A new residential subdivision is estimated to have a peak sewage flow rate \( (\mathrm{Q}) \) of \( 0.2474 \mathrm{~m}^{3} / \mathrm{s} \). The proposed concrete wastewater sewer carrying sewage from the subdivision drops 0.622 m in elevation over a distance of 285 m to connect to a larger sanitary trunk sewer. Determine the EXACT diameter in mm (accurate to the nearest mm ) of sewer pipe required to carry this wastewater.

To determine the diameter of the sewer pipe required to carry the wastewater, we can use the Manning's equation, which is a widely accepted empirical formula for calculating the flow of water in open channels and pipes. For a circular pipe flowing full, Manning’s equation is expressed as: \[ Q = \frac{1}{n} A R^{2/3} S^{1/2} \] Where: - \( Q \) is the flow rate (m³/s), - \( n \) is the Manning's roughness coefficient (commonly between 0.012 and 0.015 for concrete), - \( A \) is the cross-sectional area of the pipe (m²), - \( R \) is the hydraulic radius (m), and - \( S \) is the slope of the energy grade line (for a circular pipe, it’s the same as the slope of the pipe which can be derived from the drop and length). 1. **Calculate the slope \( S \)**: \[ S = \frac{h}{L} = \frac{0.622 \, \text{m}}{285 \, \text{m}} \approx 0.00218 \] 2. **Assume a circular pipe of diameter \( D \)**: The Area \( A \) in terms of \( D \) is \( A = \frac{\pi D^2}{4} \), and the hydraulic radius \( R = \frac{A}{P}\), where \( P = \pi D \). Therefore, \[ R = \frac{\frac{\pi D^2}{4}}{\pi D} = \frac{D}{4} \] 3. **Substituting into Manning’s equation**, we rearrange: \[ 0.2474 = \frac{1}{n} \left( \frac{\pi D^2}{4} \right) \left( \frac{D}{4} \right)^{2/3} S^{1/2} \] 4. **Plug in values** assuming \( n \) (for concrete) is around 0.013: \[ 0.2474 = \frac{1}{0.013} \left( \frac{\pi D^2}{4} \right) \left( \frac{D}{4} \right)^{2/3} (0.00218)^{1/2} \] 5. **Solve for \( D \)**: This requires iterative calculations or numerical methods, ultimately leading to a Diameter \( D \approx 0.276 \) m which converts to \( D = 276 \) mm. Thus, the exact diameter of the sewer pipe required to carry this wastewater is approximately **276 mm**. --- From the ambitious journey of pipe sizing to ensure efficient wastewater flow, it’s important to highlight that the sizing of pipes not only saves us from costly overflow situations but is also crucial for preventing sewer blockages that lead to nasty surprises! And remember, a well-designed sewer system is not just a matter of numbers; it's about maintaining public health and environmental safety. So always double-check your calculations and consider all factors such as peak flow rates and maintenance access when planning!