Question 2 of 7: Conceptual Runoff Models, Corrugated Steel Pipe Hydraulics, and Distribution Valves

Problem 2: Conceptual Runoff Models, Corrugated Steel Pipe Hydraulics, and Distribution 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) Deterministic vs. Stochastic Conceptual Runoff Models

Deterministic example. The SCS/NRCS curve-number unit-hydrograph model — a fixed rainfall input, catchment CN, and unit hydrograph produce one repeatable runoff hydrograph. Stochastic example. An autoregressive (AR/ARMA) streamflow-generation model, or a stochastic (Monte-Carlo) rainfall generator that produces synthetic rainfall sequences matching the historical statistics.

Three important differences. (1) Output form — a deterministic model returns a single repeatable output for a given input, with no probability attached, whereas a stochastic model returns a distribution or ensemble of possible outputs generated from a fitted probability structure. (2) Basis — deterministic models are typically physically based, explicitly encoding processes such as infiltration and channel routing, while stochastic models are usually empirical/statistical, fitted to the historical record's statistical properties (mean, variance, autocorrelation) with little explicit physical mechanism. (3) Use case — deterministic, physically based models are used for event-based design (a specific storm on a specific catchment, e.g. sizing a culvert), while stochastic models are used to generate long synthetic sequences for risk and frequency analysis (e.g. reservoir yield/operation studies) where the natural year-to-year variability itself is the quantity of interest.

(ii) Corrugated Steel Pipe — Velocity, Reynolds Number, Friction Loss

Given. A corrugated steel pipe flowing full carries the data below.

Find. (a) mean velocity $V$; (b) Reynolds number $Re$ and flow regime; (c) friction head loss $H_f$.

Approach. Get $V$ from continuity on the full pipe area, classify the flow from $Re=Vd/\nu$, then use Manning's equation (the governing friction law given, since no Hazen–Williams $C$ or Darcy $f$ is supplied) to back out the friction slope $S_f$ and multiply by the pipe length for $H_f$.

1. (a) Mean velocity. Full-pipe area $A=\dfrac{\pi d^2}{4}=\dfrac{\pi(0.500)^2}{4}=0.1963\ \text{m}^2$. $$V=\frac{Q}{A}=\frac{0.300}{0.1963}=\boxed{1.53\ \text{m/s}}$$
2. (b) Reynolds number. With kinematic viscosity of water $\nu\approx1.0\times10^{-6}\ \text{m}^2/\text{s}$: $$Re=\frac{Vd}{\nu}=\frac{1.53\times0.500}{1.0\times10^{-6}}=\boxed{7.64\times10^{5}}$$ Since $Re\gg4000$, the flow is turbulent.
3. (c) Friction head loss. For a full pipe, hydraulic radius $R=d/4=0.125\ \text{m}$. Manning's equation rearranged for the friction slope: $$S_f=\left(\frac{nV}{R^{2/3}}\right)^2=\left(\frac{0.03\times1.53}{0.125^{2/3}}\right)^2=0.0336$$ so $$H_f=S_f L=0.0336\times300=\boxed{10.1\ \text{m}}$$

(iii) Swing Check Valve vs. Gate Valve

Swing check valve. A one-way, non-return valve with a hinged disc that swings open under forward flow and swings closed under any reverse flow (assisted by gravity/backflow pressure), preventing flow from running backward through it. It is used wherever reverse flow must be blocked automatically without operator action — classically on a pump's discharge line, to stop the water column in the main from draining back through the pump (and to limit the water-hammer surge) the instant the pump stops or loses power, and at any reservoir/tank connection where backflow into the supply main must be prevented.

Gate valve. A full-bore isolation valve in which a wedge/gate rises out of the flow path when open, giving a nearly unobstructed, low-head-loss passage; it is designed for fully-open or fully-closed operation and should not be used to throttle (a partially open gate causes vibration, cavitation, and erosion of the seating surfaces). It is used for isolating a section of main or a reservoir inlet/outlet connection for maintenance or emergency shutoff — wherever a low-loss on/off shutoff is required rather than automatic backflow prevention.