- Generated by
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 reactive infiltration kinetics, as a solid \((S)\) material is infiltrated by a liquid \((L)\) to create a product \((P)\). The framework and usage of the model is explained below, with both mathematical formulations and example input files.
The infiltration process for a material considers a cylindrical Representative Volume Element (RVE) with radius \(R\) and span \(H\), depicting a solid capillary of radius \(r_o\), corresponding to the initial porosity of \(\varphi_0\). The mathematical description of the model uses cylindrical coordinates and assumes axisymmetry around \(r = 0\).
As the liquid enters the cylinder, it reacts with a solid to form a product with thickness \(\delta_P\), as shown schematically in the figure below.
Throughout this process, the solid radius, \(r_o\), decreases while the product thickness, \(\delta_P\), increases. Let \(r_i\) denote the inner radius of the product. In addition, define \(\alpha_i\) (in units of mole per volume) as the amount of substance in the RVE, and \(\Omega_i = \dfrac{M_i}{\rho_i}\) as the molar volume of a material with molar mass \(M_i\) (in units of amu, or gram per mole) and density \(\rho_i\) (mass per volume), with subscripts taking \(L\), \(S\), and \(P\), respectively.
The volume fraction, \(\varphi_i\) of each material is then
\[ \varphi_i = \alpha_i \Omega_i \]
and the RVE porosity (void) is
\[ \varphi_v = 1 - \varphi_L - \varphi_P - \varphi_S \]
Key assumptions made throughout the derivation are:
The following nondimensionalization is applied to the constitutive model for the reactive infiltration process.
\begin{align*} &\ \bar{\delta}_P = \frac{\delta_P}{R}, \quad \bar{r}_o = \dfrac{r_o}{R} = \sqrt{1-\varphi_S}, \quad \bar{r}_i = \dfrac{r_i}{R} = \sqrt{1-\varphi_S-\varphi_P}. \end{align*}
The complete state of the RVE is denoted by the tuple \(\left( \varphi_L, \varphi_S, \varphi_P\right)\), with \(\alpha_L\) as the prescribed force.
Mathematically, it is possible that \( \varphi_L + \varphi_P + \varphi_S \ge 1 \). Physically, this implies "overflow", aka the prescribed \(\alpha_L\) is larger than the available voids. Care must be taken at the macroscopic model to avoid or resolve this issue.
The initial-value problem (IVP) corresponding to the constitutive model is
\[ \mathbf{r} = \begin{Bmatrix} r_{\varphi_P} \\ r_{\varphi_S} \end{Bmatrix} = \mathbf{0}, \]
where \(r_{\varphi_i}\) is the residual corresponding to the volume fraction of the liquid, solid and product. The residuals are defined below. Note an implicit backward-Euler time intergration scheme is used.
\[ r_{\varphi_P} = \dfrac{\alpha_{P, n+1} - \alpha_{P, n}}{t_{n+1}-t_n} - \dfrac{2}{\Omega_L}\dfrac{D}{l_c^2}\dfrac{\bar{r}_o+\bar{r}_i}{\bar{r}_o-\bar{r}_i} R_L R_S \]
Here \(R_L(\varphi_L), R_S(\varphi_S) \in [0, 1]\) are the reactivity of liquid and solid, represented by a step function of the void fraction, \(\varphi_i\).\[ r_{\varphi_S} = \dfrac{\alpha_{S, n+1} - \alpha_{S, n}}{t_{n+1}-t_n} + \dfrac{k_P}{k_S} \dfrac{\alpha_{P, n+1} - \alpha_{P, n}}{t_{n+1}-t_n} \]
The above governing equations are fairly general for a wide range of reactive infiltration systems within this framework given the chemical reaction kinetics with appropriate reaction coefficient ( \(k_i\)) and the corresponding molar volume \(\Omega_i\) of the materials.
Example for determining \(k_i\) and \(\Omega_i\): Consider a reactive infiltration process of liquid (chemical formula \(L_x\)) and solid ( \(S_y\)), with the molar mass \(M_L\) and \(M_S\). The chemical reaction creates a product with chemical formula \(L_m S_n\), with reaction coefficients \(k_S\) and \(k_P\).
\[ L_x + k_S S_y \rightarrow k_P L_mS_n \]
For stoichiometric balance, the reaction coefficients are:
\begin{align*} k_S = \dfrac{xn}{ym}, \quad k_P = \dfrac{x}{m} \end{align*}
The densities of the liquid, solid, and product are respectively \(\rho_L, \rho_S, \rho_P\).
Then, the molar volume is:
\[ \Omega_L = \dfrac{x M_L}{\rho_L}. \quad \Omega_S = \dfrac{y M_S}{\rho_S}, \quad \Omega_P = \dfrac{mM_L+nM_S}{\rho_P}. \]
The instantaneous balance also implies that
\[ \dot{\alpha}_S = \dfrac{k_P}{k_S} \dot{\alpha}_P, \]
Since this constitutive model considers \(\alpha_L\) as the forcing function for the IVP problem, \(\dot{\alpha}_L \ne k_P \dot{\alpha}_P\).
The following table summarizes the relationship between the mathematical expressions and the NEML2 models.
| Expression | Syntax |
|---|---|
| \( \bar{r}_i = \sqrt{1-\varphi_S-\varphi_P}; \quad \bar{r}_o = \sqrt{1-\varphi_S}\) | CylindricalChannelGeometry |
| \( \dfrac{2}{\Omega_L}\dfrac{D}{l_c^2}\dfrac{\bar{r}_o+\bar{r}_i}{\bar{r}_o-\bar{r}_i} R_L R_S\) | DiffusionLimitedReaction |
| \( R_L; \quad R_S \) | HermiteSmoothStep |
| \( \varphi_i = \alpha_i \Omega_i \) | Linear Combination |
| \( r_{\varphi_P} = \dfrac{\alpha_{P, n+1} - \alpha_{P, n}}{t_{n+1}-t_n} - \dfrac{2}{\Omega_L}\dfrac{D}{l_c^2}\dfrac{\bar{r}_o+\bar{r}_i}{\bar{r}_o-\bar{r}_i} R_L R_S\) | Linear Combination, Variable Rate |
| \(r_{\varphi_S} = \dfrac{\alpha_{S, n+1} - \alpha_{S, n}}{t_{n+1}-t_n} + \dfrac{k_P}{k_S} \dfrac{\alpha_{P, n+1} - \alpha_{P, n}}{t_{n+1}-t_n}\) | Linear Combination, Variable Rate |
A complete example input file for the reactive infiltration is shown below with the appropriate model composition.
This example demonstrates the use of the chemical reactions physics module to compose models for reactive infiltration kinetics, as a solid \((S)\) material is infiltrated by a liquid \((L)\) to create a product \((P)\). The framework and usage of the model is explained below, with both mathematical formulations and example input files.
The infiltration process for a material considers a cylindrical Representative Volume Element (RVE) with radius \(R\) and span \(H\), depicting a solid capillary of radius \(r_o\), corresponding to the initial porosity of \(\varphi_0\). The mathematical description of the model uses cylindrical coordinates and assumes axisymmetry around \(r = 0\).
As the liquid enters the cylinder, it reacts with a solid to form a product with thickness \(\delta_P\), as shown schematically in the figure below.
Throughout this process, the solid radius, \(r_o\), decreases while the product thickness, \(\delta_P\), increases. Let \(r_i\) denote the inner radius of the product. In addition, define \(\alpha_i\) (in units of mole per volume) as the amount of substance in the RVE, and \(\Omega_i = \dfrac{M_i}{\rho_i}\) as the molar volume of a material with molar mass \(M_i\) (in units of amu, or gram per mole) and density \(\rho_i\) (mass per volume), with subscripts taking \(L\), \(S\), and \(P\), respectively.
The volume fraction, \(\varphi_i\) of each material is then
\[ \varphi_i = \alpha_i \Omega_i \]
and the RVE porosity (void) is
\[ \varphi_v = 1 - \varphi_L - \varphi_P - \varphi_S \]
Key assumptions made throughout the derivation are:
The following nondimensionalization is applied to the constitutive model for the reactive infiltration process.
\begin{align*} &\ \bar{\delta}_P = \frac{\delta_P}{R}, \quad \bar{r}_o = \dfrac{r_o}{R} = \sqrt{1-\varphi_S}, \quad \bar{r}_i = \dfrac{r_i}{R} = \sqrt{1-\varphi_S-\varphi_P}. \end{align*}
The complete state of the RVE is denoted by the tuple \(\left( \varphi_L, \varphi_S, \varphi_P\right)\), with \(\alpha_L\) as the prescribed force.
Mathematically, it is possible that \( \varphi_L + \varphi_P + \varphi_S \ge 1 \). Physically, this implies "overflow", aka the prescribed \(\alpha_L\) is larger than the available voids. Care must be taken at the macroscopic model to avoid or resolve this issue.
The initial-value problem (IVP) corresponding to the constitutive model is
\[ \mathbf{r} = \begin{Bmatrix} r_{\varphi_P} \\ r_{\varphi_S} \end{Bmatrix} = \mathbf{0}, \]
where \(r_{\varphi_i}\) is the residual corresponding to the volume fraction of the liquid, solid and product. The residuals are defined below. Note an implicit backward-Euler time intergration scheme is used.
\[ r_{\varphi_P} = \dfrac{\alpha_{P, n+1} - \alpha_{P, n}}{t_{n+1}-t_n} - \dfrac{2}{\Omega_L}\dfrac{D}{l_c^2}\dfrac{\bar{r}_o+\bar{r}_i}{\bar{r}_o-\bar{r}_i} R_L R_S \]
Here \(R_L(\varphi_L), R_S(\varphi_S) \in [0, 1]\) are the reactivity of liquid and solid, represented by a step function of the void fraction, \(\varphi_i\).\[ r_{\varphi_S} = \dfrac{\alpha_{S, n+1} - \alpha_{S, n}}{t_{n+1}-t_n} + \dfrac{k_P}{k_S} \dfrac{\alpha_{P, n+1} - \alpha_{P, n}}{t_{n+1}-t_n} \]
The above governing equations are fairly general for a wide range of reactive infiltration systems within this framework given the chemical reaction kinetics with appropriate reaction coefficient ( \(k_i\)) and the corresponding molar volume \(\Omega_i\) of the materials.
Example for determining \(k_i\) and \(\Omega_i\): Consider a reactive infiltration process of liquid (chemical formula \(L_x\)) and solid ( \(S_y\)), with the molar mass \(M_L\) and \(M_S\). The chemical reaction creates a product with chemical formula \(L_m S_n\), with reaction coefficients \(k_S\) and \(k_P\).
\[ L_x + k_S S_y \rightarrow k_P L_mS_n \]
For stoichiometric balance, the reaction coefficients are:
\begin{align*} k_S = \dfrac{xn}{ym}, \quad k_P = \dfrac{x}{m} \end{align*}
The densities of the liquid, solid, and product are respectively \(\rho_L, \rho_S, \rho_P\).
Then, the molar volume is:
\[ \Omega_L = \dfrac{x M_L}{\rho_L}. \quad \Omega_S = \dfrac{y M_S}{\rho_S}, \quad \Omega_P = \dfrac{mM_L+nM_S}{\rho_P}. \]
The instantaneous balance also implies that
\[ \dot{\alpha}_S = \dfrac{k_P}{k_S} \dot{\alpha}_P, \]
Since this constitutive model considers \(\alpha_L\) as the forcing function for the IVP problem, \(\dot{\alpha}_L \ne k_P \dot{\alpha}_P\).
The following table summarizes the relationship between the mathematical expressions and the NEML2 models.
| Expression | Syntax |
|---|---|
| \( \bar{r}_i = \sqrt{1-\varphi_S-\varphi_P}; \quad \bar{r}_o = \sqrt{1-\varphi_S}\) | CylindricalChannelGeometry |
| \( \dfrac{2}{\Omega_L}\dfrac{D}{l_c^2}\dfrac{\bar{r}_o+\bar{r}_i}{\bar{r}_o-\bar{r}_i} R_L R_S\) | DiffusionLimitedReaction |
| \( R_L; \quad R_S \) | HermiteSmoothStep |
| \( \varphi_i = \alpha_i \Omega_i \) | Linear Combination |
| \( r_{\varphi_P} = \dfrac{\alpha_{P, n+1} - \alpha_{P, n}}{t_{n+1}-t_n} - \dfrac{2}{\Omega_L}\dfrac{D}{l_c^2}\dfrac{\bar{r}_o+\bar{r}_i}{\bar{r}_o-\bar{r}_i} R_L R_S\) | Linear Combination, Variable Rate |
| \(r_{\varphi_S} = \dfrac{\alpha_{S, n+1} - \alpha_{S, n}}{t_{n+1}-t_n} + \dfrac{k_P}{k_S} \dfrac{\alpha_{P, n+1} - \alpha_{P, n}}{t_{n+1}-t_n}\) | Linear Combination, Variable Rate |
A complete example input file for the reactive infiltration is shown below with the appropriate model composition.
1.12.0