Analytical Relationships Among Thermal Penetration Depth Definitions in Heat Conduction

Thermal penetration depth is a widely used concept in heat-transfer analysis for characterizing the spatial and temporal extent of thermal influence. Over time, multiple penetration-depth definitions have emerged throughout the heat-transfer literature in response to differing analytical objectives and application domains, complicating comparisons among reported results. This analysis examines the principal penetration-depth formulations reported in the literature, develops a unified taxonomy comprising diffusion-length, thermal-front, threshold-isotherm, attenuation-based, penetration-time, and resistance-corrected definition families, and derives analytical relationships among them. The results indicate that many commonly used formulations may be interpreted as criterion-dependent representations of a common diffusion-based transport scale. The analysis further suggests that apparent discrepancies among reported penetration depths arise primarily from differences in extraction criteria rather than differences in the underlying transport physics. By clarifying the relationships among definitions that have evolved across diverse heat-transfer applications, the resulting framework provides a unified perspective for interpreting, comparing, and communicating penetration-depth concepts throughout the heat-transfer literature.