Download - What do these processes have in common? 1) Hydrogen embrittlement of pressure vessels in nuclear
Engineering Innovation--Engineering Innovation--DiffusionDiffusion
What is Engineering?
What do these processes have in common?
1) Hydrogen embrittlement of pressure vessels in nuclearpower 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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What is Engineering?
They all depend on
Diffusion (conduction)
What is diffusion? The transport of material--atomsor molecules--by random motion
What is conduction? The transport of heat or electronsby random motion.
Engineering Innovation--Engineering Innovation--DiffusionDiffusion
What is Engineering?
Place a drop of ink into a glass of water. What happens?
Brownian motion causes the ink particles to move erraticallyin all directions. A concentration of ink particles willdisperse.
This is NOT diffusion. How canYou tell?
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What is Engineering?
Why does random motion cause spreading of a concentrationof particles?
Because there are more ways for the particles to drift apartthan there are for the particles to drift closer together.
DIFUS.HTM
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What is Engineering?In one dimension. . .
NjNj-1 Nj+1
j-1 j j+1
Net change of particles in box j per time step is
ΔNj = Nj-1 Pa - Nj Pa +Nj+1Pb - Nj Pb
ΔNj = (Nj+1- Nj) Pb - (Nj - Nj-1)Pa
Let δNa = Nj - Nj-1 δNb = Nj+1 - Nj
Then ΔNj = Pb δNb - Pa δNa = δ(PδN)
Pa Pb
Δ is a change in timeδ is a change in space
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What is Engineering?If the P’s are constant, i.e., the probabilities are the samefrom box to box, then
ΔNj = P δ2(N)
In three dimensions and in the continuous limit, this equationbecomes the diffusion equation where C is concentrationand κ is the diffusivity of the medium.
Cz
C
y
C
x
C
t
C 22
2
2
2
2
2
or, C is temperature and κ is thermal conductivity
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What is Engineering?Consider diffusion in only one dimension. Then we have
2
2
x
C
t
C
Consider now the condition of “steady-state”, i.e., concentration C no longer changes with time. Then,
002
2
2
2
dx
Cd
x
C
t
C
This can be integrated to .constdx
dC
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What is Engineering?What can one learn from this equation?
Here’s a heat-conducting bar with a fixed temperature C at each end:C(t,0)=0; C(t,100)=100. 2k1 = k2 .
X=0 X=100C(t,0)=0 C(t,100)=100
κ1 κ2
At steady-state: 21
21 .kinkin dx
dCkconst
dx
dCk
Therefore, the ratios of the temperature gradients in each sectionmust equal the inverse ratios of the k’s.
Engineering Innovation--Engineering Innovation--DiffusionDiffusion
What is Engineering?
2. Heat transfer—Fourier’s Law
heat flux in z-direction q
A
d c T
dzz p
( ); is thermal
diffusivity, is density, cp is heat capacity, T is thermal energy(heat).
3. Mass transfer—Fick’s Law
mass flux of A in z-direction J Ddc
dzAz ABA ; D is molecular
diffusivity of A in B, CA is the concentration of A.
1. Momentum transfer—Newton’s Law
flux of x-momentum in z direction
zx
xdv
dz
( ), vx is velocity
in x-direction, is density, is viscosity.
Gradient transport
Engineering Innovation--Engineering Innovation--DiffusionDiffusion
What is Engineering?
Heat conductionConduction-1D
Diffusion processesDiffusion-2D
Diffusion-limited aggregation
Setup: ρgolf ball = 1.15 ρsalt 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?