Simulation of a Painting Arc Connecting Surface by Moving the Nozzle Based on a Sliding Mesh Model

Abstract

The paper proposes a sliding mesh model-based method to conduct a dynamic painting simulation for an arc connecting surface. In order to meet the requirements for using a sliding mesh model, the computational domain was designed with three parts—a stationary part, a translational part and a rotational part. The film-forming model based on the Euler–Euler model and the conservation equation for a sliding mesh model were established. The dynamic painting of arc connecting surface with the angles of 60, 90 and 120 degrees under the nozzle speed of 100 mm/s were simulated. It was found that the thickness on the arc surface section was larger than those on the plane sections, and through an analysis of the vector distribution, the reason for this was that more paint would be deposited on the arc surface section than on the plane sections due to the concavity of the arc surface. The results obtained from the simulation were in a reasonable agreement with the experimental results, which indicates the proposed method would be effective and applicable in simulating a dynamic painting process.

1. Introduction

An arc connecting surface exists largely in industrial products, such as the air-inlet duct of an airplane [1,2,3]. The film thickness simulation of is a critical part in the path planning of painting this type of surface by a robot.

In our previous work [1,2,3], a film thickness simulation of painting this type of surface was conducted based on an empirical function: empirical functions that describe film thickness growth rate with respect to the position on a plane were firstly obtained by experiment, which is also called as static film thickness, and then the dynamic film thickness (i.e., the coating thickness resulting from a moving nozzle driven by the robot under a certain speed) were computed by integrating the empirical function along the path. Although this method has already been applied in the manufacturing of great many products [2,3,4,5,6], it still has the disadvantages such as over-reliance on experiment and incapability in analysis of flow field.

The Computational Fluid Dynamic (CFD) technique has been adopted to simulate the spray-painting process, which can save the time of experiment for empirical function, be helpful for the studies on the mechanism and characteristic of spray flow and assist the path planning for painting robot [4,7,8,9,10,11,12,13,14,15,16]. However, few studies have reported the CFD simulation of painting arc connecting surface.

How to conduct dynamic painting simulation (i.e., simulation of the painting process with a moving nozzle) for this type of surface is an important problem to be solved. In the literature, the CFD-based strategies for dynamic painting simulation can be grouped into two categories.

One scheme is the numerical integration of the static film thickness growth rate obtained through CFD simulation [14]. It seems a simple and time-saving method for simulating the thickness of painting a regular surface, but numerical integration brings about great errors in the dynamic film thickness by accumulating the little errors existing in the static film thickness growth rate.

Another scheme is the CFD simulation of painting process based on dynamic mesh models [7,13,15,16]. By adopting this scheme, the mesh around the nozzle for computational domain deforms according to the prescribed motion and the specific models that are adopted, and thus a dynamic painting process can be simulated continuously based on the combination of the conservation equations for film-forming and dynamic mesh models. Theoretically, this scheme can be applied to any painting scenario whatever path the nozzle moves along, which makes it a better method to simulate painting process. Nevertheless, when this method is applied, the mesh for painting simulation must be carefully treated, and the parameters should be carefully set and tested repeatedly lest the mesh quality deteriorates leading to a convergence problem during the simulation. Furthermore, much more time would increase the cost for the simulation by this scheme due to the additional computation for the mesh update based on dynamic mesh models.

The sliding mesh model is a special case of dynamic mesh models where all the nodes and boundaries in the dynamic mesh zones move rigidly together. When this model is adopted, additional computation for updating the mesh based on the dynamic mesh models can be saved and no cell zone will deform so there is no need to worry about the convergence problem caused by the potential deterioration of mesh quality. As a consequence, compared to the dynamic mesh model, the sliding mesh model, in theory, can ensure the correctness and efficiency of simulation.

Therefore, in this work, the computational domain for the simulation of painting the arc connecting surface was designed with three parts based on the requirement for the sliding mesh model. The mathematical model was constructed, consisting of the film-forming model and the sliding mesh model. Simulations of painting three kinds of arc connecting surfaces were conducted based on the mathematical model, and the fluid fields as well as the film thickness distributions were obtained. The results showed a good agreement between the simulations and experiment, indicating that the method proposed in this work is effective.

2. Scheme for Computational Domain Geometry Establishment and Mesh Generation

2.1. Path of the Nozzle

Figure 1 shows the sketch of the cross-section of an arc connecting surface. It consists of two planes (A and C) and one arc surface (B). The angle between plane A and plane C is α, and these planes are both tangential to arc surface B. The nozzle geometry adopted in this study is shown in Figure 2a. When spraying, the axis of the nozzle (Z-axis of the nozzle in Figure 2) should be perpendicular to the plane being sprayed or aligned with the normal of the surface being sprayed, the motion direction is usually kept aligned with the short axis of the spray pattern (X-axis of the nozzle in Figure 2), and the Y-axis is kept normal to the nozzle path. During the painting process, the intersection of the nozzle axis and the workpiece moves linearly to the tangent point connecting the plane A and the arc surface B, then rotates to the tangent point where arc surface B and plane C meets around the center of the arc surface B and moves linearly when the plane C is painted. In this way, the path for a nozzle (as the end-effector of a painting robot) to paint an arc connecting surface combines both linear and rotational motion; that is to say the motion of the nozzle includes both translation and rotation.

2.2. Establishment of Computational Domain for Sliding Mesh Model

When the sliding mesh model is applied to simulate a dynamic painting process, the computational domain and the related setup should meet the following requirements [17,18,19]: (1) the interface that associates the cell zones in relative motion should have the same geometry; (2) the motion of the cell zones normal to the interfaces is not allowed—in other words, they should only slide along the interfaces. In order to meet these requirements, we propose that the geometry for the dynamic painting simulation based on the sliding mesh model should consist of multiple parts if the path of nozzle includes different types of motion.

As for the simulation of painting the L-shaped surface, the geometry for the computational domain was established comprising two sliding parts and a stationary part in order to meet the requirements mentioned above. As shown in Figure 3, the sliding parts include a cylindrical part (green) and a translational part (blue), which is in correspondence to the rotational and linear motion of the nozzle. The cylindrical part with the nozzle wall is fitted in the translational part with the cylindrical surface as the interface between the two parts, and the translational part is associated with the stationary part. The stationary one was designed with a groove which has the same geometrical features with the translational part. Figure 4a–e exhibits how every part moved during the simulation with the painting process for the arc connecting surface with 90 degrees as an example.

3. Mathematical Model and Simulation Setup

3.1. Film-Forming Model

3.1.1. Two-Phase Governing Equation

In the painting process, air and liquid paint exist in the spray flow. In this work, the Euler–Euler approach was used to describe the two-phase flow, where both air and paint are treated as a continuum.

Mass conservation equation for phase q is:

$$\frac{\partial {\alpha }_q{\rho }_q}{\partial t}+\nabla \cdot ({\alpha }_q{\rho }_q{\mathit{v}}_q)=0$$

(1)

where the subscript q represents the gas phase or the liquid phase when it is g or d respectively, ${\alpha }_q$ is the volume fraction of phase q, ${\rho }_q$ is the density of phase q and ${\mathit{v}}_q$ is the velocity of the phase q.

Momentum conservation equation:

$$\frac{\partial }{\partial t}\left({\alpha }_q{\rho }_q{\mathit{v}}_q\right)+\nabla \cdot ({\alpha }_q{\rho }_q{\mathit{v}}_q{\mathit{v}}_q)=-{\alpha }_q\nabla p+\nabla \cdot {\tau }_q+{\alpha }_q{\rho }_q\mathit{g}+{\mathit{F}}_{td,q}$$

(2)

where p is the pressure shared by both the phases, ${\tau }_q$ is the viscous stress of the phase q, $\mathit{g}$ is the gravity acceleration and ${\mathit{F}}_{td,q}$ is the turbulent dispersion forces of the phase q.

3.1.2. Turbulence Model

In this work, the turbulent transport of both the phases was modeled by the standard k-ε turbulence model, which is one of the RANS (Reynolds-averaged Navier–Stokes) models [20]. The model equations are

$$\frac{\partial }{\partial t}\left({\rho }_{m}k\right)+\nabla ⦁\left({\rho }_m{\mathit{v}}_{m}k\right)=\nabla ⦁\left(\frac{{\mu }_{t,m}}{{\sigma }_k}\nabla k\right)+G_{k,m}-{\rho }_m\epsilon +{\prod }_{k,m}$$

(3)

$$\frac{\partial }{\partial t}\left({\rho }_m\epsilon \right)+\nabla ⦁\left({\rho }_m{\mathit{v}}_m\epsilon \right)=\nabla ⦁\left(\frac{{\mu }_{t,m}}{{\sigma }_{\epsilon }}\nabla \epsilon \right)+\frac{\epsilon }{k}\left(C_{1\epsilon }{G}_{k,m}-C_{2\epsilon }{\rho }_m\epsilon \right)+{\prod }_{\epsilon ,m}$$

(4)

where ${\prod }_{k,m}$ and ${\prod }_{\epsilon ,m}$ are the source terms to include the turbulent interactions between the phases into the model.

3.1.3. Paint Deposition Model

When the paint liquid phase from the spray flow field impinges and sticks onto the workpiece, its mass and momentum are removed from the spray flow and added as source terms to the governing equations for the film on workpiece.

The conservation of mass for the film [21] is:

$$\frac{\partial h}{\partial t}+{\nabla }_s\cdot \left[h\cdot {\overrightarrow{V}}_l\right]=\frac{{\dot{m}}_s}{{\rho }_l}$$

(5)

$${\dot{m}}_s={\alpha }_d{\rho }_d{V}_{dn}A$$

(6)

where ${\rho }_l$ is the liquid density, h the film height, ${\nabla }_s$ the surface gradient operator, $\overrightarrow{V}$ the mean film velocity, ${\dot{m}}_s$ the mass source, where ${\alpha }_d$ is the volume fraction of the paint, ${\rho }_d$ is the paint phase density, $V_{dn}$ is the paint phase velocity normal to the wall surface and $A$ is the wall surface area.

The conservation of momentum for the film is written as:

$$\frac{\partial h{\overrightarrow{V}}_l}{\partial t}+{\nabla }_s\cdot (h{\overrightarrow{V}}_l{\overrightarrow{V}}_l)=-\frac{h{\nabla }_s{P}_L}{{\rho }_l}+\frac{3}{2{\rho }_l}{\overrightarrow{\tau }}_{fs}-\frac{3v_l}{h}{\overrightarrow{V}}_l+\frac{{\dot{q}}_s}{{\rho }_l}$$

(7)

$${\dot{\overrightarrow{q}}}_s={\dot{m}}_s{\overrightarrow{V}}_d\mathrm{s}$$

(8)

The terms on the left-hand side of Equation (7) represent transient and convection effects, respectively. On the right-hand side, the first term includes the effects of gas-flow pressure, the gravity component normal to the wall surface (known as spreading) and surface tension; the second and third terms represent the net viscous shear force on the gas-film and film-wall interfaces, based on the quadratic film velocity profile representation; and the last term is the effect of the momentum source. In Equation (8), ${\dot{\overrightarrow{q}}}_s$ is the momentum source and ${\overrightarrow{V}}_d$ is the velocity vector of the paint liquid.

3.2. Conservation Equation for Sliding Mesh Model

The paint deposition model can only solve the film-forming process where the control volume is static. Therefore, it is necessary to establish a model to describe the situation where the control volume is moving. The integral form of the conservation equation for a general scalar $\varphi$ on an arbitrary control volume, $V$, whose boundary is moving prescribed by sliding mesh model as translational or rotational motion can be written as:

$$\frac{d}{dt}{\displaystyle \underset{V}{\int }\rho \varphi dV}+{\displaystyle \underset{\partial V}{\int }\rho \varphi \left(\overrightarrow{u}-{\overrightarrow{u}}_g\right)}\cdot d\overrightarrow{A}={\displaystyle \underset{\partial V}{\int }\mathsf{\Gamma }\nabla \varphi }\cdot d\overrightarrow{A}+{\displaystyle \underset{V}{\int }{S}_{\varphi }dV}$$

(9)

where $\rho$ represents the fluid density, $\overrightarrow{u}$ is the flow velocity vector, ${\overrightarrow{u}}_g$ is the mesh velocity of the moving mesh, $\mathsf{\Gamma }$ is the diffusion coefficient, $S_{\varphi }$ is the source term of $\varphi$ and $\partial V$ is used to represent the boundary of the control volume, $V$.

3.3. Simulation Setup

Table 1. The properties of the materials in numerical simulation.

Properties	Air	Paint Phase
Density (kg/m3)	1.23	1.2 × 103
Viscosity (kg/m∙s)	1.7894 × 10−5	9.686 × 10−2
Surface Tension Coefficient (N/m)	--	2.87194 × 10−2

shows the properties of the fluid phases involved in this study, concerning liquid paint and air. The paint hole (shown in Figure 2a) was set to be a paint mass inlet, where the value of mass flow rate was 1.32 × 10⁻³ kg/s, and the turbulent intensity and hydraulic diameter were given as 5% and 1.1 mm, respectively. The atomization air hole, the assisting air hole and the shaping air hole were set to be air pressure inlets having pressure values of 110 kPa, 110 kPa, and 150 kPa, respectively. The workpiece and the nozzle (shown in Figure 3) were set as wall boundaries. The contacting faces (such as the face where the static part and the translational part contact) were set as interfaces. Moreover, other faces were set as outlets.

The center point at the bottom of the spray cone was kept at 100 mm/s, so that the translational speed of the sliding part I was set as 100 mm/s when the nozzle painting the sections A and C, and the angular speed of the sliding part II was set as 0.556 rad/s when the nozzle painting the arc section B.

The simulation was implemented on a workstation with ten processors, and the calculation was achieved with the software—ANSYS Fluent, where the model was discretized by the second-order upwind scheme and the discretized equations were solved by the PC-SIMPLE algorithm. The time-step size Δt for iteration is set as 1 × 10⁻⁴ s.

The stationary part and the rotational part were meshed with polyhedral cells, and the translational part was meshed with hexahedrons. The grids around the paint and air holes and the workpiece were refined. A mesh independence study was carried out based on three sets of meshes (coarse, medium, fine) for each type of the arc connecting surface. The numbers of the cells for all the meshes are shown in

Table 2. The cell numbers of meshes for independent study.

Mesh Quality	60-Degree	90-Degree	120-Degree
coarse	902,540 cells	824,670 cells	784,560 cells
medium	1,565,807 cells	1,443,500 cells	1,259,783 cells
fine	3,751,240 cells	3,420,109 cells	2,987,569 cells

. In each case, the simulation was run for a total number of 10⁴ time steps, which means the nozzle moved 1 s. Then, results of the air and paint velocities along the nozzle axis of the three sets of meshes were compared as shown in Figure 5. Because the results for the surfaces with different angles are almost the same, only those for 90-degree arc connecting surface are presented in Figure 5. It can be seen that as the mesh quality was improved from coarse to medium, both gas and liquid velocities changed greatly. However, as the mesh quality was improved from medium to fine, the changes in both the velocities were very small, which indicates that the calculation has been convergent for the medium type of mesh. In this work, the meshes with medium quality were selected for simulation.

4. Result and Analysis

The information of spray flow field and coating thickness can be obtained by a numerical solution. The distribution characteristic of coating thickness on the arc connecting surfaces with different angles is studied first. The characteristics of film thickness distribution are then explained by analyzing the spray flow field.

Figure 6 presents the contours of spray flow field and coating thickness when the nozzle is moving to different sections of the arc connecting surfaces. The average paint film thickness distributions along the Y-axis of the arc connecting surfaces with different angles are given in Figure 7.

Both Figure 6 and Figure 7 show that for all the cases simulated, the film thickness was larger on the arc surface (section B) than those on the planes (sections A and C), and the film thicknesses on the planes (sections A and C) of different arc connecting surfaces were approximate.

In order to explain the causes, analysis of distributions of paint velocity vectors when the nozzle is moving to different sections of the 90-degree arc connecting surface (Figure 8) were carried out.

Figure 8a,b shows that during the period when the nozzle paints the section A, the vectors of paint on both sides of the spray are parallel to the plane, while vectors of paint impinge on the arc surface B due to the concavity, which means that some of the paint deposit on the arc surface instead of the plane during this period. During the period when the section B (the arc surface) is being painted as shown as Figure 8c, more paint would deposit on the arc surface than the plane due to the concavity. As for the period of painting section C, the situation would be the same as the period of painting section C because of the symmetry of this type of surface. To sum up, more paint would impinge on the arc surface (section B) than on the planes (sections A and C) resulting to the larger film thickness on the arc surface than on the planes of the arc connecting surface.

Figure 9 shows the thickness distributions along the middle line on the 90-degree arc surface (section B) when the nozzle moves for 1 s, 1.5 s, 2 s, 2.5 s and 3 s, on which times the nozzle still paints the plane (section A). It can be seen that even when the nozzle paints section A, the paint thickness would be built on B, and the thickness would be higher when the nozzle is nearer to the section B. This result can also justify the above analysis of velocity vectors.

5. Experiment

The experiments of painting arc connecting surfaces with angles of 60, 90 and 120 degrees were conducted in order to verify the simulation results. The operating conditions of the nozzle were set to be consistent with the simulation, and the nozzle paths and speeds were set so that the motions of the nozzle in the experiments were the same with those in simulation cases. After being painted under the predetermined paths, the workpieces were dried out under room temperature. The obtained coating patterns are shown in Figure 10. Then the coating thickness measured by a coating thickness gauge. For each section, five positions along the nozzle motion direction were chosen, and the measured points were set at an interval of 1 cm wide for each position. After the measurement, the average coating thickness for each section was obtained through calculating the mean of the thickness data on the five positions.

In order to compare the experiment and the simulation, the wet paint film thickness distributions obtained through simulation were converted into dried ones based on volume shrink factor (due to the evaporation of the solvent component) of the paint. The comparison between the experimental and simulated dry coating thickness distributions on the surface with an angle of 90 degree is shown as Figure 11. A good agreement between the experiment and the simulation is clearly exhibited, and this indicates that the sliding mesh model and the governing equations are effective.

6. Conclusions

The paper proposed a sliding mesh model-based method to conduct a dynamic painting simulation for arc connecting surfaces. In order to meet the requirements for using the sliding mesh model, the computational domain was designed with three parts—a stationary part, a translational part and a rotational part. The film-forming model based on the Euler–Euler model and conservation equation for sliding mesh model were established. The dynamic painting of arc connecting surfaces with the angles of 60, 90 and 120 degrees and a nozzle speed of 100 mm/s were simulated. Through the simulation, the spray flow fields and the film thicknesses for painting the different surfaces when the nozzle moved to the different sections were obtained. It was found that the thickness on the arc surface section was larger than those on the plane sections, and the film thicknesses on the planes (sections A and C) of different arc connecting surfaces were approximate. Through an analysis of the vector distribution, the reason for this was that more paint would deposit on the arc surface section than on the plane sections due to the concavity of the arc surface. The evolution of paint film thickness distribution on a position of the arc surface (section B) during the period of the nozzle painting along the plane (section A) was obtained. It showed that when the nozzle paints section A, the paint thickness would be built on B, and the thick-ness would be higher when the nozzle is nearer to the section B, which also justifies the analysis of velocity vectors. The results obtained from the simulation were in a reasonable agreement with the experimental results, which indicates the proposed method to be effective and applicable in simulating dynamic painting process.

The method proposed in this paper can provide a theoretical basis for robotic painting of arc connecting surfaces and other complex surfaces. Our possible future work would include the study of automatically varying painting parameters such as nozzle speed and path and air and paint flowrate, in order to save paint and achieve a uniform thickness over the whole arc connecting surface.