Extrusion International 2-2023

45 Extrusion International 2/2023 range is impossible in most cases. Grey-boy models com- bine the advantages of white- and black-box models. A well known example of grey box modelling is linear re- gression, which establishes the functional, statistical re- lationship between the input and output variables with - out using physical modelling. For this, an influence of a variable on the target variable must be assumed, e.g. in the context of a sensitivity analysis. If there is a basic un - derstanding of the process, but the model quality is not sufficient, a black box model can be used. In this case, the main influences are first described mathematically (phys- ically or statistically) and then optimized by data that is not part of the white or grey box model [NN19b]. Follow- ing Ohlendorf, the suitability of linear regression is first investigated to model the tensile strength. For a linear regression there is the requirement that the coefficients must be linearly independent or not corre- lated. Table 4 shows the correlation matrix of the coeffi- cients. Correlation coefficient of 1 or -1 means that there is a strictly linear relationship between the parameters. For a linear regression, a regression coefficient between -0.2 and 0.2 is desirable for (marked in grey). Due to the high correlation of the input parameters, they are not suitable for multiple regression, as suffi- cient accuracy cannot be achieved to predict the tensile strength. None of the parameters have a significant influ- ence on the tensile strength in the longitudinal or trans- verse direction. The property model can therefore not be extended by adding further lin- ear terms in the sense of linear regression. An alternative is the method of Partial Last Square Regression (PLS). The advantage of this method is that the input parameters of the model may be highly correlated or intercor- related. If the PLS is carried out exclusively with the derived co - efficients, without considering the "classical" parameters of the property model, a model quality of r2 = 0.602 and r2 = 0.537 re - sults for the calculation of the tensile strength in the longitudi- nal and cross direction. Larger values of r2 (close to 1) indicate a high model quality, low values (close to 0) indicate a lowmodel quality. Considered on its own, the model is therefore not yet suitable for prediction. However, this finding must be set in re- lation to the fact that, in addition to the added coefficients, no further elemen- tary process parameters, such as the melt temperature or the take-off ratio, have yet been included in the PLS model. In this respect, an enormous potential of the collected parameters for the improve - ment of model quality can possibly be as - sumed. Another possibility is the use of the conventional model in combination with an artifactial neural network, which is likely able to represent also non-linear correlations of the derived coefficients with the tensile strength in a higher model quality (black box model as booster). A pure black box model including the parameters derived can also be considered. Conclusions and outlook The prediction of mechanical parameters during the blown film process is an important prerequisite for prod- uct development and quality optimization. Statistical or data-driven models are possibly suitable for modelling the influence of the process parameters on the mechani- cal properties. The problem of physically based models is the complex numerical solution with a high required number of input parameters. Ohlendorf derives a prop - erty model by an empirical-statistical approach, which al- ready allows a high prediction accuracy for the prediction of the Young’s modulus and the film shrinkage. However, a prediction of the tensile strength using this model is not yet possible with sufficient accuracy. This is due to an in- complete description of the tube formation zone, inwhich the mechanical properties are set. An extended model ap - proach therefore should describe the stretching and cool- Picture 4: Non-linear, non-quadratic behaviour of bubble temperature Picture 5: Detection of the bubble contour and approximation by a 5th degree polynomial