concepts

Transport Across a Membrane

A membrane separating two chambers can permit a solute to move through it by diffusion. A substance diffusing through a medium with diffusivity D satisfies the equation where C(x,t) is the concentration at position x, at time t.

Initial Condition: At t = 0, the concentration takes on some value as a function of x. This will depend on what was done to the membrane before the flow process starts.

Boundary Condition: Membrane operation occurs when the membrane is placed between two masses of fluid with different concentrations of solute. Then


where L is the thickness of the membrane.

Steady-State: After a long time, the solution should approach the steady-state solution, which is In steady state, the flux through the membrane is

Transients: Unsteady solutions of the diffusion equation approach the steady state exponentially fast, in a time proportional to L2/2D. In this case the flux of solute depends on x and the membrane will gain or lose solute before it reaches steady state.


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