Question 2 of 5: Pump characteristic curves

Question 2: Pump characteristic curves (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A single-pump test: discharge $Q=0,200,400,600,800,1000$ gpm at dynamic head $H=150,145,135,120,90,50$ ft respectively.

Find. The single-, series- and parallel-pump curves, and the pump arrangement that meets (a) 1,700 gpm at 80 ft and (b) 1,700 gpm at 160 ft.

Approach. Fit the test points to a head–discharge curve; build the series curve by adding heads at equal discharge and the parallel curve by adding discharges at equal head; then read which composite curve passes through (or above) each duty point.

1. Plot and fit the pump curve (Part 1). Plotting $H$ against $Q$ for the six test points gives the falling characteristic shown. A least-squares parabola through all six is $H = 148.4 + 0.0167\,Q - 1.138\times10^{-4}\,Q^{2}$ (ft, $Q$ in gpm), which reproduces every tabulated head within about 2.6 ft. Shut-off head is the measured 150 ft; extrapolating the curve to $H=0$ gives a single-pump maximum discharge of $\boxed{Q_{max}\approx 1{,}220\ \text{gpm}}$.
2. Series curve (Part 2). Two identical pumps in series pass the same discharge but each adds its head, so heads double at every $Q$: $H=300,290,270,240,180,100$ ft at $Q=0,200,400,600,800,1000$ gpm. This doubles the delivered head but does not raise the maximum discharge (still $\approx1{,}220$ gpm).
3. Parallel curve (Part 3). Two identical pumps in parallel share the head but add discharge, so discharges double at every $H$: $Q=0,400,800,1200,1600,2000$ gpm at $H=150,145,135,120,90,50$ ft. This doubles the deliverable flow but cannot exceed the 150 ft shut-off head.
4. Part 4 — 1,700 gpm at 80 ft. A single pump cannot do it: read off the curve at $H=80$ ft and it delivers only $Q\approx850$ gpm (linear interpolation between the printed 800 gpm/90 ft and 1,000 gpm/50 ft points gives exactly 850 gpm; the fitted parabola gives 852). Series does not help — two pumps in series still pass at most $\approx1{,}220$ gpm. Two in parallel each carry 850 gpm at the common head $H\approx80\ \text{ft}$, so the pair delivers 1,700 gpm at 80 ft: the duty point lies on the parallel curve. $\Rightarrow$ use $\boxed{\text{two pumps in parallel}}$.
5. Part 5 — 1,700 gpm at 160 ft. This point needs both high flow ($1700>1220$ gpm, so parallel is required) and high head ($160>150$ ft shut-off, so series is required). Neither two-pump arrangement alone suffices: two in parallel give 1,700 gpm at only $\approx80$ ft, and two in series cap at $\approx1{,}220$ gpm. A series–parallel combination is needed — two parallel branches, each of two pumps in series (four pumps). Each branch carries 850 gpm and produces $2\times80 = 160\ \text{ft}$; the two branches together deliver 1,700 gpm. $\Rightarrow \boxed{\text{two series pairs in parallel (4 pumps)}}$.