A Unified Analytical Solution for Heat Conduction in Non‐Basic Geometries Using a Generalized Shape Factor

Heat conduction within solid particles is a fundamental phenomenon governing the performance of diverse unit operations, including catalytic reactors, solid fuel combustion, and ore processing. This study advances beyond steady‑state shape‑factor methods to the first unified transient framework. We propose that the transient conduction behavior of any convex geometry can be mapped onto a generalized one‑dimensional coordinate system through a unique shape factor ( F p ), defined by the particle characteristic volume‑to‑surface ratio. By solving the resulting generalized differential equation, we derive a unified exact solution expressed in terms of Bessel functions of fractional order, where the order is directly determined by F p . The validity of this approach is rigorously verified against high‑precision numerical solutions and exact analytical benchmarks for a wide spectrum of geometries, including finite cylinders, hollow cylinders, cones, and parallelepipeds. In all cases, the maximum deviation in the calculated dimensionless volumetric average temperature is less than 0.07. This work provides a mathematically robust and computationally efficient framework for predicting the thermal evolution of arbitrary solid shapes, offering significant utility for chemical reactor design and process optimization.