The alveoli in your lungs can be modeled as conventional pin fins. Each fin is \(10 \mu \mathrm{m}\) in diameter and \(100 \mu \mathrm{m}\) long. The fin is designed to pick up oxygen from the air and transport it to the base where is reacts instantaneously with hemoglobin. The fin tip is impermeable to oxygen. Assuming air at \(25^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\), a mass transfer coefficient, \(k_{c}\), of \(1 \times 10^{-5} \mathrm{~m} / \mathrm{s}\), a diffusion coefficient for \(\mathrm{O}_{2}\) in the fin of \(3 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\) :

a. What is the governing equation for this problem?

b. What are the boundary conditions?

c. Solve the equation to obtain the concentration profile.

d. What is the effectiveness for this fin?