Extrusion International 2-2023

42 Extrusion International 2/2023 BLOWN FILM EXTRUSION – FROM THE RESEARCH The statistical process model according to Ohlendorf The basis of the process model accord - ing to Ohlendorf is the statistical design of experiments, according to which ma- chine parameters are systematically var - ied and lead to different process states and thus also to different mechanical properties of the film. Based on the re- sulting process state, the process pa- rameters are collected and converted into dimensionless numbers in order to enable a machine-independent model- ling. Subsequently, the film samples are examined for the required mechanical properties in longitudinal and transverse direction, e.g. tensile strength. Linear re- gression is used to determine the influ- ence of the dimensionless numbers on the tensile strength. As a result, a linear model for the calculation of the tensile strength is es- tablished. The procedure for deriving the model can be taken from Picture 2. First, the well-known, dimensionless numbers take- up ratio (TUR) and blow-up ratio (BUR) are introduced, which are regarded as ameasure of the elongation in the extrusion direction and in the circumferential direction. The ratioofmelt and frost line temperaturedescribes the cooling of the tube forming zone in extrusion direction. In conjunction with the volume flow and frost line ratio, this ratio quantifies the effectiveness of the cooling ring. The influence of the film thickness on the mechanical properties is taken into account by the thickness ratio of the film thickness and the outlet gap width of the die. Take-up ratio: i l = film line speed / melt speed (3.1) Blow-up ration: i q = bubble diameter / die outlet diameter (3.2) Temperature ratio: i t = melt temperature / frost line temperature (3.3) Volume flow ratio: i v = melt volume flow / cooling air flow (3.4) Frost line ratio: i f = frost line height / die outlet radius (3.5) Thickness ratio: i d = film thickness / gap width (3.6) Ohlendorf establishes a linear relationship between the established dimensionless numbers and the me - chanical characteristic values (Youngs’s modulus, ten- sile strength, shrinkage). Applied to a specific material like LDPE LD 150 AC of ExxonMobil Corp., Inving, Texas, United States, the following linear calculation equation can be established for calculating the mechanics: Mechanical parameter = l ∙ i l + q ∙ i q + t ∙ i t + v ∙ i v + f ∙ i f + d ∙ i d + k (3.7) Provided that the resulting process characteristics are linearly independent and the residuals are normally dis - tributed, the regression coefficients l, q, t, v, f, d and k can be determined from the experimental space using linear regression. To evaluate the quality of the model, the coefficient of determination r2 is used, which can assume values between zero and one and assesses the quality the model. A coefficient of determination of one indicates that the linear relationship assumed by the model is matched perfectly by real world data. Furthermore, the average percentage deviation (a) between the forecast and the measured value is used as a further evaluation criterion. Table 1 compares the coefficient of determination and the percentage deviation for predicting the mod- ulus of elasticity, tensile strength and shrinkage in the longitudinal and transverse directions. When analysing the model quality, it becomes clear that the model has the lowest accuracy when predict - ing tensile strength. In particular, the model does not provide sufficient accuracy transverse to the extrusion direction. It must therefore be stated that the statistical property model is suitable at most for rough estimation of the tensile strength. Due to the modelling with process parameters, the model must be transferable to other machines. Ohlen - dorf therefore evaluates the transferability of the Young’s modulus and the shrinkage behaviour to other production machines. Provided that the process condi - tion is described sufficiently by the key figures intro- duced, no significant change in the model quality is to be expected. Table 2 shows an example of the coefficient Picture 2: Procedure of the statistical process model [Ohl04]