Determine the full flow velocity of wastewater (accurate to 2 dec.pl.) in a concrete sewer with a diameter of 425 mm and a slope of \( 0.017 \mathrm{~m} / \mathrm{m} \).

To determine the full flow velocity of wastewater in a concrete sewer, we can use the Manning's equation, which is commonly used for open channel flow. The equation is given by: \[ V = \frac{1}{n} R^{2/3} S^{1/2} \] where: - \( V \) = flow velocity (m/s) - \( n \) = Manning's roughness coefficient (dimensionless) - \( R \) = hydraulic radius (m) - \( S \) = slope of the energy grade line (dimensionless), which is equal to the slope of the channel in this case. ### Step 1: Calculate the Hydraulic Radius \( R \) The hydraulic radius \( R \) is defined as: \[ R = \frac{A}{P} \] where: - \( A \) = cross-sectional area of flow (m²) - \( P \) = wetted perimeter (m) For a circular pipe flowing full, the area \( A \) and wetted perimeter \( P \) can be calculated as follows: 1. **Diameter of the pipe**: \( D = 0.425 \, \text{m} \) 2. **Cross-sectional area \( A \)**: \[ A = \frac{\pi D^2}{4} = \frac{\pi (0.425)^2}{4} \approx 0.140 \, \text{m}^2 \] 3. **Wetted perimeter \( P \)** (for a full circular pipe): \[ P = \pi D = \pi (0.425) \approx 1.336 \, \text{m} \] 4. **Hydraulic radius \( R \)**: \[ R = \frac{A}{P} = \frac{0.140}{1.336} \approx 0.105 \, \text{m} \] ### Step 2: Determine the Manning's Roughness Coefficient \( n \) For concrete pipes, the Manning's roughness coefficient \( n \) is typically around \( 0.013 \). ### Step 3: Calculate the Slope \( S \) The slope \( S \) is given as \( 0.017 \, \text{m/m} \). ### Step 4: Substitute Values into Manning's Equation Now we can substitute the values into Manning's equation: \[ V = \frac{1}{0.013} (0.105)^{2/3} (0.017)^{1/2} \] Calculating each part: 1. **Calculate \( R^{2/3} \)**: \[ R^{2/3} = (0.105)^{2/3} \approx 0.069 \] 2. **Calculate \( S^{1/2} \)**: \[ S^{1/2} = (0.017)^{1/2} \approx 0.130 \] 3. **Now substitute back into the equation**: \[ V = \frac{1}{0.013} \times 0.069 \times 0.130 \] Calculating: \[ V \approx \frac{1}{0.013} \times 0.00897 \approx 0.691 \, \text{m/s} \] ### Final Result The full flow velocity of wastewater in the concrete sewer is approximately: \[ \boxed{0.69 \, \text{m/s}} \]