Liñán's diffusion flame theory

The theory is well explained in the simplest possible model. Thus, assuming a one-step irreversible Arrhenius law for the combustion chemistry with constant density and transport properties and with unity Lewis number reactants, the governing equation for the non-dimensional temperature field ${\displaystyle T(y)}$ in the stagnation point flow reduces to

${\displaystyle {\frac {d^{2}T}{dy^{2}}}+y{\frac {dT}{dy}}=-\mathrm {Da} \ y_{F}y_{O}e^{-T_{a}/T},\quad Z={\frac {1}{2}}\mathrm {erfc} \left({\frac {y}{\sqrt {2}}}\right)}$

where ${\displaystyle Z}$ is the mixture fraction, ${\displaystyle \mathrm {Da} }$ is the Damköhler number, ${\displaystyle T_{a}=E/R}$ is the activation temperature and the fuel mass fraction and oxidizer mass fraction are scaled with their respective feed stream values, given by

${\displaystyle {\begin{aligned}y_{F}&=Z+T_{o}-T\\y_{O}&=(1-Z)/S+T_{o}-T\end{aligned}}}$

with boundary conditions ${\displaystyle T(-\infty )=T(\infty )=T_{o}}$. Here, ${\displaystyle T_{o}}$ is the unburnt temperature profile (frozen solution) and ${\displaystyle S}$ is the stoichiometric parameter (mass of oxidizer stream required to burn unit mass of fuel stream). The four regime are analyzed by trying to solve above equations using activation energy asymptotics and Damköhler number asymptotics. The solution to above problem is multi-valued. Treating mixture fraction ${\displaystyle Z}$ as independent variable reduces the equation to

${\displaystyle {\frac {d^{2}T}{dZ^{2}}}=-2\pi e^{y^{2}}\mathrm {Da} \ y_{F}y_{O}e^{-T_{a}/T}}$

with boundary conditions ${\displaystyle T(0)=T(1)=T_{o}}$ and ${\displaystyle y={\sqrt {2}}\mathrm {erfc} ^{-1}(2Z)}$.

The reduced Damköhler number is defined as follows

${\displaystyle \delta =8\pi Z_{s}^{2}e^{y_{s}^{2}}\left({\frac {T_{s}^{2}}{T_{a}}}\right)^{3}\mathrm {Da} \ e^{-T_{a}/T}}$

where ${\displaystyle y_{s}={\sqrt {2}}\mathrm {erfc} ^{-1}(2Z_{s}),\ Z_{s}=1/(S+1)}$ and ${\displaystyle T_{s}=T_{o}+Z_{s}}$. The theory predicted an expression for the reduced Damköhler number at which the flame will extinguish, given by

${\displaystyle \delta _{E}=e\left[(1-\gamma )-(1-\gamma )^{2}+0.26(1-\gamma )^{3}+0.055(1-\gamma )^{4}\right]}$

where ${\displaystyle \gamma =1-2(1-\alpha )(1-Z_{s})}$.

• Liñán's equation
• Emmons problem
• Clarke–Riley diffusion flame
• Burke–Schumann flame