Question 7 of 7: Pipe Network Analysis, the Hardy-Cross Method and Centrifugal Pump System Design

Problem 7: Pipe Network Analysis, the Hardy-Cross Method and Centrifugal Pump System Design (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) The Two Governing Conditions in Pipe Network Analysis

Any closed (looped) pipe network analysis must satisfy two conditions, directly analogous to Kirchhoff's laws for electrical circuits: (1) continuity (mass conservation) at every node — the sum of flows into a junction must equal the sum of flows out, because water cannot accumulate or be created at a junction; and (2) energy conservation around every closed loop — the algebraic sum of head losses around any closed loop must equal zero (going around back to the starting point, elevation and pressure must return to their original values), because head loss is a path-independent function of pipe flow and there is only one true pressure at each node regardless of which route is used to compute it. Both conditions must hold simultaneously for every pipe in the network, which is exactly why looped-network analysis requires an iterative method (Hardy-Cross, or a modern gradient/Newton solver) rather than the simple series/parallel algebra sufficient for a single branched pipeline.

(ii) Hardy-Cross Method: Steps, Assumption, and Practical Consideration

Two fundamental steps: (1) assume a trial set of pipe flows that satisfies continuity at every node (an initial guess, distributed so inflow=outflow at each junction) — this can be an arbitrary but conservative starting distribution; and (2) compute and apply a flow correction to each loop, $\Delta Q=-\dfrac{\sum h_L}{n\sum(h_L/Q)}$ (with $h_L=KQ^n$, $n\approx1.85$–2 depending on the loss formula used), iterating loop-by-loop and repeating over the whole network until the head-loss imbalance $\sum h_L$ in every loop is acceptably close to zero.

Key assumption: the head-loss exponent $n$ in $h_L=KQ^n$ is treated as constant within an iteration (e.g., $n=1.85$ for Hazen-Williams or $n=2$ for Darcy-Weisbach), which linearizes the correction formula; the method converges because this approximation is locally accurate near the true solution even though the underlying relation is nonlinear.

Practical step for full-scale application: because hand (or spreadsheet) Hardy-Cross iteration becomes impractical for a network with more than a handful of loops, real distribution-system models solve the same continuity+energy conditions with computer software (e.g., EPANET) using a gradient/Newton algorithm, and are calibrated against field-measured pressures and flows (fire-flow/hydrant tests) before being trusted for design decisions — an uncalibrated model, however elegant the solution method, does not represent the actual as-built network's roughness and minor losses.

(iii) Centrifugal Pump System Selection and Design

Given. Lake-shore water treatment plant; maximum static head 100 m; maximum distribution length 5 km; centrifugal pump system to be selected.

Approach/Steps. Four key steps, worked through with the given static head and length as an illustrative example:

1. Determine the system's total dynamic head (TDH) requirement. $\text{TDH}=H_{\text{static}}+H_{f}+H_{\text{minor}}+H_{\text{residual}}$, where $H_{\text{static}}=100$ m (given), $H_f$ is friction loss over the 5 km main (from Darcy–Weisbach/Hazen-Williams at the design flow and selected pipe diameter), $H_{\text{minor}}$ covers fittings/valves, and $H_{\text{residual}}$ is the minimum required pressure at the town's service connections (commonly ≥140 kPa, roughly 14 m, under Canadian design guidelines). This produces the system curve, $H_{\text{sys}}(Q)=H_{\text{static}}+KQ^2$, since friction/minor losses grow with $Q^2$.
2. Establish the design flow. Size the required flow from the town's maximum day demand (plus fire flow, checked as a separate lower-head/higher-flow operating condition per NFPA/AWWA fire-flow criteria) — analogous to the pump-capacity approach used in Problem 4(ii).
3. Select a pump (or pumps) whose characteristic curve intersects the system curve at the design flow. Plot the manufacturer's $H$–$Q$ characteristic curve against the system curve; their intersection is the operating point. Prefer a pump whose best-efficiency point (BEP) falls at or very near this operating point, and check the Net Positive Suction Head available (NPSH$_A$, generous here since the plant draws directly from the lake) exceeds the pump's NPSH required.
4. Check operability across the full demand range and provide standby. Because demand varies from average to peak day (and fire flow), verify the selected pump(s) operate without excessive throttling, cavitation, or motor overload across that whole range — commonly by staging multiple parallel pumps (so combinations can be switched in/out) rather than one pump sized only for the single peak point — and provide at least one standby unit (N+1) sized to maintain full duty with the largest pump out of service, consistent with the redundancy principle used for the high-lift station in Problem 4(ii).