A parametric simulation model for HVOF coating thickness control

Abstract

High-velocity oxygen-fuel (HVOF) thermal spraying is a coating process involving multidisciplinary aspects, e.g., fuel–oxidant combustion, flame–particle jet, particle deposition, mass and heat transfer, and even robotic kinematics. Like most coating processes, in HVOF processes, coating thickness is a significant property determining the coating performance; hence, this property should be accurately controlled during the process. In view of green, smart, and digital manufacturing, the coating thickness prediction model is demanded to produce high-quality coatings efficiently. This paper presents an approach to parametrically simulate the coating thickness in HVOF processes via an integrated numerical model. Firstly, an axisymmetric computational fluid dynamics (CFD) model is constructed to compute the behaviors of the fuel–oxidant combustion, flame–particle jet, and particle deposition distribution. Secondly, based on the particle distribution in a 2D axisymmetric model, a 3D single coating thickness profile model is developed by constructing a circular pattern using the axis of the nozzle. Further, this profile is smoothened by a Gaussian model, and its mathematical expression is obtained. Finally, a numerical model couples spray paths with the mathematical expression to model the coating thickness distribution on a substrate surface under industrial scenarios. At the end of this paper, to verify the proposed model’s effectiveness, four sets of operating parameters with a single straight path were experimentally implemented. The width and height of the bead-like shape coating were in good agreement with the simulated results. The normalized root-mean-square errors of the cross-sectional profile heights were around 10%.

Appendix. Mathematical representation of the HVOF in-flight behavior model

The flame in HVOF is a high-Reynolds-number turbulent compressible flow, so Reynold and Favre averaging are used to simplify the small-scale turbulent fluctuations [20, 47]:

$$ \phi =\overline{\phi}+{\phi}^{\prime },\overline{\phi}=\left(\frac{1}{\Delta t}\right){\int}_{t_0}^{t_0+\Delta t}\phi \mathrm{d}t,{\overline{\phi}}^{\prime }=0 $$

(15)

$$ \phi =\overset{\sim }{\phi }+{\phi}^{\prime \prime },\overset{\sim }{\phi }=\frac{\int_{t_0}^{t_0+\Delta t}\rho (t)\phi (t)\mathrm{d}t}{\int_{t_0}^{t_0+\Delta t}\rho (t)\mathrm{d}t}=\frac{\overline{\rho \phi}}{\overline{\rho}},\overset{\sim }{\phi^{\prime \prime }}=0 $$

(16)

The gas phase is solved by Reynolds-averaged and Favre-averaged governing equations as follows [20, 47]:

$$ \frac{\partial \overline{\rho}}{\partial t}+\frac{\partial }{\partial {x}_j}\left(\overline{\rho}{\overset{\sim }{\upsilon}}_j\right)=0, $$

(17)

$$ \frac{\partial }{\partial t}\left(\overline{\rho}{\overset{\sim }{\upsilon}}_i\right)+\frac{\partial }{\partial {x}_j}\left(\overline{\rho}{\overset{\sim }{\upsilon}}_i{\overset{\sim }{\upsilon}}_j\right)=-\frac{\partial \overline{p}}{\partial {x}_i}+\frac{\partial }{\partial {x}_j}\left[\mu \left(\frac{\partial {\overset{\sim }{\upsilon}}_i}{\partial {x}_j}+\frac{\partial {\overset{\sim }{\upsilon}}_j}{\partial {x}_i}-\frac{2}{3}{\delta}_{ij}\frac{\partial {\overset{\sim }{\upsilon}}_l}{\partial {x}_l}\right)\right]+\frac{\partial }{\partial {x}_j}\left(-\overline{\rho {\upsilon}_i^{\prime \prime }{\upsilon}_j^{\prime \prime }}\right),i=1,2,3, $$

(18)

where ρ is the density, p is the pressure, x is the coordinate, μ is the molecular viscosity, and δ_ij is the Kronecker delta. According to the Boussinesq hypothesis, the Reynolds stress term representing the effect of turbulence can be related to the mean velocity gradients:

$$ -\overline{\rho {\upsilon}_i^{\prime \prime }{\upsilon}_j^{\prime \prime }}={\mu}_t\left(\frac{\partial {\overset{\sim }{\upsilon}}_i}{\partial {x}_j}+\frac{\partial {\overset{\sim }{\upsilon}}_j}{\partial {x}_i}\right)-\frac{2}{3}\left(\overline{\rho}k+{\mu}_t\frac{\partial {\overset{\sim }{\upsilon}}_l}{\partial {x}_l}\right){\delta}_{ij}, $$

(19)

where μ_t is the turbulent viscosity and k is the turbulence kinetic energy.

Because of the supersonic flow and the large pressure gradients in the nozzle, the renormalization group (RNG) k-ε turbulence model is used to estimate the turbulent eddy viscosity with the nonequilibrium wall function treatment used to enhance the wall shear and heat transfer [2, 48, 49]:

$$ \frac{\partial }{\partial t}\left(\overline{\rho}k\right)+\frac{\partial }{\partial {x}_i}\left(\overline{\rho}{\overset{\sim }{\upsilon}}_ik\right)=\frac{\partial }{\partial {x}_j}\left[{\alpha}_k\left(\mu +{\mu}_t\right)\frac{\partial k}{\partial {x}_j}\right]+{G}_k-\overline{\rho}\varepsilon -{Y}_M, $$

(20)

$$ \frac{\partial }{\partial t}\left(\overline{\rho}\varepsilon \right)+\frac{\partial }{\partial {x}_i}\left(\overline{\rho}{\overset{\sim }{\upsilon}}_i\varepsilon \right)=\frac{\partial }{\partial {x}_j}\left[{\alpha}_{\varepsilon}\left(\mu +{\mu}_t\right)\frac{\partial \varepsilon }{\partial {x}_j}\right]+{C}_{1\varepsilon}\frac{\varepsilon }{k}{G}_k-{C}_{2\varepsilon}\overline{\rho}\frac{\varepsilon^2}{k}-{R}_{\varepsilon }, $$

(21)

where ε is the turbulence dissipation rate, G_k is the generation of turbulent kinetic energy arising from the mean velocity gradients, and Y_M is the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. α_k and α_ε are inverse effective Prandtl numbers for the turbulent kinetic energy and its dissipation. R_ε is an additional term in the ε equation. C_1ε = 1.42, C_2ε = 1.68. Within the chemical reaction, the convection–diffusion equation governs the mass fraction of each species, Y_i [2]:

$$ \frac{\partial }{\partial t}\left(\overline{\rho}{Y}_i\right)+\frac{\partial }{\partial {x}_j}\left(\overline{\rho}{Y}_i{\overset{\sim }{\upsilon}}_j\right)=\frac{\partial }{\partial {x}_j}\left({J}_i\right)+{R}_i,\kern1.25em i=1,\dots, N-1, $$

(22)

where J_i is the diffusion flux of species i calculated by Maxwell–Stefan equations, R_i is the net rate of production of species i by chemical reaction, and N is the total number of species involved in the reaction. The energy conservation is represented by

$$ \frac{\partial }{\partial t}\left(\overline{\rho}H\right)+\frac{\partial }{\partial {x}_i}\left[{\overset{\sim }{\upsilon}}_i\left(\overline{\rho}H+\overline{p}\right)\right]=\frac{\partial }{\partial {x}_j}\left[\alpha {c}_p\left(\mu +{\mu}_t\right)\frac{\partial T}{\partial {x}_j}+{\overset{\sim }{\upsilon}}_i\left(\mu +{\mu}_t\right)\left(\frac{\partial {\overset{\sim }{\upsilon}}_j}{\partial {x}_i}+\frac{\partial {\overset{\sim }{\upsilon}}_i}{\partial {x}_j}-\frac{2}{3}{\delta}_{ij}\frac{\partial {\overset{\sim }{\upsilon}}_l}{\partial {x}_l}\right)-{\sum}_{i=1}^N{J}_i{H}_i\right]+{S}_E, $$

(23)

where T is the temperature, H is the total enthalpy, and S_E is the source term.

On the particle dynamics side, owing to the very low particle loading (less 4% usually) [32], a one-way coupling between the gas phase and the particle phase is assumed; in other words, the momentum and heat of the particle are solved by Lagrangian approach after the gas flow fields are determined, and the particles have no influence on the gas phase [6, 36]. According to this analysis in a previous study [39], it is reasonable to assume that the particle coagulation process is negligible and the powder size distribution does not change during the process. The motion of the particles is governed by Newton’s law with the major drag force [19], which can be described as

$$ {m}_p\frac{d{\upsilon}_p}{dt}=\frac{1}{2}{C}_D{\rho}_g{A}_p\left({\upsilon}_g-{\upsilon}_p\right)\left|{\upsilon}_g-{\upsilon}_p\right|,\kern0.5em $$

(24)

where m_p and υ_p are the mass and velocity of the particle, υ_g and ρ_g are the velocity and density of the gas, A_p is the projected area of the particles on the plane perpendicular to the flow direction, and C_D is the drag coefficient representing the effect of the particle shape. With the assumption of negligible particle vaporization and heat transfer via radiation and oxidation, the energy equation for a single particle can be described as follows:

$$ {m}_p{c}_{p_{\mathrm{p}}}\frac{d{T}_p}{dt}=h{A}_p^{\prime}\left({T}_g-{T}_p\right),\kern0.5em $$

(25)

where m_p, T_p, A_p’, and C_pp are the mass, temperature, surface area, and heat capacity of the particle, respectively. T_g is the temperature of the gas. The heat transfer coefficient h can be obtained by the Ranz–Marshall empirical equation [2]. The melting ratio of the particles with the iterations can be calculated by

$$ {f}_p^{i+1}={f}_p^i+\frac{c_{p_p\left({T}_g-{T}_m\right)}}{\Delta {H}_m}\frac{\Delta t}{\omega_p},0<{f}_p^i,{f}_p^{i+1}<1 $$

(26)