Course Background

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Title Topics Covered Summary Examples Outline of Problem Set
Introduction Scope of Transport Problems, Transport and its relationship to engineering and science, range of fields. The connection to computer science, and to experimental science.� The relationship to theory� The connection to conservation laws.� Necessary background, mathematics, physics, chemistry, thermodynamics. The goal of the course, relation to numerical modeling. Transport equations are the consequence of local conservation laws.� A wide variety of phenomena are described by transport equations, transport of materials, energy, momentum, entropy.� Materials can be broken up into subspecies which are separately conserved.� Similar rules apply for energy, which can take on the forms of heat energy, chemical energy, radiative energy,or kinetic and potential energy.� In addition to conservation laws, one will have sources and sinks, and interactions between different species to consider.�� Hydrodynamics, gas dynamics, plasma physics, magnetohydrodynamics, chemical reactions all are governed by transport laws.�� Engineered machines, from car engines to the aerodynamics of an airplane follow from transport equations.�� Natural phenomena from ocean currents, to atmospheric chemistry, from porous flow to lightning follow the same rules.� The purpose of the class is to highlight these issues develop an understanding of the underlying science and develop the ability to apply these ideas in a wide variety of circumstances.   Mathematics to be discussed:
Test prior knowledge, with multiple choice fast quiz.
Concepts to be tested include Basic Math
(complex numbers, elementary functions, calculus
(derivative, second derivative, chain rule,� integrals) ,
ODE's� radioactive decay, harmonic oscillator,�
Multidimensional calculus and vector calculus
(partical derivative, gradient, divergence, curl, potential field)
Operator knowledge Laplace operator, Nabla, tensors),
Linear Algebra, (Matrix, Vector, conjugate, orthonormal, trace)�
PDEs (equation types, wave equation, hydroequation, Schroedinger Equation,)�
Numerical tools and algorithms, Linear Solvers (Conjugate Gradient, ... )��
Global Conservation Laws Mass, Energy, Momentum, Electric Charge, Other Charges, Forms of Energy,� Heat, The First Law of Thermodynamics.� Chemical reactions, as a sidebar, we will introduce units of mass, energy, momentum and currents,� discuss the MKSA system and its relationship to other unit systems. At the core of the physical sciences are conservation laws.� In spite of the complex dynamics of a system, certain quantities are not changed, if the systems scope is defined sufficiently large.� We will discuss examples from physics, and show the power of these laws for constraining engineering designs, and the dynamics of physical systems.� We will emphasize the distinction between open and closed systems. Billiard Balls, Combustion Reactions, Hydrogen Economy, Pendulum, Work done by pressure, heat energy in a gas, microscopic picture, ideal gas Physics Background Test,� + First problems to be solved.�
Elastic Collisions and scattering.� Energy content of chemicals,
High Explosives, Gasoline, Biomass,� Coal, Hydrogen, nuclear fuels,
provide standard, statements about energy content and translate
them onto a normalized basis, kg, m3, mole.� Translate eV, cal, kJ, on a molar basis.		   
Local Conservation Laws The concept of space, of spatially distributed quantitites, extensive and intensive quantities.� Density formulations, currents, dynamical equations.� Taking the continuum limit. Transport of material, control volumes,. Euler and Lagrange Forumulations. The Navier Stokes equations, the Euler Equations. Space and time are at the root of describing the physical world.�� Rather than just considering the mass of an object, one needs to descibe the distribution of mass, i.e. density distribution.� In moving from a decription of objects that are described by coordinates, we move on to describing filling space with materials and other properties.� We will take the continuum limit, discuss its power and short comings�� From this we will derive the hydrodynamic equations.� Continuum Limit, Transport, sources and sinks, the hydrodynamic equations.� Simple applications, sound waves, material flow, Bernoulli equations.� Multi-Phase Flow concepts. Ideal gas, fluids, control volumes. Flow in a pipe.� Flow in a channel.� The principle of a rocket, energy momentum conservation, the rocket equation.�� Stephan Boltzmann Laws and the
temperature of the sun, the heat flow
at Earth, the temperature of the black
body Earth.� Correcting for Albedo.�
Water flow in a simplified Hudson.		  
Continuum Description - Extended Various energy densities, beyond kinetic energy, other densities and currents, entropy.�� Hydro-equations conserve entropy, extensive and intensive quantities, transporting extensive and extensive quantities. Transporting heat and entropy, what about temperature? Fluid flow equations conserve and transport entropy with mass.���� Chemical Energy transport �     
Dissipative Processes and Entropy Friction, heat dissipation, irreversible processes.�� Chemical entropy, relation to the number of states, hidden degrees of freedom, relation between entropy, heat capacity and temperature. Concept of Dissipation, Heat transfer, momentum transfer, diffusion.      
Breakdown of the hydroequations, shocks rarefaction and other discontinuous solutions Air motion in front of a cylinder, formation of a shock or a rarefaction, the Hugoniot jump conditions Shocks, rarefactions, the concept of waves, sound speed etc., ultrasonic planes, shock waves, dissipation, Hugoniot jump conditions      
Navier Stokes equation      
Porous Flow Equations Fluid flow in the non-inertial limit, momentum is not conserved, transport of materials, tracers vs. active participants.� Stochastic descriptions,� instabilities Porous flow equations describe the flow of fluids in porous media, there could be one or more different fluid phases involved.� Porous flows tend to be slow and they are dominated by viscous forces.� Flow rates are proportional to pressure gradients.� Porosity, and transfer and vectors are      
Mathematical Similarity Other equations with similar properties, bores, tidal bores, the water faucet, shallow water, equations, water surface equations, shallow water equations. Discussion of a Bore, tidal bores and the water faucet, radiation transport and discontinuities      
Different types of PDEs, hyperbolic, ellipitc, parabolic Wave equations, heat equations, Schroedinger's equation, characteristic equations, boundary conditions, completeness, uniqueness etc. Mathematical characterisation of various PDE's, boundary conditions hyperbolic vs. ellipitic,��      
Dissipation, Brownian Motion and the Second Law of Thermodynamics Viscosity & Brownian Motions, Breaking the continuum limit, the second law of thermodynamics, equations with built in dissipation, chemical irreversibility, chemical reactions and entropy      
Maxwell's Daemon and the second law applied to nanomachines Maxwell's Demon, Nanomachines, Stochastic Processes, Skirting the second law, muscles, ionic pumps in biological membranes      
Multiphase Flow, multicompositional flow      
Fluidized Beds   Separation of Solid and Gases afterwards, the concepts of fluidization,      
Viscosity, Turbulence, and Dissipation Reynolds numbers, stability issues, increased heat transfer, similarity solutions, models of turbulence, k-e model, Trefethen et. Al. Reynolds numbers, stability issues, increased heat transfer, similarity solutions      
Mixing      
Heat Exchangers laminar and turbulent flow, phase changes, boundary layers, diffusion equations, turbulent vs. laminar diffusions,� eddy diffusion laminar, turbulent flow, phase change      
Rotational Flows Accelerating Coordinate Frames,� Coriolis Forces, Ekmann Layers,   A rocket design -    
Ocean Flows        
The Greenhouse Effect Radiation balance, heat balance, water cycle, solar effects direct and indirect Radiation Balance,
water cycle, solar effects direct and indirect		  
     
Hydrogen Production   Gasification, electrochemistry      
Fuel Cells, Transport Issues      
Dimensional Groups  

Reynolds numbers, other numbers

      
Separation Phenomena      
Catalysts   Refining, Stratosphere,
radiation influence
 		    
Carbon dioxide and methane hydrates   Formation, stability, uations of state      
Explosives   Detonation Waves,
energy balances		      
Boundary Layers   Ocean, Aerodynamics,
heat exchangers, ion exchanger		      
Steel making      
Refineries      
Cooling Tower      
Supernovae