In this screencast I want to describe the process of reverse osmosis, in which we use
a membrane to obtain a pure solvent from a solution of a solvent and a solute.
In particular this is the process that is used to obtain water from salt water.
And the idea is that we have a membrane that is permeable to water but almost impermeable
to the salt.
So before talking about reverse osmosis let's look at what the process of osmosis is.
If we were to put water on one side of a membrane and let's say we have them at the same height,
so the same static pressure on both sides.
So we have water plus a solute on the left side and we have pure water on the other side.
This means that the chemical potential of the water on side 1 is less than the chemical
potential on side 2 because the water concentration, the mole fraction of water, on side 1 is less
than the mole fraction of water on side 2.
As a result we have a driving force for permeation.
What we re going to see is that the water, we are going to have a flux of water across
the membrane from the right side to the left side to try and equalize the chemical potentials.
We could also write this as the fugacity of water on the left side is less than the fugacity
of water on the right side where I have used this symbol to indicate that we have a solution
and on the right side is pure water.
So if we let this system go to equilibrium this drawing now represents equilibrium which
means the chemical potential of water on the left side is the same as the chemical potential
of water on the right side, therefore there is no flux.
The way this happens is now that the pressure on the left side is greater than the pressure
on the right side.
And we increase the pressure, we increase the chemical potential.
And continue to have water moving from the right to the left, until we get to this equilibrium
situation.
So this delta P is the osmotic pressure.
So this would be the equilibrium situation.
In reverse osmosis what we do is increase the pressure by an external means on the left
side in order to permeate water to the right side.
So what I have represented here in the third figure now is the idea of reverse osmosis.
The idea is that we have created additional pressure, I have drawn it as a hyper static
pressure, but in reality we are going to pressurize up the liquid with a pump.
So now the chemical potential of the water on the left side is greater than the chemical
potential on the right side, therefore we are going to have flux of water.
So the water flux is going in this direction.
Remember, this membrane is permeable to water but it is almost impermeable to the salt.
So we have water and salt here.
what we are going to form here is essentially pure water on the right side.
This is the dominant process for desalinating salt water or brackish water meaning water
with a lower concentration.
Seawater, seawater is about 3.5 wt% of solute.
This is also used to treat some food products to keep water in them.
So for seawater the osmotic pressure is about 24 bar.
So we are going to do reverse osmosis, that means we have to have a pressure on the feed
side that is higher than 24 bar.
And typical values are 55 to 75.So let's look at our equilibrium situation again.
We have a chemical potential on the left side, equal the chemical potential on the right
side.
We could also write this as the fugacity on the left side is equal to fugacity on the
right side.
And the fugacity, be the mole fraction of water and activity coefficient and the fugacity
of pure water, 0 indicating pure water, and this is at pressure P1.
The right side is now we will assume pure water, but it is at a different pressure.
And then typically this is going to be close to 1 because this is close to 1 if this were
0.98 for example that is a pretty good approximation that the activity coefficient is close to
1.
So we have the mole fraction of water times the fugacity of pure water at this pressure
is equal to the fugacity of a different pressure.
So what we are going to use to relate the fugacites at the different pressures is the
poynting correction.
So this is the poynting correction.
It says the fugacity at pressure P1 for pure water is equal to fugacity of pressure P2
times an exponential pressure difference.
This term is the volume per mole of the liquid.
Gas constant and absolute temperature.
We have been talking here about equilibrium which means P1-P2 is our osmotic pressure.
So now let's look at reverse osmosis.
So we are going to have water permeating through the membrane in this direction.
The flux of the water, so this is flux, moles per area per time.
Flus of the water, so the flux it turns out is proportional to a permeability, this is
a property of the membrane, thickness, thicker membrane has a lower flux, and then the pressure,
on the feed side minus this quantity, pressure on the permeate side plus the osmotic pressure,
because, remember, if this pressure difference, so this we will call this pressure difference
across the membrane was equal to the osmotic pressure we would have no flux.
So we have to have a pressure difference greater than the osmotic pressure to have reverse
osmosis.
So here is an equation for the water flux.
Higher pressure on the feed side causes water to permeate through.
We could have a small amount of salt, it is not a perfect membrane.
So we could also look at the flux of the salt which would be related to the diffusivity
of the salt in the membrane.
Again divided by the membrane thickness.
Now we would have the concentration of the salt, this would be on, in the membrane right
at the edge.
so the membrane here, side 1 and side 2, that would be the concentration right here, call
that prime, and then the concentration right here.
And what we can do is relate this to the partition coefficient for the solute.
And then write the concentrations in solution.
So these now are the concentrations, this is liquid salt plus water side 1.This is almost
pure water but a very low concentration salt.
So the concentration here is this concentration and the concentration here is this concentration.
So this is a partition coefficient that is, you can see from the equation, related to
the salt concentration in our solute in general concentration at the membrane.
The solute concentration in the liquid right next to the membrane.
Now this is independent of delta P. So as the pressure difference becomes larger you
would increase the flux of the water.
You wouldn't increase the flux of the salt diffusing through the membrane so you would
actually increase the purity of the water on the right side.
This idea of reverse osmosis is what is used to create high purity water from salt water.
And it certainly is much more energy efficient than the distillation of salt water.