Proceedings of Proceedings of the ASME 2012 Summer Heat Transfer Conference HT2012 July 8-12, 2012, Rio Grande, Puerto Rico HT2012-58347 A TRANSIENT MODIFIED FOURIER-BASED APPROACH FOR THERMAL TRANSPORT MODELLING IN SUB-CONTINUUM REGIME Vivek Mishra 1,2 , Aydin Nabovati 2 , Daniel P. Sellan 2 , and Cristina H. Amon 2 1 Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, West Bengal, 721302, India 2 Department of Mechanical & Industrial Engineering, University of Toronto, Toronto, Ontario, M5S 3G8, Canada Email: vivek.mech.iitkgp@gmail.com, a.nabovati@utoronto.ca, dan.sellan@utoronto.ca and cristina.amon@utoronto.ca ABSTRACT The presence of sub-continuum effects in nano-scale sys- tems, 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 de- veloping 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 or- der to capture the steady-state temperature jumps at the system boundaries, and (ii) a size-dependent effective thermal diffusiv- ity, 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. INTRODUCTION Sub-continuum thermal effects are observed when the char- acteristic size of a system becomes comparable to the mean free path of energy carriers [1, 2]. In this work, we focus on semi- conductor materials, where the main energy carriers are atomic lattice vibrations, quantum of which is called a phonon. At small length scales, ballistic transport of phonons occurs, i.e., phonons can travel through the material without scattering. As a result of the ballistic transport of phonons through the material, phonon scattering at the boundaries becomes significant. The boundary effects impact the thermal transport, and are referred to as sub- continuum effects in the literature. The conventional continuum- based physical relations that describe heat and fluid flow in bulk systems are incapable of capturing these sub-continuum effects. An in-depth understanding of thermal transport in small scales is required for proper thermal management of nano- structured devices. Several atomistic-level approaches have been developed [3, 4, 5, 6, 7] that can well predict the thermal transport in such scales. Atomistic-level modelling techniques for ther- mal transport prediction, however, are complex, computation- ally intensive, and cannot be integrated into existing commercial softwares. These complications can potentially be eliminated by modifying Fourier-based approaches. Previous works on this ap- proach have used solutions to the Boltzmann transport equation (BTE), which considers phonon-level transport, to better under- stand sub-continuum thermal transport, and appropriately mod- ify the Fourier-based heat equation [11, 12, 13, 14]. In this work, we extend a methodology previously devel- oped by Nabovati et al. [14], where the phonon lattice Boltz- mann method (LBM) is used to solve the BTE and predict tran- sient thermal transport in the sub-continuum regime [8, 9, 10]. The LBM is a discretized representation of the BTE, where the phonon are assumed to be particles following the Bose-Einstein statistics. Here, the LBM with the Gray approximation is used for small scale simulations of thermal transport [10]. In the Gray approximation, the phonon transport is simplified by using only one representative average phonon mode. Thus, average phonon properties (e.g., frequency, group velocity, and relaxation time) are used for modelling phonon transport. 1 Copyright c  2012 by ASME