## One-Dimensional Flames¶

Cantera includes a set of models for representing steady-state, quasi-one- dimensional reacting flows.

These models 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 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 three-dimensional 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:

\begin{equation*} \frac{\partial\rho u}{\partial z} + 2 \rho V = 0 \end{equation*}

\begin{equation*} \rho u \frac{\partial V}{\partial z} + \rho V^2 = - \Lambda + \frac{\partial}{\partial z}\left(\mu \frac{\partial V}{\partial z}\right) \end{equation*}

Energy:

\begin{equation*} \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 \end{equation*}

Species:

\begin{equation*} \rho u \frac{\partial Y_k}{\partial z} = - \frac{\partial j_k}{\partial z} + W_k \dot{\omega}_k \end{equation*}

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 $$\Lambda$$:

\begin{equation*} \frac{d\Lambda}{dz} = 0 \end{equation*}

### 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:

\begin{equation*} j_k^* = - \rho \frac{W_k}{\overline{W}} D_{km}^\prime \frac{\partial X_k}{\partial z} \end{equation*}
\begin{equation*} j_k = j_k^* - Y_k \sum_i j_i^* \end{equation*}

where $$\overline{W}$$ is the mean molecular weight of the mixture, $$D_{km}^\prime$$ 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 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:

\begin{equation*} 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} \end{equation*}

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 $$z = z_0$$:

\begin{equation*} T(z_0) = T_0 \end{equation*}
\begin{equation*} V(z_0) = V_0 \end{equation*}
\begin{equation*} \dot{m}_0 Y_{k,0} - j_k(z_0) - \rho(z_0) u(z_0) Y_k(z_0) = 0 \end{equation*}

If the mass flow rate is specified, we also solve:

\begin{equation*} \rho(z_0) u(z_0) = \dot{m}_0 \end{equation*}

Otherwise, we solve:

\begin{equation*} \Lambda(z_0) = 0 \end{equation*}

### Outlet boundary¶

For a boundary located at a point $$z_0$$ where there is an outflow, we solve:

\begin{equation*} \Lambda(z_0) = 0 \end{equation*}
\begin{equation*} \left.\frac{\partial T}{\partial z}\right|_{z_0} = 0 \end{equation*}
\begin{equation*} \left.\frac{\partial Y_k}{\partial z}\right|_{z_0} = 0 \end{equation*}
\begin{equation*} V(z_0) = 0 \end{equation*}

### Symmetry boundary¶

For a symmetry boundary located at a point $$z_0$$, we solve:

\begin{equation*} \rho(z_0) u(z_0) = 0 \end{equation*}
\begin{equation*} \left.\frac{\partial V}{\partial z}\right|_{z_0} = 0 \end{equation*}
\begin{equation*} \left.\frac{\partial T}{\partial z}\right|_{z_0} = 0 \end{equation*}
\begin{equation*} j_k(z_0) = 0 \end{equation*}

### 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:

\begin{equation*} \rho(z_0) u(z_0) = 0 \end{equation*}
\begin{equation*} V(z_0) = 0 \end{equation*}
\begin{equation*} T(z_0) = T_0 \end{equation*}
\begin{equation*} j_k(z_0) + \dot{s}_k W_k = 0 \end{equation*}

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 $$\dot{s}_i = 0$$.

## The Drift-Diffusion Model¶

This feature is only available when using class IonFlow. To account for the transport of charged species in a flame, the drift term is added to the diffusive fluxes of the mixture-average formulation according to [Ped1993],

\begin{equation*} j_k^* = \rho \frac{W_k}{\overline{W}} D_{km}^\prime \frac{\partial X_k}{\partial z} + s_k \mu_k E Y_k, \end{equation*}

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:

\begin{equation*} j_k = j_k^* - \frac {1 - |s_k|} {1 - \sum_i |s_i| Y_i} Y_k \sum_i j_i^*. \end{equation*}

In addition, Gauss's law is solved simultaneously with the species and energy equations,

\begin{equation*} \frac{\partial E}{\partial z} = \frac{e}{\epsilon_0}\sum_k Z_k n_k , \end{equation*}
\begin{equation*} n_k = N_a \rho Y_k / W_k, \end{equation*}
\begin{equation*} E|_{z=0} = 0, \end{equation*}

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, 94.4:433-448, 1993. DOI: https://doi.org/10.1016/0010-2180(93)90125-M.