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1.12.0
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NEML2 2.0.0
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This example demonstrates the use of the chemical reactions physics module to compose models for pyrolysis kinetics of a thermoset material, where a binder is burnt into char. The framework and usage of the model is explained below, with both mathematical formulations and example input files.
The thermosetting pyroslysis processs of a binder-particle composite is shown schematically in the figure below. This is a temperature-controlled process in which the precursor (the binder) gets decomposed into gas and a residue solid. A portion of the gas is trapped inside the binder and/or the solid. It is assumed that the particle phase does not participate in the reaction and its mass remains constant. It is also assumed that the residue does not further decompose or react.
Let \( \omega_i \) with \(i=p,b,g,c\) denote the current weight fraction of the particle, binder, gas, and char, respectively. When the reaction is complete, all of the binder gets converted to char with final yield (often experimentally measured using thermogravimetric analysis, or TGA).
In this example, the reaction kinetics is defined by the contracting geometry model, i.e.
\[ \dot{\alpha} = k \left( 1-\alpha \right)^n, \]
where the reaction coefficient \(k\) is temperature dependent following an Arrhenius relation.
To couple with other physics based on control volume, we consider the control volume shown in figure below,
Within the RVE, we track four independent state variables: \( \omega_b, \omega_c, \omega_g, \varphi_{o} \), where \(\omega_b\), \(\omega_c\), and \(\omega_g\) are the mass fraction of the binder, char, and gas, respectively; \(\varphi_{o}\) is the volume fraction of the open pore.
Thus, the effective volume of the RVE can be calculated as
\[ V = \dfrac{M_\mathrm{ref}}{1-\varphi_o} \left( \dfrac{\omega_b}{\rho_b} + \dfrac{\omega_p}{\rho_p} + \dfrac{\omega_c}{\rho_s} + \dfrac{\omega_g}{\rho_g} \right), \]
with \(\rho\) as the mass density.
We define \(\mu \in [0, 1]\) as the instantaneous ratio between the amount of gas trapped in closed pores and the total amount of gas produced, i.e.
\[ \dot{\omega}_g = \mu \left( \dot{\omega}_g + \dot{\omega}_{g, o} \right). \]
Following conservation of mass, the gas production rate can be expressed as
\[ \dot{\omega}_g = -\mu \left( \dot{\omega}_b + \dot{\omega}_c \right). \]
Finally, the production rate of open pore is given as
\[ \dot{\varphi}_{o} = \zeta \dot{\alpha}, \]
where \( \zeta \) is total volume fraction of the open pore upon reaction completion.
The following table summarizes the relationship between the mathematical expressions and the NEML2 models.
| Expression | Syntax |
|---|---|
| \( \dot{\alpha} = k \left( 1-\alpha \right)^n \) | ContractingGeometry |
| \(k = k_0 \exp{\dfrac{-Q}{RT}} \) | ArrheniusParameter |
| \(V = \dfrac{M_\mathrm{ref}}{1-\varphi_o} \left( \dfrac{\omega_b}{\rho_b} + \dfrac{\omega_p}{\rho_p} + \dfrac{\omega_c}{\rho_s} + \dfrac{\omega_g}{\rho_g} \right) \) | EffectiveVolume |
The state variables are integrated in time using the backward-Euler scheme.
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