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18-Env-A2 Hydrology and Municipal Hydraulics Engineering · May 2013

Question 4 of 7: Closed-Pipe Hydraulics, High-Lift Pump Capacity and Runoff Models

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — May 2013 — 04-Env-A2 / Hydrology and Municipal Hydraulics Engineering. 3 hours duration; closed book with an 8×11 in double-sided aid sheet; Casio or Sharp approved calculator only. Any five questions constitute a complete paper (first five answers marked, 20 marks each, 100 marks total); all seven are solved below for completeness.

Reference texts. Davis & Cornwell, Introduction to Environmental Engineering (6th ed.) — hydrology, stormwater management and water-demand chapters; Metcalf & Eddy, Wastewater Engineering: Treatment and Resource Recovery (5th ed.) — sanitary sewer hydraulics; MWH’s Water Treatment: Principles and Design (3rd ed.) — distribution systems and pumping; Chow, Open-Channel Hydraulics — Manning's n tables and specific-energy theory; Chow, Maidment & Mays, Applied Hydrology — frequency analysis; Guidelines for Canadian Drinking Water Quality (Health Canada).

Problem 4: Closed-Pipe Hydraulics, High-Lift Pump Capacity and Runoff Models (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) PVC Pipe Flow: Rate, Reynolds Number, Head Loss

Given.

Given data
QuantitySymbolValue
Pipe length$L$1000 m
Pipe diameter$d$500 mm = 0.500 m
Flow velocity$V$2 m/s
Material—PVC ($\varepsilon\approx0.0015$ mm)
L = 1000 m, d = 500 mm V = 2 m/s
PVC pipe schematic — length $L$, diameter $d$, full-flow velocity $V$.

Find. (a) $Q$ in m³/min, (b) $Re$ and flow regime, (c) head loss $H_f$.

Approach. Get $Q$ from continuity ($Q=VA$); get $Re$ from $Re=Vd/\nu$ to classify the flow; get the Darcy friction factor $f$ from the Swamee–Jain explicit correlation (PVC's roughness is very small, so $f$ is close to the smooth-pipe limit) and apply the Darcy–Weisbach equation for $H_f$.

Check: water temperature is not stated; a standard design value of 20°C ($\nu\approx1.0\times10^{-6}$ m²/s) is assumed. Colder water (e.g. 10°C, $\nu\approx1.3\times10^{-6}$ m²/s) would lower $Re$ slightly but the flow would remain fully turbulent either way.
  1. Flow rate (continuity). $$A=\frac{\pi d^2}{4}=\frac{\pi(0.5)^2}{4}=0.1963\ \text{m}^2,\qquad Q=VA=2\times0.1963=0.3927\ \text{m}^3/\text{s}=\boxed{23.6\ \text{m}^3/\text{min}}.$$
  2. Reynolds number. $$Re=\frac{Vd}{\nu}=\frac{2\times0.5}{1.0\times10^{-6}}=\boxed{1.0\times10^{6}}\ \Rightarrow\ Re\gg4000,\ \textbf{turbulent flow}.$$
  3. Friction factor (Swamee–Jain). $$f=\frac{0.25}{\left[\log_{10}\!\left(\dfrac{\varepsilon/d}{3.7}+\dfrac{5.74}{Re^{0.9}}\right)\right]^2}=\frac{0.25}{\left[\log_{10}(8.1\times10^{-7}+1.28\times10^{-5})\right]^2}=\boxed{0.0117}.$$
  4. Head loss (Darcy–Weisbach). $$H_f=f\frac{L}{d}\frac{V^2}{2g}=0.0117\times\frac{1000}{0.5}\times\frac{2^2}{2\times9.81}=\boxed{4.76\ \text{m}}.$$
QuantityValue
Flow rate, $Q$23.6 m³/min (0.393 m³/s)
Reynolds number, $Re$1.0×10&sup6; — turbulent
Friction factor, $f$0.0117
Head loss, $H_f$4.76 m over 1000 m

(ii) High-Lift Pump Rated Capacity

Given. Community served, 10,000 population equivalent (PE); pump located at the water treatment plant (post-treatment, distribution duty).

Approach. High-lift pump capacity is sized from the projected average day demand (ADD) and a peaking factor to obtain maximum (peak) day demand (MDD); the pump station is then rated (with standby) to deliver MDD, since it must be able to refill distribution storage even on the highest-demand day without relying on fire flow being drawn simultaneously (fire flow is normally checked separately, added to MDD, for the largest single unit out of service).

Check: no per-capita demand or peaking factor is given in the problem, so this uses typical Canadian municipal design values: average per-capita demand 450 L/(capita·day) (a commonly used mid-range Canadian design figure including residential + ICI use) and a maximum-day peaking factor of 2.0 (typical range 1.5–3.0 for a community this size) — both should be replaced with the municipality's own water-use study/design-guideline values if available.

  1. Average day demand (ADD). $$\text{ADD}=10{,}000\ \text{PE}\times450\ \tfrac{\text{L}}{\text{PE}\cdot\text{d}}=4.5\times10^{6}\ \text{L/d}=\boxed{4500\ \text{m}^3/\text{d}}=52.1\ \text{L/s}.$$
  2. Maximum (peak) day demand (MDD). Applying a peaking factor $PF_{\max\text{-day}}=2.0$: $$\text{MDD}=PF\times\text{ADD}=2.0\times4500=\boxed{9000\ \text{m}^3/\text{d}}=104.2\ \text{L/s}.$$
  3. Rate the pump station. The high-lift station is rated to deliver MDD (104 L/s $\approx$ 375 m³/h) continuously with the largest pump out of service (N+1 standby), i.e. the remaining duty pump(s) alone must still cover MDD; total installed capacity is therefore somewhat larger than 104 L/s once standby is included.
QuantityValue
Average day demand4500 m³/d (52.1 L/s)
Maximum day demand (design basis for pump rating)9000 m³/d (104.2 L/s)

(iii) Conceptual vs. Analytical Runoff Models

Two important properties of conceptual runoff models: (1) they represent the watershed with simplified, lumped storage-and-routing elements (e.g., a linear or nonlinear reservoir for the unit hydrograph, or the SCS Curve Number for losses) whose parameters are calibrated against observed rainfall-runoff data rather than derived purely from first principles; and (2) they are computationally efficient and require comparatively little input data, trading some physical realism for practicality, which is why they dominate everyday design practice (e.g., Rational Method, SCS unit hydrograph).

These differ from analytical (physically based) models, which solve the governing equations of fluid motion and mass conservation directly (e.g., the kinematic-wave or full Saint-Venant equations for overland/channel flow, Richards' equation for infiltration) using measured physical parameters (slope, roughness, soil hydraulic properties) rather than calibrated lumped coefficients. Analytical models are more data- and computation-intensive and are usually distributed (spatially varying) rather than lumped, giving better process fidelity and transferability to ungauged conditions, at the cost of requiring detailed physical characterization that conceptual models can substitute with a single calibrated coefficient.