The thermal impedance integral and the pathway decomposition of supercritical turbulent heat transfer. Part I. Framework

Turbulent heat transfer to fluids at supercritical pressure is dominated by a thin zone of strongly varying properties, the pseudocritical barrier, which sweeps radially through the boundary layer as the bulk temperature rises along the pipe.
This changes the character of wall-to-bulk transport: theconstant-property Lyon integral, a thermal resistance, becomes a thermal impedance, distinguished by capacitive storage at the specific-heat peak, history dependence through the moving barrier, and response lag of the perpetually developing thermal profile. The generalised Lyon integral is derived from the Favre-averaged energy equation without the classical weak-property assumptions. First, the barrier sweep resets the fully developed equilibrium underlying the classical $\mathrm{Nu}\sim\mathrm{Re}/\ln\mathrm{Re}$ scaling, replacing it with the penetration-depth scaling $\mathrm{Nu}\sim\sqrt{\mathrm{Re}}$ in the barrier-dominated
regime. Second, the Prandtl and density correction ratios of conventional Nusselt correlations are traced to the molecular and turbulent parts of the impedance integrand, while the integrated specific-heat ratio arises outside the integral, in the enthalpy-to-temperature conversion, and is an eliminable artefact. ... mehrTen classical correlations map onto this classification. Entanglement of the artefact with the Prandtl correction through the modified Prandtl number produces spurious behaviour near the pseudocritical point. Separating the impedance integral at the pseudocritical barrier yields wall-side and bulk-side pathways, each characterised by its own thermodynamic state; a two-point quadrature gives a linearised pathway correlation. It depends only on the bulk Reynolds and Prandtl numbers and a bounded wall-to-bulk Prandtl-number ratio, is smooth across the pseudocritical transition, and carries no fitted parameter. Experimental validation is provided in the companion paper.