# Optimization of Polymer Processing: A Review (Part I—Extrusion)

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## Abstract

**:**

## 1. Introduction

- Use the simulation tools on a trial-and-error basis. This is obviously expensive and inefficient and relies on the capability of the user to input progressively more appropriate boundary conditions.
- Adopt an optimization procedure, whereby the process modelling package is used judiciously by an optimization algorithm, in order to define a “best” solution, or a Pareto optimal solution (see below). Practical polymer processing problems generally involve multiple, often conflicting, criteria (for example, maximizing output while minimizing viscous dissipation and mechanical energy consumption in plasticating single-screw extrusion); hence this approach is usually labelled as multi-objective optimization.

## 2. Need for Optimization in Polymer Processing

_{1}) and the maximum (f

_{2}) errors between the optimal thickness distribution and the thickness distribution of the parison must be minimized. More details on this optimization can be found elsewhere [15,16].

## 3. Multi-Objective Optimization

_{i}, g

_{j}are the J (J ≥ 0) inequality constraints, and h

_{k}are the K (K ≥ 0) equality constraints.

_{i}). From this model, an approximation to the optimal solution can be found both graphically and mathematically. This simple approach relates linearly the objectives with the decision variables. The quality of the solution depends strongly on the number of solutions available to construct the model. Consequently, more elaborated models can be deduced if more solutions are available. This type of regression method has different forms of being identified in the literature: design of experiments, response surface, statistical analysis, data fitting, etc.

## 4. Optimization Algorithms in Polymer Processing

#### 4.1. Methodology

- Objective function. It can be Single Objective (SO), Aggregated Product (AP), Aggregated Sum (AS), or Multi-Objective (MO).
- Optimization algorithm, e.g., Empirical, Regression, Direct, Gradient, Augmented Lagrangian (AL), Pattern Search (PS), Expert System (ES), Evolutionary Algorithm (EA), Differential Evolution (DE), Ant Colony Optimization (ACO), Stochastic Local Search (SLS), or Two-Phase Local Search (TPLS).
- Modelling approach: unidimensional (1D), two-dimensional (2D) and three-dimensional (3D), using Analytical (A), Finite Differences (FD), Finite Volumes (FV) or Finite Elements (FE) approaches; whenever relevant, the actual software used is identified.
- Decision variables, i.e., parameters to optimize. The aim can be to define the Operating Conditions (OC), Screw Design (SD), Screw Configuration (SC) (the last two will be explained below), or Die Geometry (DG). The number of variables considered in the problem is indicated between brackets in the tables below.
- Other characteristics, related with the process/modelling, the optimization, or others.

#### 4.2. Single-Screw Extrusion

Objective Function | Optimization Algorithm | Modelling Approach | Decision Variables | Other Characteristics | Authors (Year) Reference |
---|---|---|---|---|---|

SO | Direct | 1D-A | SD | Step-by-step | Helmy and Parnaby (1976) [38] |

SO | Empirical | 1D-A | SD | Grooves | Potente et al. (1992) [39] |

SO | ES | 1D-A | OC + SD | Worteberg et al. (1994) [40] | |

SO | Empirical | 1D-A | SD | Step-by-step | Chung (1998, 2016) [8,41] |

SO | Empirical | 1D-A | SD | Zone-by-zone | Rauwendaal (1986) [42] |

SO | AL | 3D-N | SD | Altinkaynak (2010) [43] | |

AP | Empirical | 1D-A | OC | Potente et al. (1993, 1994, 1996) [44,45,46] | |

AP | Regression | 1D-A | SD | Statistical | Potente and Zelleröhr (1997) [47] |

AP | Regression | 1D-A | SD | DOE | Potent and Krell (1997) [48] |

AP(3) | Regression | 1D-A | OC(2) + SD(1) | Wilczyński et al. (2001, 2003) [49,50] | |

AP(3) | Regression | 1D-A | OC(2) + SD(1) | Wilczyński et al. (2004) [51] | |

AS(3) | Regression | 1D-A | SD | Thibodeau and Lafleur (2000) [52,53] | |

AS(2) | EA | 1D-A | OC(2) + SD(1) | Nastaj and Wilczyński (2018) [54] | |

AS(2) | EA | 1D-A | OC(2) + SD(1) | Starve-feed | Nastaj and Wilczyński (2020) [55] |

AS(2) | DE + PS | Experimental | OC(1) | Various techniques | Abeykoon et al. (2011) [56] |

AS(4) | EA | 2D-N | OC(4) | Gaspar-Cunha et al. (1998) [57] | |

AS(4) + MO(4) | EA | 2D-N | OC(4) | Covas et al. (1999) [58] | |

MO(7) | EA | 2D-N | SD(6) | Gaspar-Cunha et al. (2001) [59] | |

MO(5) | EA | 2D-N | SD(5) | Barrier screws | Covas et al. (2004) [60] |

MO(2) | EA | 2D-N | OC(4) + SD(6) | Mixing | Domingues at al. (2012) [61] |

MO(5) | EA | 2D-N | SD(4) | Barrier screws | Gaspar-Cunha et al. (2006) [62] |

MO(19) | EA | 2D-N | OC(3) | Scale-up | Covas and Gaspar-Cunha (2009) [63] |

MO(9) | EA | 2D-N | SD(4) | Scale-up | Gaspar-Cunha and Covas (2014) [64] |

MO(3) | EA | 2D-N | SD(4) | Robustness + DM | Denysiuk et al. (2018) [65] |

MO(5) | EA | 2D-N | OC(4) + SD86) | Innovization | Deb et al. (2014) [66] |

#### 4.3. Twin-Screw Extrusion

_{b}-barrel temperature profile); (ii) the determination of the geometry of individual screw elements; and/or (iii) the determination of the screw configuration, (i.e., finding the best location along the screw axis of existing screw elements). These problems arise within the context of compounding, reactive extrusion, extrusion, or scale-up and can involve co-rotating or counter-rotating intermeshing twin-screw extruders. Table 2 summarizes the features of the previous studies on these topics.

Objective Function | Optimization Algorithm | Modelling Approach | Decision Variables | Other Characteristics | Authors (Year) Reference |
---|---|---|---|---|---|

Not defined | Empirical | 1D-A | Not defined | Potente et al. (1994, 1999) [67,68] | |

SO | Empirical | Experimental | Not-defined | Mixing | Vainio et al. (1995) [69] |

SO(2) | Regression | 1D-Ludovic | OC(3) + SD(1) | Reactive Extrusion | Berzin et al. (2007) [70] |

SO | Regression | Experimental | OC(2) | Counter-rotating | Maridass and Gupta (2004) [71] |

SO(2) | Regression | Experimental | OC | Reactive Extrusion | Ulitzsh et al. (2020) [72] |

SO(2) | Regression | Experimental | OC(2) | Scale-up | Fukuda et al. (2015) [73] |

AP(3) | Gradient | 1D-A | SD(2) | Conv. elements | Potente and Thümen (2006) [74] |

AS(2) | EA | 2D-numerical | OC(1) + SD(1) | Reactive Extrusion | Zhang et al. (2015) [75] |

AS + MO(6) | EA | 1D-Ludovic | OC(4) | Gaspar-Cunha et al. (2002) [76] | |

MO(7+2) | EA | 1D-Ludovic | OC(4) + SC(10) | Reactive Extrusion | Gaspar-Cunha et al. (2005) [78] |

MO(5)(7) | EA | 1D-Ludovic | SD(4) + SC(10) | Robustness | Covas et al. (2004) [60] |

MO(3) | SLS | 2D-FD | SC(14) | Teixeira et al. (2011) [79] | |

MO(3) | EA + ACO + SLS + TPLS | 2D-FD | SC(14) | Teixeira et al. (2012) [81] | |

MO(3) | ACO + TPLS | 2D-FD | SC(14) | Teixeira et al. (2014) [82] | |

MO(3) | EA | 1D-Ludovic | OC(1) + SC(14) | Reactive Extrusion | Teixeira et al. (2011) [83] |

MO(3) | EA | 2D-FD | SD(1) + SC(8) | Scale-up | Gaspar-Cunha and Covas (2011) [84] |

#### 4.4. Dies and Calibrators

- (i)
- Using a manifold, i.e., use a larger channel upstream to distribute the flow transversally, prior to its progress downstream. The die geometry is such that a central flow stream has a shorter path in the manifold and a longer path in the shallower parallel zone, while the reverse occurs for a flow stream near to the edges. This approach is frequently adopted for the production of cast film and sheet, wire insulation, and in extrusion blow moulding.
- (ii)
- Using a cylindrical mandrel to convert the circular flow from the extruder into an annular flow. Since the classical torpedo-type solution with its supports (known as spider legs) creates unbalanced flow and strong weld lines, it was progressively replaced by basket-type dies and spiral mandrel dies. The mandrels of the latter are designed in such a way that the flow from the extruder is divided into individual melts that feed helical channels with decreasing depth along their length in the mandrel. Thus, the helical flow is gradually converted into an axial annular flow.
- (iii)
- Change gradually from the inlet circular channel into the desired cross-section. The design of dies for hollow profiles, or for profiles containing thickness differences in their cross-section is particularly challenging.

#### 4.4.1. Manifold Dies

#### 4.4.2. Mandrel Dies

#### 4.4.3. Profile Dies

**Table 5.**Previous publications on the optimization of profile dies (KP—key points (see text); MP—mesh parameterization; GP—geometry parameterization).

Objective Function | Optimization Algorithm | Modelling Approach | Decision Variables | Other Characteristics | Authors (Year) Reference |
---|---|---|---|---|---|

SO | Empirical | 3D-N | GP | IEP | Legat and Marchal (1993) [127] |

SO | Empirical | 3D-N | GP | IEP | Tran-Cong and Phan-Thien (1988) [129] |

SO | Empirical | A | GP | Hurez et al. (1996) [130] | |

SO | Empirical | 3D-N | GP | Švábík et al. (1999) [131] | |

SO | Empirical | 3D-N | GP | IEP | Gifford (2003) [132] |

SO | Empirical | 3D-N | GP(3) | Rezaei Shahreza et al. (2010) [133] | |

SO | Simplex | 3D-N | GP | Coupez et al. (1999) [134] | |

SO | Regression | 3D-N | MP | Ready and Schaub (1999) [135] | |

SO | Regression | 3D-N | GP(22) | Elgeti et al. (2012) [136] | |

SO | Regression | 3D-N | GP(171) | IEP | Pauli et al. (2013) [137] |

SO | Gradient | 3D-N | MP | Sienz et al. (1998, 2010) [138,139] | |

SO | Gradient | 3D-N | GP | Szarvasy et al. (2000) [140] | |

SO | ES | 3D-N | MP | Sienz et al. (1999) [141] | |

SO | Gradient | 2D-N | KP | Ettinger et al. (2004, 2004) [142,143] | |

SO | Gradient | 2D-N | KP(2-46] | Sienz et al. (2012) [144] | |

SO | SA | 3D-N | GP(3) | Yilmaz et al. (2014) [145] | |

SO | Feedback Control | 3D-N | GP | IEP | Spanjaards et al. (2021) [146] |

WS(2) | Simplex | 3D-N | GP | Nóbrega et al. (2002, 2003) [147,148,149] | |

WS(2) | Simplex | 3D-N | GP | Carneiro et al. (2004) [150] | |

WS(4) | Gradient | 3D-N | GP(8) | Zhang et al. (2019) [154] |

#### 4.4.4. Calibrators

Objective function | Optimization Algorithm | Modelling Approach | Decision Variables | Other Characteristics | Authors (Year) Reference |
---|---|---|---|---|---|

SO | Simplex | 3D-N | GP(5) | - | Nóbrega and Carneiro (2005) [156] |

AS(2) | Empirical | 3D-N | GP(n) | - | Duan and Zhang (2014) [158] |

AS(2) | Gradient | 3D-N | GP(48) | - | Fradette et al. (1996) [155] |

AS(2) | EA | 3D-N | GP(n) | - | Ren et al. (2010) [159] |

MO | EA | 3D-N | GP(8) | - | Nóbrega et al. (2008) [157] |

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Optimization of Injection Stretch Blow-Moulding. (

**A**–

**F**) illustrate the process steps. Open arrows follow the process sequence; curved arrows follow the optimization sequence.

**Figure 2.**Typical results for a blow-moulding optimization: (i) optimization of bottle thickness profile; (ii) optimization of pre-form thickness profile before blowing. For example: solution S3-i is selected for step (ii), from which a new Pareto set is obtained.

**Figure 5.**Concept of robustness in multi-objective environment. Solution 1 is more robust than Solution 2, as the same variation in the decision variables domain (x

_{1},x

_{2}) produces less variation in the objectives’ domain (f

_{1},f

_{2}).

**Figure 6.**Polymer processing sequences targeted by the present review. (

**A**) Single-screw extrusion of profiles (

**A1**), flat film/sheet for thermoforming; (

**A2**), extrusion blow moulding (

**A3**); (

**B**) co-rotating twin-screw compounding and pelletizing (

**B1**); (

**C**) injection moulding: (

**C1**) mould (

**C2**); injection blow moulding. Left: plasticating units; Right: shaping and cooling.

**Figure 7.**Pareto curves after optimization of the operating conditions of an SSE in order to maximize output and mixing, and minimize the length of screw required for melting.

**Figure 8.**Optimization of co-rotating twin-screw extruders (TSE): (

**a**) operating conditions—screw speed (N), feed rate (Q) and barrel and die set temperatures (Tb); (

**b**) geometry of individual screw elements; (

**c**) position of a set of individual screw elements (5 conveying elements, 3 kneading blocks, and 1 left-handed element) along the screw shaft.

**Figure 9.**Optimization of a die for the production of a hollow profile: (

**A**) required fixed-window profile cross section and (

**B**) evolution of the objective function versus function call for automatic and manual optimization (adapted with permission from [144]).

**Table 3.**Previous publications on the optimization of manifold dies (manifold type: CH-Coat Hanger, TCH-Tapered Coat Hanger, Blow–blow moulding).

Objective Function | Optimization Algorithm | Modelling Approach | Decision Variables | Other Characteristics | Authors (Year) Reference |
---|---|---|---|---|---|

Not defined | Empirical | 1D-A | DG | Various dies | Rakos and Sebastian (1990) [85] |

SO | Empirical | 1D-A | DG(1) | CH | Matsubara (1979, 1980) [86,87] |

SO | Empirical | 1D-A | DG(1) | T-die | Matsubara (1980, 1988) [88,89] |

SO | Empirical | 1D-A | DG(3) | CH | Winter and Fritz (1986) [90] |

SO | Empirical | 3D-N | DG(3) | CH | Liu et al. (1988, 1994) [91] |

SO | Empirical | 3D-N | DG(4) | TCH, 2 cavities | Lee and Liu (1989) [92] |

SO | Empirical | 3D-N | DG(3) | CH | Liu et al. (1988, 1994) [93] |

SO | Empirical | 3D-N | DG(4) | TCH | Yu and Liu (1998) [94] |

SO | Empirical | 3D-N | DG(3) | CH | Na and Kim (1995) [95] |

SO | Empirical | 2D-N | DG(2) | CH | Huang et al. (2004) [96] |

SO | Regression | 1D-A | OC(1) + DG(3) | CH | Chen et al. (1997) [97] |

SO | Regression | 3D-N | DG(5) | CH | Razeghiyadaki et al. (2020, 2021) [98,99] |

SO | SQP + Regression | 3D-N | DG(1) | CH | Lebaal et al. (2006) [100] |

SO | SQP + Regression | 3D-N | DG(4) | CH | Lebaal et al. (2009) [101] |

SO | SQP + Regression | 3D-N | OC(3) + DG(1) | CH | Lebaal et al. (2010) [102] |

SO | SQP + Regression | 3D-N | DG(4) | CH (wire) | Lebaal et al. (2012) [103] |

SO | Gradient | 3D-N | DG(2) | CH | Smith et al. (1998, 1998) [104,105] |

SO | Gradient | 3D-N | OC(1) + DG(2) | CH | Smith (2003) [106] |

SO | Gradient | 3D-N | DG(811) | CH, Robustness | Smith (2003) [107] |

SO | Gradient | 3D-N | DG(9) | CH | Sun and Gupta (2004) [108] |

SO | Gradient | 3D-N | DG(5) | CH, Restrictor | Bates et al. (2003) [109] |

SO | Regression + Gradient + EA | 3D-N | DG(5) | CH, Restrictor | Siens et al. (2006) [110] |

SO | EA | 3D-N | DG(n) | CH | Michaeli and Kaul (2004) [111] |

SO | EA | 3D-N | DG(2) | CH | Meng and Zhao (2011) [112] |

SO | EA | 3D-N | DG(4) | Slot die | Sun and Wang (2010) [113] |

SO | EA | 3D-N | DG(2) | Blow: 2-CH | Meng et al. (2009, 2012) [114,115] |

AS(2) | Regression | 3D-N | DG(3) | CH | Han and Wang (2012) [116] |

AS(n) | Gradient | 3D-N | OC(1) + DG(2) | CH, Robustness | Smith and Wang (2004) [117] |

AS(n) | Gradient | 3D-N | OC(1) + DG(2) | CH | Smith and Wang (2005) [118] |

AS(n) | SQP | 3D-N | OC(1) + DG(2) | CH | Wang and Smith (2006) [119,120] |

AS(3) | EA | 3D-N | OC() + DG() | CH | Zhang et al. (2020) [121] |

MO(2) | DOE, RSM, EA | 3D-N | DG(3/8/12) | CH | Lee et al. (2015) [122] |

MO(2) | EA | 3D-N | DG(3) | CH | Han and Wang (2012) [123] |

AS(2) & MO(2) | Regression + EA | 3D-N | DG(1) | Blow: 2-CH | Han and Wang (2014) [124] |

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Gaspar-Cunha, A.; Covas, J.A.; Sikora, J.
Optimization of Polymer Processing: A Review (Part I—Extrusion). *Materials* **2022**, *15*, 384.
https://doi.org/10.3390/ma15010384

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Gaspar-Cunha A, Covas JA, Sikora J.
Optimization of Polymer Processing: A Review (Part I—Extrusion). *Materials*. 2022; 15(1):384.
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Gaspar-Cunha, António, José A. Covas, and Janusz Sikora.
2022. "Optimization of Polymer Processing: A Review (Part I—Extrusion)" *Materials* 15, no. 1: 384.
https://doi.org/10.3390/ma15010384