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We develop a solution to the two-temperature diffusion equation in axisymmetric cylindrical coordinates to model heat transport in thermoreflectance experiments. Our solution builds upon prior solutions that account for two-channel diffusion in each layer of an N-layered geometry, but adds the ability to deposit heat at any location within each layer. We use this solution to account for non-surface heating in the transducer layer of thermoreflectance experiments that challenge the timescales of electron-phonon coupling. A sensitivity analysis is performed to identify important parameters in the solution and to establish a guideline for when to use the two-temperature model to interpret thermoreflectance data. We then fit broadband frequency domain thermoreflectance (BB-FDTR) measurements of SiO2 and platinum at a temperature of 300 K with our two-temperature solution to parameterize the gold/chromium transducer layer. We then refit BB-FDTR measurements of silicon and find that accounting for non-equilibrium between electrons and phonons in the gold layer does lessen the previously observed heating frequency dependence reported in Regner et al. [Nat. Commun. 4, 1640 (2013)] but does not completely eliminate it. We perform BB-FDTR experiments on silicon with an aluminum transducer and find limited heating frequency dependence, in agreement with time domain thermoreflectance results. We hypothesize that the discrepancy between thermoreflectance measurements with different transducers results in part from spectrally dependent phonon transmission at the transducer/silicon interface.

Composites made with high thermal conductivity meshes embedded in phase change materials (PCMs) increase charge/discharge rates of latent heat energy storage systems. We study the benefits of spatially-dependent enhancements to thermal conductivity on the charge/discharge rates of PCMs in both one-dimensional Cartesian and one-dimensional cylindrical coordinates. Our nondimensionalized quasi-steady (Stefan number <∼ 0.1) solution indicates that the average charge (discharge) rate in a spatially-enhanced PCM outperforms the uniformly-enhanced case by maximizing the enhancement near the heat source and therein reducing the time averaged thermal resistance to melting (solidifying). Relative to a uniformly-enhanced thermal conductivity, the optimal charge/discharge rate enhancement is a modest 12% in one-dimensional Cartesian coordinates but as high as 140% in one-dimensional cylindrical coordinates. Our analytical solutions are a design guide for graded mesh structures that can be realized by advanced fabrication techniques such as additive manufacturing and applied in applications ranging from telecommunications to buildings, where PCMs are employed to harness rapidly varying energy sources.

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