Quantifying Thermal Transport in Three-Dimensional Printed Fractal Structures

Heat transport through three-dimensional printed fractal media is investigated by comparing a fractal diffusion model to infrared measurements using Bayesian uncertainty quantification. The delayed rejection adaptive metropolis (DRAM) algorithm, based on the Markov Chain Monte Carlo (MCMC) sampling technique, is used to infer parameter uncertainty, quantify parameter correlation, and compute error propagation of the temperature distributions. The results demonstrate that fractal operators improve modeling thermal transport through complex fractal structures and help understand fractal structure–property relationships. For example, correlations among fractal spatial and temporal scaling parameters, diffusion coefficients, and fractal dimensions are quantified. We find a scaling relationship between the diffusion coefficient D and the temporal fractal time derivative order $α$ that scales nominally as $D∝e−α$ based on constraints from the second law of thermodynamics. The results have implications for building a stronger understanding of heat transport in complex materials beyond random media and models based on Gaussian probability homogenization.