A single-tank liquid-level system is shown in Figure 7.45, in which water flows into the tank at a volume flow rate \(q_{\mathrm{i}}\) and out of the tank through two valves at points 1 and 2 . The linear resistances of the two valves are \(R_{1}\) and \(R_{2}\), respectively. Assuming that \(h>h_{1}\), derive the differential equation relating the liquid height \(h\) and the volume flow rate \(q_{\mathrm{i}}\) at the inlet. The cross-sectional area of the tank \(A\) is constant. The density \(ho\) of the liquid is constant.

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Pa FIGURE 7.45 Problem 2. A