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# Reverse Osmosis

This Demonstration models a reverse osmosis process in which salt water at high pressure is fed to a chamber with a membrane that is only permeable to water. The outlet permeate stream is pure water at low pressure, and the retentate stream is water enriched in salt.
Use the sliders to set the pressure drop across the membrane, the inlet salt mass fraction and the stirring bar speed. Increasing the stirring speed decreases concentration polarization on the feed side of the membrane and increases water flux through the membrane. The inlet mass flow rate is fixed and the process is isothermal.
The water flux through the membrane is plotted versus the pressure drop across the membrane. The blue line represents the water flux when pure water is fed to the system. The green line represents the water flux when salt water is fed to the system. The dashed black line indicates the maximum water flux through the membrane. The values in the diagram on the right correspond to the conditions represented by the black dot, whose location is selected by the pressure drop slider. If the pressure drop is less than the osmotic pressure (orange dot on the axis), the system exhibits osmosis, and pure water permeates into the salt water feed.
Uncheck the "show diagram" box to remove the flow diagram on the right and display a larger graph of flux versus pressure drop.

### DETAILS

First, solve for the mass transfer coefficient using the equation:
,
where is the mass transfer coefficient,
is the rotational speed of the stirring bar (rad/s),
is the tank diameter (m),
is kinematic viscosity (),
is the Schmidt number (dimensionless) and
is the diffusivity of the solute in water ().
Then solve for the maximum solvent flux:
,
where is maximum water flux through the membrane () and is the density of water ().
Solve for the solvent flux:
,
where is mechanical pressure (atm),
is osmotic pressure (atm) and
is permeability of the membrane to the solvent ().
A material balance is done for salt across the system to get the value for the mass fraction of salt in the outlet stream, which is the same as the retentate mass fraction inside the tank (given the tank is well mixed):
,
where is inlet feed (kg/s), is the mole fraction of salt in the inlet stream, and is the area of the membrane ().
To find how much of the feed exits as the permeate, an overall mass balance is done:
,
where is the fraction of the feed that is purified and exits as the permeate, i.e., cut (dimensionless).

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