ARTICLE IN PRESS

0304-3886/$ - se

doi:10.1016/j.el

�CorrespondE-mail addr

Cairncross@dr

Journal of Electrostatics 64 (2006) 234–246

www.elsevier.com/locate/elstat

A discrete droplet transport model for predicting spray coatingpatterns of an electrostatic rotary atomizer

S.A. Colberta,b, R.A. Cairncrossb,�

aMaterials & Process Engineering Department, Thomson, United StatesbChemical and Biological Engineering Department, Drexel University, Philadelphia, PA 19104, United States

Received 12 August 2004; received in revised form 26 May 2005; accepted 10 June 2005

Available online 18 July 2005

Abstract

Electrostatic spray (E-spray) coating is widely used for coating conductive substrates. The combination of a high-velocity shaping

air, an imposed electric field and charged droplets, leads to higher transfer efficiency than conventional spray coating. In this paper,

a mathematical model of droplet transport in E-spray is presented which enables simulating the coating deposition rate profile.

A dilute spray assumption (no particle–particle interactions) allows modeling single-droplet trajectories resulting from a balance of

electrostatic force, drag and inertia. Atomization of liquid droplets is not modeled explicitly—rather an empirical correlation is used

for the mean droplet size while individual droplet sizes and starting locations are determined using random distributions. Strong

coupling requires the electrostatic field and droplet trajectories be determined iteratively by successive substitution with relaxation.

The influences of bell-cup voltage and atomization constant on the coating deposition rate profile, mass transfer efficiency and

droplet trajectories are also shown. Using individually predicted droplet trajectories and impact locations, a static coating deposition

rate profiles is determined. For the parametric values considered in this paper, the predicted spray is a cone hollow with no

deposition in the center, a heavy ring near the center, and a tapering of thickness toward the outer edge.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Electrostatic spray; Computer simulation; Axisymmetric jet; Rotary atomizer; Particle tracking

1. Introduction

Electrostatic spraying (E-spraying) refers to the use ofan electric field to assist in the spraying of liquiddroplets onto a grounded substrate. The droplets in thespray are charged as they exit the spray nozzle and areattracted to the grounded substrate. By providing thiselectric potential difference, the driving force of dropletsis accentuated, thereby increasing the transfer efficiencyof the spray, the fraction of sprayed liquid that impactsthe desired substrate. It has been shown that optimizedE-spray setup can have transfer efficiencies (Z) of75–85%, whereas conventional spray applications can

e front matter r 2005 Elsevier B.V. All rights reserved.

stat.2005.06.003

ing author.

esses: sac36@drexel.edu (S.A. Colbert),

exel.edu (R.A. Cairncross).

have efficiencies as low as 20–30% [1]. Most of thisincreased efficiency is the result of the finer spraydroplets being electrostatically attracted to the target.Otherwise, smaller droplets would not have the mo-mentum to reach the target. In many instances, a streamof focused air, called ‘‘shaping air’’, is used to augmenttransfer efficiency.

In one of its earliest industrial uses, E-spraying wasused to apply paint to metal parts in the automotiveindustry, where it is still widely used today [2]. Becauseof efficiency of material usage and completeness ofcoverage, this technique has been applied to many otherareas. One such area is crop dusting. The ability ofcharged droplets to ‘‘turn a corner’’ and coat theunderside of leaves makes electrostatic application ofpesticides highly effective at reducing pest populations[3]. Another innovation in the use of electrostatics is in

ARTICLE IN PRESSS.A. Colbert, R.A. Cairncross / Journal of Electrostatics 64 (2006) 234–246 235

the pharmaceutical industry with charged inhalers [4].However, the bulk of the E-spray industry is still theapplication of coatings. The model developed in thispaper has been applied to a non-aqueous non-conduc-tive paint, in the form of a xylene/polystyrene solution,applied to a conductive substrate using a rotary bellE-spray gun.

The primary goal of the model presented in this paperis to establish a mathematical model of an E-sprayprocess capable of predicting the coating thickness anduniformity by accurately describing the spray distribu-tion. Such knowledge would enable users of E-sprayequipment to attain high levels of cost savings in theform of reduced material usage and lower lead times toproduction. This goal is approached by numericallysolving the equations that describe the flow of theentraining air stream, the electrostatic field, and theresultant droplet trajectories. The numerical techniqueused is a combination of three models—an axisymmetricfinite element method solution of a k � � turbulencemodel for the continuum velocity field, an axisymmetricfinite element method solution of the Poisson equationfor the electrostatic field, and Newton’s equation ofmotion for the droplet tracking of the sprayed dropletsin three-dimensional (3D) cylindrical coordinates. Theelectrostatics and droplet trajectories are coupled by aprojection mapping of the 3D solutions of the droplettrajectories to determine a time-averaged axisymmetricspace charge. The simulation represents a slice in time ofa pseudo-steady-state spraying operation in which theelectrostatic field is treated as steady during predictionsof motions of individual particles. The material proper-ties and operating conditions of the E-spray gun are theinputs to the model. The model, by predicting the spatialdistribution of the droplets in the spray and the rate ofdroplet deposition on the substrate, is able to also gaugethe effect of operating parameters on localized filmthickness, transfer efficiency, and coating uniformity.

2. Literature review of E-spray modeling

Various aspects of the E-spray coating process havebeen the subjects of recent research. Hakberg et al. [5]and Filippov [6] developed models of electrostaticallycharged particles in flight through a quiescent domain(i.e. no shaping air involved) [5,6]. Elmoursi [7,8]developed techniques for modeling the Laplacian fieldand electrical characterization of the bell-cup geometry;however, his models applied to transport of ions, notdroplets (i.e. drag forces, etc. are ignored). A particletransport model presented by Meesters et al. [2] did notaccount for the effects of distributions of particle size orcharge. In addition, Meesters model did not involve arotary atomizer.

Ellwood and Braslaw [9] assembled a comprehensivemodel using an iterative particle source in cell approach[9], which forms the basis for the model in this paper.The Ellwood model incorporates a torsional axisym-metric flow field in which all three velocity componentsare independent of the azimuthal position. Ellwood alsomakes the assumption that the droplet charge-to-massratio is constant, and uses an empirical correlationdeveloped by Bell and Hochberg [10] to determine thisvalue. However, not only did Bell and Hochberg [10]used a different bell-cup to develop their relationships,but they also reported variations in the charge-to-massratio within the spray cloud indicating a dependence ondroplet size. In this paper, the droplet charge isdistributed based on droplet surface area rather thanon droplet mass.

In general, most of the prior models have focused onthe electrostatic aspects of E-spray, with little or noattention paid to the multiphase transport phenomenainvolved [5–8]. Others have had simplifying assumptionsthat have significantly limited the applicability of thesemodels to typical industrial use [2,9]. One primaryreason for the necessity of these assumptions in priorwork is limited computational capability. With theproliferation of faster processors in recent years,detailed models such as the one in this paper shouldbecome more routine. By restricting the use ofempirically based equations to the atomization phaseof the spray process, this model will provide valuableinsight into how individual operating parameters affectthe spray plume development and the resultant deposi-tion patterns.

3. Physics of E-spray

A Ransburg Aerobell 33 model electrostatic rotaryatomizer was used as the basis for this model and sub-sequent laboratory work [11]. This E-spray gun incor-porates a rotating bell-cup and an annular shaping air tofacilitate the atomization of the liquid (see Fig. 1). Thebell-cup has a serrated lip, which facilitates droplet for-mation, and rotates at very high speeds (10–50 kRPM).In addition, the bell-cup also has a conductive coatingalong its outer surface, which supplies the charge tothe spray material via induction. While some E-sprayguns have additional high voltage sources near thenozzle to modify the electric field and repel the dropletsforward, the Aerobell spray assembly modeled in thispaper does not. The high voltage (30–90 kV) applied onthe bell-cup provides a substantial electrostatic drivingforce.

On the Aerobell 33, a shroud circumscribes the bell-cup. This shroud provides an annulus through whichthe shaping air is supplied to the spray. The shapingair pushes the spray toward the target; otherwise, the

ARTICLE IN PRESS

High Potential on Bell Cup

Exterior

Serrated EdgeBreaks Liquidinto Droplets

Shaping Air Inlet(Annulus of Gun)

Momentum fromCup Rotation

Flings DropletsRadially

Electric Field ChargesDroplets Inductively

Resultant Vectorof Drag & Radial

Velocity

Drag fromShape AirVelocity

Bell Cup Rotates at High RPM (~ 25 KRPM)

Fluid FlowInlet

FG

FE/SFDRAG

Fig. 1. Cross-section of a rotary bell electrostatic atomizer with

schematic representation of dominant forces acting on an individual

droplet.

S.A. Colbert, R.A. Cairncross / Journal of Electrostatics 64 (2006) 234–246236

droplets would primarily leave the bell-cup perpendi-cular to the axis of the rotation. While the shaping airassists the droplets in reaching the target, excessiveairflow can form highly focused streams of material,dominated by hydrodynamic interaction. Such over-focusing can lead to running paint and a general non-uniformity in coating thickness, thus losing some of thebenefit of electrostatic attraction.

The performance of E-spray applicators is highlysensitive to droplet size. This becomes evident whenconsidering the forces acting on the droplet duringflight. Fig. 1 pictorially shows the three primary forcesexperienced by the droplet: fluid drag (FDRAG), cou-lombic force (FE/S), and gravity (FG). While the dropletmass scales with the volume of the droplet, for a givensimulation the coulombic force, being proportional todroplet charge, and the drag both scale with the surfacearea of the droplet.

The bell-cup voltage, rotational speed, and shapingair velocity are key parameters that affect the size,charge, and trajectory of the spray droplets. Bell andHochberg [10] demonstrated a power-law relationshipfor the mean droplet size ðDpÞ vs. the bell voltage (F),rotation speed (o), fluid feed rate ð _VLÞ, and feedviscosity (mL) [10]:

Dp ¼ f ðF;o; _VL; mLÞ

¼ CF�0:2o�0:7 _V0:4L m�0:2L . ð1Þ

The constant, C, is dependent upon the geometry ofthe bell-cup used. For example, in the Bell paper, C

equals 12,500 for a bell-cup diameter of 72.5mm.The amount of charge that each droplet is capable of

holding is also a strong function of the droplet size. Inthis paper, the total charge delivered to the nozzle perunit time (i.e. the ‘‘gun current’’) is assumed to be aknown parameter, measured from experiments. Thischarge is distributed to droplets according to theirsurface area—i.e. the charge to surface area ratio isassumed constant. This procedure differs from priormodels where the charge-to-mass ratio was heldconstant [9]. In future work, a correlation will beestablished for how the gun current depends onoperating parameters similar to the droplet size correla-tion above.

In practice, the sprayed droplets contain solvent thatcan evaporate during flight. However, the surroundingair is heavily laden with solvent vapor from thegrounded disk, limiting the driving force for dropletevaporation [13]. Mass transfer calculations for theevaporation of the solvent from a droplet show that thesolvent evaporation is also minimal due to the briefflight time. For example, we estimate that a 75 mmdiameter droplet of toluene traveling at 10m/s throughsolvent-free air at 300K would only lose 0.0185% of itsinitial volume (or 0.005% of its initial diameter) toevaporation in the 20ms it takes to reach the groundeddisk. Based on these calculations, droplet evaporationduring flight is negligible.

4. Mathematical model of E-spray

An axisymmetric representation of the E-spray systemmodeled in this project, including some of the boundaryconditions is shown in Fig. 2. The model geometryincludes the gun pointing directly upward and sprayingat a circular grounded disk that is a fixed distance away.The entire apparatus is surrounded by a groundedphysical boundary with the exception of an exhaust ventlocated at the top of the domain and an air inlet locatedat the bottom of the domain.

The goal of the mathematical model in this paper is topredict coating thickness profiles in deposited coatingsby predicting the trajectories of individual dropletsbetween the bell-cup and the substrate. The modelcontains three coupled components described below: adroplet trajectory model, a turbulent fluid mechanicsmodel, and an electrostatic field model. The rotationalspeed and voltage of the bell-cup, the flow rates of theliquid and shaping air streams, the current draw onthe bell-cup, along with several material properties andthe system geometry, define the variables necessary tocalculate the droplet transport in an E-spray systemfrom bell-cup to grounded disk.

ARTICLE IN PRESS

55 cm

15 cm

121cm

Axis of

Symmetry

GROUNDED CEILING

Grounded DiscΦ = 0 V

20 cm

G

R

O

U

N

D

E

D

W

A

L

L

GROUNDED FLOOR

Bell-cup

Φ = 45 kVradius=1.5 cm

6.75 cm

Exhaust

Vent

Access Hole 45.7 cm

Fig. 2. Axisymmetric schematic of E-spray system.

S.A. Colbert, R.A. Cairncross / Journal of Electrostatics 64 (2006) 234–246 237

4.1. Droplet trajectory model

Despite the complexities of this model, the momen-tum of droplets is affected by only two dominant forces:drag force from the surrounding turbulent airflow andcoulombic force from the electrostatic field on thecharged droplets. The droplet trajectory model is basedon Newton’s 2nd law of motion, where the rate ofchange of inertia of the droplet equals the sum of(three forces) the steady-state drag force, FD, and thecoulombic force, FE=S:

mdv

dt¼X

F ¼ FD þ FE=S. (2)

In general multiphase flow models, other forces can beimportant, such as gravitational, buoyancy, virtualmass, and the Basset forces; these forces are notsignificant in the E-spray system discussed here.

The drag force is predicted from a generalization ofthe Stokes law:

FD � 3pmgDpf ðu� vÞ. (3)

Here, mg, Dp, f , and ðu� vÞ are the viscosity of the air,the droplet diameter, a drag factor, and the relativevelocity between that of the surrounding air, u, and

droplet,v, respectively. The drag factor depends on therelative Reynolds number (Re). Based on Putnam [14],the drag factor can be calculated for a wide range of Re:

f ¼ 1þRe

2=3r

6; Rero1000,

f ¼ 0:0183Rer; 1000pRero3� 105, ð4Þ

Rer �Dpju� vjr

mg.

The mean gas velocity, u, and turbulence intensity, k,are predicted by a k � � model discussed below. Theturbulence intensity is the square of the magnitude ofthe turbulent velocity, u0. The intensity magnitude isdefined by interpolation with the particle position, afterwhich, a random direction is assigned to this velocityusing a random unit vector generated in sphericalcoordinates. An instantaneous velocity, u, is obtainedby summing the mean and turbulent velocities in threedimensions, which simulates the eddies of a turbulentgas flow. It is this instantaneous velocity that is used inthe droplet trajectory calculations. Because the turbu-lent velocity is time dependent, a newly chosen randomvector is determined at each time-step in the droplettrajectory calculation.

The coulombic force is the product of the dropletcharge, q, and the electrostatic field, E, which is thegradient of the electrostatic potential:

mdv

dt¼ 3pmgDpf ðu� vÞ � qrF. (5)

A droplet trajectory (i.e. evolution of droplet posi-tion with time) is predicted by integrating the three2nd-order ordinary differential equations of motionrepresented by vector Eq. (2). The droplet trajec-tory equations are solved by forward Euler timeintegration with a variable time-step, Dt. Because ofthe coupling between droplet trajectories and theelectrostatic field, which is predicted on a finiteelement mesh, the time-step sizes are chosen suchthat there are always several time-steps in each finiteelement.

To solve these equations, the initial position andvelocity of the droplet are needed. The droplettrajectories are predicted for a dilute droplet loadingwhere inter-droplet interactions are neglected. Using thecriteria set forth by Crowe et al. [12], this assumption isvalid for the majority of the solution domain, with theexception of the first few millimeters of spray comingfrom the nozzle. To account for the conditions in closeproximity to the bell-cup, a model describing atomiza-tion dynamics would be necessary. In lieu of such amodel, the droplets modeled here are released fromarbitrary locations near the lip of the bell-cup with aninitial theta velocity equal to the tangential speed of thebell-cup lip and an initial axial velocity equal to the

ARTICLE IN PRESS

STOP

SOLVE FORELECTROSTATICFIELD WITHOUTSPACE CHARGE

SOLVE FORPARTICLE

TRAJECTORIES

SOLVE FORTURBULENT AIRVELOCITY FIELD

START

BUILD GRID FORAIR VELOCITY

FIELD

NO

ESTIMATE SPACECHARGE

ISSYSTEM

CONVERGED?

ALLPARTICLESMODELED

?YES

DEFINE GRID FORELECTROSTATIC

FIELD & PARTICLETRAJECTORIES

INPUT INITIAL &BOUNDARYCONDITIONS

UPDATE COATINGTHICKNESS &

CHARGE

DEFINEGEOMETRY

OFF-LINE

FEMLAB

EXPORT SOLUTIONTO NEW GRID

DEFINE PARTICLEINITIAL VELOCITIES

AND SIZEDISTRIBUTION

SOLVE FORELECTROSTATIC

FIELD WITHSPACE CHARGE

PAR

TIC

LE L

OO

P

CO

NV

ER

GE

NC

E L

OO

P

AC

CU

MU

LAT

ION

LO

OP

NO

YES

CUSTOM-WRITTENC++ CODE

Fig. 3. Iterative solution algorithm for E-spray calculation.

S.A. Colbert, R.A. Cairncross / Journal of Electrostatics 64 (2006) 234–246238

velocity obtained over the distance of 12of Dz of an

element by the droplet under the influence of drag in theaxial direction.

Droplet size is a significant factor in determin-ing the drag and electrostatic effects of this model [15].In the literature, particle size distributions forspray coating systems are typically reported to beGaussian or lognormal distribution [16–18]. In thispaper, the droplet size distribution is assumed to belognormal.

4.2. Turbulent fluid mechanics model

For a turbulent air stream, the gas flow fluctuates.The instantaneous velocity is the sum of the time-averaged velocity, u, and perturbations from the averagevelocity, known as the turbulent velocity, u0:

u ¼ uþ u0. (6)

As the Reynolds number of the airflow increases andthe flow becomes turbulent, perturbations in the flowvelocity do not dissipate and eddies are formed. Theformation of these eddies makes the turbulent contribu-tion to the instantaneous velocity significant.

In this paper, we use a commercially available finiteelement software package (FEMLAB) to solve for theturbulent air velocities via an axisymmetric implementa-tion of the turbulence energy-dissipation model, alsoknown as the two-equation or (k � �) model [19–21]. Inthe (k � �) model, the turbulent velocity field is assumedto be made up of individual eddies which have discretevelocities and lifetimes. Two additional differentialequations are used to determine the turbulence energy(k) and the eddy dissipation rate (e), which are used todetermine the Reynolds stress. The turbulence energy isthe time-averaged square of the magnitude of turbulentvelocity.

The geometry of the E-spray system modeled in thispaper is shown in Fig. 2. There are two inlets where airenters into the domain: the annular region surroundingthe bell-cup where the shaping air is injected and aneutral access hole behind the gun. There is one outlet,an active exhaust port behind the substrate. The shapingair velocity profile at the base of an annular inlet regionis assumed parabolic with a prescribed total flow rate.The active exhaust is imposed with a uniform axialvelocity that matches velocities measured on an experi-mental apparatus. All other boundaries are either theaxis of symmetry or solid walls. Along all solid walls, thelaw of the wall is used [22].

4.3. Electrostatic field model

The electric field in the presence of charged dropletsis described by the Poisson equation as shown belowin cylindrical coordinates using an axisymmetric

assumption:

r2F ¼1

r

d

drrdFdr

� �þ

d2Fdz2¼ �

r�0. (7)

Here F, r, and �0 are the electrostatic potential, spacecharge (or charge density), and permitivity of free space,respectively. This equation is readily solved by manytechniques including the finite element method [23,24].Because the Poisson equation is linear, the stiffnessmatrix, once calculated, remains unchanged. Reusingthe stiffness matrix results in a reduction in overallcomputation time.

To establish the droplet trajectories, the electrostaticfield is calculated initially without the contribution ofcharged droplets (i.e. q ¼ 0). Once a set of dropletpaths has been calculated based on the current electro-static field, a new estimate for the space charge iscalculated. The electrostatic field and droplet trajec-tories are updated iteratively in a sequential solutionprocedure described later (see Fig. 3). Because there is astrong coupling between the two fields, relaxation of thenewly calculated electrostatic fields is used to aid inconvergence.

ARTICLE IN PRESSS.A. Colbert, R.A. Cairncross / Journal of Electrostatics 64 (2006) 234–246 239

The Poisson equation (7) is used to determine theelectrostatic field in the domain shown in Fig. 2 subjectto boundary conditions around the perimeter of thedomain. The electrostatic potential at the bell-cupand grounded substrate are imposed as Dirichletconditions. However, the choice of boundary conditionsat all of the external boundaries was more difficult.Some commercial software packages (e.g. MAZE) use aNeumann boundary condition for unspecified bound-aries in electrostatic FEM models [23]. After testingDirichlet (e.g. grounded) and Neumann conditionsusing a finite-difference model, it was decided that aDirichlet boundary condition does not distort theelectrostatic field near the edges of the groundeddisk as much as a Neumann boundary condition.In addition, incorporating the Dirichlet boundarycondition is easier to implement in a laboratoryenvironment [11].

4.4. Space charge estimation

The droplets emitted from the nozzle are charged andthe presence of the spray modifies the electrostatic fieldbetween the bell-cup and the grounded disk. It isnecessary to estimate a time-averaged space charge dueto the droplets passing through the domain; forconvenience, values of the space charge are assigned toevery node in the finite element mesh used for theelectrostatic field calculation. Essentially, the spacecharge at a node is the average charge contributed bydroplets passing through the vicinity of the node dividedby a volume assigned to the node, Vi, in the electrostaticdomain corresponding to the area of the element inwhich it resides weighted by a bi-quadratic basisfunction ðciÞ:

V i ¼ 2pZOcirdrdz. (8)

The average charge in the vicinity of a node is basedon the path and speed of the droplet as it passes throughan element and the charge of the droplet:

hqii ¼

Pnj¼1qj

R t1t0ciðxjðtÞÞdt

tspray. (9)

The numerator of Eq. (9) contains the product of thedroplet charge, qj , and a time integral of the basisfunction summed over all droplets. The time integral isassociated with node i along the path taken by thedroplet, xjðtÞ. This integration is accomplished by Eulerintegration using the mid-point method. The basisfunction provides a weighting that emphasizes nodesthat are closer to the droplet path. The denominator isthe droplet generation time, tspray, i.e. the time requiredto generate n droplets at the prescribed fluid flow rate.The nodal average values of the space charge are used asa source term in the next iteration of electrostatic field

calculations:

hrii ¼hqii

Vi

¼

Pnj¼1qj

R t1t0ciðxjðtÞÞdt

2ptsprayRO cirdrdz

. (10)

4.5. Coupled model of the E-spray system

The fluid velocity field, the electrostatic field, and thedroplet trajectories are all coupled, but the coupling isassumed weak enough that the fields can be calculatedseparately in an iterative procedure as shown in Fig. 3.Because the turbulent fluid mechanics are assumed to beunaffected by the drag of the droplets, the turbulent flowfield is predicted first. The axisymmetric turbulent flowfield is determined using a commercial finite elementpackage, FEMLAB, but the electrostatic field and thedroplet trajectories are predicted with a custom-builtcomputer program. The finite element meshes used forthe fluid mechanics and the electrostatics are differentdue to the separate resolution requirements in each meshfor accurate solutions of the flow and electrostatic fields.The mesh generated in FEMLAB for the fluidmechanics is made up of 10,280 irregular triangularelements with three nodes per element (5499 nodes intotal), whereas the mesh generated by the custom-builtC++ program for the electrostatic field is comprises20,765 regular rectilinear elements with nine nodes perelement (83,685 nodes in total). Interpolation of theturbulent velocity field solution to the electrostatic meshis done using the postinterp function in Matlab(FEMLAB is implemented within Matlab). The averageradial and axial velocities as well as the values for theturbulence intensity, k, are provided in this manner andit is these interpolated values that are used in the droplettrajectory calculations. Because the electrostatic fieldstrongly affects the droplet trajectories, and vice versa,relaxed iterations between these two solutions isrequired. In each iteration, the electrostatic field anddroplet trajectories are updated to account for changesin the electrostatic field to bring the entire system toglobal convergence, typically less than 100 iterations arerequired.

The electrostatic field and individual droplet paths arepredicted by a custom-written code shown in the lowerbox in Fig. 3. There are three loops shown in the figurewhich correspond to (1) a loop over each of the 10,000droplets simulated, (2) a loop for iterating the electro-static field to convergence and (3) a loop for timeintegration of charge accumulation and decay on thesubstrate. The electrostatic field is initialized by assum-ing the domain is droplet free which translates to a zerocharge density. In each iteration of the electrostatic fieldand droplet trajectories, a new charge density field isdetermined as input to the next iteration. The electro-static force vectors obtained from these calculations,

ARTICLE IN PRESS

Table 1

Outline of simulation details

Parameter Typical value

Number of particles 10,000

Number of elements in fluid mechanics mesh 10,280

Number of nodes in fluid mechanics mesh 5499

Number of elements in electrostatics mesh 20,765

Number of nodes in electrostatics mesh 83,685

Relaxation parameter 0.05

Number of iterations to convergence 93

Table 2

Experimental parameters for base case and ranges exploreda

Parameter Base case

value

Range

explored

Units

Mean droplet size 10.9 5.4–21.7 mmBell-cup voltage ðF0Þ 45 22.5–90.0 kV

Atomization constant 2500 1250–5000

Dimensionless

Coating fluid

viscosity

7.5 N/A cP

Coating fluid flow

rate

2.5 N/A mL/s

Bell-cup rotational

speed

29.3 N/A KRPM

Current supplied to

bell-cup

0.1 N/A mA

Shaping air inlet

speed

1.65 N/A m/s

Exhaust air speed 0.685 N/A m/s

aGeometry shown in Fig. 2.

S.A. Colbert, R.A. Cairncross / Journal of Electrostatics 64 (2006) 234–246240

as well as the instantaneous velocities of the air streamproduced in the prior step, are both used as inputs to thedroplet trajectory calculations.

Apart from the operating parameters of the gun(spray material feed rate, shaping air pressure, etc.),other inputs to the droplet trajectory calculations are thesize distribution and initial trajectory of the droplets,which are a distribution of droplets being released froma random location at random initial velocities in thevicinity of the bell-cup lip. The droplet trajectory makesuse of the same grid as the electrostatic solution;however, it also incorporates an azimuthal component,y, to accommodate the 3D nature of the turbulentvelocity.

Each droplet path is modeled independently, whichcorresponds to the droplet loop shown in Fig. 3. Thetime of flight of the droplet is defined by the summationof all elemental residence times from droplet release tolanding. To ensure that multiple steps are taken in eachelement, a maximum flight time-step is set (e.g. 10�3);however, this value is reduced when a droplet is about toleave an element. Then, the flight time-step is the time ittakes a droplet to reach the border of an element basedon the velocity at that point in the flight path. Thesecalculations are repeated per droplet until the positionof the droplet coincides with a physical barrier in thesystem. Once a droplet makes contact with a physicalbarrier, a contribution to the local charge and coatingthickness accumulation is made. The next droplet path ismodeled until all droplets have landed on someimpenetrable boundary or passed through an outflowboundary, which represents the spray accumulationtime-step.

To account for coupling between the spatial chargedensity and the electrostatic potential, the electro-static solution and droplet trajectories are iterated untilglobal convergence is achieved. In this model, globalconvergence is considered to be attained when theroot-mean-squared difference in the electrostaticpotential values normalized by the number of nodesbecomes lower than a specified convergence criterion. Aconvergence criterion of 10�3 was used for simulationsin this paper. Because of the significant affect that thespace charge has on the electrostatic field, particularlynear the spray nozzle, updates in the electrostaticpotential at each iteration, Fi, are relaxed based on thepotential from the previous iteration, Fi�1 and thepotential calculated using the current space chargefield, Fcalc:

Fi ¼ ð1� aÞFi�1 þ aFrec. (11)

A relaxation parameter, a, between 0 and 1 is chosento represent the weight of the new solution vs. the oldsolution. In the simulations in this paper, a is set to 0.05(Table 1).

5. Results

5.1. Model base case

The model described in the previous sections wassolved to predict E-spray coating of a xylene/polystyr-ene solution. The simulations correspond to a singlerotary bell E-spray gun mounted vertically within acylindrical grounded chamber, as depicted in Fig. 2. Aset of base case conditions was chosen that produceresults typical of the model, and these parameters areshown in Table 2. The predicted results are sensitive tothe operating conditions. Two parametric studies areincluded after detailed analysis of the base case resultsthat show how bell-cup voltage and atomizationconstant affect the predicted deposition rate profiles ofE-spray coatings.

In Fig. 4, the effect of charged droplets on theelectrostatic field is displayed. Fig. 4A shows theelectrostatic field without droplets (i.e. r ¼ 0) in whichthe contour lines around the bell-cup show a predomi-nantly spherical shape with only a slight distortion nearthe edge of the grounded disk. Fig. 4B shows a set of

ARTICLE IN PRESS

Fig. 4. Contour plots of the normalized electrostatic potential (F=F0) along with select droplet paths (A) prior to spray (B) after spray solution has

converged (C) close-up view of the domain in 7B between bell-cup and grounded disk for base case conditions (F0 ¼ 45 kV).

S.A. Colbert, R.A. Cairncross / Journal of Electrostatics 64 (2006) 234–246 241

individual droplet paths is overlaid onto the equipoten-tial contour map of the converged base case solution.The droplet paths shown in Figs. 4B and C differ due todroplet size, starting position, and turbulence. Thedroplet size is chosen randomly from a lognormaldistribution about the mean droplet size, as calculated inEq. (1). The starting locations from the droplet pathsshown in Figs. 4B and C are uniformly distributedacross the range of starting locations, which has beenchosen as the gap width of the shaping air annulus. Thepredicted droplet paths comprise a tightly focusedhollow cone-shaped spray distribution, which turnssharply to follow a path parallel to the surface of thegrounded disk.

Figs. 5A and B show the time-averaged air speedintensity (juj from Eq. (6)) and streamlines of theairflow. The streamlines originating at the dropletlaunch point strongly resemble the droplet pathsindicated in Fig. 4C, which implies that the majorityof these droplets are driven by drag forces. As drag isdirectly proportional to the surface area of the droplet,the diameter of the droplets greatly affects its path. Nearthe shaping air annulus, the air velocity is highestcreating a significant drag force. Only the largestdroplets in the distribution are capable of overcomingthis drag because they possess adequate momentumfrom the bell-cup rotation while the remaining dropletsare caught up in the stream of the jet. Not until theseentrained droplets make the sharp turn near thestagnation point at the center of the grounded disk dothey begin to divert from the streamlines. Here,momentum and electrostatic forces guide the larger ofthe entrained droplets toward the grounded disk withthe heaviest ones landing closest to the center point.Meanwhile, the finer droplets continue to be carriedaway by the air stream. While most of these fine dropletsform the over-spray, some succumb to electrostatic

attraction and are capable of wrapping around the edgeof the grounded disk and coating the edge or back sideof the disk. Because drag plays such a significant role inthe paths of the droplets, droplets with a similardiameter and launch site would have nearly identicalpaths in a steady-state air stream. However, dropletshaving similar characteristics do not share the samepaths. This spreading of paths is from the incorporationof turbulence in the force balance for droplet motion,which causes a randomization in the droplet trajectories.

In addition to turbulence, this spreading in dropletpaths is caused by the contribution of the dropletcharges on the electrostatic potential in the vicinity ofthe flight paths. The average density of droplet chargesas calculated by Eq. (10) is displayed in Figs. 5C and D,with the same droplet paths from Fig. 4. In Fig. 5D, themaximum charge density (roughly 1� 10�3 c/m3) islocated at the lip of the bell-cup and decreases rapidlyfrom there. The charge density allows for an ensembleview of the spray distribution as opposed to individualdroplet paths. The spray depicted in Fig. 5D appearssomewhat wider than indicated by the droplet paths inFig. 4C. The spray that does not follow the airstreamlines results from larger droplets penetrating thefast-moving annular air curtain by virtue of their highermomentum and traveling into the slower movingentrained air stream shown in Fig. 5B. Droplets alsodeviate from the streamlines because of the elevatedelectrostatic potential along the central axis, whichaugments the radial component in the electrostatic fieldand drives droplets away from the axis of symmetry.

The intense radial velocities induced by air stagnationon the disk cause a sheet of high charge density in theplane of the disk extending all the way to the chamberwall. Many droplets are entrained in this sheet formingover-spray that misses the grounded disk. This chargedover-spray produces a distortion in the electrostatic field

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Fig. 5. (A) Contour map of wind speed intensities ðjujÞ and velocity streamlines for entire domain. (B) Close-up view of 5A between bell-cup and

grounded disk. (C) Contour map of charge density distribution and select droplet paths for base case. (D) Close-up view of 5C between bell-cup and

grounded disk.

S.A. Colbert, R.A. Cairncross / Journal of Electrostatics 64 (2006) 234–246242

between the edge of the grounded disk and the outerradius of the domain. As these droplets travel towardthe exhaust vent, the electrostatic potential is signifi-cantly elevated in the region behind the substrate.However, Fig. 4C, a close-up view of 4B, shows themost significant effect of the presence of chargeddroplets on the trajectories. Between the bell-cup andthe grounded disk, the once nearly spherical contourlines of Fig. 4A become stretched toward the disk. Thisstretching is caused by the charge carried by the dropletsas they travel toward the grounded disk, which raises thelocal electrostatic potential. The electrostatic fieldintensity (i.e. gradient of electrostatic potential) alsochanges due to presence of charged particles, which willchange the coulombic force on the droplets (Eq. (5)).The axial component of the electrostatic field contri-butes to droplet motion toward the grounded disk.

Fig. 6 displays the axial component of the electrostaticfield for the base case both with and without thepresence of droplets. The electrostatic field strengthdecreases monotonically when there are no chargeddroplets to distort the field. However, the presence ofdroplets causes a decrease in electrostatic field near thebell-cup and an increase near the grounded disk (Fig. 6).The increased field intensity near the disk causes a driftof droplets across the air streamlines (which are parallelto the disk) toward the substrate. Because of highersurface area to volume ratio in smaller droplets, a highercharge-to-mass ratio is present causing these smallerdroplets to drift more quickly toward the grounded disk.

In addition, the increase in electrostatic potentialalong the droplet path produces an increase in the fieldintensity near the grounded disk with the final resultbeing an order of magnitude net increase in field

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Fig. 6. Change in axial electrostatic field strength, qF=qz, at the bell-

cup radius vs. distance to grounded disk with and without droplets at

the base case conditions.

Fig. 7. Mass percentage of droplets striking grounded disk as a

function of launch position normalized to total mass launched from

each position in the base case. The line is a best fit to the data using

linear regression.

Fig. 8. Size distributions of droplets striking target (gray bars) and

missing the target (black bars). The initial size distribution of droplets

is indicated by the curve.

S.A. Colbert, R.A. Cairncross / Journal of Electrostatics 64 (2006) 234–246 243

strength near the grounded disk. It is this increased fieldstrength in conjunction with the momentum of thedroplets that causes charged droplets to drift across theair stream and contributes to the higher transferefficiency (Z) of E-spray vs. conventional spraying.

For the base case, a majority of the mass sprayed hitsthe face of the grounded disk with a transfer efficiency(Z) of 77%. Transfer efficiency is affected by variationsin the trajectory of the droplets. The variability of thetrajectories of different droplets is affected by severalfactors: starting position, droplet size, and turbulence.Fig. 7 displays the fraction of particles that hit the targetas a function of their radial starting location. Thestarting location of each droplet is assigned based on thedimensions of the 1.5mm annulus of the E-spray gun.The first droplet is released from the inner lip of theannulus and the last is released from the outer lip. Thedroplets were launched sequentially at locations equal-ing: nGN�1, where n is the particle number and N is thetotal number of particles released, and G is the annulargap. In Fig. 7, data points show the percent of dropletsthat strike the 150mm target as a function of startinglocation. There is a general trend that droplets releasednear the inner wall of the shaping air annulus are morelikely (by �18%) to strike the target than those releasednear the outer wall of the annulus. The predictedtransfer efficiency is sensitive to starting locations of theparticles, which is affected by the atomization process asthey travel the 200mm from the lip of the gun to thetarget.

Fig. 8 shows the size distribution of the dropletsmodeled in the base case. The mean droplet size is10.9 mm, around which a lognormal distribution of sizeswas assumed using a random number generator. Theresult of this method of obtaining droplet sizes is thatthere is a concentration of smaller droplets and a tailin the larger droplet sizes. Fig. 8 also displays size

distributions for particles that hit the target and forparticles that miss the target as over-spray. In this figure,the distribution of droplets is divided into two groups:droplets that strike the target (Hits) and droplets thatstrike other surfaces (Misses). The shape of the distri-bution for ‘‘Hits’’ is similar to, but slightly narrowerthan, the initial droplet size distribution. This indicatesthat the droplets closest to the mean diameter are mostlikely to strike the grounded disk, while large andsmaller droplets are more likely to miss the target. The‘‘Misses’’ exhibit a bimodal distribution with concentra-tions at the two extremes of the total droplet sizedistribution. There are two phenomena responsible forthe presence of these two peaks. For the small particlepeak, the droplets are dominated by drag, lacking themass needed to cross the air stream, and therefore areswept away to the exhaust vent. The large particle peakresults from the initial momentum allowing these

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droplets to ‘‘pass through’’ the high-speed annular aircurtain into the slower moving entrained air streamdepicted in Fig. 5B. As these droplets near the target,they lack the speed to penetrate the fast-moving airstream that passes along the surface of the disk resultingin these droplets being swept away as well.

As the droplets strike the grounded disk, a coating isformed. To calculate the deposition rate of the coatingvs. radial location, the radius of the disk is divided into100 sections. When a droplet lands on a particularsection, the volume of the droplet is spread over the areaof the section, producing a thickness. Once a set ofdroplets has been simulated, the coating deposition rateis calculated by dividing the thickness of each section bythe droplet generation time. Fig. 9 shows the coatingdeposition rate as a function of radial position on thegrounded disk. The overall roughness of the plot is anartifact of the number of droplets simulated (104) andthe number of sections into which the disk is divided(1 0 0). Increasing the number of droplets would providefor smoother plots as would reducing the number ofdivisions. The sharp spike in deposition rate at the edgeof the disk (near the 100% mark) is the result of edgeeffects in the electrostatic field attracting droplets.About 1.5% of the total mass sprayed in the base caselanded on the vertical edge of the disk, and there is acorresponding increase of deposition on the outer 2% ofthe disk radius. The remainder of the disk shows moremoderate deposition changes with respect to radius.While no droplets land on the center of the disk, a well-defined peak exists at roughly 50%. The difference inslope on either side of the peak may be related to thelognormal droplet size distribution. As mentionedearlier, for the droplets entrained in the annular jet aircurtain, the larger droplets sizes would be more likely toseparate from the air stream after the stagnation pointat the center of the disk by virtue of their momentum;however, the large droplets only represent a tail of

Fig. 9. Thickness deposition rate as a function of radial position on

the grounded disk for three realizations at the base case conditions.

the distribution (see Fig. 8), whereas the overalldistribution is skewed toward the smaller dropletssizes. It is these mid-range and smaller droplets thatwould be striking the target further from the centerpoint. The rate at which these droplets land on thegrounded disk is directly related to the frequency withwhich they occur.

In summary, the base case is characterized by a tightcone of spray in which the droplet paths are dominatedby the drag forces imparted on the droplets from theannular jet of the shaping air. This jet impinges on thetarget center causing a stagnation point, the flow fromwhich diverts the droplets radially. The main mechanismby which these droplets leave this stagnation flow airstream is either through their momentum or by theincreased electrostatic field strength caused by thepresence of the charged droplets near the grounded disk.

5.2. The effect of voltage on thickness deposition rate

In Fig. 10, three coating deposition rate profiles areshown which correspond to different values of thebell-cup voltage. The input parameters used to predictthese different profiles are the same as the base case (seeTable 2), with the exception of a variation in the voltagesupplied to the bell-cup. All three of the voltage levelsproduce no deposition near the central axis and a thickrim at the edge of the substrate (i.e. ‘‘fatty edges’’)similar to those noted in the discussion of Fig. 9. Each ofthe deposition rate profiles exhibits a peak thicknessnear 50% of the radius of the substrate. As voltage isincreased, the thickness profile becomes narrower andmore peaked. By the atomization model in Eq. (1),increasing the voltage causes a slight drop in meandroplet diameter (as listed in the inset table). Lowervoltages correspond to larger droplets, which depositcloser to the center of the grounded disk. Surprisingly,there is relatively little change in the mass transfer

Fig. 10. Coating deposition rate as a function of position on grounded

disk. Inset table contains the resultant mean droplet diameters and

transfer efficiencies at various bell-cup voltages.

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efficiency, Z, as the voltage increases. In fact, themaximum efficiency appears to be near the base casevoltage of 45 kV. Because Z is relatively constant overthe range of voltages explored at this set of operatingparameters, other optimization criteria can be used. Forexample, if coating uniformity is a high priority, a lowervoltage is preferred over a higher one. In cases wherecost drives the process, lower voltage processes are lesscostly to equip and operate. Finally, if coating finish isthe parameter to be optimized, lower voltages again arethe preferred mode. By Eq. (1), higher voltages producesmaller mean droplet diameters. These smaller dropletsare more prone to evaporation during flight, which leadsto defects on the finished surface (i.e. ‘‘fish-eye’’ or‘‘orange-peel’’) due to semi-evaporated droplets landingon the substrate.

6. Conclusions

A computer model capable of simulating the paths ofdroplets in an E-spray system has been presented. Thismodel solves for the turbulent air velocity field, theelectrostatic field, and the droplet trajectories individu-ally. FEMLAB was used to obtain the average airvelocities and turbulence intensities a priori using a k � �model. A random vector whose magnitude was based onthe turbulence intensity was used to simulate thestochastic nature of the turbulence. These velocitiesand turbulence intensities were interpolated into a meshgenerated by the custom-written C++ program used tosolve for the electrostatic field and droplet trajectories.However, the significant contribution of the chargeddroplets to the electrostatic potential necessitated therelaxed iteration of these two calculations to arrive at aconverged solution. A simplified atomization model wasused which incorporates an empirical correlation [10] tocalculate a mean droplet size around which a rando-mized lognormal distribution was taken.

Analysis of a base case shows that drag forcesfrom the shaping air, and not electrostatic forces,dominate most droplet trajectories. The exception tothis case would be in the instance of very large drop-lets whose initial momentum would allow these dropletsto penetrate the high-speed annular air curtain andenter the slower moving entrained air mass outside thejet. These large, slower moving droplets lack themomentum to penetrate the stagnation flow thatsweeps across the face of the disk and become over-spray. Similarly, smaller droplets follow the air stream-lines thus forming over-spray by being carried alongwith the stagnation flow. However, the larger drop-lets that are entrained in the air stream are capableof depositing on the target through their momentumand the increased electrostatic field strength near thetarget.

This model allows for parametric studies of how theindividual variables involved in E-spray affect theoverall spray pattern and transfer efficiency, which canhelp operators of such equipment define their optimumspray setup without the need of costly trial-and-errorempirical development. While some characteristics ofthe coating thickness deposition rate remained constantregardless of parametric settings (i.e. no coating incenter, increased accumulation on the edge, and a singlepeak in coating thickness) the shape and location of thecoating peak could be changed. From the parametricstudies presented in this paper, it appears that anoptimum droplet size of roughly 10 mm provides thehighest mass transfer efficiency. This optimum could bethe result of a beneficial balance between momentumand drag, both near the shaping air annulus at thelaunch point and near the stagnation point on the disk.Eq. (1) shows how this optimum droplet size can beachieved via multiple routes through different para-metric combinations of bell-cup voltage, bell-cup rota-tional speed, spray material viscosity, or spray materialvolumetric flow rate. This model enables the predictionof the spray distribution (or coating deposition rateprofile) and how it can be modified through thechanging of operating parameters while still maintainingoptimal transfer efficiency.

Acknowledgments

The authors wish to thank Drexel University andThomson Inc. for their support.

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