What do these processes have in common

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What do these processes have in common? 1)
Hydrogen embrittlement of pressure vessels in
nuclear power plants 2) Flow of electrons
through conductors 3) Dispersion of pollutants
from smoke stacks 4) Transdermal drug
delivery 5) Influenza epidemics 6) Chemical
reactions 7) Absorption of oxygen into the
bloodstream
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They all depend on Diffusion (conduction)
What is diffusion? The transport of
material--atoms or molecules--by random
motion What is conduction? The transport of
heat or electrons by random motion.
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Place a drop of ink into a glass of water. What
happens? Brownian motion causes the ink
particles to move erratically in all directions.
A concentration of ink particles will disperse.
This is NOT diffusion. How can You tell?
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Why does random motion cause spreading of a
concentration of particles? Because there are
more ways for the particles to drift apart than
there are for the particles to drift closer
together. DIFUS.HTM
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In one dimension. . .
Pa
Pb
Nj
Nj-1
Nj1
j
j-1
j1
Net change of particles in box j per time step
is ?Nj Nj-1 Pa - Nj Pa Nj1Pb - Nj Pb
? is a change in time d is a change in space
?Nj (Nj1- Nj) Pb - (Nj - Nj-1)Pa Let
dNa Nj - Nj-1 dNb Nj1 -
Nj Then ?Nj Pb dNb - Pa dNa d(PdN)
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If the Ps are constant, i.e., the probabilities
are the same from box to box, then ?Nj P
d2(N)
In three dimensions and in the continuous limit,
this equation becomes the diffusion equation
where C is concentration and ? is the diffusivity
of the medium.
or, C is temperature and ? is thermal conductivity
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Consider diffusion in only one dimension. Then we
have
Consider now the condition of steady-state,
i.e., concentration C no longer changes with
time. Then,
This can be integrated to
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?1
What can one learn from this equation? Heres a
heat-conducting bar with a fixed temperature C at
each end C(t,0)0 C(t,100)100. 2k1 k2 .
?2
X0
X100
C(t,0)0
C(t,100)100
At steady-state
Therefore, the ratios of the temperature
gradients in each section must equal the inverse
ratios of the ks.
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Gradient transport
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Diffusion processes Diffusion-2D
Heat conduction Conduction-1D
Diffusion-limited aggregation
Setup ?golf ball 1.15 ?salt
water 1.13 Conc(sat) 1.20
Dsalt 1.4 x 10-5 cm2/sec Initial
condition Dry salt at bottom of
cylinder. Drop in ball. Add water. What
happens? How long does it take?