# Simulation of Reinforced Reactive Injection Molding with the Finite Volume Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Models and Implementations

#### 2.1. Governing Equations

#### 2.2. Phase-Dependent Boundary Condition

**U**is implemented as an interpolation between a Dirichlet boundary condition, defining the absolute value of

**U**to be zero, and a Neumann boundary condition, defining the gradient of

**U**to be zero. The interpolation depends on the VoF factor α, in such a way that a cell completely filled with polymer has a pure Dirichlet BC and a cell completely filled with air has a pure Neumann BC. Consequently, the boundary face changes from an outlet to a wall during filling, represented by:

#### 2.3. Model for Curing Kinetics

#### 2.4. Model for Viscosity

#### 2.5. Model for Fiber Orientation

**v**is the flux of the velocity vector.

## 3. Test Setup for Experimental Validation

#### 3.1. Test Structure and Process Conditions

^{3}of the dosage volume remains in the plasticizing unit. After the switchover, a constant pressure (hold pressure) acts on the inlet. After the hold pressure stages, the mold is separated from the plasticizing unit and no material flow in the mold is possible. The curing time ensures the shape stability of the composite part.

#### 3.2. Numerical Model, Boundary Conditions and Model Parameters

_{rgh}. In this study it is equal to the pressure p, because gravity is neglected.

^{3}/s and 250 cm

^{3}/s, because only one quarter of the mold is simulated. According to the experiments, the filling simulation is initially volume-flux controlled (Table 4), and pressure-controlled after the first switchover point. Two switchover points are also used in the FEM and FVM simulations, where the FVM settings are chosen according to the FEM simulations: After the first switchover point, the pressure is set constant at the actual values, calculated for this time step. After the second one, the pressure is set according to the holding pressure of the experiments, see Table 1. At the first switchover point, the inlet boundary condition for the velocity is changed to pressureInletOutletVelocity, so that the velocity is calculated relating to the fixed pressure. The switchover points are calculated from the experimental parameters and the mold volume. They are given in Table 7.

#### 3.3. Numerical Model for Verification of Fiber Orientations

^{3}/s is chosen, leading to a filling time of 1 s. The inlet and outlet are squares with an edge length of 5 mm. The FEM mesh is built up in Moldflow with tetrahedral elements, while hexahedral volumes created with BlockMesh are used for the FVM mesh. In both meshes a global edge length of 1 mm is set.

## 4. Results

#### 4.1. Results of the Filling Simulation

^{3}/s and a mold temperature of 170 °C. Displayed are (from top to bottom) the filling level (via VoF), the velocity in m/s and the pressure in MPa at 1 s (Figure 5a), where the filling is velocity-controlled, and at 1.5 s (Figure 5b), where it is pressure-controlled and should be just filled.

^{3}/s in the experimental studies (see Section 4.2).

^{3}/s based on the inlet area. Furthermore, it can be seen that the velocity is nearly zero after 1.5 s (Figure 5b middle), although there is still a pressure gradient at that moment (Figure 5 bottom). This aspect approves the phase-dependent boundary condition, which stops the flow when the mold is filled.

#### 4.2. Comparisson of Pressure

^{3}/s is shown in Figure 7a for a mold temperature of 170 °C and Figure 7b for 190 °C. The experiments at 170 °C are more reproducible than the ones at 190 °C, as visualized by the scatter beams in Figure 7. The large scatter at 190 °C might be caused by the fact that 190 °C is the maximum temperature for injection molding according to the manufacturer. Every curve of sensor position 1 shows a continuous pressure growth during filling (0–1.46 s). After filling, the experimental curves of sensor 1 and 2 are nearly identical, which shows good process control and filling behavior. The simulation results of FEM and FVM considerably differ from each other. Compared to measurements at sensor 1, FEM predicts a higher pressure, while the FVM results fit better to the experiments and are just slightly higher for 170 °C. At sensor 2, a significant pressure rise after switchover (1.46 s) is detectable in experiments as well as in FEM and FVM simulations. However, both simulations show a too fast pressure rise at sensor 2 compared to the measurements.

^{3}/s, there is also a greater scatter in pressure measurement for 190 °C (Figure 8b) than for 170 °C (Figure 8a) and the pressure is again lower at 190 °C because of the lower viscosity. Up to the first switchover, the experimental data of a filling rate of 250 cm

^{3}/s show a higher pressure growth during filling than the data of the filling rate of 137.5 cm

^{3}/s (Figure 7), which is a consequence of the higher velocity. Hence, there is no visible pressure rise at switchover (0.8 s) for sensor 1. After switchover, the experimental curves of sensor 1 and 2 do not immediately fit to each other as well as for 137.5 cm

^{3}/s. For 190 °C the measured pressure at sensor 2 is even higher than the one at sensor 1 between 0.85 and 1 s.

#### 4.3. Comparison of Curing Kinetics

#### 4.4. Comparison of Fiber Orientation

## 5. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**U**file of the 0 folder (or other corresponding time folders) in the boundary field block. In this case it is specified for a patch called “outlet”.

outlet |

{ |

type groovyBC; |

aliases {alpha1 alpha.polymer;} |

valueExpression “vector(0,0,0)”; |

gradientExpression “vector(0,0,0)”; |

fractionExpression “alpha1”; |

value uniform (0 0 0); |

} |

Furthermore, the relevant library must be read at the end of the ControlDict file: |

libs (“libgroovyBC.so”); |

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**Figure 5.**FVM-simulation results of polymer filling state alpha (top), velocity

**U**in m/s (middle) and pressure p in MPa (bottom) after 1 s (

**a**) and 1.5 s (

**b**) for a filling rate of 137.5 cm

^{3}/s and a mold temperature of 170 °C.

**Figure 7.**Pressure over time for a filling speed of 137.5 cm

^{3}/s and a mold temperature of 170 °C (

**a**) and 190 °C (

**b**). Comparing measurement (black), FEM (red) and FVM (blue) at sensor position 1 (solid line) and 2 (dotted line).

**Figure 8.**Pressure as a function of time for a filling speed of 250 cm

^{3}/s and a mold temperature of 170 °C (

**a**) and 190 °C (

**b**). Comparing measurement (black), FEM (red) and FVM (blue) at sensor Position 1 (solid line) and 2 (dotted line).

**Figure 9.**Degree of cure as a function of time for FEM (black) and FVM (red) for a filling speed of 137.5 cm

^{3}/s and a mold temperature of 170 °C.

**Figure 10.**Comparison of predicted fiber orientation between FEM and FVM in x-direction (

**a**), y-direction (

**b**) and z-direction (

**c**).

Parameter | Value | Unit |
---|---|---|

Plasticizing unit temperature profile from inlet to nozzle | 60–70, 70–80, 80–90 | °C |

Screw speed | 40 | 1/min |

Back pressure | 40 | bar |

Switchover point | 5 | cm^{3} |

Hold pressure stage 1 | 800 | bar |

Time period of stage 1 | 30 | s |

Hold pressure stage 2 | 800–15 (linear) | bar |

Time period of stage 2 | 10 | s |

Curing time period | 40 | s |

Parameter | Value Min | Value Max | Unit |
---|---|---|---|

Mold temperature | 170 | 190 | °C |

Injection speed | 137.5 | 250 | cm^{3}/s |

Dosage volume | 230 | 232 | cm^{3} |

Field Variable | Value | Unit |
---|---|---|

A (fiber tensor) | $\left(\begin{array}{ccc}0.33& 0& 0\\ 0& 0.33& 0\\ 0& 0& 0.33\end{array}\right)$ | - |

α | 0 | - |

c | 0.001 | - |

cure rate | 0 | 1/s |

$\dot{\gamma}$ | 0 | 1/s |

p and p_{rgh} | 10^{5} | Pa |

T | 443.15 or 463.15 | K |

U | (0 0 0) | m/s |

Field Variable | Type | Value | Unit |
---|---|---|---|

A (fiber tensor) | fixedValue | $\left(\begin{array}{ccc}0.33& 0& 0\\ 0& 0.33& 0\\ 0& 0& 0.33\end{array}\right)$ | - |

α | fixedValue | 1 | - |

c | zeroGradient | - | - |

cure rate | zeroGradient | - | 1/s |

$\dot{\gamma}$ | zeroGradient | - | 1/s |

p and p_{rgh} | fixedfluxPressure | 10^{5} | Pa |

T | fixedValue | 423.15 | K |

U | flowRateInletVelocity | 34.375 × 10^{−6} or 62.5 × 10^{−6} | m^{3}/s |

Field Variable | Type | Value | Unit |
---|---|---|---|

A (fiber tensor) | zeroGradient | - | - |

α | zeroGradient | - | - |

cure | zeroGradient | - | - |

cure rate | zeroGradient | - | 1/s |

gammadot | zeroGradient | - | 1/s |

p and p_{rgh} | zeroGradient | - | Pa |

T | fixedValue | 443.15 or 463.15 | K |

U | BC (see Section 2.2) | - | m/s |

Field Variable | Type | Value | Unit |
---|---|---|---|

A (fiber tensor) | zeroGradient | - | - |

α | zeroGradient | - | - |

cure | zeroGradient | - | - |

cureRate | zeroGradient | - | 1/s |

gammadot | zeroGradient | - | 1/s |

p and p_{rgh} | fixedfluxPressure | 10^{5} | Pa |

T | fixedValue | 443.15 or 463.15 | K |

U | noSlip | - | m/s |

Filling Rate | First Point | Second Point |
---|---|---|

137.5 cm^{3}/s | 1.46 s | 1.5 s |

250 cm^{3}/s | 0.8 s | 0.82 s |

Parameter | Value | Unit |
---|---|---|

R | 8.3144598 | J/(K∙mol) |

A1 | 1.9454 × 10^{12} | 1/s |

A2 | 3041.4 | 1/s |

E1 | 2,878,805.64 | J/mol |

E2 | 38,425.6452 | J/mol |

m | 1.643 | - |

n | 0.4893 | - |

Parameter | Value | Unit |
---|---|---|

τ * | 0.79 | Pa |

n | 0.5 | - |

${c}_{1}$ | 17 | - |

${c}_{2}$ | 17 | - |

B | 1.123 × 10^{−7} | Pa∙s |

T_{b} | 13.750 | K |

c_{g} | 0.4 | - |

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**MDPI and ACS Style**

Wittemann, F.; Maertens, R.; Bernath, A.; Hohberg, M.; Kärger, L.; Henning, F.
Simulation of Reinforced Reactive Injection Molding with the Finite Volume Method. *J. Compos. Sci.* **2018**, *2*, 5.
https://doi.org/10.3390/jcs2010005

**AMA Style**

Wittemann F, Maertens R, Bernath A, Hohberg M, Kärger L, Henning F.
Simulation of Reinforced Reactive Injection Molding with the Finite Volume Method. *Journal of Composites Science*. 2018; 2(1):5.
https://doi.org/10.3390/jcs2010005

**Chicago/Turabian Style**

Wittemann, Florian, Robert Maertens, Alexander Bernath, Martin Hohberg, Luise Kärger, and Frank Henning.
2018. "Simulation of Reinforced Reactive Injection Molding with the Finite Volume Method" *Journal of Composites Science* 2, no. 1: 5.
https://doi.org/10.3390/jcs2010005