The catalytic liquid phase irreversible reaction $(AB)$ is performed using a non$-$porous catalyst. Several experiments have been performed and it was shown that the intrinsic reaction kinetics can be described according to:

$-r_A=\frac{c_1{C}_{As}}{1+c_2{C}_{As}},-r_{A}inmol{m}^{-2}{s}^{-1}$ and $C_{A}inmol{m}^{-3}$

Where $C_{AS}$ is the concentration of $A$ on the catalyst surface. The constants $c_1$ and $c_2$ were fitted and found to be $1\mathrm{.}5*{10}^{-3}m{s}^{-1}$ and $1\mathrm{.}0*{10}^{-3}{m}^{3}mo{l}^{-1}$ respectively.

To further study the reaction, experiments are carried out in a long pipe $(L=10m,$ where the inside of the pipe $)=(0\mathrm{.}025m$ is coated with a thin layer of the non$-$porous catalyst. The feed to the solution is $1\mathrm{.}0L{s}^{-1}(1\mathrm{.}0*{10}^{-3}{m}^3{s}^{-1})$ and the inlet concentration of $A$ is $1000mol{m}^{-3}.$

a$)$ Determine the conversion of $A$ in the pipe, if external mass transfer limitations can be neglected.

The operator of the "pipe experiments" starts to doubt the assumption of neglecting external mass transfer effects and does some calculations and finds that the external mass transfer coefficient, in the used pipe and under the experimental flow conditions, equals $1\mathrm{.}0*{10}^{-3}\frac{m}{s}.$

b$)$ Show that the concentration of A near the surface of the entrance of the pipe $(C_{\text{A}\text{s}\text{o}})$ is half of the bulk concentration. Is the operator right in doubting the absence of external mass transfer limitations$?$ Motivate!

c$)$ Determine the conversion of $A$ in the pipe, taking external mass transfer limitations into account.