Question $3.$

Consider that you secure employment as a Process Engineer with Company XYZ after the successful completion of your

M$.$Sc chemical engineering programme. Your fist assignment at work concerns the development of the dynamic model of a variable volume CSTR for the oxidation of carbon monoxide, and the determination of the system steady$-$state$($s$)$ for given operating conditions. You are told that the oxidation of carbon monoxide is carried out in excess oxygen in an isothermal CSTR containing platinum catalyst particles as follows:

$CO+\frac{1}{2}{O}_{2}C{O}_2$

$-r_A=\frac{k{C}_A}{1+K{C}_A^2}$

The rate of $CO$ disappearance is

where $CA$ is the concentration of $CO.$ The volu:ne of the reactor is $V,$ while the volumetric flowrate of the reactant is $F.$

$($a$)$ Develop the dynamic model of the variable volume isothermal CSTR system $(8$ marks$)$

$($b$)$ If the volume is held constant, show that at steady$-$state, the system can be described by the equation $(2$marks$)$

$\frac{C_A}{}-\frac{C_{A0}}{}+\frac{k{C}_A}{1+K{C}_A^2}=0$

where $$ is the residence time $(\frac{V}{F})$

$($c$)$Find the steady state solution for the case: $K=10,CA0=1,=100,k=0\mathrm{.}2(10$marks$)$

$($d$)$ Show that if the value of $k$ is increased to $0\mathrm{.}3,$ three real steady states can be found. Find these multiplesteady$-$states. Comment on the nature of these steady$-$states. $(13$ marks$)$