Question 4 of 7

Problem 4 (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) Concrete pipe: flow rate, Reynolds number, head loss (6 marks)

Given.

Find. $Q$ in m³/min; $Re$ and flow type; head loss $H_f$.

Approach. Continuity gives $Q=VA$; $Re=Vd/\nu$ classifies the flow; the Darcy–Weisbach equation with a friction factor from the explicit Swamee–Jain fit (equivalent to reading the Moody chart at this $Re$ and relative roughness) gives $H_f$.

1. (a) Flow rate. $$A=\frac{\pi d^2}{4}=\frac{\pi(0.400)^2}{4}=0.1257\ \text{m}^2 \qquad Q = VA = 2.5\times0.1257 = 0.3142\ \text{m}^3/\text{s}$$ $$Q = 0.3142\times60 = \boxed{18.85\ \text{m}^3/\text{min}}$$
2. (b) Reynolds number. $$Re = \frac{Vd}{\nu} = \frac{2.5\times0.400}{1.0\times10^{-6}} = \boxed{1.00\times10^{6}}$$ Since $Re \gg 4000$, the flow is turbulent.
3. (c) Head loss. Relative roughness $\varepsilon/d = 0.3/400 = 0.00075$; the Swamee–Jain friction factor is $$f = \frac{0.25}{\left[\log_{10}\!\left(\dfrac{\varepsilon/d}{3.7}+\dfrac{5.74}{Re^{0.9}}\right)\right]^2} = 0.0188$$ $$H_f = f\,\frac{L}{d}\,\frac{V^2}{2g} = 0.0188\times\frac{1200}{0.400}\times\frac{2.5^2}{2(9.81)} = \boxed{17.97\ \text{m}}$$

(ii) Waterhammer: causes and solutions (8 marks)

Waterhammer is the transient pressure surge generated when a moving column of water is suddenly decelerated, converting its momentum into a pressure wave that travels back and forth through the pipe at (near) the speed of sound in water. Two potential causes are: (1) rapid valve closure — a valve (including a check valve slamming shut) closed faster than the pipeline's critical closure time reflects the full momentum of the flowing column into a pressure spike; and (2) sudden pump trip/power failure — loss of power stops the pump essentially instantaneously while the water column in the discharge main continues moving under its own inertia, and the resulting flow reversal against a closing check valve produces a severe surge.

Two potential solutions are: (1) controlled (slow) valve operation and surge-anticipating/soft-closing check valves — extending the closure time beyond the pipeline's critical period (based on the pressure-wave travel time) allows the surge to dissipate gradually rather than reflect as a sharp spike; and (2) surge-control devices such as air/vacuum-relief valves, surge (relief) valves set to open above a threshold pressure, or a surge tank/standpipe/air chamber on the discharge main, which absorb or vent the transient energy before it can damage the pipe, joints or fittings.

(iii) Conceptual vs. analytical runoff models (6 marks)

Conceptual models of runoff represent a catchment's rainfall-runoff response as a simplified system of interconnected linear or non-linear storage elements (e.g., a cascade of linear reservoirs, or the widely used unit-hydrograph approach itself) whose parameters are calibrated against observed rainfall and streamflow records rather than derived purely from first-principles physics; they are used operationally for design-storm hydrograph generation, flood forecasting and reservoir/detention-pond routing because they are computationally simple and only need a modest calibration dataset. Two important differences from analytical (physically-based) models are: (1) basis of the governing equations — conceptual models use empirical storage–discharge relationships calibrated to fit observed behaviour, while analytical models solve the actual physical equations of motion (e.g., the kinematic-wave or full St. Venant equations for overland and channel flow) from first principles using measured physical parameters (slope, roughness, geometry); and (2) data and transferability — a conceptual model's calibrated parameters are specific to the gauged catchment they were fit to and transfer poorly to an ungauged basin, whereas an analytical model's physically-based parameters can, in principle, be estimated from maps and surveys and applied to an ungauged catchment, at the cost of much greater data and computational demand.