Problem $1$:

Consider the system of three tanks shown below. All flow rates are volumetric, and the crosssectional areas of the three tanks are $A_1,A_2,$ and $A_3,$ respectively. The flow rate $q_5$ is constant and does not depend on $h_3,$ while all other effluent flow rates are proportional to the corresponding hydrostatic liquid pressure that cause the flow $($i$.$e$.,q=h).$

$($a$)$ Develop a mathematical model for the system. $($Write a mass balance for each tank$)$

$($b$)$ Solve the system of equations and plot the levels as a function of time.

Data: $A_1=10m^2$;$A_2=A_3=5m^2$;$h_1(0)=3m$;$h_2(0)=5m$;$h_3(0)=3m$;$q_1=2\frac{m^3}{s}$; $q_5=0\mathrm{.}5\frac{m^3}{s}$;${}_2=1\frac{m^2}{s}$;${}_3=1\mathrm{.}5\frac{m^2}{s}$;${}_4=1\frac{m^2}{s}$;

Problem $2$:

In an enzyme$-$catalyzed reaction with stoichiometry $AB,A$ is consumed at a rate given by an expression of the Michaelis$-$Menten form:

$r[mo\frac{l}{L*s}]=\frac{k_1{C}_A}{1+k_2{C}_A}$

where $C_A(mo\frac{l}{L})$ is the reactant concentration, and $k_1=0\mathrm{.}0028\frac{L}{m}ol.s$ and $k_2=0\mathrm{.}115\frac{L}{m}ol$ The reaction is carried out in an isothermal batch reactor with constant reaction mixture volume $V($liters$),$ beginning with pure $A$ at a concentration $C_{A0}.$ Write a balance on A and integrate it to obtain an expression for the time required to achieve a specified concentration $C_A.$ beginning with $A$ at a concentration $C_{A0}=5\mathrm{.}00mo\frac{l}{L}.$