02-10-2023-0828 - burning timescale activation energy

 In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction.^[1] The activation energy (E_a) of a reaction is measured in joules per mole (J/mol), kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol).^[2] Activation energy can be thought of as the magnitude of the potential barrier (sometimes called the energy barrier) separating minima of the potential energy surface pertaining to the initial and final thermodynamic state. For a chemical reaction to proceed at a reasonable rate, the temperature of the system should be high enough such that there exists an appreciable number of molecules with translational energy equal to or greater than the activation energy. The term "activation energy" was introduced in 1889 by the Swedish scientist Svante Arrhenius.^[3]

^{https://en.wikipedia.org/wiki/Activation_energy}

Flame physics

The underlying flame physics can be understood with the help of an idealized model consisting of a uniform one-dimensional tube of unburnt and burned gaseous fuel, separated by a thin transitional region of width ${\displaystyle \delta }$ in which the burning occurs. The burning region is commonly referred to as the flame or flame front. In equilibrium, thermal diffusion across the flame front is balanced by the heat supplied by burning.^[3][4][5][6]

Two characteristic timescales are important here. The first is the thermal diffusion timescale ${\displaystyle {\tau }_d}$, which is approximately equal to

${\displaystyle {\tau }_d\simeq {\delta }^2/\kappa }$,

where ${\displaystyle \kappa }$ is the thermal diffusivity. The second is the burning timescale ${\displaystyle {\tau }_b}$ that strongly decreases with temperature, typically as

${\displaystyle {\tau }_b\propto \exp [\mathrm{\Delta }U/(k_B{T}_f)]}$,

where ${\displaystyle \mathrm{\Delta }U}$ is the activation barrier for the burning reaction and ${\displaystyle T_f}$ is the temperature developed as the result of burning; the value of this so-called "flame temperature" can be determined from the laws of thermodynamics.

For a stationary moving deflagration front, these two timescales must be equal: the heat generated by burning is equal to the heat carried away by heat transfer. This makes it possible to calculate the characteristic width ${\displaystyle \delta }$ of the flame front:

${\displaystyle {\tau }_b={\tau }_d}$,

thus

${\displaystyle \delta \simeq \sqrt{\kappa {\tau }_b}}$.

Now, the thermal flame front propagates at a characteristic speed ${\displaystyle S_l}$, which is simply equal to the flame width divided by the burn time:

${\displaystyle S_l\simeq \delta /{\tau }_b\simeq \sqrt{\kappa /{\tau }_b}}$.

This simplified model neglects the change of temperature and thus the burning rate across the deflagration front. This model also neglects the possible influence of turbulence. As a result, this derivation gives only the laminar flame speed—hence the designation ${\displaystyle S_l}$. 

https://en.wikipedia.org/wiki/Deflagration