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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,195 @@ | ||
| Wall Degradation Modeling | ||
| ========================= | ||
|
|
||
| Conserved Quantities | ||
| ^^^^^^^^^^^^^^^^^^^^ | ||
| .. autoclass:: mirgecom.wall_model.PorousFlowDependentVars | ||
|
|
||
| Equations of State | ||
| ^^^^^^^^^^^^^^^^^^ | ||
| .. autoclass:: mirgecom.wall_model.WallEOS | ||
| .. autoclass:: mirgecom.wall_model.WallDependentVars | ||
|
|
||
| Model-specific properties | ||
| ^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
| The properties of the materials are defined in specific files. These files | ||
| are required by :class:`~mirgecom.wall_model.WallEOS`. | ||
|
|
||
| Carbon fiber | ||
| ------------ | ||
| .. automodule:: mirgecom.materials.carbon_fiber | ||
|
|
||
| TACOT | ||
| ----- | ||
| .. automodule:: mirgecom.materials.tacot | ||
|
|
||
| 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 | ||
| :class:`~mirgecom.materials.carbon_fiber.Oxidation` progresses, | ||
| the temporal evolution of the fibers mass is given by | ||
|
|
||
| .. math :: | ||
| \frac{\partial \rho_s}{\partial t} = \dot{\omega_s} \mbox{ .} | ||
| This process creates gaseous species following | ||
|
|
||
| .. math :: | ||
| \frac{\partial \rho_g}{\partial t} = - \dot{\omega}_s \mbox{ ,} | ||
| where the | ||
| :attr:`~mirgecom.fluid.ConservedVars.mass` | ||
| is $\rho_g$. The source term indicates that the fibers become gas in order | ||
| to satisfy mass conservation. | ||
|
|
||
| The | ||
| :attr:`~mirgecom.fluid.ConservedVars.energy` | ||
| of the bulk material is given by | ||
|
|
||
| .. math:: | ||
| \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 | ||
|
|
||
| .. math:: | ||
| \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 | ||
| :attr:`~mirgecom.wall_model.WallDependentVars.tau`. | ||
| As a consequence, the instantaneous material properties will change due to | ||
| the mass loss. | ||
|
|
||
| The | ||
| :attr:`~mirgecom.eos.GasDependentVars.temperature` | ||
| is evaluated using Newton iteration based on both | ||
| :attr:`~mirgecom.eos.PyrometheusMixture.get_internal_energy` and | ||
| :attr:`~mirgecom.wall_model.WallEOS.enthalpy`, | ||
| as well as their respective derivatives, namely | ||
| :attr:`~mirgecom.eos.PyrometheusMixture.heat_capacity_cv` and | ||
| :attr:`~mirgecom.wall_model.WallEOS.heat_capacity`. | ||
| Note that :mod:`pyrometheus` is used to handle the species properties. | ||
|
|
||
| Composite Materials | ||
| ^^^^^^^^^^^^^^^^^^^ | ||
|
|
||
| This section covers the response of composite materials made of phenolic | ||
| resin and carbon fibers. | ||
| The carbon fibers are characterized as a highly porous material, | ||
| with void fraction $\approx 90\%$. Phenolic resins are added to rigidize | ||
| the fibers by permeating the material and filling partially the gaps between | ||
| fibers. As the material is heated up by the flow, the resin pyrolysis, 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 | ||
| :class:`~mirgecom.materials.tacot.Pyrolysis` progresses, the mass of each | ||
| $i$ constituents of the resin, denoted by | ||
| :attr:`~mirgecom.wall_model.PorousFlowDependentVars.material_densities`, | ||
| is calculated as | ||
|
|
||
| .. math :: | ||
| \frac{\partial \rho_i}{\partial t} = \dot{\omega}_i \mbox{ .} | ||
| This process creates gaseous species following | ||
|
|
||
| .. math :: | ||
| \frac{\partial \rho_g}{\partial t} + \nabla \cdot \rho_g \mathbf{u} = | ||
| - \sum_i \dot{\omega}_i \mbox{ ,} | ||
| where the | ||
| :attr:`~mirgecom.fluid.ConservedVars.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 | ||
|
|
||
| .. math:: | ||
| \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 | ||
| :attr:`~mirgecom.fluid.ConservedVars.energy` | ||
| of the bulk material is given by | ||
|
|
||
| .. math:: | ||
| \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 | ||
|
|
||
| .. math:: | ||
| \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 | ||
| :attr:`~mirgecom.wall_model.WallDependentVars.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 | ||
| :attr:`~mirgecom.eos.GasDependentVars.temperature`. | ||
| It is evaluated using Newton iteration based on both | ||
| :attr:`~mirgecom.materials.tacot.GasProperties.enthalpy` (tabulated | ||
| data) or | ||
| :attr:`~mirgecom.eos.PyrometheusMixture.get_internal_energy` (Pyrometheus) | ||
| and | ||
| :attr:`~mirgecom.wall_model.WallEOS.enthalpy`, | ||
| as well as their respective derivatives, namely | ||
| :attr:`~mirgecom.materials.tacot.GasProperties.heat_capacity` and | ||
| :attr:`~mirgecom.wall_model.WallEOS.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 | ||
| :mod:`~mirgecom.materials.tacot`. | ||
| Different materials can be incorporated as separated files. | ||
|
|
||
| The | ||
| :class:`~mirgecom.materials.tacot.GasProperties` are obtained based on | ||
| tabulated data, assuming chemical equilibrium, and evaluated with splines | ||
| for interpolation of the entries. However, the data is currently obtained | ||
| by PICA. | ||
|
|
||
| .. 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 | ||
| Helper Functions | ||
| ---------------- | ||
| .. autofunction:: mirgecom.multiphysics.phenolics.initializer |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
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|
@@ -9,3 +9,4 @@ Operators | |
| artificial_viscosity | ||
| gas-dynamics | ||
| thermally_coupled | ||
| composite_materials | ||
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1 @@ | ||
| ablation-workshop-mpi.py |
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