Figure 7.18 shows a liquid-level system in which two tanks have cross-sectional areas \(A_{1}\) and \(A_{2}\), respectively. The volume flow rate into tank 1 is \(q_{\mathrm{i}}\). A pump is connected to the bottom of tank 1 , and the pressure of the fluid increases by \(\Delta p\) when crossing the pump. Tank 2 is located higher than tank 1, and the vertical distance between the two tanks is \(H\). The liquid is pumped from tank 1 to tank 2 through a valve of linear resistance \(R_{1}\) and leaves tank 2 through a valve of linear resistance \(R_{2}\). The density \(ho\) of the liquid is constant. Derive the differential equations in terms of the liquid heights \(h_{1}\) and \(h_{2}\). Write the equations in second-order matrix form.

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FIGURE 7.18 Problem 6. Ji Pa Pa A H R A2 R