Physics:Emmons problem

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In combustion, Emmons problem describes the flame structure which develops inside the boundary layer, created by a flowing oxidizer stream on flat fuel (solid or liquid) surfaces. The problem was first studied by Howard Wilson Emmons in 1956.[1][2][3] The flame is of diffusion flame type because it separates fuel and oxygen by a flame sheet. The corresponding problem in a quiescent oxidizer environment is known as Clarke–Riley diffusion flame.

Burning rate

Source:[4]

Consider a semi-infinite fuel surface with leading edge located at [math]\displaystyle{ x=0 }[/math] and let the free stream oxidizer velocity be [math]\displaystyle{ U_\infty }[/math]. Through the solution [math]\displaystyle{ f(\eta) }[/math] of Blasius equation [math]\displaystyle{ f'''+ff''=0 }[/math] ([math]\displaystyle{ \eta }[/math] is the self-similar Howarth–Dorodnitsyn coordinate), the mass flux [math]\displaystyle{ \rho v }[/math] ([math]\displaystyle{ \rho }[/math] is density and [math]\displaystyle{ v }[/math] is vertical velocity) in the vertical direction can be obtained

[math]\displaystyle{ \rho v = \rho_\infty \mu_\infty \sqrt{\frac{2\xi}{U_\infty}} \left(f'\rho \int_0^\eta \rho^{-1} \ d\eta - f\right), }[/math]

where

[math]\displaystyle{ \xi = \int_0^x \rho_\infty \mu_\infty \ dx. }[/math]

In deriving this, it is assumed that the density [math]\displaystyle{ \rho \sim 1/T }[/math] and the viscosity [math]\displaystyle{ \mu \sim T }[/math], where [math]\displaystyle{ T }[/math] is the temperature. The subscript [math]\displaystyle{ \infty }[/math] describes the values far away from the fuel surface. The main interest in combustion process is the fuel burning rate, which is obtained by evaluating [math]\displaystyle{ \rho v }[/math] at [math]\displaystyle{ \eta=0 }[/math], as given below,

[math]\displaystyle{ \rho_o v_o = \rho_\infty \mu_\infty \left[\frac{2U_\infty}{\mu_\infty^2}\int_0^x \rho_\infty \mu_\infty \ dx\right]^{-1/2} [-f(0)]. }[/math]

See also

References

  1. Emmons, H. W. (1956). The film combustion of liquid fuel. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 36(1‐2), 60-71.
  2. Clarke, J. F. (1969). Emmons' problem according to the Oseen approximation. The Physics of Fluids, 12(1), 241-243.
  3. Baum, H. R., & Atreya, A. (2015). The Elliptic Emmons Problem. In ICHMT DIGITAL LIBRARY ONLINE. Begel House Inc.
  4. Williams, F. A. (2018). Combustion theory. CRC Press.