Question 1 of 7: Closed-Pipe Hydraulics and Water Distribution Components

Question 1: Closed-Pipe Hydraulics and Water Distribution Components (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) Deterministic vs. Stochastic Conceptual Models

A deterministic model produces exactly one output for a given set of inputs: it is built from physical laws (conservation of mass, momentum and energy; infiltration and routing equations) and, run twice with identical inputs, always returns the identical hydrograph. A stochastic model instead treats one or more inputs, parameters or the process itself as random variables described by a probability distribution, so its output is a probability distribution of possible outcomes (or an ensemble of synthetic traces) rather than a single trace. Two main differences follow from this: (1) treatment of uncertainty — deterministic models assume the input-output relationship is exactly known and repeatable, while stochastic models explicitly encode the natural randomness of climate and rainfall and report a range/likelihood of outcomes; and (2) basis of the relationship — deterministic models are physically based (e.g. a kinematic-wave or SCS curve-number rainfall-runoff transform), whereas stochastic models are built from statistical analysis of historical records (frequency distributions, time-series/Markov structure) without necessarily encoding the governing physics.

A deterministic model is preferred when a single defensible answer is needed for a specific, known design storm — for example, sizing a culvert or storm sewer for the design rainfall hyetograph of a stated return period, where the engineer needs one traceable, physically justified hydrograph. A stochastic model is preferred when the question is fundamentally about long-term risk or variability rather than one event — for example, a flood-frequency analysis of a gauge's annual maximum series (fitting a Log-Pearson III distribution) to estimate the discharge associated with a 100-year return period for floodplain mapping, or a stochastic rainfall generator used to test a reservoir's long-term yield reliability against decades of natural rainfall variability that no single deterministic storm can represent.

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

Given. Concrete 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, then a Colebrook-family friction factor (Swamee–Jain explicit form) to evaluate Darcy–Weisbach head loss.

1. (a) Average velocity. Cross-sectional area and continuity: $$A = \frac{\pi d^2}{4} = \frac{\pi (0.500)^2}{4} = 0.1963\ \text{m}^2, \qquad V = \frac{Q}{A} = \frac{1.000}{0.1963} = \boxed{5.09\ \text{m/s}}.$$
2. (b) Reynolds number. $$Re = \frac{Vd}{\nu} = \frac{(5.09)(0.500)}{1.0\times10^{-6}} = \boxed{2.55\times10^{6}}.$$ Since $Re \gg 4000$, the flow is turbulent.
3. (c) Friction factor. With relative roughness $\varepsilon/d = 0.0003/0.500 = 6.0\times10^{-4}$, the Swamee–Jain explicit friction factor is $$f = \frac{0.25}{\left[\log_{10}\left(\dfrac{\varepsilon/d}{3.7} + \dfrac{5.74}{Re^{0.9}}\right)\right]^2} = 0.0176.$$
4. (c) Friction head loss. Darcy–Weisbach: $$H_f = f\,\frac{L}{d}\,\frac{V^2}{2g} = (0.0176)\left(\frac{1000}{0.500}\right)\frac{(5.09)^2}{2(9.81)} = \boxed{46.6\ \text{m}}.$$

(iii) Relief and Air/Vacuum Valves

A combination air-and-vacuum valve mounted at a pipeline high point performs two functions: a large orifice opens to admit air into the pipe whenever internal pressure drops below atmospheric (during draining, or during a main break or pump shutdown that lets the column of water separate), preventing a vacuum from forming; the same or a companion small orifice continuously bleeds off air that collects at the high point during normal pressurized operation (entrained air released from solution, or air pockets from filling), and admits/expels large volumes of air rapidly during initial filling or draining of the main. A relief (pressure-relief) valve is a separate, spring- or pilot-operated device that opens automatically once internal pressure exceeds a set threshold — typically triggered by a transient/water-hammer surge from rapid valve closure, pump trip or power failure — discharging water to relieve the excess pressure before it can burst the pipe or damage fittings.

Two important reasons these valves are necessary: (1) protection of pipe integrity against both extremes of transient pressure — a rigid pipe (especially large-diameter concrete or PVC) can collapse under external soil/atmospheric pressure once an internal vacuum forms, while the same pipe can rupture under a positive transient surge, so both air/vacuum and relief valves are needed to bound the pressure envelope the pipe experiences; and (2) maintenance of hydraulic capacity and operational safety — trapped air pockets at high points reduce the effective flow area and increase head loss (and can produce violent, damaging surging as a large air pocket is suddenly expelled, known as "air valve slam"), so continuous air release during normal operation keeps the main running at its design capacity and avoids transient damage from sudden air expulsion.