The liquid$-$phase, elementary reaction

$2A+\frac{1}{2}BC+2D$

Will be conducted in an isothermal batch reactor equipped with cooling coils. The reactor is

initially charged with a stoichiometric ratio of pure A and B$.$ The reaction is moderately exothermic

and the heat of reaction will be removed via cooling water. The cooling water will enter at a $T_{a1}$

and leave at $T_{a2}.$ The following data is available:

Cooling water:

Cooling water inlet temperature $=T_{at}$

Cooling water outlet temperature $=T_{a2}$

Cooling water heat capacity $=$ CPoocl

Reactor conditions:

Reactor volume $=V$

Heat transfer coefficient $=U$

Heat transfer area $=A$

Reactor pressure $=P_0=P$

Material properties:

Heat capacities: $CP{A}_{PA},C{P}_{PB},C{P}_{PC},CP{D}_{PD}$

Heat of reaction $=H_R=$ constant

Molecular weight of reactants and products: $M{W}_A,M{W}_B,MW,MW{D}_D$

Rate constant at reaction temperature $=k_1$

a$)$ Plot the expected flow rate of coolant required to maintain isothermal conditions versus the

reaction conversion. Please note the figure is on the following page. $(5$ points$)$

b$)$ Develop the expression$($s$)$ that will allow the determination of the required flow rate of

cooling water versus the reaction conversion. Please use an extra sheet of paper while doing

this problem. Remember to show all of your steps. $(25$ points$)$