The compound tank system shown in Figure consists of a spherical tank of radius $R_1$ and a

cylindrical tank of diameter $D_2.$ A liquid of constant density is fed at a volumetric rate $F_{1in}$ into the

top of a spherical tank and volumetric rate $F_{in}$ into the top of the cylindrical tank. The spherical and

cylindrical tanks interact through the pipe connecting them. The flow rates into the connecting pipe

depend on the heights of the liquid in the tanks. The volumetric flow rate out of the spherical tank

into the pipe is given by $F_{10ut}=k_1\sqrt[\phantom{2}]{h_1},$ while the volumetric flow rate out of the cylindrical tank into

the pipe is given by $F_{20ut}=k_1\sqrt[\phantom{2}]{h_2}$ where $h_1$ and $h_2$ are the heights of the liquid in the spherical and

cylindrical tanks respectively and $k_1$ is the common valve $a,$ cient. The cylindrical tank also has a

drain on the right$-$hand side which has volumetric flow rate $F_{30ut}=k_2\sqrt[\phantom{2}]{h_2}$ where $k_2$ is the valve

coefficient for the right$-$hand side drain.

a$)$ Obtain a dynamic model that describes the heights of the liquid in the tanks. Is this a linear or

nonlinear model?

b$)$ For constant input flow rates, $F_{in}$ and $F_{in},$ analytically determine the steady$-$state values of $h_1$ and

$h_2.$ Do the shapes and dimensions of the tanks affect the steady$-$state values?

Note that you do not need to solve the differential equations for the steady state analysis.

c$)$ Simulate the system and plot the heights of the liquid in the tanks versus time for constant input

flow rates using the values given in the table below. $($Run the simulation for $2000sec)$