Separate sewer, check self-cleansing

Given

Peak flow $Q_{max}=0.5$ m³/s, $S=10/10000=0.001$, $n=0.012$, $d/D=0.7$ at peak flow, peak factor = 3.

Design for peak flow

For $d/D=0.7$: $\theta = 2\cos^{-1}(1-2d/D)=227.16^\circ$, $A/A_f=\dfrac{\theta-\sin\theta}{2\pi}=0.7477$, $R/R_f=1-\dfrac{\sin\theta}{\theta}=1.1849$, so $v/v_f=(R/R_f)^{2/3}=1.1198$ and $Q/Q_f=(A/A_f)(v/v_f)=0.8372$ (n taken constant).

Size of the sewer

Discharge when running full:

$$Q_f=\frac{Q}{Q/Q_f}=\frac{0.5000}{0.8372}=0.5972\ \text{m}^3/\text{s}$$

Manning's formula for a full circular pipe, $Q_f=\dfrac{0.3117}{n}D^{8/3}S^{1/2}$, with $S=0.00100$ and $n=0.012$:

$$D=\left(\frac{Q_f\,n}{0.3117\,S^{1/2}}\right)^{3/8}=\left(\frac{0.5972\times0.012}{0.3117\times0.03162}\right)^{3/8}=0.887\ \text{m}$$

Adopt the next standard size, D = 900 mm.

Check of velocity and capacity

For the adopted pipe: $v_f=\dfrac{0.3969}{n}D^{2/3}S^{1/2}=0.975$ m/s and $Q_f=0.6202$ m³/s.

$$\begin{aligned} v&=v_f\times(v/v_f)=0.975\times1.1198=1.09\ \text{m/s}\\ Q_{capacity}&=Q_f\times(Q/Q_f)=0.6202\times0.8372=0.5193\ \text{m}^3/\text{s}\ (\ge 0.5000) \end{aligned}$$

The velocity at maximum flow, 1.09 m/s, lies between the minimum (0.6 m/s) and the maximum non-scouring limit (3.0 m/s), so it is satisfactory.

Check in dry weather flow

Average (dry weather) flow $=\dfrac{Q_{max}}{\text{peak factor}}=\dfrac{0.5}{3}=0.1667$ m³/s. For the adopted pipe ($D=0.900$ m, $Q_f=0.6202$ m³/s, $v_f=0.975$ m/s):

$$\frac{Q}{Q_f}=\frac{0.1667}{0.6202}=0.2687\ \Rightarrow\ \frac{d}{D}=0.354,\ \ \frac{v}{v_f}=0.848$$ $$v=0.848\times0.975=0.83\ \text{m/s}$$

At the minimum flow (about one-third of the average, $0.0556$ m³/s), $d/D=0.202$ and $v=0.60$ m/s.

The self-cleansing velocity is 0.6 m/s (0.75 m/s is preferred for sewage). The velocity at the average dry weather flow, 0.83 m/s, is well above 0.6 m/s, and even at the minimum flow the velocity is just about 0.6 m/s.

Answer: Provide a 900 mm sewer at 1 in 1000. Velocity at peak flow = 1.09 m/s; at dry weather (average) flow = 0.83 m/s, so the self-cleansing velocity is maintained.