# Fundamentals of Global Modeling for Polymer Extrusion

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## Abstract

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## 1. Introduction

## 2. Modeling of the Extrusion Process

#### 2.1. Solid Conveying Section

#### 2.2. Melting Section

#### 2.3. Melt Conveying Section

## 3. Computer Models of Extrusion

## 4. Global Modeling of the Extrusion Process

## 5. Future Concepts

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Scheme of the extrusion process: 1—solid polymer, 2—hopper, 3—barrel, 4—screw, 5—heaters, 6—die, 7—extrudate, A—solid conveying zone, B—pre-melting zone (delay zone), C—melting zone, D—melt conveying zone, E—melt flow zone in the die, X—width of the solid bed, W—width of the screw channel, H—height of the screw channel, h

_{f}—clearance between the screw flights and the barrel, and h—polymer melt thickness.

**Figure 2.**CSM melting mechanism (Contiguous Solid Melting) observed for flood fed single screw extrusion of polypropylene [59].

**Figure 3.**Melting mechanism for counter-rotating twin-screw extrusion [120].

**Figure 4.**Melting mechanism for starve fed extrusion of polypropylene [59].

**Figure 5.**Melting of a wood-plastic composite of polypropylene PP and wood flour WF of a different composition in the single screw extrusion: (

**a**) 25% WF, (

**b**) 50% WF, (

**c**) 75% WF, A—molten material, B—solid material [129] (with permission from Int. Polym. Process. 2015, 30, 113-120, by Wilczyński, K.; Nastaj, A., Lewandowski, A., Wilczyński, K.J., Buziak, K. © Carl Hanser Verlag GmbH & Co. KG, Muenchen).

**Figure 6.**Melting of polyblends: (

**a**) high density polyethylene/polystyrene blend (HDPE/PS)—starve fed extrusion, (

**b**) polypropylene/polymethyl methacrylate blend (PP/PMMA)—starve fed extrusion, and (

**c**) polypropylene/polystyrene blend (PP/PS)—flood fed extrusion [131].

**Figure 7.**Melting of polymer blends in starve fed single-screw extrusion: (

**a**) melting visualization, (

**b**) melting model: A—major component of polyblen (HDPE), B—minor component of polyblend (PS), A/B—polyblend (HDPE/PS), MELTING_I—by heat conduction, MELTING_II—by energy dissipation [131].

**Figure 8.**Melting mechanism for injection molding [141].

**Figure 10.**Screw flow simulations: pressure/velocity distributions for the power law model at slip/no slip conditions [251].

**Figure 11.**Slip effects and melting: (

**a**) velocity distribution without a slip, and with slipping, and (

**b**) possible melting mechanisms.

**Figure 12.**Screw flow simulations: pressure/velocity distributions for Bingham model [251].

**Figure 13.**Modeling concepts: (

**a**) classical modeling: A—solid conveying model, B—pre-melting model, C—melting model, D—melt conveying model, E—die flow model, (

**b**) continuum modeling: G—continuous model.

**Figure 14.**A forward scheme of computations for flood fed extrusion, and a backward scheme of computations.

**Figure 15.**Scheme of computation for flood fed single screw extrusion: (

**a**) Δp

_{die}< 0, (

**b**) Δp

_{die}> 0, (

**c**) ׀Δp

_{die}׀ < δ

_{p}, 1—start of melting, 2—start of compression section, 3—pressure max, 4—end of pressure drop, 5—end of melting, 6—pressure at screw exit (die inlet), Q—flow rate, Q

_{i}

_{+1}—next iteration flow rate, p—pressure, Δp

_{die}—die pressure at die exit, δ

_{p}—accuarcy of pressure computation, T—temperature, T

_{m}—melting point, T

_{die}—die melt temperature, M—solid fraction (melting), E—power consumption.

**Figure 16.**Computation scheme: (

**a**) one-stage melting mechanism, computation discrepancy, (

**b**) two-stage melting mechanism, computation discrepancy, (

**c**) computation convergency: (I), forward computations in the melting section, (II) forward computations in the die section, (III), backward computations in the melt conveying section, (M), melting (SF), (F), filling (fill factor), (T), temperature, (P), pressure, (1), start of melting, (2) end of melting, (3) transfer of computations to the die, (4) start of die melt temperature computation, (5) start of die pressure computation, (6) zero pressure location, (7) beginning of filling computation (partly filled region starts), ΔP

_{DIEi}, die pressure, T

_{DIEi}, presumed melt temperature, T

_{m}, melting point, i, number of iterations, ΔT = |T

_{m}− T

_{i}|, convergency checking, and δT, computation accuracy [206].

**Figure 17.**Conventional and non-conventional screw configurations: (

**a**) conventional section, (

**b**) Maddock section, (

**c**) mixing section, (

**d**) Maillefer section, (

**e**) Barr section, and (

**f**) Rheotoc section.

**Figure 18.**Example of modeling: (

**a**) geometrical model of the melting mechanism, and (

**b**) temperature and velocity distribution in the cross-section of the screw channel.

**Figure 19.**Various concepts of implementation of screw characteristics into the global model of the extrusion process: (

**a**) screw pumping characteristics implemented into the melt region, (

**b**) total (continuous) screw characteristics implemented into the entire area of the screw. Example of modeling.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wilczyński, K.; Nastaj, A.; Lewandowski, A.; Wilczyński, K.J.; Buziak, K.
Fundamentals of Global Modeling for Polymer Extrusion. *Polymers* **2019**, *11*, 2106.
https://doi.org/10.3390/polym11122106

**AMA Style**

Wilczyński K, Nastaj A, Lewandowski A, Wilczyński KJ, Buziak K.
Fundamentals of Global Modeling for Polymer Extrusion. *Polymers*. 2019; 11(12):2106.
https://doi.org/10.3390/polym11122106

**Chicago/Turabian Style**

Wilczyński, Krzysztof, Andrzej Nastaj, Adrian Lewandowski, Krzysztof J. Wilczyński, and Kamila Buziak.
2019. "Fundamentals of Global Modeling for Polymer Extrusion" *Polymers* 11, no. 12: 2106.
https://doi.org/10.3390/polym11122106