A water reservoir is supplied by two sources, rainwater and groundwater, while a single main water pipe leaves the reservoir, with water pumped out...

A water reservoir is supplied by two sources, rainwater and groundwater, while a single main water pipe

leaves the reservoir, with water pumped out at a rate q to supply the local community. The rate at which

rainwater enters the reservoir, qr, of course depends on how much rain has fallen recently, while the

groundwater is pumped directly into the reservoir, so its flow rate, qg, can be controlled. Your goal as the

engineer operating the reservoir is to maintain the reservoir at a steady state in which the depth of water in

the reservoir, h, is constant at hs (or as close to constant as possible).

(a) Sketch the system and define the control volume over which you should set up a mass balance on the

water.

(b) Develop the mass balance and simplify the equation by making suitable assumptions.

(c) Specialize the mass balance to the form that will apply at steady state (flow rates qrs, qgs, qs).

(d) All the flow rates actually vary with time, so you decide to implement a feedback control system in

which the qg is varied depending on the value of h. What constitutive relation (control law) would

you implement?

(e) The variation of the different flow rates is approximated as following sinusoidal trends, i.e., qi = qis +

ai sin kit, where qi is either qr or q. Specialize the differential equation to reflect this form and your

result from (d).

(f) Use the Matlab M-file solve_de.m, available on Canvas, to solve the differential equation with the

numerical parameters below. You may use any initial condition that you like. Based on your

simulations with different values of any adjustable parameter in your control law, what value of the

parameter would you suggest using?