Diffusion: Lecture 1

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Diffusion Lecture 1

• Physics and applications

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From EM fields to diffusion

• Fields will be now be of temperature or
  concentration (not EM).
• Fields will be typically be out of equilibrium.
• We will study the evolution of the fields toward
  steady state.
• Equations and vector calculus are our main
  connections to the past.
• Diffusion will provide a first order system to
  put us on track to study waves which are second
  order (First semester op-amps, hot-wire,
  oscillator)

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The diffusion equation

• Mass (Mass diffusivity - Length2/time)
• How should we mix fuel and air in an engine to
  reduce pollution?
• How can gases change the properties of materials
  (semiconductors or carburizing steel) ?
• How long does it take to smell your feet?
• Heat (thermal diffusivity - Length2/time)
• How long do we cook a steak to get medium rare?
• How much heat is wasted out the window?
• How do we keep the Pentium cool?
• Momentum (viscosity - Length2/time)
• How do we predict the drag on an airplane?
• Why do golf balls have dimples?
• How do sperm swim?

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Questions

• What is diffusion?
• the process whereby particles of liquids, gases,
  or solids intermingle as the result of their
  spontaneous movement caused by thermal agitation
  (from Webster)
• If there were only molecular diffusion, how long
  would it take to smell your feet after removing
  your sock?
• 1 year!

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So why do we smell our feet?
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Mass diffusion in gases

• Easy to understand gas molecules move and
  collide with neighbors resulting in randomness.
• Mean free path is 0.1 microns in standard air.
• In air molecules undergoes about 1010 collisions
  per second.
• Kinetic theory can predict everything quite well.
• However, convection dominates most of our
  everyday experience in gases.

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Particle view of diffusion
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Assume some fluid flow
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No molecular diffusion
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Diffusion is 1/100 of convection
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Diffusion is 1/20 of convection
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Without convection, mixing in liquids is slow
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Convective diffusive behavior
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Mass Diffusion Coefficients

• Time to move distance L t L2/D
• O2 in N2 D 0.18 cm2/s (0.42 cm/sec)
• Caffeine in H2O D 630 µm2/s (25 micron/sec)
• H2 in Fe D 0.25 µm2/s (0.5 micron/sec)
• Al in Cu D 0.5 angstroms2/million years

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Return of the glob of particles
How fast does it spread? How do we measure?
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Mean distance traveled
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Histogram of x location
numbers
-3 -2 -1 0 1
2 3
location
How does the std. vary with time ?
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Particle location histogram on semi-log plot
Probability
location
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Diffusion of momentum in a fluid
How does the boundary layer thickness change with
time?
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Heat conduction in solids

• Lattice vibrations in non-metals.
• Electrons participate in metals similar
  mechanism as electrical conductivity.
• No convection pure diffusion problem.
• Can be understood from continuum formulation.

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Thermal diffusivity

• Aluminum a 0.9 cm2/s
• Iron a 0.12 cm2/s
• Air a 0.18 cm2/s
• Water a 0.0017 cm2/s
• Brick a 0.005 cm2/s
• Glass a 0.005 cm2/s

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Cooking a steak - medium
Constant temperature heat source
HOT!
Constant temperature heat source
HOT!
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How long do we cook in order to achieve the same
center temperature when the steak is two times
thicker?
Constant temperature heat source
Meat thermometer
Constant temperature heat source
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Experimental data
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Hot/Cold in contact
----------- Insulated------------------
----------- Insulated------------------
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Fouriers Law

• Empirical Law
• Derivable for gases

K thermal conductivity W/mK
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Heat flow
dT/dx
Distance
Temp. or temp. gradient
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Hot/cold (new condition)
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Heat flow
dT/dx
Distance
Temp. or temp. gradient
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Conservation of energy
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Heat equation