What is the concentration profile at steady state?

[ZintheDestroy]

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) \frac{dC}{dt} = 0 = \frac{D}{r^{2}}\frac{∂}{∂r}(r^{2}\frac{dC}{dr})

2) Boundary conditions

r = R_{1} C = C(0)

r = R_{2} C = C(L)

Where C(0) and C(L) are the concentrations at the boundaries of the membrane

Integrate to get:

C(r) = (\frac{C(L)-C(0)}{R_{2}-R_{1}})\frac{1}{r}+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

 

[LawrenceC]

Question 1 requests a steady state profile across the spherical membrane. If you integrate the original steady state equation, you get two constants of integration. They are evaluated based on the boundary conditions at R1 and R2.

You will get two equations and two unknowns for C1 and C2. What I work out differs from yours. The way to check it is to plug in r=R1 to see what the value of the function for C is at R1. It must reduce to the boundary condition. Do the same at R2.