How heat transfer coefficient definitions shape local thermal non-equilibrium in porous media: A Biot number perspective

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Abstract
The fluid–solid heat transfer coefficient in porous media is not uniquely defined. Its magnitude depends fundamentally on whether the solid temperature is treated as a phase-averaged bulk value or an interface-averaged surface value. Engineering Nusselt–Reynolds correlations assume negligible intra-particle thermal gradient with small particle Biot numbers (Bi << 1), making the solid temperature constant throughout the entire single grain. When this assumption breaks down, intra-grain thermal gradients develop and the two temperature definitions for the solid phase diverge. Yet, this distinction is rarely acknowledged when engineering correlations are imported into continuum-scale local thermal non-equilibrium formulations. To quantify this inconsistency, we combine granular-scale laboratory heat transport experiments with experimentally informed two-dimensional numerical simulations across 36 configurations. Grain diameters range from 5 to 30 mm and Darcy velocities from 3 to 23 m d$^{-1}$, yielding Reynolds numbers between 0.5 and 21.5. The low thermal conductivity of the glass beads (1 W~m$^{-1}$~K$^{-1}$) ensures non-negligible Biot numbers (Bi > 0.1), creating a controlled regime in which the two definitions of the solid temperature diverge. We set Bi = 0.1 as the grain-isothermal threshold, beyond which the lump capacitance method begins to cause significant error. All investigated configurations exceed the isothermal threshold of 0.1. When the two solid temperature definitions are compared, interface-averaged Nusselt numbers exceed phase-averaged values by factors of 3.6 to 17.5 and remain effectively time-invariant. However, phase-averaged coefficients exhibit pronounced transient decay toward smaller steady-state magnitudes, reflecting intra-grain conduction resistance. Existing empirical correlations show systematic bias within the configuration range with this coefficient definition, underestimating Nusselt numbers at Reynolds numbers below 7.5 and overestimating above it. Based on the results, we suggest the phase-averaged heat transfer coefficient for solid materials with Bi > 0.1, as it is consistent with the phase-averaged formulation in the model and captures the intra-grain thermal gradient. These findings demonstrate that physically consistent continuum-scale modelling requires explicit alignment between the solid temperature definition in the constitutive law and that in the governing energy equations. These results can be interpreted for groundwater applications, since the Biot-number framework provides a transferable quantitative diagnostic. Practitioners can evaluate Bi for their aquifer material and flow conditions: Bi < 0.1 validates engineering correlations; Bi > 0.1 signals that a phase-averaged coefficient is required. The grain-size thresholds reported here (15 mm for glass beads) represent order-of-magnitude estimates that would shift toward coarser sizes in 3D configurations due to reduced intra-grain conduction resistance. The exact 3D thresholds remain to be validated in future studies.

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