Question 5 of 7

Problem 5 (20 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.

(i) Conservation laws governing pipe-network analysis (6 marks)

Two conservation laws must be satisfied simultaneously at every point in a pipe network: (1) conservation of mass (continuity) at every node — the sum of flows entering a junction must equal the sum leaving it, since water cannot accumulate or be created at a node; this is used to constrain the flow distribution assumed in every trial solution (e.g., in Hardy-Cross balancing, every initial guess must already satisfy nodal continuity before loop corrections are applied). (2) conservation of energy around every closed loop — the algebraic sum of head losses around any closed loop of pipes must equal zero (equivalently, the head loss travelling clockwise around a loop equals the head loss travelling counter-clockwise), since pressure at any node is single-valued regardless of which path is used to reach it; this is used as the correction criterion in loop-balancing methods, where an assumed flow distribution that satisfies continuity but not loop energy balance is iteratively adjusted (via a correction flow $\Delta Q=-\dfrac{\sum KQ|Q|}{\sum 2K|Q|}$ per loop, in the Hardy-Cross method) until both laws hold together, at which point the resulting flows and the corresponding nodal residual pressures are the network's true solution.

(ii) Continuity between a 200 mm and a 150 mm pipe (6 marks)

Given.

Find. Velocity $V_{150}$ in the 150 mm pipe; volumetric flow rate $Q$.

Approach. Assuming steady, incompressible flow through a single pipeline (no branching flow lost or gained between the two diameters), continuity $Q_{150}=Q_{200}$ gives $V_{150}$ directly from the area ratio.

1. Assumption. Steady, incompressible (constant-density) flow with no intermediate branch, so the same volumetric flow rate passes through both diameters: $A_{200}V_{200}=A_{150}V_{150}$.
2. Solve for $V_{150}$. $$V_{150} = V_{200}\left(\frac{D_{200}}{D_{150}}\right)^2 = 10\left(\frac{0.200}{0.150}\right)^2 = \boxed{17.78\ \text{m/s}}$$
3. Volumetric flow rate. $$A_{200}=\frac{\pi(0.200)^2}{4}=0.03142\ \text{m}^2 \qquad Q = V_{200}A_{200} = 10\times0.03142 = \boxed{0.3142\ \text{m}^3/\text{s}}$$

(iii) System pump-head curve: operating point, NPSH, shutoff head (8 marks)

The system-head curve plots the total head the pipeline demands as a function of flow rate, $H_{sys}(Q)=H_{static}+kQ^2$ — a constant static-lift term plus a friction-loss term that rises with the square of flow — while the pump curve plots the head a given pump can deliver, which falls as flow increases. The operating point is where the two curves intersect: the unique $(Q,H)$ pair at which the head the pump produces exactly equals the head the system demands, and is therefore where the pump will actually run once installed. The shutoff head is the pump curve's value at $Q=0$ (the head produced with the discharge valve fully closed, no flow) and represents the maximum head/pressure the pump can ever generate against that system. The net positive suction head compares the available energy at the pump suction above the liquid's vapour pressure, $\text{NPSH}_A$ (a property of the installation — suction lift, pipe losses, atmospheric pressure, fluid temperature), against $\text{NPSH}_R$ (a property of the pump itself, rising with flow, published by the manufacturer); the pump must always be operated with $\text{NPSH}_A > \text{NPSH}_R$ at the chosen operating point, or the pressure at the impeller eye will fall below vapour pressure and cause cavitation.

To select a proper pump configuration, the engineer overlays the calculated system-head curve on the manufacturer's family of pump curves (for different impeller trims or staging options) and reads off the operating point for each candidate pump: the selected unit should place the operating point at or near its curve's best-efficiency point (for economical, low-vibration operation over the facility's service life), deliver the required design flow at that point, and maintain $\text{NPSH}_A > \text{NPSH}_R$ with an adequate margin across the full range of anticipated system conditions (e.g., a fouled/aged pipe system curve that has shifted upward, or a lowered supply water level that reduces $\text{NPSH}_A$); where the single-pump operating point cannot meet the demand, pumps are staged in series (to add head) or in parallel (to add flow) and the combined curve is re-checked against the system curve the same way.