A pump curve is H_p(Q) = 60 - 0.8 Q (m). System head loss is H_f(Q) = 20 + 0.5 Q^2 (m). Find the operating point Q.

• A. Q ≈ 8.18 m^3/s; H ≈ 53.46 m
• B. Q ≈ 6.0 m^3/s; H ≈ 40 m
• C. Q ≈ 10.0 m^3/s; H ≈ 20 m
• D. Q ≈ 2.0 m^3/s; H ≈ 60 m

Answer: (A)

The operating point is found at the intersection of the pump curve and the system head curve, where the pump can just overcome the system losses. Set the two heads equal: 60 − 0.8Q = 20 + 0.5Q^2. Rearranging gives 0.5Q^2 + 0.8Q − 40 = 0, or multiplying by 2, Q^2 + 1.6Q − 80 = 0. Solving the quadratic, Q = [-1.6 ± sqrt(1.6^2 − 4(1)(−80))]/2 = [-1.6 ± sqrt(322.56)]/2. The positive root is Q ≈ (−1.6 + 17.95)/2 ≈ 8.18 m^3/s (the negative root is not physically meaningful). The corresponding head is H = 60 − 0.8×8.18 ≈ 53.46 m. So the operating point is about Q ≈ 8.18 m^3/s and H ≈ 53.46 m, matching the given option.