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ZintheDestroy
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I have a sphere of radius 4.8mm surrounded by a membrane .2 mm that has a drug in the inner sphere and is diffusing out through the membrane. I know the saturation concentration, the partition coefficient and diffusivity of the drug. I also know that the concentration outside of the membrane is essentially zero as the drug is absorbed quickly I need to do the following:
1) Derive a one dimensional steady state balance on the membrane
2) Solve for the concentration profile at steady state
3) Calculate the flux
4) Determine how often the drug needs to be administered
So far I have the following:
1) [itex]\frac{dC}{dt}[/itex] = 0 = [itex]\frac{D}{r^{2}}[/itex][itex]\frac{∂}{∂r}[/itex]([itex]r^{2}[/itex][itex]\frac{dC}{dr}[/itex])
2) Boundary conditions
r = [itex]R_{1}[/itex] C = C(0)
r = [itex]R_{2}[/itex] C = C(L)
Where C(0) and C(L) are the concentrations at the boundaries of the membrane
Integrate to get:
C(r) = ([itex]\frac{C(L)-C(0)}{R_{2}-R_{1}}[/itex])[itex]\frac{1}{r}[/itex]+C(0)
This is where I'm stuck as I'm not sure this is the correct integration. Does this work then I just set C(L) equal to 0 and C(0) equal to the partition coefficient times the saturation concentration?
3) Not sure what to set equal to J
4) How would i even solve for the time it takes?
Any help would be appreciated
1) Derive a one dimensional steady state balance on the membrane
2) Solve for the concentration profile at steady state
3) Calculate the flux
4) Determine how often the drug needs to be administered
So far I have the following:
1) [itex]\frac{dC}{dt}[/itex] = 0 = [itex]\frac{D}{r^{2}}[/itex][itex]\frac{∂}{∂r}[/itex]([itex]r^{2}[/itex][itex]\frac{dC}{dr}[/itex])
2) Boundary conditions
r = [itex]R_{1}[/itex] C = C(0)
r = [itex]R_{2}[/itex] C = C(L)
Where C(0) and C(L) are the concentrations at the boundaries of the membrane
Integrate to get:
C(r) = ([itex]\frac{C(L)-C(0)}{R_{2}-R_{1}}[/itex])[itex]\frac{1}{r}[/itex]+C(0)
This is where I'm stuck as I'm not sure this is the correct integration. Does this work then I just set C(L) equal to 0 and C(0) equal to the partition coefficient times the saturation concentration?
3) Not sure what to set equal to J
4) How would i even solve for the time it takes?
Any help would be appreciated