From 530d503848f06403206b6639814e1a168ea19b5f Mon Sep 17 00:00:00 2001
From: Ray Speth <speth@mit.edu>
Date: Sun, 30 Oct 2016 00:27:38 -0400
Subject: [PATCH] [Doc] Describe governing equations for 1D flames

---
 doc/sphinx/flames.rst | 129 ++++++++++++++++++++++++++++++++++++++++++
 doc/sphinx/index.rst  |   1 +
 2 files changed, 130 insertions(+)
 create mode 100644 doc/sphinx/flames.rst

diff --git a/doc/sphinx/flames.rst b/doc/sphinx/flames.rst
new file mode 100644
index 000000000..5aec96877
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+++ b/doc/sphinx/flames.rst
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+.. default-role:: math
+
+.. py:currentmodule:: cantera
+
+**********************
+One-Dimensional Flames
+**********************
+
+Cantera includes a set of models for representing steady-state, quasi-one-
+dimensional reacting flows, which can be used to simulate a number of common
+flames, such as:
+
+- freely-propagating premixed laminar flames
+- burner-stabilized premixed flames
+- counterflow diffusion flames
+- counterflow (strained) premixed flames
+
+Additional capabilities include simulation of surface reactions, which can be
+used to represent processes such as combustion on a catalytic surface or
+chemical vapor deposition processes.
+
+All of these configurations are simulated using a common set of governing
+equations within a 1D "flow" domain, with the differences between the models
+being represented by differences in the boundary conditions applied. Here, we
+describe the governing equations and the various boundary conditions which can
+be applied.
+
+Stagnation Flow Governing Equations
+===================================
+
+Cantera models flames which are stabilized in an axisymmetric stagnation flow,
+and computes the solution along the stagnation streamline (`r=0`), using a
+similarity solution to reduce the three-dimensional governing equations to a
+single dimension.
+
+The governing equations for a steady axisymmetric stagnation flow follow those
+derived in Section 6.2 of [KCG2003]_:
+
+*Continuity*:
+
+.. math::
+
+    \frac{\partial\rho u}{\partial z} + 2 \rho V = 0
+
+*Radial momentum*:
+
+.. math::
+
+    \rho u \frac{\partial V}{\partial z} + \rho V^2 =
+        - \Lambda
+        + \frac{\partial}{\partial z}\left(\mu \frac{\partial V}{\partial z}\right)
+
+
+*Energy*:
+
+.. math::
+
+    \rho c_p u \frac{\partial T}{\partial z} =
+        \frac{\partial}{\partial z}\left(\lambda \frac{\partial T}{\partial z}\right)
+        - \sum_k j_k c_{p,k} \frac{\partial T}{\partial z}
+        - \sum_k h_k W_k \dot{\omega}_k
+
+*Species*:
+
+.. math::
+
+    \rho u \frac{\partial Y_k}{\partial z} = - \frac{\partial j_k}{\partial z}
+        + W_k \dot{\omega}_k
+
+where `\rho` is the density, `u` is the axial velocity, `v` is the radial
+velocity, `V = v/r` is the scaled radial velocity, `\Lambda` is the pressure
+eigenvalue (independent of `z`), `\mu` is the dynamic viscosity, `c_p` is the
+heat capacity at constant pressure, `T` is the temperature, `\lambda` is the
+thermal conductivity, `Y_k` is the mass fraction of species `k`, `j_k` is the
+diffusive mass flux of species `k`, `c_{p,k}` is the specific heat capacity of
+species `k`, `h_k` is the enthalpy of species `k`, `W_k` is the molecular weight
+of species `k`, and `\dot{\omega}_k` is the molar production rate of species
+`k`.
+
+The tangential velocity `w` has been assumed to be zero, and the fluid has been
+assumed to behave as an ideal gas.
+
+To help in the solution of the discretized problem, it is convenient to write a
+differential equation for the scalar `\Lambda`:
+
+.. math::
+
+    \frac{d\Lambda}{\dz} = 0
+
+Diffusive Fluxes
+----------------
+
+The species diffusive mass fluxes `j_k` are computed according to either a
+mixture-averaged or multicomponent formulation. If the mixture-averaged
+formulation is used, the calculation performed is:
+
+.. math::
+
+    j_k^* = \rho \frac{W_k}{\bar{W}} D_{k,m} \frac{\partial X_k}{\partial z}
+
+    j_k = j_k^* - Y_k \sum_i j_i^*
+
+where `\bar{W}` is the mean molecular weight of the mixture, `D_{k,m}` is the
+mixture-averaged diffusion coefficient for species `k`, and `X_k` is the mole
+fraction for species `k`. The diffusion coefficients used here are those
+computed by the method :ct:`GasTransport::getMixDiffCoeffs`. The correction
+applied by the second equation ensures that the sum of the mass fluxes is zero,
+a condition which is not inherently guaranteed by the mixture-averaged
+formulation.
+
+When using the multicomponent formulation, the mass fluxes are computed
+according to:
+
+.. math::
+
+    j_k = \frac{\rho W_k}{\bar{W}^2} \sum_i W_i D_{ki} \frac{\partial X_i}{\partial z}
+          - \frac{D_k^T}{T} \frac{\partial T}{\partial z}
+
+where `D_{ki}` is the multicomponent diffusion coefficient and `D_k^T` is the
+Soret diffusion coefficient (used only if calculation of this term is
+specifically enabled).
+
+
+References
+==========
+
+.. [KCG2003] Kee, Coltrin, Glarborg: *Chemically Reacting Flow*.
+             Wiley-Interscience, 2003
+
diff --git a/doc/sphinx/index.rst b/doc/sphinx/index.rst
index cc35c56a5..ff46cc8f8 100644
--- a/doc/sphinx/index.rst
+++ b/doc/sphinx/index.rst
@@ -29,6 +29,7 @@ Documentation
    language-interfaces
    cti/index
    reactors
+   flames
    cython/index
    matlab/index
    cxx-guide/index