Governing Equations for One-dimensional Flow#
Cantera models flames that are stabilized in an axisymmetric stagnation flow, and
computes the solution along the stagnation streamline (
Axisymmetric Stagnation Flow#
The governing equations for a steady axisymmetric stagnation flow follow those derived in Section 7.2 of Kee et al. [2017] and are implemented by class Flow1D.
Continuity:
Radial momentum:
Energy:
Species:
where the following variables are used:
is the axial coordinate is the radial coordinate is the density is the axial velocity is the radial velocity is the scaled radial velocity is the pressure eigenvalue (independent of ) is the dynamic viscosity is the heat capacity at constant pressure is the temperature is the thermal conductivity is the mass fraction of species is the diffusive mass flux of species is the specific heat capacity of species is the enthalpy of species is the molecular weight of species is the molar production rate of species .
The tangential velocity
Added in version 3.0: Support for real gases in the flame models was introduced in Cantera 3.0.
To help in the solution of the discretized problem, it is useful to write a
differential equation for the scalar
When discretized, the Jacobian terms introduced by this equation match the block diagonal structure produced by the other governing equations, rather than creating a column of entries that would cause fill-in when factorizing as part of the Newton solver.
Diffusive Fluxes#
The species diffusive mass fluxes
where
When using the multicomponent formulation, the mass fluxes are computed according to:
where soret_enabled
property (Python).
Boundary Conditions#
Inlet boundary#
For a boundary located at a point
The following equations are solved at the point
If the mass flow rate is specified, we also solve:
Otherwise, we solve:
These equations are implemented by class Inlet1D.
Outlet boundary#
For a boundary located at a point
These equations are implemented by class Outlet1D.
Symmetry boundary#
For a symmetry boundary located at a point
These equations are implemented by class Symm1D.
Reacting surface#
For a surface boundary located at a point
where
These equations are implemented by class ReactingSurf1D.
The Drift-Diffusion Model#
To account for the transport of charged species in a flame, class IonFlow adds the drift term to the diffusive fluxes of the mixture-average formulation according to Pedersen and Brown [1993],
where
In addition, Gauss’s law is solved simultaneously with the species and energy equations,
where
Counterflow Two-Point Flame Control#
A two-point temperature control feature is available for counterflow diffusion flames. This feature allows users to set a control point on each side of a flame and incrementally lower the flame temperature. This allows for the simulation of the stable burning branch as well as the unstable burning branch of the standard flamelet “S-curve”. The implementation is based on the method discussed in Nishioka et al. [1996] and Huo et al. [2014]. The diagram below shows the general concept of the two-point flame control method, with control points located on either side of the peak flame temperature. An initial flame solution is used as a starting point, and the temperatures at the control points are lowered to produce a new flame solution that satisfies the governing equations and passes through the new temperatures at the control points.
For the two-point control method, one governing equation is modified (
and
At the left control point the residual for the
At the left control point the residual for the
Where
The values of
At the left boundary, the boundary condition for the continuity equation is imposed by using the value of the axial velocity at the left boundary. At the right boundary, the boundary condition for the continuity equation is imposed by using the solution from the oxidizer velocity equation.