OneDimensional Flames¶
Cantera includes a set of models for representing steadystate, quasione dimensional reacting flows.
These models can be used to simulate a number of common flames, such as:
freelypropagating premixed laminar flames
burnerstabilized 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 that are stabilized in an axisymmetric stagnation flow, and computes the solution along the stagnation streamline (\(r=0\)), using a similarity solution to reduce the threedimensional governing equations to a single dimension.
The governing equations for a steady axisymmetric stagnation flow follow those derived in Section 7.2 of [Kee2017]:
Continuity:
Radial momentum:
Energy:
Species:
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. The model is applicable to both ideal and nonideal fluids, which follow idealgas or realgas (RedlichKwong and PengRobinson) equations of state. The realgas support for the flame models has been newly implemented as a part of Cantera 3.0.
To help in the solution of the discretized problem, it is convenient to write a differential equation for the scalar \(\Lambda\):
Diffusive Fluxes¶
The species diffusive mass fluxes \(j_k\) are computed according to either a mixtureaveraged or multicomponent formulation. If the mixtureaveraged formulation is used, the calculation performed is:
where \(\overline{W}\) is the mean molecular weight of the mixture, \(D_{km}^\prime\) is the mixtureaveraged 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 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 mixtureaveraged formulation.
When using the multicomponent formulation, the mass fluxes are computed according to:
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 freelypropagating flame).
The following equations are solved at the point \(z = z_0\):
If the mass flow rate is specified, we also solve:
Otherwise, we solve:
Outlet boundary¶
For a boundary located at a point \(z_0\) where there is an outflow, we solve:
Symmetry boundary¶
For a symmetry boundary located at a point \(z_0\), we solve:
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:
where \(\dot{s}_k\) is the molar production rate of the gasphase species \(k\) on the surface. In addition, the surface coverages \(\theta_i\) for each surface species \(i\) are computed such that \(\dot{s}_i = 0\).
The DriftDiffusion Model¶
To account for the transport of charged species in a flame, class IonFlow adds the drift term to the diffusive fluxes of the mixtureaverage formulation according to [Ped1993],
where \(s_k\) is the sign of charge (1,1, and 0 respectively for positive, negative, and neutral charge), \(\mu_k\) is the mobility, and \(E\) is the electric field. The diffusion coefficients and mobilities of charged species can be more accurately calculated by IonGasTransport::getMixDiffCoeffs and IonGasTransport::getMobilities. The following correction is applied instead to preserve the correct fluxes of charged species:
In addition, Gauss's law is solved simultaneously with the species and energy equations,
where \(Z_k\) is the charge number, \(n_k\) is the number density, and \(N_a\) is the Avogadro number.
References
 Kee2017

R. J. Kee, M. E. Coltrin, P. Glarborg, and H. Zhu. Chemically Reacting Flow: Theory and Practice. 2nd Ed. John Wiley and Sons, 2017.
 Ped1993

T. Pederson and R. C. Brown. Simulation of electric field effects in premixed methane flames. Combustion and Flames, 94.4:433448, 1993. DOI: https://doi.org/10.1016/00102180(93)90125M.