CFD simulation of paint deposition in an air spray process.

This work analyzes the mechanism of spray deposition by means of computational fluid dynamics (CFD) in order to reproduce virtually the spraying of a paint gun adopted for use in the automotive industry and to predict paint drop trajectories and film builds on the target surface. The prediction of the flow of the continuous phase was obtained by solving the time averaged Navier-Stokes equations in connection with suitable closure models for turbulence (RNG and Realizable k-[epsilon]). The dispersed phase was treated by a Lagrangian approach, by tracking numerically a large number of representative particles from the gun exit to the target surface. The initial conditions for the droplets were estimated from a detailed simulation of the paint jet at the exit of the nozzle. In this way one could evaluate positions and velocities of droplets at impact and estimate the properties of the deposited layer of paint. The method was validated by comparison with experimental data obtained by phase doppler anemometry and, subsequently, the approach was applied to different geometries and operating conditions.

Keywords: Spray application, solvent-based, process modeling, simulation, computational fluid dynamics

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Spray deposition processes are used for many industrial applications, mostly in the area of surface coating, but also for the manufacturing of new materials with peculiar properties. One of the reasons for their widespread use is the ability to provide a finish with very fine microstructure that results in remarkable improvement of protective and aesthetic properties.

In order to obtain the appropriate characteristics of the coats, the application process of the liquid paint must be well-controlled and reproducible. As illustrated in this work, significant help can be provided by computational fluid dynamics (CFD), which can offer a detailed view of the operation and show the effect of different operating conditions by simulating the flow field generated by the spraying device. CFD can provide significant insight into the painting process and has the ability to show how changes in operating conditions, applicator type, or workpiece geometry may affect performance. A number of advances can be expected from the better understanding of the basic processes of spray painting provided by the computational methods, including: environmental issues, with improvement of paint transfer efficiency and reduction of the paint that escapes to the environment; quality issues, with more uniform coating deposition and easier identification of the reasons for maldistribution of paint; and safety issues, by assessing workers' exposure to paint as a function of work practices and local ventilation.

This work will show an example of applying CFD to the simulation of an air spray painting process and to the prediction of film builds on simple surfaces. The results reveal the type of information provided by such an analysis. The approach is quite general and applicable to other conventional paint spray systems.

Few authors have tried to measure and model paint sprays originated from pneumatic atomizers so far. Kwok (1) made measurements of the air flow near the nozzle (without paint flow) using hot wire and Pitot tube methods. Domnick et al. (2) measured details of the spray structure at 10-cm distance below the spray nozzle by using phase doppler anemometry. Similar measurements were performed by Morikita and Taylor (3) using shadow doppler velocimetry at 30-cm distance from the gun nozzle. Some papers about the modeling of this process can be found in the field of occupational hygiene: they are finalized to the estimation of the amount of contaminant inhaled by workers during spray painting. In most cases these works were based on semiempirical models of the spray process, (4-6) but recently the CFD approach was extended to this field, too. (7)

A typical problem of the CFD approach in the simulation of sprays is that a general model for the prediction of primary atomization does not yet exist. In order to circumvent this difficulty, the atomization zone is not usually considered in the simulation, and initial droplet characteristics, such as velocity and size distribution, have to be provided in a region sufficiently far from the nozzle, where the break-up of the liquid jet exiting the nozzle can be regarded as completed. Since the conditions for air and droplets in this region are normally obtained by experimental measurements on unconfined sprays, (8) the region must be far from the impacting surface too, so that the distortion of the fluid stream created by the wall is negligible.

[FIGURE 1 OMITTED]

The application of this "traditional" spray modeling approach to paint deposition is summarized in the work by Hicks and Senser, (9) who considered the operation of a gun at 25 cm from the target wall. Initial conditions for the dispersed phase were prescribed at 14 cm from the gun cap. Clearly, this method is not feasible when the distance between gun and wall is small. Furthermore, its implementation requires detailed experimental data for the simulated configuration.

[FIGURE 2 OMITTED]

Ye et al. (10) proposed a different set of initial conditions for the dispersed phase. They demonstrated that the complete air flow field between nozzle and target surface could be calculated by applying air inlet conditions directly at the atomizer and that the initial droplet conditions necessary for the simulation could be prescribed very close to the nozzle. In their approach the interactions between air and droplets were already considered immediately after the liquid exit. This is an important point for spray simulation since these interactions can modify significantly the spray cone shape.

The aim of our investigation is to model a spray painting process by means of CFD, using simplified inlet boundary conditions for both air and liquid and introducing the droplets as close as possible to the nozzle; a general procedure for spray modeling is given, finalized to determine thickness and morphology of the deposited paint.

EXPERIMENTAL SET-UP

The spray originated from a DeVilbiss "Compact Transtech" gun analyzed experimentally by means of phase doppler anemometry. Water was used as the liquid and air as the carrier gas. A pressurized tank at 1.5 bar gauge supplied the water, while air was provided by the internal line of the laboratory at 1.2 bar gauge. The gun was regulated in order to give the maximum spray spread. In these conditions the water volumetric flow rate was 180 [cm.sup.3]/min.

The studied gun is shown in Figure 1. It has a coaxial jet arrangement, as depicted in Figure 2, composed of a central nozzle 2 mm in diameter for the liquid jet, which is surrounded by an annular ring for the primary air. This air accelerates the liquid jet in the axial direction and breaks it into tiny droplets. Around this zone there are six additional holes for air with a diameter of about 0.5 mm to prevent paint contamination of the air cap (cleaning air). Two horns are present at each side of the cap with two holes each, with diameters of 1 mm and 0.5 mm, respectively. These are the so-called shaping nozzles: the air exiting from these holes flattens the spray along the Y direction and gives droplets a component of velocity perpendicular to the centerline of the spray, in order to further contribute to drop breakage. In the studied conditions the spray assumed a flat fan shape and presented an elliptical section, which was approximately 30 cm long (in the Y direction) and 8 cm large (in the Z direction) at 20 cm from the nozzle. More details on the configuration and the operation of a paint gun can be found in the paper by Micheli. (11)

[FIGURE 3 OMITTED]

The continuous phase plays a fundamental role in spray paint applications. Coatings of good quality require small droplets and high impact velocities to maximize drop spreading. For this reason the continuous phase is ejected with the highest possible velocity. Indeed, sonic conditions are often achieved in pneumatic atomizers.

The air deviation in the proximity of the wall should also be considered. Part of the droplets deviates with the air and does not reach the wall. As a consequence, a certain amount of paint is wasted. The term overspray usually indicates this phenomenon, which, obviously, is more intense at higher gas velocity.

MODELING OF THE SPRAY BEHAVIOR

The prediction of the flow of the continuous phase was obtained by solving the time-averaged Navier-Stokes equation in connection with a suitable closure model for turbulence. (12)

The most employed model of this kind is the so-called k-[epsilon], proposed by Launder and Spalding. (13) It assumes that the Reynolds stresses are proportional to the mean velocity gradients through an eddy viscosity depending on the turbulent kinetic energy k and on the turbulent dissipation rate of k, [epsilon]. Many refinements of the k-[epsilon] model have been proposed in the literature, including the RNG (14) and the Realizable (15) models. Both show substantial improvements over the standard k-[epsilon] if the flow features include strong streamline curvature, vortices, or rotation. Since the models are still relatively new, it is not clear in exactly which instances the Realizable k-[epsilon] model is superior to the RNG and vice versa. In the present work these two models were applied to the simulation of the spray and compared to experiments.

The dispersed phase was treated by the Lagrangian approach, where a large number of droplet parcels, representing a number of real droplets with the same properties, were traced through the flow field. By representing droplets by parcels, one can consider size distribution and simulate the measured liquid mass flow rate at the injection locations by a reasonable number of computational droplets. The trajectory of each droplet parcel was calculated by solving the equation of motion for a single droplet. (16)

The equation of motion is the result of the force balance on the particle written in a Lagrangian reference frame. A basic equation of motion can be written neglecting the forces due to virtual mass and history terms that are usually small in spray deposition processes. This equation includes drag, gravity, and buoyancy force, and has the following form, for the component along the i-th direction:

[[rho].sub.p][v.sub.p][[d[U.sub.p,i]]/[dt]] = [C.sub.D][[[rho][a.sub.p]]/2]|[U.sub.i] - [U.sub.p,i]|([U.sub.i] - [U.sub.p,i])+([[rho].sub.p] - [rho])[g.sub.i][v.sub.p] (1)

Here, [v.sub.p] and [a.sub.p] are particle volume and cross-sectional area, respectively: [v.sub.p] = [pi][D.sub.p.sup.3]/6, [a.sub.p] = [pi][D.sub.p.sup.2]/4.

The drag coefficient [C.sub.D] was evaluated from the following equation, valid for spherical droplets:

[C.sub.D] = [24/[Re.sub.p]](1 + 0.15[Re.sub.p.sup.0.687]) (2)

where the particle Reynolds number was defined as:

[Re.sub.p] = [[rho][D.sub.p]|[[right arrow].U.sub.p] - [[right arrow].U]|]/[mu] (3)

In order to model droplet dispersion in turbulent flows and to obtain a representation of the local velocity, the so-called eddy lifetime concept was applied. (16) This model assumes that the droplet interacts with a sequence of turbulent eddies with randomly sampled fluctuations.

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The solution of the flow field for both phases can be obtained by an iterative calculation. Initially a solution of the gas field is computed without considering the dispersed phase. Afterwards, a large number of discrete parcels are traced through the flow field and averaged values of interphase transfer terms of momentum are calculated. (8) At this point, the gas flow field is recalculated considering the influence of droplets by means of the transfer terms just computed. Then the discrete phase trajectories are calculated again in the modified continuous phase flow field and new transfer terms are obtained. The last two steps are repeated until convergence.

[FIGURE 7 OMITTED]

Simulations have been performed with the CFD code FLUENT, version 6.1.18. Considering the symmetry of the system with respect to the planes XY and XZ, simplified computational domains relative to only one-half or one-quarter of the whole system have been used, according to the case analyzed. Nonstructured meshes were created with a total number of cells varying from 115,000 for the simplest geometry, relative to the normal wall case, up to 188,000 for the case of oblique surfaces. The region close to the gun and along the spray axis had higher mesh resolution to model accurately the interaction between air and droplets immediately after the nozzle. The details of the structure of the mesh on the symmetry planes close to the gun are shown in Figure 3.

Velocity inlets have been prescribed for the continuous phase as boundary conditions at the nozzles: a velocity magnitude of 300 m/sec, with a temperature of 300 K and a direction perpendicular with respect to the cross-section of the nozzles, was set for the air. Due to the high gas velocity at the exit of the gun, compressibility effects were taken into account in the simulation. Turbulence intensity at air inlets was assumed to be 10%, while the turbulence length scale was assumed to be equal to the diameter of the considered air outlet.

A crucial aspect in spray deposition processes is the definition of the initial conditions for droplets. As said before, the generally adopted approach is to start the simulation of the droplets from a zone where the breakage of the liquid jet exiting from the nozzle can be regarded as definitely completed, defining size distribution, position, and velocity of the droplets on the basis of experimental data taken at that zone. A different approach was followed in this work. In order to make simulations less dependent on experiments, drops were injected very close to the nozzle of the gun and it was assumed that initial droplet velocities were the same as that of the liquid jet at the moment of breakage. In this way, drop velocities could be estimated simply from the reduction of the jet section.

A preliminary simulation based on the VOF model (17) was performed to estimate the thinning of the liquid jet in the region close to the nozzle, before the occurrence of breakage. A 2D axisymmetric domain constituted by a structured mesh of 2120 cells was adopted. (18) The liquid phase leaves the central nozzle with a velocity of about 1 m/sec, which is sensibly different from the surrounding air that flows under sonic conditions. This creates strong instability at the liquid surface that imposes the need to carry on simulations with very small time steps, from [10.sup.-8] to [10.sup.-9] sec. By means of this simulation we calculated the shape assumed by the jet at the exit of the nozzle: the section of the jet was reduced by the primary air to approximately one-quarter of its initial dimension at 3.5 mm from the nozzle and was accelerated to a velocity of about 16 m/sec. From this point on the liquid jet started to break, generating the spray.

[FIGURE 8 OMITTED]

The grid relative to the whole spray simulation was modified by introducing a truncated cone at the liquid nozzle (visible in Figure 3), in order to take into account the deformation of the liquid jet. The lateral surface of the cone was modeled as a frictionless wall. This little cone stabilized the numerical computation considerably and modeled better the outflow of the primary air.

Particle injections were located at the top of the truncated cone; that is, at the same position and with the same velocity as the thinnest section of the liquid jet (Figure 4), just before the maximum turbulence zone where the shaping air interacts with the primary air. This assumption made it possible to simulate the interaction between secondary air and droplets and, as a consequence, to model the shape of the spray immediately after the atomizer.

On the basis of experimental data, particle size was characterized by a Rosin-Rammler distribution, with a mean value of 36 [micro]m and spread factor of 2.11. The actual considered range for particle size was between 10 and 90 [micro]m.

For every fourth of symmetry of the spray section, 118 injections representing groups of particles with the cited size distribution were considered. Velocity, temperature, and mass flow of the droplets were prescribed for each inlet position. For every injection and for every drop diameter, a number of individual trajectories were used to resolve the effects of the turbulent air flow on drop transport. A total of 4720 droplet trajectories were tracked through the computational domain for each simulation. The liquid mass flow rate and the number of droplets of a single injection were considered proportional to the local axial velocity.

[FIGURE 9 OMITTED]

SPRAY ANALYSIS AND VALIDATION

Hereafter, the results predicted by simulation, both with the RNG and Realizable model, were compared with data obtained during the experimental investigation to validate the method. The analysis was performed for a spray that is not confined by walls. In the considered reference system X is the axial coordinate, Y the coordinate along the maximum width of the spray section, and Z that along the minor axis of the spray section.

In Figure 5, predicted and experimentally measured dimensionless axial velocity U/[U.sub.max] was reported as a function of the dimensionless coordinate Y/Y([U.sub.50%]) for constant value of axial coordinate X. In the figures [U.sub.max] is the maximum axial velocity at the considered X value, while Y([U.sub.50%]) is the Y coordinate where the spray velocity has a value equal to the half of the maximum velocity. Therefore, Y([U.sub.50%]) describes the spread of the spray.

Figure 5A reports the situation close to the nozzle, at a distance of 40 mm from the gun. The experimental curve presents a maximum at the centerline of the spray, while, in the same zone, those of the models have a smaller velocity. Far from the axis the predicted trend was similar for all the curves, even though the velocities predicted by the simulations were slightly higher than the experimental ones. The two turbulence models behaved in the same way. The only difference was a smoother response from the Realizable model in comparison with RNG.

By increasing the distance from the nozzle, the difference between experiment and simulations slowly disappeared. At 100 mm from the nozzle the predicted flow field was in good agreement with the experiments (Figure 5B). Here the droplets followed the air stream with the same velocity and direction. A good prediction was achieved with both turbulence models: the Realizable predicted slightly better the zone close to the spray axis, while the RNG the external one.

By further increasing the distance from the nozzle, the axial velocity predicted by the simulations were consistent with those measured experimentally.

Figure 6 shows the profiles of transverse velocity scaled in the same way as for the axial profiles. At a 40-mm distance from the nozzle, the transverse velocities predicted by the Realizable model were in perfect agreement with those measured experimentally. The RNG model was less satisfactory than the Realizable one, especially far from the spray axis, where it greatly overpredicted the velocity.

[FIGURE 10 OMITTED]

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At greater distances from the nozzle the transverse velocity strongly decreased. The RNG model captured the decay of this component better than the Realizable. At 100 mm from the nozzle, the RNG model simulated a significant component of transverse velocity for a wide region around the spray axis, more in agreement with experiments than the Realizable one. Both models underpredicted transverse velocities far from the spray centerline. It should be mentioned that at this distance (100 mm from the nozzle), RNG simulated a maximum absolute velocity slightly higher than experiments while Realizable predicted a lower value.

The spreading of the spray in the Y direction can be evaluated by plotting Y([U.sub.50%]) as a function of the axial coordinate. As it appears in Figure 7, the shape of the simulated sprays is larger than the real one. By increasing the distance from the nozzle, the Realizable model predicted a value of spread closer to experiments than that given by RNG. This fact further outlines its better aptitude at simulating jet spreading.

The profiles of concentration of the dispersed phase are plotted in Figures 8 and 9. The figures compare the average concentration of droplets with photographs taken from experiments. The results obtained by the two turbulence models are quite similar for the XY plane. The only difference is a more uniform concentration of droplets predicted by the Realizable model. Actually, in the RNG model, drop trajectories seem straighter and poorly mixed by air turbulence. A greater difference between the two models is apparent when looking at the drop concentration on the plane XZ (Figure 9). Here it can be noticed that the spray predicted by the Realizable model is larger than that obtained by using RNG and closer to the real one, as shown in the photograph.

As a conclusion, we can say that both the RNG and the Realizable models predicted well the main flow field of the spray. The Realizable model predicted more uniform droplet distribution and a spread of the spray that is closer to the real one (especially along the Z direction), while RNG estimated better the velocity decay. A possible reason could be that the Realizable model computes higher values of turbulent viscosity, which lead to smoother velocity profiles, larger dispersion of droplets, and higher decay of velocity along the axial coordinate.

Since the spread of the spray is a fundamental parameter for the determination of film morphology and since this seemed to be better predicted by the Realizable model than by RNG, the Realizable model was adopted for the study of film thickness, as reported in the next section.

MORPHOLOGY AND THICKNESS OF THE COAT

In this section the painting of surfaces with different orientations (90[degrees], 60[degrees], 45[degrees] with respect to the spray axis) was considered. In all the studied cases the surface was at a distance of 24 cm from the gun and the paint flow rate was 3 g/sec (180 [cm.sup.3]/min).

In order to determine the thickness of the deposited paint layer, drops that impact the surface were assumed to stick and their mass and deposition locations were recorded. In this way, one can reconstruct the distribution of paint thickness on the studied surface from the calculation of the trajectories of the drops, as shown in the "Modeling of the Spray Behavior" section. The criteria proposed by the groups of Mundo (19) and of Cossali (20) exclude riatomization of the impacting droplets for the considered range of size and velocity, confirming the assumption of sticking.

Normal Wall

The air flow distribution is one of the most interesting pieces of information for the operation of a paint gun. Therefore, in Figure 10, the simulated air velocity vectors lying on the planes XY and XZ were reported for the coating of a planar workpiece perpendicular to the spray axis. The deviation of the air flow in the proximity of the solid wall, which originates overspray, is clearly visible.

Figure 11 plots the paint thickness predicted by a simulation of the coating of a wall perpendicular to the spray axis. The diagram is relative to a motionless gun operating under its design conditions, as reported in the "Experimental Set-Up" section.

The simulated coat presents an ellipsoidal shape very similar to that observed in the laboratory. The major axis had a value of about 30 cm while the minor axis was approximately 8 cm. The surface was calculated from the CFD simulation, using the information on the locations of drop impacts and drop size as described above. The image refers to a pre-leveling situation; this is why the obtained deposit shows a very indented configuration, which, in a real system, is immediately smoothed by surface tension.

In spray deposition processes, the gun usually moves at a fixed distance from the wall with a constant velocity. If the wall is flat and regular the shape of the film can be considered uniform in the direction of gun movement. For the case analyzed, the mark left by the gun had the shape shown in Figure 12 (again, for the pre-leveling case), which was relative to a gun speed of 0.1 m/sec. Neglecting the peaks and the valleys of the coat that was rapidly levelled, the film thickness could be approximated well by a beta function, as suggested by Balkan and Arikan (21):

H(y)/[H.sub.max] = [1 - (2Y/W)[.sup.2]][.sup.[beta]-1] (5)

From a best fitting procedure the exponent result is [beta] = 1.2, while the width of the mark left by the gun is W = 33 cm. The average paint thickness over this width is 80 [micro]m.

The distribution of the colliding droplets, according to impact velocity, is reported in Figure 13. The majority of the drops collided with the surface at a velocity of about 12 m/sec. As it can be seen, a large amount of particles approached the surface with very small velocities. This situation is typical of the smallest droplets, which have very low inertia. In addition, most of the smallest particles were deviated in the proximity of the wall by the air flux and did not even reach the target surface. This phenomenon, indicated as overspray, is one of the major causes of the waste of paint. As shown by previous researchers, (9,22) one potential route to overspray control is the manipulation of the size distribution of atomized droplets. If it was possible to eliminate the particle fraction finer than 80 [micro]m, the results of Hicks and Senser (9) indicate that transfer efficiency should approach 100%. Unfortunately such a large droplet size would reduce the uniformity and the quality of the sprayed coating.

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Wall at 60[degrees]

The movements of a spraying gun are normally calibrated in such a way as to maintain the direction of the spray perpendicular to the wall. This is the best condition since it ensures uniform paint distribution and minimizes overspray. Nevertheless, with elements of complex geometry, it may happen that the spray axis is oblique with respect to the surface. Figure 14 refers to the case of a wall inclined at 60[degrees] with respect to the centerline of the spray and reports the contours of paint concentration in the air phase on the symmetry plane. The intersection point between the surface and the centerline of the spray was again at 24 cm from the paint nozzle.

One half of the surface, due to its inclination, was sensibly closer to the gun than the other half. This implies a paint distribution less uniform than in the case of the normal wall. Now the paint is more accumulated on the part of the surface closer to the gun, as shown in Figure 15. On the other portion of the surface the film thickness is smaller, since many of the droplets are blown away from the air stream.

Wall at 45[degrees]

In this case more than half of the atomized paint does not reach the wall. Looking at Figure 16, it appears that many droplets were deviated and blown away by the air flux. Practically only the part of the surface closest to the nozzle receives a sufficient amount of paint. This fact is made apparent by Figure 17, which shows the thickness of the coat deposited by a motionless gun.

By comparing Figures 15 and 17, relative to walls at 60[degrees] and 45[degrees], respectively, one can argue that this is the critical range of orientation at which overspray (and thus wasted paint) suddenly increases.

Edge

Finally, the case of a 90[degrees] edge inclined at 45[degrees], with respect to the spray axis, was considered. The coating of an edge is quite common in spray application. Here the goal is to coat the corner without creating a too thick film.

The contours of paint concentration in the air relative to this case are reported in Figure 18. As in the case of oblique surfaces, a large amount of paint is also wasted here. The morphology of the obtained film is shown in Figure 19. The quantity of the deposited paint was high, very close to the edge, and rapidly decreased along the surfaces.

CONCLUSIONS

This work concerns the simulation of paint transfer in an air spray process. Attention was paid to the characterization of the flow field of the continuous phase, the determination of droplet trajectories, and the prediction of the deposited film for different coating scenarios.

Both the RNG and Realizable k-[epsilon] models were tested for modeling turbulence. The RNG model showed better prediction of the velocity decay, while the Realizable estimated better the shape of the spray. The Realizable model also gave smoother profiles, in better agreement with experiments.

A procedure for the simulation of the spray was developed and validated. Different from previous works, initial conditions for droplets were prescribed very close to the gun cap, where the liquid jet of paint breaks. The procedure was based on the analysis of the thinning process of the jet exiting from the atomizer and on the characteristics of the air flow field without droplets. This made it possible to model the strong interactions between the continuous and discrete phases immediately after the gun and to extend the study of spray behavior to different operating conditions without additional experiments.

The method was applied to the simulation of paint deposition over walls with different inclinations with respect to the spray gun. The effect on the thickness of the deposited layer was discussed.

References

(1) Kwok, K.C., "A Fundamental Study of Air Spray Painting," Ph.D. Thesis, University of Minnesota, 1991.

(2) Domnick, J., Lindenthal, A., Tropea, C., and Xu, T.-H., "Application of Phase-Doppler Anemometry in Paint Sprays," Atomization and Sprays, 4, 437 (1994).

(3) Morikita, H. and Taylor, A.M.K.P., "Application of Shadow Doppler Velocimetry to Paint Spray: Potential and Limitations in Sizing Optically Inhomogeneous Droplets," Meas. Sci. Technol., 9, 221 (1998).

(4) Carlton, G.N. and Flynn, M.R., "A Model to Estimate Worker Exposure to Paint Spray Mist," Appl. Occup. Environ. Hyg., 12, 375 (1997).

(5) Flynn, M.R., Gatano, B.L., McKernan, J.L., Dunn, K.H., Blazick, B.A., and Carlton, G.N., "Modeling Breathing Zone Concentrations of Airborne Contaminants Generated During Compressed Air Spray Painting," Ann. Occup. Hyg., 43, 67 (1999).

(6) Brouwer, D.H., Semple, S., Marquart, J., Cherrie, J.W., "A Dermal Model for Spray Painters. Part I: Subjective Exposure Modeling of Spray Paint Deposition," Ann. Occup. Hyg., 45, 15 (2001).

(7) Flynn, M.R., Sills, E.D., "On the Use of Computational Fluid Dynamics in the Prediction and Control of Exposure to Airborne Contaminants--An Illustration Using Spray Painting," Ann. Occup. Hyg., 44, 191 (2000).

(8) Ruger, M., Hohmann, S., Sommerfeld, M., and Kohnen, G., "Euler-Lagrange Calculation of Turbulent Sprays: The Effect of Droplet Collision and Coalescence," Atomization and Sprays, 10, 47 (2000).

(9) Hicks, P.G. and Senser, D.W., "Simulation of Paint Transfer in an Air Spray Process," J. Fluids Eng., 117, 145 (1995).

(10) Ye, Q., Domnick, J., and Khalifa, E., "Simulations of the Spray Coating Process Using a Pneumatic Atomizer," Proc. ILASS-Europe 2002, Zaragoza, Spain, 2002.

(11) Micheli, P., "Understanding How a Spray Gun Atomizes Paint," Metal Finishing, 59 (October 2003).

(12) Wilcox, D.C., Turbulence Modeling for CFD, DWC industries, La Canada, CA, 1993.

(13) Launder, B.E., and Spalding, D.B., "The Numerical Computation of Turbulent Flows," Comput. Meth. Appl. Mech. Eng., 3, 269 (1974).

(14) Yakhot, V. and Orszag, S.A., "Renormalization Group Analysis of Turbulence," J. Sci. Comput, 1, 3 (1986).

(15) Shih, T.H., Liou, W.W., Shabbir, A., and Zhu, J., "A New k-[epsilon] Eddy-Viscosity Model for High Reynolds Number Turbulent Flows--Model Development and Validation," Computers Fluids, 24, 227 (1995).

(16) Crowe, C.T., Sommerfeld, M., and Tsuji, Y., Multiphase Flows with Droplets and Particles, CRC Press, Boca Raton, FL, 1997.

(17) Hirt, C.W. and Nichols, B.D., "Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries," J. Comput. Phys., 39, 201 (1981).

(18) Garbero, M., "Modeling of Spray Deposition Processes," Ph.D. Thesis, Politecnico di Torino, Torino, Italy, 2004.

(19) Mundo, C., Sommerfeld, M., and Tropea, C., "On the Modeling of Liquid Sprays Impinging on Surfaces," Atomization and Sprays, 8, 625 (1998).

(20) Cossali, G.E., Coghe, A., and Marengo, M., "The Impact of a Single Drop on a Wetted Solid Surface," Exp. Fluids, 22, 463 (1997).

(21) Balkan, T. and Arikan, M.A.S., "Modeling of Paint Flow Rate Flux for Circular Paint Sprays by Using Experimental Paint Thickness Distribution," Mechanics Research Commun., 26, 609 (1999).

(22) Kwok, K.C. and Liu, B.Y.H., "How Atomization Affects Transfer Efficiency," Industrial Finishing, 28 (May 1992).

M. Fogliati, D. Fontana, M. Garbero, M. Vanni,** and G. Baldi -- Politecnico di Torino*

R. Donde -- Istituto per l'Energetica e le Interfasi--Consiglio Nazionale delle Ricerche ([dagger])

* Dipartimento di Scienza dei Materiali e Ingegneria Chimica, corso Duca degli Abruzzi 24, 10129 Torino, Italy.

([dagger]) via R. Cozzi 53, 20125 Milano, Italy.

** Author to whom correspondence should be addressed. Email: [e-mail].

NOTATION

[C.sub.D]      drag coefficient
[D.sub.p]      drop diameter
[g.sub.i]      i-th component of gravitational acceleration
t              time
[Re.sub.p]     particle Reynolds number, equation (2)
[U.sub.i]      i-th component of gas velocity
[U.sub.p,i]    i-th component of drop velocity
U              velocity component along X
V              velocity component along Y
W              velocity component along Z
X              spatial coordinate along spray axis
Y              spatial coordinate normal to X and along the larger axis
               of the spray
Z              spatial coordinate normal to X and along the smaller axis
               of the spray
[rho]          gas density
[[rho].sub.p]  drop density