A Deterministic Limit for the Boiling Crisis: Deriving the Critical Vapor Patch Radius via Multi-Phase Hydrodynamic Quenching

The prediction of the Critical Heat Flux (CHF) and the Departure from Nucleate Boiling (DNB) remains heavily reliant on empirical correlations, most notably the classical Zuber hydrodynamic instability model. While these statistical thermodynamics models effectively estimate macro-scale heat dissipation limits, they fail to deterministically define the exact spatial boundary where a localized vapor patch triggers a cascading transition into catastrophic film boiling. This paper introduces a strict continuum framework for multi-phase micro-scale scaling. By modeling the heated boundary layer as a dynamic balance between macroscopic coolant quenching (re-wetting diffusivity) and localized vapor generation at nucleation sites, we derive a universal critical boiling radius (RDNB). We demonstrate that the Boiling Crisis is not a statistical anomaly of high temperatures, but an exact deterministic limit where localized vapor generation strictly overpowers the advective hydrodynamic quenching capacity of the surrounding liquid. We contrast this geometric invariant with traditional CHF probability matrices and propose a blueprint for Active Phase-Change Monitoring (APM) using high-frequency acoustic telemetry.