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Multiscale simulation of electrostatic powder coating sprays

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Abstract

Powder coating is a finishing process used in various industries where electrostatically charged particles are sprayed onto metal surfaces. High-fidelity simulation is becoming increasingly popular to optimize this technique. However, the current standard approaches based on modeling the individual particles require substantial computational effort, which limits their practical application. To alleviate this drawback, the multiscale pseudo-direct numerical simulation method (P-DNS) is extended for the study of the turbulent flow of charged particles. In an offline process, representative volume elements (RVEs) are simulated considering the particles and their interactions with the surrounding gas and electrostatic field, and the dynamic response of the mixture is homogenized. On the global scale, where a continuous field for the particulate phase is used, the effects of turbulence and coupling forces are obtained by evaluating synthetic models of the RVEs with local dimensionless parameters. Additionally, a coat film model is presented to quantify the coating layer thickness on the target surface. The numerical tool is first validated with experimental data from the literature. Then, it is applied to the coating of a surface of complex geometry, analyzing the impact of particle diameter, applied voltage, and spray gun trajectory. The reliable results obtained with modest resources confirm the potential of the methodology as a rapid predictive tool in processes involving this coating technology.

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Abbreviations

\(\textrm{H}\) :

Length [m]

\(\textrm{r}\) :

Particle to fluid density ratio [−]

\(\epsilon \) :

Volume fraction of the carrier phase [−]

\(\epsilon _{\textrm{p}}\) :

Volume fraction of the particulate phase [−]

\(\textbf{u}\) :

Velocity of the carrier phase [m/s]

\({\textbf{u}}_{\textrm{p}}\) :

Velocity of the particulate phase [m/s]

\(\textbf{u}_{\textrm{r}}\) :

Relative velocity between phases [m/s]

\(\textbf{u}^{\textrm{M}}\) :

Velocity of the moving frame [m/s]

\(\textbf{v}_\textrm{p}\) :

Velocity of a particle [m/s]

\(\mathrm p\) :

Pressure [kg/(m s\(^2\))] = [Pa]

\(\rho \) :

Density of the carrier phase [kg/m\(^3\)]

\(\rho _{\textrm{p}}\) :

Density of the particulate phase [kg/m\(^3\)]

\(\mu \) :

Dynamic viscosity of the carrier phase [kg/(m s\(^2\))]

\(\mathrm g\) :

Gravitational acceleration magnitude [m/s\(^2\)]

\({\textbf {T}}^\mu \) :

Viscous stress tensor [m\(^2\)/s\(^2\)]

\({\textbf {T}}^\rho \) :

Fine inertial stress tensor from the fine scale [m\(^2\)/s\(^2\)]

\({\textbf {s}}\) :

Fine-scale dispersion field [m/s]

\(\textbf{f}\) :

Momentum transfer between phases by mass unit [m/s\(^2\)]

\(\textrm{V}_{\textrm{p}}\) :

Volume of a particle [m\(^3\)]

\({{\varvec{f}}}_{\textrm{d}}\) :

Drag force acting over a particle [kg m/s\(^2\)] = [N]

\({{\varvec{f}}}_{\textrm{b}}\) :

Mass-dependent forces acting over a particle [kg m/s\(^2\)] = [N]

\({{\varvec{f}}}_{\textrm{g}}\) :

Gravitational force acting over a particle [kg m/s\(^2\)] = [N]

\({{\varvec{f}}}_{\textrm{e}}\) :

Electric force acting over a particle [kg m/s\(^2\)] = [N]

\({\textbf {G}}\) :

Coarse velocity symmetric gradient [m\(^2\)/s\(^2\)]

\({\textbf {b}}\) :

Body acceleration [m/s\(^2\)]

\(\varphi \) :

Electrostatic potential [kg m\(^2\)/(s\(^3\) A)] = [kV]

\(\textbf{e}\) :

Electric field [kg m/(s\(^3\) A)] = [kV/m]

\(\textrm{q}_{\textrm{p}}\) :

Specific charge of a particle [A s/kg]

\(\rho _{\textrm{e}}\) :

Density of charge [A s/m\(^3\)]

\(\epsilon _0\) :

Permittivity of void [A\(^2\) s\(^4\)/(kg  m\(^3\))]

\(\textrm{Id}_1\) :

Dimensionless instability index 1 [−]

\(\textrm{Id}_2\) :

Dimensionless instability index 2 [−]

\(\textrm{b}\) :

Body acceleration magnitude [m/s\(^2\)]

\(\theta \) :

Polar angle of the body acceleration [−]

\(\psi \) :

Azimuth angle of the body acceleration [−]

\(\textrm{d}_{\textrm{p}}\) :

Mean of particle diameters [−]

\(\eta \) :

Dispersion of particle diameters [−]

\(\textrm{r}\) :

Particle to fluid density ratio [−]

\({\textrm{k}}\) :

Turbulent kinetic energy [m\(^2\)/s\(^2\)]

\(\textbf{K}\) :

Turbulent diffusivity of mass [m\(^2\)/s]

\(\textbf{D}\) :

Turbulent diffusivity of momentum [m\(^2\)/s]

\(\delta _{\textrm{p}}\) :

Coating thickness [m]

\(\delta \) :

First-cell height [m]

\(\textbf{n}\) :

Surface normal [−]

Normal:

Scalar

Bold-lowercase:

Vector

Bold-uppercase:

Tensor

\({(.)}^{\textrm{c}}\) :

Coarse-scale field

\({(.)}^{\textrm{f}}\) :

Fine-scale field

\(\mathrm {(.)}^{\textrm{R}}\) :

RVE field

\(\widetilde{(.)}\) :

Dimensionless field

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Acknowledgements

This work was supported by the CERCA program of the Generalitat de Catalunya, and the Spanish Ministry of Economy and Competitiveness through the "Severo Ochoa Programme for Centres of Excellence in R &D" (CEX2018-000797-S). Also, the author acknowledges MCIN / AEI and FEDER Una manera de hacer Europa for funding this work via project PID2021-122676NB-I00. by CIENCIA FEDER Una manera de hacer Europa via project PID2021-122676NB-I00. Last but not least, the author kindly thanks Prof. Sergio Idelsohn for the fruitful meetings and discussions. His distinct viewpoint was helpful in crafting this work.

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  1. Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Edifici C1 Campus Nord UPC C/ Gran Capità, S/N 08034, Barcelona, Spain

    Juan M. Gimenez

  2. Centro de Investigaciones de Métodos Computacionales (CIMEC), UNL/CONICET, Predio CONICET Colectora Ruta Nac Nro 168, Km 0, 3000, Santa Fe, Argentina

    Juan M. Gimenez

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Gimenez, J.M. Multiscale simulation of electrostatic powder coating sprays. Comp. Part. Mech. (2024). https://doi.org/10.1007/s40571-023-00703-w

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