A Deterministic Limit for Exothermic Runaway: Deriving the Critical Hot-Spot Radius via Diffusive Reactor Quenching

The prevention of thermal runaway in industrial chemical reactors relies heavily on classical thermal explosion theory, primarily utilizing Semenov and Frank-Kamenetskii empirical criteria. While these lumped-capacitance and steady-state models effectively estimate macroscopic stability margins, they fail to deterministically define the exact spatial boundary where a localized reactive hot-spot triggers a cascading transition into catastrophic exponential self-heating. This paper introduces a strict continuum framework for macroscopic chemical kinetic scaling. By modeling the reactor volume as a dynamic thermodynamic balance between the spatial capacity for advective cooling (jacket diffusivity) and the localized rate of Arrhenius heat generation, we derive a universal critical runaway radius (Rrunaway). We demonstrate that thermal runaway is not a random statistical threshold of bulk fluid temperature, but an exact deterministic limit where localized exothermic kinetics strictly overpower the advective thermal quenching capacity of the reactor infrastructure. We contrast this geometric invariant with traditional Damköhler probability matrices and propose a blueprint for Active Reactor Quenching (ARQ) using multipoint fiber-optic spatial telemetry.