Rate increase in chemical reaction and its variance under turbulent equilibrium

Abstract

As contrast to the Maxwellian distribution, equilibrium distribution of the second kind or turbulent equilibrium is shown to exist under tertiary molecular chaos hypothesis to replace the classical binary chaos by Boltzmann. It is expressed as bimodal Maxwellians each mode differing by plus/minus root-mean-squares of fluctuations in macroscopic variables. Chemical reaction rates calculated using the turbulent-equilibrium are obtained in a closed form, accounting for discrepancy between experiments and classical theory based on Arrhenius' law that underestimates the burning rate considerably. The key issue is the correct estimation of the high-energy tail of the distribution function that is minor in population, yet playing a major role for reactions with high activation energy. It is extremely sensitive to turbulence level in the temperature, causing slightly subcritical molecules to clear the potential barrier to cause reactions just like quantum particles undergoing tunnelling effect owing to the uncertainty principle. Variance of the fluctuating turbulent chemical reaction rate is also calculated, verifying that relative variance based on the turbulent equilibrium is low, whereas its classical counterpart (Arrhenius) is pathologically high. A closed set of equations governing reactive turbulent gases is presented on this sound basis.