4.8 A surge tank in Fig. E4.8 is designed with a slotted weir so that the outflow rate, w, is proportional to the liquid level to the 1.5 power, that is, wRh15 where R is constant. If the single stream enters the tank with flow rate w, find the transfer function H(s)/W(s). Identify the gain and all time constants. Verify units The cross-sectional area of the tank is A. Density ρ is constant. -Weir Figure E4.8

Show the changes (dynamic model) of the water levels (h1, h2, h3) of the tanks connected as in the figure below. Find the transfer function that connects the change in flow rate Q'3 (s) from the third tank to the change in flow rate in the first tank, and Q (s). Show how this transfer function relates to other transfer functions (including changes in inlet-outlet flows and tank liquid heights).(H'1 (s), H'2 (s) and H'3 (s) show the deviations in the liquid levels of tank 1, tank 2 and tank 3, respectively.)Valve resistances are R1 = R2 = R3 = R. The density of the liquid is constant and the pressures in the tank are equal to the ambient pressure.