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In the study of
diffusion flame In combustion, a diffusion flame is a flame in which the oxidizer and fuel are separated before burning. Contrary to its name, a diffusion flame involves both diffusion and convection processes. The name diffusion flame was first suggested by ...
, Liñán's equation is a second-order nonlinear
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
which describes the inner structure of the diffusion flame, first derived by
Amable Liñán Amable Liñán Martínez (born 1934 in Noceda de Cabrera, Castrillo de Cabrera, León, Spain) is a Spanish aeronautical engineer working in the field of combustion. Biography He holds a PhD in Aeronautical Engineering from the Technical Uni ...
in 1974. The equation reads as :\frac =(y^2-\zeta^2)e^ subjected to the boundary conditions : \begin \zeta\rightarrow -\infty : &\quad \frac=-1,\\ \zeta\rightarrow +\infty : &\quad \frac=+1 \end where \delta is the reduced or rescaled Damköhler number and \gamma is the ratio of excess heat conducted to one side of the reaction sheet to the total heat generated in the reaction zone. If \gamma>0, more heat is transported to the oxidizer side, thereby reducing the
reaction rate The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per u ...
on the oxidizer side (since reaction rate depends on the temperature) and consequently greater amount of fuel will be leaked into the oxidizer side. Whereas, if \gamma<0, more heat is transported to the fuel side of the diffusion flame, thereby reducing the reaction rate on the fuel side of the flame and increasing the oxidizer leakage into the fuel side. When \gamma\rightarrow 1 (\gamma\rightarrow -1), all the heat is transported to the oxidizer (fuel) side and therefore the flame sustains extremely large amount of fuel (oxidizer) leakage. The equation is, in some aspects, universal (also called as the canonical equation of the diffusion flame) since although Liñán derived the equation for
stagnation point flow In fluid dynamics, a stagnation point flow refers to a fluid flow in the Neighbourhood (mathematics), neighbourhood of a stagnation point (in two-dimensional flows) or a stagnation line (in three-dimensional flows) with which the stagnation point/ ...
, assuming unity Lewis numbers for the reactants, the same equation is found to represent the inner structure for general laminar flamelets, having arbitrary Lewis numbers.


Existence of solutions

Near the extinction of the diffusion flame, \delta is order unity. The equation has no solution for \delta<\delta_E, where \delta_E is the extinction Damköhler number. For \delta>\delta_E with , \gamma, <1, the equation possess two solutions, of which one is an unstable solution. Unique solution exist if , \gamma, >1 and \delta>\delta_E. The solution is unique for \delta>\delta_I, where \delta_I is the ignition Damköhler number. Liñán also gave a correlation formula for the extinction Damköhler number, which is increasingly accurate for 1-\gamma \ll 1, :\delta_E = e 1-\gamma)-(1-\gamma)^2+0.26(1-\gamma)^3 + 0.055(1-\gamma)^4


Generalized Liñán's equation

The generalized Liñán's equation is given by :\frac =(y-\zeta)^m (y+\zeta)^ne^ where m and n are constant reaction orders of fuel and oxidizer, respectively.


Large Damköhler number limit

In the
Burke–Schumann limit In combustion, Burke–Schumann limit, or large Damköhler number limit, is the limit of infinitely fast chemistry (or in other words, infinite Damköhler number), named after S.P. Burke and T.E.W. Schumann, due to their pioneering work on Burke� ...
, \delta\rightarrow\infty. Then the equation reduces to :\frac = (y-\zeta)^m(y+\zeta)^n, \quad \zeta\rightarrow\pm\infty:\, \frac=\pm 1. An approximate solution to this equation was developed by Liñán himself using integral method in 1963 for his thesis, :y(\zeta)=y_m + (\zeta-\zeta_m)\operatorname (\zeta-\zeta_m)- \frac\left -e^\right where \mathrm is the
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integrat ...
and :\begin \zeta_m &= \frac y_m,\\ y_m &= \frac\left frac\left(\sqrt-1\right)\right,\\ k &= \frac m^m n^n\left(\frac\right)^. \end Here \zeta=\zeta_m is the location where y(\zeta) reaches its minimum value y(\zeta_m)=y_m. When m=n=1, \zeta_m=0, y_m=0.8702 and k=0.6711.


See also

*
Liñán's diffusion flame theory Liñán diffusion flame theory is a theory developed by Amable Liñán in 1974 to explain the diffusion flame structure using activation energy asymptotics and Damköhler number asymptotics.Liñán, A., Martínez-Ruiz, D., Vera, M., & Sánchez, A. ...


References

{{reflist, 30em Fluid dynamics Combustion Ordinary differential equations