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# Derive the differential equations in terms of the liquid heights h1 and h2

Figure 7.43 shows a liquid-level system in which two tanks have hydraulic capacitances

C1 and C2, respectively. The volume flow rate into tank 1 is qi. The liquid flows from tank 1 to tank 2 through a valve of linear resistance R1 and leaves tank 2 through a valve of linear resistance R2. The density ρ of the liquid is constant.

a. Derive the differential equations in terms of the liquid heights h1 and h2. Write the equations in matrix form.

b. Assume that the volume flow rate qi is the input and the liquid heights h1 and h2 are the outputs. Determine the state-space form of the system.

c. Construct a Simulink block diagram to find the outputs h1(t) and h2(t) of the liquid-level system. Assume that ρ = 1000 kg/m3= 9.81 m/s2C1 = 0.15 kg・m2/N, C2 = 0.25 kg・m2/N,

R1 = R2 = 100 N・s/(kg・m2), and initial liquid heights h1(0) = 1 m and h2(0) = 0 m. The volume flow rate qi is a step function with a magnitude of 0 before = 0 s and a magnitude of

0.25 m3/s after = 0 s.

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