# Melt Conveying in Single-Screw Extruders: Modeling and Simulation

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## Abstract

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## 1. Introduction

## 2. Modeling Fundamentals

#### 2.1. Screw Geometry

#### 2.2. Conservation Equations

#### 2.3. Constitutive Equations

#### 2.4. Fully Developed Flows

#### 2.5. Developing Flows

#### 2.6. Boundary Conditions and Mathematical Constraints

## 3. Exact Analytical Approaches

#### 3.1. Flow Pattern and Pumping Capability

#### 3.2. Dissipation and Power Consumption

## 4. Numerical Approaches

#### 4.1. One-Dimensional Non-Newtonian Down-Channel Flows

#### 4.2. Two-Dimensional Non-Newtonian Flows in Screw Channels of Infinite Width

#### 4.2.1. Fully Developed Flows

#### 4.2.2. Developing Flows

#### 4.3. Three-Dimensional Non-Newtonian Flows in Screw Channels of Finite Width

#### 4.3.1. Fully Developed Flows

#### 4.3.2. Developing Flows

## 5. Approximate Methods

## 6. Leakage Flow

## 7. Curved Channel Systems

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${a}_{t}$ | Temperature shift factor | $\mathrm{v}$ | Characteristic velocity |

${c}_{\mathrm{p}}$ | Specific heat capacity (constant pressure) | ${\mathrm{v}}_{b}$ | Barrel velocity |

${c}_{\mathrm{v}}$ | Specific heat capacity (constant volume) | ${\mathrm{v}}_{b,x}$ | Barrel velocity in the x-direction |

$Br$ | Brinkman number | ${\mathrm{v}}_{b,z}$ | Barrel velocity in the z-direction |

${D}_{b}$ | Barrel diameter | ${\mathrm{v}}_{i}$ | Velocities |

${D}_{s}$ | Screw core diameter | $\mathrm{v}$ | Velocity vector |

$D$ | Rate-of-deformation tensor | $\dot{V}$ | Volume flow rate |

$e$ | Flight width | ${\dot{V}}_{d}$ | Drag flow rate |

$f$ | Degree of filling | ${\dot{V}}_{p}$ | Pressure flow rate |

${f}_{L}$ | Correction factor for leakage flow | ${w}_{b}$ | Channel width at barrel surface |

${F}_{d}$ | Shape factor (drag flow) | ${w}_{b,f}$ | Width of filled channel |

${F}_{d,f}$ | Shape factor (drag flow), partially filled | ${w}_{b,uf}$ | Width of unfilled channel |

${F}_{p}$ | Shape factor (pressure flow) | $x$ | Cross-channel coordinate |

$g$ | Gravity vector | $y$ | Up-channel coordinate |

$h$ | Channel depth | $z$ | Down-channel coordinate |

$i$ | Number of screw flights | $Z$ | Unwound length |

$K$ | Consistency | $\alpha $ | Temperature coefficient |

$L$ | Characteristic length | $\delta $ | Flight clearance |

$L$ | Velocity gradient tensor | $\dot{\gamma}$ | Shear rate |

$MAE$ | Mean absolute error | $\eta $ | Viscosity |

$n$ | Power-law index | ${\eta}_{f}$ | Viscosity in the flight clearance |

$N$ | Screw speed | $\lambda $ | Heat conductivity |

${R}^{2}$ | Coefficient of determination | ${v}_{d}$ | Dimensionless velocity (drag flow) |

$Re$ | Reynolds number | ${v}_{i}$ | Dimensionless velocities |

$p$ | Pressure | ${v}_{p}$ | Dimensionless velocity (pressure flow) |

${P}_{Drive}$ | Drive power | ${\Pi}_{p,i}^{}$ | Dimensionless pressure gradients |

$Pe$ | Péclet number | ${\Pi}_{Q}^{}$ | Dimensionless dissipation |

${\dot{q}}_{Diss}$ | Viscous dissipation | ${\Pi}_{V}^{}$ | Dimensionless flow rate |

$t$ | Screw pitch | $\rho $ | Density |

$T$ | Temperature | ${\tau}_{ij}$ | Shear stresses |

${T}_{0}$ | Reference temperature | $\tau $ | Stress tensor |

${T}_{b}$ | Barrel temperature | ${\phi}_{b}$ | Pitch angle |

${T}_{s}$ | Screw temperature |

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**Figure 1.**Model categorization A: Classification of melt-conveying models according to the geometric and physical conditions under consideration.

**Figure 2.**Model categorization B: Classification of melt-conveying models according to the mathematical methodology applied.

**Figure 4.**Flat-plate model of the helical screw channel (

**a**). For shallow screw channels with $h/{w}_{b}<0.1$, the effect of the screw flights can be ignored (

**b**).

**Figure 5.**Partially filled screw channel. The widths of the filled and unfilled channel sections are ${w}_{b,f}$ and ${w}_{b,uf}$, respectively.

**Figure 6.**Dimensionless down-channel velocity distribution for a pressure-generating screw section (

**a**), a pressure-neutral screw section (

**b**), and a pressure-consuming screw section (

**c**).

**Figure 7.**Influence of the power-law index on the fully developed one-dimensional down-channel flow of a power-law fluid under isothermal conditions. The dimensionless volume flow rate ${\Pi}_{V}$ as a function of the dimensionless pressure gradient ${\Pi}_{p,z}$ (

**a**), and the dimensionless dissipation ${\Pi}_{Q}$ as a function of the dimensionless pressure gradient ${\Pi}_{p,z}$ (

**b**).

**Figure 8.**Influence of the power-law index on a fully developed two-dimensional flow of a power-law fluid in a screw channel of infinite width under isothermal conditions. The dimensionless volume flow rate ${\Pi}_{V}$ as a function of the dimensionless pressure gradient ${\Pi}_{p,z}$ (

**a**), and the dimensionless dissipation ${\Pi}_{Q}$ as a function of the dimensionless pressure gradient ${\Pi}_{p,z}$ (

**b**).

**Figure 9.**Influence of the power-law index on a fully developed two-dimensional flow of a power-law fluid in a screw channel of infinite width under isothermal conditions. The dimensionless volume flow rate ${\Pi}_{V}$ as a function of the dimensionless pressure gradient ${\Pi}_{p,z}$ (

**a**), and the dimensionless dissipation ${\Pi}_{Q}$ as a function of the dimensionless pressure gradient ${\Pi}_{p,z}$ (

**b**).

**Figure 10.**Influence of the power-law index on a fully developed three-dimensional flow of a power-law fluid in a screw channel of finite width under isothermal conditions. The dimensionless volume flow rate ${\Pi}_{V}$ as a function of the dimensionless pressure gradient ${\Pi}_{p,z}$ (

**a**), and the dimensionless dissipation ${\Pi}_{Q}$ as a function of the dimensionless pressure gradient ${\Pi}_{p,z}$ (

**b**).

**Figure 11.**Influence of the aspect ratio on a fully developed three-dimensional flow of a power-law fluid in a screw channel of finite width under isothermal conditions. The dimensionless volume flow rate ${\Pi}_{V}$ as a function of the dimensionless pressure gradient ${\Pi}_{p,z}$ (

**a**), and the dimensionless dissipation ${\Pi}_{Q}$ as a function of the dimensionless pressure gradient ${\Pi}_{p,z}$ (

**b**).

**Table 1.**Mathematical models of extruder flow of Newtonian fluids with temperature-independent viscosity.

No. | Flow | Equations | Boundary Conditions |
---|---|---|---|

1D_a | One-dimensional isothermal down-channel flow of a Newtonian fluid | (27) ($\eta =const.$) | (37) |

1D_b | One-dimensional isothermal down-channel flow of a Newtonian fluid with wall effects | (22) ($\eta =const.$) | (37) |

1D_c | One-dimensional isothermal cross-channel flow of a Newtonian fluid | (25) ($\eta =const.$) | (35), (42) |

2D_a | Two-dimensional isothermal recirculating cross-channel flow of a Newtonian fluid | (19)–(21) ($\eta =const.$) $\left(\partial {\mathrm{v}}_{x}/\partial x=\partial {\mathrm{v}}_{y}/\partial x=0\right)$ | (35), (36), (42) |

No. | Flow | Equations | Boundary Conditions |
---|---|---|---|

1D_d | One-dimensional isothermal down-channel flow of a power-law fluid | (16), (27), (31) | (37) |

1D_e | One-dimensional isothermal down-channel flow of a power-law fluid with wall effects | $\left(16\right),\text{}\left(22\right),\phantom{\rule{0ex}{0ex}}\left(23\right)\text{}({\mathrm{v}}_{x}={\mathrm{v}}_{y}=0)$ | (37) |

1D_f | One-dimensional non-isothermal down-channel flow of a power-law fluid | (16), (18), (27), (30), (31) | (37), (39), (40) or (41) |

**Table 3.**Mathematical models of a two-dimensional flow of a power-law fluid in a screw channel of infinite width.

No. | Flow | Equations | Boundary Conditions |
---|---|---|---|

2D_b | Fully developed two-dimensional isothermal flow of a power-law fluid in a screw channel of infinite width | (16), (25)–(27), (29) | (35), (37), (42) |

2D_c | Fully developed two-dimensional non-isothermal flow of a power-law fluid in a screw channel of infinite width | (16), (18), (25)–(29) | (35), (37), (39), (40) or (41), (42) |

2D_d | Developing two-dimensional flow of a power-law fluid in a screw channel of infinite width | (16), (18), (25)–(27), (29), (34) | (35), (37), (39), (40) or (41), (42) |

**Table 4.**Mathematical models of a three-dimensional flow of a power-law fluid in a screw channel of finite width.

No. | Flow | Equations | Boundary Conditions |
---|---|---|---|

3D_a | Fully developed three-dimensional isothermal flow of a power-law fluid in a screw channel of finite width | (16), (19)–(22), (23) | (35)–(37), (42) |

3D_b | Fully developed three-dimensional non-isothermal flow of a power-law fluid in a screw channel of finite width | (16), (18), (19)–(23) | (35)–(37), (39), (40) or (41), (42) |

3D_c | Developing three-dimensional flow of a power-law fluid in a screw channel of finite width | (16), (18), (19)–(22), (23), (33) | (35)–(37), (39), (40) or (41), (42) |

Year | Author | Target Variables | Flow Situation | Section |
---|---|---|---|---|

1969 | Krüger | Flow rate | 1D_d | Section 4.1 |

1981 | Potente | Flow rate and power consumption | 1D_d | Section 4.1 |

1981 | Booy | Flow rate | 2D_b | Section 4.2.1 |

1983 | Potente | Flow rate and power consumption | 2D_b | Section 4.2.1 |

1986 | Rauwendaal | Flow rate | 2D_b | Section 4.2.1 |

1995 | Kim and Kwon | Flow rate | 3D_a | Section 4.3.1 |

1996 | Effen | Flow rate | 2D_b | Section 4.2.1 |

1999 | Obermann | Power consumption | 3D_a | Section 4.3.1 |

2011 | Spalding and Campbell | Flow rate | 3D_a | Section 4.3.1 |

2017 | Pachner et al. | Flow rate | 2D_b | Section 4.2.1 |

2017 | Marschik et al. | Flow rate | 3D_a | Section 4.3.1 |

2018 | Roland and Miethlinger | Viscous dissipation | 1D_d and 2D_b | Section 4.1/Section 4.2.1 |

2019 | Roland | Flow rate | 1D_d | Section 4.1 |

2019 | Roland et al. | Flow rate and viscous dissipation | 2D_b | Section 4.2.1 |

2019 | Roland et al. | Viscous dissipation | 3D_a | Section 4.3.1 |

No. | Model | Literature | Flow Situation | Section | Modifications |
---|---|---|---|---|---|

1 | Marschik et al. | [116] | 3D_a | Section 4.3.1 | - |

2 | Rauwendaal | [125] | 2D_b | Section 4.2.1 | Shape factors |

3 | Effen | [130] | 2D_b | Section 4.2.1 | Shape factors |

4 | Roland et al. | [106] | 2D_b | Section 4.2.1 | Shape factors |

5 | Roland | [136] | 1D_d | Section 4.1 | Shape factors |

6 | Newtonian pumping model | [1] | 1D_b | Section 3.1 | Shape factors |

No. | $\mathit{n}$ | $\mathit{t}/{\mathit{D}}_{\mathit{b}}\text{}$ | $\mathit{h}/{\mathit{w}}_{\mathit{b}}\text{}$ | ${\mathit{\Pi}}_{\mathit{p},\mathit{z}}\text{}$ | ${\mathit{\Pi}}_{\mathit{V}}$ | Models |
---|---|---|---|---|---|---|

Dataset 1 | 0.2–1.0 | 0.6–2.0 | 0.05–0.5 | $-1.0$–var. | - | 1, 4, 5, 6 |

Dataset 2 | 0.2–1.0 | 0.8–2.0 | 0.05–0.5 | $-1.0$–var. | 0.1–2.0 | 1, 3, 4, 5, 6 |

Dataset 3 | 0.2–1.0 | 0.84–1.46 | 0.05–0.5 | $-1.0$–var. | 0.1–2.0 | 1–6 |

**Table 8.**Quality measures of the approximations: mean absolute error ($MAE$ ) and coefficient of determination (${R}^{2}$ ).

No. | Model | Dataset 1 | Dataset 2 | Dataset 3 | |
---|---|---|---|---|---|

1 | Marschik et al. | $MAE$ | 0.00719 | 0.00673 | 0.00555 |

${R}^{2}$ | 0.99973 | 0.99967 | 0.99980 | ||

2 | Rauwendaal | $MAE$ | - | - | 0.05290 |

${R}^{2}$ | - | - | 0.97291 | ||

3 | Effen | $MAE$ | - | 0.10934 | 0.18908 |

${R}^{2}$ | - | 0.02351 | −1.07304 | ||

4 | Roland et al. | $MAE$ | 0.02681 | 0.02465 | 0.02363 |

${R}^{2}$ | 0.99433 | 0.99244 | 0.99344 | ||

5 | Roland | $MAE$ | 0.11079 | 0.09800 | 0.09974 |

${R}^{2}$ | 0.90623 | 0.90105 | 0.99344 | ||

6 | Newtonian pumping model | $MAE$ | 0.17595 | 0.14418 | 0.14149 |

${R}^{2}$ | 0.83683 | 0.84890 | 0.85426 |

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**MDPI and ACS Style**

Marschik, C.; Roland, W.; Osswald, T.A.
Melt Conveying in Single-Screw Extruders: Modeling and Simulation. *Polymers* **2022**, *14*, 875.
https://doi.org/10.3390/polym14050875

**AMA Style**

Marschik C, Roland W, Osswald TA.
Melt Conveying in Single-Screw Extruders: Modeling and Simulation. *Polymers*. 2022; 14(5):875.
https://doi.org/10.3390/polym14050875

**Chicago/Turabian Style**

Marschik, Christian, Wolfgang Roland, and Tim A. Osswald.
2022. "Melt Conveying in Single-Screw Extruders: Modeling and Simulation" *Polymers* 14, no. 5: 875.
https://doi.org/10.3390/polym14050875