Diffusion
What is Engineering
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
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.
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. DIFUS.HTM
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.
We can also explain the spreading of a concentration
by entropy.
The second law of thermodynamics says that systems tend
towards maximum entropy – or maximum disorder.
Area of high concentration and low/zero concentration is an
ordered state and the mixed state is the disordered state!
Other examples?
Why do metal cooking spoons have plastic handles?
Other examples?
What happens if someone across the room sprays perfume?
Perfume diffusion simulation
After adding milk and
sugar, why do we stir our
coffee?
Diffusion is slow!
Agitation (or stirring) can move fluids much larger distances in the same amount
of time, which can accelerate the diffusion process.
Values for Diffusivity D
Temperature
(°C)
CO2-N20
0
(gas)
Ar-O2
20
Ethanol(5%)-Water
25
(liquid)
Water(13%)-Butanol
30
H2-Ni
85
(solid)
Al-Cu
20
Greater the diffusivity, greater the flux!
Diffusivity
(cm2/s)
0.096
0.2
1.13E-05
1.24E-05
1.16E-08
1.30E-30
In each of these examples, molecules
(or heat) are moving down a gradient!
(From an area of high concentration to an area of low concentration)
Fick’s Law:
Ji is called the flux. It has units of
dci
Ji  D
dz
amount of material diffused
(l 2 )(t )
l2
D is called the diffusion coefficient. It has units of
t
Do our definitions of flux make sense?
N2
mass
 amount of gas removed  J 

(carbon dioxide flux)  
time  length 2
 time  area capillary 
• If double area of capillary, expect the amount of gas
transported to double.
• Want flux independent of apparatus – normalize by area.
 carbon dioxide concentration difference 

(carbon dioxide flux)  
capillary
length


Ji  D
CO2
(constant T & P)
dci
dx
• Flux is proportional to the concentration gradient –
steeper the gradient, more material transported.
• Flux is inversely proportional to capillary length –
increasing the distance to travel will decrease the flux.
Steady diffusion across a thin film
Now let’s use our diffusion equation to predict the concentration profile
of a material diffusing across a thin film!
Thin film
ci,0
ci,l
Well-mixed dilute
solution with
concentration ci,0
l
Well-mixed dilute
solution with
concentration ci,l
If we are at steady-state (the concentration profile has no time dependence, or in other words,
there is no accumulation of i in the film), we have a linear concentration profile.
Concentration-dependent diffusion
Consider two neighboring thin
films with a separation at ci,c:
ci,0
D1
D2
ci,c
ci,l
z=0
Which diffusivity is greater? How do you know?
z=zc
z=l
Unsteady state diffusion
Back to a drop of ink in a glass of water…
If consider diffusion in the z-direction only:
How does the concentration profile change with time?
t=0
(add ink drop – all ink
located at z = 0)
t
z
z=0
A measure of the spread due to diffusion is the diffusion length Ld = (4Dt)0.5,
where D is the diffusivity coefficient and t is time. Note: for small time,
spreading is quick, but for long times it slows down. That’s why you
stir your coffee after adding cream. Diffusion doesn’t work fast enough
over long distances.
Heat Transfer
Occurs by three means:
1.
2.
3.
Conduction:
•
Occurs between two static objects
•
Heat flows from the hotter to the cooler object
•
For example, holding a cup of hot coffee
Convection:
•
Transport of heat via a fluid medium
•
Currents caused by hot air rising, fan circulating air
Radiation:
•
Transport of energy as electromagnetic waves; the
receiving body absorbs the waves and is warmed
•
For example, warmth of a fire
Heat moves down a temperature gradient!
(From an area of high temperature to an area of low temperature)
Fourier’s Law:
dT
q z  k
dz
energy
qz is called the heat flux. It has units of
(l 2 )(t )
energy
k is called the thermal conductivity. It has units of
(l )(t )(T )
k
α is called the thermal diffusivity. It is defined as
(  )(Cˆ p )
and has units of
l2
t
Thermal Conductivity Values
(gas)
(liquid)
(solid)
H2
O2
Benzene
Water
Steel
Wood
T
(°C)
27
27
23
60
100
--
k
(cal/cm s C)
4.23E-04
6.35E-05
3.78E-04
1.56E-03
9.08E-01
9.00E-05
Greater the thermal conductivity, greater the heat flux!
Heat Conduction
Consider a two-paneled door:
TH
Tc
z
wood
metal
What will the steady-state temperature profile look like? Why?
kmetal > kwood
Here’s a heat-conducting bar with a fixed temperature T at each end:
T(t,0)=0; T(t,100)=100. 2k1 = k2 .
κ1
κ2
z=0
z=100
T(t,0)=0
T(t,100)=100
At steady-state:
dC
dC
k1
 const .  k 2
dzin k1
dzin k2
(Constant flux)
Therefore, the ratios of the temperature gradients in each section
must equal the inverse ratios of the k’s.
Gradient transport summary
1. Momentum transfer—Newton’s Law
flux of x-momentum in z direction
t zx  
m d (v x  )
, vx is velocity
 dz
in x-direction,  is density, m is viscosity.
2. Heat transfer—Fourier’s Law
d ( c pT )
qz
 a
heat flux in z-direction
; a is thermal diffusivity,
A
dz
 is density, cp is heat capacity, T is thermal energy (heat).
3. Mass transfer—Fick’s Law
dc A
mass flux of A in z-direction J A z
; D is molecular
dz
diffusivity of A in B, CA is the concentration of A.
  D AB
Diffusion processes
Heat conduction
Diffusion-limited aggregation