Question $1$

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

$$

$$

$->$

The dynamic equations that govern the system are given by carrying out the molar balance

for component A for each CSTR: Accumulation $=$ In $-$ Out $+$ Generation$-$Consumption

$()$

$()$

$=(1)()()()=1,2,3.$

Using the residence time $=$

$$

$$

the equation can be transformed to

$()$

$=$

$1$

$$

$(1)($

$1$

$$

$+)()=1,2,3.$

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.

$$: $1=1$; $2=2$; $3=3$; $0=100$

$$

$$

; $1=1$

$1$

; $2=2$

$1$

; $3=3$

$1$

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

$=$

$1\mathrm{.}2$ Use the Gauss$-$Jordan reduction method to solve the three equations with

three unknowns CA$1,$ CA$2,$ and CA$3.[20$ Marks$]$