A liquid$-$phase isomerisation reaction is to be conducted such that $1000$ tonnes of the

desired product are produced per year of operation. Accounting for start$-$up$,$ shutdown

and maintenance periods this can be taken as $8000h$ of reactor operation, regardless

of reactor type. The reaction is operated isothermally at $150C.$ Over the course of

reaction $95\%$ of all of the initial reactant is converted into the desired product. The

following additional parameters are known:

Rate constant: $0\mathrm{.}2h^{-1}$

Density of reactant: $0\mathrm{.}75gc{m}^{-3}$

Density of product: $0\mathrm{.}75gc{m}^{-3}$

Enthalpy of reaction: $-300k\frac{J}{k}g$

Heat capacity of reactant: $2kJk{g}^{-1}{K}^{-1}$

Heat capacity of product: $2kJk{g}^{-1}{K}^{-1}$

$($a$)$ The design equation for a batch reactor can be expressed as:

$r_A=-\frac{1}{V}\frac{d{N}_A}{dt}$

Starting from this, show the reaction time to achieve a given conversion for the

reaction in a constant volume batch reactor can be expressed as:

$t=-\frac{1}{k}ln(1-x_A)$

$($b$)$ If start up and shut$-$down procedures, including heating and cooling the reactor,

take $3h$ per batch, then what is the volume of the batch reactor? Assume no time

is lost due to maintenance.

$($c$)$ If the reaction was instead to be conducted in a continuous reactor, what total

volumetric flow rate would be required?

$($d$)$ The design equation for a PFR can be expressed as:

$r_A=-\frac{d{n}_A}{dV}$

Starting from this, determine the volume of a PFR used for the reaction. Comment

on the relative value of this compared with your answer for $($b$).$