Accessible Unlicensed Requires Authentication Published by De Gruyter November 16, 2021

Predicting the Non-Linear Conveying Behavior in Single-Screw Extrusion: A Comparison of Various Data-Based Modeling Approaches used with CFD Simulations

W. Roland, C. Marschik, M. Kommenda, A. Haghofer, S. Dorl and S. Winkler


The traditional approach to modeling the polymer melt flow in single-screw extruders is based on analytical and numerical analyses. Due to increasing computational power, data-driven modeling has grown significantly in popularity in recent years. In this study, we compared and evaluated databased modeling approaches (i. e., gradient-boosted trees, artificial neural networks, and symbolic regression models based on genetic programming) in terms of their ability to predict – within a hybrid modeling framework – the three-dimensional non-linear throughput-pressure relationship of metering channels in single-screw extruders. By applying the theory of similarity to the governing flow equations, we identified the characteristic dimensionless influencing parameters, which we then varied to create a large dataset covering a wide range of possible applications. For each single design point we conducted numerical simulations and obtained the dimensionless flow rate. The large dataset was divided into three independent sets for training, interpolation, and extrapolation, the first being used to generate and the remaining two to evaluate the models. Further, we added two features derived from expert knowledge to the models and analyzed their influence on predictive power. In addition to prediction accuracy and interpolation and extrapolation capabilities, we evaluated model complexity, interpretability, and time required to learn the models. This study provides a rigorous analysis of various data-based modeling approaches applied to simulation data in extrusion.

Wolfgang Roland, Institute of Polymer Extrusion and Compounding, Johannes Kepler University Linz, Altenbergerstraße 69, 4040 Linz, Austria


This research was funded by the Austrian Science Fund (FWF), grant number: I 4872-N. The authors additionally acknowledge support from the Christian Doppler Research Association and the Federal Ministry for Digital and Economic Affairs under the aegis of the Josef Ressel Center for Symbolic Regression.

The computational results presented were achieved in part by using the Vienna Scientific Cluster (VSC).


Affenzeller, M., Wagner, S., Winkler, S. and Beham, A.: Genetic Algorithms and Genetic Programming: Modern Concepts and Practical Applications, CRC Press, Boca Raton (2009), DOI:10.1201/978142001132610.1201/9781420011326Search in Google Scholar

Aigner, M., “Computational and Experimental Modelling of Transport Phenomena in Single Screw Plasticating Units under Consideration of the Melt Quality", PhD Thesis, JKU Linz, Linz, Austria (2014)Search in Google Scholar

Bird, R. B., Stewart, W. E. and Lightfoot, E. N.: Transport Phenomena, 2nd Edition, John Wiley & Sons, New York (2002)Search in Google Scholar

Bishop, C. M.: Neural Networks for Pattern Recognition, Oxford University Press, New York (1995), DOI:10.1201/9781420050646.ptb610.1201/9781420050646.ptb6Search in Google Scholar

Booy, M. L., “The Influence of Non-Newtonian Flow on Effective Viscosity and Channel Efficiency in Screw Pumps", Polym. Eng. Sci., 21, 93–99 (1981), DOI:10.1002/pen.76021020710.1002/pen.760210207Search in Google Scholar

Bre, F., Gimenez, J. and Fachinotti, V., “Prediction of Wind Pressure Coefficients on Building Surfaces Using Artificial Neural Networks", Energy Build., 158, 1429–441 (2018), DOI:10.1016/j.enbuild.2017.11.04510.1016/j.enbuild.2017.11.045Search in Google Scholar

Breiman, L., “Random Forests", Machine Learning, 45, 5–32 (2001), DOI:10.1023/A:101093340432410.1023/A:1010933404324Search in Google Scholar

Chen, T., Guestrin, C., “XGBoost: A Scalable Tree Boosting System", in Proceedings of the 22nd acm sigkdd International Conference on Knowledge Discovery and Data Mining, San Francisco, p. 785–794 (2016), DOI:10.1145/2939672.293978510.1145/2939672.2939785Search in Google Scholar

Chollet, F., “Keras: The Python Deep Learning Library" (2015), https://keras. ioSearch in Google Scholar

Durst, F.: Fluid Mechanics: An Introduction to the Theory of Fluid Flows, Springer, Berlin (2008), DOI:10.1007/978-3-540-71343-210.1007/978-3-540-71343-2Search in Google Scholar

Fenner, R. T., “Developments in the Analysis of Steady Screw Extrusion of Polymers", Polymer, 18, 617–635 (1977), DOI:10.1016/0032-3861(77)90066-010.1016/0032-3861(77)90066-0Search in Google Scholar

Freund, Y., Schapire, R. E., “A Short Introduction to Boosting", Journal of the Japanese Society for Artificial Intelligence, 14, 771–780 (1999)Search in Google Scholar

Friedman, J. H., “Greedy Function Approximation: A Gradient Boosting Machine", Annals of Statistics, 29, 1189–1232 (2001), DOI:10.1214/aos/101320345110.1214/aos/1013203451Search in Google Scholar

Ghoreishy, M. H. R., Razavi-Nouri, M., “Finite Element Analysis of Thermoplastic Melts Flow through the Metering and Die Regions of Single Screw Extruders", J. Appl. Polym. Sci., 74, 676–689 (1999), DOI:10.1002/(SICI)1097-4628(19991017)74 : 3<676::AID-APP22>3.0.CO;2-%2310.1002/(SICI)1097-4628(19991017)74 : 3<676::AID-APP22>3.0.CO;2-%23Search in Google Scholar

Ghoreishy, M. H. R., Razavi-Nouri, M. and Naderi, G., “Finite Element Analysis of a Thermoplastic Elastomer Melt Flow in the Metering Region of a Single Screw Extruder", Comp. Mat. Sci., 34, 389–396 (2005), DOI:10.1016/j.commatsci.2005.01.01110.1016/j.commatsci.2005.01.011Search in Google Scholar

Griffith, R. M., “Fully Developed Flow in Screw Extruders. Theoretical and Experimental Study", Ind. Eng. Chem., 1, 180–187 (1962), DOI:10.1021/i160003a00410.1021/i160003a004Search in Google Scholar

Hawkins, D. M., “The Problem of Overfitting", J. Chem. Inf. Comput. Sci., 44, 1–2 (2004), DOI:10.1021/ci034247210.1021/ci0342472Search in Google Scholar

Ioffe, S., Szegedy, C., “Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift", in 32nd International Conference on Machine Learning, Lille (2015)Search in Google Scholar

Juszczak, P., Tax, D. and Duin, R. P., “Feature Scaling in Support Vector Data Description", in Proceeding of Asci, p. 95–102 (2002)Search in Google Scholar

Kadyirov, A., Gataullin, R. and Karaeva, J., “Numerical Simulation of Polymer Solutions in a Single-Screw Extruder", Appl. Sci., 9, 5423 (2019), DOI:10.3390/app924542310.3390/app9245423Search in Google Scholar

Kim, S. J., Won, T. H., “A Simple Approach to Determining Three-Dimensional Screw Characteristics in the Metering Zone of Extrusion Processes Using a Total Shape Factor", Polym. Eng. Sci., 35, 274–283 (1995), DOI:10.1002/pen.76035030810.1002/pen.760350308Search in Google Scholar

Kommenda, M., Kronberger, G., Winkler, S., Affenzeller, M. and Wagner, S., “Effects of Constant Optimization by Nonlinear Least Squares Minimization in Symbolic Regression", Proceedings of the 15th Annual Conference Companion on Genetic and Evolutionary Computation, Amsterdam, p. 1121–1128 (2013), DOI:10.1145/2464576.248269110.1145/2464576.2482691Search in Google Scholar

Kommenda, M., Burlacu, B., Kronberger, G. and Affenzeller, M., “Parameter Identification for Symbolic Regression Using Nonlinear Least Squares", Genetic Programming and Evolvable Machines, 21, 471–501, (2020), DOI:10.1007/s10710-019-09371-310.1007/s10710-019-09371-3Search in Google Scholar

Koza, J. R.: Genetic Programming: on the Programming of Computers by Means of Natural Selection, MIT Press, Cambridge (1992)Search in Google Scholar

Luger, H. J., Roland W., Löw-Baselli, B. and Miethlinger, J., “A Network-Analysis-Based Comparative Study of the Throughput Behavior in Double Wave Screw Geometries", SPE ANTEC Tech. Papers, (2018)Search in Google Scholar

Manas-Zloczower, I.: Mixing and Compounding of Polymers – Theory and Practice, Hanser, Munich (2006)Search in Google Scholar

Marschik, C., Roland, W., Löw-Baselli, B. and Miethlinger, J., “Modeling Three-Dimensional Non-Newtonian Flows in Single-Screw Extruders", SPE ANTEC Tech. Papers, 1125–1130 (2017a), DOI:10.1016/j.jnnfm.2017.08.00710.1016/j.jnnfm.2017.08.007Search in Google Scholar

Marschik, C., Roland, W., Löw-Baselli, B. and Miethlinger, J., “A Heuristic Method for Modeling Three-Dimensional Non-Newtonian Flows of Polymer Melts in Single-Screw Extruders", J. Non-Newtonian Fluid Mech., 248, 27–39 (2017b), DOI:10.1016/j.jnnfm.2017.08.00710.1016/j.jnnfm.2017.08.007Search in Google Scholar

Marschik, C., Roland, W. and Miethlinger, J., “A Network-Theory-Based Comparative Study of Melt-Conveying Models in Single-Screw Extrusion: A. Isothermal Flow", Polymers, 10, 929 (2018), DOI:10.3390/polym1008092910.3390/polym10080929Search in Google Scholar

Marschik, C., Roland, W., Dörner, M., Schaufler, S., Schöppner, V. and Steinbichler, G., “Application of Network Analysis to Flow Systems with Alternating Wave Channels: Part B. (Superimposed Drag-Pressure Flows in Extrusion)", Polymers, 12, 1900 (2020a), DOI:10.3390/polym1209190010.3390/polym12091900Search in Google Scholar

Marschik, C., Roland, W., Löw-Baselli, B. and Steinbichler, G., “Application of Hybrid Modeling in Polymer Processing", SPE ANTEC Tech. Papers, 811–818 (2020b)Search in Google Scholar

Montans, F. J., Chinesta, F., Gomez-Bombarelli, R. and Nathan-Kutz, J., “Data-Driven Modeling and Learning in Science and Engineering", C. R. Mec., 347, 845–855 (2019), DOI:10.1016/j.crme.2019.11.00910.1016/j.crme.2019.11.009Search in Google Scholar

Mori, Y., Matsumoto. T. K., “Analytical Study of Plastics Extrusion", Rheol. Acta, 1, 240–242 (1958), DOI:10.1007/BF0196887410.1007/BF01968874Search in Google Scholar

Narkis, M., Ram, A., “Extrusion Discharge Rate Equations for Non-Newtonian Fluids", Polym. Eng. Sci., 7, 161–167 (1967), DOI:10.1002/pen.76007030610.1002/pen.760070306Search in Google Scholar

Ng, A. Y., “Feature Selection, L1 vs. L2 Regularization, and Rotational Invariance", Twenty-First International Conference on Machine Learning, Banff, Alberta (2004), DOI:10.1145/1015330.101543510.1145/1015330.1015435Search in Google Scholar

Pachner, S., Löw-Baselli, B., Affenzeller, M. and Miethlinger, J., “A Generalized 2D Output Model of Polymer Melt Flow in Single-Screw Extrusion", Int. Polym. Proc., 32, 209–216 (2017), DOI:10.3139/217.332610.3139/217.3326Search in Google Scholar

Potente, H., “Auslegung von Schmelzeextrudern für Kunststoffschmelzen mit Potenzverhalten", Kunststoffe, 71, 474–478 (1981)Search in Google Scholar

Potente, H., “Approximationsgleichungen für Schmelzeextruder", Rheol. Acta, 22, 387–395 (1983), DOI:10.1007/BF0133376910.1007/BF01333769Search in Google Scholar

Potente, H., Hanhart, W. and Schöppner, V., “Potential Applications for Computer-Aided Extruder Design", Int. Polym. Proc., 8, 335–344 (1993), DOI:10.3139/217.93033510.3139/217.930335Search in Google Scholar

Rauwendaal, C., “Throughput-Pressure Relationship for Power Law Fluids in Single Screw Extruders", Polym. Eng. Sci., 26, 1240–1244 (1986), DOI:10.1002/pen.76026180310.1002/pen.760261803Search in Google Scholar

Rauwendaal, C., “Finite Element Studies of Flow and Temperature Evolution in Single Screw Extruders", Plast. Rubber, Comp., 33, 390–396 (2013), DOI:10.1179/174328904X2488010.1179/174328904X24880Search in Google Scholar

Rotem, Z., Shinnar, R., “Non-Newtonian Flow between Parallel Boundaries in Linear Movement", Chem. Eng. Sci., 15, 130–143 (1961), DOI:10.1016/0009-509(61)85006-910.1016/0009-509(61)85006-9Search in Google Scholar

Roland, W., Kommenda, M., Marschik, C. and Miethlinger, J., “Extended Regression Models for Predicting the Pumping Capability and Viscous Dissipation of Two-Dimensional Flows in Single-Screw Extrusion", Polymers, 11, 334 (2019a), DOI:10.3390/polym1102033410.3390/polym11020334Search in Google Scholar

Roland, W., Marschik, C., Krieger, M., Löw-Baselli, B. and Miethlinger, J., “Symbolic Regression Models for Predicting Viscous Dissipation of Three-Dimensional Non-Newtonian flows in Single-Screw Extruders", J. Non-Newtonian Fluid Mech., 268, 12–29 (2019b), DOI:10.1016/j.jnnfm.2019.04.00610.1016/j.jnnfm.2019.04.006Search in Google Scholar

Roland, W., Marschik, C., Hammer, A. and Steinbichler, G., “Modeling the Non-Isothermal Conveying Characteristics in Single-Screw Extrusion by Application of Network Analysis", SPE ANTEC Tech. Papers, 605–612 (2020)Search in Google Scholar

Rowell, H. S., Finlayson, D., “Screw Viscosity Pumps", Engineering, 114, 606–607 (1922)Search in Google Scholar

Rowell, H. S., Finlayson, D., “Screw Viscosity Pumps", Engineering, 126, 249–387 (1928)Search in Google Scholar

Spalding,M. A., Dooley, J., Hyun, K. S. and Strand, S. R., “Three Dimensional Analysis of the Metering Section of a Single-Screw Extruder", SPE ANTEC Tech. Papers, 1533–1541 (1993)Search in Google Scholar

Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I. and Salakhutdinov, R., “Dropout: A Simple Way to Prevent Neural Networks from Overfitting", Journal of Machine Learning Research, 15, 1929–1958 (2014)Search in Google Scholar

Stijven, S., Minnebo, W. and Vladislavleva, K., “Separating the Wheat from the Chaff: on Feature Selection and Feature Importance in Regression Random Forests and Symbolic Regression", in Proceedings of the 13th Annual Conference Companion on Genetic and Evolutionary Computation, Dublin, p. 623–630 (2011), DOI:10.1145/2001858.200205910.1145/2001858.2002059Search in Google Scholar

Tadmor, Z., Gogos, C. G.: Principles of Polymer Processing, 2nd Edition, Wiley, New Jersey (2006)Search in Google Scholar

Tadmor, Z., Klein, I.: Engineering Principles of Plasticating Extrusion, Van Nostrand Reinhold, New York (1970)Search in Google Scholar

Vachagina, E. K., Kadyirov, A. I. and Karaeva, J. V., “Simulation of Giesekus Fluid Flow in Extruder Using Helical Coordinate System", IOP Conf. Ser.: Mater. Sci. Eng., 733, 1–5 (2020), DOI:10.1088/1757-899X/733/1/01203310.1088/1757-899X/733/1/012033Search in Google Scholar

Wagner, S., Kronberger, G., Beham, A., Kommenda, M., Scheibenpflug, A., Pitzer, E. and Affenzeller, M.: “Architecture and Design of the Heuristiclab Optimization Environment", Advanced Methods and Applications in Computational Intelligence, p. 197–261, Springer, Heidelberg (2014), DOI:10.1007/978-3-319-01436-4_1010.1007/978-3-319-01436-4_10Search in Google Scholar


The best performing symbolic regression model is given by:

(34) ΠV(Πp,Z,n,t/Db,h/w)=a00+a01ea02F1+A1(A2+A3)+A4A5.

with the sub-functions:

(35) A1=a03ea04h/w+a05Πp,za06( e a07F2+a08h/w+a09n+a10Πp,z )n,
(36) A2=a11(ea12Πp,z+a13n)(a14h/w+Πp,z)+a21h/w+a22n+a23Πp,z,
(37) A3=ea14F2ea16h/wea17ea18t/Db+a19F1+a20Πp,2n
(38) A4=a24e(ea25h/w+a26Πp,z)(a27F1+a28Πp,z)+a29Πp,z
(39) A5=a30+ea31t/Db(ea32h/w+a33Πp,znA6+a34t/Db),
(40) A6=a35+eea36t/Db(a37h/w+a38Πp,2)+a39F1+a40n+a41t/Db.

The derived features F1 and F2 are according to Eqs. 31 and 32, respectively. The model coefficients are given by Table 8.

Received: 2021-02-02
Accepted: 2021-04-24
Published Online: 2021-11-16

© 2021 Walter de Gruyter GmbH, Berlin/Boston, Germany