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. It is named after S.P. Burke and T.E.W. Schumann,[1][2] who were able to predict the flame height and flame shape using their simply analysis of infinitely fast chemistry (which is now called as Burke–Schumann limit) in 1928.
Mathematical description[3][4]
Consider a cylindrical duct with axis along
direction with radius
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
outside the fuel tube. 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:
- The average velocity is parallel to axis (
direction) of the ducts, ![{\displaystyle \mathbf {v} =v\mathbf {e} _{z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4442f9cc2fdecdac6114aaf5e290193b65588b31)
- The mass flux in the axial direction is constant,
![{\displaystyle \rho v=\mathrm {constant} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8d8db636366e31243c61f33e1e2d04d9e2c0cc)
- Axial diffusion is negligible compared to the transverse/radial diffusion
- The flame occurs infinitely fast (Burke–Schumann limit), therefore flame appears as a reaction sheet across which properties of flow changes
- Effects of gravity has been neglected
Consider a one-step irreversible Arrhenius law,
, where
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. If
is the number of moles of fuel burned per unit volume per unit time and introducing the non-dimensional fuel and mass fraction and the Stoichiometry parameter,
![{\displaystyle y_{F}={\frac {Y_{F}}{Y_{Fo}}},\quad y_{O}={\frac {Y_{O}}{Y_{Oo}}},\quad S={\frac {sY_{Fo}}{Y_{Oo}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce972d0ab7be0674bf46ad887168595b15aeb90c)
the governing equations for fuel and oxidizer mass fraction reduce to
![{\displaystyle {\begin{aligned}{\frac {\rho D_{T}}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial y_{F}}{\partial r}}\right)-\rho v{\frac {\partial y_{F}}{\partial z}}={\frac {\omega }{Y_{Fo}}}\\{\frac {\rho D_{T}}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial y_{O}}{\partial r}}\right)-\rho v{\frac {\partial y_{O}}{\partial z}}=S{\frac {\omega }{Y_{Fo}}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3a299fca9d5616bb28bf542e116c90a4a579ce7)
where Lewis number of both species is assumed to be unity and
is assumed to be constant, where
is the thermal diffusivity. The boundary conditions for the problem are
![{\displaystyle {\begin{aligned}{\text{at}}\,&z=0,\,0<r<a,\,y_{F}=1,\,y_{O}=0,\\{\text{at}}\,&z=0,\,a<r<b,\,y_{F}=0,\,y_{O}=1,\\{\text{at}}\,&r=b,\,0<z<\infty ,\,{\frac {\partial y_{F}}{\partial r}}=0,\,{\frac {\partial y_{O}}{\partial r}}=0.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bade12dac3d6fcc5c440f383154eb1b4c22e00f)
The equation can be linearly combined to eliminate the non-linear reaction term
and solve for the new variable
,
where
is known as the mixture fraction. 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. The equation satisfied by
is
![{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial Z}{\partial r}}\right)-{\frac {\rho v}{\rho D_{T}}}{\frac {\partial Z}{\partial z}}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5fe9aaff3dfe37584a3d0869bcf872e87ecdfa4)
Introducing the following coordinate transformation
![{\displaystyle \xi ={\frac {r}{b}},\quad \eta ={\frac {\rho D_{T}}{\rho v}}{\frac {z}{b^{2}}},\quad c={\frac {a}{b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f19efe330dad8c5d81a7c7e10cfa176d7de027eb)
reduces the equation to
![{\displaystyle {\frac {1}{\xi }}{\frac {\partial }{\partial \xi }}\left(\xi {\frac {\partial Z}{\partial \xi }}\right)-{\frac {\partial Z}{\partial \eta }}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe3dbc9dfdb5d243b7cc813dfeef3c3d4d586fa)
The corresponding boundary conditions become
![{\displaystyle {\begin{aligned}{\text{at}}\,&\eta =0,\,0<\xi <c,\,Z=1,\\{\text{at}}\,&\eta =0,\,c<\xi <1,\,Z=0,\\{\text{at}}\,&\xi =1,\,0<\eta <\infty ,\,{\frac {\partial Z}{\partial \xi }}=0.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/deeede6e5d50f484cb3612f3349e1ca8c7282d89)
The equation can be solved by separation of variables
![{\displaystyle Z(\xi ,\eta )=c^{2}+2c\sum _{n=1}^{\infty }{\frac {1}{\lambda _{n}}}{\frac {J_{1}(c\lambda _{n})}{J_{0}^{2}(\lambda _{n})}}J_{0}(\lambda _{n}\xi )e^{-\lambda _{n}^{2}\eta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d819e1a8afbf76a889842e4fbb872e00912078)
where
and
are the Bessel function of the first kind and
is the nth root of
Solution can also be obtained for the planar ducts instead of the axisymmetric ducts discussed here.
Flame shape and height
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
, in other words, where
![{\displaystyle Z_{s}={\frac {1}{S+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a3c1846cf88e87d62d842d48745a0c69b40aa63)
where
is the stoichiometric mixture fraction. Therefore, for given values of
and
, the flame shape
is determined by the following equation
![{\displaystyle {\frac {1}{S+1}}=c^{2}+2c\sum _{n=1}^{\infty }{\frac {1}{\lambda _{n}}}{\frac {J_{1}(c\lambda _{n})}{J_{0}^{2}(\lambda _{n})}}J_{0}(\lambda _{n}\xi )e^{-\lambda _{n}^{2}\eta }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7440ceddd4f05a805921cd086e2eaf1089d276)
The reaction sheet separates fuel and oxidizer region. On the fuel side of the reaction sheet
and on the oxidizer side
When
, the flame extends from the mouth of the inner tube to some height at the outer tube (under-ventilated case) and when
, the flame stars from the mouth of the inner tube and joins at the axis at some height away from the mouth (over-ventilated case). Similar results are obtained when
. The flame height can be estimated approximately by keeping only the first term of the series (it give accurate results because it appears in the exponential) by setting
for the over-ventilated case and
for the under-ventilated case. This approximation predicts the flame height to be
![{\displaystyle {\begin{aligned}\eta &={\frac {1}{\lambda _{1}^{2}}}\ln \left[{\frac {2cJ_{1}(c\lambda _{1})}{(Z_{s}-c^{2})\lambda _{1}J_{0}(\lambda _{1})}}\right],\quad {\text{under-ventilated}}\\\eta &={\frac {1}{\lambda _{1}^{2}}}\ln \left[{\frac {2cJ_{1}(c\lambda _{1})}{(Z_{s}-c^{2})\lambda _{1}J_{0}^{2}(\lambda _{1})}}\right],\quad {\text{over-ventilated}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf6ee3f878b5258213a600baf4e245fd3d583c7b)
where
References
- ^ Burke, S. P., and T. E. W. Schumann. "Diffusion flames." Industrial & Engineering Chemistry 20.10 (1928): 998–1004.
- ^ Zeldovich, I. A., Barenblatt, G. I., Librovich, V. B., & Makhviladze, G. M. (1985). Mathematical theory of combustion and explosions.
- ^ Williams, F. A. (2018). Combustion theory. CRC Press.
- ^ Williams, F. A. (1965). Combustion Theory: the fundamental theory of chemical reacting flow systems. Addison-Wesley.