The following elementary irreversible reaction is in liquid phase.

$A+BC$

An equal molar of $A$&$B$ enters at $300K$ with $v_0=2d\frac{m^3}{s}.$ For the given data below:

$H_A(273K)=-20\frac{\text{k}\text{c}\text{a}\text{l}}{\text{m}\text{o}\text{l}},H_c(273K)=-41\frac{\text{k}\text{c}\text{a}\text{l}}{\text{m}\text{o}\text{l}}$

$H_B(273K)=-15\frac{\text{k}\text{c}\text{a}\text{l}}{\text{m}\text{o}\text{l}},$

Activation energy, $E=10$kca$\frac{l}{m}ol$

$k=0\mathrm{.}01d\frac{m^3}{m}ol.s$ at $300K$

$C_{A0}=0\mathrm{.}1$kmo$\frac{l}{m^3}$

$C_{PC}=30$kca$\frac{l}{m}ol.K$

$C_{PA}=C_{PB}=15$kca$\frac{l}{m}ol.K$

$($a$)$ Determine the PFR volumes necessary to achieve $50\%$ conversion if the system are

operated adiabatically.

$($b$)$ Plot the conversion and temperature profiles if the temperature was maintained by using

heat exchanger where $Ua=20ca\frac{l}{m^3}.\frac{s}{K}$ and the coolant temperature is constant at $T_a$

$=450K.$

$($c$)$ Repeat for $($b$)$ for a co$-$current heat exchanger for a coolant flow rate of $50\frac{g}{s}$ and $C_{P,C}$

$=1ca\frac{l}{g}.K,$ and inlet coolant temperature of $T_{a0}=450K.$