$(2)$

$(20$p$.)$ CSTR reactor design

$($a$)$ Show that recycling the effluent of a CSTR reactor $($as done by Uppal, Ray and Poore$)$ has no effect on the conversion of the reactant for a $n-$th order irreversible reaction.

$(10$p$)$

$($b$)$ Consider two CSTR reactors in series. For a fixed total volume $V=V_1+V_2,$ determine the optimal distribution of the two volumes that maximizes the overall conversion of a reactant undergoing a first$-$order irreversible reaction. $(10$p$)$