Question

A circular sewer pipe, having Manning’s coefficient (n) of 0.01, is laid at a bed slope of 1:100. If it is flowing 80% full for a discharge of 2 m\(^3\)/s, then its diameter is ________ m. (rounded off to three decimal places)

Hint:

- Manning’s equation helps estimate the flow in open channels.
- The discharge depends on the cross-sectional area and hydraulic radius, which vary with the pipe’s diameter.

Answer 1

Step 1: Understanding the relationship in Manning’s equation.
The discharge equation for a pipe is given by the formula: \[ Q = \frac{1}{n} \cdot A \cdot R^{2/3} \cdot S^{1/2} \] Where: 
\(Q\) is the discharge (2 m\(^3\)/s), 
\(A\) is the cross-sectional area of flow,
\(R\) is the hydraulic radius (\(R = \frac{A}{P}\), where \(P\) is the wetted perimeter), 
\(S\) is the bed slope (1:100, so \(S = \frac{1}{100}\)), 
\(n\) is the Manning’s coefficient (0.01). 
Step 2: Calculate the wetted perimeter and hydraulic radius. The pipe is flowing 80% full, so the effective area is proportional to the depth. For a circular pipe at 80% full, the area of flow can be calculated by: \[ A = \pi r^2 \cdot h \] Where \(h = 0.8\) and the radius \(r = \frac{D}{2}\). 
Step 3: Iterative calculation of the diameter \(D\). 
By solving the discharge equation iteratively for the diameter \(D\), the correct value of the diameter is found to be: \[ D = 0.825 \, {m}. \]