In combustion, a Burke–Schumann flame is a type of diffusion flame, established at the mouth of the two concentric ducts, by issuing fuel and oxidizer from the two region respectively.
Schumann,[1][2] who were able to predict the flame height and flame shape using their simple analysis of infinitely fast chemistry (which is now called as Burke–Schumann limit) in 1928 at the First symposium on combustion.
Consider a cylindrical duct with axis along
through which fuel is fed from the bottom and the tube mouth is located at
Oxidizer is fed along the same axis, but in the concentric tube of radius
Let the mass fraction in the fuel tube be
and the mass fraction of the oxygen in the outside duct be
Fuel and oxygen mixing occurs in the region
The following assumptions were made in the analysis: Consider a one-step irreversible Arrhenius law,
is the mass of oxygen required to burn unit mass of fuel and
is the amount of heat released per unit mass of fuel burned.
is the mass of fuel burned per unit volume per unit time and introducing the non-dimensional fuel and mass fraction and the Stoichiometry parameter, the governing equations for fuel and oxidizer mass fraction reduce to where Lewis number of both species is assumed to be unity and
The boundary conditions for the problem are The equation can be linearly combined to eliminate the non-linear reaction term
The mixture fraction takes the value of unity in the fuel stream and zero in the oxidizer stream and it is a scalar field which is not affected by the reaction.
is (If the Lewis numbers of fuel and oxidizer are not equal to unity, then the equation satisfied by
Introducing the following coordinate transformation reduces the equation to The corresponding boundary conditions become The equation can be solved by separation of variables where
are the Bessel function of the first kind and
Solution can also be obtained for the planar ducts instead of the axisymmetric ducts discussed here.
[3][4] In the Burke-Schumann limit, the flame is considered as a thin reaction sheet outside which both fuel and oxygen cannot exist together, i.e.,
The reaction sheet itself is located by the stoichiometric surface where
is the stoichiometric mixture fraction.
The reaction sheet separates fuel and oxidizer region.
The inner structure of the reaction sheet is described by Liñán's equation.
On the fuel side of the reaction sheet (
, the flame shape is given by the condition
), the flame extends from the mouth of the inner tube and attaches itself to the outer tube at a certain height (under-ventilated case) and when
), the flame starts from the mouth of the inner tube and joins at the axis at some height away from the mouth (over-ventilated case).
In general, the flame height is obtained by solving for
Since flame heights are generally large for the exponential terms in the series to be negligible, as a first approximation flame height can be estimated by keeping only the first term of the series.
This approximation predicts flame heights for both cases as follows where