# Approaches for Numerical Modeling and Simulation of the Filling Phase in Injection Molding: A Review

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## Abstract

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## 1. Introduction

## 2. Modeling of the Filling Phase

#### 2.1. 1-Dimensional Model

#### 2.2. 2-Dimensional Model

#### 2.3. 2.5-Dimensional Model

#### 2.4. 3-Dimensional Model

#### 2.5. Rheological Model

#### 2.5.1. Power-Law Model

#### 2.5.2. Second-Order Model

#### 2.5.3. Herschel–Bulkley Model

#### 2.5.4. Bingham Plastic Model

#### 2.5.5. Temperature Shift Factors

#### 2.5.6. Carreau Model

#### 2.5.7. Bird–Carreau Model

#### 2.5.8. Cross Model

## 3. Overview

#### 3.1. Overview of Commercial Software

#### 3.2. Overview of Research with Commercial Software

#### 3.3. Overview of Research Approaches

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Center Gated Disc | Tube | Strip | |
---|---|---|---|

Volume flow $\dot{V}$ | $\dot{V}=2\pi R\underset{-h/2}{\overset{h/2}{\int}}\dot{\gamma}y\phantom{\rule{0.166667em}{0ex}}\mathrm{d}y$ | $\dot{V}=\pi \underset{0}{\overset{R}{\int}}\dot{\gamma}{r}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}r$ | $\dot{V}=b\underset{-h/2}{\overset{h/2}{\int}}\dot{\gamma}y\phantom{\rule{0.166667em}{0ex}}\mathrm{d}y$ |

Shear rate $\dot{\gamma}$ | $\dot{\gamma}={\displaystyle \frac{\Lambda y}{\eta}}$ | $\dot{\gamma}={\displaystyle \frac{\Lambda r}{2\eta}}$ | $\dot{\gamma}={\displaystyle \frac{\Lambda y}{\eta}}$ |

Pressure gradient $\Lambda $ | $\Lambda ={\displaystyle \frac{\dot{V}}{2\pi RS}}$ | $\Lambda ={\displaystyle \frac{2\dot{V}}{\pi S}}$ | $\Lambda ={\displaystyle \frac{\dot{V}}{bS}}$ |

Fluidity S | $S=\underset{-h/2}{\overset{h/2}{\int}}{\displaystyle \frac{{y}^{2}}{\eta}}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}y$ | $S=\underset{0}{\overset{R}{\int}}{\displaystyle \frac{{r}^{3}}{\eta}}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}r$ | $S=\underset{-h/2}{\overset{h/2}{\int}}{\displaystyle \frac{{y}^{2}}{\eta}}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}y$ |

Supplier | Software | Type of Model Approach | Rheological Model | References |
---|---|---|---|---|

Autodesk, Inc. (San Fransico, CA, USA) | Moldflow | 2.5D, 3D | Cross-WLF, second-order | [101] |

CoreTech System Co., Ltd. (Zhubei City, Taiwan) | Moldex3D | 2D, 3D | Power-Law, Cross-WLF, Carreau, Herschel–Bulkley | [5,101,123,124] |

C-Solution, Inc. (Boulder, CO, USA) | Simuflow | 2D | Carreau | [5,125] |

Hexagon AB (Stockholm, Sweden) | VISI-Flow | 2.5D (only surface) | Cross-WLF | [5,126] |

SIGMA Engineering GmbH (Aachen, Germany) | Sigmasoft | 3D | Cross-WLF, Herschel–Bulkley | [5,101,127] |

Simcon kunststofftechnische Software GmbH (Würselen, Germany) | Cadmould 3D-F/3D-V | advanced 2.5D, 3D | Carreau-WLF, Cross-WLF | [5,101] |

Toray Engineering Co., Ltd. (Tokyo, Japan) | 3D TIMON | pseudo 3D | Cross-WLF | [1,5,27,128] |

Tanslavor S.A. (Biot, France) | Rem3D | 3D | Cross-WLF, Cross-Arrhenius | [1,5,27,129] |

Year | Author | Commercial Software | Rheological Model | References |
---|---|---|---|---|

2008 | Zhou et al. | Moldflow | Cross-WLF | [130] |

2019 | Yu et al. | Moldflow | Cross-WLF | [131] |

2022 | Lin et al. | Moldflow | Cross-WLF | [132] |

2023 | Yu et al. | Moldflow | Cross-WLF | [133] |

2019 | Tran and Gehde | Moldex3D | Cross-WLF | [134] |

2019 | Islam et al. | Moldex3D | Cross-WLF | [135] |

2022 | Tsai et al. | Moldex3D | Cross-WLF | [136] |

2011 | Mannella et al. | VISI-Flow, Moldflow | Cross-WLF | [126] |

2011 | Mulser et al. | Sigmasoft | Cross-WLF | [137] |

2013 | Ariff and Khang | Cadmould 3D-F | Carreau-WLF | [138] |

2017 | Ou et al. | Cadmould 3D-F | Carreau-WLF | [139] |

2017 | Othman et al. | Cadmould 3D-F | N/A | [140] |

2020 | Sahli et al. | Cadmould 3D-F | Carreau-WLF | [141] |

2009 | Shin et al. | 3D Timon | Cross-WLF | [128] |

2005 | Silva et al. | Rem3D | Cross-WLF | [129] |

2014 | Fang et al. | ANSYS CFX | Power-Law-Arrhenius | [142] |

2016 | Zhuang et al. | ANSYS CFX, Moldflow | Cross-WLF | [143] |

2016 | Mukras and Al-Mufadi | ANSYS CFX | Newtonian | [144] |

2021 | Anders et al. | ANSYS CFX, Cadmould 3D-F | Carreau-WLF | [145] |

2021 | Baum and Anders | ANSYS CFX, Cadmould 3D-F | Carreau-WLF | [146] |

2022 | Baum et al. | ANSYS CFX | Carreau, Carreau-WLF | [147] |

2016 | Rusdi et al. | ANSYS Fluent | Cross-Arrhenius | [148] |

2022 | Zaki et al. | ANSYS Fluent | Cross | [149] |

2023 | Abdullah et al. | ANSYS Fluent | Cross-Arrhenius | [150] |

Year | Author | Model Approach | Rheological Model | References |
---|---|---|---|---|

1971 | Barie | 1D | Power-Law | [111] |

1972 | Kamal and Kenig | 1D | Power-Law | [7,8] |

1973 | Berger and Gogos | 1D | Power-Law | [205] |

1973 | Broyer et al. | 1D | Newtonian | [21] |

1974 | Wu et al. | 1D | Power-Law | [9] |

1975 | Lord and Williams | 1D | second-order | [12] |

1975 | Williams and Lord | 1D | second-order | [13] |

1977 | Nunn and Fenner | 1D | Power-Law | [10] |

1978 | Stevenson | 1D | Power-Law | [11] |

1979 | Stevenson and Chuck | 1D | Power-Law | [206] |

1974 | Tadmor et al. | 2D | Power-Law | [23] |

1978 | Hieber and Shen | 2D | Power-Law | [24] |

1980 | Hieber and Shen | 2D | Power-Law | [25] |

1986 | Wang et al. | 2D | Cross-Arrhenius | [37] |

1991 | Chiang et al. | 2D | Cross-WLF | [31,32] |

1993 | Chiang et al. | 2D | Cross-WLF | [46] |

1994 | Chen and Liu | 2D | Cross-Arrhenius, Cross-WLF | [67] |

1995 | Chung and Kwon | 2D | Cross | [41] |

1996 | Chung and Kwon | 2D | Cross | [207] |

1997 | Han and Im | 2D | Cross-WLF | [208] |

1999 | Holm and Langtangen | 2D | Power-Law | [196] |

2007 | Estacio and Mangiavacchi | 2D | Cross-Arrhenius | [188] |

2008 | Wang et al. | 2D | Cross-WLF | [209] |

2017 | Xu and Yu | 2D, 3D | Newtonian, Cross | [210] |

1995 | Kwon and Ahn | 2.5D | Cross | [211] |

1997 | Rajupalem et al. | 2.5D | N/A | [164] |

1998 | Talwar et al. | 2.5D | N/A | [165] |

2001 | Zhou and Li | 2.5D | Cross-Arrhenius | [52] |

2001 | Zhou et al. | 2.5D | Cross-Arrhenius | [28] |

2001 | Chang and Yang | 2.5D, 3D | Newtonian | [178] |

2002 | Zhou and Li | 2.5D | Cross-Arrhenius | [53] |

1998 | Hétu et al. | 3D | Newtonian, Bird–Carreau, Bird–Carreau–Arrhenius | [176] |

1998 | Pichelin and Coupez | 3D | Bird–Carreau | [212] |

1999 | Pichelin and Coupez | 3D | Bird–Carreau | [213] |

1999 | Zheng et al. | 3D | Cross | [214] |

2000 | Ilinca and Hétu | 3D | Cross-WLF | [215] |

2001 | Haag et al. | 3D | Cross-WLF | [168] |

2001 | Khayat et al. | 3D | Newtonian | [192] |

2002 | Hwang and Kwon | 3D | Cross | [216] |

2003 | Ilinca and Hétu | 3D | Carreau-WLF | [100] |

2004 | Yang et al. | 3D | Cross-Arrhenius | [179] |

2005 | Cao et al. | 3D | Cross-WLF | [217] |

2005 | Zhou et al. | 3D | Cross-Arrhenius | [167] |

2006 | Kim and Turng | 3D | Power-Law, Cross-WLF | [166] |

2006 | Zhou and Turng | 3D | Power-Law | [181] |

2012 | Wang et al. | 3D | Cross-Arrhenius, second-order | [74] |

2017 | Liang et al. | 3D | Cross-WLF | [218] |

2017 | He et al. | 3D | Power-Law | [219] |

1D Model | 2D Model | 2.5D Model | 3D Model | |
---|---|---|---|---|

Assumptions for a thin-walled laminar flow (Hele–Shaw) | X | √ | √ | X |

Assumption of incompressibility | √ | √ | √ | √ |

Heat transfer in flow direction | X | √ | √ | √ |

Constant physical variables | √ | √ | √ | √ |

Planar flow front | X | √ | √ | X |

Mesh Size | Coarse | Medium | Fine | Ultrafine |

Complexity of numerical model | Very simple | Simple | Complex | Very complex |

Solution time | Very short | Short | Moderate | Long |

Viscosity Model | Shear Rate | Temperature | Pressure |
---|---|---|---|

Power-Law | √ | X | X |

Second-order | √ | √ | X |

Herschel–Bulkley | √ | X | X |

Bingham | √ | X | X |

Carreau | √ | X | X |

Bird–Carreau | √ | X | X |

Cross | √ | X | X |

Power-Law-Arrhenius | √ | √ | √ |

Carreau-WLF | √ | √ | √ |

Bird–Carreau–Arrhenius | √ | √ | X |

Cross-WLF | √ | √ | √ |

Cross-Arrhenius | √ | √ | √ |

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## Share and Cite

**MDPI and ACS Style**

Baum, M.; Anders, D.; Reinicke, T.
Approaches for Numerical Modeling and Simulation of the Filling Phase in Injection Molding: A Review. *Polymers* **2023**, *15*, 4220.
https://doi.org/10.3390/polym15214220

**AMA Style**

Baum M, Anders D, Reinicke T.
Approaches for Numerical Modeling and Simulation of the Filling Phase in Injection Molding: A Review. *Polymers*. 2023; 15(21):4220.
https://doi.org/10.3390/polym15214220

**Chicago/Turabian Style**

Baum, Markus, Denis Anders, and Tamara Reinicke.
2023. "Approaches for Numerical Modeling and Simulation of the Filling Phase in Injection Molding: A Review" *Polymers* 15, no. 21: 4220.
https://doi.org/10.3390/polym15214220