smaller than the integral of the plasma velocity. Details of a calculation for an assumed plasma gas velocity of the form of, for example Up1
=
Up10
exp(—Gz)
(12)
are given in Appendix B. However, if the velocity of one powder particle fraction with a defined mean diameter is known, velocities of other particle fractions can be calculated with a good accuracy, as will be shown. 4. Results and discussion The simple model described here and the formula for estimating particle velocities in a very straightforward way are tested by comparison with
121 Pressure: 4000 Pa Power: 22.4 kW (Mo 28.6 kW) 1 Gas flow: 130 standard I min
400 0
I: : t0o.o.~/tg. 3
0.0
F 0.0
40.0
I
go.o
120.0
T
tSO.O
Distance to Nozzle Exit (mm) Fig. 1. Centre-line velocity distribution.
measurements carried out for different powder materials with different particle diameters. The injection of the powder particles can be influenced by the carrier gas flow. This process parameter was adjusted such that the powder trajectories follow the jet axis exactly. This can be checked by measuring the mean flow angle of the powder particles downstream of the plasma jet. Measured particle velocities are plotted as a function of the distance to the nozzle exit in Fig. 1. The particle velocities for the different types of powder vary from 50 to 450 m s~. The particles are accelerated up to a distance of 200 mm from the nozzle exit. Here the visible part of the plasma flame (described by an isotherm of approximately 9000 K for an argon plasma jet) ends. The results of the model are shown in Fig. 2. The calculated velocities are plotted vs. distance from the nozzle exit. The computations were carried out for the same values of density and particle diameter as used for the measurements. The initial particle velocity has been taken from L2F measurements close to the nozzle exit. Measured and calculated particle velocities are in excellent agreement for high density, large and spherical particles. The model presented fails for broken, irregularly-shaped (crushed) low density materials such as small-sized alumina. A major reason for this is the rather arbitrary choice of the plasma
122 Pressure: 8000 Pa Power: 22.4 1 kW Argon Gas flow: 130 standard I min Pl.-Vel.: Splines Grid points: >2000
400.0
~4
A1203r 35j~m/3.6
300.0
~0
U
NiCr~: 26~m/8.O
200 . 0 100
W:
0
-
0.0
-
0.0 Fig. 2. Simulation
TABLE
F
I
I
40.0
260rn/19.3
W: 6.8~im/19,3
7~~~4O~m/1O.2
I
I
F
I
I
80.0 120.0 160.0 Distance to Nozzle Exit (mm)
of centre-line
velocity
distribution.
I
Calculated
and measured
Powder (diameter)
particle velocities
Velocity
(m
Distance to nozzle, 100 mm Experiment
tungsten (68 pm)
s’)
Theory
—.______________________________
Distance to nozzle, 160 mm Experiment
Theory
94
70
~09
80
molybdenum (40 pm)
163
163
187
187
tungsten (26 pm)
193
182
215
209
NiCrAl (26 pm)
283
283
333
324
Al203 (35 pm)
378
314
450
360
123 TABLE 2 Calculated values for the simulated and measured velocity relations (eqn. 11)
Experiment
Theory
4.6
4.5
1.8
1.75
1.1
1.1
v(A1
203, 35 pm)
v(W, 68 pm) u(NiCrA1, 26 pm) v(Mo, 40 pm) v(W, 26 pm) v(Mo, 40 pm)
gas velocity. Small and light particles are more influenced by variations in the plasma gas velocity than are particles with a high specific weight. Table 1 gives some values for calculated and measured velocities. Table 2 shows a few results for calculated and measured velocity relations (eqn. (11)) which are also in excellent agreement. In the case of small and light particles, e.g. alumina, it turns out that the assumed plasma gas velocity has to be corrected. The results indicate how the decrease in plasma gas velocity along the jet axis should be modified. The calculated particles velocities become closer to measured values if the following assumptions are made. (1) Lower initial value Up~0. (2) Faster decrease in the plasma gas velocity for distances z <100 mm and a slower decrease for z> 100 mm. (3) Correction of the parameters depending on pressure (e.g. the viscosity has a lower value for low pressure). Furthermore, the differences between measured and calculated particle velocities can be partly explained by different deviations of the sphericity for the different powder types.
5. Conclusions For a mathematical simulation of particle velocities the plasma gas velocity field, the dynamic viscosity and the injection velocity of the particles have to be known. The model presented neglects the influence of thermophoresis and evaporation of particles. Nevertheless, it is in good agreement with measurements for large, spherical particles with high specific weights. The differences in velocity between calculated and measured values for the very light particles (e.g. alumina, 26 pm) may be used to recalculate the assumed plasma gas velocity. Particle velocities in a low pressure plasma can be related to the density and diameter of powder particles. It is therefore easy to estimate dwell times
124
of powder particles in a plasma jet, if the velocity of a certain powder fraction is known. Acknowledgment This work has been supported by the Stiftung Volkswagenwerk, Hannover, F.R.G. References 1 H.-D. Steffens and M. Dvorak, Arc and Plasma Spraying Today and in the 90th SIMAP ‘88, Trans. JWRI, 17(1988) 57 - 70. 2 Y. C. Lee and E. Pfender, Plasma Chem. Plasma Proc., 7 (1987) 1ff. 3 E. Fleck, Y. C. Lee and E. Pfender, Experimental and numerical studies of particle velocities in thermal plasmas. In Proc. 8th mt. Symp. on Plasma Chemistry. Vol. 1, Tokyo, Japan, 1982, p. 392. 4 M. F. Smith and R. C. Dykhuisen, Surf. Coat. Technol., 34 (1988) 25 - 31. 5 A. Vardelle, M. Vardelle and P. Fauchais, Plasma Chem. Plasma Proc., 2 (1982) 255 291. 6 M. F. Smith, Laser measurement of particle velocities in vacuum plasma spray deposition. In Proc. 1st Plasma-Technik-Symp., Vol. 1, Plasma-Technik, Wohlen, Switzerland, 1988, pp. 77 - 84. 7 J. M. Houben, Relation of the adhesion of plasma sprayed coatings of the process parameters size, velocity and heat content of spray particles, Dissertation, Technische Hoogeschool Eindhoven, 1988. 8 H.-D. Steffens, M. Mack and R. Lauterbach, Measurement of particle velocities for an analytical model of low pressure plasma jets. In Proc. 1st Plasma-Technik-Symp.. Vol. 1, Lucerne, Switzerland, 1988, pp. 67 - 76. 9 R. Schodl, Laser-two-focus velocimetry (L2F) for use in aero engines, DFVLRInstitut für Luftstrahlantriebe, Linder Hbhe, Köln, F.R.G., July, 1977. 10 I. Groma and Veto, ml. J. Heat Mass Transf., 29(1988)549-554. 11 H. Nuss, Bestimmung der Plasmaparameter des Niedervoltbogens in Edelgasen, D. Sc. Thesis, University of Stuttgart, 1970. 12 C. H. Chang and E. Pfender, Non-equilibrium behaviour of low pressure plasma jets. In Proc. Conf. on the High Temperature Dust Laden Jets in the Processes of Treatrnent of Powder, Novosibirsk, USSR, 1988. 13 N. El-Kaddah, J. McKelliget and J. Szekely, Metall. Trans. B. 15 (1984) 59.
Appendix A: nomenclature Cd
g(Re) m r Re tdw
drag coefficient correction factor for slip flow correction factor for variable boundary layer properties correction factor of Reynolds number mass of particles radius of particle Reynolds number dwell time
125
velocity of plasma gas position, velocity and acceleration of particles step size viscosity of plasma gas density of plasma gas
Up~
z,
Pp,
Appendix B For fast estimates of the particle velocity an analytical approximation may be used. The starting point is the reduced drag equation (4) (Bi) In this, it is assumed that the relation between drag coefficient Cd and the Reynolds number Re is of the form Cd = 24/Re. Then k is given by 1 8i~ 2g(Re) p0D0
k=f1f2
(B2)
with correction factors f and f2 for slip flow and variable boundary layer properties which may be taken from the literature [Bi]. The viscosity is taken at the film temperature (average of the particle surface temperature and the temperature of the plasma adjacent to the boundary layer). The solution of eqn. (Bi) in position space, given by ~,
2
/‘vP1~
dz
—
/‘~~
(B3)
= (2k)”
is evaluated under the assumption f~dz~Jvpith
(B4)
which holds for a wide range of distances from the injection point. Then the second integral in eqn. (B3) may be replaced by
(2k)1~’2~f(
f
vP 1dz)dz’~
(B5)
This estimate leads to a particle velocity which tends to be slightly too low at larger distances. Assuming that the decrease in plasma gas velocity is exponential, e.g. of the form Vp1(Z)
=
v,,,0 exp(—Gz)
(B6)
126
the integrals may he evaluated analytically. Then it turns out that the approximate particle velocity depends only on two parameters, namely a parameter ct, independent of z and defined by 2k = (B7) G
Vp 10
and the integral of the plasma gas velocity decrease, here introduced as = 1 exp(—Gz) —
(B8)
The particle velocity, relative to the initial plasma gas velocity at the starting point, is then 2—~
-2~<1
(BlO)
Under normal conditions, eqn. (BlO) will hold for almost the whole particle acceleration region. Then for this region, the particle velocity is approximately proportional to the square root of the integral of the plasma gas velocity and the square root of the constant k (see eqn. (B2)). The maximum particle velocity ~max is reached at the distance z,, where the plasma gas velocity equals the particle velocity. Usually Zn5ax ~ ~ holds; therefore, series expansion renders a crude but rather reliable estimate for ~ given by 1 —In 1+—I—lnct (Bli) * G 21 Finally, the dwell time of particles in the plasma jet is estimated. The dwell time is most critical at the start of the acceleration phase, where the particle traverses the hottest regions of the jet. Here, tdv., may be estimated by ln(1 +~3/1--—j3) tdw
(2kGv~10)
(B12)
,>
Equation (B12) will underestimate tdw for larger distances. The approximations presented in eqns. (Bi) -(B12) are only intended to demonstrate how simple, fast and reliable scaling of parameter dependencies may be obtained when using different powder materials, different powder fractions or different plasma gas velocities (e.g. by changing the chamber pressure). More elaborate calculations are certainly needed, e.g. to predict the decrease of the plasma gas velocity.
Reference for Appendix B BI
a
Pfender, Pure ,4pp/. (‘hem., .57 (1985) 1179
-
1195.