Abstract
When selecting a melt-filtration system, the initial pressure drop is a critical parameter. We used heuristic optimization algorithms to develop general analytical equations for estimating the dimensionless pressure loss of square and Dutch woven screens in polymer processing and recycling. We present a mathematical description – without the need for further numerical methods – of the dimensionless pressure loss of non-Newtonian polymer melt-flows through woven screens. Applying the theory of similarity, we first simplified, and then transformed into dimensionless form, the governing equations. By varying the characteristic independent dimensionless influencing parameters, we created a comprehensive parameter set. For each design point, the nonlinear governing equations were solved numerically. We subsequently applied symbolic regression based on genetic programming to develop models for the dimensionless pressure drop. Finally, we validated our models against experiments using both virgin and slightly contaminated in-house and post-industrial recycling materials. Our regression models predict the experimental data accurately, yielding a mean relative error of MRE = 13.7%. Our modeling approach, the accuracy of which we have proven, allows fast and stable prediction of the initial pressure drop of polymer-melt flows through square woven and Dutch weave screens, rendering further numerical simulations unnecessary.
Acknowledgements
This work was supported by Erema Engineering Recycling Maschinen und Anlagen Ges.m.b.H and funded by the Austrian Research Promotion Agency (FFG; project number: 867202) and by the Austrian Science Fund (FWF; project number: P 29545-N34).
Appendix
Subfunctions for ΠP,zDutch Eq. 39.
Nomenclature
- DScreen
screen diameter
- MW
mesh width or mesh opening
- MN
number of wires per inch
- dW
wire diameter
- ΔPScreen
initial pressure drop of woven screen
- η
viscosity
- ṁ
mass flow rate
- γ
shear rate
- ρ
melt density
- n
power-law exponent
- K
consistency index
- DK
diameter of the warp wire
- PK
pitch in warp direction
- DS
diameter of the weft wire
- PS
pitch in weft direction
- x, y, z
spatial coordinates
- v
velocity vector
- vx, vy, vz
velocity components
- τ
viscous stress tensor
- D
rate-of-deformation tensor
- vref
undisturbed velocity
- ⟇cell
screen-specific volume flow rate
- ṁcell
screen-specific mass flow rate
- ⟇total
volume flow rate
- AScreen
overall screening surface
- ACell
characteristic surface of the elementary cell
- ξ, ψ, μ
dimensionless coordinates
- vx, vy, vz
dimensionless velocity components
- ΠV
dimensionless volume flow rate
- ΠP,i
dimensionless pressure gradient
- γ
dimensionless shear rate
- η
dimensionless viscosity
- Repl
Reynolds number for power-law fluid-based flows
- PD
dimensionless pitch-to-diameter ratio
- A
dimensionless pitch ratio
- α
dimensionless overlapping ratio
- D
dimensionless diameter ratio
- dpz
pressure loss
- ΠP,zSquare
dimensionless pressure gradient (square woven screens)
- A1 – A6
subfunctions (square woven)
- a1 – a33
constants (square woven)
- ΠP,zDutch
dimensionless pressure gradient (Dutch weave screens)
- B1 – B10
subfunctions (Dutch weave)
- b1 – a54
constants (Dutch weave)
- R2
coefficient of determination
- MAE
mean absolute error
- MRE
mean relative error
- REmax
maximum relative error
References
Affenzeller, M., Wagner, S., Winkler, S. and Beham, A.: Genetic Algorithms and Genetic Programming: Modern Concepts and Practical Applications, CRC Press, Boca Raton, FL, USA (2009), DOI:10.1201/978142001132610.1201/9781420011326Search in Google Scholar
Affenzeller, M., Winkler, S. M., Kronberger, G., Kommenda, M., Burlacu, B. and Wagner, S., "Chapter 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), DOI:10.1007/978-1-4939-0375-7_1010.1007/978-1-4939-0375-7_10Search in Google Scholar
Bailey, J.: Processing and Finishing of Polymeric Materials, Wiley, Hoboken (2011)Search in Google Scholar
Bird, R. B., Stewart, W. E. and Lightfoot, E. N.: Transport Phenomena, 2nd Edition, Wiley, New York (2007)Search in Google Scholar
Brackett-Rozinsky, N., Mondal, S., Fowler, K. R. and Jenkins, E. W., "Analysis of Model Parameters for a Polymer Filtration Simulator", Modell. Simul. Eng., 2011, 1–11 (2011), DOI:10.1155/2011/13814310.1155/2011/138143Search in Google Scholar
Campbell, G. A., Spalding, M. A.: Analyzing and Troubleshooting Single-Screw Extruders, Hanser Publishers, Munich (2013), DOI:10.3139/9783446432666.fm10.3139/9783446432666.fmSearch in Google Scholar
Carley, J. F., Smith, W. C., "Design and Operation of Screen Packs", Polym. Eng. Sci, 18, 408–415 (1978), DOI:10.1002/pen.76018051310.1002/pen.760180513Search in Google Scholar
Deb, K., Pratap, A., Agarwal, S. and Meyarivan, T., "A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II", IEEE Trans. Evol. Comput., 6, 182–197 (2002), DOI:10.1109/4235.99601710.1109/4235.996017Search in Google Scholar
Edle, D., Gooding, C., "Prediction of Pressure Drop for the Flow of Polymer Melts through Sintered Metal Filters", Industrial and Engineering Chemisty Process Design and Development, 1985, °–12 (1985), DOI:10.1021/ᐳ200028a00210.1021/ᐳ200028a002Search in Google Scholar
European Comission: A European Strategy for Plastics in a Circular Economy (2018), online: Search in Google Scholar
Giles, H. F, Mount, E. M and Wagner, J. R.: Extrusion: The Definitive Processing Guide and Handbook, William Andrew, New York (2005)Search in Google Scholar
Gu, F., Guo, J., Zhang, W., Summers, P. A. and Hall, P., "From Waste Plastics to Industrial Raw Materials: A Life Cycle Assessment of Mechanical Plastic Recycling Practice Based on a Real-World Case Study", Sci. Total Environ., 601–602, 1192–1207 (2017), PMid:28605837, DOI:10.1016/j.scitotenv.2017.05.27810.1016/j.scitotenv.2017.05.278Search in Google Scholar
Kommenda, M., Kronberger, G., Affenzeller, M., Winkler, S. and Burlacu, B., "Chapter 1 Evolving Simple Symbolic 1004 Regression Models by Mulit-Objective Genetic Programming", in Genetic and Evolutionary Computation, Genetic Programming Theory and Practice XIII, Riolo, R., Worzel, W. P., Kotanchek, M. and Kordon, A. (Eds.), Springer International Publishing, Switzerland, p. 1–19 (2016), DOI:10.1007/978-3-319-34223-°_110.1007/978-3-319-34223-°_1Search in Google Scholar
Koza, J. R.: Genetic Programming, 3rd Edition, MIT Press, Cambridge (1993)Search in Google Scholar
Langtangen, H. P., Pedersen, G. K.: Scaling of Differential Equations, Springer (2016), DOI:10.1007/978-3-319-32726-610.1007/978-3-319-32726-6Search in Google Scholar
Lazarevic, D., Aoustin, E., Buclet, N. and Brandt, N., "Plastic Waste Management in the Context of a European Recycling Society: Comparing Results and Uncertainties in a Life Cycle Perspective", Resour. Conserv. Recycl., 55, 246–259 (2010), DOI:10.1016/j.resconrec.2010.09.01410.1016/j.resconrec.2010.09.014Search in Google Scholar
Lughofer, E., Kronberger, G., Kommenda, M., Saminger-Platz, S., Promberger, A., Nickel, F., Winkler, S. and Affenzeller, M., "Robust Fuzzy Modeling and Symbolic Regression for Establishing Accurate and Interpretable Prediction Models in Supervising Tribological Systems", Proceedings of the °th International Joint Conference on Computational Intelligence: SCITEPRESS – Science and Technology Publications, Porto, Portugal, p. 51–63 (2016), DOI:10.5220/000606840051006310.5220/0006068400510063Search in Google Scholar
Markarian, J., "Choosing a Melt Filtration System", Plastics, Additives and Compounding, 10, 32–35 (2008), DOI:10.1016/S1464-391X(08)70093-X10.1016/S1464-391X(08)70093-XSearch 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 (2017), 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), PMid:30960854, DOI:10.3390/polym1008092910.3390/polym10080929Search in Google Scholar
Müller, M., Piesche, M., "Ähnlichkeitsgesetze zur Beschreibung des Anfangsdruckverlustes metallischer Drahtgewebe bei der Filtration nicht-Newtonscher Fluide", F&S Filtrieren und Separieren, 27, 284–291 (2013)Search in Google Scholar
Pachner, S., Aigner, M. and Miethlinger, J., "Modeling and Optimization of Melt Filtration Systems In Polymer Recycling", AIP Conference Proceedings, 1914, 80004 (2017), DOI:10.1063/1.501674410.1063/1.5016744Search in Google Scholar
Pachner, S., Aigner, M. and Miethlinger, J., "Modeling the Operating Performance of Melt Filtration in Polymer Recycling", SPE ANTEC Tech. Papers (2018), DOI:10.1063/1.501674410.1063/1.5016744Search in Google Scholar
Pachner, S., Aigner, M. and Miethlinger, J., "A Heuristic Method for Modeling the Initial Pressure Drop in Melt Filtration using Woven Screens in Polymer Recycling", Polym. Eng. Sci, 59, 1105–1113 (2019), DOI:10.1002/PEN.2508810.1002/PEN.25088Search in Google Scholar
PlasticsEurope, Plastics – the Facts 2019: An Analysis of European Plastics Production, Demand and Waste Data (2019), online: Search in Google Scholar
Poli, R., Langdon, W. B., McPhee, N. F. and Koza, J. R.: A Field Guide to Genetic Programming. Lulu Press, Morrisville (2008)Search in Google Scholar
Rauwendaal, C.: Polymer Extrusion, 5th Edition, Hanser Publishers, Munich (2014), DOI:10.3139/9781569905395.fm10.3139/9781569905395.fmSearch in Google Scholar
Roland, W., Miethlinger J., "Heuristic Analysis of Viscous Dissipation in Single-Screw Extrusion", Polym. Eng. Sci., 58, 2055–2017 (2018), DOI:10.1002/pen.2481710.1002/pen.24817Search 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: 0.3390/ polym11020334, PMid:30960318, 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
Schöppner, V., Meilwes, P., "Modelling the Contamination Behavior of Polymer Melt Filters and Pressure Loss Simulations of Filtration Media", SPE ANTEC Tech. Papers (2018), DOI:10.1063/1.512164810.1063/1.5121648Search in Google Scholar
Tadmor, Z., Gogos, C. G.: Principles of Polymer Processing, 2nd Edition, Wiley-Interscience, Hoboken (2006)Search in Google Scholar
Todd, D. B., "Determining Pressure Drop in Extrusion", Plastics Compounding, 17, 23 (1994)Search in Google Scholar
Verein Deutscher Ingenieure, Gesellschaft Kunststofftechnik: Filtration of Polymer Melts, VDI-Verlag, Düsseldorf (1981)Search in Google Scholar
Vergnes, B., Lafleur, P. G.: Polymer Extrusion, Wiley, Hoboken (2014), DOI:10.1002/978111882712310.1002/9781118827123Search in Google Scholar
Wagner, S., Kronberger, G., Beham, A., Kommenda, M., Scheibenpflug, A., Pitzer, E., Vonolfen, S., Kofler, M., Winkler, S., Dorfer, V. and Affenzeller, M.: "Chapter 10 Architecture and Design of the HeuristicLab Optimization Environment", in Topics in Intelligent Engineering and Informatics, Advanced Methods and Applications in Computational Intelligence, Klempous, R., Nikodem, J., Jacak, W. and Chaczko, Z. (Eds.), Springer International, Heidelberg, p. 197–261 (2014), DOI:10.1007/978-3-319-01436-4_1010.1007/978-3-319-01436-4_10Search in Google Scholar
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