Question 3 of 6: Two pumps and a 6-km pipeline over a hill
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper: National Exams — 98-Civ-A5 Hydraulic Engineering, May 2013 · 3 hours, closed book (one aid sheet). Six questions; any five constitute a complete paper — all six are solved here as a study resource. All parts of a question are of equal value.
Reference texts. Mays, Water Resources Engineering, 3rd ed. (pipe systems, pumps, network analysis); Chow, Open-Channel Hydraulics (Manning flow, compound sections); Crowe, Elger & Roberson, Engineering Fluid Mechanics (energy/continuity). Exam-supplied relations used throughout: Hazen–Williams $Q=0.278\,C\,D^{2.63}\,S^{0.54}$ with $S=h_f/L$ (SI), Manning $Q=\tfrac{1}{n}A\,R^{2/3}\,S^{1/2}$, and total dynamic head $\text{TDH}=H_s+H_f$. Unless stated, local losses and velocity head are neglected, and water has $\rho=1000\ \text{kg/m}^3$.
Check (Q4 data consistency): the pipe/valve data in Question 4 are internally inconsistent — the stated “initial valve flow = 400 L/s” corresponds to a node head of only 9.5 m, but the two supply pipes driven by the 96 m and 89 m tank levels deliver far more than that at 9.5 m. Continuity at the node fixes a network-consistent initial discharge of ≈693 L/s at a node head of ≈28.4 m. The simulation below is run from that physically consistent state, with the discrepancy noted (per Note 1, candidates may state assumptions).
Question 3: Two pumps and a 6-km pipeline over a hill (20 marks)
Given. Pipeline A(50 m)→C(40 m), $L=6000$ m, $D=0.800$ m, $C=140$; crest B at ground 55 m, 1200 m from A. Each pump: $H=20-9\,Q^{1.8}$.
Find. (a) pipeline flow for two pumps in series and in parallel; (b) whether the pressure head at B exceeds 14 m under series operation.
Figure 3 (Q3). Pump station at A lifts water over crest B (55 m) and down to reservoir C (40 m). Net static head A→C is favourable (−10 m); the crest sets the pressure-head constraint.
Approach. Build the system curve $H_{\text{req}}=(z_C-z_A)+h_f(Q)$ and intersect it with the combined pump curve — head-additive for series, flow-additive for parallel. Then, at the series operating point, walk the HGL to the crest B and compare its pressure head with 14 m.
Pipeline friction. With $C=140$, $D=0.800$ m: $0.278\,C\,D^{2.63}=21.64$, so $$h_f=\frac{6000}{21.64^{\,1.852}}Q^{1.852}=20.2\,Q^{1.852}.$$
System curve. Static lift $H_s=z_C-z_A=40-50=-10$ m (C is below A, so gravity assists). $$H_{\text{req}}(Q)=-10+20.2\,Q^{1.852}.$$
Series operation (heads add). Two identical pumps in series deliver $H_{\text{pump}}=2(20-9Q^{1.8})=40-18Q^{1.8}$ at the common flow $Q$. Setting $H_{\text{pump}}=H_{\text{req}}$: $$40-18Q^{1.8}=-10+20.2\,Q^{1.852}\ \Rightarrow\ \boxed{Q_{\text{series}}\approx 1.16\ \text{m}^3/\text{s}}\ (1160\ \text{L/s}),$$ at a pump head of 16.5 m.
Parallel operation (flows add). Each pump passes $Q/2$ at the common head, so $H_{\text{pump}}=20-9(Q/2)^{1.8}$. Setting equal to $H_{\text{req}}$: $$20-9(Q/2)^{1.8}=-10+20.2\,Q^{1.852}\ \Rightarrow\ \boxed{Q_{\text{parallel}}\approx 1.16\ \text{m}^3/\text{s}}\ (1161\ \text{L/s}).$$
Interpret series vs. parallel. The two configurations give essentially the same flow here (≈1.16 m³/s). This is expected: the system curve is steep (6 km of friction), so it intersects both combined curves at nearly the same discharge; series buys extra head but the low static lift means little extra flow results. On a friction-dominated line, doubling head (series) and doubling flow capacity (parallel) trade off to a near-identical operating point.
Pressure head at crest B (series). Series pump head $=40-18(1.16)^{1.8}=16.5$ m and $h_f(\text{A}\to\text{B})=\tfrac{1200}{6000}(20.2)(1.16)^{1.852}=5.3$ m. Energy from A: $$\text{HGL}_B=z_A+H_{\text{pump}}-h_{f,A\to B}=50+16.5-5.3=61.2\ \text{m}.$$ Pressure head at B: $$p_B/\gamma=\text{HGL}_B-z_B=61.2-55=6.2\ \text{m}.$$ Since $6.2\ \text{m}\lt 14\ \text{m}$: $$\boxed{\text{No — the pressure head at B }(\approx 6.2\text{ m})\text{ is below the 14 m minimum.}}$$
Question 3 — results
Quantity
Value
Flow, two pumps in series
≈ 1.16 m³/s (1160 L/s)
Flow, two pumps in parallel
≈ 1.16 m³/s (1161 L/s)
Pressure head at crest B (series)
≈ 6.2 m
Meets 14 m minimum at B?
No (6.2 m < 14 m)
Check: the crest B controls the design. To lift the pressure head at B to 14 m the pipeline would need a higher HGL there — e.g. an intermediate booster or a smaller-diameter (higher-loss is the wrong way; rather a larger-diameter downstream leg / air-valve and a re-sited or additional pump). The near-equal series/parallel flows also mean the choice of configuration should be made on the crest-pressure and NPSH grounds, not on delivered flow.
