Warning
This documentation is for an old version of Cantera. You can find docs for newer versions here.
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:
Radial momentum:
Energy:
Species:
where ρ is the density, u is the axial velocity, v is the radial velocity, V=v/r is the scaled radial velocity, Λ is the pressure eigenvalue (independent of z), μ is the dynamic viscosity, cp is the heat capacity at constant pressure, T is the temperature, λ is the thermal conductivity, Yk is the mass fraction of species k, jk is the diffusive mass flux of species k, cp,k is the specific heat capacity of species k, hk is the enthalpy of species k, Wk is the molecular weight of species k, and ˙ωk 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 Λ:
Diffusive Fluxes¶
The species diffusive mass fluxes jk are computed according to either a mixture-averaged or multicomponent formulation. If the mixture-averaged formulation is used, the calculation performed is:
where ¯W is the mean molecular weight of the mixture, Dk,m is the mixture-averaged diffusion coefficient for species k, and Xk is the mole fraction for species k. The diffusion coefficients used here are those computed by the method 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:
where Dki is the multicomponent diffusion coefficient and DTk 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 z0 where there is an inflow, values are supplied for the temperature T0, the species mass fractions Yk,0 the scaled radial velocity V0, and the mass flow rate ˙m0 (except in the case of the freely-propagating flame).
The following equations are solved at the point z=z0:
If the mass flow rate is specified, we also solve:
Otherwise, we solve:
Outlet boundary¶
For a boundary located at a point z0 where there is an outflow, we solve:
Symmetry boundary¶
For a symmetry boundary located at a point z0, we solve:
Reacting surface¶
For a surface boundary located at a point z0 on which reactions may occur, the temperature T0 is specified. We solve:
where ˙sk is the molar production rate of the gas-phase species k on the surface. In addition, the surface coverages θi for each surface species i are computed such that ˙si=0.