# Investigation of a Non-Equilibrium Energy Model for Resin Transfer Molding Simulations

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## Abstract

**:**

## 1. Introduction

^{®}, is extended to include coupled transport equations for the resin (fluid) and fiber (solid) phases, wherein the fluid equation contains a heat release source term to account for resin curing, and the fluid and solid equations are linked through a bulk (interstitial) heat exchange term. Model verification is achieved using a 1D representative geometry to ensure model accuracy. A non-dimensional analysis is then presented to determine the appropriate material, geometric, and operating conditions where the LTNE energy model is worth the added computational cost and complexity. Finally, a sample case of a complex geometry simulated using both thermal equilibrium and LTNE models is presented to demonstrate the value of the LTNE approach.

## 2. Governing Equations and Models for Mold Filling Simulations

#### 2.1. Conservation of Mass and Momentum

**K**is the permeability tensor, $\mu $ is the dynamic fluid viscosity and $\nabla p$ is the pressure gradient in the mold. This formulation of Darcy’s law is applicable for flows where the Reynolds number based on the pore diameter is small ($R{e}_{d}$ < 1). When considering flows at higher Reynolds numbers, the flow transfers into the Forchheimer flow regime [28], where the pressure drop is quadratically related to the fluid velocity. This is shown as

#### 2.2. Conservation of Energy

#### 2.2.1. Thermal Equilibrium

#### 2.2.2. Local Thermal Non-Equilibrium

#### 2.3. Chemo-Rheological Models

#### 2.3.1. Reaction Kinetics

#### 2.3.2. Glass Transition Temperature

#### 2.3.3. Viscosity

## 3. Results

#### 3.1. Grid Independence

#### 3.2. Temporal Resolution Study

#### 3.3. Non-Dimensional Analysis

#### 3.4. Temperature Distribution and Development

#### 3.5. Fluid Property Development

#### 3.6. Complex Floor Geometry

#### 3.6.1. Infiltration

#### 3.6.2. Curing

## 4. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CFRP | Carbon fiber reinforced plastic |

HP-RTM | High-pressure resin transfer molding |

LTNE | Local thermal non-equilibrium |

PID | Process-induced distortion |

RTM | Resin transfer molding |

VOF | Volume of fluid |

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**Figure 2.**Temperature profiles at (

**a**) x = 25 mm and (

**b**) x = 75 mm from the inlet after the domain was fully infiltrated. Note that the legend in (

**b**) is applicable to both figures.

**Figure 3.**Reduction in temperature rise error between simulation and analytical results for decreasing time-step size.

**Figure 4.**(

**a**–

**p**) Non-dimensional temperature and fluid–solid temperature difference for varying $Pe$ and $Da$ numbers. Values on the left hand side of the vertical dashed line (in black) are for a conductivity ratio of 0.2 and values on the right hand side of the dashed line (in grey) are for a conductivity ratio of 125.

**Figure 5.**Fluid viscosity development during infiltration for the thermal non-equilibrium and thermal equilibrium model at a location of (

**a**) 25% and (

**b**) 75% in the domain. Note that the legend in (

**b**) is applicable to both figures.

**Figure 6.**Cure degree development during infiltration for the thermal non-equilibrium and thermal equilibrium model at a location of (

**a**) 25% and (

**b**) 75% in the domain. Note that the legend in (

**b**) is applicable to both figures.

**Figure 7.**Complicated geometry used for energy modelling comparison. Image (

**a**) shows an isometric view of the domain, while image (

**b**) shows a surface mesh to illustrate the grid distribution with respect to geometric features.

**Figure 8.**Predicted cure degree development during infiltration using different energy modelling approaches.

**Figure 9.**Predicted cure degree distribution at 10$\mathrm{s}$ into the curing stage using different energy modelling approaches.

**Figure 10.**Predicted cure rate distribution at 10$\mathrm{s}$ into the curing stage using different energy modelling approaches.

Location | Field Type | Value |
---|---|---|

inlet | Velocity | fixedValue |

$cure$ | 0% | |

${T}_{f}$ | $363.15\text{}\mathrm{K}$ | |

${T}_{s}$ | $393.15\text{}\mathrm{K}$ | |

upperMold | Velocity | noSlip |

${T}_{f}$ | $393.15\text{}\mathrm{K}$ | |

${T}_{s}$ | $393.15\text{}\mathrm{K}$ | |

lowerMold | Velocity | noSlip |

${T}_{f}$ | $393.15\text{}\mathrm{K}$ | |

${T}_{s}$ | $393.15\text{}\mathrm{K}$ | |

wall | Velocity | noSlip |

${T}_{f}$ | $393.15\text{}\mathrm{K}$ | |

${T}_{s}$ | $393.15\text{}\mathrm{K}$ | |

symmetry | Velocity | symmetry |

${T}_{f}$ | ||

${T}_{s}$ |

Model Parameter | Value |
---|---|

${A}_{1}$ | 3,862,141.7 |

${A}_{2}$ | 105,920,589,010.0 |

${E}_{1}$ | 62,877.7 |

${E}_{2}$ | 321,915.1 |

m | 1.571 |

n | 1.63 |

R | 8.314 |

Model Parameter | Value |
---|---|

${T}_{g,0}$ | 243.0 |

${T}_{g,\infty}$ | 406.09 |

$\lambda $ | 0.390 |

Model Parameter | Value |
---|---|

${C}_{1}$ | 3.91 |

${C}_{2}$ | $2.12\times {10}^{-13}$ |

B | $1.414\times {10}^{-12}$ |

${T}_{b}$ | $8.489\times {10}^{3}$ |

R | 8.314 |

${c}_{g}$ | 0.72 |

**Table 5.**Minimum and maximum cure degree values in the domain after infiltration for each energy modelling approach.

Energy Modelling Approach | Maximum Cure Degree | Maximum Cure Rate |
---|---|---|

Isothermal 363 $\mathrm{K}$ | 0.96% | $0.3$%/$\mathrm{s}$ |

Isothermal 393 $\mathrm{K}$ | 4.6% | $1.7$%/$\mathrm{s}$ |

Thermal Equilibrium | 4.5% | $1.7$%/$\mathrm{s}$ |

Thermal Non-Equilibrium | 3.5% | $1.6$%/$\mathrm{s}$ |

Energy Modelling Approach | Cure Degree Range (Min–Max) | ||
---|---|---|---|

10 s | 100 s | 180 s | |

Isothermal 363 $\mathrm{K}$ | 3.4–4.3% | 26.9–27.5% | 40.9–41.3% |

Isothermal 393 $\mathrm{K}$ | 14.9–18.5% | 68.5–69.2% | 81.8–82.1% |

Thermal Equilibrium | 14.5–18.4% | 68.4–69.2% | 81.8–82.1% |

Thermal Non-Equilibrium | 12.5–17.6% | 68.0–69.0% | 81.6–82.0% |

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**MDPI and ACS Style**

Sherratt, A.; Straatman, A.G.; DeGroot, C.T.; Henning, F.
Investigation of a Non-Equilibrium Energy Model for Resin Transfer Molding Simulations. *J. Compos. Sci.* **2022**, *6*, 180.
https://doi.org/10.3390/jcs6060180

**AMA Style**

Sherratt A, Straatman AG, DeGroot CT, Henning F.
Investigation of a Non-Equilibrium Energy Model for Resin Transfer Molding Simulations. *Journal of Composites Science*. 2022; 6(6):180.
https://doi.org/10.3390/jcs6060180

**Chicago/Turabian Style**

Sherratt, Anthony, Anthony G. Straatman, Christopher T. DeGroot, and Frank Henning.
2022. "Investigation of a Non-Equilibrium Energy Model for Resin Transfer Molding Simulations" *Journal of Composites Science* 6, no. 6: 180.
https://doi.org/10.3390/jcs6060180