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Topics Discussed: Conservation laws and local conservation laws;
continuum dynamics; dimensional analysis; dispersion; dissipation;
diffusion; Brownian motion; fluid flow, supersonic flow and shock
waves; viscosity and heat transfer; turbulence and mixing; radiation
flow; thermodynamic laws and their applications in chemistry and physics; thermodynamic
equilibria; chemical equilibria; chemical reactions, reaction kinetics, activation
thresholds, high explosives, gas-solid reactions, Navier Stokes equations,
Euler equations, heat equations; diffusion equations; wave equations;
greenhouse effect; radioactive waste disposal and transport phenomena;
carbon sequestration; carbon cycle, nitrogen cycle; stratospheric
ozone; acid rain; chemical processing, solid gas reactions; air mixing,
.pdf files can be found at the bottom of the syllabus.
| 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. |
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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. |
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| 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. |
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| 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 � |
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| 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. |
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| 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 |
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| Navier Stokes equation |
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| 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 |
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| 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 |
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| 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, |
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| 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 |
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| 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 |
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| Multiphase Flow, multicompositional
flow |
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| Fluidized Beds |
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Separation
of Solid and Gases afterwards, the concepts of fluidization, |
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| 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 |
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| Mixing |
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| Heat Exchangers |
laminar and turbulent flow, phase
changes, boundary layers, diffusion equations, turbulent vs. laminar
diffusions, eddy diffusion |
laminar, turbulent flow, phase
change |
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| Rotational Flows |
Accelerating
Coordinate Frames, Coriolis Forces, Ekmann Layers, |
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A rocket
design - |
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| Ocean Flows |
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The Greenhouse Effect |
Radiation balance, heat balance,
water cycle, solar effects direct and indirect |
Radiation Balance,
water cycle, solar effects direct and indirect |
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| Hydrogen Production |
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Gasification, electrochemistry |
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| Fuel Cells, Transport Issues |
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| Dimensional Groups |
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Reynolds numbers, other numbers
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| Separation Phenomena |
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| Catalysts |
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Refining, Stratosphere,
radiation influence |
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| Carbon dioxide and methane hydrates |
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Formation, stability, uations
of state |
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| Explosives |
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Detonation Waves,
energy balances |
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| Boundary Layers |
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Ocean, Aerodynamics,
heat exchangers, ion exchanger |
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| Steel making |
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| Refineries |
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| Cooling Tower |
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| Supernovae |
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pdf's
Conservation of Energy
Chapter 3: The Concept of Viscocity
Chapter 5: The Differential Equations of Flow
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