finite rate chemistry

Signed-off-by: jtneedels <jneedels@stanford.edu>

Author: jtneedels

_docs_v7/Thermochemical-Nonequilibrium.md

@@ -8,6 +8,7 @@ This page contains a summary of the physical models implemented in the NEMO solv
 ---
 
 - [Thermodynamic Model](#thermodynamic-model)
+- [Finite Rate Chemistry](#finite-rate-chemistry)
 
 ---
 
@@ -17,10 +18,9 @@ This page contains a summary of the physical models implemented in the NEMO solv
 | --- | --- |
 | `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 |
 
-A rigid-rotor harmonic oscillator (RRHO) two-temperature model is used to model the thermodynamic state of continuum hypersonic flows. Through the independence of the energy levels, the~total energy and vibrational--electronic energy per unit volume can be expressed as
+A rigid-rotor harmonic oscillator (RRHO) two-temperature model is used to model the thermodynamic state of continuum hypersonic flows. Through the independence of the energy levels, the total energy and vibrational--electronic energy per unit volume can be expressed as
 $$   \rho e = \sum_s \rho_s \left(e_s^{tr} + e_s^{rot} + e_s^{vib} + e_s^{el} + e^{\circ}_s + \frac{1}{2} \bar{v}^{\top} \bar{v}\right),
 $$
-
 and
 $$
         \rho e^{ve} = \sum_s \rho_{s} \left(e_s^{vib} + e_s^{el}\right).
@@ -33,27 +33,65 @@ $$
     0 & \text{for electrons,}
     \end{cases}
 $$
-\begin{equation}
+$$
     e^{rot}_s =\begin{cases}
     \frac{\xi  }{2} \frac{R}{M_s} T & \text{for polyatomic species,}\\
     0 & \text{for monatomic species and electrons,}
     \end{cases}
-\end{equation}
+$$
 where $\xi$ is an integer specifying the number of axes of rotation,
-\begin{equation}
+$$
     e^{vib}_s =\begin{cases}
     \frac{R}{M_s} \frac{\theta^{vib}_s}{exp\left( \theta^{vib}_s / T^{ve}\right) - 1} & \text{for polyatomic species,}\\
     0 & \text{for monatomic species and electrons,}
     \end{cases}
-\end{equation}
-where $\theta^{vib}_s$ is the characteristic vibrational temperature of the species, and~\begin{equation}
+$$
+where $\theta^{vib}_s$ is the characteristic vibrational temperature of the species, and 
+
+
+$$
     e^{el}_s =\begin{cases}
-    \frac{R}{M_s}\frac{\sum_{i=1}^{\infty} g_{i,s}{\theta^{el}_{i,s} exp(-\theta^{el}_{i,s}/T_{ve})}}{\sum_{i=0}^{\infty} g_{i,s} exp(-\theta^{el}_{i,s}/T_{ve})} & \text{for polyatomic and monatomic species,}\\
+    \frac{R}{M_s}\frac{\sum_{i=1}^{\infty} g_{i,s}{\theta^{el}_{i,s} \exp(-\theta^{el}_{i,s}/T_{ve})}}{\sum_{i=0}^{\infty} g_{i,s} exp(-\theta^{el}_{i,s}/T_{ve})} & \text{for polyatomic and monatomic species,}\\
     \frac{3}{2} \frac{R}{M_s} T^{ve} & \text{for electrons,}
     \end{cases}
-\end{equation}
+$$
+
+where $\theta^{el}_s$ is the characteristic electronic temperature of the species and $g_i$ is the degeneracy of the $i^{th}$ state.
+
+---
+
+# Finite Rate Chemistry #
+
+| Solver | Version | 
+| --- | --- |
+| `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 |
+
+The source terms in the species conservation equations are the  volumetric mass production rates which are governed by the forward and backward reaction rates, $R^f$ and $R^b$, for a given reaction $r$, and can be expressed as
+$$
+    \dot{w}_s = M_s \sum_r (\beta_{s,r} - \alpha_{s,r})(R_{r}^{f} - R_{r}^{b}). 
+$$
+
+From kinetic theory, the forward and backward reaction rates are dependent on the molar concentrations of the reactants and products, as well as the forward and backward reaction rate coefficients, $k^f$ and $k^b$, respectively, and can be expressed as
+$$
+    R_{r}^f = k_{r}^f \prod_s (\frac{\rho_s}{M_s})^{\alpha_{s,r}},
+$$
+and
+$$
+    R_{r}^b = k_{r}^b \prod_s (\frac{\rho_s}{M_s})^{\beta_{s,r}}.
+$$
+
+For an Arrhenius reaction, the forward reaction rate coefficient can be computed as
+$$
+    k_{r}^f = C_r(T_r)^{\eta_r} exp\left(- \frac{\epsilon_r}{k_B T_r}\right),
+$$
+where $C_r$ is the pre-factor, $T_r$ is the rate-controlling temperature for the reaction, $\eta_r$ is an empirical exponent, and $\epsilon_r$ is the activation energy per molecule.
+
+The rate-controlling temperature of the reaction is calculated as a geometric average of the translation--rotational and vibrational--electronic temperatures,
+$$
+    T_r = (T)^{a_r}(T^{ve})^{b_r},
+$$
+
 
-\noindent where $\theta^{el}_s$ is the characteristic electronic temperature of the species and $g_i$ is the degeneracy of the $i^{th}$ state.
 
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