A thin cylinder, \(2 \mathrm{~mm}\) in diameter and \(0.3 \mathrm{~m}\) long, is to be coated via a variant of chemical vapor deposition. The wire is spun on its axis at 20 revolutions per second. A hot, polymer vapor flows over the wire and reacts on the surface at a rate that has been measured \(\left(\overline{S h_{d}}=0.15 R e_{d}^{0.7} S c^{1 / 3}\right)\) to be:

\[r_{p}\left(\frac{m o l e s}{m^{2} s}\right)=k_{s}^{\prime \prime} c_{p s}^{1.35}\]

where \(c_{p s}\) is the concentration of the polymer in the vapor at the surface of the cylinder. The concentration of polymer vapor in the chamber is \(10 \mathrm{moles} / \mathrm{m}^{3}\). The diffusivity of the polymer within the vapor is \(D_{p a}=1 \times 10^{-7} \frac{\mathrm{m}^{2}}{\mathrm{~s}}\) and \(k_{s}^{\prime \prime}=1 \times 10^{-3} \frac{\mathrm{m}}{\text { moles }^{0.35} \mathrm{~s}}\). The rest of the vapor properties can be assumed to be that of air at \(375 \mathrm{~K}\).

a. Derive an expression for the mass transfer coefficient needed if the process is operating at a steady state with a reaction rate given by the equation above.

b. If the deposition rate is to be \(2 \times 10^{-5} \mathrm{moles} / \mathrm{s}\) on the wire, at what velocity do we need to flow the vapor over the wire? A correlation for this situation was developed and found to be:

\[\overline{S h}_{d}=0.15 \operatorname{Re}_{d}^{0.7} S c^{1 / 3}\]