Diffusion Relations
Many flow processes can be treated as diffusion problems. Chemical diffusion,
heat flow, allele spread in populations and others. All of these depend on a
gradient of some kind driving the motion of the diffusing quantity.
Quantity:
heat
chemical
alleles
population
Electric
charge
Gradient
T
concentration Allele frequency Population Electric
of
in population
density
potential
energy
A special case of chemical diffusion is of equal importance in both biology and
chemical engineering is osmosis. In this situation, a semi-permeable
membrane separates two volumes and diffusion must happen through the
membrane. The membrane changes the diffusion constants so that, for large
molecules which cannot pass through, the diffusion constant is zero. For other
molecules, diffusion is from the side with higher concentration to the side with
lower concentration, as usual.
From the particle model standpoint, these gradient driven processes can be
seen as the outcome of the laws of probability. As the particles jostle around
and collide, the most probable distribution of particles is to have them
spread essentially uniformly. Consider a situation with a high concentration
of material in one volume surrounded by lower concentration regions. As the
particles move about, there are many more particles in the concentrated region
to move away than there are in the low concentration regions to move in. Thus,
random motions are more likely to carry a particle away from an area of high
concentration than to carry one back towards high concentration. We will
return to this idea later on when we talk about entropy.
For our purposes, the general ideas of use are:
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Diffusion processes are driven by concentration gradients.
Probability explains why this happens without additional energy
input.
For a fixed amount of starting material, the concentration decays
exponentially toward the equilibrium value.
The flux of material (amount per area per second) is proportional to
the negative gradient of the concentration.
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Mathematical model
The basic starting point for the mathematical model is to model the flux or flow
of a quantity across an area due to its concentration gradient:
Flux = D A ∆C(q) where A is the area
∆x
The version of this to give the flux per unit area is known as Fick’s first law;
http://en.wikipedia.org/wiki/Fick%27s_law_of_diffusion that the flow or flux of the
quantity per unit area is proportional to the negative gradient of the
concentration:
J = -D ∆C(q)
∆x
where J is the “current of the quantity q in units of q/m2s and C(q) is the
concentration or density of q in q/m3. The constant D is the diffusion
coefficient and may depend on temperature and location as well as the nature
of the quantity diffusing and the medium it is diffusing through.
If the concentration gradient is somehow maintained (by supplying fresh q as it
diffuses), then this is the whole story. This works for things like heating a
house to a constant temperature while the temperature outside remains the
same as well; the furnace continually supplies heat and the heat leaked out is
continually removed. In the case where you have a limited supply of quantity
q, there is the mass balance to consider.
Mass balance:
Consider a cube of area A and depth ∆x with a net flux J outward to the right.
The flux J caries away some of the quantity q,
changing the concentration. Since the units of J are
(amount of q)/A∙t , the amount of material removed
would be
∆q = J A ∆t
And the change in concentration, C(q), would be
∆C(q) = (J A ∆t)/Volume
= J (A ∆t)/(A ∆x)
Canceling the Areas and rearranging gives
∆C(q)/∆t = J/∆x = -D ∆C(q)/(∆x)2
Using Fick’s first law for the second step. This is the general diffusion
equation. Basically what it says is that the rate of change of the concentration
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is proportional to how fast the concentration gradient changes (sort of like the
acceleration of the concentration).
The mathematical consequence of this is that the concentration decreases
exponentially over time and varies as a sinusoidal wave over space.
The details of a particular case may involve many factors which can affect the
diffusion “constant” D. Temperature almost always affects D for diffusion of
matter. Diffusion may be different in different directions. Thermal diffusion in
wood, for example, is faster along the grain than across the grain. The nature
of the medium through which the quantity is diffusing and the nature of the
quantity itself influence the diffusion constant. It is frequently easier to
determine the diffusion constant experimentally than to calculate it from a
model, which is invariably simplified.
Osmosis: an important example.
Consider a “U tube” with a membrane across the bottom:
One side has pure water, the other
a solution of some species which
cannot pass through the
membrane, a protein in this
picture. Since the dissolved
species (solute) cannot pass
through the membrane, its
diffusion constant is zero in this
situation. The water, on the other
hand, passes through and has
some diffusion constant D whose
value is determined largely by the
membrane. Normally, we think of concentration as the concentration of the
solute, but here, the solute isn’t diffusing. The water concentration, however,
is higher on the pure water side than on the solution side (solute molecules do
take up space). This means that diffusion of water through the membrane
proceeds from the pure water side to the solution side.
So when does this stop? When is equilibrium reached? Look again at the
picture. Notice that the solution level is higher than the level of pure water.
Recall from our study of fluids that this means there is an increased pressure
on the solution side with value
P = ρsghs - ρwghw
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Where ρs is the density of the solution and hs the height of the top of the
solution from the bottom of the tube. Similar variables are used for the water
side. Now, consider what this pressure means at the particle level; this is the
force per unit area caused by particle collisions. This means on the side with
greater pressure, there are more collisions with particles, increasing the
probability that water molecules are driven back across the membrane (and
decreasing the probability that others make it through). Under such
conditions, the rate of water molecules crossing the membrane into the
solution side is decreased as the height of solution in the column rises and
equilibrium is reached before the water concentrations are equal on both sides.
The expression which captures this is the expression for osmotic pressure:
Π = cRT
Where Π is the osmotic pressure and c the concentration of the solution at
equilibrium. Π is calculated as ρsghs - ρwghw or, if the densities are nearly the
same, as
Π = ρsg∆hs
using the difference in column height.
You may have noticed the striking resemblance of this to the ideal gas law:
P = (n/V) RT
Π = cRT
Where n/V is a concentration of particles per volume for the gas. The
similarity is due to the dependence of both on particle collisions as the key
factor in the underlying processes.
The seemingly contradictory aspect is that the water is flowing toward the side
with large concentration; but remember, this is concentration of solute, not
water.
Thermal conduction:
Q = -σ∆T A/l
where Q = heat flow, in Watts
A = area through which heat flows,
l = thickness through which heat flows
σ = thermal conductivity
Rearranging things a bit gives
J = Q/A = -σ/l ∆T
It is now apparent that σ/l is the diffusion constant for this situation by
comparing the above with Fick’s Law.
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Transport phenomena, diffusion.
We have discussed diffusion and several similar transport phenomena this term. Let’s look at these from a different perspective. The common
structure of these models is that
[1] The flux of a quantity is proportional to the concentration gradient for that quantity.
[2] The process runs on the random motion of particles.
[3] The concentration gradient serves as the force driving the flow.
[4] Equilibrium is attained when the gradient is zero.
[5] At equilibrium, particles flow in all directions equally giving no net flux.
[6] No additional energy is required to drive the process.
Process
Quantity
concentration
gradient
flow
Diffusion (Fick’s Law) molecules
Concentration
ΔC/Δx
moles/s
Osmosis
molecules
Solvent concentration
Π=cRT
moles/s
Heat conduction
heat
Temperature
ΔT/d
Electric conduction
electric charge
Potential or Voltage
ΔV/L
Joules/s
relation
1 C
C
D
A t
t
1 Q
T
 kT
A t
d
Amps = C/s
I
V I
1 V
; 
R
A  L
This depends on R = ρL/A and V = kq/r, so a larger concentration of charge gives a larger potential.
Notice that flow described by Bernoulli’s equation doesn’t fit here since it isn’t driven by a concentration gradient nor is that flow
produced by the random motion of the particles.
Another note, the expressions for electrical and thermal conductivity are more similar than they look. The electrical conduction equation can be
written with a conductivity σ = 1/ρ instead of a resistivity ρ. Similarly, the thermal conductivity kT can be replaced with a thermal resistance, R =
1/ kT. This is the R value for insulation.
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