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 of 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 corona-

charging 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 to simulate the powder particle trajectories

for variations of particle size, corona voltage, charge to 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 ake this opportunity t o express my sincere

grat i tude to my chief advisor Professor Kazirnierz Adamiak for h i s

invaluable guidance and suppor t throughout the project .

1 would also like t o thank Professor G 5 . P Cast le , Professor Terry

E. Base, and Dr. Sergey Primak for their advise and helpful discussions

on the project.

Finally, I want t o thank my parents, my family and al1 my friends

for their support and everlast ing encouragement through al1 these years.

TABLE OF CONTENTS

CHAPTE R 1 Introduction ------------------------*---------------------------------------------....- 1

1 - 1 Background .------. * - - * * .*.- - -..*----------.----.--------------- * - * - - - . * - - - - - 1

1 . 2 Objectives ----------------.-------.-------.------+-+--+.-------------.--------------.---------------..---- 3

CHAPTER 3 Mathematical Mode1 of Powder Coating

4 . 3 . 1 interpolation o f Space Charge Density ... 44

4.4 Calculating the Powder Particle Trajectories --------------------.. ... 46

4.5 Convergence of the Iterative Process ----...............----.---------------- 49

CHAPTER 6 Conclusions and Recommenda tions . - - - - - 78

6 . 1 Conclusions ---------,,.--....-.--------...-..-----.-----.---------.--------------------*-----.------.--- 78

6 . 2 Recommendations --------...--....,.-------.--.----.------------------.------*------------------- 80

REFERENCES .---.----.-----------------------+-+..-.------------.--------------*-------------------------.--.-------.-- 83

APPENDIX ---------.--------------.-..----.----------------------------------.-------------------------------.--------------- 8 9

CURRICULUM VITAE - - - - - - . - - - - - - - - - - - - - - - - - - - - . - - - - * - - - - * - * - * * - - - - - - - - - - - . - - - - - - - - - - - - - - . - - ~ * - - - - - - . - - . - - 94

LIST OF FIGURES

............................ Figure 1 . 1 Schematic diagram of an electrostatic powder coating unit 2

............................................................. Figure 2.1 Particle charging by corona discharge 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 ~ g unit 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 of elehc 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 -57

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 5.1 1 Electric field intensity dong the furthest trajectory .................................... 68

Figure 5.1 2 Axial component of air velocity dong axial axis ......................................... 69

Figure 5.12 Particle tnjectories as finction of applied voltage ................................... -73

Figure 5.13 Particle trajectories as funaion of charge to mass ratio .............................. 74

....................................... Figure 5.14 Particle trajectories as fiinction of particle radius 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 ....... 92

Figure A5 nowchart of a subrouthe evaluating the particle trajectories ...................... 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 D e s c r i ~ t i o n Unit

ion mobility m Z / ~ 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 of 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 by 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 of 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 to

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 to ground. The powder is usually an epoxy resin

ground to a very small mean size of 10-100 pm 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

Free ions L-

Fluidized bed hopper

Corona wire - Q -0 \ 0-- - Q - O -

Elecmc

- High voltage generat or

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 to form a continuous coating.

1.2 Objectives

The objective of this project is to develop a full numerical

algorithm of a corona-charging EPC process. I t 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 to 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 of 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. In 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 to study the charging mechanism of the two systems. A

cornparision of the tribo versus corona charging systems was reviewed

by Kleber [21] , Moyle and Hughes [29], and Reddy [32] . 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 known 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. In 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 to 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 due 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 or ot her mechanisms are attracted to the

corona wire while electrons are repelled away by the h igh 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 to gain

sufficient energy from the electric field between collisions to 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 Limit

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 f ield

a s shown in Figure 2 . 1 . The charging then ceases and the particle is

said to 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 the 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 to 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 .

Table 2.1 Advaotages and disadvantages of different types of powder in E P C .

Disadvan tages Material

Acrylic

polyethers 1 mechanical and electrical 1 h igh cost, fair adhesion

Advantages

Cellulose acetate

Chlorinated

1 resistance, good I

High durability, good hardness, excellent

Poor f low, moderate adhesion, brittle

chemical resistance Excellent flow, good gloss and colour, excellent chernical resistance Excellent corrosion,

1 alkali, abrasion and / in exterior applications.

Moderate adhesion, soft film, discolours at high temperature Poor exterior durability,

Epoxies dimensional stability Excellent adhesion, high

Fluorocarbons

Chalking and yellowing

and service temperature, low coefficient of friction, Excellent chemical and

high dielectric strength, ( difficult, poor adhesion, chernical and abrasion 1 high cost, low service

corrosion resistance High exterior durability

high curing temperature, high cost

Nylon

high cost Poor adhesion, requires

electrical resistance 1

Polyester

and electrical insulation, durability, poor exterior excellent flexibility, low / durability, and abrasion

L o w coefficient of friction,

t hermoset avaiiable, intermediate cost

Polyethylene , Good chernical resistance

1 water absorption, resists 1 resistance, low service

Films beiow 100 pm

resistance T hermoplastic and

Poor adhesion and

temperature High curing temperature

Polyurethane

long-term durability, good Abrasion resistance,

Vinyls

1 intermediate cost

low temperatures, low cost High chernical resistance, Excellent hardness. good gloss and flow

temperature Poor adhesion, h igh water resistivity,chalks and yellows in exterior

High chernical resistance, high dielectric strength,

use Poor adhesion, only thick fi lms 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 bu t also their adherence

to the workpiece. If the charging is too low, the powder may not

adhere to t h e workpiece; if too high, i t can give rise to 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. i t

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 a l . [ 3 5 ] reported that a minimum charge t o mass

ra t io o f 2x10-' C/kg is required before the powder par t ic les will adhere

to any grounded workpiece . A charge to mass ra t io o f 5 x 1 O-' C/kg is

usually necessary fo r adequate adhesion in the coat ing process .

2.4.4 Powder Adhesion

When t h e cha rged powder particles a r e carried near the grounded

workpiece, they a r e a t t rac ted to the workpiece by the Coulomb force.

The powder par t ic les build u p a layer covering the workpiece. The

workpiece is then t ranspor ted t o an oven to fuse the par t ic les and form

a cont inuous coa t ing . Meanwhile. t h e powder must re ta in i ts charge so

it doesn' t fa11 off t h e workpiece. The time constant fo r the discharge of

the powder is given as

where 5 is the powder resist ivity, EQ is the permittivity o f f ree space and

E~ i s t he relative permitt ivity of the powder material. For insulating

powder with resis t ivi ty exceeding 10" am, it can retain i t s charge for

several minutes while conduct ive particles or materials with low

resistivity tend to lose their charge rapidly and fa11 off the workpiece.

Therefore, the powder resistivity must be 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 to 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

C harged particles Uncharged

Figure 2.2 Schematic representation of back-ionization in a corona-charged powder layer

2.5 Low Voltage Corona Gun

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 of 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 gun nozzle. Al1 the powder emanating from the gun will pass

through the corona region and will be 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 to 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 Schematic of a low voltage corona gun

2.6 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 to the target. The particle and the

space charge density are dependent on 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, Ang 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 [ I l . 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 to 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. [ 38 ]

developed a mathematical model to 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 he 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 to 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 and 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 pm

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 in 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

by 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 the 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 by the electrostatic and the

aerodynamic forces. In normal operating condition, the particle motion

i s governed by the air velocity in the early s tage of the trajectories and

then carried to the grounded workpiece by the electrostatic forces. For

small particles, these forces can be modeled by 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)

--

r -(af -q + 60' JGJ dr dr (Basset memory term)

O ,/=

- -

" 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 upon a srnall spherical particle moving in 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 the 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 in the order of 10' (kg /m3) compared

to 1.2 for air, we can simplify the 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 Uj are 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 the 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 as

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

One 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 of 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 of 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 one 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 l ine g iven by equation ( 3 -22 ) .

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,) of equal thickness as shown in

Figure 3 .2 . Al1 the particles in the toroidal volume are represented by a

super particle and its equivalent charge can be calculated as

At any time step, the charge of the super particle is assigned to 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 rave is an average radius of the

triangular element.

FEM mesh

,. 6 7 . m . - 6

Y . .

S . * * 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 by introduction of two new scalar

variables: the stream function (v and the vorticity w . The stream

function is defined as

and t h e vorticity

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 was assumed t o exit the nozz le at a uniform veloci ty , V.O.

Neglect ing 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 FEM 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 of 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 FEM

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

- -

Determine the trajectories of powder particles

r

Update the charge densiîy d i s t r i i on

+

Recompute E solving the Poisson equation -1

Appiy MOC to caicuiate the space c b g e disaibution

post pmcesing of output data A

4

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 tar t s by calculating the air velocity distribution. It

then applies the FEM t o calculate the initial e lectr ic field distribution by

solving the Laplace equat ion . From the calculated field. the MOC is

applied to calculate t h e ion density and the B B O ' s equation is integrated

to determine the powder particle trajectories. The next step is to

interpolate the space cha rge density frorn both the free ions and the

powder charge. With an updated space charge distribution, the electric

field is recalculated a long with the space charge and the powder

trajectories. This process constitutes the inner loop o f the program and

i s iterated until a self-consistent solution for t h e electr ic 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 he electric field and the charge density satisfy

Kaptzov conditions [4]. According to this hypothesis, t he electric field

on the corona wire sur face 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 the air density which

i s related to the absotu te room temperature and pressure as

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 in 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 the 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 f ie ld)

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 half-

bandwidth 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 be 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 .12) are solved to 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 ine

begins, po,, is estimated f irst . 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 .22) 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 be 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) to 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 the 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 gun nozzle at a known velocity and

contains no net charge. The particle continues to 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 FEM. 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 i tb 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. I t

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 be 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 kCh

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 to 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,

ion mobil ity, 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 the electrostatic forces. Therefore, it

is very important to know the distribution of 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 of both the applied (Laplacian) field and the space charge

field. It is observed that the electric field is very high near t h e corona

wire and decreases rapidly when moving away from the wire.

The intensity of the applied electric field is directly related to the

applied voltage between the 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 to 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 the

values used by Hughes [ 2 0 ] , Cross [14] and Bright et. al. [IO].

Figure 5 .2 shows the component o f the electric field in the axial

direction E,. while Figure 5.3 shows the component of the electric field

in the radial direction Er. Figure 5.4 shows the contour plot of 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 6.5

Figure 5.3 Radial component of electnc field in EPC systern

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. The 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 three-

dimensional 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), rP=30pm, 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 to 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 the 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 mC/kg (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 the 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 -3 mC/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 is .

Figure 5 .13 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. In 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 .14 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) to -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 5.16 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 lg / 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 in t h e particle trajectories.

In Figure 5.17, 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. (@W- 1 .3 mC/kg, rp=3 O ~ ~ r n / i = 1 g/s,S=O. 3 m)

Figure 5.14 Puticle trajectories as function of charge to mass ratio. ( U p 1 OOkV, 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= 1 OOkV,Qf M=l .3mC/kg,rp=30pm,S=0.3 m)

Figure 5.1 7 Particle traject ories 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 the 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 self-

consistent 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 cone-

like 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 the

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 the 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. The

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 in 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 f low. For most commercial coating

processes, the Reynolds number is much higher and the flow

develops into a turbulent f low. The full solution for such cases is

very difficult because o f the 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 T11rbufent Jets, MIT Press, Cambridge, Mass..

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Figure A 1 An eiectros tatic powder coating unit manufactured b y Wagner.

Figure ~ 2 ' Powder flow of a Bat spray gun nozzle.

pp

* These pictures are obtained from internet wirh premission from Wagner (www.wagner-systems.com).

90

get local &ces for each eiemem

calculate the elecbic fidd

post P-g 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

( determine the charge density. p 1

1 Uneqmiate the charge deasi~y I

pst processing of charge density data -

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 calculate the particle velocities a

1 caldate the ne= particle pWtim I

Figure A5 Flowchart of a subroutine evaluating the particle trajectones.