Figure 7.12 shows a liquid-level system in which two tanks have cross-sectional areas \(A_{1}\) and \(A_{2}\), respectively. A pump is connected to the bottom of tank 1 through a valve of linear resistance \(R_{1}\). The liquid flows from tank 1 to tank 2 through a valve of linear resistance \(R_{2}\) and leaves tank 2 through a valve of linear resistance \(R_{3}\). The density \(ho\) of the liquid is constant.

a. Derive the differential equations in terms of the liquid heights \(h_{1}\) and \(h_{2}\). Write the equations in second-order matrix form.

b. Assume the pump pressure \(\Delta p\) as the input and the liquid heights \(h_{1}\) and \(h_{2}\) as the outputs. Determine the state-space form of the system.

Transcribed Image Text:

Pa Ap FIGURE 7.12 A two-tank liquid-level system. Pa R Pa A R3 90