COPOLYMERIZATION $($CSTR$)$

Ethylene and $1-$octene are copolymerized with a coordination catalyst in a continuous$-$stirred tank reactor $($CSTR$)$ at $140$ oC$.$ The reactor system is assumed ideally mixed, and has a reactor volume of $600$ ml$.$ The ethylene and $1-$octene feeds into the reactor are $5\mathrm{.}38$ g$/$min and $2\mathrm{.}41$ g$/$min$,$ respectively.

The concentration of the titanium catalyst used in the reactor is $15$mol$/$dm$3.$ Due to strong catalyst deactivation, the catalysts$$ mean residence time in the reactor is $4$ min. The reactivity ratios have been previously found to be r$1=7\mathrm{.}9($ethylene$)$ and r$2=0\mathrm{.}099(1-$octene$).$ The polymer yield from the reactor is $5\mathrm{.}49$ g$/$min$.$

a$)$ Calculate the concentrations of ethylene and $1-$octene when the reactor is operated at steady state.

The polymer yield from the reactor may be calculated using the following equation

$[($C$1,$in$-$C$1)$M$1+($C$2,$in$-$C$2)$M$2]$v $=$ amount of copolymer coming out of the reactor $[$g$/$min$](1)$

where:

v$=$ volumetric flow rate

M$1,$ M$2=$ comonomer molar masses

C$1,$ C$2=$ comonomer steady state concentrations

C$1,$in$,$ C$2,$in $=$ comonomer feed concentrations

The copolymer compositions can be expressed using the following equations

dC$1/$dC$2=$ C$1($r$1$C$1+$ C$2)/$C$2($C$1+$r$2$C$2)$

dC$1/$dC$2=($C$1,$in $-$ C$1)/($C$2,$in $-$ C$2)$

b$)$ Draw plots showing ethylene and $1-$octene reactor concentrations versus time $($from the initial concentrations until steady state concentrations are reached$).$ At t$=0$ the reactor is full of solution. The consumption of ethylene and $1-$octene during the reaction may be neglected.

To solve equations for monomer concentrations, apply mole balances on both species:

b$)$ Draw plots showing ethylene and $1-$octene reactor concentrations versus time $($from the initial concentrations until steady state concentrations are reached$).$ At t$=0$ the reactor is full of

solution. The consumption of ethylene and $1-$octene during the reaction may be neglected.

To solve equations for monomer concentrations, apply mole balances on both species:

dNj$/$dt $=$ Fj$0-$ Fj

where:

Nj $=$ represents the number of moles of species j in the system at time t

Fj$0=$ rate of flow of j into the system $[$mol$/$s$]$

Fj $=$ rate of flow of j out of the system $[$mol$/$s$]$

Please note that the hints for solving the problem are provided along with the data in the picture attached to the problem. Use those hints and data from the picture attached to it in this question to solve it's both parts. the text elow is just some of the formulas typed from the hints

The polymer yield from the reactor may be calculated using the following equation

$[(C_{1,in}-C_1)M_1+(C_2,in-C_2)M_2]v=$ amount of copolymer coming out of the reactor g$/$m$in$

where:

$v,=$ volumetric flow rate

$M_1,M_2,=$ comonomer molar masses

$C_1,C_2,=$ comonomer steady state concentrations

$C_{1in},C_{2in}$ comonomer feed concentrations

The copolymer compositions can be expressed using the following equations

$\frac{d{C}_1}{d{C}_2}=\frac{C_1(r_1{C}_1+C_2)}{C_2(C_1+r_2{C}_2)}$

$\frac{d{C}_1}{d{C}_2}=\frac{C_{1,n}-C_1}{C_{2,n}-C_2}$

Values to solve this problem are: