1.2. Wall Degradation

1.2.1. Carbon Fiber Oxidation

This section covers the response of carbon fiber when exposed to oxygen. The carbon fibers are characterized as a highly porous material, with void fraction \(\approx 90\%\). As the material is exposed to the flow, oxygen can diffuse inside the wall and result in oxidation, not only limited to the surface but also as a volumetric process. For now, convection inside the wall will be neglected and the species are only allowed to diffuse.

The temporal evolution of mass density is solved in order to predict the material degradation. As the Oxidation progresses, the temporal evolution of the fibers mass is given by

\[\frac{\partial \rho_s}{\partial t} = \dot{\omega_s} \mbox{ .}\]

This process creates gaseous species following

\[\frac{\partial \rho_g}{\partial t} = - \dot{\omega}_s \mbox{ ,}\]

where the mass is \(\rho_g\). The source term indicates that the fibers become gas in order to satisfy mass conservation.

The energy of the bulk material is given by

\[\frac{\partial \rho_b e_b}{\partial t} = \nabla \cdot \left( \bar{\boldsymbol{\kappa}} \nabla T + h_\alpha \mathbf{J}_{\alpha} \right) \mbox{ .}\]

The first term on the RHS is modeled as an effective diffusive transfer using Fourier’s law, where the thermal conductivity is given by \(\bar{\boldsymbol{\kappa}}\). The second term on the RHS is due to the species diffusion, where the species specific enthalpy \({h}_{\alpha}\), and the species diffusive flux vector \(\mathbf{J}_{\alpha}\). The sub-index \(b\) is the bulk material consisting of both solid and gas phases, where

\[\rho_b e_b = \epsilon_{g} \rho_g e_g + \epsilon_s \rho_s h_s \mbox{ .}\]

The internal energy of the gas is given by \(e_g(T)\) and the solid energy/enthalpy is \(h_s(T)\).

From the conserved variables, it is possible to compute the oxidation progress, denoted by tau. As a consequence, the instantaneous material properties will change due to the mass loss.

The temperature is evaluated using Newton iteration based on both get_internal_energy and enthalpy, as well as their respective derivatives, namely heat_capacity_cv and heat_capacity. Note that pyrometheus is used to handle the species properties.

1.2.2. Composite Materials

This section covers the response of composite materials made of phenolic resin and carbon fibers. Phenolic resins are added to rigidize the fibers by permeating the material and filling partially the gaps between fibers, reducing the porousity to \(\approx 80\%\). As the material is heated up by the flow, the resin pyrolyses, i.e., it degrades and produces gaseous species.

The temporal evolution of wall density is solved in order to predict the material degradation. As the Pyrolysis progresses, the mass of each \(i\) constituents of the resin, denoted by material_densities, is calculated as

\[\frac{\partial \rho_i}{\partial t} = \dot{\omega}_i \mbox{ .}\]

This process creates gaseous species following

\[\frac{\partial \rho_g}{\partial t} + \nabla \cdot \rho_g \mathbf{u} = - \sum_i \dot{\omega}_i \mbox{ ,}\]

where the mass is \(\rho_g\). The source term indicates that all solid resin must become gas in order to satisfy mass conservation. Lastly, the gas velocity \(\mathbf{u}\) is obtained by Darcy’s law, given by

\[\mathbf{u} = \frac{\mathbf{K}}{\mu \epsilon} \cdot \nabla P \mbox{ .}\]

In this equation, \(\mathbf{K}\) is the second-order permeability tensor, \(\mu\) is the gas viscosity, \(\epsilon\) is the void fraction and \(P\) is the gas pressure.

The energy of the bulk material is given by

\[\frac{\partial \rho_b e_b}{\partial t} + \nabla \cdot (\epsilon_{g} \rho_g h_g \mathbf{u}) = \nabla \cdot \left( \bar{\boldsymbol{\kappa}} \nabla T \right) + \mu \epsilon_{g}^2 (\bar{\mathbf{K}}^{-1} \cdot \vec{v} ) \cdot \vec{v} \mbox{ .}\]

The first term on the RHS is modeled as an effective diffusive transfer using Fourier’s law, where the thermal conductivity is given by \(\bar{\boldsymbol{\kappa}}\). The second term on the RHS account for the viscous dissipation in the Darcy’s flow. The sub-index \(b\) is the bulk material consisting of both solid and gas phases, where

\[\rho_b e_b = \epsilon_{g} \rho_g e_g + \epsilon_s \rho_s e_s \mbox{ .}\]

The energy of the gas is given by \(e_g(T) = h_g(T) - \frac{P}{\rho_g}\), where \(h_g\) is its enthalpy.

From the conserved variables, it is possible to compute the decomposition status, denoted by tau. This yields the proportion of virgin (unpyrolyzed material) to char (fully pyrolyzed) and, consequently, the different thermophysicochemical properties of the solid phase. Thus, the instantaneous material properties depend on the current state of the material, as well as the temperature. It is evaluated using Newton iteration based on get_internal_energy and enthalpy, as well as their respective derivatives heat_capacity_cv and heat_capacity.

In MIRGE-Com, the solid properties are obtained by fitting polynomials to tabulated data for easy evaluation of the properties based on the temperature. The complete list of properties can be find, for instance, in tacot. Different materials can be incorporated as separated files.

Important

The current implementation follows the description of [Lachaud_2014] for type 2 code. Additional details, extensive formulation and references are provided in https://github.com/illinois-ceesd/phenolics-notes