Figure $1$: Membrane reactor. The concentration $C_A=C_0$ shown for the feed stream is assumed to apply also inside

the membrane at $x=L.$

$($a$)$ State the equations $($both governing equations and boundary conditions$)$ that govern the reactant and

product concentrations in the membrane, first in dimensional form and then using the following dimensionless

quantities:

$=\frac{x}{L},{}_A=\frac{C_A}{C_0},{}_B=\frac{C_B}{C_0},Da=\frac{k_V{L}^2}{D},=\frac{k_{m}L}{D}.$

$($b$)$ Determine ${}_A()$ and ${}_B().$

$($c$)$ Two measures of reactor performance are the purity of the product stream $({}_1)$ and the fractional recovery

of $B$ in the product stream $({}_2).$ With the composition of the product stream determined by the fluxes at

$x=0.$ these quantities are defined as