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Numerical modelling of tribo-charge powder coating systems
Journalof
ELECTROSTATICS ELSEVIER
Journal of Electrostatics 40&41 (1997) 395-400
N u m e r i c a l Modelling o f Tribo-Charge P o w d e r Coating Systems Kazimierz Adamiak Department of Electrical Engineering, The University of Western Ontario London, Ontario, Canada N6A 5B9 A numerical algorithm for simulating the electric field distribution and particle trajectories in tribo-eharge powder coating systems is presented. It combines the finite element method for the electric field analysis and the numerical integration of the equation for the particle trajectoties. The particle-in-cell technique is used to couple both methods, which are applied iteratively until a self-consistent solution is obtained. I. INTRODUCTION Two electrostatic powder coating systems are commonly used in commercial applications: the corona charging system and the tribo charging system [I]. They differ not only in the way the particles are charged but also the electrical conditions in the transport and deposition zones are quite distinct. In the corona system, the particles are charged by the ionic bombardment so they can be charged intensively and the whole process is more reliable and predictable. The high voltage supply is needed for generating the corona discharge, but it also generates a strong electric field which drives the powder particles towards the target. Not all ions are collected by the particles so they also move towards the coated object and can produce the back corona discharge which often deteriorates the finish quality. In the tfibo-charge powder coating the powder particles are charged by the frictional contact with a gun. A cloud of the charged particles ejected from the gun forms a space charge which in turn produces an electric field which drives the particles towards the target. Therefore, the whole system does not require an external voltage supply and is preferred in some applications [2]. This technique can produce a better finish qual/ty, especially for targets with complex shapes. Both reasons increase interest in the tribo-charge powder coating systems. However, the magnitude of the electric field in the tribo systems is usually much weaker than that in the corona systems resulting in a decrease in the electric force and greater sensitivity to different kinds of disturbing factors. Modelling of the tribo-charge coating process is much more difficult than that of the conventional corona system. In both cases the electrostatic field and the equation for the particle motion must be solved. However, the electric field is a function of the space charge density. In the corona coating the majority of the space charge comes from the corona discharge and the field generated by the particles can be neglected. In tfibo-charge coating, the space charge is associated only with the particle cloud. The particle movement results from the balance of electrical, inertial, air-drag and gravitational forces. Conversely, the electric field distn'bution is a function of the particle concentration. Therefore, both problems are mutually coupled [3]. 0304-3886/97/$17.00 © Elsevier Science B.V. All fights reserved. S0304-3886(97)00077-6
PI1
K. Adamiak/dournal of Electrostatics 40&41 (1997) 395-400
396
2. MATHEMATICAL MODEL ~r
The process is simulated for the simplified model of the problem shown in Fig. 1. A cylindrical gun made of a dielectric material (permittivity ~) is at some distance from a target in the form of the infinite plane, assumed to be conZ ductive and grounded. The powder particles are Spray . -., ejected from the gun towards the target. All the gun Panrt~cle L~ particles are spherical and initially move with identical velocity, but their specific parameters can vary. A number of different particle species can be defined and each is characterized by mass Figure 1. Schematic view of the tribe-pow(mp), charge to mass ratio (q/m), radius (rp) and der coating system the mass outflow (m/t). The numerical algorithm is iterative and begins with predicting the initial particle trajectoties, assuming that there is no space charge. This is done by integrating the equation of motion (Newton's equation) for all particles. Then, the particle-in-cell technique is used to evaluate the space charge density corresponding to a given concentration of particles. As the third step in the algorithm, the electric field is determined by means of the finite element method. All problems are solved iteratively until a self-consistent solution is found for all relevant parameters. The computer program compares the particle trajectories after each iteration and when they change at each point less than an assumed limit iterations are terminated.
2.1 Particle trajectories It was assumed that the particle motion results from the balance of the inertial, air drag, electrical and gravitational forces. Therefore, the particle trajectory is governed by the following vector differential equation [4]
dw_ m-di
CaRer -~ Fs+Fg+F e
(1)
where w - the particle velocity, m - the particle mass, F e - the electric force, F s - the air drag force, C d - the non-Stokesian drag coefficient, Rep - the Reynolds number for the particle and Fg - the gravity force. The air drag force results from the difference between the particle velocity and velocity of the ambient air
Fs-
6x~gRp
Cc (u-w)
(2)
where: "qg is the kinematic viscosity of gas, Rp - the particle radius, u - the vector of gas velocity and Cc is the Cunningham factor. The electric force is caused by the electric field. The gravity is assumed to act along the z axis and can be positive (the gun above the target) or negative (the gun below the target). The horizontal arrangement of the gun and object would spoil the axial symmetry of the problem and cannot be simulated by this algorithm. Eq, (l) was solved using the Runge-Kutta technique. It was assumed that inside the gun all particles move along linear trajectories with an initial velocity which is equal to the velocity of the assisting air. The particles begin to diverge after leaving the gun. Some number of trajecto-
K. Adamiak/Journal of Electrostatics 40&4I (1997) 395-400
397
des were determined and they started from different points distributed uniformly between the system axis and internal surface of the gun. 2.2 Space charge density Each particle carries some electric charge, so the cloud of particles moving from the gun to the target forms the space charge. Evaluation of its density is a crucial point of the algorithm, as this provides a coupling between the particle motion and the electric field distribution. In paper [3], the space charge density was calculated from the condition of the current continuity. This method worked well but was applicable only to cases with one species of particles. An objective of this project was to investigate systems with multiple particle species and therefore a different technique needs to be found. Evaluation of the space charge density was based on the particle-in-cell technique [5]. A large number of particles are launched from the initial cross section of the gun. The trajectory of each particle is calculated, solving Eq.(1), for the real particle parameters (size, charge and mass). However, for evaluation of the space density it is assumed that each particle represents some number of particles. The equivalent electric charge of the whole group of particles is assigned to an element of the finite element mesh (cell). 2.3 Air flow distribution The assisting air can significantly affect the particle trajectories as it produces the drag force. In reality, this air forms a turbulent jet. However, for simplification of the algorithm, a potential flow model was assumed. In such a case and after introducing the stream function, W, the problem is governed by the following equation
02
1 0~
~r 2
r Or + -
02~ 02
= 0
(3)
This equation was solved by means of the finite element method, using triangular discretization of the domain and linear approximation of the stream function over each element. Both components of the air velocity vector are then calculated from ~/, by simple differentiation. 2.4 Electric field distribution When the space charge density is known the electric potential distribution can be calculated from the Poisson equation. The boundary conditions result from the assumption that the coated object is conducting and grounded. Again, this boundary value problem was solved using the finite element method. The details of the calculations are slightly different in the air flow case as the equation for the electric potential is different than that for the stream function but the general approach is identical. It even uses the same discretization of the domain and similarly, as for the air flow, the components of the electric field vector, needed for the particle trajectories calculation, are determined by differentiation.
3. RESULTS The algorithm described above was used to simulate the process with parameters as given in Table 1. The results show particle trajectories and their sensitivity to some parameters, as charge to mass ratio, gravity, velocity of the assisting air and the particle size. Calculations
K. Adamiak l,lournal of Electrostatics 40&41 (1997) 395-400
398
were done for uniform parameters of particles and then for mixtures of different partitles.
Table 1 Parameters of the simulated system
Monodispersepowdersystem particle initial velocity 1.0 m/s Fig. 2 shows selected particle trajecto- powder particle density 1500 kg/m 3 ries when all parameters of the system air velocity 1.0 m/s were equal to the basic values shown in diameter of powder particle 100 tan Table I. As particles leave the gun they distance to target 0.3 m form a concentrated cloud. The density of powder charge/mass ratio 0.3 ttC/g the resultant space charge is quite high and powder flow rate 0.6 g/s produces the strong radial component of the electric field deflecting particles in the radial direction. The axial electric field first slightly retards the particles but then drives them towards the target. At distances closer to the target the radial electric field decreases and the axial electric force begins to dominate the other forces. As a result, trajectories bend toward the target. The distance between adjacent trajectories slowly increases, which indicates that the coating layer is thickest on the axis and becomes thinner as the distance to the system axis increases. The influence of the charge to mass ratio (q/m) is shown in Fig. 3, In order to simplify the graph only the last trajectories (those closest to the gun surface) are presented. For the same mass outflow, the space charge density monotonically depends on the charge to mass ratio. Therefore, for large q/m the electric field is stronger and this clearly causes stronger repulsion of the particle trajectories. The significance of the gravity force depends on the particle charging and particle size. For relatively small q/m there is a big difference between positive, negative and no gravity conditions (Fig. 4). For positive gravity (gun above the target) particles form a narrow beam. When the gun is below the target this'beam is much wider. Some outer trajectories (close to the gun edge) do not reach the target at all and simply fall down. This picture is quite different for large 3.1
0.20 ..............................
0.25
0.15
0 . 2 0 "0.2 uC/9 . . . ~
0.3 u c l g ._
•.0~
J
"
0.1 uC/cj. ~
..-.%
E
E
0.15
O.lO 0.I0
0.05
0.05,
0.00
0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.30 z oxis
(m)
Figure 2. Particle trajectories in the tribopowder coating system
. . . .
i
. . . .
i
. . . .
i
. . . .
i
. . . .
i
. . . .
0.00 0.05 0.10 0.15 0.20 0.25 0.30 Z Oxis (m)
Figure 3. The shape of the powder cone for different values of charge/mass ratio
399
K. Adamiak/Journal of Electrostatics 40&41 (1997) 395-400
0.40 '' "-'~' . . . . . . . . . . . . . . . . . . . . . . ~ j = V g = 9 . ~ E"-" ~
0
="- " 3
E m= 0.20 ~5 2 0.10
0.10 0.00 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.00 0.05 0.10 0.15 0.20 0.25 0.30 z oxls (m)
Figure 4. Influence of gravity on particle trajectories for small q/m (0.3 p.C/g) . . . .
,
. . . .
: v=0.5 i v= 1.0 0.15 v=2.0 i v=3.0
,
. . . .
,
. . . .
,
. . . .
,
. . . .
'
. . . .
'
. . . .
'
. . . .
'
. . . .
'
. . . .
9=-9.81 0.30
0
0.20
0.20
0 . 4 0
9 =0 = "
0.00 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.00 0.05 0.10 0.15 0.20 0.25 0.30 z oxis (m)
Figure 5. Influence of gravity on particle trajectories for large q/m (1.0 ~tC/g)
. . . .
rn/s m/s m/s m/s
q/m (Fig. 5). Here, gravity does not play an important role and the difference between the configura~ . tion with the gun above the target and the gun below the target is not very significant. The velocity of the assisting air is also quite 0.10 important in shaping the powder cone (Fig. 6). Strong air flow quickly transports the powder 0.05 towards the target and the effective powder cone is narrow. The smaller the velocity of air, the weaker the influence of air drag and dispersion by O.OC 0.00 0.05 O. 10 0.15 0.20 0.25 0.30 the electric forces is stronger. The air flow influz axis (m) ence on the particle motion practically stops when Figure 6. Shaping role of the assisting air its velocity drops below the 0.5 m/s level. . . . .
i
. . . .
i
. . . .
i
. . . .
i
. . . .
i
. . . .
3.2 Polydisperse powder system The algorithm presented above can also be applied to systems where powder particles are not uniform but there is some variation in their parameters. Of course, such a model is much more realistic. In the first simulation, it was assumed that all parameters of particles are identical except for their size. The powder is composed of four different particle sizes: 40, 60, 80 and 100 pro, and the mass outflow for each group is identical. Results of calculations (Fig. 7) show that smaller particles are deflected more in the area close to the gun. However, closer to the target they are more attracted, so they cover approximately the same area as the larger one. Therefore, surprisingly, the "sorting" effect does not occur in this case; powder composition in the deposited layer is almost uniform. In another simulation of this kind, powder particles were identical, but they carried a different charge (Fig. 8). As all particles had the same size, trajectories of different particles were different. As expected, the particles with the largest q/m were deflected most significantly.
400
K. Adamiak/Journal of Electrostatics 40&41 (1997) 395-400
0.25
. . . .
-
0.20 E 0.15
-
,
. . . .
i
. . . .
r
. . . .
,
. . . .
,
. . . .
100 um
--- 80 um - 60 urn - - 40 um ..-"
0.25
- - 0.2 u C / g
0.20
0.3 u C / g 0.4 u C / g 0.5 uC/9
-
vE 0.15
-
o~
• /
~o 0.10 0.05
.,' • / . ~
. / /
0.10 .~.~--'-
1
o
o.oo
0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.30 z o×is (rn)
Figure 7. Trajectories for mixture of particles with different diameters
,,-"" ."" .. "~ , , " , - " .,-~ ."" - - " , - ' " . . " ] 1
ft",~
j...-
,s
J
".s
. , /; . / ~ ~.., , / ~, ........ , , .~-
2:'-::.
0.00 0.05 O. 10 O. 15 0.20 0.25 0.30
z o,;s (m)
Figure 8. Influence of charge/mass ratio on particle trajectories
4. CONCLUSIONS The numerical algorithm for simulating the particle trajectories in the tribo-powder coating systems was presented. In this algorithm the assisting air velocity is calculated first using the finite element method. Then, calculations are done iteratively in three steps: prediction of particle trajectories, evaluation of the space charge density and determination of the electric field. Iterations are repeated until convergence for the particle trajectories is reached. The algorithm converges quickly: typically about ten iterations are sufficient to reach convergence. The finite element mesh used for the air flow and electric field analysis had about 3000 elements, but this part is not very time consuming. The most computationally intensive part is related to determination of the particle trajectories. ACKNOWLEDGMENTS This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada. REFERENCES 1. J.F. Hughes, Electrostatic Powder Coating, J. Wiley & Sons, 1984. 2. K. Adamiak, G.S.P. Castle, I.I. Inculet and E. Hem, IEEE Trans. on Industry Appl., vol. 30 (1994) 215-222. 3. K. Adamiak and J. Mao, Conference Record of the IEEE Industry Applications Society annual Meeting, Orlando, Florida (1995) 1273-1279. 4. A. Jaworek, A. Krupa and K. Adamiak:, Conference Record of the IEEE Industry Applications Society Annual Meeting, San Diego, California, 1996 (in print). 5. R.W. Hockney and J.W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, 1981.
ELECTROSTATICS ELSEVIER
Journal of Electrostatics 40&41 (1997) 395-400
N u m e r i c a l Modelling o f Tribo-Charge P o w d e r Coating Systems Kazimierz Adamiak Department of Electrical Engineering, The University of Western Ontario London, Ontario, Canada N6A 5B9 A numerical algorithm for simulating the electric field distribution and particle trajectories in tribo-eharge powder coating systems is presented. It combines the finite element method for the electric field analysis and the numerical integration of the equation for the particle trajectoties. The particle-in-cell technique is used to couple both methods, which are applied iteratively until a self-consistent solution is obtained. I. INTRODUCTION Two electrostatic powder coating systems are commonly used in commercial applications: the corona charging system and the tribo charging system [I]. They differ not only in the way the particles are charged but also the electrical conditions in the transport and deposition zones are quite distinct. In the corona system, the particles are charged by the ionic bombardment so they can be charged intensively and the whole process is more reliable and predictable. The high voltage supply is needed for generating the corona discharge, but it also generates a strong electric field which drives the powder particles towards the target. Not all ions are collected by the particles so they also move towards the coated object and can produce the back corona discharge which often deteriorates the finish quality. In the tfibo-charge powder coating the powder particles are charged by the frictional contact with a gun. A cloud of the charged particles ejected from the gun forms a space charge which in turn produces an electric field which drives the particles towards the target. Therefore, the whole system does not require an external voltage supply and is preferred in some applications [2]. This technique can produce a better finish qual/ty, especially for targets with complex shapes. Both reasons increase interest in the tribo-charge powder coating systems. However, the magnitude of the electric field in the tribo systems is usually much weaker than that in the corona systems resulting in a decrease in the electric force and greater sensitivity to different kinds of disturbing factors. Modelling of the tribo-charge coating process is much more difficult than that of the conventional corona system. In both cases the electrostatic field and the equation for the particle motion must be solved. However, the electric field is a function of the space charge density. In the corona coating the majority of the space charge comes from the corona discharge and the field generated by the particles can be neglected. In tfibo-charge coating, the space charge is associated only with the particle cloud. The particle movement results from the balance of electrical, inertial, air-drag and gravitational forces. Conversely, the electric field distn'bution is a function of the particle concentration. Therefore, both problems are mutually coupled [3]. 0304-3886/97/$17.00 © Elsevier Science B.V. All fights reserved. S0304-3886(97)00077-6
PI1
K. Adamiak/dournal of Electrostatics 40&41 (1997) 395-400
396
2. MATHEMATICAL MODEL ~r
The process is simulated for the simplified model of the problem shown in Fig. 1. A cylindrical gun made of a dielectric material (permittivity ~) is at some distance from a target in the form of the infinite plane, assumed to be conZ ductive and grounded. The powder particles are Spray . -., ejected from the gun towards the target. All the gun Panrt~cle L~ particles are spherical and initially move with identical velocity, but their specific parameters can vary. A number of different particle species can be defined and each is characterized by mass Figure 1. Schematic view of the tribe-pow(mp), charge to mass ratio (q/m), radius (rp) and der coating system the mass outflow (m/t). The numerical algorithm is iterative and begins with predicting the initial particle trajectoties, assuming that there is no space charge. This is done by integrating the equation of motion (Newton's equation) for all particles. Then, the particle-in-cell technique is used to evaluate the space charge density corresponding to a given concentration of particles. As the third step in the algorithm, the electric field is determined by means of the finite element method. All problems are solved iteratively until a self-consistent solution is found for all relevant parameters. The computer program compares the particle trajectories after each iteration and when they change at each point less than an assumed limit iterations are terminated.
2.1 Particle trajectories It was assumed that the particle motion results from the balance of the inertial, air drag, electrical and gravitational forces. Therefore, the particle trajectory is governed by the following vector differential equation [4]
dw_ m-di
CaRer -~ Fs+Fg+F e
(1)
where w - the particle velocity, m - the particle mass, F e - the electric force, F s - the air drag force, C d - the non-Stokesian drag coefficient, Rep - the Reynolds number for the particle and Fg - the gravity force. The air drag force results from the difference between the particle velocity and velocity of the ambient air
Fs-
6x~gRp
Cc (u-w)
(2)
where: "qg is the kinematic viscosity of gas, Rp - the particle radius, u - the vector of gas velocity and Cc is the Cunningham factor. The electric force is caused by the electric field. The gravity is assumed to act along the z axis and can be positive (the gun above the target) or negative (the gun below the target). The horizontal arrangement of the gun and object would spoil the axial symmetry of the problem and cannot be simulated by this algorithm. Eq, (l) was solved using the Runge-Kutta technique. It was assumed that inside the gun all particles move along linear trajectories with an initial velocity which is equal to the velocity of the assisting air. The particles begin to diverge after leaving the gun. Some number of trajecto-
K. Adamiak/Journal of Electrostatics 40&4I (1997) 395-400
397
des were determined and they started from different points distributed uniformly between the system axis and internal surface of the gun. 2.2 Space charge density Each particle carries some electric charge, so the cloud of particles moving from the gun to the target forms the space charge. Evaluation of its density is a crucial point of the algorithm, as this provides a coupling between the particle motion and the electric field distribution. In paper [3], the space charge density was calculated from the condition of the current continuity. This method worked well but was applicable only to cases with one species of particles. An objective of this project was to investigate systems with multiple particle species and therefore a different technique needs to be found. Evaluation of the space charge density was based on the particle-in-cell technique [5]. A large number of particles are launched from the initial cross section of the gun. The trajectory of each particle is calculated, solving Eq.(1), for the real particle parameters (size, charge and mass). However, for evaluation of the space density it is assumed that each particle represents some number of particles. The equivalent electric charge of the whole group of particles is assigned to an element of the finite element mesh (cell). 2.3 Air flow distribution The assisting air can significantly affect the particle trajectories as it produces the drag force. In reality, this air forms a turbulent jet. However, for simplification of the algorithm, a potential flow model was assumed. In such a case and after introducing the stream function, W, the problem is governed by the following equation
02
1 0~
~r 2
r Or + -
02~ 02
= 0
(3)
This equation was solved by means of the finite element method, using triangular discretization of the domain and linear approximation of the stream function over each element. Both components of the air velocity vector are then calculated from ~/, by simple differentiation. 2.4 Electric field distribution When the space charge density is known the electric potential distribution can be calculated from the Poisson equation. The boundary conditions result from the assumption that the coated object is conducting and grounded. Again, this boundary value problem was solved using the finite element method. The details of the calculations are slightly different in the air flow case as the equation for the electric potential is different than that for the stream function but the general approach is identical. It even uses the same discretization of the domain and similarly, as for the air flow, the components of the electric field vector, needed for the particle trajectories calculation, are determined by differentiation.
3. RESULTS The algorithm described above was used to simulate the process with parameters as given in Table 1. The results show particle trajectories and their sensitivity to some parameters, as charge to mass ratio, gravity, velocity of the assisting air and the particle size. Calculations
K. Adamiak l,lournal of Electrostatics 40&41 (1997) 395-400
398
were done for uniform parameters of particles and then for mixtures of different partitles.
Table 1 Parameters of the simulated system
Monodispersepowdersystem particle initial velocity 1.0 m/s Fig. 2 shows selected particle trajecto- powder particle density 1500 kg/m 3 ries when all parameters of the system air velocity 1.0 m/s were equal to the basic values shown in diameter of powder particle 100 tan Table I. As particles leave the gun they distance to target 0.3 m form a concentrated cloud. The density of powder charge/mass ratio 0.3 ttC/g the resultant space charge is quite high and powder flow rate 0.6 g/s produces the strong radial component of the electric field deflecting particles in the radial direction. The axial electric field first slightly retards the particles but then drives them towards the target. At distances closer to the target the radial electric field decreases and the axial electric force begins to dominate the other forces. As a result, trajectories bend toward the target. The distance between adjacent trajectories slowly increases, which indicates that the coating layer is thickest on the axis and becomes thinner as the distance to the system axis increases. The influence of the charge to mass ratio (q/m) is shown in Fig. 3, In order to simplify the graph only the last trajectories (those closest to the gun surface) are presented. For the same mass outflow, the space charge density monotonically depends on the charge to mass ratio. Therefore, for large q/m the electric field is stronger and this clearly causes stronger repulsion of the particle trajectories. The significance of the gravity force depends on the particle charging and particle size. For relatively small q/m there is a big difference between positive, negative and no gravity conditions (Fig. 4). For positive gravity (gun above the target) particles form a narrow beam. When the gun is below the target this'beam is much wider. Some outer trajectories (close to the gun edge) do not reach the target at all and simply fall down. This picture is quite different for large 3.1
0.20 ..............................
0.25
0.15
0 . 2 0 "0.2 uC/9 . . . ~
0.3 u c l g ._
•.0~
J
"
0.1 uC/cj. ~
..-.%
E
E
0.15
O.lO 0.I0
0.05
0.05,
0.00
0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.30 z oxis
(m)
Figure 2. Particle trajectories in the tribopowder coating system
. . . .
i
. . . .
i
. . . .
i
. . . .
i
. . . .
i
. . . .
0.00 0.05 0.10 0.15 0.20 0.25 0.30 Z Oxis (m)
Figure 3. The shape of the powder cone for different values of charge/mass ratio
399
K. Adamiak/Journal of Electrostatics 40&41 (1997) 395-400
0.40 '' "-'~' . . . . . . . . . . . . . . . . . . . . . . ~ j = V g = 9 . ~ E"-" ~
0
="- " 3
E m= 0.20 ~5 2 0.10
0.10 0.00 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.00 0.05 0.10 0.15 0.20 0.25 0.30 z oxls (m)
Figure 4. Influence of gravity on particle trajectories for small q/m (0.3 p.C/g) . . . .
,
. . . .
: v=0.5 i v= 1.0 0.15 v=2.0 i v=3.0
,
. . . .
,
. . . .
,
. . . .
,
. . . .
'
. . . .
'
. . . .
'
. . . .
'
. . . .
'
. . . .
9=-9.81 0.30
0
0.20
0.20
0 . 4 0
9 =0 = "
0.00 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.00 0.05 0.10 0.15 0.20 0.25 0.30 z oxis (m)
Figure 5. Influence of gravity on particle trajectories for large q/m (1.0 ~tC/g)
. . . .
rn/s m/s m/s m/s
q/m (Fig. 5). Here, gravity does not play an important role and the difference between the configura~ . tion with the gun above the target and the gun below the target is not very significant. The velocity of the assisting air is also quite 0.10 important in shaping the powder cone (Fig. 6). Strong air flow quickly transports the powder 0.05 towards the target and the effective powder cone is narrow. The smaller the velocity of air, the weaker the influence of air drag and dispersion by O.OC 0.00 0.05 O. 10 0.15 0.20 0.25 0.30 the electric forces is stronger. The air flow influz axis (m) ence on the particle motion practically stops when Figure 6. Shaping role of the assisting air its velocity drops below the 0.5 m/s level. . . . .
i
. . . .
i
. . . .
i
. . . .
i
. . . .
i
. . . .
3.2 Polydisperse powder system The algorithm presented above can also be applied to systems where powder particles are not uniform but there is some variation in their parameters. Of course, such a model is much more realistic. In the first simulation, it was assumed that all parameters of particles are identical except for their size. The powder is composed of four different particle sizes: 40, 60, 80 and 100 pro, and the mass outflow for each group is identical. Results of calculations (Fig. 7) show that smaller particles are deflected more in the area close to the gun. However, closer to the target they are more attracted, so they cover approximately the same area as the larger one. Therefore, surprisingly, the "sorting" effect does not occur in this case; powder composition in the deposited layer is almost uniform. In another simulation of this kind, powder particles were identical, but they carried a different charge (Fig. 8). As all particles had the same size, trajectories of different particles were different. As expected, the particles with the largest q/m were deflected most significantly.
400
K. Adamiak/Journal of Electrostatics 40&41 (1997) 395-400
0.25
. . . .
-
0.20 E 0.15
-
,
. . . .
i
. . . .
r
. . . .
,
. . . .
,
. . . .
100 um
--- 80 um - 60 urn - - 40 um ..-"
0.25
- - 0.2 u C / g
0.20
0.3 u C / g 0.4 u C / g 0.5 uC/9
-
vE 0.15
-
o~
• /
~o 0.10 0.05
.,' • / . ~
. / /
0.10 .~.~--'-
1
o
o.oo
0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.30 z o×is (rn)
Figure 7. Trajectories for mixture of particles with different diameters
,,-"" ."" .. "~ , , " , - " .,-~ ."" - - " , - ' " . . " ] 1
ft",~
j...-
,s
J
".s
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0.00 0.05 O. 10 O. 15 0.20 0.25 0.30
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Figure 8. Influence of charge/mass ratio on particle trajectories
4. CONCLUSIONS The numerical algorithm for simulating the particle trajectories in the tribo-powder coating systems was presented. In this algorithm the assisting air velocity is calculated first using the finite element method. Then, calculations are done iteratively in three steps: prediction of particle trajectories, evaluation of the space charge density and determination of the electric field. Iterations are repeated until convergence for the particle trajectories is reached. The algorithm converges quickly: typically about ten iterations are sufficient to reach convergence. The finite element mesh used for the air flow and electric field analysis had about 3000 elements, but this part is not very time consuming. The most computationally intensive part is related to determination of the particle trajectories. ACKNOWLEDGMENTS This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada. REFERENCES 1. J.F. Hughes, Electrostatic Powder Coating, J. Wiley & Sons, 1984. 2. K. Adamiak, G.S.P. Castle, I.I. Inculet and E. Hem, IEEE Trans. on Industry Appl., vol. 30 (1994) 215-222. 3. K. Adamiak and J. Mao, Conference Record of the IEEE Industry Applications Society annual Meeting, Orlando, Florida (1995) 1273-1279. 4. A. Jaworek, A. Krupa and K. Adamiak:, Conference Record of the IEEE Industry Applications Society Annual Meeting, San Diego, California, 1996 (in print). 5. R.W. Hockney and J.W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, 1981.