A cascade of three isothermal CSIRs in series is used to carry out a liquid$-$phase reaction.

$AkB$

The dynamic equations that govern the system are given by carrying out the molar balance for component A for each CSTR: Accumulation $=$ ln $-$ Out $+$ Generation$-$Consumption

$V_{(i)}\frac{d{C}_{A(i)}}{dt}=F{C}_{A(i-1)}-F{C}_{A(i)}-k{C}_{A(i)}{V}_{(i)},$ for $i=1,2,3.$

Using the residence time $=\frac{V}{F}$ the equation can be transformed to

$\frac{d{C}_{A(i)}}{dt}=\frac{1}{{}_i}{C}_{A(i-1)}-(\frac{1}{{}_i}+k_i)C_{A(i)},$ for $,i=1,2,3.$

If we set up the left$-$hand side of each equation to zero, then we will end up with a set of three linear algebraic equations that describe the three CSTRs under a state$-$state condition. Notice that each CSTR has its own residence time and rate constant.

Assume: ${}_1=1s$;${}_2=2s$;${}_3=3s$;$C_{A0}=100\frac{\text{m}\text{o}\text{l}}{L}$;$k_1=1s^{-1}$;$k_2=2s^{-1}$;$k_3=3s^{-1}$

$1\mathrm{.}1$ Compute the set of linear equations in a matrix$/$vector form as $[10$ Marks$]$

$Ax=b$

$1\mathrm{.}2$ Use the Gauss$-$Jordan reduction method to solve the three equations with three unknowns $C_{A1},C_{A2},$ and $C_{A3}.20$ Marks$]$