NUMERICAL SIMULATION OF CORONA

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NUMERICAL SIMULATION OF CORONA-CHARGING
POWDER COATING SYSTEM
by
Thanh Lam
Department of Electrical and Cornputer Engineering
Submitted in partial fulfilment
of the requirements for the degree o f
Master o f Engineering Sciences
Faculty o f Graduate Studies
The University o f Western Ontario
London, Ontario
August 1998
OThanh Lam, 1998
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ABSTRACT
This thesis describes a numerical algorithm to calculate the
electric field. charge density and the particle trajectories in a coronacharging powder coating system. The system consists of an
electrostatic powder gun, powder particles and a target plane. The
algorithm employs an iterative technique where the Finite Element
Method is used for cornputing the electric field strength i n conjunction
with the Method of Characteristics for determining the ionic charge
density and the Particle-In-Cell Method for simulating the powder
particle trajectories from the gun to the target plane. The airflow
between the electrostatic gun and the target plate is calculated by
solving the Navier-Stokes equation for steady viscous laminar flow.
The particle trajectories are modeled using the Basset, Boussineq and
Oseen equation and integrated using the Euler's method. The process is
ccrnputed recursively until a self-consistent solution for the electric
field, particle trajectories and t h e space charge density distribution is
obtairied.
The algorithm is used t o simulate the powder particle trajectories
for variations of particle size, corona voltage, charge t o mass ratio,
mass transfer rate and the gun position relative to the target plane. The
results provide a further understanding of the electric field. powder
iii
trajectories and the space charge density distribution in the electrostatic
coating process.
ACKNOWLEGEMENTS
First, 1 would like t o t a k e this opportunity t o express my sincere
gratitude to my chief advisor Professor Kazirnierz Adamiak for h i s
invaluable guidance and s u p p o r t throughout the project.
1 would also like t o thank Professor G 5 . P Castle, Professor Terry
E. Base, and Dr. Sergey Primak f o r their advise and helpful discussions
o n t h e project.
Finally, I want t o thank my parents, my family and al1 my friends
for their support and everlasting encouragement t h r o u g h al1 these years.
T A B L E OF CONTENTS
CHAPTE R 1
Introduction ------------------------*------------------------------1
1
- -..*----------.----.---------------
1- 1
Background .------.
1. 2
Objectives ----------------.-------.-------.------+-+--+.-------------.--------------.------3
*--*
* .*.-
*-*---.*-----
CHAPTER 3
Mathematical Mode1 of Powder Coating
4 . 3 . 1 interpolation o f Space Charge Density
...4 4
4.4
Calculating t h e Powder Particle Trajectories --------------------..
...46
4.5
Convergence of the Iterative Process ----...............----.---------------49
CHAPTER 6
Conclusions and Recommenda tions
-
.----
78
6.1
78
Conclusions ---------,,.--....-.--------...-..-----.-----.---------.------------
6.2
Recommendations
--------...--....,.-------.--.----.------------------.---80
83
REFERENCES .---.----.-----------------------+-+..-.------------.--------------*---------------
89
APPENDIX ---------.--------------.-..----.----------------------------------.----
CURRICULUM VITAE
------.--------------------.----*----*-*-**-----------.--------------.--~*------.--.--
94
LIST OF FIGURES
Figure 1 . 1 Schematic diagram of an electrostatic powder coating unit
............................
Figure 2.1 Particle charging by corona discharge.............................................................
2
9
Figure 2.2 Schernaîic representation of back-ionization in corona-charged powder layer 14
Figure 2.3 Schematic of a low voltage curona gun
....................................................... 16
Figure 3.1 Geornetry of the electrostatic powder c o a ~ unit
g
.......................................
21
Figure 3.2 Charge assigrnent in Particle-In-CeiJ method .............................................. 32
Figure 3 -3 Boundary conditions for airflow mode1 ....................................................... -34
Figure 4.1 Flowchart of cornputer program ................................................................. - 3 7
Figure 4.2 Electric field at tip of wrona wire surfiice .................................................... 41
Figure 4.3 Trajectory seps for characteristic hes .........................................................
43
Figure 4.4 Interpolation of charge density at FEM mesh nodes .................................... -46
Figure 5.1 Electric field distribution in EPC system .......................................................
54
Figure 5.2 Axial component ofelehc field in EPC system .......................................... - 5 5
Figure 5.3 Radial component of electric field in EPC systern ........................................ -56
Figure 5.4 Contour plot of the electric field near corona wire ....................................... - 5 7
Figure 5.5 Ions demity distn'bution produced by corona discharge ............................... -59
Figure 5.6 Space charge deosity distributioa in EPC system .......................................... 60
Figure 5.7 Powder tmjectories in EPC system .............................................................. -64
Figure 5.8 Particle velocity vecton dong trajectories in EPC system
(Pauthenia LVna Q/W-1- 3 mClkg) .........*.............................................*..................... -65
Figure 5.9 Particle velocity vecton with smaller charge to mass ratio
(Singh minimum Q M=.0.2 rnC/kg) ...............................................................................
66
Figure 5.10 Electric field vecton dong particle trajectories in EPC system ................... 67
Figure 5.11 Electric field intensity d o n g the furthest trajectory .................................... 68
Figure 5.12 Axial component of air velocity dong axial axis ......................................... 69
Figure 5.12 Particle tnjectories as finction of applied voltage
...................................
Figure 5.13 Particle trajectories as funaion of charge to mass ratio
Figure 5.14 Particle trajectories as fiinction of particle radius
-73
..............................
74
.......................................
75
Figure 5.15 Particle trajectories as fùnction of mass transfer rate
..................................
76
Figure 5.16 Particle trajectories for d e r gun-to-target separation ............................ 77
Figure A l An electrostatic powder coating unit manufactured by Wagner................... 90
Figure A2 Powder flow of a flat spray gun nonle ....................................................... 90
Figure A3 Flowchart of Finite Element subrouthe ....................................................... 91
Figure A4 Flowchart of a subroutine evaluating space charge density using MOC
Figure A5 nowchart of a subrouthe evaluating the particle trajectories
....... 92
......................
93
LIST OF TABLES
Table 2.1 Advantages and disadvantages of different types of powder in EPC ............... I I
Table 5.1 Parameters of the EPC Process......................................................................
52
NOMENCLATURE
Svrnbol
Descri~tion
Unit
ion mobility
mZ/~s
absolute room temperature
K
operating temperature
K
normal atmospheric pressure
Pa
operating pressure
Pa
roughness factor
t i me
axial position
radial position
permittivity
permittivity of free space ( 8 . 8 5 4 8 4 2 )
relative permittivity
Peek's corona onset electric field
electric field
axial component o f electric field
radial component of electric field
electric field at corona wire surface
current density
space charge density
electric potential
electric potential associated with node i
shape function associated with node i
charge density associated with node i
characteristic matrices of triangular mesh
energy function associated with one triangle
energy function
powder discharge time constant
powder resistivity
powder particle radius
mass of particle
powder density
fluid density
maximum charge carried b y a particle
charge to mass ratio
charge density coefficient
ernitting angle
space-time trajectory of ions
number of trajectory segments
initial incrernent step
trajectory step constant
force acting on particle
air drag force
gravity force
electrostatic force
particle velocity
fluid velocity
drag coefficient
gravitational constant (9.81)
diffusion coefficient
equivalent charge of super particle
number o f toroidal rings
mass transfer rate
area of triangle
average radius of a triangle
length of corona wire
radius of corona wire
length o f gun nozzle
radius of gun nozzle
radius of target plane
gun to target distance
electric potential at corona wire
electric potential at target plane
mean radius of particle
particle eject velocity
radial component of fluid velocity
axial component of fluid velocity
axial component of particle velocity
radial component o f particle velocity
kinematic viscosity
stream function
vorticity
axial acceleration of particle
radial acceleration of particle
axial position of particle
radial position of particle
error of inner loop
error of outer loop
CHAPTER 1
Introduction
1.1
Background
Electrostatic Powder Coating (EPC) is a process whereby
electrostatic forces are applied to increase the transfer efficiency of
powder particles onto a grounded workpiece. The high deposition rate
and low operating costs have attracted many industrial manufacturers to
apply EPC to their finishing products such as household appliances and
auto body parts. The coating process generally consists of three main
stages:
1 . Charge and transport the powder to the workpiece
2 . Deposit the powder to the workpiece
3 . Transport the workpiece to an oven and fuse powder t o
form a continuous coating
A diagram of a corona charging EPC system is shown in Figure
1 . 1 . It consists of a fluidized bed hopper, an electrostatic g u n and a
workpiece connected t o ground. T h e powder is usually an epoxy resin
ground to a very small mean size of 10-100 p m diameter. The powder is
first fluidized in the bed hopper and then pumped to the electrostatic
gun. A very thin corona wire connected to a high voltage supply is
L-
Free ions
-
Corona wire
\
Q
0--
-
Q-O
-0
-
Elecmc
Fluidized
bed hopper
High
voltage
generator
8 a
Control unit
Figure 1.1. Schematic diagram of an electrostatic powder coating unit
placed at the gun nozzle. When the applied voltage is higher than the
onset voltage, a phenornenon known as corona discharge is created
which produces a cloud of unipolar ions. These ions attach to the
passing-by powder. The powder particles become charged and are
carried to the grounded workpiece and adhered to i t by the electrostatic
forces. T h e workpiece is then transported to an oven where the powder
is fused t o form a continuous coating.
1.2
Objectives
T h e objective of this project is t o develop a full numerical
algorithm of a corona-charging EPC process. It applies the Finite
Element Method and the Method of Characteristics to compute the
electric field and the space charge distribution coupled with the
numerical integration of the Basset, Boussinesq and Oseen's equation
and the Particle-In-Cell Method to determine the trajectories of the
powder particles. The mode1 is used t o study the trajectories of powder
particles for various operating conditions such as different applied
voltages, mass transfer rates, particle sizes, charge to mass ratios and
gun-to-target separations.
CHAPTER 2
Electrostatic Powder Coatiog
2.1 Introduction
There are two cornmon EPC systems used in the commercial
applications: the tribo charging system and the corona charging system.
They differ not only in the way the particles are charged, but also in the
electrical conditions o f the transport and the deposition zones. In the
corona charging system. the powder particles are charged by
bombardment with the ions produced in the corona discharge. A high
voltage supply is required to create the corona discharge but it also
generates a strong electric field which drives the powder particles
towards the target. I n the tribo charging system, the powder particles
are charged by frictional contact between the powder particles and the
material of the gun body. The cloud of charged particles ejected from
the gun forms a space charge, which in turn produces an electric field
conveying the particles towards the target. Considerable research has
been undertaken t o study the charging mechanism of the t w o systems. A
cornparision o f the tribo versus corona charging systems was reviewed
by Kleber [ 2 1 ] , Moyle and Hughes [29], and Reddy [ 3 2 ] . Many
different gun designs and powder materials were investigated and tested
to optimize the transfer efficiency of the coating process.
2.2 Tribo Charging
Frictional or tribo charging has been k n o w n for more than 2000
years. When two insulating materials are rubbed together repetitively,
the material with the higher work function will be charged negatively
while the other is charged positively. For some powder materials. they
can experience considerable frictional charging during transport from
the bed hopper to the electrostatic gun. This charging usually depends
on the temperature, density, conductivity, permittivity, duration of
contact, surface treatment, and number of contact points. I n some
instances, the powder acquires an amount of charge in the same order of
magnitude as in the corona charging. The amount of charge acquired by
the powder is very important i n the EPC as it establishes not only the
intensity of the electric field which drives the particles towards the
workpiece, but also the ability of the particles t o adhere to the
grounded workpiece.
The advantage of the tribo system is that it does not need a high
voltage supply to generate the ions for charging the powder particles
such as in the corona charging system. Because of this. the system
produces no free ions and gives a superior powder penetration into
targets that have corners and cavities. The drawback of the system is
the steady decrease in t h e mean charge to mass ratio d u e to the
accumulation of a powder layer on the interna1 surface of the gun barrel.
This occasionally makes t h e tribo charging system unreliable and
unpredictable during a long coating operation.
2.3 Corona Charging
2.3.1
Corona Discharge
The most important property of the corona discharge is its ability
to generate a space charge in the gas filled regions. This ionic space
charge can be used for charging small solid particles, liquid droplets, or
dielectric surfaces making possible numerous applications such as
powder coating, paint spraying, crop spraying, electrophotography.
particle separation and precipitation. The corona discharge is produced
when a high voltage is applied between two electrodes with different
curvatures. In the corona charging EPC system, the electrostatic
powder gun consists of a very thin corona wire connected to a high
voltage supply and the workpiece connected to ground as shown in
Figure 1 . 1 . When a high negativel voltage is applied to the corona wire,
it produces a very high electric field region near the corona wire known
as the ionization region. Positive ions created by ionization from
electromagnetic radiation o r ot her mechanisms are attracted t o the
corona wire while electrons are repelled away by the h i g h electric field.
The electrons accelerate away from the negative corona wire and gain
sufficient energy to ionize any molecules with which they collide. These
collisions produce an avalanche of electrons and positive ions until new
electrons are formed far enough away that they are unable t o gain
sufficient energy from the electric field between collisions t o ionize
another atom. The electrons emerging from the ionization region are
attached to air molecules t o form a space charge cloud of negative ions
drifting from the corona wire towards the grounded target. These ions
help suppress the electric field near the corona wire making t h e corona
discharge stable and accessible for charging powder particles in the EPC
process.
2.3.2
Pauthenier L i m i t
When the spraying powder particles enter the space charge cloud
produced by the corona discharge, the particles are charged by the
negative ions impinging on the particle surfaces untii the repelling
1
For more information about positive and negative corona. set White[ll] and Hughes[ZO].
electric field produced by the particles is equal to the surrounding field
a s shown in Figure 2 . 1 . The charging then ceases and the particle is
said t o have acquired a saturation charge, known as the Pauthenier
limit. Pauthenier predicted that the tirne variation of the charge on a
spherical particle by corona charging is given by
where a is the particle radius and E is the electric field intensity. From
the above equation, the maximum charge that can be acquired by the
particle depends on the magnitude of the electric field and t h e square of
t h e particle radius. In normal coating operation, the charging tirne is
very small, usually about 0.01-0.1 rns for a particle in the charging zone
with a sufficiency strong electric field E [IO].
The maximum charge
acquired by the powder particle is therefore equal t o
For a spherical particle with density p,, t h e saturation charge to
mass ratio is
elearic field
source of ions
(a) no charge
(b) saturation charge
Figure 2 . 1 Particle charging by corona discharge.
2.4 Parameters of EPC System
The effectiveness o f the EPC. system is measured by the transfer
efficiency and the quality of the finish coating. The transfer efficiency
may be defined as the ratio of the powder mass deposited on the target
to the mass of the powder emitted from the gun. The primary goals are
to obtain a transfer efficiency as high as possible while meeting the
aesthetic requirements for the coating applications concerned. These
goals can be achieved by optimizing the following factors: powder
materiais. flow rate. powder charging and the adhesion ont0 the
workpiece.
2.4.1
Powder Material
AH materiais can be electrostatically applied in an EPC process
provided they are electrical insulating and can b e produced in powder
form of approximately 10
-
100 pm mean diameter. The powders used
are normally made of synthetic materials with high resistivity such as
polyester, epoxy. vinyls. etc ...
The mean particle size and the
geometric standard deviation of the size distribution of the powder
particles have a dominant effect on the final coating appearance and the
charging characteristics. Smaller particle sizes and deviation are found
to give a more uniformly deposited powder layer and greater coating
consistency while larger particle sizes usually acquire a bigger charge
and are more efficiently deposited on the grounded workpiece. Singh
1361 observed that the mean charge to mass ratio ( Q M) is inversely
proportional t o the particle radius for tribo charging where as the
powder pigmentation and the surface chemistry have a dominant effect
on the mean charge to mass ratio for corona charging. Basic data of
some commercial powders and their applications obtained from Cross
[ 1 5 ] are compared in Table 2 . 1 .
T a b l e 2.1
Advaotages a n d d i s a d v a n t a g e s of different types of
powder i n E P C .
Material
Acrylic
Cellulose
acetate
Chlorinated
polyethers
Epoxies
Fluorocarbons
Nylon
Polyester
Polyethylene
Polyurethane
Vinyls
Disadvan tages
Advantages
High durability, good
hardness, excellent
chemical resistance
Excellent flow, good gloss
and colour, excellent
chernical resistance
Excellent corrosion,
mechanical and electrical
1 resistance, good
dimensional stability
Excellent adhesion, high
alkali, abrasion and
corrosion resistance
High exterior durability
and service temperature,
low coefficient of friction,
Excellent chemical and
electrical resistance
L o w coefficient of friction,
high dielectric strength,
chernical and abrasion
resistance
T hermoplastic and
t hermoset avaiiable,
intermediate cost
Good chernical resistance
and electrical insulation,
excellent flexibility, low
1 water absorption, resists
low temperatures, low cost
High chernical resistance,
Excellent hardness. good
gloss and flow
Poor f l o w , moderate
adhesion, brittle
Moderate adhesion, soft
film, discolours at high
temperature
Poor exterior durability,
h i g h cost, fair adhesion
1
1
1
/ in exterior applications.
,
High chernical resistance,
high dielectric strength,
long-term durability, good
Abrasion resistance,
1 intermediate cost
I
Chalking and yellowing
high cost
Poor adhesion, requires
high curing temperature,
high cost
1
Films beiow 100 pm
( difficult, poor adhesion,
1 high
cost, low service
temperature
High curing temperature
Poor adhesion and
durability, poor exterior
durability, and abrasion
1 resistance, low service
temperature
Poor adhesion, h i g h
water resistivity,chalks
and yellows in exterior
use
Poor adhesion, only thick
films 175 pm
/
2.4.2
Powder Flow
The powder flowability and fluidization are also very important in
the powder coating process. In order to have uniform coating, the
powder must flow through the gun and be well dispersed while
maintaining a uniform mass transfer rate. Even minor changes in
dispersion and flow characteristics alter the film properties. Mazumder
et al.[26] found that powder with a wider size distribution has a higher
fluidity while powder with a surface treatment additive gives a better
flowability. The flow rate usually depends on the work environment and
the equipment used. A flow rate of 1-3 g/s is commonly applied in many
industrial EPC processes.
2.4.3
Powder Charging
The charging of powder is considered to be the most important
step i n the EPC process. The amount of charge acquired by the powder
particles not only determines their trajectories b u t also their adherence
to the workpiece.
If the charging is too low, the powder may not
adhere t o t h e workpiece; if too high, i t can give rise t o poor quality
coating. Singh [36] found that the deposition efficiency is a function of
the mean charge to mass ratio ( Q / M )of the powder.
The higher the
mean charge to mass ratio, the higher the deposition rate. Therefore. it
is very important that the powder particles are charged consistently with
a sufficient charge during the coating process in order to maximize the
efficiency. Singh e t al. [ 3 5 ] reported that a minimum charge t o mass
ratio o f 2x10-' C / k g is required before the powder particles will adhere
to any grounded w o r k p i e c e . A charge to mass ratio o f 5 x 1O-' C/kg is
usually necessary f o r a d e q u a t e adhesion in t h e coating process.
2.4.4
Powder Adhesion
When t h e c h a r g e d powder particles a r e carried near t h e grounded
workpiece, they a r e a t t r a c t e d t o the workpiece by t h e Coulomb force.
The powder particles build u p a layer covering the workpiece. The
workpiece is then t r a n s p o r t e d t o an oven t o fuse t h e particles and form
a continuous c o a t i n g . Meanwhile. t h e powder must retain its charge so
it doesn't fa11 off t h e workpiece. The time constant f o r t h e discharge o f
the powder is given a s
where
E~ i s
5 is
the p o w d e r resistivity,
EQ
is t h e permittivity o f free space and
t h e relative permittivity of the p o w d e r material. For insulating
powder with resistivity exceeding 10"
am,it
can retain i t s charge for
several minutes while conductive particles o r materials with l o w
resistivity tend to l o s e their charge rapidly and fa11 off t h e workpiece.
Therefore, the powder resistivity must b e selective so that the particle
will retain its charge long enough to fuse and form the finishing coating.
As more and more particles and ions move from the gun to the
workpiece, the potential across the deposited layer increases. As the
layer thickness increases, the field strength increases until it reaches a
critical field strength ( - 3 x 1 0 ~ V/m) sufficient t o cause a break down in
the surrounding air. The break down produces bipolar ions. which
neutralize the incoming charge particles and create a non-uniform
coating on the workpiece. This is known as back ionization. Back
ionization often Iimits the thickness of the deposited layers and can
produce a poor quality finishing by creating surface blemishes such as
pinholes. moon craters, and the orange peel effect in the finishing
coating.
Free ions
w
Charged
particles
Uncharged
Figure 2.2 Schematic representation of back-ionization in a corona-charged
powder layer
2.5 Low Voltage Corona G u n
As discussed earlier. the tribo system is unreliable and the corona
system produces excessive free ions giving rise to the early onset of
back ionization. One approach for eliminating these free ions is to
modify the corona gun design as shown in Figure 2 . 2 . Instead of
placing the corona wire outside the gun nozzle, the corona wire is
withdrawn within the confines o f the gun notzle [ 2 9 ] . If an additional
electrode. called an attractor electrode, is grounded and placed near the
corona wire, the corona discharge can be generated at a much lower
voltage. Corona charging is carried out in the interelectrode spacing
within the g u n nozzle. Al1 the powder emanating from the g u n will pass
through the corona region and will b e charged, while the free ions are
attracted to the attractor electrode and very few ions are ejected from
the gun nozzle. With these low voltage guns, there are some
operational difficulties due t o the powder deposition on the interna1
attractor electrode and the gun nozzle. As the powder deposition
increases, it decreases the charging and can also cause the onset of back
ionization within the gun. Thus, these guns would require regular
maintenance after a period of coating operation.
Figure 2 . 3
2.6
Schematic of a low voltage corona gun
Literature Review of EPC Simulation
The numerical simulation of the corona charging EPC system
consists of calculating the electric field, space charge and the powder
particle distribution from the gun t o the target. The particle and the
space charge density are dependent o n the electric field. On the other
hand, t h e electric field is dependent on the particle and the space charge
density. Therefore, both distributions are mutually coupled and should
be solved simultaneously.
In 1987, A n g and Lloyd [6] presented a computational mode1 for
calculating the trajectories of charged particles in an EPC system. The
equation of motion of the particle was formulated by performing a force
balance of the aerodynamic and the electrostatic forces involved. The
airflow from the gun nozzle was measured using a hot wire anemometer
and found to be reasonably approximated by Tallien's solution for an
axisyrnmetric jet
[Il.
Using a point-to-plane mode1 developed by Wu
[42]. the electric field was calculated by solving the one-dimensional
Poisson's equation assuming the charge density distribution was
constant. The trajectories of the particles were calculated and
compared with the experimental measurements using photographic
techniques. The accuracy between the computed and the experimental
results showed improvement when moving towards the centerline of the
air jet. The results showed that the airflow from the spraying gun was
responsible for the initial particle transport, with increasing dominance
of the electrostatic forces near the grounded workpiece mainly due the
field enhancement effect of the space charge.
Artamonov and Vereshchagin [7] described a mathematical mode1
for calculating the electric field and the space charge density
distribution of an electrostatic spraying process. The electric field and
the space charge density from the gun t o the target plane were found to
be interconnected and described by the Poisson and the continuity
equation. The Finite Difference Method was used to solve both
equations simultaneously. The calculated results showed that the
particle size affects greatly the deposition efficiency. Higher efficiency
was obtained when larger particles were used.
Using the Boundary Element Method and the Finite Element
Method to solve for the electric field distribution, Tanasescu et al. [ 3 8 ]
developed a mathematical model t o calculate the charged particle
trajectories in an electrostatic field. Neglecting the spatial charge. the
particle trajectories were deterrnined by integrating two non-linear
ordinary differential equations governing the axial and the radial
acceleration of the particles. The air velocity was computed by solving
the Navier-Stokes equation for small flow rates. The particle
trajectories were computed for charge-to-mass ratios both from the
Pauthenier charging equation and the experimental charge measurement.
Ali 151 presented a mathematical mode1 for t h e trajectories of
charge powder particles in an EPC system by considering the
electrostatic and the aerodynamic forces acting on the particles. The
model employed an iterative technique wherein the Charge Simulation
Method was used t o compute the electric field and the Method of
Characteristics was used to compute the charge density distribution in
the interelectrode space between the g u n a n d the target plane. The
airflow was interpolated €rom experimental rneasurements using a hot
wire anemometer. Neglecting the space charge from powder particles,
several powder particles were simulated for size range of 10-40 p m
diameter and charge-to-mass ratios of 0.01-0.1 rnClkg.
In this thesis, a new numerical technique is developed employing
the Finite Element Method (FEM), the Method of Characteristics (MOC)
and the Particle-In-Cell (PIC) Method to simulate t h e motions of t h e
charged powder particles in an EPC process. The powder charge and
the ionic charge density are included i n the numerical mode1 by
calculating the combined electric field, space charge and the trajectories
of powder particle recursively until a self-consistent solution is
obtained.
CHAPTER 3
Mathematical Model of Powder Coating
Process
3.1
Model and Assumptions
As usual in numerical simulation, some idealizing assumptions
have to be made in order to simplify the problem. Using the cylindrical
coordinate system and the axial symmetry of the model. only one half of
the geometry of the EPC unit from the g u n nozzle to the target needs to
be rnodeled, as shown in Figure 3 . 1 .
The mode1 consists of a 2-cm
diameter dielectric g u n nozzle' located at a distance S away from the
target plane. A small corona wire of 0.5-mm diameter is placed at t h e
center of the gun nozzle. The wire is assumed to be connected to a high
voltage supply and t h e corona discharge is produced continuously along
the hemispherical tip of the corona wire. The target is considered as an
infinite circular plate connected to ground. A plate radius of 0.6-rn
(twice the gun-to-target distance) is found sufficient to represent the
infinite target plate. The powder particles are assumed to be fluidized
b y the bed hopper and uniformly distributed. Each particle has identical
' Measuremests are taken h m an electrostatic powder gun in the W O AERC Iaboratory.
size and characteristics and al1 exit the gun nozzle with the same
velocity.
Gunnovfe
Free space
I
H.V.corona wire
Figure 3.1 Geometry of the electrostatic powder coating unit.
The electrostatic fieid in the space charge area is governed by the
following subset of Maxwell equations
where E is the vector of t h e electric field intensity,
E
is t h e permittivity
and p is the space charge density.
The set of partial differential equations can be rewritten by
introducing a scalar function called the electric potential CI defined as
The electric potential U satisfies the Poisson equation shown
below
The space charge from the corona wire moves towards t h e
grounded target plane and this creates an electric current with the
magnitude related to the electric field and the space charge density as
where p is t h e charge mobility and J is the vector current density. The
movement of the space charge must satisfy the current continuity as
The trajectories of the powder particles are calculated by
considering al1 the forces acting on the particle. These forces are
expressed by Newton's second equation
where Fr is the drag force, Fe is the electrostatic force and F, is the
gravitational force. Because the particles are charged, they contribute to
the space charge density that affects the electric field in equation ( 3 . 4 ) .
Equations ( 3 . 4 ) to (3.7) govern the particle trajectories in an EPC
system. The problem can be considered as a simultaneous set of partial
differential equations. Because of t h e non-linear character of the
problem, it is very difficult to obtain an analytical solution and it must
be solved numerically. The following sections discuss several numerical
techniques for calculating the particle trajectories, electric field and the
space charge distribution of the EPC system.
3.2
Particle Trajectories
Ang and Lloyd [6] have suggested that the motion of the powder
particles in the EPC process is governed b y the electrostatic and t h e
aerodynamic forces. In normal operating condition, the particle motion
i s governed by t h e air velocity in the early stage of the trajectories and
then carried t o the grounded workpiece by the electrostatic forces. For
small particles, these forces can be modeled b y the Basset. Boussinesq
and Oseen's ( ~ ~ 0 ) " c p a t i o n as
(air drag)
(gravity)
(electrostatic)
(local pressure gradient effect)
)
(apparent mass acceieration)
--
-(af -q
dr
+ 60' JGJ
dr
r
O
-
,/=
(Basset memory term)
-
" Baw[8] proposed a modined BBO equation based upon the addition of a t e m for the Magnus effect
abich is a force perpendicular to the velocity of the particle due to spia
The BBO's equation is a complete equation describing the forces
acting u p o n a srnall spherical particle moving i n a fluid. In our case, it
is the powder particles travelling through air. The solution of the full
BBO's equation is very complicated and requires integration in t h e last
two terms. Soo (371 and Klinzing [22] stated that if the density of the
particles is much greater than the density of the fluid, then the pressure
gradient, the apparent mass acceleration and the memory terms would
contribute very little to the net force acting on the particle. Since the
density of most powder material is i n the order of 10' ( k g / m 3 ) compared
to 1.2 for air, we can simplify t h e BBO's equation to just the first three
terms: air drag, gravity and electrostatic forces.
3.3 Finite Element Method
The Finite Element Method (FEM) is a numerical technique for
solving partial differential equations such as the Laplace or Poisson
equation. The method can provide numerical results with good accuracy
assuming that the discretization is sufficiently fine.
The basic idea of
the FEM is to divide the problem domain into some number of srna11
elements and interpolate the solution over each element by a simple
function. The linear interpolation function for a triangular element is
given as
where N I , N2,and N3 are the shape functions for the nodal points 1 to 3
U , . UI and Ujare the potential values at the nodes. The space charge
density distribution can also b e approxirnated in the same way as
According to the variational principle, the problem of solving the
Poisson equation is equivalent to t h e problem of finding an extremum of
the energy functional. The element energy of the triangle can be defined
as
where the second term represent the force function (source) within the
element. The gradient of the potential given by equation (3.9) can be
calculated from the following equation
Assuming that the charge density is known for each node. the
source integral may be written as
Substituting equations ( 3 . 9 ) and ( 3 . 1 2 ) into ( 3 . 1 1). the element
energy equation becomes
Equation ( 3 . 1 4 ) can be rewritten in matrix form as
are two new characteristic matrices defining the triangular element. The
total energy associated with the whole domain is the sum of al1 the
individual element energies
By assembling al1 the individual matrices into one global matrix,
the energy function of the whole domain is defined as
Applying the energy extremum principle, W should be
differentiated with respect to al1 nodal potentials a s
for i=1,2,
...
n
This gives a set of n equations with unknown nodal values of
potential
The relation between the electric field and the potential is given i n
equation ( 3 . 3 ) . Having evaluated the potential for al1 the nodes within
the domain, it is then possible to evaluate the electric field using
equat ion (3.12).
3.4
Method of Characteristics
O n e of the most efficient techniques for space charge evaluation is
the Method of Characteristics (MOC). The method was first derived by
Waters et al. [39],who investigated the drifting o f an unipolar ion cloud
with mobility p. Perhaps the unique thing about the MOC is that it
transforms the partial differential equation governing the evolution of
charge density into a first order ordinary differential equation along a
specific space-time trajectory.
This space-time trajectory, or so-called
characteristic line, is a path on which the ions are drifting in the electric
field region. The ordinary differential equation can easily be integrated
yielding an analytical solution in terms o f t h e initial charge density and
the time needed by the ions to migrate from one point to another.
The law governing the conservation of charge is given as:
where J is the current density and defined as
Assuming that the diffusion coefficient D is negligible,
substituting J into equation ( 3 . 1 9 ) and simplifying o n e obtains
W e can define the characteristic line as
and the partial differential equation ( 3 . 2 1 ) is transformed into an
ordinary differential equation
along the characteristic line g i v e n by equation ( 3 - 2 2 ) .
Integration o f the differential equation ( 3 . 2 3 ) yields
where po is the charge density at the starting of the characteristic line
and t is the elapsed time. By making an estimate of the charge density
po, we can evaluate the space charge density in the interelectrode space
from the corona wire towards the target plane.
A s discussed in chapter 3.1, the motion of each particle can be
determined by balancing al1 the forces acting o n the particles along its
trajectories from the gun to the workpiece. The calculations usually
require large memory and are very time consuming for large number of
particles such as in an EPC systern. Methods such as the Particle-In-Ce11
(PIC) [18] are used to irnprove the computing process while maintaining
the accuracy requirements. The advantage of t h e PIC method is that i t
couples the electric field and the powder particle distribution. At every
point along the particle trajectory, the powder charge is calculated and
added to the ionic charge density produced by the corona discharge.
The electric field and the particle trajectories are calculated repeatedly
until a self-consistent particle trajectory is obtained.
In order to simplify the problem, the powder particles are assumed
to have equal size and exit the gun nozzle at a uniform speed. An
elernentary cylindrical volume dv containing many particles, is divided
into sorne number of toroidal rings (n,) o f equal thickness as s h o w n i n
Figure 3 . 2 . Al1 the particles in the toroidal volume are represented b y a
super particle and its equivalent charge can be calculated as
At any time step, t h e charge of the super particle is assigned t o a
triangle mesh according to the formula
where p represents the powder charge density, i is the toroid ring
number, A is the triangle area, and raveis an average radius of the
triangular element.
FEM
mesh
,
.
7
Y
S
m . -
.
.
.
.
*
*
6
6
i l -
Figure 3.2 Charge assignment using Particle-In-Cell method
3.6
Airflow Aoalysis
The airflow emanating from the gun nozzle was computed by
solving the Navier-Stokes equation [3 O] by assuming steady viscous
laminar flow. The Navier-Stokes equation governing the airflow in the
cylindrical CO-ordinates is given as
where v , and v, are two unknown components of the velocity vector.
The equation can be simplified b y introduction of two new scalar
variables: t h e stream function
function is defined a s
and t h e vorticity
(v
and the vorticity w . The stream
Using the above definitions, we can rewrite the Navier-Stokes
equations in term o f the vorticity and the stream function a s
The air w a s assumed t o exit the n o z z l e at a uniform velocity,
V.O.
Neglecting the corona wire, the boundary conditions can b e formulated
as shown in Figure 3 . 3 .
Figure 3 -3 Boundary conditions for airflow model.
The two equations ( 3 . 3 2 ) and ( 3 . 3 3 ) can b e solved iteratively using
the F E M by first assuming the vorticity o = O and solving equation ( 3 . 3 2 )
for y. Then vis substituted into equation ( 3 . 3 3 ) and soived for o. The
process is repeated continuously until a converging process is obtained
for y and o.
CHAPTER 4
Numerical Simulation of EPC System
4.1
Description of Corn puter Program
I n order to simulate the corona charging EPC system, a cornputer
software has been developed incorporating the algorithrn previously
described in chapter 1.2. The software consists o f a main program and
several subroutines'. which functions are to carry out a specific task.
There are three basic tasks in the algorithm:
Calculate the electric field using F E M
Caiculate the space charge using MOC
Determine the powder particles trajectories
The programs are compiled on an IBM 850 PowerPC with 100 Mhz
clock speed. The main program acts as an interface and iteratively calls
these subroutines according to the flowchart shown in Figure 4.1.
There
are two iterative processes in the program: an inner and an outer loop.
See subroutine flowcharts in the apgendiu
1
Calculate the air velocity distribution
1
Apply FEM to sobe Laplace equation,
ampute E
+
Appiy MOC to caicuiate the space c b g e disaibution
-
4
-
Determine the trajectories of powder particles
r
Update the charge densiîy d i s t r i i o n
1 -
Recompute E
solving the Poisson equation
post pmcesing of output data
A
Figure 4 . 1 Flowchan of cornputer program.
E r r l f l % , , e r r t f S % ) : assumed errors for inner and outer loops
Ec: electric field on the surface of corona wire
En: onset electric field eom Peek's formula
The program s t a r t s by calculating t h e air velocity distribution. It
then applies the F E M t o calculate the initial electric field distribution by
solving the Laplace e q u a t i o n . From the calculated field. the M O C is
applied to calculate t h e ion density and the B B O ' s equation is integrated
t o determine the powder particle trajectories. The next step is t o
interpolate the space c h a r g e density frorn both t h e free ions and the
powder charge. With an updated space charge distribution, the electric
field is recalculated a l o n g with the space charge and t h e powder
trajectories. This process constitutes the inner l o o p o f t h e program and
i s iterated until a self-consistent solution for t h e electric field and the
space charge density i s obtained.
I n the outer loop, t h e space charge density o n the corona wire
surface is iterated until t h e electric field and t h e c h a r g e density satisfy
Kaptzov conditions [4]. According to this hypothesis, t h e electric field
o n the corona wire s u r f a c e remains constant at t h e value resulting from
Peek's formula defined a s
where Rw is the radius of t h e corona wire and 6 is t h e air density which
is related t o the a b s o t u t e room temperature and pressure a s
If the surface of t h e corona wire is not smooth as in most practical
cases, it is necessary to introduce a roughness factor f, which has a value
of less than unity. Then equation (4.1) becomes
4.2
Electric Field Calculation using FEM
The geometry mode1 shown in Figure 3.1 is initially discretized
into a number of triangular elements (approximately 3200 nodes). The
Delaunay Triangulation Method [1 I l is used for the refinement process
to produce a set of triangles close to equilateral ones as possible.
Because the electric field changes very rapidly around the corona wire,
very fine mesh is allocated i n this region and the triangle sizes increase
gradually approaching the target plane. A potential Il,,, is assigned to t h e
boundary nodal points at t h e surface of t h e corona wire and zero
potential is assigned to the boundary nodal points at the target plane.
The domain is assumed to be initially free of charge (Laplacian field)
and calculated using t h e FEM. Since a11 the matrices involved in the
FEM are symmetric. sparse. and banded. they can be stored in a halfbandwidth format. This technique reduces the storage requirement for
the matrix. In order to minimize the bandwidth, the Cuthill-McKee
algorithm was applied to number the nodes. The minimum bandwidth
can b e achieved by minimizing the difference between the lowest and the
highest node number of any single element in the domain. Then the
equations ( 3 . 9 ) and ( 3 . 1 2 ) are solved t o calculate the nodal potentials
and the electric field distribution within the domain.
4.3
Space Charge Calculation using MOC
The ionic charge distribution of the corona discharge is calculated
using the MOC described in chapter 3 . 4 . The electric field along t h e tip
of the corona wire is shown in Figure 4 . 2 - Wherever the electric field
exceeds the onset field on the surface of the corona wire, the surface is
assumed to go into corona and emits ions. These ions then forrn a space
charge cloud propagating towards the target plane.
Figure 4 . 2 . Electric field at t i p of corona wire surface
Initially, twenty characteristic lines are set at the emitting surface
but this changes as the iteration process proceeds. The space charge
density at the corona wire surface where the first characteristic l i n e
begins, po,, is estimated first. Then, the charge density at al1 other points
i s calculated using the following formula
pot
= po!cosB,
The initial estimation of the charge density is very important for
the quick convergence of the process. I t requires some previous
experience and general knowledge of relationship between the current
and the surface charge density at the corona wire. While a good estimate
of the charge density, po,, closer to the physical value generally gives a
faster convergence and reduces the computing tirne; a poor estimate may
slow down the convergence and in some cases, it can lead to instability.
Using the calculated electric field distribution, the charge density is
computed along each characteristic line using the equations (3.22) and
(3.24). The characteristic line in ( 3 . 2 2 ) can be approximated by a finite
difference equation as
In t h e above equation, we can select a fixed time or a fixed
distance for the trajectory step. Because the distance from the corona
wire to the target plane is given, it was found that it is easier to set the
step along the z-axis, A t . Then, the time increment can be calculated as
NOTE TO USERS
Page(s) not included in the original manuscript and are
unavailable from the author or university. The manuscript
was microfilmed as received.
This reproduction is the best copy available.
where a, 1 and y are three unknown coefficients which can be
determined from the known charge density at the nodes of the
constructed triangle element. At the vertices of each triangle. the
following equations can b e written:
Treating these equations as a set of 3 equations with 3 unknowns,
we can calculate a, and y as functions of p,, p,. pi and substitute these
values back into equation (4.8) t o calculate the charge density p(r.z)
Any node outside the last characteristic line is considered free of ions
drifting toward the area and is assigned a zero charge density. This
process is carried out for al1 t h e nodal points of the FEM mesh to
determine t h e charge density distribution produced by the corona
discharge.
...._.--
-.._.
-.._.
.a..
point on the
,
-
.-..
a *.:
Figure 4 . 4 Interpolation of charge density at FEM mesh nodes
4.4
Calculating the Powder Particle Trajectories
The powder particle trajectories are calculated using the %BO'S
equation in ( 3 . 8 ) . Because of the large number of particles with many
different sizes and shapes, it is very difficult, if not impossible, to
sirnulate the motion of every particle in an EPC process. Therefore, the
following simplifying assumptions are made in the calculation:
the particles are spherical and uniformly distributed at the gun nozzle
al1 particles have identical size and characteristics
there are no particle-particle or particle-fluid interactions
the particle exit velocity is given
the particles are charge-free when they exit the gun (no tribo-charge)
the particles acquire an unipolar charge instantly at a charging zone
near the corona wire
the particles experience constant acceleration and velocity over each
time step
From the BBO's equation. the particle acceleration at any given
point can be calculated as
where CD is the drag coefficient and v, and v , are the air velocities. The
air velocities are interpolated from the airflow calcuIation described in
chapter 3.5 (this airflow distribution is assumed to b e constant
throughout the calculation process). Another assumption made is that
the powder particle exits the g u n nozzle at a known velocity and
contains no net charge. The particle continues t o move until it cornes
into contact with the ions produced by the corona discharge. I t then
acquires al1 of the charge instantly and is transported to the target.
Using the calculated charge density from the MOC, the diameter
of the ionic charge cloud is found equal to the diameter of the gun
nozzle at 5-mm away from the corona wire. The particles are assumed to
have made contacts with the ions at that point and acquire an unipolar
charge instantly. The electric field at the charging point is found to be
approximately 1 x 1 o6 Vlm. This value is substituted into the Pauthenier
formula in (2.3) to determine the magnitude of the particle charge to
mass ratio.
The particle velocity can be found by integrating the accelerarion
equation using Euler's method as
and similarly for the particles trajectories
= pzi
+ U p z , At
PC+i = Pr, + U r ,
4.5
Convergence of the Iterative Process
As mentioned earlier, the algorithm consists of two iterative loops.
The inner loop requires the convergence of the electric field and the
charge density distribution, and the outer loop requires the convergence
of the electric field and the space charge to satisfy Kaptzov condition at
the corona wire surface. After the charge density has been interpolated
and updated, the electric field is recalculated using the F E M . For every
nodal point within t h e domain, the electric field is compared with the
previously calculated field. The process is repeated iteratively until a
self-consistent solution for the electric field and the space charge is
obtained. The criterion used for the convergence is
for i=l,2,3 .. . . n
where E,'*' is the electric field at node itb in the krh iteration step. In
general, the process converges very fast and requires on average 3-4
iterations for accuracy better than 1%.
The handling of the outer loop is a little more difficult. It
requires the adjusting of the space charge density on the corona wire
surface to satisfy the Kaptzov condition. The boundary condition
requirement is that the electric field on the emitting surface of the
corona wire b e equal to the Peek's onset field. En. With the electric
field on the corona wire surface shown in Figure 4.2. the field can be
suppressed to Peek's onset field by altering the charge density at the
starting points of the characteristic lines. For any lines whose electric
field at the starting point is greater than En. the charge density will be
modified according to t h e formula
where p,, is the charge density at the starting points of line i in the k C h
iteration. K is the charge density coefficient and Ec, is the electric field
at the starting point of the line. For any lines whose electric field is
smaller than Eo, the charge density is assigned zero and considered not
to be emitting any space charge. The iterations are stopped when the
electric field of al1 the points at the corona wire surface are within 5%
of Peek's onset field.
CHAPTER 5
Simulation Results and Discussions
5.1 Simulation of EPC System
Using the geometry model shown i n Figure 3.1 and the
assumptions made in the previous chapter, t h e developed numericai
algorithm is used to calculate the trajectories of the powder particles for
a typical operating condition of an EPC process. The dimensions of the
model and other system variables are given in Table 5 . 1 . The particle
trajectories are calculated and compared for different operating
parameters such as:
applied voltages
charge to mass ratios
particle sizes
mass transfer rates
g u n position relative t o target plane.
-
--
-
Table 5.1
Parameters of the EPC System
--
length of corona wire, L w
radius of corona wire, R ,
length o f gun nozzle, L ,
radius of gun nozzle. R,
radius of target plate. Rt
distance from gun to target. S
corona wire voltage, CI,
i o n mobility, p
mass transfer rate, rn , t
mean radius o f powder particle, r,
powder density, p,
charge to mass ratio. Q A4
particle eject velocity. v , ~
gravitational constant, g
5.2
Electric Field Distribution
As discussed earlier. the trajectories of the powder particles are
governed by the aerodynamic and t h e electrostatic forces. Therefore, it
is very important t o know the distribution o f the electric field within t h e
system. The electric field is presented in Figures 5 . 1 to 5 . 4 . Figure 5 . 1
shows the electric field distribution within the problem domain. The
field consists o f both the applied (Laplacian) field and the space charge
field. It is observed that t h e electric field is very high near t h e corona
wire and decreases rapidly when moving away from the wire.
The intensity o f the applied electric field is directly related to the
applied voltage between t h e corona wire and the grounded target. If the
applied voltage increases, it increases t h e applied field which in turn
increases the space charge field. After a rapid decrease near the corona
wire, the electric field begins t o level off and decreases slowly for much
of the remaining area. The electric field strength in t h e particle
trajectory region is in the order of 10' Vlm which coincides with t h e
values used by H u g h e s [ 2 0 ] , Cross [14] and Bright et. al. [IO].
Figure 5 . 2 shows the component o f the electric field in t h e axial
direction E,. while Figure 5.3 shows the component o f the electric field
in t h e radial direction Er. Figure 5.4 s h o w s the contour plot o f al1 field
components in t h e high field region near the corona wire.
Figure 5.1 Electric field distribution in EPC system
Figure 5.2 Axial component of electnc field in EPC system.
3
3.5
4
4.5
5
5.5
6
Figure 5.3 Radial component of electnc field in EPC systern
6.5
Figure 5.4 Contour plot of the electric field near corona wire.
5.3
Charge Density Distribution
The space charge density distribution is shown in Figures 5 . 5 and
5 . 6 . Figure 5.5 shows a contour plot of the ions density produced by the
corona discharge. The free ions spread out to form a cone-shaped cloud
drifting towards the grounded target. T h e magnitude of the space charge
density is found to be near the estimated charge density of 2 x 1 0 ' ~c h 3 .
This indicates that the initial estimate of the charge density at the corona
wire is good and the program converges rather quickly.
The magnitude of the space charge density depends on the applied
voltage at the corona wire. An increased applied voltage produces a
higher ion density as required to suppress the electric field to Peek's
onset field at t h e corona wire surface. Figure 5 . 6 shows a threedimensional view of the space charge density distribution. The charge
density consists of both the free ionic charge and the powder charge.
The powder charge density is found to b e small as compared to the ionic
charge density.
Moyle & Hughes [29] estimated that i n normal corona
charging operations, the space charge is made u p of 0.5% powder charge
and 99.5% of free ions.
Figure 5.5 Ions density distribution produced by corona discharge.
Figure 5.6 Space charge density distribution of EPC system
5.4
Particle Trajectories
Figure 5 . 7 shows the trajectories of the powder particles from the
gun to the target plane separated by 0 . 3 m (other parameters are: LI,=-
100kV. Q M=-1 . 3 m U k g (Pauthenier limit), r P = 3 0 p m , m t= lgls). As
discussed in chapter 3.5, each trajectory represents a number of charged
powder particles. As the particle leaves the gun, it becomes charged and
experiences a strong electrical force acting along t h e trajectory. The
axial component of the electric field is responsibIe for driving the
particles to the target while the radial component of the electric field
diverges the particles in the radial direction. A higher field intensity
causes the powder particles to spread out further in the radial direction
and increases the coating area on the target plane. The results of the
simulation show the radius of the coating area approaches the coating
distance from the gun to the target plane. The particles form a layer
covering an area of approximately 0.2m radius when the gun-to-target
separation is 0.3m. This is relative to Ali [ 5 ] measurements of 0 . h
radius for 0.25111
gun-to-target separation.
Figure 5 . 8 shows the velocity vectors of the particles along the
trajectories. It is observed that the particle travels at a similar velocity
for most of the distance from the gun t o the target. The particle is
ejected from the gun at a constant velocity until it is charged by the
negative ions produced by the corona discharge. The particle is then
accelerated and the velocity increases rapidly because of the high field
intensity near t h e corona wire. As the particle moves away from the
corona wire toward the target, the field begins to decay and the velocity
decreases until it reaches the terminal velocity. For this particular
operating condition. it is noted thst the terminal velocity is higher than
the ejecting velocity of 1.0 m/s at the gun nozzle. This can be explained
by the high electrical force exerted on the particle because of the high
charge to mass ratio (Pauthenier limit). If the particle charge to mass
ratio decreases, then the electrical force acting on the particle also
decreases and lowers the terminal velocity. This is shown in Figure 5 . 9
when the particle charge to mass ratio is reduced to -0.2 m C / k g (Singh
minimum Q ,M for particle adhesion to grounded workpiece).
Figure 5.10 shows the electric field vectors along the particle
trajectories. The field vectors are pointing towards the gun nozzle
because of the negative polarity of the applied voltage at the corona
wire. Figure 5 . 1 1 shows the electric field (E, E,, and E r ) of the farthest
particle trajectory from the axis. The field inteosity E, decreases
when t h e particle moves towards the grounded target. The radial
component of t h e electrical field, Er also shows a similar relationship
and decreases when moving towards the target. Meanwhile, the axial
component of the electric field, E, decreases along t h e trajectory and
then increases near the target. This electric field behavior is typical for
al1 wire-plane corona systems.
Figure 5.7 Powder trajectories in EPC sy stem.
Figure 5.8 Particle velocity vectors dong trajectories in EPC syaem.
(Pauthenier Q/M =- 1- 3mC/kg)
Figure 5.9 Velocity vectors with smaller charge to m a s ratio.
(Minimum QM=-O.lrnC!kg)
Figure 5.10 Electric field vecton dong pamcle uajectories in EPC ?stem.
Figure S. 1 1 Electric field dong the tan trajectories from g m to taqet.
Figure 5.12 Axial component of air velocity along axial a i s .
Figure 5 . 1 3 shows the particle trajectories as function of the
applied voltage. Only the furthest trajectories for each simulation are
shown in order to compare the powder cone shape from the gun to the
target plane. The results show a direct relationship between the cone
shape and the applied voltage. An increase of 10 kV enlarges the
coating radius of the target area by approximately 2 cm. This
relationship can be explained by analyzing the electric field and the
charge density distribution. When the applied voltage increases. it
elevates the electric field intensity. which in turn increases the powder
charging and the space charge density produced by the corona discharge.
Higher charge density produces a higher space charge field and causes
stronger dispersion of the powder. A stronger dispersion indicates that
the particles are more spread out in the radial direction and thus build up
a wider layer covering the target plane. I n some EPC processes. the
increasing of applied voltage may be the easiest way to increase the
charging but this also produces more free ions that can cause earlier
onset back ionization and leads to poor quality finishing.
Figure 5 . 1 4 shows the particle trajectories as function of the
charge to mass ratio. The trajectories are simulated for particle charge
to mass ratios €rom -1.3 rnClkg (Pauthenier limit) t o -0.2 mC/kg (Singh
minimum QIM requirement for particle adhesion to any grounded
workpiece).
The results show the particles with higher charge to mass
ratio experience a higher electrical force and are spread out further along
the target plane.
Figure 5 . 1 5 shows the particle trajectories as function of the
particle size. The particle trajectories are calculated for panicle radii
from 10 to 50 prn. Because of low inertia, the small particles are more
affected by the aerodynamic forces in the early stage of the trajectories.
The results show that the 10 pm particles are more dispersed at the gun
nozzle, but then form a relatively focused beam and give a smaller cone
radius than larger particles.
Figure 5 . 1 6 shows the particle trajectories as function of the mass
transfer rate. There is a little difference in the particle trajectories when
the mass transfer rate is increased frorn l g / s to 3 g/s. The computation
process tries to simulate more particles travelling from the gun towards
the target. This creates a shift i n the charge density distribution within
the system. More ions are transferred to the powder particles as it
increases the powder charge density while lowers the ioaic charge.
There is little change in the electric field distribution and this is
portrayed i n t h e particle trajectories.
In Figure 5 . 1 7 , the target has been moved closer towards the gun
nozzle (from O.3m to 0.2). The result shows a similar trajectory pattern
as in Figure 5.7. The powder particles build a denser layer covering a
smaller area on the target plane.
Figure 5.13 Particle trajectories as function of applied voltage.
(@W1.3mC/kg, rp=3O ~ ~ r n / 1ig/s,S=O.
=
3 m)
Figure 5.14 Puticle trajectories as function of charge to mass ratio.
( U p1OOkV, rP=30pm,m/t=l g/s,S=O.3 m)
Figure 5.15 Particle trajectories as function of particle radius.
(&= iOOW,Q;M=-O.SmClkg,m~lg/s,S=û.3rn)
Figure 5.16 Particle trajectories as function of mass transfer rate.
(UW=1OOkV,QfM=l .3mC/kg,rp=30pm,S=0.3m)
Figure 5.17 Particle trajectories for smaller gun-to-target separation.
(UW=100kV,Q,W- 1. 3 m C l k g , r p = 3 0 p ~ m
lg/s,S=O.Zm)
'~
CHAPTER 6
Conclusions and Recommendations
6.1
Conclusions
The numerical algorithm for simulating the particle trajectories in
the corona-charging powder coating system has been presented in this
thesis. The Finite Element Method for calculating the electric field
strength and the Method of Characteristics for determining the ionic
charge density were used in conjunction with t h e Particle-In-Ce11
Method to simulate the powder particle trajectories from the g u n to the
target plane. The problem was computed iteratively until a selfconsistent solution for the electric field, particle trajectories and the
space charge density distribution was obtained.
The simulation results showed a highly non-uniform electric field
distribution near the corona wire. The field decreased rapidly near the
wire and then leveled off for much of the particle trajectory area. The
space charge density produced by the corona discharge forrned a conelike shape in the interelectrode space between the corona wire and the
target plane. The magnitude of the charge density was related to the
applied voltage between the corona wire and t h e grounded target. A
higher voltage produced more ionic charge which in turn, increased the
space charge field and dispersed the powder particles Further in the
radial direction. The particles form a layer covering a target area with
radius close to the gun-to-target separation.
The algorithm was used to study the powder particle trajectories as
functions of several application parameters o f the coating process. The
following relationships were obtained:
Increased applied voltage:
It has a strong effect on the particle
trajectories as it produces a higher electric field and charge density.
The particles experience a higher electrostatic force, which causes
more dispersion and increases the coating area.
Increased charge to mass ratio:
The particles experience a higher
electrostatic force because of the higher particle charge. It increases
the particle terminal velocity and spreads the particles further i n the
radial direction.
Decreased the particle size:
Small particles (10pm)are dispersed
more when emanating from the gun and are influenced more by t h e
mechanical force.
fncreased mass transfer rate:
It increases the powder charge density
but decreases the ionic charge density. There is little change in the
electric field and the particle trajectories.
Decreased gun to target distance:
Powder particles deposit across a
smaller target area and produce a thicker layer.
6.2 Recommendations
Because of the complexity of the EPC process involving many
parameters that govern t h e electrostatic and the mechanical forces acting
on the powder particles, an exact model of the practical coating process
was difficuit to achieve. The numerical algorithm presented in this
thesis provided a further understanding of the electric field, space
charge density and the powder trajectories in the EPC system. However.
there are still many modifications that can be implemented to improve
the accuracy of the simulation process. Here are some recornrnendations
that can be considered for further development of the numerical model.
Include the non-uniform particle size and shape distribution. T h e
current model assumes that the particles are sphericai with
identical sites.
Include the influences by the environment conditions such as
.
temperature, humidity, surface roughness of corona wire etc . . . as
these affect the charge density produced by the corona discharge.
Model the particles with different charge and include the charge
decay. Hughes [ 2 0 ] suggests that about 60% of the powder
particles are charged in a normal EPC process.
Carry out experimental measurements of the particle trajectories
t o compare with the simulation results.
Include the booth (recovering) airflow and the ion wind i n the
model.
Model the target with corners and cavities or use different shapes
such as sphere or cylinder. The different target shapes would
create different electric field and airflow distribution.
One of the most difficult tasks in the simulation is t o calculate
the aerodynamic forces acting on the particles.
The current
simulation is carried out based on the assumptions that there is no
interaction between the particles and the carrying airflow. The
air velocity is calculated by solving the Navier-Stokes equation
assuming viscous laminar flow. For most commercial coating
processes, the Reynolds number is much higher and the flow
develops into a turbulent flow. The full solution for such cases is
very difficult because o f t h e turbulent characteristic of the flow
and the distortion from the particle interactions. This challenge
is left for future development.
[ 11 Abramovich, G. N., Tne 7heory of T11rbufentJets, MIT Press, Cambridge, Mass..
1963.
[ 21 Abuaf, N. and Gutfinger C ., Trajectones of Charged Solid Parttcles in an Air Jet
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University of Western Ontario, 1994.
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M.L.and Lloyd, P . J., Imestigutim of Charged Particle Trajectories in
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of Electrostatic Air
Spraying rmd S m e Numerical Experimenz Results, J o d of Electrostatics, vol. 23
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Applications, John Wiley & Sons hc.,Toronto, 1997.
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, E., G d o l i d Transport, McGraw-Hill Book Co.. Inc.. New York,
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Recipes in C,Cambridge University Press, 1990.
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d 7heir Selectim. Plating and Surface
Finishing, June 1989, pp. 34-38.
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(1970).
R T.,Rikard, T.E.
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[ 431 The Student Edition of Matlab. The MathWorks Inc., 1992.
Figure A 1 An eiectrostatic powder coating unit manufactured b y Wagner.
Figure ~ 2 'Powder flow of a Bat spray gun nozzle.
p
p
* These pictures are obtained from internet wirh premission from Wagner
(www.wagner-systems.com).
90
get local &ces
each eiemem
for
calculate the elecbic fidd
postg-P
of
elecîric field &ta
Figure A3 Flowchart of Finite Elemem subroutine.
1 calculate the emiîting angle at comna wire 1
caldate the initial charge density. PO,
I
&
malute the electric field E
calculate the auial trajeztory disbnce. ct!
I
(
1
determine the charge density. p
1
Uneqmiate the charge deasi~y
pst processingof charge density data
I
-
Figure A4 Flowchart of a subroutine evduating space charge density using M W .
1 caicuiate the equivalern charge
a d air velocities
1
calculate the paruele acceleratim
I
a
calculate the particle velocities
1 caldate the ne= particle pWtim I
Figure A5 Flowchart of a subroutine evaluating the particle trajectones.

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