Activation energy asymptotics

In combustion processes, the reaction rate ${\displaystyle \omega }$ is dependent on temperature ${\displaystyle T}$ in the following form (Arrhenius law),

${\displaystyle \omega (T)\propto \mathrm {e} ^{-E_{\rm {a}}/RT},}$

where ${\displaystyle E_{\rm {a}}}$ is the activation energy, and ${\displaystyle R}$ is the universal gas constant. In general, the condition ${\displaystyle E_{\rm {a}}/R\gg T_{b}}$ is satisfied, where ${\displaystyle T_{\rm {b}}}$ is the burnt gas temperature. This condition forms the basis for activation energy asymptotics. Denoting ${\displaystyle T_{\rm {u}}}$ for unburnt gas temperature, one can define the Zel'dovich number and heat release parameter as follows

${\displaystyle \beta ={\frac {E_{\rm {a}}}{RT_{\rm {b}}}}{\frac {T_{\rm {b}}-T_{\rm {u}}}{T_{\rm {b}}}},\quad q={\frac {T_{\rm {b}}-T_{\rm {u}}}{T_{\rm {u}}}}.}$

In addition, if we define a non-dimensional temperature

${\displaystyle \theta ={\frac {T-T_{\rm {u}}}{T_{\rm {b}}-T_{\rm {u}}}},}$

such that ${\displaystyle \theta }$ approaching zero in the unburnt region and approaching unity in the burnt gas region (in other words, ${\displaystyle 0\leq \theta \leq 1}$), then the ratio of reaction rate at any temperature to reaction rate at burnt gas temperature is given by^[9][10]

${\displaystyle {\frac {\omega (T)}{\omega (T_{\rm {b}})}}\propto {\frac {\mathrm {e} ^{-E_{\rm {a}}/RT}}{\mathrm {e} ^{-E_{\rm {a}}/RT_{\rm {b}}}}}=\exp \left[-\beta (1-\theta ){\frac {1+q}{1+q\theta }}\right].}$

Now in the limit of ${\displaystyle \beta \rightarrow \infty }$ (large activation energy) with ${\displaystyle q\sim O(1)}$, the reaction rate is exponentially small i.e., ${\displaystyle O(e^{-\beta })}$ and negligible everywhere, but non-negligible when ${\displaystyle \beta (1-\theta )\sim O(1)}$. In other words, the reaction rate is negligible everywhere, except in a small region very close to burnt gas temperature, where ${\displaystyle 1-\theta \sim O(1/\beta )}$. Thus, in solving the conservation equations, one identifies two different regimes, at leading order,

• Outer convective-diffusive zone
• Inner reactive-diffusive layer

where in the convective-diffusive zone, reaction term will be neglected and in the thin reactive-diffusive layer, convective terms can be neglected and the solutions in these two regions are stitched together by matching slopes using method of matched asymptotic expansions. The above-mentioned two regime are true only at leading order since the next order corrections may involve all the three transport mechanisms.

• Zeldovich–Frank-Kamenetskii equation
• Burke–Schumann limit