Consider the single-tank liquid-level system shown in Figure 7.11, where a pump is connected to the bottom of the tank through a valve of linear resistance \(R\). The inlet to the pump is open to the atmosphere, and the pressure of the fluid increases by \(\Delta p\) when crossing the pump. Derive the differential equation relating the liquid height \(h\) and the volume flow rate \(q_{\mathrm{o}}\) at the outlet. The tank's cross-sectional area \(A\) is constant. The density \(ho\) of the liquid is constant.

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1 R Pa Pa P1 P2 A FIGURE 7.11 A single-tank liquid-level system with a pump. 90