A HYBRID S N -P N FORMULATION FOR SOLUTION OF THE BOLTZMANN TRANSPORT EQUATION FOR PHONONS

ABSTRACT A generalized form of the Ballistic-Diffusive Equations (BDE) for approximate solution of the Boltzmann Transport Equation (BTE) for phonons is formulated. The formulation presented here is new and general in the sense that, unlike previously published formulations of the BDE, it does not require a priori knowledge of the specific heat capacity of the material. Furthermore, it does not introduce artifacts such as media and ballistic temperatures. As a consequence, the boundary conditions have clear physical meaning. In formulating the BDE, the phonon intensity is split into two components: ballistic and diffusive. The ballistic component is traditionally determined using a viewfactor formulation, while the diffusive component is solved by invoking spherical harmonics expansions. Use of the viewfactor approach for the ballistic component is prohibitive for complex large-scale geometries. Instead, in this work, the ballistic equation is solved using two different established methods that are appropriate for use in complex geometries, namely the discrete ordinates method (DOM), and the control angle discrete ordinates method (CADOM). Results of each method for solving the BDE are compared against benchmark Monte Carlo results, as well as solutions of the BTE using standalone DOM and CADOM for a two-dimensional transient heat conduction problem at various Knudsen numbers. It is found that standalone CADOM (for BTE) and hybrid CADOM-P 1 (for BDE) yield the best accuracy. The hybrid CADOM-P 1 is found to be the best method in terms of computational efficiency. NOMENCLATUR

Isomorphism Theorems on Quasi Module

A quasimodel is an algebraic axiomatisation of the hyperspace structure based on a module. We initiated this structure in our paper [2]. It is a generalisation of the module structure in the sense that every module can be embedded into a quasi module and every quasi module contains a module. The structure a quasimodel is a conglomeration of a commutative semigroup with an external ring multiplication and a compatible partial order. In the entire structure partial order has an intrinsic effect and plays a key role in any development of the theory of quasi module. In the present paper we have discussed order-morphism which is a morphism like concept. Also with the help of the quotient structure of a quasi module by means of a suitable compatible congruence, we have proved order-isomorphism theorem

First Principles Modeling of Phonon Heat Conduction in Nanoscale Crystalline Structures

The inability to remove heat efficiently is currently one of the stumbling blocks toward further miniaturization and advancement of electronic, optoelectronic, and micro-electro-mechanical devices. In order to formulate better heat removal strategies and designs, it is first necessary to understand the fundamental mechanisms of heat transport in semiconductor thin films. Modeling techniques, based on first principles, can play the crucial role of filling gaps in our understanding by revealing information that experiments are incapable of. Heat conduction in crystalline semiconductor films occurs by lattice vibrations that result in the propagation of quanta of energy called phonons. If the mean free path of the traveling phonons is larger than the film thickness, thermodynamic equilibrium ceases to exist, and thus, the Fourier law of heat conduction is invalid. In this scenario, bulk thermal conductivity values, which are experimentally determined by inversion of the Fourier law itself, cannot be used for analysis. The Boltzmann Transport Equation (BTE) is a powerful tool to treat non-equilibrium heat transport in thin films. The BTE describes the evolution of the number density (or energy) distribution for phonons as a result of transport (or drift) and inter-phonon collisions. Drift causes the phonon energy distribution to deviate from equilibrium, while collisions tend to restore equilibrium. Prior to solution of the BTE, it is necessary to compute the lifetimes (or scattering rates) for phonons of all wave-vector and polarization. The lifetime of a phonon is the net result of its collisions with other phonons, which in turn is governed by the conservation of energy and momentum during the underlying collision processes. This research project contributed to the state-of-the-art in two ways: (1) by developing and demonstrating a calibration-free simple methodology to compute intrinsic phonon scattering (Normal and Umklapp processes) time scales with the inclusion of optical phonons, and (2) by developing a suite of numerical algorithms for solution of the BTE for phonons. The suite of numerical algorithms includes Monte Carlo techniques and deterministic techniques based on the Discrete Ordinates Method and the Ballistic-Diffusive approximation of the BTE. These methods were applied to calculation of thermal conductivity of silicon thin films, and to simulate heat conduction in multi-dimensional structures. In addition, thermal transport in silicon nanowires was investigated using two different first principles methods. One was to apply the Green-Kubo formulation to an equilibrium system. The other was to use Non-Equilibrium Molecular Dynamics (NEMD). Results of MD simulations showed that the nanowire cross-sectional shape and size significantly affects the thermal conductivity, as has been found experimentally. In summary, the project clarified the role of various phonon modes - in particular, optical phonon - in non-equilibrium transport in silicon. It laid the foundation for the solution of the BTE in complex three-dimensional structures using deterministic techniques, paving the way for the development of robust numerical tools that could be coupled to existing device simulation tools to enable coupled electro-thermal modeling of practical electronic/optoelectronic devices. Finally, it shed light on why the thermal conductivity of silicon nanowires is so sensitive to its cross-sectional shape

Faster-Than-Real-Time Simulation of Lithium Ion Batteries with Full Spatial and Temporal Resolution

A one-dimensional coupled electrochemical-thermal model of a lithium ion battery with full temporal and normal-to-electrode spatial resolution is presented. Only a single pair of electrodes is considered in the model. It is shown that simulation of a lithium ion battery with the inclusion of detailed transport phenomena and electrochemistry is possible with faster-than-real-time compute times. The governing conservation equations of mass, charge, and energy are discretized using the finite volume method and solved using an iterative procedure. The model is first successfully validated against experimental data for both charge and discharge processes in a LixC6-LiyMn2O4 battery. Finally, it is demonstrated for an arbitrary rapidly changing transient load typical of a hybrid electric vehicle drive cycle. The model is able to predict the cell voltage of a 15-minute drive cycle in less than 12 seconds of compute time on a laptop with a 2.33 GHz Intel Pentium 4 processor