Question 5 of 5: Sanitary sewer trunk design

Question 5: Sanitary sewer trunk design (25 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.

Given. $P_0=5000$; growth $r=5\%/\text{yr}$; design period $n=20$ yr; domestic production $q=450\ \text{L/(cap}\cdot\text{d)}$; inflow & infiltration $50\ \text{L/(cap}\cdot\text{d)}$; trunk slope $S=0.02$, $n=0.011$ (concrete).

Find. Design population, average domestic flow, min/max design flows, required diameter, and the velocities at max/min flow (with an acceptability check).

Approach. Project population geometrically; convert to average flow; apply a Harmon peak factor for the maximum and a minimum-flow factor, adding constant I&I; size the pipe from the full-flow Manning equation; then use partial-flow hydraulics to get the velocities.

1. Design population (Part 1). Geometric growth $P_n=P_0(1+r)^n = 5000(1.05)^{20}=5000(2.653)= \boxed{13{,}266}$ persons.
2. Average domestic flow (Part 2). $Q_{avg}=P_n\,q = 13{,}266\times450\ \text{L/d}=5.97\times10^{6}\ \text{L/d}= \boxed{5{,}970\ \text{m}^3/\text{d}}$. Constant infiltration/inflow adds $Q_{II}=13{,}266\times50=663\ \text{m}^3/\text{d}$.
3. Max and min design flows (Part 3). Harmon peak factor $M=1+\dfrac{14}{4+\sqrt{P}}=1+\dfrac{14}{4+\sqrt{13.27}}=2.83$ ($P$ in thousands). Peaking the domestic flow and adding constant I&I: $Q_{max}=2.83(5{,}970)+663= \boxed{17{,}570\ \text{m}^3/\text{d}}$. A minimum-flow factor $f_{min}=0.2\,P^{0.16}=0.30$ gives $Q_{min}=0.30(5{,}970)+663= \boxed{2{,}470\ \text{m}^3/\text{d}}$.
4. Required diameter (Part 4). $Q_{max}=17{,}570\ \text{m}^3/\text{d}=0.2034\ \text{m}^3/\text{s}$. For a pipe flowing full, $Q_{full}=\dfrac{0.312}{n}D^{8/3}S^{1/2}$, so $D=\left(\dfrac{Q_{max}\,n}{0.312\,S^{1/2}}\right)^{3/8}=\left(\dfrac{0.2034}{28.36\times0.1414}\right)^{3/8}=\boxed{0.327\ \text{m}}$. Adopt the next commercial size, $D=375$ mm.
5. Velocities (Part 5). For $D=0.375$ m: $Q_{full}=\dfrac{0.312}{0.011}(0.375)^{8/3}(0.02)^{1/2}=0.293\ \text{m}^3/\text{s}$ and $V_{full}=Q_{full}/(\pi D^2/4)=2.66\ \text{m/s}$. At $Q_{max}/Q_{full}=0.69$ the pipe runs $\approx61\%$ full and (partial-flow Manning) $V_{max}= \boxed{2.87\ \text{m/s}}$; at $Q_{min}/Q_{full}=0.097$ it runs $\approx21\%$ full and $V_{min}= \boxed{1.68\ \text{m/s}}$. Both lie between the self-cleansing minimum ($\ge0.6\ \text{m/s}$) and the scour limit ($\le3\ \text{m/s}$), so the velocities are acceptable.