vib relaxation and start transport coeffs

Signed-off-by: jtneedels <[email protected]>

Author: jtneedels

_docs_v7/Thermochemical-Nonequilibrium.md

@@ -9,6 +9,10 @@ This page contains a summary of the physical models implemented in the NEMO solv
 
 - [Thermodynamic Model](#thermodynamic-model)
 - [Finite Rate Chemistry](#finite-rate-chemistry)
+- [Vibrational Relaxation](#vibrational-relaxation)
+- [Transport Coefficients](#transport-coefficients)
+    -[Wilkes-Blottner-Eucken](#wilkes-blottner-eucken)
+    -[Gupta-Yos](#gupta-yos)
 
 ---
 
@@ -86,12 +90,82 @@ $$
 $$
 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,
+The rate-controlling temperature of the reaction is calculated as a geometric average of the translation-rotational and vibrational-electronic temperatures,
 $$
-    T_r = (T)^{a_r}(T^{ve})^{b_r},
+    T_r = (T)^{a_r}(T^{ve})^{b_r}.
 $$
 
+The value of he equilibrium constant $K_{eq}$ is expressed as 
 
+$$
+    K_{eq} = \exp( A_0 T_m + A_1 + A_2 \log(1/T_m) + A_3 (1/T_m) + A_4 (1/T_m)^2  ),
+$$
+
+where $T_m$ is a modified temperature and $A_1 - A_4$ are constants dependent on the reaction. These reaction constants, the rate constrolling temperature and Arrhenius parameters are stored within the fluid model class in SU2 NEMO.
+
+---
+
+# Vibrational Relaxation #
+
+| Solver | Version | 
+| --- | --- |
+| `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 |
+
+Vibrational relaxation is computed using a standard Landau-Teller relaxation time with a Park high-temperature correction
+$$
+     \dot{\Theta}^{tr:ve} = \sum _s \rho_s \frac{de^{ve}_{s}}{dt} = \sum _s \rho_s \frac{e^{ve*}_{s} - e^{ve}_{s}}{\tau_s},
+$$
+where $\tau_s$ is computed using a combination of the Landau-Teller relaxation time, $\langle \tau_s \rangle_{L-T}$, and a limiting relaxation time from Park, $\tau_{ps}$ using
+$$
+    \tau_s = \langle \tau_s \rangle_{L-T} + \tau_{ps},
+$$
+and
+$$
+    \langle \tau_s \rangle_{L-T} = \frac{\sum_r X_r}{\sum_r X_r/\tau_{sr}}.
+$$
+The interspecies relaxation times are taken from experimental data from Millikan and White, expressed as
+$$
+    \tau_{sr} = \frac{1}{P}exp\left[A_sr\left(T^{-1/3} - 0.015\mu_{sr}^{1/4}\right) - 18.42\right].
+$$
+A limiting relaxation time, $\tau_{ps}$, is used to correct for under-prediction of the Millikan--White model at high temperatures. $\tau_{ps}$ is defined as
+$$
+    \tau_{ps} = \frac{1}{\sigma_s c_s n},
+$$
+
+where $\sigma_s$ is the effective collision~cross-section.
 
 ---
 
+# Transport Coefficients #
+
+| Solver | Version | 
+| --- | --- |
+| `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 |
+
+
+Mass, momentum, and  energy transport in fluids are all governed by molecular collisions, and  expressions for these transport properties can be derived from the kinetic theory. The  mass diffusion fluxes, $\mathbf{J}_s$, are computed using Fick's Law of Diffusion:
+$$
+    \mathbf{J}_s = \rho D_s \nabla(c_s),
+$$
+
+where $c_s$ is the species mass fraction and $D_s$ is the species multi-component diffusion coefficient. The  values of $D_s$ are computed as a weighted sum of binary diffusion coefficients between all species in the mixture. These are obtained by solving the Stefan--Maxwell equations under the Ramshaw approximations. The  viscous stress tensor is written as
+$$
+   \boldsymbol{\sigma} = \mu \left( \nabla \mathbf{u} + \nabla {\mathbf{u}}^\mathsf{T} - \frac{2}{3} \mathbf{I} (\nabla \cdot \mathbf{u}) \right),
+$$
+where $\mu$ is the mixture viscosity coefficient. The  conduction heat flux for each thermal energy mode, $\mathbf{q}^{k}$, is assumed to be given by Fourier’s Law of heat conduction:
+$$
+\mathbf{q}^{k} = \kappa^{k}  \nabla(T^k),
+$$
+
+where $\kappa^{k}$ is the thermal conductivity associated with energy mode $k$.
+
+$D_s$, $\mu$, and $\kappa$ can be evaluated using either a Wilkes-Blottner-Eucken or Gupta-Yos transport models, with the implemntation details and reocmmendations on use given in the sections below.
+
+
+## Wilkes-Blottner-Eucken ##
+
+
+
+## Gupta-Yos ##
+
+