Flicks diffusion equation - metabolism rate

[gladerider]

So i have an organic chemical in a bio-film reactor being diffused into a bio-film and also being metabolized at constant rate R by bacteria.

the concentration into the reactor Cin = 2mg/L with f-in at 20 m^3/hr
the concentration in the reactor is C1
C3 is the concentration in the bio-film

flicks law of diffusion:

e*dc/dt = d*(d^2C/Dz^2) - R where c = C3

where e, d, and R are constants.
e-porosity
d - diffusion coefficient

with bc's ;
dC3/dz = 0 @ z = 2micrometers
C3 = C1(.005mg/l) @ z = 0


I've was trying to solve with separation of variables but I'm rusty and running into some problems... Please help!

 

[Eynstone]

The general solution will be a particular solution + the general solution of the heat equation. I hope we can find a particular solution (even approximate) in polynomials is z,t .