A Generalized 2D Output Model of Polymer Melt Flow in Single-Screw Extrusion

REGULAR CONTRIBUTED ARTICLES S. Pachner 1 *, B. Löw-Baselli 1, M. Affenzeller 2, J. Miethlinger 1 1 Institute of Polymer Etrusion and Compounding, Johannes Kepler University Linz, Linz, Austria 2 School of Informatics, Communications and Media, University of Applied Sciences Upper Austria, Hagenberg, Austria A Generalized 2D Output Model of Polymer Melt Flow in Single-Screw Etrusion The multidimensional output-pressure behavior of non-newtonian fluids in single-screw etrusion can only be determined by using numerical methods. We present two methods which employ mathematical models building on analytic equations developed using an evolutionary heuristic optimization algorithm. Both allow fast and stable calculation of the 2-dimensional throughput pressure gradient relationship of singlescrew etruders, rendering cost-intensive CFD simulations of the output-pressure behavior redundant. A performed error analysis showed that our methods yield good approimations of the numerically determined data. 1 Introduction With increasing heli angle of a single-screw etruder and decreasing power-law inde of a polymer melt, the differences in the results of one- and two-dimensional analyses of the throughput pressure gradient characteristics of the metering section increase: For a power-law inde below 0.5, the difference in the dimensionless volume throughput amounts to 10 % for small pressure gradients and to about 40 % for large pressure gradients. The differences decrease to less than 10 % with increasing power-law eponent. Large heli angles and strongly non-newtonian fluids therefore require two-dimensional analysis (Rauwendaal, 2014). Unlike the one-dimensional calculation, two-dimensional flow analysis considers both down-channel and cross-channel flow. Considering steady-state and isothermal flow conditions, wall adherence, an infinite channel width (i. e., ignoring the effect of flight flanks) and using the flat-plate model (i.e., ignoring channel curvature) while ignoring body and inertia forces, the equations of motion in the down-channel and cross-channel directions can be reduced to: 0 ¼ qp qz þ qs yz qy ; 0 ¼ qp q þ qs y qy : ð1þ ð2þ * Mail address: Sophie Pachner, Institute of Polymer Etrusion and Compounding, Johannes Kepler University Linz, Altenbergerstr. 69, 4040 Linz, Austria E-mail: sophie.pachner@jku.at For power-law fluids, with power-law inde n and consistency inde K, Eqs. 1 and 2 can be rewritten as: 8" 9 #n 1 0 ¼ qp qz þ K q < qv 2 þ qv 2 2 z qv = z qy : qy qy qy ; ; ð3þ 8 0 ¼ qp q þ K q < qy : " #n 1 2 qv 2 þ qv 2 z qy qy 9 qv = qy ; : ð4þ By introducing the dimensionless characteristic numbers n and w i : n ¼ y h ; w z ¼ v z v 0z and w ¼ v v 0 ; the equations of motion in the down-channel (Eq. 3) and crosschannel directions (Eq. 4), can be converted into dimensionless forms: qp h 1þn qz v 0z n ¼ p pz 8 < ¼ q : qp h 1þn q v 0z n ¼ p p 8 < ¼ q : " #n 1 2 qw 2 tan 2 þ qw 2 z " #n 1 2 qw 2 tan 2 þ qw 2 z qw z 9 = ð5þ ð6þ ; ; ð7þ 9 qw = tan : ð8þ ; Since no analytic solutions to this problem have been found to date, numerical approaches such as finite-difference, finite-element or finite-volume methods must be applied to describe the 2D dimensionless throughput pressure gradient relationship. Intern. Polymer Processing XXXII (2017) 2 Ó Carl Hanser Verlag, Munich 209

The dimensionless volume throughput p V and the pressure gradient p p are defined as: p V ¼ 2 _V i v 0z ; p p ¼ h1þn qp 6 v n 0z qz : ð9þ ð10þ Figure 1 shows the dimensionless throughput pressure gradient relationship of a square-pitched screw for different powerlaw eponents. 2 State of the Art Various approimation methods eist that describe the 2-dimensional throughput pressure gradient relationship without the need for comple numerical calculations; for instance, Steller (Steller, 1990; Steller and Iwko, 2001) developed an analytical solution for determining the 2-dimensional flow of powerlaw and Ellis fluids. These approaches considerably simplify the numerical procedures but, these methods require numerical analysis to evaluate the solution. Rauwendaal (2014) introduced correction factors for non-newtonian behavior of the polymer melt. These correction factors are valid for heli angles within a range of 158 to 258 respectively 0.8 to 1.5 D. The dimensionless volume throughput can be calculated as: p V ¼ 8 þ 2 10 3 1 þ 2 p p: ð11þ Figure 2A shows the 2-dimensional throughput pressure gradient relationship of a square-pitched screw for a range of power-law eponents computed using (i) numerical calculations and (ii) Rauwendaal s method. Figure 2B illustrates the influence of the screw pitch on the dimensionless volume throughput for different power-law eponents. The linear approimation method according to Rauwendaal is independent of the screw pitch, but the dimensionless volume throughput clearly depends on the heli angle of the screw (see Fig. 2B). Especially for small power-law eponents, the influence of the heli angle on the dimensionless throughput becomes more important. For eample, for a polymer melt with a power-law eponent of 0.2, the difference in the dimensionless volume throughput between screw pitches of 1 D and 1.5 D is almost 15%. Nevertheless, due to its simplicity, the linearization method is widely used in industry. Potente in Potente (1983) and in White et al. (2003) introduced a method for describing the 2-dimensional flow behavior of polymer melts in single-screw etrusion. This approimation is only valid when: 0:8 t=d 2; 0:1 p V 2; 0:2 n 1: A) Fig. 1. 2D throughput (p V Þ pressure gradient (p p ) behavior of a square-pitched screw for different power-law eponents n B) Fig. 2. Dimensionless throughput (p V ) versus dimensionless pressure gradient (p p ) of a square-pitched screw computed using numerical methods (solid lines) and the approimation method according to Rauwendaal (dashed lines); n denotes the power-law eponent (A); influence of the screw pitch on the dimensionless volume throughput shown for pitches of 1D (solid lines) and 1.5D (dashed lines); n denotes the power-law eponent and t the screw pitch (B) 210 Intern. Polymer Processing XXXII (2017) 2

The dimensionless volume throughput as a function of the dimensionless pressure gradient can be calculated according to Eq. 12. The correction factors U 1 and U 2 must be determined according to Eqs. 13 and 14. The coefficients A to F are constants and are chosen to match the scope of application. The total application range of this approimation method is split into 8 regions, see Table 3. In each region, the constants A to F are separately defined and can thus assume 8 different values (White et al., 2003). p V ¼ U 1 U 2 p p ; U 1 ¼ na cos n ; ð12þ ð13þ 3 Modeling We introduce two generally valid analytic equations for fast and stable computational modeling of the isothermal 2-dimensional flow of polymer melts. The equation builds on a mathematical model employing an evolutionary heuristic optimization algorithm. This method is based on numerically determined data for different heli angles of the screw and power-law eponents of the polymer melt. In total more than 10.000 data pairs of p p and p V in the total range of application were generated. For calculating the generally valid analytic equation we used the HeuristicLab (Heuristic and Evolutionary Algorithms Laboratory) open-source software, which is a heur- cos D U 2 ¼ C : nesin þfcos ð14þ The application range of the approimation method according to Potente has undefined regions, and the method is not continuous across the full range. Consider, for eample, the case of a screw with 1.5 D pitch and a melt with a flow eponent of 0.6 (see Fig. 3). Functions 1 and 2 are within both the defined scope of application and their duration limits. However, as the illustration clearly shows, there is by definition no way to determine the dimensionless volume throughput for a dimensionless pressure gradient of 0.25, and the defined functions are not continuous across the whole application range. As previously mentioned, there are considerable differences between 1- and 2-dimensional flow analyses of the metering section of a single-screw etruder, and optimal design of a screw requires 2-dimensional calculations. Since both stateof-the-art methods have their disadvantages, we present a novel approach to describing the 2-dimensional throughput pressure gradient relationship. Fig. 3. Dimensionless throughput (p V ) versus dimensionless pressure gradient (p p ) of a screw with pitch of 1.5D and a polymer melt with power-law eponent of n = 0.6 with function 1 and 2 according to Potente s approimation method; solid lines represent the application range of the functions; dashed lines are eceeding the valid range Coefficient Value Coefficient Value Coefficient Value c 0 1.1157 c 17 0.7675 c 34 0.2469 c 1 0.4709 c 18 0.6082 c 35 3.2199 c 2 0.0109 c 19 0.9863 c 36 5.3695 c 3 1.5684 c 20 1.7486 c 37 1.5250 c 4 0.0504 c 21 0.0001 c 38 2.9937 c 5 1.904 c 22 0.1917 c 39 2.0815 c 6 0.0280 c 23 0.7037 c 40 1.3206 c 7 0.9602 c 24 0.5965 c 41 0.2401 c 8 1.3160 c 25 0.2943 c 42 0.3233 c 9 1.1135 c 26 1.2244 c 43 0.3055 c 10 0.0017 c 27 0.5774 c 44 0.0341 c 11 0.7164 c 28 0.7556 c 45 3.7904 c 12 0.0101 c 29 2.6227 c 46 13.2660 c 13 0.5809 c 30 1.8483 c 47 0.8534 c 14 0.7398 c 31 0.1583 c 48 0.6312 c 15 1.0754 c 32 0.0090 c 49 1.2899 c 16 1.6654 c 33 3.3499 c 50 0.1453 Table 1. Values of constants for equation C50 Intern. Polymer Processing XXXII (2017) 2 211

istic optimization system that features several metaheuristic algorithms for a number of optimization problems (Kronberger et al., 2012). We applied symbolic regression based on genetic programming to determine the analytic equations in order to describe the throughput pressure gradient relationship as a function of screw pitch and power-law eponent. An error analysis was performed to check the applicability of the generated equations and to determine the accuracy of the functions. The sequence of modelling is summarized in Fig. 4. 3.1 Symbolic Regression Symbolic regression is the induction of mathematical epressions on data. Among modern data-based modeling techniques, symbolic regression (Koza, 1993) distinguishes itself by its ability to identify nonlinear white-bo models without the need to specify the structure of the models. This makes a big difference to other methods of regressions where a specific model is assumed and only the compleity of this model can be varied. So the main goal of this method is to determine the relationship of a target variable to a set of specified independent input variables (Affenzeller et al., 2009) in terms of a mathematical epression (Fig. 5). Compared to conventional data based modeling techniques symbolic regression is considered to be especially suited for the given task due to the following reasons: On the one hand the set of considered models (space of hypothesis) is significantly more powerful compared to polynomial models for eample. On the other hand symbolic regression is able to identify simpler and interpretable models which are capable to handle real-time modeling tasks which is major benefit compared to Gauss regression for eample. A unique feature of symbolic regression is its ability to simultaneously optimize the model structure and its parameters. By considering a huge variety of mathematical epressions in various combinations in terms of considered input variables and structures, symbolic regression problems can be solved by means of genetic programming (Affenzeller et al., 2009; Koza, 1993). Genetic programming is a widely applicable (White et al., 2013) metaheuristics for the automated synthesis of computer programs or as in our case mathematical formulae in synta tree representation. Similar to other evolutionary algorithms like genetic algorithms (Holland, 1992) or evolution strategies (Schwefel, 1975) genetic programming handles the search process in a space of potential solution candidates by iteratively applying the evolutionary operators selection, crossover and mutation with the aim to continuously optimize solution quality based on fitness. In symbolic regression the fitness function is to minimize for eample the mean square error based on training data. The basic principle of genetic programming is shown in Fig. 6. For the task of modeling the isothermal 2-dimensional flow of polymer melts genetic programming based symbolic regression appears especially interesting due to its ability to model non-linear corellations without the need to specify the structure of the formula a priori. At the same time the detected models can be interpreted, validated and transformed (Affenzeller et al., 2014). As the model detection stage is performed offline and the evalutaion of the models can be performed very efficiently for new data the approach seems very well suited for real world online applications. 3.2 Developing Analytic Equations Determining an analytic equation that describes the 2-dimensional throughput pressure gradient relationship first requires a numerical calculation, for instance, according to Rauwendaal (2014), of the dimensionless pressure gradient and throughput values for different screw pitches and power-law eponents. Considering the dimensionless representation of the equations of motion, Eq. 7 and 8, it can be seen that by varying the dimensionless pressure gradient p p the dimensionless throughput p V is only a function of the power law eponent n and the heli angle of the screw u. The analytic equation obtained must cover the most important range of application in single-screw etrusion, that is, screw pitches ranging from 0.75 to 2D and heli angles of 138 Fig. 4. Flow-chart; 2D Calculation and generation of data pairs, heuristic analysis by genetic programming, development of generally valid analytic equations, error analysis and proof of accuracy Fig. 5. Symbolic regression model in mathematical notation as well as in synta tree representation 212 Intern. Polymer Processing XXXII (2017) 2

to 338. In addition, the equation must be applicable to all common types of polymers (i. e., to flow eponents ranging from 0.2 to 0.9). We developed two analytic equations with different model application ranges (see Table 4). The first method, called \C50", is a generally valid analytic equation that can be applied to the whole range of the dimensionless pressure gradient p p. The second method, called \C23", is based on a polynomial function of the 3 rd order. Determining this function requires four sub-functions that can easily be calculated. These subfunctions can also be used to calculate characteristic values for polymer etrusion. Sub-function 1 describes the dimensionless drag flow in relation to the power-law eponent n and the heli angle u of the screw. The dimensionless dam-up pressure Fig. 6. Symbolic regression based on genetic programming can be determined using sub-function 2. Sub-function 3 describes the dimensionless volume throughput as a function of screw pitch and power-law eponent at a defined dimensionless pressure gradient of 0.325, which roughly corresponds to the location of the inflection point. Finally, sub-function 4 describes the dimensionless volume throughput at the fulcrum, where the dimensionless pressure gradient is 0.15. 3.2.1 Analytic Equation C50 The analytic equations we developed for describing the 2-dimensional throughput pressure gradient relationship are generally valid, continuous across the full application range, and depend on the heli angle of the screw u and the power-law eponent n. The overall generally valid Equation C50 can be written as follows: p V ¼ ða 1 þ A 2 þ A 4 þ A 5 Þc 49 þ c 50 ; ð15þ where c 0 to c 50 are constants and A 1 to A 9 are sub functions all listed in the appendi (see Table 1 and Eqs. 21 to 29). 3.2.2 Analytic Equation C23 The second method for describing the 2-dimensional throughput pressure gradient relationship is the polynomial function C23: p V ¼ A p p 3 þ B p p 2 þ C p p 1 þ D p p 0 ; Coefficient Value Coefficient Value Coefficient Value c 0 0.0001 c 8 0.0058 c 16 2.0980 c 1 0.0027 c 9 4.4171 c 17 0.0266 c 2 0.0023 c 10 1.1234 c 18 0.0777 c 3 1.0104 c 11 0.0007 c 19 11.7740 c 4 0.0131 c 12 0.1703 c 20 10.0921 c 5 0.2282 c 13 0.1709 c 21 37.3060 c 6 6.6792 c 14 0.0007 c 22 1.1370 c 7 1.3885 c 15 2.1730 Table 2. Values of constants for Eq. C23 ð16þ Screw pitch t Flow eponent n Dim. throughput p V Constants Region 1 0.8D t 1.2D 0.2 n 0.5 0.1 p V 0.55 A 1 to F 1 Region 2 0.55 p V 2 A 2 to F 2 Region 3 0.5 n 1 0.1 p V 0.55 A 3 to F 3 Region 4 0.551 p V 2 A 4 to F 4 Region 5 1.2D t 2D 0.2 n 0.5 0.1 p V 0.55 A 5 to F 5 Region 6 0.55 p V 2 A 6 to F 6 Region 7 0.5 n 1 0.1 p V 0.55 A 7 to F 7 Region 8 0.551 p V 2 A 8 to F 8 Table 3. Application range split into 8 regions for method according to (White et al., 2003) Intern. Polymer Processing XXXII (2017) 2 213

here, the coefficients A to D depend only on the power-law eponent n and the heli angle u, and must be determined using individual sub-functions such that the polynomial includes the four given points. Equations for determining A to D are listed in the appendi (Eqs. 34 to 37.) Sub-function (Sub) 1 describes the dimensionless drag flow p V,Drag : p V;Drag ¼ C 0 þ C 1 þ C n 2 2 þ C 3 ; n ð17þ where C 0 to C 3 are constants. The dimensionless dam-up pressure p p,dup can be determined using sub-function (Sub) 2: p p;dup ¼ C 4 þ C 5 þ C 6 C 7 þ C 8 þ C 9 þ C 10 ; ð18þ where C 4 to C 10 are, again, constants listed in the appendi. Sub-function (Sub) 3 describes the dimensionless volume throughput at a defined dimensionless pressure gradient of 0.325: ð ð Þ ¼ C 11 þ C 12 ÞðC 13 þ C 14 Þ þ C 17 ; ð19þ C 15 þ C 16 p V0:325 and sub-function 4 at the fulcrum, where the dimensionless pressure gradient is 0.15: p V0:150 ð Þ ¼ C 18 n þ C 19 1 n 1 C 20 n þ C þ C 22 : 21 ð20þ Constants c 0 to c 22 are again listed in the appendi, see Table 2. Figure 7 shows the polynomial function C23, the numerically determined data, and the locations of the four sub-functions. Sub1 represents the dimensionless drag flow p V,Drag, Sub2 the dimensionless dam-up pressure p p,dup, and Sub3 and Sub4 the dimensionless volume flow rates at defined dimensionless pressure gradients p V(0,325) and p V(0,150), respectively. 4 Results and Discussion We conducted a test for different screw geometries and various materials to validate our analytic equations. Figure 8 presents the resulting characteristic throughput pressure gradient curves for a square-pitched screw and different power-law eponents, comparing the numerically determined data (num.) to that of the generally valid analytic equations (C50 and C23). Figure 9 shows a normalized representation of the numerically determined and the calculated data for screw pitches ranging from 1 to 2D and power-law eponents from 0.2 to 0.8. As can be seen, using the analytic equations to describe the isothermal 2-dimensional throughput pressure gradient relationship, we achieved errors of less than 5%. We subsequently assessed the accuracy of the analytic equations by means of an error analysis, determining the following characteristic values:. the coefficient of determination (R 2 ),. the mean absolute error (MAE), and. the mean relative error (MRE). The results of the error analysis, summarized in Table 5, show that with our analytic equations, R 2 values of 0.9993 and above can be achieved. The sub-functions for drag flow rate and damup pressure of equation C23 provide additional information about the throughput pressure gradient relationship. These characteristic values can be determined very quickly, and R 2 values of 0.9994 and above can be achieved. 5 Conclusion We have shown that the generally valid analytic equations we introduced approimate the numerically determined data very well and can be used for any melt-conveying screw geometry. It is therefore possible to determine quickly and reliably the 2D output-pressure data for any channel height and any screw pitch in the range from 0.75 D to 2 D. The equations we presented are continuous across the full range of application and can be used for polymeric materials with flow eponents ranging from 0.2 to 0.9. Fig. 7. Dimensionless throughput (p V ) versus dimensionless pressure gradient (p p ) for different screw pitches t and a power-law eponent of n = 0.2; num. denotes the numerically determined data (solid lines); C23 denotes the calculated data according to Equation C23 (dashed lines); Sub1 represents the dimensionless drag flow p V,Drag, Sub2 the dimensionless dam-up pressure p p,dup, Sub3 and Sub4 the dimensionless volume flow rates at defined dimensionless pressure gradients p V(0,325) and p V(0,150) Fig. 8. Throughput-pressure gradient curves for a square-pitched screw for different power-law eponents n, calculated numerically (solid lines) and by using our two general valid analytic equations (dashed lines) 214 Intern. Polymer Processing XXXII (2017) 2

An error analysis showed that, using our analytic equations, coefficients of determination (R 2 ) of 0.9993 and above can be achieved. Further, we have demonstrated that very small errors of about 5 % can be realized. The equations presented in this paper represent a comprehensive description of the isothermal 2-dimensional throughput-pressure characteristics. The illustrated models are not very simple but they are removing the need for cost- and timeintensive CFD simulations and numerical calculations of the output-pressure behaviour. In addition, the presented equations are continuous throughout the entire range of application, and even simple programs such as Microsoft Ecel can be used for the calculation. Our equations enable fast and stable computational modeling of the 2D output-pressure characteristics. Equation C23 also offers the possibility to obtain important information on the isothermal throughput-pressure characteristics by means of a very simple calculation. Thus, characteristic values such as drag flow rate and dam-up pressure can be determined by means of a pocket calculator. The analytic equations, which we developed on the basis of evolutionary heuristic optimization algorithms, are a result of \smart engineering" and A) could be used in conjunction with screw segmentation for non-isothermal simulation. References Affenzeller, M., Wagner, S., Winkler, S. and Beham, A.: Genetic Algorithms and Genetic Programming: Modern Concepts and Practical Applications, CRC Press, Boca Raton (2009) Affenzeller, M., Winkler, S. M., Kronberger, G., Kommenda, M., Burlacu, B. and Wagner, S.: \Chapter No. 10 Gaining Deeper Insights in Symbolic Regression", in Genetic and Evolutionary Computation. Genetic Programming Theory and Practice XI, Riolo, R., Moore, J. H. and Kotanchek, M. (Eds.), Springer, New York, p. 175 190 (2014) Heuristic and Evolutionary Algorithms Laboratory. HeuristicLab. Retrieved from http://dev.heuristiclab.com/ Holland, J. H.: Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence (1st MIT Press Ed.), Comple Adaptive Systems, MIT Press, Cambridge, Mass. (1992) Koza, J. R.: Genetic Programming (3. Print), Genetic Programming Series, MIT Press, Cambridge, Mass. (1993) Kronberger, G., Wagner, S., Kommenda, M., Beham, A., Scheibenpflug, A. and Affenzeller, M.: \Chapter System Demonstrations Track, Knowledge Discovery through Symbolic Regression with HeuristicLab", in Machine Learning and Knowledge Discovery in Databases, Flach, P. A., De Bie, T. and Cristianini, N. (Eds.), Volume 7524, Springer, Berlin, Heidelberg, p. 824 827 (2012), DOI:10.1007/978-3-642-33486-3_56 Potente, H., \Approimationsgleichungen für Schmelzeetruder", Rheol. Acta, 22, 387 395 (1983), DOI:10.1007/BF01333769 Rauwendaal, C.: Polymer Etrusion, Hanser, Munich (2014), DOI:10.3139/9781569905395 Schwefel, H. P.: Evolutionsstrategie und numerische Optimierung, Technische Universität Berlin, Berlin (1975) Steller, R. T., \Theoretical Model for Flow of Polymer Melts in the Screw Channel", Polym. Eng. Sci., 30, 400 407 (1990), DOI:10.1002/pen.760300704 Steller, R. T., Iwko, J., \Generalized Flow of Ellis Fluid in the Screw Channel", Int. Polym. Proc., 16, 249 256 (2001), DOI:10.3139/217.1650 White, D. R., McDermott, J., Castelli, M., Manzoni, L., Goldman, B. W., Kronberger, G. and Luke, S., \Better GP Benchmarks: Community Survey Results and Proposals", Genetic Programming and Evolvable Machines, 14, 3 29 (2013), DOI:10.1007/s10710-012-9177-2 White, J. L., Potente, H. and Berghaus, U.: Screw Etrusion: Science and Technology. Progress in Polymer Processing, Hanser, Munich, Cincinnati (2003) Acknowledgements Financial support by the Linz Center of Mechatronics GmbH (LCM) is gratefully acknowledged. LCM is funded by the Austrian Government and the Provincial Government of Upper Austria within the COMET program. Further financial support within the COMET K-Project Heuristic Optimization in Production and Logistics (HOPL) is gratefully acknowledged. Date received: July 07, 2016 Date accepted: November 01, 2016 B) Fig. 9. Normalized representations of the results from Equations C50 (A) and C23 (B) for pitches of 1 D, 1.5 D and 2 D and power-law eponents of 0.2, 0.4, 0.6 and 0.8 Bibliography DOI 10.3139/217.3326 Intern. Polymer Processing XXXII (2017) 2; page 209 216 ª Carl Hanser Verlag GmbH & Co. KG ISSN 0930-777X Intern. Polymer Processing XXXII (2017) 2 215

Appendi Sub functions for equation \C50" A 1 ¼ c 0 p p þ c 48 ; ð21þ A 2 ¼ A 3 þ sin c 7 þc 8 pp c c 10 9 c 11 þ c 12 ; ð22þ c 13 2 A 3 ¼ c 1 p p þ c 2 þ c 3 p p c4 þ sin c 5 p p c6 ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 14 A 4 ¼ c 18 ; ep c 15þc 16 p p c 17 ð23þ ð24þ logðc 19 Þ c 20 p p þ c21 þc22pp c 23 c 24 A 5 ¼ ; ð25þ A 6 c 47 A 6 ¼ ep 2! c 25 p p þ c 26 sinðc 27 Þc 28 A7 ; ð26þ A 8 A 7 ¼ c 29 þ c 30 p p þ sin c 31 p p þ sin c 2 33 p p þ c 35; c 32 ð27þ A 8 ¼ c 36 þ c 37 p p þ sin c 38 p p þ A 9 ; ð28þ c 39 sin c 40 p p A 9 ¼ 2; þ c 44 þ c 45 p p ð29þ c 41 þ sin c 42 p p c43 Sub functions for equation \C23" Sub-Function 1 (dimensionless drag flow): p V;Drag ¼ C 0 n 2 þ C 1 þ C 2 þ C 3 ; n c 34 Sub-Function 2 (dimensionless dam-up pressure) p p;dup ¼ C 4 þ C 5 þ C 6 C 7 þ C 8 þ C 9 þ C 10 ; ð30þ ð31þ Sub-Function 3 (dimensionless volume flow rate at p p = 0.325) ð ð Þ ¼ C 11 þ C 12 ÞðC 13 þ C 14 Þ þ C 17 ; ð32þ C 15 þ C 16 p V0;325 Sub-Function 4 (dimensionless volume flow rate at p p = 0.150) p V0;150 ð Þ ¼ C 18 n þ C 19 1 n 1 C 20 n þ C þ C 22 ; 21 Coefficients A to D can be determined as follows: y 0 A ¼ 1 2 3 ð 1 2 Þ 2 ð 2 3 Þ þ y 3 y 2 ð33þ ; ð34þ ð 1 3 Þð 2 3 Þ 3 ð B ¼ 1 þ 2 þ 3 Þy 0 1 2 3 ð 1 þ 3 Þy 2 þ ð 1 2 Þ 2 ð 2 3 Þ ð 1 þ 2 Þ y 3 þ ; ð35þ ð 1 3 Þð 3 2 Þ 3 C ¼ 1 2 þ 3 1 3 y 2 y 0 1 2 3 ð 1 2 Þ 2 ð 2 3 Þ 1 2 y 3 ; ð36þ ð 1 3 Þð 3 2 Þ 3 D ¼ p V;Drag ; with: 0 ¼ 0 and y 0 ¼ p V;Drag ; 1 ¼ p p;dup and y 1 ¼ 0; ð37þ ð38þ ð39þ 2 ¼ 0:325 and y 2 ¼ p V0;325 ð Þ ; ð40þ 3 ¼ 0:150 and y 3 ¼ p V0;150 ð Þ : ð41þ C50 C23 Mean R 2 0.9999 0.9993 MAE 1 0.0181 0.0069 MRE 2 9.92 % 4.86 % 1 mean absolute error; 2 mean relative error Table 5. Results of the error analysis C50 C23 Screw pitch t [D] 0.75 to 2 0.75 to 2 Heli angle u [8] 13.4 to 32.5 13.4 to 32.5 Power-law eponent n [ ] 0.2 to 0.9 0.2 to 0.9 Dim. throughput p V [ ] >0 >0 Dim. pressure gradient p p [ ] > 0 Table 4. Application range for equations C50 and C23 216 Intern. Polymer Processing XXXII (2017) 2