The alveoli of the lungs can be modeled as pin fins with a diameter of \(20 \mu \mathrm{m}\) and a length of \(250 \mu \mathrm{m}\). The asthma drug isoproterenol is administered by inhalation. The patient inhales fully and holds his breath for \(\sim 30 \mathrm{~s}\). The normal does of isoproterenol is \(120 \mu \mathrm{g}(M w=247.7)\) and the volume of the average male's lungs aside from the alveoli is 2.25 liters. The tissue of the alveoli is approximately \(0.1 \mu \mathrm{m}\) thick and has an isoproterenol diffusivity of \(1 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\). The diffusion coefficient of isoproterenol inside an alveolus is \(3 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\). The drug is picked up by the blood and rapidly removed so the concentration of isoproterenol at the base of the alveolus is zero. The mass transfer coefficient to the alveolus is \(1 \times 10^{-5} \mathrm{~m} / \mathrm{s}\). The alveolus tip is impermeable.

a. Find the initial flux of isoproterenol into the blood assuming pseudo-steady-state operation

b. Find the effectiveness and efficiency of an alveolus.

c. If the human lungs were simply two spheres with no extended surfaces, what radius would they need to have to achieve the same mass transport rate? There are some \(700 \times 10^{6}\) alveoli in the lungs.