Speaker | Transcript |
| Marcello FIDALEO | Hi, my name is Marcello Fidaleo, and my presentation is about the functional data analysis and design of experiments applied to food milling processes. |
| The aim of this research was to study the milling step of hazelnut and cocoa bean base used the in the manufacturing of ice creams. |
| These process is called refining and is typically carried out in stirred ball mills in the batch mode, like this one report to the here on the right. |
| The aim of this process is to reduce the particle size of the solid ingredients, such as cocoa and sugar, so that the final product is not gritty. |
| To this aim we designed our central composite design like one report here on the right. We did three factors, N, D and S, where |
| N the shaft the rotation speed, D is the ball diameter, and S is the overall mass of balls. |
| As the responses, we consider that the size of largest solid particles, so that is the fineness, and the milling energy. Both responses were measured as a function of time, so this was actually a |
| functional designed experiment, because the responses were not ???, but were functions, in this particular case, functions of time. We used JMP Pro to design the experiment and to analyze the results. |
| For the analysis of the results, we followed two approaches. The first approach is the classical analysis of designed experiments. |
| So we use the response surface models to regressive the responses as a function of the process parameters at two different time instance. In the second approach, which is a functional data analysis in design of experiments, we |
| were able to include also the effect of time in the final model, that is, we considered the |
| nature of the functional responses so...that's the functional nature of the responses. So in this presentation, I will talk about the second approach. |
| Functional data analysis in design of experiments |
| requires some intermediate steps to build the final model and also to obtain the design space of the system. |
| Basically, we start with the functional data analysis to smooth the functional data. |
| Then we apply functional principal component analysis to the smoothing functions. And so, at this point, by retaining a just a few components of this system we developed a model for the functional responses. So |
| let's see the results... |
| the results of this |
| case studies. |
| These are the results of the smoothing procedure. We applied the beta splines and we considered the ??? fineness and the energy. We can see that by using data transformation and also data filtering simple |
| fitting functions, we're able to fit well the experimental data. In fact, we used one knot and a cubic spline and the one knot and a linear spline for fineness and for energy. |
| So then we applied the function principal component analysis to the smoothing function. Here on the left, we can see that for fineness, |
| the first two components explained 96% of the variance, while for energy, just the first component explained 98% of the varience. So in the final models, |
| we used the two components for fineness and one component for energy. Here on the right, I reported the scores calculated for the 16 trials of the experiment. |
| And we found from our experience that the scores are useful to understand the behavior of the system. For example, in this case, the first component acted as a |
| grinding intensity axis with the high grinding intensity runs on the left and low grinding intensity runs on the right. And so, at this point, we |
| regressed the scores as a function of the process parameters by using a responsiveness(?) model |
| by including the linear quadratic and the interaction effects. Also these these models were very useful to understand the effect of the process parameters on the functional |
| responses. |
| So finally, |
| here, we can see that finally, we were able to build the final models, as I said. And here I reported the fineness and the energy as a function of time for a few runs. And we can see that the agreement between the experimental data and the predictio...prediction ones |
| are really good. As I said before, we use the two components for fineness and one component for energy. |
| So, finally, we were able to build the design space and to study...to understand the effect of the process parameters, that is N, D and S, |
| on the functional responses here. On the profiler, we can see that, under these conditions of N, D and S, |
| we could predict the profile of fineness as a function of time and the profile of energy as a function of time. At the bottom...at the bottom here, I reported two contour plots |
| of the system. The one on the left was obtained with the 55 rpm rotation speed and 6.5 millimeter of ball diameter. |
| While on the right, the contour plot was used with the mass of spheres of 29 kilograms and an operating time of 80 minutes. |
| So, for example here on the left, we can see that in the boundary that we have the mass of spheres when we increase the the operating time, |
| the energy, of course, increases, while the fineness decreases. |
| The white areas in both plots is the design space, and it is the the area and in which fineness is between 20 and 30 microns, so it's the area where the final product is not gritty and it's not over milled. |
| So from these results, we can conclude that the functional data analysis applied to functional designed experiments appears to be a straightforward, robust, and easy to use approach to build the dynamical design space of a batch process. |
