Q$2.(25$ marks$)$

A mixing tank has a transmitter that measures the molar concentration $($in $mo\frac{l}{m^3})$ of $A$ in the outlet stream $(C).$ The total volume of the liquid in the tank is $1m^3$ with density as $300k\frac{g}{m^3}.$ These are constants throughout the process. $C_m$ is the concentration measured by the transmitter and the transmitter can be approximated by a first order transfer function $\frac{1}{0\mathrm{.}1s+1}$ with respect to $C.$ The respective volumetric flowrates and molar concentrations of the inlet streams are given as shown in the figure. Assume that the volumetric flowrates are constant as indicated in the figure.

a$.$ Show the overall mass balance and component balance. $(5$ marks$)$

b$.$ Find the transfer function $\frac{C_m^{\text{'}}}{C_1^{\text{'}}}$ where $C_1$ is the concentration of the inlet stream. $(5$ marks$)$

c$.$ For a step change of $3mo\frac{l}{m^3}$ in $C_1,$ what is the $C_m^{\text{'}}(t)?$ Also find the final steady state response of $C_m^{\text{'}}(t).(10$ marks$)$

d$.$ If $\frac{C_m^{\text{'}}}{C^{\text{'}}}$ can be approximated as $1,$ how does the $C_m^{\text{'}}(t)$ change for the same input change in $C_1?(5$ marks$)$

For parts $($b$)$ and $($c$),$ assume there is no change in $C_2.$