Numerical simulation of thin paint film flow
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DOI: 10.1186/2190-5983-2-1
- Cite this article as:
- Figliuzzi, B., Jeulin, D., Lemaître, A. et al. J.Math.Industry (2012) 2: 1. doi:10.1186/2190-5983-2-1
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Abstract
Purpose
Being able to predict the visual appearance of a painted steel sheet, given its topography before paint application, is of crucial importance for car makers. Accurate modeling of the industrial painting process is required.
Results
The equations describing the leveling of the paint film are complex and their numerical simulation requires advanced mathematical tools, which are described in detail in this paper. Simulations are validated using a large experimental data base obtained with a wavefront sensor developed by Phasics™.
Conclusions
The conducted simulations are complex and require the development of advanced numerical tools, like those presented in this paper.
Keywords
thin filmsnumerical simulationindustrial painting processroughnesslubrication approximation1 Introduction
The visual appearance of painted steel sheets forming the body of a car is a prominent factor in appreciating its quality. Being able to predict it is thus of crucial importance to car makers, while remaining a serious mathematical challenge requiring accurate modeling of the industrial painting process.
the leveling of the film (flow and evaporation) which occurs during the flash time, i.e. the time period just following the end of the deposit,
baking in an oven, which favors evaporation.
The leveling process has received considerable attention in the literature, although not in the context of the industrial paints used in the automotive industry. In 1961, Orchard [1] was the first to note that the leveling dynamics is controlled by an interplay between surface tension, with capillary forces tending to reduce surface irregularities, and the fluid viscosity limiting the flow induced by that leveling. Orchard’s model is mainly based on two assumptions: the paint exhibits a Newtonian behavior and evaporation effects are negligible. To take into account the effects of evaporation, Overdiep [2] considered a fluid made of a resin and a solvent, where only the solvent can evaporate, demonstrating the potential importance of the surface tension spatial variations. Surface tension indeed depends on the paint composition, in particular on the respective proportions of resin and solvent. In the presence of evaporation, thinner regions tend to dry faster, and therefore to have lower solvent concentrations, which causes surface tension gradients, a physical phenomenon known as Marangoni effect, hence a shearing effect at the film surface, understood as the main physical effect involved in the leveling of the paint film by Overdiep. This approach was taken up and developed in several subsequent articles. Wilson [3] and later Howison et al. [4] analyzed and generalized Overdiep’s model, performing numerical simulations that showed good agreement with experimental data collected for simple deposit geometries.
The topography of the substrate on which the coating is deposited plays an important role in the flow dynamics. In 1995, Weidner et al. [5] studied the effect of substrate curvature on the film flow in a two-dimensional context. Subsequently Eres et al. [6] and later Schwartz et al. [7] generalized the work to the three-dimensional case. In these papers, numerical models have been implemented for specific topographies, showing good agreement with experimental measurements. Gaskell et al. [8, 9] finally considered the generalization of the different models to the case of inclined substrates, where gravity plays a significant physical role in the flow dynamics.
Industrial paints used in the context of the automotive industry are complex media that have not been extensively studied. Their detailed rheology is not well known, though its effects on the leveling are a key issue. In view of the complexity of the phenomena, experiments aiming at the identification of the physical effects within the film and the evaluation of their relative importance appear to be a prerequisite to film flow modeling. Using a wavefront sensor developed by Phasics™ [10], we could determine the evolution of rough surfaces accurately and with a high temporal resolution throughout the whole painting process [11]. In Section 2, we describe the mathematical model used to model the evolution of the painted film topography and its numerical simulation. Section 3 is devoted to the presentation of the experimental data obtained with the wavefront sensor. Rheological parameters extracted from the experimental data are used in Section 4 to perform a simulation of the topography evolution during the painting process. Conclusions are drawn in Section 5.
2 The mathematical model and its implementation
Following the accepted practice, we study the leveling process within the framework of a lubrication approximation, but more elaborate theories can be developed from the Navier-Stokes equations [12–17]. The lubrication approximation builds on two observations: firstly, the thin film flow is very slow, so that it becomes possible to neglect the inertia terms in the Navier-Stokes equation; secondly, the thickness of the film is much smaller than the wavelength of the modulations along the surface, which also implies that the fluid velocity is essentially directed parallel the surface. All this allows a substantial simplification of the equations describing the flow of the thin paint film.
2.1 Physical model
2.1.1 Lubrication approximation
where ${\mathbf{\nabla}}_{\mathrm{h}}$ is the gradient along the plane $(x,y)$.
2.1.2 Paint rheology
Equations 6-8 have been derived without making any assumptions about the paint rheology. To close these equations, we have to prescribe how the mass flux q depends on the local pressure gradient.
Estimating the left hand side of Equation 9 indeed allows the access to the local values of the mass flux by solving the Poisson equation, and hence permits us to test the rheological model. The so-obtained data showed that for the space and time scales involved in the problem, the film can be considered as Newtonian.
2.1.3 Newtonian model equation
We will assume that the paint is composed of a resin in concentration $1-c$ and a solvent in concentration c. Only the solvent can evaporate, while the evaporation rate will essentially depend on the solvent concentration. Accordingly, we shall assume that the largest scales patterns attenuation is mainly caused by evaporation, for a leveling caused by surface tension would suppose a huge mass transport which would be unrealistic considering the geometric characteristics of the painted film. A method based on this idea is presented in [11], which allows a determination of the evaporation rate as a function of c. If we neglect the local variations of the solvent concentration, the evaporation rate will consequently be spatially constant, and will only vary with time.
2.1.4 Marangoni effect
The combination of Equations 18 and 20 completely describes the evolution of the film topography. The physical parameter γ is related to the solvent concentration by the law presented later on Figure 7.
2.2 Numerical implementation
where F is a non-linear function of the spatial derivatives. The method of lines [18] is used to solve Equation 21, in combination with a pseudo-spectral method: Function F is evaluated in the Fourier space and Equation 21 is integrated using an adaptative step size Runge-Kutta scheme.
2.2.1 Evaluation of spatial gradients
This inequality ensure that the quantities $k-M-{k}_{1}$ and $k+M-{k}_{1}$ fall into the intervals $]-\frac{M}{2},-\frac{N}{2}[$ and $]\frac{N}{2},\frac{M}{2}[$. The argument is easily extended to higher degree nonlinearities. Since Equation 15 involves fourth-degree monomials, full desaliasing requires $M=\frac{5N}{2}$.
2.2.2 Integration of the equation
To specify a particular method, one simply has to set the coefficients ${a}_{ij}$, ${b}_{i}$ and ${c}_{i}$ which characterize the discretization of the equation for $i=1,2,\dots ,N$ and $j=1,2,\dots ,i$. The selected coefficients can be represented in a table called the Butcher table.
A first evaluation ${\psi}_{n+1}={\psi}_{n}+\mathrm{\Delta}t{\sum}_{i=1}^{N}{b}_{i}{k}_{i}$ accurate at order N.
A second evaluation ${\psi}_{n+1}={\psi}_{n}+\mathrm{\Delta}t{\sum}_{i=1}^{N-1}b{\ast}_{i}{k}_{i}$ accurate at order $N-1$, which uses an other ponderation $\{b{\ast}_{i}\}$, $i=1,2,\dots ,N$.
General Butcher table of a Runge-Kutta scheme.
Butcher table of the Heun scheme.
Butcher table of the Bogacki-Shampine scheme.
Butcher table of the Cash-Karp scheme.
The dynamics of the paint levelling varies considerably during the painting process, and it is then of interest to use an adaptive stepsize integration scheme. A method described in [22] is used to adjust the time step, which uses the error estimate returned by the integration scheme.
2.3 Validation of the numerical scheme
Physical parameters used for the numerical scheme validation.
Parameter | Symbol | Value | Unit |
---|---|---|---|
Surface tension | γ | 3.0 × 10^{−2} | N/m |
Paint viscosity | η | 1.0 | Pa.s |
Initial thickness | ${e}_{0}$ | 20.0 | μ m |
3 Experimental measurements
Paint is deposited over a sample of metal sheet (polished or already covered with an electrophoresis layer) in a painting cabin using a paint gun.
The sheet is then placed on a baking plate. During the first few minutes, complete samplings of the surface are performed at regular time intervals (typically 2.5 Hz), in order to record the evolution of the painted layer topography at the beginning of the flash time in detail.
After two minutes the sampling rate is decreased to 0.1 Hz, for the flow dynamic next slows down considerably.
The baking cycle starts after 10 minutes, with the sampling frequency reincreased to 1.25 Hz.
Chemical bonds begin to form within the paint 5 minutes after the beginning of the baking. Cross-linking then stops the evolution so that the sampling frequency can be decreased to 0.1 Hz.
The wavefront sensor collects information over a surface of $18\times 18{\text{mm}}^{2}$. The topography is analyzed as a $128\times 128$ square image. Each pixel represents the mean altitude over a $60\times 60{\mu \text{m}}^{2}$ surface. The precision of vertical measurements is up to ${10}^{-2}\mu \text{m}$.
During the flash time, the outside temperature is 25^{∘}C. The baking cycle is divided into two stages: a linear temperature rise during 300 sec until reaching 150^{∘}C, followed by a 15 minutes plateau at this temperature.
3.1 Surface evolution
The following figures show the evolution of the topography of a lacquer layer during the whole painting process. The lacquer is deposited on a smooth substrate. Altitudes are given in μ m. On each surface, during measurements, the minimum is arbitrarily set to zero since only relative but not absolute altitudes can be obtained from the device.
3.2 Evolution of the roughness
Roughness evolution during the painting process helps us quantifying the paint leveling capability. Since the physical effects involved develop at different scales, it is of interest to play with tools able to separate the different roughness scales. The surface is sampled with a 60 μ m horizontal step, yielding a $128\times 128$ image $S=S[{n}_{1},{n}_{2}]$. An algorithm based on the wavelet packet transform [23] and the reconstruction formula is used, that allows a decomposition of the roughness into a sum of contributions [24–26].
The scale-by-scale study of the surface roughness provides valuable information on the dynamics of the leveling. The curves in Figure 6 show little leveling during the baking, the difference being mainly due to evaporation since the two curves are quite similar. In the next section we therefore focus only on the simulation of the surface dynamics during flash time, when both flow and evaporation are involved.
4 Direct simulation
Simulation parameters.
Parameter | Symbol | Value | Value in [2] | Unit |
---|---|---|---|---|
Resin surface tension | ${\gamma}_{r}$ | - | 3.0 × 10^{−2} | N/m |
Solvent surface tension | ${\gamma}_{s}$ | - | 2.5 × 10^{−2} | N/m |
Initial paint viscosity | ${\eta}_{0}$ | - | 0.55-1.59 | Pa.s |
Initial solvent concentration | ${c}_{0}$ | 0.58 | 0.5 | - |
Viscosity exponent | a | 18 | 15 [5] | - |
Evaporation parameter | λ | 4.0 × 10^{−9} | 2.0 × 10^{−9} | m/s |
Initial rheological parameter | ${\gamma}_{0}/3{\eta}_{0}$ | 1.0 × 10^{4} | 1.0 × 10^{4} | μ m/s |
Using the experimental data obtained with the wavefront analyzer and the evaporation law deduced from these measurements, simulations were performed with the two models described in Section 2. These simulations start from the first reconstructed topography and aim at reproducing the entire evolution of the film during the flash time. Parameters used are given in Table 6 obtained as explained above. We consider that the substrate is completely smooth. The numerical resolution code was described at the end of Section 2. The simulations are performed using a 3.40 GHz Intel(R) Xeon(TM) processor, and last about five hours.
5 Conclusion
Painting of steel sheets is a complex phenomenon that depends on many physical processes. With the wavefront sensor developed by Phasics™, it was possible to perform experiments allowing an accurate monitoring of the topography of a film during its deposition. The fast response time of the wavefront sensor allowed us to access the rheological parameters of the paint in an original way by solving an inverse problem. The obtained parameters were used to perform a complete simulation of the film evolution during the painting process, which demonstrated that the Newtonian model was able to reproduce the leveling of the paint layer accurately and that Marangoni effect could be neglected at the beginning of the flash time, when significant flow occurs. At the end of the flash time, the flow rates decreases and it is clear that the film then exhibits a more complex rheology due to the solvent evaporation, but the leveling dynamic is then considerably attenuated, and the influence on the surface topography is negligible. The conducted simulations are however complex and require the development of advanced numerical tools, like those presented in this paper.
Supplementary material
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