Initial droplet conditions in numerical spray paintingby electrostatic rotary bell sprayers

A framework for optimization of injection model coefficients

Nico Guettler, Philipp Knee, Qiaoyan Ye, Oliver Tiedje

� The Author(s) 2020

Abstract In computational fluid dynamics, the mod-eling of paint application processes by electrostaticrotary bell sprayer is mostly performed using an Euler–Lagrange approach. The initial conditions of thediscrete phase—position, velocity, size, and charge—have an essential influence on the resulting filmthickness distribution and the total charge transferredto the object. Typically, so-called injection models areused to specify these initial conditions, whereby thedetermination of the injection model coefficients iscrucial. In this paper, a framework is proposed thatcombines experimental input data, an injection model,and a metamodel-based optimization. The paintingtests for the generation of input and validation datawere carried out in a technical center in the industrialstandard. The simulations were performed usingANSYSFluent. Initial droplet conditions could effi-ciently be determined via the framework so that thepainting-specific objectives were achieved with reason-able accuracy. In addition to the framework, a turbu-lence study of the strongly swirled shaping air of thisatomizer was carried out, whereby a substantialunderestimation of the axial air velocity was found inthe turbulence models being investigated. The initialdroplet conditions were also used in this study to drawconclusions about the accuracy of the airflow simula-tion. The proposed framework can be adapted to othersolvers and efficiently finds injection model coefficientsfor other paint applicators.

Keywords Spray painting simulation, Rotary bellatomizer, Electrostatic painting simulation,Metamodel

Introduction

Numerical simulations of paint application processes areused for virtual operating trials to perform paintabilitystudies in early design phases, to optimize processparameters, and to reduce the number of paintingprototypes. To reduce the computational effort andincrease the accuracy of painting simulations, bothnumerical and painting-specific models and methodsmust be continuously enhanced. Particularly in spraysimulations, the Euler–Lagrange approach requiresinjection models to determine the initial condi-tions—position, velocity, size, and charge of the parti-cles. In this study, an electrostatic rotary bell sprayer(ERBS) was used, which is being used in the automotiveindustry to paint car bodies and plastic parts likebumpers. When using ERBS, paint is stationary addedinside the bell-cup through a small orifice on therotation axis. Due to the rotational speed of the bell-cup, the paint is distributed radially on the inner surfaceby centrifugal forces. At the bell-edge, the paint isatomized because of the high relative velocity betweenthe liquid and gaseous phases. The shaping air causes achange in the direction of the droplet movement by astrong axial component toward the object to be painted.The electrostatic field between the charged bell-cup,shaping air ring, and the grounded substrate increasesthe transfer efficiency of the paint by applying anadditional force to the charged particles (see Fig. 1).

In general, rotary atomizers have been investigatedin various fields of application, such as spray drying,spray painting, or granulate production, where differ-ent types of disks and bell-cups both with and withoutshaping air were being used. Investigations on the film

This paper was presented at the 15th Coatings Science Inter-national Conference on June 24–29, 2019, in Noordwijk, theNetherlands.

N. Guettler (&), P. Knee, Q. Ye, O. TiedjeFraunhofer Institute for Manufacturing Engineering andAutomation IPA, Nobelstrasse 12, 70569 Stuttgart,Germanye-mail: nico.guettler@ipa.fraunhofer.de

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thickness of a liquid within a rotating conical bodywere performed by references (1), (2), and (3) whoderived a formula to calculate the film flow velocity onthe inner bell-cup surface. Reference (2) also showedthat the disintegration process on rotating cups couldbe divided into three modes: direct drop formation,ligament formation, and film formation. The transi-tions between these modes were also experimentallyinvestigated by references (4) and (5) on a rotating diskas well as by references (6) and (7) on a rotating bell-cup. However, there are very few numerical works thathave investigated the primary breakup of paint liquidsby ERBS in a full computational framework, forexample, references (8) and (9). In their work, paintfilm distribution on the bell cup and the primarybreakup of liquid quite close to the bell edge wereexperimentally and numerically investigated, whichdelivers some useful information for the particletrajectory calculation using the Euler–Lagrange meth-od. Due to the complex properties of paint materials(non-Newtonian, shear-thinning, and thixotropic fluid),the simulation of the primary atomization is onlypossible with very high computational effort and well-known material properties, which is why the Euler–Lagrange approach is preferred for paint simulations inthe industry and industrial applied science.

Detailed spray simulations have already been per-formed and evaluated by reference (10). The authors alsodescribed that the spray behavior strongly depends on theinitial conditions of the discrete phase and must becalibrated to experimental data. However, the dependen-cies of the initial droplet conditions were not directlyshown, and no framework was derived to determine theinjection parameters. Recently published work by refer-ence (11) presents an injection model for ERBS in whichthe injection model consists of multiple rings withdifferent amounts of injection points, radii, angles, andmass flow rates. It was shown that simulation results couldbe improved with steeper injection angles to the object,respectively, higher axial velocity. However, in thismodel, no dependence on the charge of the droplets wasshown.Reference (12) also presents a parameter study onan injection model by systematically investigating injec-tion positions and velocities. The author concludes that

the axial injection position, as well as the radial andtangential injection velocity, has no significant influenceon the spray pattern, whereas the radial position and theaxial velocity clearly affect the simulation result.

Numerical investigations on the effects of electro-statically assisted paint application have already beencarried out by references (10), (11), (12), (13), and (14),using different models to distribute the charge on thedroplets as a function of droplet diameter. A study onthe influence of different models of particle chargedistribution on the resulting film thickness distributionwas performed by reference (13). The authors con-cluded that the transfer efficiency increases withincreasing charge on the droplet surface, but noproposal to the use of a specific model was derived.

This paper proposes an injection model and aframework to calibrate the injection model coefficientsby a metamodel-based optimization in combinationwith experimental painting-specific data. Afterwards asimulation with a complex workpiece including movingatomizers along a robot path will be simulated with theoptimized injection parameters.

The paper is divided into sevenmain sections. First, anoverview of the entire framework is presented in‘‘Structure and objective of the framework’’ section. ‘‘Ex-perimental materials and methods’’ and ‘‘Numericalmethods’’ sections describe the experimental, numericalmaterials and methods used in this framework. Theresults of a turbulence study of the shaping air flow arediscussed in ‘‘Simulation of the airflow field’’ section. Aninjection model for this ERBS is proposed in ‘‘Modelinginitial droplet conditions’’ section. The effective deter-mination of the injection model coefficients is thendescribed and discussed in ‘‘Metamodel-based optimiza-tion of injection model coefficients’’ section. The resultsandprocedure are finally discussed, and a conclusionwithan outlook on further research work is given.

Structure and objective of the framework

The considerable variation of paint properties incombination with system-specific or customer-specificERBS equipment and process parameters makes itdifficult to define a generic model for the initial dropletconditions in painting simulations. The framework ispresented using an ERBS but can also be usedefficiently for other atomizers, such as high volumelow pressure or airless atomizers. Figure 2 illustratesthe workflow of the proposed framework in form of ahigh-level sketch. The majority of the workflow aims tocreate metamodels. The metamodels are used in thesubsequent step of multiobjective optimization tomake the very high number of function evaluationsmore efficient. First, a feasible range for the injectionmodel coefficients must be defined. Using a Latin-Hypercube experiment design, a set of sampling pointsis generated within the feasible space. Based on this, aset of simulations will be carried out and evaluated

Shaping Air1 & 2

Grounded Subst rate

E

Bell-cup

AtomizerHousing

Shaping Air Ring

Fig. 1: Functional principle of paint application usingelectrostatic rotary bell sprayer

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with regard to the painting-specific objectives. Corre-lation coefficients are used to identify the principalcomponents of the model.

The framework assumes that experimental data onthe paint material (e.g., density and solids content), aswell as process-related data (e.g., spray pattern,discharge current, and particle size distribution), areavailable. Often a large part of this data is alreadyavailable or can be generated relatively inexpensivelyin paint shops.

Experimental materials and methods

This section describes the equipment and processsettings used as well as the experimental methods inmore detail.

Spray booth, atomizer, and paint material

The experimental investigations were carried out inan environmentally controlled spray booth with an

ambient temperature of 23 �C, a relative humidity of60%, and a vertical booth airflow of 0:3ms�1. TheERBS picoBell HiBlow III of the company EISEN-MANN-LACTEC with a 50mm bell-cup (no serration)and a direct charging mechanism was chosen andmounted on a 6-axis ABB robot. The shaping air ringhas 40 outer and 32 inner nozzles, which are referred toin the following as SAout and SAin. The outer and innernozzles are oriented tangentially to the axis of rotationbut have different axial angles (see Fig. 3). The differentorientation of SAout and SAin enable a purposefulforming of the airflow to paint complex geometries, e.g.,bumpers, efficiently. An application setting was chosen,which is used in industrial paint shops. In addition to theshaping air, the process settings are determined by thepaint flow rate PF, the revolutions per minute of thebell-cup nbell, the high voltage HV applied to the bell-cup and the shaping air ring, and the painting distance d(also referred as the distance to bell-edge). The paintapplication process setting is shown in Table 1. A two-component solvent-borne industrial coating with a wet

density of 1109:5 kg m�3, a dry density of

1293:8 kg m�3, and a nonvolatile content of 67:9%was used as painting material.

Determination of shaping air velocity

The airflow velocity of the shaping air was determinedusing a DANTEC 2-component fiber Laser-DopplerAnemometry (LDA) system. As tracer particles, glyc-erine mist was injected into the intake flow above the

Feasable Space of Injection Parameters

Design of Experiment(Sampling Points)Injectionfiles

CFD Simulations

Metamodel Current

Metamodel Spraypattern

Principle Component Analysis

Multi-Objective Optimization

Experimental InputWet Density of PaintParticle Size Distribution

Experimental InputSpraypatternTransferred Current

Injection Model

Experimental InputDry Density of PaintSolid Content of Paint

Evaluation of Simulations

Fig. 2: High-level sketch of the framework for thedetermination of the optimized initial droplet conditions.Gray boxes indicate experimental input data are needed inthis step

SAin SAout

Bell-Cup

Shaping Air Ring

Fig. 3: Simulation-relevant components of the atomizer.Needles were drilled into the shaping air nozzles tovisualize their orientation

Table 1: Paint application process setting

PF SAout SAin nbell HV dml min�1 ls min�1 ls min�1 min�1 kV mm

300 300 240 50000 65 200

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ERBS. To determine the velocity components in threespatial directions, the flow field was scanned horizon-tally in two spatial directions at a fixed axial distance tothe bell-edge. The scanned lines were measuredselectively in an increment of 5mm. The measurementof a single position was stopped at the terminationcriterion of 10000 particles, and the mean value and themean square error were calculated. The results areused to validate a turbulence model study (See‘‘Simulation of the airflow field’’ section) focusingboth close to the bell-cup (d ¼ 10mm) and close to theplate (d ¼ 180mm) (Fig. 4).

Determination of particle size distribution

The volumetric particle size distribution was measuredusing a SPRAYTEC RTS 5001 of the companyMalvern Instruments. To minimize evaporation effectsof the paint droplets on the result, the measurementswere carried out at a 45� angle and distance of 25mmto the bell-edge. The optics of the device are protectedby an attached tube against contamination by paintoverspray. It is essential to keep the distance betweenbell-edge and measuring volume as small as possible toreduce evaporation effects, but it must be ensured that

the primary and secondary atomization is completed(Fig. 5).

Determination of painting-specific data

The paint application was executed by a painting robotmoving the TCP with 180mm s�1 at a distance of200mm parallel to the plate, which was placed hori-zontal in the center of a table (see Fig. 6). After thepaint cured, the film thickness was measured by amagneto-inductive device perpendicular to the direc-tion of painting, from which the so-called dynamicspray pattern could be obtained. At the same time, thedischarge current at the background table was mea-sured, from which the charge per time was calculatedin the steady state.

Numerical methods

This section describes the scene, setup, and settings ofthe numerical simulations.

Domain, grid and boundary settings

The numerical simulations were carried out using thecommercial computational fluid dynamics codeANSYSFluent. For this study, the ERBS was posi-tioned in the center of a cylindric domain(radius ¼ 1:1m, height ¼ 0:8m) above the plate witha radius of 0:5m. The block-structured grid of the fluiddomain contained about 17.36 million cells having amean first prism layer height of 30 lm around the bell-cup and 150 lm on the top of the plate. A velocity-inletof 0:3ms�1 was applied to the top boundary of the fluiddomain to map the downdraft velocity of the paintbooth air, whereby the sides and the bottom of thedomain were set as pressure-outlets. The measuredvolumetric flow rates of the inner and outer shaping airwere calculated into a mass flow rate using theInternational Standard Atmosphere ISO2533 and were

Fig. 4: Measurement setup to determine airflow velocitiesof shaping air by means of LDA

Fig. 5: Measurement setup to determine particle sizedistribution by means of laser diffraction

Fig. 6: Measurement setup to determine film thicknessdistribution—dynamic spray pattern

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then applied as massflow-inlets on the top of shapingair nozzles. The rotation of the bell-cup was taken intoaccount by setting the angular velocity on the walls ofthe bell-cup (Figs. 7 and 8).

Turbulence model and solver settings

In previous studies on ERBS, various 2-equationturbulence models, as well as Reynolds-Stress models,were used. In order to investigate the capability of theturbulence models for the strongly swirled airflow, astudy using different modeling approaches was com-pleted. The airflow results of the k-x SST model arecompared to the results of a Linear Pressure-StrainReynolds Stress Model (RSM), and ANSYSFluentsImproved Delayed Detached Eddy Simulation(IDDES) using the k-x SST as a sub-model. Asecond-order upwind was used as a discretizationscheme, and the SIMPLEC algorithm with a fixedtime step size of Dt ¼ 1� 10�4s was chosen as thesolver. With this time step size, a maximum CFLnumber of approximately 100 close to the shaping airnozzles was achieved on the generated grid. As theflow at the shaping air nozzles was almost linear, theerror was not considered to be significant. However,the time step size could not be further reduced due tothe high computational effort.

Discrete phase model: motion of particles

The motion of paint droplets were represented by inertand spherical particles computed through ANSYSFlu-ents discrete-phase-model (DPM) in an Euler–La-

grange framework. To capture the effects of theinteraction of the particles with the flow and theelectrostatic field, a two-way coupling was used andimplemented, respectively.10,11,14 A random walkingmodel was used to capture the turbulence effects onthe trajectories of the particles. As a boundary condi-tion, the particles were reflected on the bell-cup,trapped on the top of the plate and escaped on allother walls, inlets and outlets. A droplet depositionmodel was not used in this study because the effect onfilm thickness seemed very unlikely. The modeling ofinitial droplet conditions of the discrete phase areexplained in detail in ‘‘Modeling initial droplet condi-tions’’ section.

Simulation of the airflow field

Since the airflow has a decisive influence on theresulting spray pattern, its accurate simulation is anessential basis for paint application simulations. Thesimulation must, therefore, be able to map the typicalflow characteristics, such as vortex with backflow,distinct stagnation point on the object, and continuousair entrainment into the main flow field. Figure 9 showsthe airflow field without particles, which was simulatedusing the k-x SST turbulence model. The mean valueswere calculated over 0:5 s flow time. Qualitatively, theresult shows all flow features of an ERBS. In partic-ular, the backflow in the center of the flow field ischaracteristic of a strongly swirled shaping airflow fieldof ERBSs.15 However, it is well known that 2-equationturbulence models have limitations in predictingswirled impinging flows. In order to evaluate theaccuracy and applicability, the results of the k-x SSTare quantitatively compared with the simulation resultsof the RSM and the IDDES as well as with experi-mental LDA data (see Fig. 10). Figure 10 shows theaxial, tangential and radial velocity components at adistance of d ¼ 10mm and d ¼ 180mm. In general, allturbulence models show an acceptable agreement with

velocity-inlet

pressure-outlet

Fig. 7: Sketch of the fluid domain including the ERBS andthe plate to be painted

Fig. 8: Block-structured grid on a cross section throughthe center of the fluid domain

Mean velocity magnitude (m/s)1.0e+00 1.0e+0250501052

Fig. 9: Mean velocity magnitude of the shaping airflow fieldwithout droplets simulated using the k-x SST turbulencemodel

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the experimental data. However, each turbulencemodel has strengths and weaknesses in different flowregions and velocity components. Thus, the k-x SSTturbulence model shows significantly higher axial andtangential velocity than the IDDES near the bell-cup.Close to the plate, all simulation results show substan-tially lower axial velocity with good agreement in theradial velocity component. If the axial to radial velocityratio in the experiment and the simulation are too farapart, the particles are inevitably transported outwardsshortly before the plate, which widens the spraypattern. It can also be seen that the tangential velocityof RSM is overestimated in the center. As a result, theswirl strength of the flow increases and will be forcedapart, and thus, higher radial velocities are generated,as it can be seen in the results near the plates.

Modeling initial droplet conditions

In this section, we propose an injection model for theinitial droplet conditions—position, velocity, size, andcharge—for an ERBS.

Injection model: droplet position

This ERBS has a notably formed bell-edge, which isoften called a thick edge. Part of the paint climbs upthis thick edge so that the atomization process takesplace in a spread range of the axial position. Thisspread is visualized by a high-speed image close to the

bell-cup, see Fig. 11a. It is assumed that all fluidelements (droplets and/or ligaments) will pass theblack plane shown in Fig. 11b, and this was alsovalidated in the primary breakup simulation in the nearbell region [see reference (9)]. The radial offsetposition roff is therefore defined at a fixed value ofroff ¼ 0:75mm from the bell-edge (end of flat part). Inthis case, 180 injection points that were uniformlydistributed around the bell-cup were used. The entireparticle size distribution is injected from each point perDPM iteration. To enhance the influence of theposition on the result, the injection position wasdeliberately fixed for all particle sizes. Generally, theinjection points should be distributed randomly insmall areas to avoid overloading of the computationalcells in the injection area and thus to achieve fasterconvergence.

0 10 20 30 40 50 60

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02468

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alve

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/s]

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experimentalk-ω SSTRSMIDDES

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experimentalk-ω SSTRSMIDDES

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Fig. 10: Comparison of simulation results using three different turbulence models with experimental LDA data. Theexperimental graph shows the mean and 95% confidence. The velocity components axial (left column), tangential (midcolumn), and radial (right column) are compared at d ¼ 10mm (upper row) and d ¼ 180mm (lower row)

(a) Shadow image of disintegra-tion process

(b) Injection positions visual-ized by a black line; colorbarshows airflow velocity magnitude

Velocity magnitude (m/s)1.0e+01 1.0e+0250

Fig. 11: Visualization of the disintegration process at thebell-edge (a) and the injection position in the simulation (b)

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Injection model: droplet velocity

In paint application processes, the disintegration modeof the paint by ligament formation is preferred todirect droplet formation and sheet atomization, due tothe resulting narrow, stable, and fine particle sizedistribution. Typically, a spiral formation of the liga-ments at the bell-edge occurs visually (see Fig. 12 ingreen). The ligament formation begins at the bell-edge,where a fluid element with a velocity and direction ofthe vector v0;res from the bell-cup surface. In thefollowing time steps (visualized by dots), the bell-cuprotates by an angle u. The straight-line direction of thefluid element and the angular velocity of the bell-cupcreate a spiral-shaped ligament. The mathematicaldescription of these ligaments, assuming that no forcesact on a massless fluid element, is done by a simplegeometric model. Due to the paint viscosity and theassociated internal forces in the ligament, the directionof trajectory in the geometric model is bent. The anglea in the injection point thus increases with increasinginternal forces and shifts the ratio of radial to tangen-tial velocity components in the direction of thetangential velocity.

The initial vectors at the bell-edge are defined by therotational speed and also by the film flow velocity onthe bell. The tangential film flow velocity v0;tan of thepaint film is calculated assuming no-slip conditions onthe surface of the bell-cup and homogeneous velocitydistribution within the film. Reference (2) derived aformula for a simple cone to calculate the radial filmflow velocity on the bell-edge. Assuming a Newtonianfluid and neglecting the circumferential and sinkingspeed, the radial velocity v0;rad will be obtained by

v0;rad ¼ 9qx2sin bQ2

32p2gsrbell

� �1=3

ð1Þ

where Q denotes the quantity of paint delivery, x theangular bell-cup velocity, rbell the radius of the bell-cup, b the cone angle at the bell-edge, q the wet densityof the paint, and gs the dynamic viscosity of the paint.Usually, paints are shear thinning and thixotropicliquids so that the viscosity is a function of the shear

rate. Typically, automotive paint materials reach aviscosity plateau at approximately 20mPa s at veryhigh shear rates16 measured through a capillary vis-cosimeter.

As a result of the energy loss during atomization, theligament shape for calculating the acting forces, andthe point of droplet formation, a slipfactor is intro-duced, which was applied to the vector v0;res. Thederivation of the slipfactor and angle a from analyticalequations as well as material and process conditions isnot trivial and is currently an unsolved problem. Forthis reason, the determination of the slipfactor and awas included in the optimization process.

Injection model: droplet size distribution

Although there is some numerical study of primarybreakup in the near bell region,9 it is still quite difficultto simulate a fully developed droplet size distributionfor the spray painting application, since very fine cellsizes have to be used. Instead of using the costlyVolume-of-Fluid to DPM approach, a measured par-ticle size distribution will be used. For this paintmaterial, equipment, and process setting, the measuredparticle size distribution shows good agreement to theshape of a logarithmic normal distribution (seeFig. 13). The characteristic distribution values d32(Sauter mean diameter) and dvð50Þ (volumetric med-ian) are in a typical range of automotive paint sprays.

Injection: droplet charge

Through the contact of paint on the inner bell-cup side,its surface is getting charged. After the atomizationprocess and the resulting surface enlargement, thecharge is distributed over the paint droplets, wherecurrently two models are dominating in spray paintingsimulations. The first model assumes that the charge isdistributed proportionally to r2 of the droplet, whereasthe second model assumes that the charge is dis-

tributed proportionally toffiffiffiffir3

p. The latter is based on

the equation of the Rayleigh-limit, where the maxi-

(a) Ligament formation (b) Geometric modelx

rbellrbell

v0,res v0,resv0,radv0,tan

vrad

vtanroff

vres

xω

ϕ

α

ϕ

ω

yy

Fig. 12: Sketch of the geometrical velocity modelassuming ligament formation

0.1 1 10 100 10000.00

0.02

0.04

0.06

0.08

0.10

0.12

Vol

ume

frac

tion

Particle diameter [µm]

d32 = 13.73 µmdv(10) = 18.10 µmdv(50) = 18.53 µmdv(90) = 43.37 µm

Fig. 13: Droplet size distribution

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mum charge on a droplet is calculated before itbecomes unstable and disintegrates into smallerdroplets.

qray ¼ 8pffiffiffiffiffiffiffiffiffiffie0cr3

pð2Þ

In this equation, e0 denotes the permittivity of thevacuum, c the surface tension of the paint and r thedroplet radius. The charge of the droplets in theinjection model was calculated according to theequation of Rayleigh-limit. Since the surface chargeis only a small percentage of the Rayleigh-limit, theRayleigh-limit coefficient Coeffray will be introducedand applied on the equation of Rayleigh-limit. TheRayleigh-limit coefficient is assumed in this model tobe constant over all particle sizes. The dependence onthe radius defines the total charge on the drop.

qsim ¼ Coeffrayqray ð3Þ

From the experimentally measured discharge current,the Rayleigh-limit can be calculated at given particlesize distribution assuming 100% paint transfer effi-ciency. In this way, the experimentally determinedCoeffray of 1:1% was calculated from the measured

transferred charge per second of 11:37� 10�6 C s�1.Since the transfer efficiency in the experiment forERBS using direct charging mechanism lies in therange of 80% to 95%, it is meaningful to increaseCoeffray. To obtain a rough estimation of Coeffrayrange, the experimentally calculated value(Coeffray ¼ 1:1%) should be increased by 20% to40% so that in this study the range between Coeffray ¼1:32% and Coeffray ¼ 1:54% is assumed to be thetarget range. Due to the uncertainty in the oversprayestimation and the amount of charge it contains, theCoeffray is included in the optimization process.

Metamodel-based optimization of injection modelcoefficients

In this step of the framework, a metamodel-basedoptimization strategy will be used to efficiently deter-mine the injection model coefficients.

Design of experiments: sampling points

The framework starts with the definition of a feasiblespace of the injection model parameters, whereby thisshould cover a wide range of the parameters. In thisdesign space, the axial injection position was based onthe area of atomization mentioned in ‘‘Injectionmodel: droplet position’’ section. The axial velocity isalso defined for negative values since the thick bell-edge may lead to an upward flow of the paint film. Themaximum value for the axial velocity is defined as

30ms�1, which represents an overestimation due to thesmall airflow velocity in that area (see Fig. 11b). Sincethe internal forces in the ligament and the energy lossduring atomization are not known, the slipfactor andthe angle a are investigated over an extensive range.Table 2 shows the design space by minimum andmaximum values of the parameters as well as constantvalues. Typically, Latin-Hypercube sampling methodsare used for computer experiments to train regressionmodels. Particularly for practical applications, the»maximin« method is often used, which maximizesthe minimum distance between sampling points andtherefore avoids too narrow points. This method wasalso chosen to create 150 sample points for this study.

Conduction and evaluation of simulations

The simulations were performed on the cluster of theHigh Performance Computing Center Stuttgart(HLRS). To reduce the total duration time of thesimulations, a task-based parallelization is stronglyrecommended. In this series, ten simulations wereperformed simultaneously. In each simulation, apresimulated mean airflow field from the turbulencestudy is loaded at the beginning. Particles are theninjected over 0:75 s, whereby the first 0:25 s are nottaken into account in the evaluation, as the spray hasnot reached a steady state. With very fine computa-tional grids on the objects to be painted, the so-calledbinning effects often occur in the calculation of the filmthickness. This is caused by large particle volumedeposits in a very small computing cell. For paintingsimulations, it is recommended to decouple the com-putational cells of the airflow from cells of the filmthickness. Therefore, particles deposited on the plateare sampled calculated into the film thickness in a post-processing step. The dynamic spray pattern can becalculated on the one hand directly from the pointcloud of the sampled data via a kernel densityestimation (KDE) or on the other hand via an initiallycalculated static spray pattern (see Fig. 14) using a newgenerated structured.

In the simplest case, the conversion from static todynamic spray pattern can be done by integrating thefilm thickness along a coordinate axis. Since binningeffects create noise within the calculated spray pattern,

Table 2: Feasible space of injection model parameters

Parameter Description Min Max Unit

Axial position AX_POS �0:5 0.5 mmAxial velocity AX_VEL �5 30 ms�1

Slipfactor SLIPFACTOR 25 95 %Angle a ANGLE 76.62 85 �

Coeffray COEFF_RAY 0.5 1.5 %Constants Value UnitRadial Position 0.75 mm

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it is recommended to average the results by rotatingthe axis around the center of the static spray pattern.An inverse Radon transformation can efficiently cal-culate this rotatory averaging over 360 � around thepattern center, which is why this method is preferred tothe costly KDE. Figure 15 shows the result of the threemethods discussed for calculating a dynamic spraypattern from a static (nonmoving atomizer) simulation.

It can be seen that the inverse Radon transformationis almost identical to the result of the integral method,but due to the 360 � rotation, the curve is muchsmoother and more symmetrical. Because of the rota-tion, this method is suitable only for static andhorizontal simulation scenes using a flat plate. TheKDE also generates a continuous spray pattern curve,but the maximum differs significantly from the othertwo methods. This deviation can be attributed to theknown problem of choosing the kernel and thebandwidth of the estimator. Nevertheless, KDE meth-ods are helpful tools to calculate the film thickness oncomplex 3D parts, as it is shown in reference (17).

In addition to the important spray pattern, furtherpainting-specific parameters are calculated and evalu-ated from the simulations and partly included in theoptimization. These values are briefly explained in thefollowing enumeration:

• RESIDUAL: Sum of least squares between simu-lated and experimental film thickness distribution

• PW50: Pattern width 50, the full width at halfmaximum of the film thickness distribution

• SUM_CHARGE: Sum of the charge from thedroplets deposited on the plate

• TE_WEIGHT: Mass-based transfer efficiency, ratiobetween deposited to injected mass of paint

Principle component analysis

From the results of the statistical design of experi-ments, the correlations of the input parameters to theobjective values were calculated using a Spearman’scorrelation, which is a nonparametric measure of rank-order correlation. The measure describes the strengthand direction of a monotonic association between twovariables. Values close to � 1 (perfect negative corre-lation) and 1 (perfect positive correlation) represent astrong monotonic correlation, whereas values around 0represent a very weak monotonic relationship betweentwo variables.

Figure 16 shows that the correlation between axialposition and painting specific values is very weak and,therefore, does not represent a principal component.The same applies to the angle a. The correlationbetween the COEFF_RAY and the transferred chargeand the transfer efficiency is particularly high as well ashighly linear. The well-known expansion of the spraywith increasing charge is also found in the correlationcoefficient between COEFF_RAY and the PW50 witha moderate value of 0.43. In the following optimizationsteps, only the principal model parameters AX_VEL,SLIPFACTOR, and COEFF_RAY are considered.

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sitio

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)

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–200 –100 0 100 2000

102030405060708090

Film

thic

knes

s[µm

]

Position [mm]

IntegralInverse Radon transformationKernel density estimation

Fig. 15: Comparison of methods to calculate dynamicspray pattern

– 0.099 – 0.26 – 0.38 0.049 – 0.35

0.01 0.022 0.1 0.02 1

0.17 0.36 0.043 – 0.016 0.43

0.044 – 0.071 – 0.34 0.037 0.87

RESIDUAL

SUM_CHARGE

PW50

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0.25

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–0.50

TE_WEIGHT

AX_POS AX_VEL SLIPFACTOR ANGLE COEFF_RAY

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–1.00

Fig. 16: Correlation matrix using Spearman’s correlationcoefficient

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Training of metamodels

Global multiobjective optimizations often requiremany function evaluations to calculate a Pareto front.Since each function evaluation means a paintingsimulation, two metamodels were created first. Meta-models or surrogate models are compact, mathematicalapproximations that map the multivariate input/outputbehavior of complex systems. In this study, a meta-model was created for two objectives—RESIDUALand residual of transferred charge (RESIDUAL_CHARGE), which is the quadratic error betweensimulated and measured value. For the RESIDUAL ofthe spray pattern, a three-parameter Kriging modelfrom the »SMT—Surrogate Modeling Toolbox« Py-thon toolbox18 was used.

In common machine learning practice, the quality ofthe prediction for regression models is determined bythe coefficient of determination R2. Because of the fewavailable sample points, it is recommended to deter-mine a mean R2 via the »leave-one-out cross-valida-tion« method. In this method, the metamodel wastrained with n-1 sample points, and R2 is evaluated atthe omitted location. Afterward, a mean R2 value wascomputed from all sample point evaluations. Thelearning curve of the metamodel can thus be moni-tored for the current number of sample points, seeFig. 17. This three-parameter metamodel learns par-ticularly active within the first 60 sample points. At 100sample points, the metamodel reaches a constant levelabove a R2[0:9, which allows acceptable predictionsto be made for this application. To accelerate perfor-mance and to reduce computational effort, the learningprocess should be terminated after reaching R2[0:9for ten consecutive training sets.

Figure 18 visualizes the influence of the injectionmodel parameters on the RESIDUAL of the spraypattern. The visualization was done by evaluating theKriging predictions on a uniform grid in each dimen-sion.

The evaluation of the parameter impacting on thespray pattern result shows a clear trend toward highslipfactor, a Rayleigh-limit coefficient of approxi-

mately 1:4%, and a weak trend toward low axialvelocities. These results also follow the expectedtheoretical properties of injection in the velocityreduced zone of the thick bell-edge.

A simple 1D interpolation is a sufficient metamodelfor the second objective function because only thecharge on the droplet acts as a principal component.The transferred charge on the plate scales linearly withthe charge on the droplets, so the residual is also aquadratic function. The residual between simulatedand experimental values reaches a minimum at aRayleigh limit of 1:39% (Fig. 19).

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Fig. 17: Learning curve of the metamodel scored by R2

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Fig. 18: Pairplots of input parameters of the Kriging modeltrained with the simulation data from the RSM turbulencemodel. The colorbar visualizes the values of the RESIDUAL

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Multiobjective optimization

The Pareto optimization of the principal componentsin the injection model is based on the minimization oftwo objective functions and previously trained meta-models. The first objective function f1 is to minimizethe RESIDUAL, and the second objective function f2is to minimize the least square between measured andsimulated charge per time (discharge current) on theplate. It is possible to define further objective func-tions, e.g., based on LDA velocity data, which couldlead to more precise solutions. However, for anefficient application, it is advisable to limit the objec-tive functions to existing or easily generated charac-teristics of the painting process.

The multiobjective optimization was performedusing the Nondominated Sorting Genetic AlgorithmIII (NSGA-III) from the Python toolbox »pymoo«.19

Details on the implementation of the NSGA-IIIalgorithm into the »pymoo« toolbox can be found inreference (20). The various parameters of the NSGA-III algorithm are listed in Table 3 with the descriptionsspecified in the toolbox.

Minimizing the two objective functions f1 and f2leads to a Pareto front and a single point of Pareto-optimal solution. Due to the different scaling for f1 andf2, it is recommended to use a pseudo-weighted23

decision to determine the Pareto-optimal point. In this

study, the optimal initial droplet conditions werecomputed for both the RSM and the k-x SST turbu-lence model (see Table 4).

The optimized initial droplet conditions were vali-dated by one additional simulation to evaluate theerror. In both cases, the deviation of the simulated tothe experimental charge per time is less than 2%. Thesimulated spray pattern also shows an acceptable fit tothe experimentally determined result (see Fig. 20).

In particular, the axial velocity for RSM and k-xSST differs significantly. These discrepancies relate tothe results of the turbulence study, where in bothturbulence models the axial velocity near the plate wasclearly underestimated. However, the k-x SST simu-lations have a good agreement in the contour of thecurve and the position of the peaks, as well as in thetangential and radial velocity. Consequently, the opti-mization process tries to compensate for the error ofthe underestimated axial velocity. For the RSM, theunderestimated axial velocity applies as well, but theentire shaping airflow field is slightly narrower than thek-x SST simulations so that a too narrow shapingairflow field compensates the missing axial impulse.Furthermore, the high initial axial velocity in the k-xSST case leads to a lower Rayleigh-limit because thehigh initial impulse forces larger particles to bedeposited on the plate. For this particular ERBS witha thick bell-edge, it is unlikely that such a high axialvelocity is present for the initial drop conditions and is,therefore, an indicator for incorrect air flow prediction.However, the optimization process itself can efficiently

0.004 0.006 0.008 0.010 0.012 0.014 0.016

0

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idua

ltra

nsfe

rred

curr

ent

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Training dataInterpolation

Fig. 19: 1D metamodel for the residual of transferredcharge to the plate built using a quadratic interpolationmethod

Table 3: Parameter of NSGA-III

Parameter Method Value

Population size – 100Sampling method Latin-Hypercube –Reference points Das-Dennis21 n_partitions = 25Crossover operation Point crossover22 n_points = 2Mutation operation Polynomial mutation22 eta = 30, prob = 0.9Stopping criteria Maximum generations 1000Decision making Pseudo-weights23 f1 = 0.5, f2 = 0.5

Table 4: Optimized injection model coefficients for RSMand k-x SST turbulence model cases using themetamodel-based optimization framework

Parameter RSM k-x SST Unit

AX_POS 0 (fixed) 0 (fixed) mmAX_VEL 3.5 18.6 ms�1

SLIPFACTOR 88.0 86.8 %ANGLE 76.62 (fixed) 76.62 (fixed) �

COEFF_RAY 1.45 1.31 %

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determine the Pareto-optimum. At this point, thephysically meaningful range of initial droplet condi-tions for different ERBS types is not established. Itmust be defined by further experimental investigationsnear the bell-edge as well as by the support ofadditional sub-models in the injection modeling.

Discussion

The framework presented here to determine the initialdroplet conditions for ERBS simulations has beenrealized using a simple injection model, experimentaldata, and a metamodel-based optimization strategy.Within the framework, uncertainties occur both in thedetermination of the experimental data and in thepaint application simulation.

A statistically relevant mean value should be formedto reduce the experimental uncertainties. Experiencehas shown that variations of up to 15% can occur,especially for spray pattern and measurement ofdischarge current on the plate. However, for economicand plant availability reasons, a sufficient number ofexperiments cannot always be guaranteed, so thatthere is a risk of calibrating to a spray pattern at theboundary of the variation.

The particle size determination could alternativelybe measured using a Phase-Doppler Anemometer(PDA) by systematically evaluating the spray pointby point and converting the obtained number distri-bution into a volume distribution under considerationof droplet velocities. The disadvantage of the PDAmethod is that the measurements with opaque paintmaterials do not work, and are, respectively, veryerror-prone. Besides, under economical aspects forindustrial applications, the costs for PDA measure-ments are many times more expensive. Therefore,measuring methods with laser diffraction are recom-mended to measure the particle size distribution as aninput condition for painting simulations.

The proposed injection model is a simple first-orderapproximation of the highly complex processes duringprimary atomization at the bell-edge. With the help of

the principal component analysis and metamodels, it ispossible to adapt further and optimize the injectionmodels. In particular, the initial velocity independentfrom particle sizes is currently a significant weak point.It is known that the particle relaxation times near thebell for particle sizes larger than 100 lm are so highthat they are not distracted much and retain theirinitial direction. Particle sizes smaller than 5 lm, on theother hand, are almost strongly coupled to the flow.The range of these initial velocities as a function ofparticle size is to be included in further investigationsin the form of a crossflow submodel. In order todetermine the range of velocities, complex experimentswith optical measuring methods must be carried out inorder to investigate the correlation between particlesize and particle velocity. Furthermore, the results ofthe primary break-up volume-of-fluid simulations atthe bell-edge of references (8) and (9) will be inves-tigated and used to enhance and validate the injectionmodels for ERBS.

The most significant impact on the simulated resultis the correct prediction of the airflow. Modern ERBStend to utilize swirled shaping air to increase its rangeof application. Particularly in the simulation with lineareddy-viscosity models, overproduction of turbulence atimpinging flows is known, despite using a productionlimiter. In all turbulence models, the axial velocity wassignificantly underestimated. For simulations in anindustrial environment with commercial simulationsoftware, turbulence models such as the k-x SST willcontinue to be used in the future. Since other preim-plemented models also fail, further investigations andadjustments of the models have to be carried out inorder to achieve high accuracy of the film thicknessprediction.

It must be assumed that further changes in themodel constants, e.g., turbulence or particle models,will also lead to changes in the initial particle condi-tions. In particular, the uncertainty of solvent evapo-ration from the droplets, which was not taken intoaccount in the modeling, exists in this work. Particu-larly for solvent-based paints with low solids content, adrastic influence could be caused. The implementationof evaporation models is very complicated in the fieldof paint materials because interactions of complexsolvent mixtures have to be modeled, which createsfurther hurdles in the generation of input data.However, water-based solvents could be consideredusing existing evaporation laws. A significant influenceof droplet impingement models as well as dropletcollision models film thickness distribution is currentlyestimated to be very low and is therefore not consid-ered in these painting simulations.

The metamodel-based optimization is an efficientmethod to perform a multicriteria black-box optimiza-tion with moderate computational effort. The blackbox contains both the experimental input data and thepainting simulations themselves. The previously dis-cussed errors are wholly included. If the errors, such asthe airflow field, are substantial, the optimizer will find

–0.3 –0.2 –0.1 0.0 0.1 0.2 0.30

10

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60Fi

lmth

ickn

ess[

µm]

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Experimentalk-ω SSTRSM

Fig. 20: Comparison of experimental and simulated spraypattern using the Pareto-optimal initial droplet conditions

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solutions in the boundary area of the defined designspace. Currently, this space is not completely describedand limited by experiments, models, and assumptions,which will be investigated in further research anddevelopment.

From a holistic point of view, good agreement withthe objectives could be achieved through this frame-work while painting complex parts, the alignment ofthe atomizer to the part surface changes, and as aresult, the flow field is changed. Thus, the question ofthe validity range of the optimized parameters remainsopen. Further investigations with different distancesand angles of the atomizer to the surface shouldprovide clarity. However, the requirements for thecorrectness of the simulated airflow are also appliedhere.

Conclusions

In this paper, we have proposed a framework tosystematically address the practical problem of initialdrop conditions at ERBS for painting simulations. Theframework is a combination of experimental inputdata, an injection model, and a metamodel-basedoptimization. First, the determination of the experi-mental data was discussed and its utilization in theframework graphically outlined. Second, the numericalcases and models were explained. A turbulence studywith three turbulence models was performed andcompared with experimental LDA data. A strongunderestimation of the axial velocity component in allthree turbulence models was clearly shown. Thetangential and radial velocity, on the other hand, weresimulated with reasonable accuracy. Furthermore, aninjection model used to determine the initial dropconditions—position, particle velocity, particle size,particle charge—was introduced. A metamodel-basedoptimization then adjusted the model parameters. Theresults of the optimization show a good agreement withthe defined painting-specific objectives. However, theoptimization revealed weak points of the underesti-mated axial air velocity by setting high initial axialvelocities for the droplets to compensate for themissing impulse. Further investigations will focus onthe particle size-dependent initial velocity in theinjection model and supporting the model develop-ment and validation by experimental investigations.

Acknowledgments Open Access funding provided byProjekt DEAL. This research was supported within theproject »SelfPaint« by Fraunhofer-Gesellschaft zurFoerderung der angewandten Forschung e.V. Theauthors acknowledge for the support of the HighPerformance Computing Center Stuttgart HLRS.

Open Access This article is licensed under a CreativeCommons Attribution 4.0 International License, whichpermits use, sharing, adaptation, distribution and

reproduction in any medium or format, as long as yougive appropriate credit to the original author(s) andthe source, provide a link to the Creative Commonslicence, and indicate if changes were made. The imagesor other third party material in this article are includedin the article’s Creative Commons licence, unlessindicated otherwise in a credit line to the material. Ifmaterial is not included in the article’s CreativeCommons licence and your intended use is not per-mitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly fromthe copyright holder. To view a copy of this licence,visit http://creativecommons.org/licenses/by/4.0/.

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