Characterization of Particle Motion and Deposition Behaviour in Electro-Static Fields

The quality of electro-static powder coating processes of metal substrates is related to the uniformity of coating layer thickness. To increase this uniformity, knowledge based adaptions of process parameters and or geometry of the set-up are needed. In order to create the basis for such improvement efforts, a thorough understanding of particle motion and deposition effects within flowand electro-static fields is required. In addition to previous research (see e.g.: [2], [3] and [4]), this work presents recent advances in terms of characterization and modelling [1] of the coating process.

charting is sought.Therefore chapter 3.3 also presents the results of a numerical Euler-LaGrangian coating model [1].As the model is applied to representative example cases, the results give enough insight into the actual meaning behind the distinct, dimensionless parameter combinations, to be able to roughly structure the coating chart into zones of force dominance.
The powder coating process is basically about a coating pistol, equipped with a negatively charged high voltage electrode, a grounded metallic substrate and e.g.thermoplastic or thermoset polymer coating particles with diameters: 5µm< Dp <300µm [3].The particles are carried by an air-flow through the pistol, past the electrode and into the coating chamber towards the substrate.
Close to the negatively charged electrode, oxygen ionization processes cause the presence of spatial charges of free electrons e -, positively charged oxygen ions O + 2 and negatively charged oxygen ions O - 2 [5].Figure 1 gives an impression of typical charge carrier densities n (cm -3 ) as a function of distance r (cm) to the electrode.From Figure 1 it becomes clear that, for r > 10 -3 m, negatively charged oxygen ions are the main carriers of spatial charges.In effect it is mainly the negatively charged oxygen ions that attach themselves to the surface of the passing coating particles.
As the coating particles are charged up to a negative surface specific charge qp (C/m 2 ), they get impacted by the spatial electric field  ⃗ (x,y,z) (N/C).Upon entering the space between the charging zone and the substrate (the coating chamber), three main sorts of acting forces determine particle motion and particle-substrate deposition behaviour: gravity forces   , electric forces   and fluid-drag forces   , [1].
The equations for describing the three relevant, acting types of particle forces are well known and summed up in Table 1.Thereby spherical particles are assumed.Even though the flow situation within the coating chamber is expected to be turbulent, relative flow between air and particles is not.Thus a laminar particle-air drag relation can be used.In Table 1, Dp (m) is the particle-diameter, ρp and ρa (kg/m3) are the particle-and air densities respectively, μa (Pas) is the dynamic viscosity of air and  , (m/s) is the relative flow velocity between particles and air. ) and   = (  ).A simple evaluation of these dependencies, for an exemplary case (Figure 2, bottom) shows that for small particles (e.g.: case Figure 2, Dp < 20µm) fluid drag forces dominate; that for medium sized particles (e.g.: case Figure 2, 20µm < Dp < 230µm) electric forces dominate; and that large particle motion (e.g.: case Figure 2, Dp > 230µm) is dominated by gravity.These initial insights do correspond with findings from practical applications, which report highest coating efficiencies for medium sized particles.

Name
ρ A new descriptive method to characterize the flow path and substrate-impact-position of each coating particle is hereby introduced.On the basis of relating the maximum, individual force contributions to the sum of all acting forces, the particle state can be completely characterized within the coating chamber.Thus three dimensionless groups πf, πel and πg are obtained, which represent fluid drag, electric effects and gravity respectively.A fourth dimensionless group πm, representing the effects of inertia in relation to the acting forces, can be defined via the additional introduction of a characteristic length scale Lr (m) and a characteristic particle residence time τr (s).However, it turns out that πm is directly proportional to πg, which means that it is redundant information when characterizing the particle flow path.The definitions of πf, πel, πg and πm are found in Table 2.As a consequence of the definitions of πf, πel and πg, Eqn.1 can be written as:

Sign Definition Relation to
Figure 2 shows a direct comparison between representing an exemplary case by inserting the dimensionless groups (top) and by using absolute values (bottom).
Considering Eqn.1, as well as the definitions of πf, πel and πg, a triangle chart notation is proposed here.By defining the three dimensionless groups as dimensions, the chart can be used to visually characterize the states of particle flow paths for any set of process parameters and any combination of particle properties.Since a coating particle cloud usually features poly-disperse particle size distributions, the state of one entire coating procedure cannot simply be represented by one single dot in the triangle chart.Each particle size class will be placed at different locations within the chart, such that a non-linear, yet continuous, representative curve can be drawn to characterize the coating state of the entire cloud.Figure 3 shows a dimensionless triangle chart, with representative coating curves for three exemplary coating process parameter sets (Table 3).Note that for each case, smaller particles are located within the flow-dominated section, while larger particles tend to be found within the gravity section.Intermediate particle sizes can either be placed within the electro-statically dominated region, or at least have a tendency towards stronger electro-static impact.

Dominating Effect Coloration of Curve in
Figure 3 Process Parameter Set Electric Forces Red Fluid Drag Forces Blue

Intermediate Green
In addition to the force-relation-characterization method, presented in chapter 2, another characterization method, based on the dimensionless particle momentum equation (PME), has been developed.It amounts to the dimensionless chart of coating states and is presented in the following.
Assuming LaGrangian particle perspective, Newton's second law, in combination with the acting force relations, shown in Table 1, gives the governing PME for coating processes, as seen in Eqn. 2. Thereby  ̈p is particle acceleration and mp the particle mass. ( Combining the relations, seen in Table 4, with Eqn.2, the dimensionless version of the governing PME has been derived.It is depicted in Eqn.3.In Table 4 the sign '*' signifies the non-dimensionality of a variable, vp (m/s) is the particle velocity, max(vp) (m/s), max(va) (m/s), max(va,p) (m/s) and max(E) (N/C) are the characteristic maximum particle velocity, airflow velocity, relative air-particle velocity and maximum electric field strength respectively.Being the smallest extension of any numerical mesh cell, Δx has been brought into the considerations to create a link to related, numerical modelling efforts [1].Note that the relation for calculating maximum particle velocity in Table 4 is chosen, such that the co-located occurrence of extreme states within the flow-and the electric field is assumed.The relation for relative maximum air-particle velocity in Table 4 considers a state of equilibrium between viscose fluid drag and electro-static forces.It shows that high relative fluid-particle velocities can only be achieved in regions of high electric field strength.

Description
(3) Combining Eqn. 3 with the relations for the maximum air-particle-and particle velocity, seen in Table 4, two decisive dimensionless groups can be derived.The first, πel/g depicting the relation between electro-static and gravity effects (Eqn.4) and the second πf/el,i relating fluid viscose drag to electro-static times inertial effects (Eqn.5).
In the context of creating a suitable, LaGrangian particle model of the coating process, another dimensionless number πt to represent the relation between numerical particle-time step Δt and particle relaxation time τp is proposed according Eqn.6, [6].
While πel/g and πf/el,i reflect the physical state of the coating process, πt the quality of the numerical time step and should be kept well below 1 at all times.Choosing πt too high will result in an un-physical, merely numerical alteration of calculated particle paths.
The dimensionless groups πel/g and πf/el,i span a two-dimensional space which encompasses all possible coating states for all possible combinations of particle-and process parameters.This two dimensional space constitutes the coating chart, shown in Figure 4.
By combining the coating chart characterization methodology with results form numerical simulations [1], we were able to interpret the coating chart and its zones in terms of applicable, practical meaning.Several numerical experiments at varying parameter configurations were conducted, the dimensionless numbers πel/g and πf/el,I of these cases were determined and the results were qualitatively checked.Thereby the evaluation criteria were: the shape of the coating particle cloud as well as the distribution of particle deposition patterns on the frontand back-side of a metallic plate.On the basis of these evaluations, a qualitative distinction between chart-zones of varying force-dominance could be made.Of course it is impossible to part the individual zones by calculated curves, since transitions are blurred and strongly related to the observed effects.Thus the achieved structure in Figure 4 is relatively rough and leaves room for interpretation.Still it does constitute a first design guideline for practical application.
Being the basis of thoroughly understanding the coating process, new means of proper characterization have been worked out and presented.While the proposed triangle chart notation (chapter 2) helps to understand the acting force relations on individual particles, the chart of coating states (chapter 3) provides a more practically applicable view on the entire process.Where the triangle chart notation concentrates on individual particles, the chart of coating states focuses on the big picture.Both characterization methods have already been applied in combination with a new numerical, OpenFoam® [7] based solver.The corresponding results have on the one hand led to interpreting the coating chart (Figure 4) and have on the other hand already greatly contributed to achieving a much better understanding of cause-effect relations for the coating process as a whole.These findings are published in [1].This work will thus constitute the foundation of future, knowledge based improvement efforts in terms of optimized coating process parameter combinations.