The presence of sub-continuum effects in nano-scale systems, including size and boundary effects, causes the continuum-level relations (e.g., Fourier heat equation) to break down at such scales. The thermal sub-continuum effects are manifested as a temperature jump at the system boundaries and a reduced heat flux across the system. In this work, we reproduce transient and steady-state results of Gray lattice Boltzmann simulations by developing a one-dimensional, transient, modified Fourier-based approach. The proposed methodology introduces the following two modifications into the Fourier heat equation: (i) an increase in the sample length by a fixed length at the two ends, in order to capture the steady-state temperature jumps at the system boundaries, and (ii) a size-dependent effective thermal diffusivity, to recover the transient temperature profiles and heat flux values. The predicted temperature and heat flux values from the proposed modified Fourier approach are in good agreement with those predicted by the Gray lattice Boltzmann simulations.

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