# Modelling the Heating Process in the Transient and Steady State of an In Situ Tape-Laying Machine Head

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## Abstract

**:**

## 1. Introduction and Related Works

## 2. The ATL Machine Head

#### 2.1. Infrared Heating Element

#### 2.2. Compaction Roll

#### 2.3. Material Feeder

## 3. 1.5D Mathematical Model for the ATL Heating Process

#### 3.1. Heater

#### 3.1.1. Tungsten Filament

- ${m}_{f}$: Filament mass, kg.
- ${T}_{f}$: Filament temperature, K.
- $c{p}_{f}\left({T}_{f}\right)$: Filament specific heat (temperature dependent), $\left[\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}\right]$.
- ${P}_{el}$: Electric power, W.
- ${Q}_{r,f}$: Radiation heat, W.
- ${Q}_{c,n}$: Conduction heat, W.

#### 3.1.2. Neon

- ${Q}_{c,n}$: Conduction heat, W.
- ${K}_{n}\left({T}_{n}\right)$: Neon conductivity (temperature dependent), $\left[\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}\right]$.
- ${T}_{n}$: Neon mean temperature $\left(\frac{{T}_{f}-{T}_{q}}{2}\right)$, K.
- ${T}_{q}$: Quartz lamp temperature, K.
- ${d}_{L}$: Lamp diameter, m.
- ${d}_{coil}$: Filament coil diameter, m.
- ${l}_{q}$: Lamp length, m.

#### 3.1.3. Quartz Glass Envelope

- ${m}_{q}$: Quartz cylinder mass, kg.
- $c{p}_{q}\left({T}_{q}\right)$: Quartz specific heat, $\left[\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}\right]$.
- ${Q}_{r,f}$: Incoming radiation heat, W.
- ${Q}_{r,q}$: Outgoing radiation heat, W.
- ${Q}_{c,n}$: Conduction heat, W.
- ${h}_{2q}$: Convection coefficient, $\left[\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}\right]$.
- ${A}_{q}$: Quartz surface area, ${\mathrm{m}}^{2}$.
- ${T}_{\infty}$: Air Temperature inside the radiation cavity, K

#### 3.2. Reflector

- ${\rho}_{r}$: Reflector material density, $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right]$.
- ${T}_{r}$: Reflector temperature, K.
- $c{p}_{r}\left({T}_{r}\right)$: Reflector specific heat (temperature dependent), $\left[\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}\right]$.
- ${k}_{r}\left({T}_{r}\right)$: Reflector conductivity (temperature dependent), $\left[\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}\right]$.
- ${q}_{rad,r}^{\u2033}\left({T}_{r}\right)$: Radiation heat, $\left[\frac{\mathrm{W}}{{\mathrm{m}}^{2}}\right]$.
- ${q}_{conv,r}^{\u2033}({T}_{r},{T}_{\infty})$: Convection heat $\left[\frac{\mathrm{W}}{{\mathrm{m}}^{2}}\right]$.

- r: Reflector cell number 0,1,...,n.
- ${L}_{r}$: Length of reflector cell, m.
- ${t}_{r}$: Reflector thickness, m.
- ${T}_{\infty}$: Air temperature, K.
- ${h}_{1}({T}_{r},{T}_{\infty})$: External convection coefficient, $\left[\frac{\mathrm{W}}{{\mathrm{m}}^{2}\xb7\mathrm{K}}\right]$.
- ${h}_{2r}({T}_{r},{T}_{\infty})$: Internal convection coefficient, $\left[\frac{\mathrm{W}}{{\mathrm{m}}^{2}\xb7\mathrm{K}}\right]$.

#### 3.3. Material

- ${\rho}_{m}$: Composite material density, $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right]$.
- ${T}_{m}$: Material Temperature, K.
- $c{p}_{m}\left({T}_{m}\right)$: Composite specific heat (temperature dependent), $\left[\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}\right]$.
- ${k}_{m}\left({T}_{m}\right)$: Composite conductivity (temperature dependent), $\left[\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}\right]$.
- ${q}_{rad,m}^{\u2033}\left({T}_{m}\right)$: Radiation heat, $\left[\frac{\mathrm{W}}{{\mathrm{m}}^{2}}\right]$.
- ${q}_{\mathrm{conv},m}^{\u2033}({T}_{m},{T}_{\infty})$: Convection heat, $\left[\frac{\mathrm{W}}{{\mathrm{m}}^{2}}\right]$.
- ${q}_{\mathrm{cond},m}^{\u2033}({T}_{m},{T}_{mold})$: Conduction heat, $\left[\frac{\mathrm{W}}{{\mathrm{m}}^{2}}\right]$.

- m: Material cell number 0, 1,..., n.
- ${R}_{mold}$: Composite wall resistance for heat conduction between the mold and the material, $\left[\frac{{\mathrm{m}}^{2}\xb7\mathrm{K}}{W}\right]$.
- ${R}_{roll}$: Composite wall resistance for heat conduction between compaction roll and the material, $\left[\frac{{\mathrm{m}}^{2}\xb7\mathrm{K}}{W}\right]$.
- ${{q}^{\u2033}}_{rad}$: Net radiation from the surfaces involved in the heat exchange process, $\left[\frac{\mathrm{W}}{{\mathrm{m}}^{2}}\right]$.
- ${h}_{2m}$: Convection coefficient for the material surface facing the heating element, $\left[\frac{\mathrm{W}}{{\mathrm{m}}^{2}}\right]$.

#### 3.4. Compaction Roll and Mold

#### 3.5. Radiation

- $\lambda $: Wavelength, $\mathsf{\mu}$ m.
- T: Absolute temperature of the black body, K.
- h: Universal Planck constant, $6.626\times {10}^{-34}\phantom{\rule{0.277778em}{0ex}}\left[\mathrm{J}\xb7\mathrm{s}\right]$.
- ${k}_{b}$: Boltzmann constant, $1.381\times {10}^{-23}\phantom{\rule{0.277778em}{0ex}}\left[\mathrm{J}/\mathrm{K}\right]$.
- ${c}_{0}$: Speed of light in vacuum, $2.998\times {10}^{8}\phantom{\rule{0.277778em}{0ex}}\left[\mathrm{m}/\mathrm{s}\right]$.

#### 3.5.1. Radiative Fluxes

#### 3.5.2. View Factors

## 4. Properties

#### 4.1. Thermal Properties

#### 4.2. Optical Properties

- $\frac{\u03f5}{{\u03f5}_{0}}$: relative material i permitivity.
- ${\u03f5}_{0}$: vacuum permitivity $[\mathrm{F}\xb7{\mathrm{m}}^{-1}]$.

#### 4.3. Convection Coefficients

## 5. Materials and Methods

#### 5.1. Material Description

^{®}. This has a 50 mm width, 0.16 mm thickness, and a fiber content of 60% [46].

#### 5.1.1. Thermal Properties

#### 5.1.2. Optical Properties

#### 5.2. Methods

#### 5.2.1. Time Integration Scheme

- Step 1.
- For the given trial temperature.
- Step 2.
- Evaluate all material properties at the trial temperature.
- Step 3.
- Solve the radiation heat flux system for the trial temperature.
- Step 4.
- Evaluate the ODE system array with the temperature derivatives.

#### 5.2.2. Convergence Analysis

#### 5.2.3. Measures and Instrumentation

#### 5.2.4. Model Validation

## 6. Results

#### 6.1. Mesh Convergence

#### 6.2. Model Validation

#### Sensitivity Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Machine head assembly. (1) Material feeder. (2) Heating element. (3) Reflector. (4) Optical temperature sensor. (5) Compaction roll. (6) Mold. (7) Nip point.

**Figure 7.**Composite wall for the heat conduction balance. (

**a**) Composite wall model for the 1D compaction roll ${D}_{3}$. (

**b**) The 1D compaction roll in section ${D}_{4}$.

**Figure 8.**Surface identification. (1) Tungsten filament surface. (2) Internal quartz glass surface. (3) External quarts glass surface. (4) Reflector surface. (5) Material surface. (6) Equivalent surroundings surface.

**Figure 9.**View factor between a cylindrical surface and a plane surface. (

**a**) Rectangle not aligned with the cylinder vertical center line. (

**b**) Rectangle symmetrically aligned with the cylinder vertical center line.

**Figure 13.**Emissivity and reflectivity as functions of temperature for the composite material obtained by an in-house test.

**Figure 15.**Composite material temperature measurement scheme. (1) Pyrometer PyroNFC-K. (2) Incoming composite material. (3) Compaction roll. (4) Compaction roll axis; fluid inlet/outlet ports. (5) Inlet fluid temperature sensor; outlet temperature sensor located at the other end of the axis. (6) Mold temperature sensor.

**Figure 16.**Temperature error as a function of the mesh size for the material. (

**a**) speed value 5 mm/s. (

**b**) speed value 15 mm/s.

**Figure 17.**Results of Power consumption and temperature variation at the measuring point according to parameter set 1 from Table 2. (

**a**) test 1, electrical power. (

**b**) test 1, composite temperature response. (

**c**) test 2, electrical power. (

**d**) test 2, composite temperature response.

**Figure 19.**Results of power consumption and temperature variation at the measuring point according to parameter set 2 from Table 2. (

**a**) Test 1, electrical power. (

**b**) Test 1, composite temperature response. (

**c**) Test 2, electrical power. (

**d**) Test 2, composite temperature response.

**Figure 21.**Temperature at measuring point, predicted at measuring point, and predicted at nip point. (

**a**) First measured test. (

**b**) Second measured test.

**Figure 22.**Results obtained for the simulations according to the first simulation proposed in Table 3.

**Figure 23.**Results obtained for the simulations according to the second simulation proposed in Table 3.

Lamp Property | Measured Value |
---|---|

Lamp length | 189.02 mm ± 0.34 mm |

Lamp diameter | 10.01 mm ± 0.19 mm |

Lamp resistance (at 23 °C) | 9.6 $\Omega $ ± 6.41 ·10${}^{-6}\phantom{\rule{0.277778em}{0ex}}\Omega $ |

Filament diameter | 0.410 mm ± 0.004 mm |

Filament coil diameter | 2.890 mm ± 0.004 mm |

Filament coil length | 290.04 mm ± 0.37 mm |

Filament mass | 5.2019 g ± 0.0005 g |

Lamp glass mass | 13.8780 g ± 0.0005 g |

Parameter | Set 1 | Set 2 |
---|---|---|

Compaction roll temperature | $55{\phantom{\rule{0.277778em}{0ex}}}^{\xb0}\mathrm{C}$ | $55{\phantom{\rule{0.277778em}{0ex}}}^{\xb0}\mathrm{C}$ |

Mold temperature | $22{\phantom{\rule{0.277778em}{0ex}}}^{\xb0}\mathrm{C}$ | $22{\phantom{\rule{0.277778em}{0ex}}}^{\xb0}\mathrm{C}$ |

Process speed | $5\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}/\mathrm{s}$ | $15\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}/\mathrm{s}$ |

Simulation | Parameters | Values |
---|---|---|

1 | Compaction roll Temperature | 55 °C |

Mold Temperature | 22 °C | |

Speed | 5 mm/s | |

Voltage | 100 V | |

Voltage | 150 V | |

Voltage | 200 V | |

2 | Compaction roll Temperature | 55 °C |

Mold Temperature | 22 °C | |

Voltage | 150 V | |

Speed | 5 mm/s | |

Speed | 10 mm/s | |

Speed | 15 mm/s |

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## Share and Cite

**MDPI and ACS Style**

de Sá Rodrigues, J.; Gonçalves, P.T.; Pina, L.; Gomes de Almeida, F. Modelling the Heating Process in the Transient and Steady State of an In Situ Tape-Laying Machine Head. *J. Manuf. Mater. Process.* **2022**, *6*, 8.
https://doi.org/10.3390/jmmp6010008

**AMA Style**

de Sá Rodrigues J, Gonçalves PT, Pina L, Gomes de Almeida F. Modelling the Heating Process in the Transient and Steady State of an In Situ Tape-Laying Machine Head. *Journal of Manufacturing and Materials Processing*. 2022; 6(1):8.
https://doi.org/10.3390/jmmp6010008

**Chicago/Turabian Style**

de Sá Rodrigues, Jhonny, Paulo Teixeira Gonçalves, Luis Pina, and Fernando Gomes de Almeida. 2022. "Modelling the Heating Process in the Transient and Steady State of an In Situ Tape-Laying Machine Head" *Journal of Manufacturing and Materials Processing* 6, no. 1: 8.
https://doi.org/10.3390/jmmp6010008