We are interested in using a large surface area, pin fin based adsorbent for removing toxic material from a water solution. The system operates in batch mode with a stirred tank of contaminated fluid in which we insert the adsorbent fin system. Material that makes it from the fin surface to the fin base is swept away by solvent so that the contaminant concentration is always 0 at the base. As the tank fluid is cleared of contaminant, its contaminant concentration exponentially decays according to:

\[c_{a \infty}=c_{a \infty 0} \exp (-\alpha t)\]

a. Assuming all fins are independent, derive the differential equation governing the transient concentration of contaminant in the fin.

b. Assuming the fin tip is sealed off, what are the boundary conditions for this system? Initially the fin contains no contaminant.

c. Solve the fin equation assuming the concentration in the fin can be expressed as:

\[c_{a}(x, t)=C(x) \exp (-\beta t)\]