Numerical modeling of the spray/spin coating of the interior of metal beverage cans: complete three-dimensional simulation

Abstract

In this work we develop a numerical simulation of the spray coating of spinning beverage cans. Though the substrate of the can must be axisymmetric, the coating need not be. We start with an evolution equation, which was derived using scaling arguments and perturbation theory. We then use implicit finite differences and an ADI scheme, with periodic boundary conditions, to efficiently solve the problem numerically. The spray fan is modeled as an expanding ellipse, and we use parameters typical of the coating industry in our simulations. The simulations show that if the can rotates an exact integral number of rotations during the spray process, then the coating layer is almost axisymmetric. But when this cannot be achieved, then three-dimensional effects greatly change the coating dynamics of the thin liquid film and must be included in the analysis.

Appendix A: numerical scheme

The functions $\tilde{\mathrm{A}}$, $\tilde{\mathrm{B}}$, $\tilde{\mathrm{C}}$, $\tilde{\mathrm{D}}$, and $\tilde{\mathrm{E}}$ shall be defined explicitly in what follows. First we introduce the operators

$$\begin{aligned} \left( \varLambda ^k_{i+1/2,j}\right) _{30}= & {} \frac{\left[ \hat{h}^k_{i+2,j}-3\hat{h}^k_{i+1,j} +3\hat{h}^k_{i,j}-\hat{h}^k_{i-1,j}\right] }{(\varDelta \hat{s})^3}\\ \left( \varLambda ^k_{i+1/2,j}\right) _{20}= & {} \frac{\left[ \hat{h}^k_{i+2,j}- \hat{h}^k_{i+1,j} -\hat{h}^k_{i,j}+\hat{h}^k_{i-1,j}\right] }{(2\varDelta \hat{s})^2}\\ \left( \varLambda ^k_{i+1/2,j}\right) _{10}= & {} \frac{\left[ \hat{h}^k_{i+1,j} -\hat{h}^k_{i,j}\right] }{(\varDelta \hat{s})}\\ \left( \varLambda ^k_{i,j+1/2}\right) _{03}= & {} \frac{\left[ \hat{h}^k_{i,j+2}-3\hat{h}^k_{i,j+1} +3\hat{h}^k_{i,j}-\hat{h}^k_{i,j-1}\right] }{(\varDelta \phi )^3}\\ \left( \varLambda ^k_{i,j+1/2}\right) _{02}= & {} \frac{\left[ \hat{h}^k_{i,j+2}- \hat{h}^k_{i,j+1} -\hat{h}^k_{i,j}+\hat{h}^k_{i,j-1}\right] }{2(\varDelta \phi )^2}\\ \left( \varLambda ^k_{i,j+1/2}\right) _{01}= & {} \frac{\left[ \hat{h}^k_{i,j+1} -\hat{h}^k_{i,j}\right] }{(\varDelta \phi )}\\ \left( \varLambda ^k_{i+1/2,j+1/2}\right) _{11}= & {} \frac{\left[ \hat{h}^k_{i+1,j} -\hat{h}^k_{i,j}\right] }{(\varDelta \hat{s}\varDelta \phi )} -\frac{\left[ \hat{h}^k_{i,j+1} -\hat{h}^k_{i,j}\right] }{(\varDelta \hat{s}\varDelta \phi )}\\ \left( \varLambda ^k_{i+1/2,j+1/2}\right) _{21}= & {} \frac{\left[ \hat{h}^k_{i+2,j+1}- \hat{h}^k_{i+1,j+1} -\hat{h}^k_{i,j+1}+\hat{h}^k_{i-1,j+1}\right] }{2(\varDelta \hat{s})^2\varDelta \phi } -\frac{\left[ \hat{h}^k_{i+2,j}- \hat{h}^k_{i+1,j} -\hat{h}^k_{i,j}+\hat{h}^k_{i-1,j}\right] }{2(\varDelta \hat{s})^2\varDelta \phi }\\ \left( \varLambda ^k_{i+1/2,j+1/2}\right) _{12}= & {} \frac{\left[ \hat{h}^k_{i,j+2}- \hat{h}^k_{i,j+1} -\hat{h}^k_{i,j}+\hat{h}^k_{i,j-1}\right] }{2(\varDelta \phi )^2\varDelta s}\\&\quad -\frac{\left[ \hat{h}^k_{i+2,j}- \hat{h}^k_{i+1,j} -\hat{h}^k_{i,j}+\hat{h}^k_{i,j-1}\right] }{2(\varDelta \phi )^2\varDelta s} \end{aligned}$$

and the definitions

$$\begin{aligned} \hat{r}_{i+1/2}= & {} \frac{\hat{r}_{i+1}+\hat{r}_i}{2}\\ \left( \hat{r}'\right) _{i+1/2}= & {} \frac{\hat{r}_{i+1}-\hat{r}_i}{\varDelta h_{r}}\\ \left( \hat{r}''\right) _{i+1/2}= & {} \frac{\hat{r}_{i+2}-\hat{r}_{i+1}+\hat{r}_i-\hat{r}_{i-1}}{2(\varDelta \hat{r})^2}\\ (\varDelta \hat{r})_{i+1/2}= & {} (\hat{r}_{i+1}-\hat{r}_{i}), \end{aligned}$$

where the subscript $i+1/2$ means that the quantity is evaluated between the nodes i and $i+1$. Using these operators we make the following definitions:

$$\begin{aligned}&\tilde{\mathrm{A}}^k_{i,j} =-\left[ \lambda _2\sin \theta _i+\varGamma _2\hat{r}_i\cos \theta _i \right] \\&\tilde{\mathrm{B}}^{k+1/2}_{i,j} = -\left( \varLambda ^{k+1/2}_{i+1/2,j}\right) _{30} -\left( \varLambda ^{k+1/2}_{i+1/2,j}\right) _{20}\\&\quad \left( \frac{\left( \hat{r}'\right) _{i+1/2}}{\hat{r}_{i+1/2}}\right) .\\ -\left( \varLambda ^{k+1/2}_{i+1/2,j}\right) _{10} \left( \frac{\left( \hat{r}''\right) _{i+1/2}}{\hat{r}_{i+1/2}}\right) \\&\quad +\left( \frac{\left( \hat{r}'\right) _{i+1/2}}{(\hat{r}_{i+1/2})^2}\right) \\&\tilde{\mathrm{C}}^k_{i,j} = -\left( \varLambda ^k_{i,j+1/2}\right) _{01} \frac{\lambda _2\cos (\theta _i)}{\hat{r}_{i}}\\&\quad -\left( \varLambda ^k_{i,j+1/2}\right) _{01} \varGamma _2\sin \theta _i\\&\quad -\left( \varLambda ^k_{i,j+1/2}\right) _{11} \left[ \frac{(r')_i}{(\hat{r}_i)^2}\right] \\&\quad -\frac{\left( \varLambda ^k_{i,j}\right) _{21}}{\hat{r}_{i}}\\&\quad -\frac{\left( \varLambda ^k_{i,j}\right) _{03}}{(\hat{r}_{i})^3}\\&\tilde{\mathrm{D}}^{k+1}_{i,j} = -\left( \varLambda ^{k+1}_{i,j+1/2}\right) _{01} \frac{\lambda _2\cos (\theta _i)}{\hat{r}_{i}}\\&\quad -\left( \varLambda ^{k+1}_{i,j+1/2}\right) _{01} \varGamma _2\sin \theta _i\\&\quad -\left( \varLambda ^{k+1}_{i,j+1/2}\right) _{11} \left[ \frac{(r')_i}{(r_i)^2}\right] \\&\quad -\frac{\left( \varLambda ^{k+1}_{i,j}\right) _{21}}{\hat{r}_{i}}\\&\quad -\frac{\left( \varLambda ^{k+1}_{i,j}\right) _{03}}{(\hat{r}_{i})^3}\\&\tilde{\mathrm{E}}^{k}_{i,j} = -\left( \varLambda ^{k}_{i,j+1/2}\right) _{01} \frac{\lambda _2\cos (\theta _i)}{\hat{r}_{i}}\\&\quad -\left( \varLambda ^{k}_{i,j+1/2}\right) _{01} \varGamma _2\sin \theta _i\\&\quad -\left( \varLambda ^{k}_{i,j+1/2}\right) _{11} \left[ \frac{(r')_i}{(r_i)^2}\right] \\&\quad -\frac{\left( \varLambda ^{k}_{i,j}\right) _{21}}{\hat{r}_{i}}\\&\quad -\frac{\left( \varLambda ^{k}_{i,j}\right) _{03}}{(\hat{r}_{i})^3}.\\ \end{aligned}$$