Transport Phenomena, 1. Foundations

Abstract

Abstract Transport phenomena is concerned with the analysis of processes involving the transport of mass, momentum, and energy. The focus of this article is the underlying foundations of transport phenomena, including the mathematical background, the basic laws of mechanics, as well as the mass and energy conservation laws. Cartesian tensor representation and analysis provides the mathematical framework and basic kinematic relations including the Reynolds Transport Theorem provide the relations for developing the basic equations of change from the governing fundamental laws, specifically, in the development of the equations of continuity, motion, and energy. In addition, the thermal and mechanical energy equations are developed and their relations to the equations of energy and motion are discussed. Further, the constitutive equations (or transport flux relations) governing simple and complex momentum and conduction heat transfer in materials are described and integrated with the basic equations of change. Finally, an overview of the solution framework and approaches for analyzing transport phenomena problems is summarized and discussed. The article contains sections titled: 1. Introduction 2. Mathematical Preliminaries 2.1. Coordinate Systems 2.2. Vector and Tensor Operations 2.3 Vector and Tensor Representations – Curvilinear Coordinates 2.4. The Jacobian 2.5. Calculus of Vectors and Tensors 2.6. Divergence Theorem 2.7. Kinematic Relations 2.8. Partial and Total Derivatives 2.9. Relation Between Different Time Derivatives 3. Basic Equations for Compositionally Homogeneous Systems 3.1. The Reynolds Transport Theorem 3.2. Conservation of Total Mass 3.3. Conservation of Linear Momentum 3.4 Conservation of Angular Momentum 3.5 Mechanical Energy Equation 3.6 Conservation of Total Energy 3.7 Thermal Energy Equation 3.8 Forms of the Governing Equations 3.9 Entropy Inequality 3.10 Linear Transport Fluxes and Relations 3.11 Non‐Newtonian Fluids 3.12 Summary of Basic Equations 3.13 Boundary Conditions 3.14 Solution Philosophy 4 Dimensionless Equations of Change