From b7e490203537eb69caafc1e673457709a41f6cab Mon Sep 17 00:00:00 2001 From: Ray Speth Date: Mon, 31 Oct 2016 00:03:41 -0400 Subject: [PATCH] [Doc] Describe boundary conditions for 1D flames --- doc/sphinx/flames.rst | 92 +++++++++++++++++++++++++++++++++++++++++-- 1 file changed, 88 insertions(+), 4 deletions(-) diff --git a/doc/sphinx/flames.rst b/doc/sphinx/flames.rst index 5aec96877..b39dfa533 100644 --- a/doc/sphinx/flames.rst +++ b/doc/sphinx/flames.rst @@ -85,7 +85,7 @@ differential equation for the scalar `\Lambda`: .. math:: - \frac{d\Lambda}{\dz} = 0 + \frac{d\Lambda}{dz} = 0 Diffusive Fluxes ---------------- @@ -96,11 +96,11 @@ 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^* = \rho \frac{W_k}{\overline{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 +where `\overline{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 @@ -113,13 +113,97 @@ according to: .. math:: - j_k = \frac{\rho W_k}{\bar{W}^2} \sum_i W_i D_{ki} \frac{\partial X_i}{\partial z} + j_k = \frac{\rho W_k}{\overline{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). +Boundary Conditions +=================== + +Inlet boundary +-------------- + +For a boundary located at a point `z_0` where there is an inflow, values are +supplied for the temperature `T_0`, the species mass fractions `Y_{k,0}` the +scaled radial velocity `V_0`, and the mass flow rate `\dot{m}_0` (except in the +case of the freely-propagating flame). + +The following equations are solved at the point `z = z_0`: + +.. math:: + + T(z_0) = T_0 + + V(z_0) = V_0 + + \dot{m}_0 Y_{k,0} - j_k(z_0) - \rho(z_0) u(z_0) Y_k(z_0) = 0 + +If the mass flow rate is specified, we also solve: + +.. math:: + + \rho(z_0) u(z_0) = \dot{m}_0 + +Otherwise, we solve: + +.. math:: + + \Lambda(z_0) = 0 + +Outlet boundary +--------------- + +For a boundary located at a point `z_0` where there is an outflow, we solve: + +.. math:: + + \Lambda(z_0) = 0 + + \left.\frac{\partial T}{\partial z}\right|_{z_0} = 0 + + \left.\frac{\partial Y_k}{\partial z}\right|_{z_0} = 0 + + V(z_0) = 0 + + +Symmetry boundary +----------------- + +For a symmetry boundary located at a point `z_0`, we solve: + +.. math:: + + \rho(z_0) u(z_0) = 0 + + \left.\frac{\partial V}{\partial z}\right|_{z_0} = 0 + + \left.\frac{\partial T}{\partial z}\right|_{z_0} = 0 + + j_k(z_0) = 0 + +Reacting surface +---------------- + +For a surface boundary located at a point `z_0` on which reactions may occur, +the temperature `T_0` is specified. We solve: + +.. math:: + + \rho(z_0) u(z_0) = 0 + + V(z_0) = 0 + + T(z_0) = T_0 + + j_k(z_0) + \dot{s}_k W_k = 0 + +where `\dot{s}_k` is the molar production rate of the gas-phase species `k` on +the surface. In addition, the surface coverages `\theta_i` for each surface +species `i` are computed such that `\dot{s}_i = 0`. + References ==========