# Design and Ductile Behavior of Torsion Configurations in Material Extrusion to Enhance Plasticizing and Melting

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Computational Methodology

#### 2.1. Physical Model

#### 2.2. Governing Equations and Mesh System

#### 2.3. Boundary Conditions

#### 2.4. Characterization

_{SD}) at the outlet was used to characterize the plasticizing uniformity, which can be expressed as:

_{i}and A

_{i}are the temperature and area of the ith cell, A is the area of the whole domain, and $\overline{T}$ is the average temperature of the whole domain.

#### 2.5. Orthogonal Experiment Design

^{5}), 16 torsion configurations were designed, as shown in Figure 4.

#### 2.6. Experimental

## 3. Results and Discussion

#### 3.1. The Ductile Formation Mechanism of The Torsion Configuration

#### 3.2. Analysis of Orthogonal Test Results

_{SD}) at the outlet were studied. The orthogonal experiment result table L16(4

^{5}) involves three factors, with four levels of each factor, and the test indexes of the results are the liquid mass fraction (LMF), the Nusselt number (Nu), the shear stress (τ), and the standard deviation of temperature (T

_{SD}) at the outlet when the screw speed was 75 r/min, as shown in Table 6. All the indexes of the test results were averaged.

_{ij}(i = A, B, C; j = 1, 2, 3, 4) reflects the average of all the result indicators related to the level j; the range (R

_{i}) is the difference between minimum and maximum of K

_{ij}for the factor i in the same column, representing the importance degree of factor i.

#### 3.2.1. Liquid Mass Fraction

_{ij}is optimum. Thus, the best group is A

_{1}B

_{1}C

_{1}, corresponding to case 1, that is, when the number of flights is eight, the width of flight is 1 mm, and the height of the channel is 3 mm, the average liquid mass fraction of the whole domain is the maximum, which is 79.7%.

_{C}> R

_{B}> R

_{A}. Accordingly, the influence degree of three factors on the liquid mass fraction at 75 r/min is: the height of the channel > the width of flight > the number of flights. Thus, we can find that, in three factors affecting the liquid mass fraction, the height of the channel is the main factor, and the width of flight and the number of flights are second, respectively.

#### 3.2.2. Nusselt Number

_{ij}is optimum. Thus, the best group is A

_{1}B

_{1}C

_{1}, corresponding to case 1, that is, when the number of flights is eight, the width of flight is 1 mm, and the height of the channel is 3 mm, the average Nusselt number of the whole domain is the maximum, which is 342.2.

_{C}> R

_{A}> R

_{B}. Accordingly, the influence degree of three factors on the Nusselt number at 75 r/min is: the height of the channel > the number of flights > the width of flight. Thus, we found that, in three factors affecting the Nusselt number, the height of the channel is the main factor, and the number of flights and the width of the flight are second, respectively.

#### 3.2.3. Shear Stress

_{ij}is optimum, special attention should be paid to the fact that the shear stress should not be too large to avoid product defects caused by overheating and over-shear; fortunately, this condition did not occur in this orthogonal experiment.

_{3}B

_{2}C

_{4}, corresponding to case 10, that is, when the number of flights is 12, the width of flight is 2 mm, and the height of the channel is 1.5 mm, the average shear stress of the whole domain is the maximum, which is 3.855 kPa.

_{C}> R

_{B}> R

_{A}. Accordingly, the influence degree of three factors on the shear stress at 75 r/min is: the height of the channel > the width of flight > the number of flights. Thus, we can conclude that, in three factors affecting the shear stress, the height of the channel is the main factor, and the width of flight and the number of flights are second, respectively.

#### 3.2.4. Standard Deviation of Temperature

_{ij}is optimum. Thus, the best group is A

_{4}B

_{3}C

_{4}, which was obviously not in this orthogonal experiment scheme, reflecting that the orthogonal analysis can find the best combination, even the unplanned experiment group. According to the results of the orthogonal experiment, the best group of this test is A

_{4}B

_{3}C

_{2}, corresponding to case 15, that is, when the number of flights is 14, the width of the flight is 3 mm, and the height of the channel is 2.5 mm, the average standard deviation of temperature at the outlet is the minimum, which is 0.212 °C.

_{A}> R

_{B}> R

_{C}. Accordingly, the influence degree of three factors on the standard deviation of temperature at 75 r/min is: the number of flights > the width of flight > the height of the channel.

_{1}B

_{1}C

_{1}, whereas the optimal combination is A

_{3}B

_{2}C

_{4}for the shear stress (τ) and is A

_{4}B

_{3}C

_{4}for the standard deviation of temperature (T

_{SD}), respectively. In a word, the influence of characteristic parameters of the torsion configuration on different indexes is different. Therefore, we should consider each index comprehensively, weigh the requirements to maximize the benefits of the whole life cycle in the extrusion, including efficiency, energy consumption and quality, and finally get the optimal combination.

#### 3.3. Screw Speed Analysis

_{SD}) at various screw speeds were similar; in particular, the similar phenomenon is more obvious at high screw speeds for the standard deviation of temperature (T

_{SD}). This shows from a certain side that the above orthogonal analysis is feasible to select a certain speed, 75 r/min in the analysis.

#### 3.4. Weight Matrix Analysis

_{SD}) are A = 0.4348, B = 0.4131, C = 0.1521, the impact order of three factors for T

_{SD}is: A > B > C, the weights of factor A and factor B are close and much larger than that of factor C. The results of the weight matrix analysis and the range analysis are in good agreement. Considering the comprehensive effect of the three factors on the liquid mass fraction (LMF) and standard deviation of temperature (T

_{SD}) at the outlet, the best group is A

_{4}B

_{3}C

_{1}, and the worst combination is A

_{1}B

_{1}C

_{4}.

#### 3.5. Experimental Validation

_{4}B

_{3}C

_{1}, marked with A*) and the worst (A

_{1}B

_{1}C

_{4}, marked with B*) torsion configurations were designed and fabricated to run an extrusion experiment, and as a control, the conventional screw (marked with C*) was also employed in the experiment.

## 4. Conclusions

_{SD}) is: the number of flights ≒ the width of flight > the height of the channel (A ≒ B > C), respectively. The results indicated that the influence of characteristic parameters of torsion configuration on different indicators is different, which should be considered comprehensively.

_{SD}) are A = 0.4348, B = 0.4131, C = 0.1521. Results indicated the height of the channel is the main factor to determine the liquid mass fraction (LMF), and combinations of the number of flights and the width of flight, reflecting the width of the torsion channel, are the main factors to jointly determine the standard deviation of temperature (T

_{SD}).

_{4}B

_{3}C

_{1}; in other words, when the number of flights is 14, the width of flight is 3 mm, and the height of the channel is 3mm, i.e., the aspect ratio of the torsion channel is almost equal to 1, the plasticizing and melting capability of the polymer is the best, which was validated by the extrusion experiment in practice. Thereby, it can offer a reference for the design and optimization of torsion configurations and provide an example for energy-efficient plasticization of polymers.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

LMF | =liquid mass fraction, % |

Nu | =Nusselt number |

τ | =shear stress, kPa |

T_{SD} | =standard deviation of temperature, °C |

D | =screw diameter, mm |

D_{0} | =barrel diameter, mm |

l_{1} | =length of screw zone, mm |

L + 2b | =length of torsion configuration, mm |

l_{2} | =length of torsion zone, mm |

l_{3} | =length of polished rod area, mm |

ΔH_{f} | =pure solvent melting heat, J/kg |

T_{S} | =solidus Temperature, °C |

T_{L} | =liquidus temperature, °C |

K_{ij} | =average of all the results indicator related to the level j |

R_{i} | =range |

u | =velocity vector, m/s |

T | =temperature, K |

P | =pressure, Pa |

φ | =heat dissipation term |

$\dot{\gamma}$ | =shear rate, s^{−1} |

ρ | =density, kg/m^{3} |

C_{p} | =specific heat capacity, J/(kg·K) |

λ | =thermal conductivity, W/(m·K) |

η | =apparent viscosity, Pa·s |

η_{0} | =zero shear viscosity, Pa·s |

η_{∞} | =viscosity at an infinite shear rate, Pa·s |

t_{0} | =natural time, s |

α | =temperature sensibility coefficient, K^{−1} |

n | =non-Newtonian index |

T_{α} | =reference temperature, °C |

T_{0} | =absolute zero, °C |

δ(T) | =standard deviation of temperature, °C |

T_{i} | =temperature of the ith cell, °C |

A_{i} | =area of the ith cell, m^{2} |

A | =area of the whole domain, m^{2} |

$\overline{T}$ | =average temperature of the whole domain, °C |

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**Figure 1.**Three-dimensional model (

**a**) and the characteristic dimensions (

**b**) of the torsion configuration.

**Figure 3.**Grid independent test for data of liquid mass fraction and standard deviation of temperature (

**a**), and the discrete model with about 2,100,000 cells (

**b**).

**Figure 5.**The experimental extruder with a radial temperature measurement in the position of the die.

**Figure 6.**Streamline distributions of the axial direction (

**a**) and the vertical cross-section (

**b**) in the screw channel for case 6.

**Figure 10.**Variation curve of LMF (

**a**) and T

_{SD}(

**b**) with factors and levels at different screw speeds.

**Figure 12.**Temperature profile in the extrusion die: (

**a**) radial temperature distribution at 75 r/min; (

**b**) standard deviation of temperature at different screw speeds.

Dimension | Symbol | Value/mm |
---|---|---|

Screw diameter | D | 30 |

Barrel diameter | D_{0} | 30.4 |

Length of screw zone | l_{1} | 30 |

Length of torsion configuration | L + 2b | 14 |

Length of torsion zone | l_{2} | 42 |

Length of polished rod area | l_{3} | 10 |

Parameters | Symbol | Value |
---|---|---|

Density | ρ | 910 kg/m^{3} |

Thermal conductivity | λ | 0.2 W/(m·°C) |

Specific heat capacity | C_{p} | 2300 J/(kg·°C) |

Pure solvent melting heat |
∆H_{f} | 150,000 J/kg |

Solidus Temperature | T_{S} | 100 °C |

Liquidus temperature | T_{L} | 170 °C |

Zero shear viscosity | η_{0} | 9650 Pa·s |

Viscosity at an infinite shear rate | η_{∞} | 0 Pa·s |

Non-Newtonian index | n | 0.48 |

Natural time | t_{0} | 0.3664 s |

Coefficient of temperature sensibility | α | 2000 °C^{−1} |

Reference temperature | T_{α} | 0 °C |

Absolute zero | T_{0} | −273.15 °C |

Screw Model | Boundary | Flow Conditions | Thermal Conditions |
---|---|---|---|

Inlet wall | Velocity inflow ^{1} | 100 °C | |

Outlet wall | Pressure = 3 MPa | 185 °C | |

Barrel wall | 30, 45, 60, 75, 90, 105, 120 r/min | Heat flux, Q = 40,000W/m^{2} | |

Screw wall | Stationary surface | Insulated surface |

^{1}Velocity varied with the screw speed.

Level | Factor A | Factor B | Factor C |

1 | 8 | 1 mm | 3.0 mm |

2 | 10 | 2 mm | 2.5 mm |

3 | 12 | 3 mm | 2.0 mm |

4 | 14 | 4 mm | 1.5 mm |

Case | Factor A | Factor B | Factor C |

1 | 8 | 1 mm | 3.0 mm |

2 | 8 | 2 mm | 2.5 mm |

3 | 8 | 3 mm | 2.0 mm |

4 | 8 | 4 mm | 1.5 mm |

5 | 10 | 1 mm | 2.5 mm |

6 | 10 | 2 mm | 3.0 mm |

7 | 10 | 3 mm | 1.5 mm |

8 | 10 | 4 mm | 2.0 mm |

9 | 12 | 1 mm | 2.0 mm |

10 | 12 | 2 mm | 1.5 mm |

11 | 12 | 3 mm | 3.0 mm |

12 | 12 | 4 mm | 2.5 mm |

13 | 14 | 1 mm | 1.5 mm |

14 | 14 | 2 mm | 2.0 mm |

15 | 14 | 3 mm | 2.5 mm |

16 | 14 | 4 mm | 3.0 mm |

Case | Factor A | Factor B | Factor C | LMF/% | Nu | τ/kPa | T_{SD}/ °C |
---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 79.70 | 342.2 | 2.504 | 0.625 |

2 | 1 | 2 | 2 | 78.75 | 315.9 | 2.832 | 0.418 |

3 | 1 | 3 | 3 | 76.38 | 274.5 | 3.331 | 0.439 |

4 | 1 | 4 | 4 | 73.58 | 240.8 | 3.806 | 0.410 |

5 | 2 | 1 | 2 | 78.95 | 318.8 | 2.765 | 0.499 |

6 | 2 | 2 | 1 | 78.01 | 308.9 | 2.942 | 0.381 |

7 | 2 | 3 | 4 | 74.03 | 242.7 | 3.804 | 0.290 |

8 | 2 | 4 | 3 | 76.68 | 277.1 | 3.241 | 0.360 |

9 | 3 | 1 | 3 | 75.67 | 266.7 | 3.363 | 0.550 |

10 | 3 | 2 | 4 | 74.56 | 246.8 | 3.855 | 0.276 |

11 | 3 | 3 | 1 | 78.57 | 317.4 | 2.871 | 0.294 |

12 | 3 | 4 | 2 | 75.68 | 271.4 | 3.382 | 0.329 |

13 | 4 | 1 | 4 | 75.36 | 254.6 | 3.453 | 0.327 |

14 | 4 | 2 | 3 | 73.02 | 242.2 | 3.794 | 0.214 |

15 | 4 | 3 | 2 | 76.87 | 288.9 | 3.120 | 0.212 |

16 | 4 | 4 | 1 | 79.69 | 339.2 | 2.640 | 0.319 |

Parameters | Factor A | Factor B | Factor C |
---|---|---|---|

K_{i}_{1} | 77.10 | 77.42 | 78.99 |

K_{i}_{2} | 76.92 | 76.09 | 77.56 |

K_{i}_{3} | 76.12 | 76.46 | 75.44 |

K_{i}_{4} | 76.24 | 76.41 | 74.38 |

R_{i} | 0.98 | 1.33 | 4.61 |

Parameters | Factor A | Factor B | Factor C |
---|---|---|---|

K_{i}_{1} | 293.35 | 295.58 | 326.93 |

K_{i}_{2} | 286.88 | 278.45 | 298.75 |

K_{i}_{3} | 275.58 | 280.88 | 265.16 |

K_{i}_{4} | 281.23 | 282.13 | 246.23 |

R_{i} | 17.77 | 17.13 | 80.70 |

Parameters | Factor A | Factor B | Factor C |
---|---|---|---|

K_{i}_{1} | 3.12 | 3.02 | 2.74 |

K_{i}_{2} | 3.19 | 3.36 | 3.02 |

K_{i}_{3} | 3.37 | 3.28 | 3.43 |

K_{i}_{4} | 3.25 | 3.27 | 3.73 |

R_{i} | 0.25 | 0.34 | 0.99 |

Parameters | Factor A | Factor B | Factor C |
---|---|---|---|

K_{i}_{1} | 0.47 | 0.50 | 0.40 |

K_{i}_{2} | 0.38 | 0.32 | 0.36 |

K_{i}_{3} | 0.36 | 0.31 | 0.39 |

K_{i}_{4} | 0.27 | 0.35 | 0.33 |

R_{i} | 0.20 | 0.19 | 0.07 |

Case | Length | Height | Width | Aspect Ratio |
---|---|---|---|---|

I | 15 mm | 2 mm | 1 mm | 0.5 |

II | 15 mm | 2 mm | 2 mm | 1 |

III | 15 mm | 2 mm | 4 mm | 2 |

IV | 15 mm | 2 mm | 8 mm | 4 |

Parameters | A1 | A2 | A3 | A4 | B1 | B2 | B3 | B4 | C1 | C2 | C3 | C4 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

LMF/% | 0.03556 | 0.03548 | 0.03511 | 0.03516 | 0.04877 | 0.04793 | 0.04817 | 0.04814 | 0.17163 | 0.16853 | 0.16391 | 0.16162 |

T_{SD}/ °C | 0.08192 | 0.10127 | 0.10700 | 0.14460 | 0.07399 | 0.11484 | 0.11982 | 0.10440 | 0.03466 | 0.03851 | 0.03593 | 0.04308 |

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## Share and Cite

**MDPI and ACS Style**

Jian, R.; Yang, W.; Sain, M.; Zhang, C.; Wu, L.
Design and Ductile Behavior of Torsion Configurations in Material Extrusion to Enhance Plasticizing and Melting. *Polymers* **2021**, *13*, 3181.
https://doi.org/10.3390/polym13183181

**AMA Style**

Jian R, Yang W, Sain M, Zhang C, Wu L.
Design and Ductile Behavior of Torsion Configurations in Material Extrusion to Enhance Plasticizing and Melting. *Polymers*. 2021; 13(18):3181.
https://doi.org/10.3390/polym13183181

**Chicago/Turabian Style**

Jian, Ranran, Weimin Yang, Mohini Sain, Chuanwei Zhang, and Lupeng Wu.
2021. "Design and Ductile Behavior of Torsion Configurations in Material Extrusion to Enhance Plasticizing and Melting" *Polymers* 13, no. 18: 3181.
https://doi.org/10.3390/polym13183181