In the production of superconducting wire tube, the alloy is reacted with oxygen to produce the appropriate ceramic phase. Oxygen at a concentration co flows through the inner tube and diffuses into the metal, undergoing a chemical reaction to produce the ceramic as shown in Figure P$5\mathrm{.}31.$ The reaction rate has been measured as a function of position and follows:

Pure nitrogen flows on the outside of the tube to scavenge away any oxygen that doesn$$t react by the time it reaches the surface.

a$.$ Draw the control volume for deriving the mass balance for the system.

b$.$ Using the general balance equation, derive the differential mass balance giving the concentration profile of oxygen in the superconductor.

c$.$ What are the boundary conditions for this problem?

d$.$ Solve the differential equation for the concentration profile.

e$.$ What is the flux of a at r $=$ ri$?$