Question 1 of 7: Runoff Models, Closed-Pipe Hydraulics, and Pressure Relief Valves

Problem 1: Runoff Models, Closed-Pipe Hydraulics, and Pressure Relief Valves (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) Key Steps in Developing a Conceptual Runoff Model

Developing a conceptual model to predict peak runoff proceeds through three key steps. (1) Watershed characterization and schematization. The catchment is delineated and represented as a small number of interconnected storage/routing elements (interception storage, upper-zone soil moisture, lower-zone or groundwater storage, and a channel-routing element), with the physical drainage area, land cover, soil type and time of concentration used to set initial parameter estimates — this converts an intractable distributed physical problem into a lumped or semi-distributed structure that can actually be solved. (2) Selection and estimation of the abstraction/transfer functions. Each storage element needs a rule governing how water enters, is held, and is released (an infiltration-capacity curve, a linear-reservoir recession constant, a unit-hydrograph ordinate set); these are chosen based on watershed characteristics and, where available, regional or literature-derived coefficients. (3) Simulation and comparison against observed data. The assembled model is run against historical rainfall to generate simulated hydrographs.

The conceptual model is best validated by calibrating its free parameters against one portion of the observed streamflow record (matching simulated to observed peak discharge, time-to-peak and runoff volume), then running the calibrated model, unchanged, against an independent portion of the record it was not fitted to (a split-sample or hold-out test). A model that reproduces the peak and shape of the verification-period hydrographs without further parameter adjustment is validated; systematic bias or a badly mistimed peak in the hold-out period signals that the storage/transfer structure — not just the parameter values — needs revisiting before the model is trusted for design or forecasting.

(ii) PVC Pipe — Velocity, Reynolds Number, Friction Loss

Given. PVC pipe flowing full:

Find. The average velocity $V$, the Reynolds number $Re$ (and flow regime), and the friction head loss $H_f$.

Approach. Get $V$ from continuity, $Re$ from the pipe-flow definition, the Darcy friction factor $f$ from the Swamee–Jain explicit approximation to the Colebrook equation, then $H_f$ from the Darcy–Weisbach equation.

1. (a) Average velocity. Cross-sectional area and continuity: $$A = \frac{\pi d^2}{4} = \frac{\pi (0.600)^2}{4} = 0.2827\ \text{m}^2, \qquad V = \frac{Q}{A} = \frac{0.500}{0.2827} = \boxed{1.768\ \text{m/s}}.$$
2. (b) Reynolds number. $$Re = \frac{Vd}{\nu} = \frac{(1.768)(0.600)}{1.0\times10^{-6}} = \boxed{1.061\times10^{6}}.$$ Since $Re \gg 4000$, the flow is turbulent.
3. (c) Friction head loss via Darcy–Weisbach. Swamee–Jain friction factor: $$f = \frac{0.25}{\left[\log_{10}\left(\dfrac{\varepsilon}{3.7d}+\dfrac{5.74}{Re^{0.9}}\right)\right]^2} = \boxed{0.01156},$$ $$H_f = f\,\frac{L}{d}\,\frac{V^2}{2g} = (0.01156)\left(\frac{1000}{0.600}\right)\frac{(1.768)^2}{19.62} = \boxed{3.07\ \text{m}}.$$

(iii) Functions of Pressure Relief Valves

A pressure relief (surge-relief) valve protects a distribution main by opening automatically once line pressure exceeds a set threshold, and serves three related functions. (1) Water-hammer surge attenuation. When a pump trips or a valve closes quickly, the resulting Joukowsky pressure pulse propagates along the main; the relief valve opens within the transient's timescale and discharges a controlled volume, clipping the peak surge pressure before it can reach the pipe's or fittings' rated limit. (2) Protection of the weakest system component. Distribution networks contain joints, fittings, hydrants and older pipe sections with lower pressure ratings than the mains around them; a relief valve set just above normal operating pressure ensures a transient is bled off before it finds and ruptures the weakest link, which is usually cheaper and safer than up-rating every component in the network. (3) Leak and main-break prevention through routine over-pressure events. Diurnal demand swings, pump start/stop cycling and firefighting draws all produce smaller, repeated pressure excursions; continuously relieving these keeps the system inside its designed working-pressure envelope, which reduces the fatigue-driven joint leakage and pipe breaks that accumulate under chronic over-pressure even without a single dramatic transient event.