Theory of scaling for fluctuations in thermal explosion conditions

Abstract

AbstractWe demonstrate that fluctuations in the transient bimodality regime for gaseous thermal explosions scale with the inverse of an effective size parameter Ω. The ratio between this parameter and the volume V is obtained analytically for a nonturbulent heat conduction regime. We show that as the system departs from the laminar flow limit towards a turbulent regime, the fluctuations are enhanced since Ω decreases. We also prove that as V tends to zero, the ratio Ω/V tends to one, an intuitively expected result. The theory is inspired and confirmed by experiments on the decomposition of ethyl azide. Within the induction period, we use a corrected parabolic approximation for the potential determining the drift term in the Fokker‐Planck equation. The maximum occurs at the ignition temperature. The correction arises from size effects (Ω/V ≠ 1) due to an inhomogeneous distribution of temperature. The parabolic approximation is valid only within the induction period, until the fluctuations have macroscopic consequences. Our study reveals that the following anomalies bear a common origin that can be explained quantitatively if the correction is introduced: a) Discrepancy between the experimental and theoretical value for the Frank‐Kamenetsky constant. Early work by O.K. Rice on the gaseous thermal explosion of ethyl azide indicates that this discrepancy is enhanced when the size of the vessel increases. — b) Reduction of the induction period, or, equivalently, contraction of the time‐scale for the transient bimodality during which heat accumulates in the system. This anomaly is accompanied by an enhancement of the growth rate for the Gaussian width of the probability density distribution for the temperature due to fluctuations. This phenomenon will be interpreted by the observer as a premature ignition.