A Unified Heat Transfer Coefficient for Phase Change Time Prediction Across Biot Number Regimes

This paper introduces a novel, closed-form analytical model for predicting phase change times that effectively bridges the gap between lumped-capacitance and moving-boundary formulations. The model is valid across an extended Biot number range of 0.01≤Bi≤20.01≤Bi≤2, addressing the challenging intermediate regime where internal and external thermal resistances are comparable—a region where neither classical approximation applies and previous analytical methods fail beyond Bi<0.2Bi<0.2. At the core of the formulation is a physically derived global heat transfer coefficient U=heff/(1+ΦBi)U=heff/(1+ΦBi), obtained from first-principles thermal resistance analysis. This coefficient incorporates geometry-specific ΦΦ factors (Φ=1Φ=1 for planar, 1/21/2 for cylindrical, and 1/31/3 for spherical configurations), rigorously derived from steady-state solutions of Laplace’s equation with convective boundary conditions. Unlike empirical effective-coefficient approaches, these factors emerge from fundamental conduction theory and require no calibration. The model explicitly separates sensible and latent heat contributions, enabling both physical insight into the dominant thermal mechanisms and computational efficiency for rapid design-space exploration. Validation against high-resolution numerical solutions of the one-dimensional Stefan problem—using an enthalpy-based finite-difference scheme with mesh independence and convergence verification—demonstrates that the model maintains relative errors below 5% for Bi≤1Bi≤1 and below 15% for Bi=2Bi=2. This represents a substantial improvement over previous lumped-capacitance extensions (e.g., Liu et al., valid only for Bi<0.2Bi<0.2; Karwa et al., limited to Bi<0.5Bi<0.5). Comprehensive error decomposition identifies primary error sources (constant-UU assumption, constant-property approximation, and geometric curvature effects), with sensitivity analysis showing that model errors remain smaller than typical input parameter uncertainties (15–35%) in engineering practice. Correction factors for Bi>1Bi>1 and iterative schemes for temperature-dependent properties are provided to further enhance accuracy. The framework is further applied to water solidification with supercooling, successfully predicting the non-monotonic relationship between solidification time and ambient temperature, including a minimum near −12∘C−12∘C consistent with experimental observations (Müller et al., 2015). This is achieved through an effective phase change temperature Tf†Tf† derived from an energy balance during recalescence, with the parameter α≈0.7α≈0.7 obtained from specific-heat ratios—offering a more rigorous treatment than prior empirical correlations. Key strengths of the model include: A closed-form expression enabling rapid parameter sweeps, Monte Carlo uncertainty propagation, and multi-objective optimization without the computational expense of full numerical simulations. Applicability to multiple canonical geometries (plane wall, infinite cylinder, sphere) with validated ΦΦ factors, including guidance for non-ideal aspect ratios and extension to Bi≤5Bi≤5 with correction factors. Incorporation of supercooling effects via a thermodynamically grounded effective phase change temperature, validated against experimental droplet solidification data. Publicly available numerical codes, complete datasets, and detailed derivations to ensure full reproducibility and facilitate adoption by both researchers and practitioners. The work provides practical, ready-to-implement tools for the design and optimization of phase change material systems in thermal energy storage, cryopreservation, electronics cooling, and climate modeling. By unifying previously disparate regimes into a single, analytically tractable framework, the model fills a long-standing gap in phase change heat transfer and offers a robust foundation for future extensions to temperature-dependent properties, mushy zones, and convective melting.