A villus of diameter, \(d\), and length, \(L\), is attached to a membrane. At the base of the villus, the membrane has enzymes immobilized on it that catalyze the transformation of the substrate according to the rate law:

\[-r_{s}=\frac{k^{\prime \prime} K_{s} c_{s}}{K_{i} K_{m}+K_{i} c_{s}}\]

The tip of the villus is held at the free-stream concentration of substrate, \(c_{s \infty}\). Motion of the fluid about the villus provides for a mass transfer coefficient of \(k_{c}\). The diffusivity of substrate through the villus material is \(D_{v s}\).

a. Determine the substrate concentration profile within the villus.

b. What is its efficiency?