The crystalline product $(P)$ is synthesised from an amorphous substrate $(A)$ in aqueous sodium hydroxide $(NaOH)$ in a well agitated isothermal batch re$$ctor at $100C.$ The substrate $($A$)$ rapidly dissolves into soluble species $($S$)$ and crystalises into a crystalline product $($P$)$ following an autocatalytic reaction.

$A+P+$NaOH$S+PP+P$

The reaction rate equation is given as$,$

$-r_A=\frac{k_2(C_{\text{N}\text{a}\text{O}\text{H}}{)}_0^{k_1}{C}_A{C}_P}{k_3{C}_P+(k_3+1)C_A}$

where $C_2$ and $C_{}$ are the concentrations of amorphous substrate and crystalline kmol.$m^{-3}.$ The kinetic parameters, $k_1,k_2$ and $k_3$ are given at the reaction temperature of $100C$ as $2\mathrm{.}36,6\mathrm{.}25{10}^{-4}{m}^{7\mathrm{.}00}*kmo{l}^{-2\%}.s^{-1}$ and $0\mathrm{.}36,$ respectively.

$($a$)$ The reactor feed is slurry, containing $1kg*m^3$ of crystalline product and $24kg.m^{-3}$ of amorphous substrate. The reactor operates isothermally at $100C.$ and the concentration of sodium hydroxide is $1\mathrm{.}5$kmol.$m^{-3}.$ Determine the time required to achieve $98\%$ of the substrate conversion to zeolite.

$[24$ marks$]$

$($b$)$ Determine the reactor size and total weight of charge necessary to produce be used and it will be necessary to shut down for $0\mathrm{.}5h$ between batches for removal of product, cleaning, and startup. The product $)$ to be recycled to the reactor will come from $2000kg$ produced daily.