$($ not that write it in paper please provide clear answer with the drowing same grophes$)$Consider the two mixing tanks in series shown in Figure $1.$ The tanks have constant volumes

$V_1=V_2=V=1m^3$ and constant volumetric flowrates $F=0\mathrm{.}1\frac{m^3}{m}in.$ The feed stream may

contain a dye of concentration $C_0$ that can be adjusted by an upstream unit. Derive a math$-$

ematical model for the dynamics of the dye concentrations $C_1$ and $C_2$ in the two tanks, and

calculate their responses for the following cases.

$(1)$ The feed is pure water, $C_0(t)=0,t>0,$ but there is leftover dye in the first tank from a

previous experiment, with initial concentration of $C_1(0)=10k\frac{g}{m^3}.$ The second tank

initially contains pure water $C_2(0)=0.$

$(2)$ The feed is initially pure water, the tanks initially contain pure water, but at time $t=0,$ the

feed is suddenly switched to a solution of dye of concentration $10k\frac{g}{m^3}.$

$(3)$ The feed is initially pure water, the tanks initially contain pure water, but at time $t=0,$ the

$F$

$C_0(t)$

feed is suddenly switched to a solution of dye of concentration $10k\frac{g}{m^3},$ up to time $t=$

$=2\mathrm{.}5$min, when it is switched back to pure water.

$(4)$ The same amount of dye as in the previous question is delivered during an infinitesimally

small time period $(0).$

$(5)$ The feed is initially pure water, the tanks initially contain pure water, but starting at time

$t=0,$ the feed contains dye with linearly increasing concentration at a rate of

$0\mathrm{.}1kg*m^{-3}mi{n}^{-1}$