Frequently in the design of pump systems, a bypass line will be installed in parallel to the pump so that some of the fluid can recirculate as shown. The bypass valve then controls the flow rate in the system. Assume that the head-versus-discharge curve for the pump is given by h_p = 100 – 100Q, where h_p is in meters and Q is in m³/s. The bypass line is 10 cm in diameter. Assume the only head loss is that due to the valve, which has a head loss coefficient of 0.2. The discharge leaving the system is 0.2 m³/s. Find the discharge through the pump and bypass line.

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Valve Bypass line 0.20 m/s Pump