# Chemical Supercritical Fluid Infiltration of Pyrocarbon with Thermal Gradients: Deposition Kinetics and Multiphysics Modeling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### Experimental Materials and Methods

## 3. Experimental Results

#### 3.1. Kinetics of the Deposition Reaction on a Single Filament

#### 3.2. Thermal Study of an Infiltration Reactor

#### 3.3. Infiltrations: Kinetics and Densification Profiles

## 4. Infiltration Modeling

#### Model Setup

## 5. Numerical Results and Discussion

#### 5.1. Thermal Study

#### 5.2. Evolution of Pressure during the Infiltration Runs

#### 5.3. Infiltration Fronts

## 6. Summary and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Deposition rate on the single filament as a function of temperature and for different methane pressures. The start of the observed homogeneous nucleation is indicated by the brackets ending the tendency curves for 10, 30, and 50 bars.

**Figure 4.**Equilibrium partial pressure of methane (

**left**) and equilibrium mole fraction of methane (

**right**) for the deposition reaction (1) as a function of temperature and total pressure.

**Figure 6.**Pre-exponential factor—activation energy diagram for methane deposition at various pressures and following various authors. The dotted line is a guide for the eye to the data from this study.

**Figure 8.**Micrographs of the cross-section for 3 successive infiltrations #P2, P3, and P4, from left to right.

**Figure 10.**(

**a**–

**i**) Transverse micrographs of samples obtained after runs #P2 to P10, respectively. The scale bar is common to all micrographs. The approximate locations of the front starts and ends are indicated by red and green dashed lines, respectively.

**Figure 12.**Computation of the thermal histories of selected points of the setup: (

**left**) position of the sampled points; (

**right**) thermal history of these points compared to experimental ones.

**Figure 15.**Kinetic parameters vs. total pressure in CVD and CVI experiments. The size of the symbols indicates the experimental uncertainty.

**Figure 16.**Simulated infiltration runs #P2–4: porosity profiles: (

**top**) 2D computed profiles compared to micrographs on the same scale; (

**bottom**) comparison of the porosity profiles obtained at the half-height of the preform.

Part | Material | Dimensions |
---|---|---|

Reactor | Inconel | ${V}_{tank}=0.3$ L |

Gas inlet | Inconel | |

Sealing cap | Inconel | |

Electrodes | Inconel | |

Clamps | Steel | |

Resistor | Graphite tube | $H=60$ mm; ${d}_{ext}=6$ mm; ${d}_{int}$ = 3 mm |

Preform | Carbon fibers (${d}_{f}$ = 7 $\mathsf{\mu}$m) | $H=20$ mm; $d=$15 mm |

Run | Initial Pressure | Power | Time | Measured Temperature * |
---|---|---|---|---|

# | (bar) | (kW) | (min) | (${}^{\circ}$C) |

P1 | 50 | 1.87 | 15 | ≈850 |

P2 | 50 | 2.0 | 15 | ≈900 |

P3 | 50 | 2.0 | 15 | ≈900 |

then 2.5 | 15 | ≈1050 | ||

P4 | 50 | 2.0 | 15 | ≈900 |

then 2.5 | 15 | ≈1050 | ||

then 3.0 | 15 | ≈1200 | ||

P5 | 60 | 2.0 | 15 | ≈900 |

P6 | 60 | 2.0 | 15 | ≈900 |

then 2.5 | 15 | ≈1050 | ||

P7 | 60 | 2.0 | 15 | ≈900 |

then 2.5 | 15 | ≈1050 | ||

then 3.0 | 15 | ≈1200 | ||

P8 | 70 | 2.0 | 15 | ≈900 |

P9 | 70 | 2.0 | 15 | ≈900 |

then 2.5 | 15 | ≈1050 | ||

P10 | 70 | 2.0 | 15 | ≈900 |

then 2.5 | 15 | ≈1050 | ||

then 3.0 | 15 | ≈1200 |

Total Pressure | Pre-Exponential Constant A | Activation Energy | Ref. |
---|---|---|---|

(bar) | ($\mathsf{\mu}\mathrm{m}\mathsf{\xb7}\mathrm{min}$${}^{-1}$·bar${}^{-1}$) | (kJ·mol${}^{-1}$) | |

1 | $2.80\times {10}^{8}$ | 451.9 | [32] |

1 | $2.18\times {10}^{8}$ | 272 | [33] |

2 | $(2.3\pm 0.2)\times {10}^{7}$ | $284\pm 6$ | This work |

10 | $(1.2\pm 0.1)\times {10}^{6}$ | $246\pm 5$ | This work |

30 | $(3.2\pm 0.3)\times {10}^{2}$ | $129\pm 3$ | This work |

50 | $(4.1\pm 0.4)$ | $83\pm 2$ | This work |

Parameter | Value or Expression | Unit |
---|---|---|

Reactor | ||

Initial outer wall temperature | ${T}_{ext}=298$ | K |

Initial total pressure | ${P}_{ext}=5,\phantom{\rule{0.277778em}{0ex}}6,\phantom{\rule{0.277778em}{0ex}}\mathrm{or}7\times {10}^{6}$ | Pa |

Diffusion boundary layer thickness | $\delta =0.5$ | m |

Heat capacity of the reactor | ${m}_{tank}{c}_{p,tank}=300$ | J·K${}^{-1}$ |

Reactor volume | ${V}_{tank}=3\times {10}^{-4}$ | m${}^{3}$ |

Outer heat exchange coefficient | ${h}_{w}{S}_{w}=10$ | W·K${}^{-1}$ |

Preform/fluid heat transfer coefficient | $h=4200$ | W·m${}^{-2}$·K${}^{-1}$ |

Resistor/exterior heat transfer coefficient | ${h}_{res}=6000$ | W·m${}^{-2}$·K${}^{-1}$ |

Preform | ||

Fiber density | ${\rho}_{f}=1840$ | kg·m${}^{-3}$ |

Fiber initial diameter | ${d}_{f,0}=7\times {10}^{-6}$ | m |

Initial porosity | ${\epsilon}_{0}=0.77$ | - |

Mass transfer parameters | ||

Carbon density | ${\rho}_{c}=2180$ | kg·m${}^{-3}$ |

Carbon molar volume | ${\mathrm{\Omega}}_{c}=5\times {10}^{-6}$ | m${}^{3}$·mol${}^{-1}$ |

Internal surface area | ${\sigma}_{v}=\frac{4}{{d}_{f}}\left[\left(2-{\epsilon}_{0}\right)\left(\frac{\epsilon}{{\epsilon}_{0}}\right)-{\left(\frac{\epsilon}{{\epsilon}_{0}}\right)}^{2}\right]$ | m${}^{-1}$ |

Effective pore diameter | ${d}_{p}\left(\epsilon \right)=4\frac{\epsilon}{{\sigma}_{v}}$ | m |

Viscous flow tortuosity | ${\eta}_{v}\left(\epsilon \right)=2.76{\epsilon}^{-2/3}{\left(ln\epsilon \right)}^{2}$ | - |

Darcy Permeability | $K\left(\epsilon \right)=\epsilon \frac{{d}_{p}^{2}}{32{\eta}_{v}}$ | m${}^{2}$ |

Diffusion tortuosity | ${\eta}_{d}\left(\epsilon \right)={\epsilon}^{-2/3}$ | - |

Mutual diffusion coefficient | $D}_{mh}=\frac{\epsilon}{{\eta}_{b}}\frac{2.62\times {10}^{-8}{T}^{3/2}}{{P}_{tot}\sqrt{{M}_{12}}{\sigma}_{12}^{2}{\mathrm{\Omega}}_{d}$ | m${}^{2}$·s${}^{-1}$ |

Fluid dynamic viscosity | $\mu =1.876+0.2441T-4\times {10}^{-5}\xb7{T}^{2}$ | Pa·s |

Heat transfer parameters | ||

Conductivity, effective | ${\lambda}_{eff}=\left(1-{\epsilon}_{0}\right){\lambda}_{f}+\left({\epsilon}_{0}-\epsilon \right){\lambda}_{d}+\epsilon {\lambda}_{g}$ | W·m${}^{-1}$·K${}^{-1}$ |

Conductivity, fibers | ${\lambda}_{f}\left(T\right)=-25.671+0.22728\phantom{\rule{0.166667em}{0ex}}T-1.3159\times {10}^{-4}\phantom{\rule{0.166667em}{0ex}}{T}^{2}+2.4971\times {10}^{-8}\phantom{\rule{0.166667em}{0ex}}{T}^{3}$ | W·m${}^{-1}$·K${}^{-1}$ |

Conductivity, deposit | ${\lambda}_{d}\left(T\right)=-3.466+0.0271\phantom{\rule{0.166667em}{0ex}}T-2.05\times {10}^{-5}\phantom{\rule{0.166667em}{0ex}}{T}^{2}+5.3\times {10}^{-9}\phantom{\rule{0.166667em}{0ex}}{T}^{3}$ | W·m${}^{-1}$·K${}^{-1}$ |

Conductivity, gas | ${\lambda}_{g}\left(T\right)=-0.02329+1.1092\times {10}^{-4}\phantom{\rule{0.166667em}{0ex}}T-2.0\times {10}^{-8}\phantom{\rule{0.166667em}{0ex}}{\mathit{T}}^{2}$ | W·m${}^{-1}$·K${}^{-1}$ |

Molar heat capacities | ${c}_{f}={c}_{d}=-42.468+2.852T+0.001{T}^{2}$ | J·K${}^{-1}$·mol${}^{-1}$ |

${c}_{m}=24.38+3.3\times {10}^{-2}T+3.0\times {10}^{-5}{T}^{2}-2.0\times {10}^{-8}{T}^{3}$ | J·K${}^{-1}$·mol${}^{-1}$ |

Run | Power | TC1 | TC2 | TC3 | TC4 | Max. Temp. | Radial Flux |
---|---|---|---|---|---|---|---|

# | kW | ${}^{\circ}$C | ${}^{\circ}$C | ${}^{\circ}$C | ${}^{\circ}$C | ${}^{\circ}$C | MW·m${}^{-2}$ |

P1 | 1.87 | 715 | 843 | 783 | 608 | 1113 | 4.94 |

P2, P5, P8 | 2 | 950 | 922 | 854 | 638 | 1154 | 5.31 |

P3, P6, P9 | 2.5 | 1025 | 1124 | 1033 | 778 | 1409 | 6.64 |

P4, P7, P10 | 3 | 1200 | 1334 | 1245 | 945 | 1425 | 7.97 |

Run | ${\mathit{P}}_{\mathit{tot}}$ | ${\mathit{P}}_{\mathit{m}}-{\mathit{P}}_{\mathit{m},\mathit{eq}}$ | ${\mathit{\ell}}_{\mathit{front}}$ | ${\mathit{x}}_{\mathit{front}}$ | ${\mathit{v}}_{\mathit{front}}$ | ${\mathit{E}}_{\mathit{a}}$ | ln(A) |
---|---|---|---|---|---|---|---|

# | bar | bar | mm | mm | nm·s${}^{-1}$ | kJ·mol${}^{-1}$ | $\mathsf{\mu}$m·min${}^{-1}$·bar${}^{-1}$ |

P2 | 81 | 14.72 | |||||

P3 | 89 | 24.45 | 1.2 | 1.4 | 1.56 | 80 | 5.35 |

P4 | 96 | 25.35 | 1.0 | 1.95 | 0.61 | 81 | 4.09 |

P5 | 98 | 14.97 | |||||

P6 | 107 | 25.11 | 1.55 | 1.45 | 1.6 | 62 | 3.91 |

P7 | 116 | 26.02 | 1.15 | 1.85 | 0.45 | 71 | 2.92 |

P8 | 114 | 15.14 | 1.2 | 1.1 | 1.22 | 72 | 5.14 |

P9 | 125 | 25.59 | 0.9 | 1.2 | 1.11 | 106 | 4.48 |

P10 | 136 | 26.50 | 1.1 | 2.45 | 1.39 | 74 | 3.86 |

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**MDPI and ACS Style**

Vignoles, G.L.; Talué, G.; Badey, Q.; Guette, A.; Pailler, R.; Le Petitcorps, Y.; Maillé, L. Chemical Supercritical Fluid Infiltration of Pyrocarbon with Thermal Gradients: Deposition Kinetics and Multiphysics Modeling. *J. Compos. Sci.* **2022**, *6*, 20.
https://doi.org/10.3390/jcs6010020

**AMA Style**

Vignoles GL, Talué G, Badey Q, Guette A, Pailler R, Le Petitcorps Y, Maillé L. Chemical Supercritical Fluid Infiltration of Pyrocarbon with Thermal Gradients: Deposition Kinetics and Multiphysics Modeling. *Journal of Composites Science*. 2022; 6(1):20.
https://doi.org/10.3390/jcs6010020

**Chicago/Turabian Style**

Vignoles, Gerard L., Gaëtan Talué, Quentin Badey, Alain Guette, René Pailler, Yann Le Petitcorps, and Laurence Maillé. 2022. "Chemical Supercritical Fluid Infiltration of Pyrocarbon with Thermal Gradients: Deposition Kinetics and Multiphysics Modeling" *Journal of Composites Science* 6, no. 1: 20.
https://doi.org/10.3390/jcs6010020