Sewer diameter for a 2.5 sq km area

The sewer is taken as a combined sewer (sewage + storm water), running half full at maximum flow.

Step 1: Sanitary sewage

Area = 2.5 km² = 250 ha, so population $=400\times250=100000$. Assume 80% of the supply becomes sewage; peak factor = 2.7.

$$Q_{sew}=\frac{100000\times110\times0.8\times2.7}{86400\times1000}=0.2750\ \text{m}^3/\text{s}$$

Step 2: Storm water

Weighted coefficient: $C=0.5(0.5)+0.5(0.6)=0.55$. No intensity formula is given, so $i=\dfrac{760}{t+10}$ mm/hr is assumed for the given storm duration of 21 min (the time of concentration is 18 min, so a 21 min storm covers the whole catchment).

$$\begin{aligned} i&=\frac{760}{21+10}=24.52\ \text{mm/hr}\\ Q_{storm}&=\frac{0.55\times24.52\times250}{360}=9.364\ \text{m}^3/\text{s} \end{aligned}$$

Step 3: Total design discharge

$$Q=0.2750+9.364=9.639\ \text{m}^3/\text{s}$$

Step 4: Sewer design (half full, $S=1/500$, $n=0.013$)

For $d/D=0.5$: $\theta = 2\cos^{-1}(1-2d/D)=180.00^\circ$, $A/A_f=\dfrac{\theta-\sin\theta}{2\pi}=0.5000$, $R/R_f=1-\dfrac{\sin\theta}{\theta}=1.0000$, so $v/v_f=(R/R_f)^{2/3}=1.0000$ and $Q/Q_f=(A/A_f)(v/v_f)=0.5000$ (n taken constant).

Size of the sewer

Discharge when running full:

$$Q_f=\frac{Q}{Q/Q_f}=\frac{9.6388}{0.5000}=19.2776\ \text{m}^3/\text{s}$$

Manning's formula for a full circular pipe, $Q_f=\dfrac{0.3117}{n}D^{8/3}S^{1/2}$, with $S=0.00200$ and $n=0.013$:

$$D=\left(\frac{Q_f\,n}{0.3117\,S^{1/2}}\right)^{3/8}=\left(\frac{19.2776\times0.013}{0.3117\times0.04472}\right)^{3/8}=2.955\ \text{m}$$

Adopt the next standard size, D = 3000 mm.

Check of velocity and capacity

For the adopted pipe: $v_f=\dfrac{0.3969}{n}D^{2/3}S^{1/2}=2.840$ m/s and $Q_f=20.0739$ m³/s.

$$\begin{aligned} v&=v_f\times(v/v_f)=2.840\times1.0000=2.84\ \text{m/s}\\ Q_{capacity}&=Q_f\times(Q/Q_f)=20.0739\times0.5000=10.0370\ \text{m}^3/\text{s}\ (\ge 9.6388) \end{aligned}$$

The velocity at maximum flow, 2.84 m/s, lies between the minimum (0.6 m/s) and the maximum non-scouring limit (3.0 m/s), so it is satisfactory.

Answer: Diameter required = 2.955 m; provide 3000 mm circular sewer (running half full, v = 2.84 m/s).