Final answer:
To determine the channel bottom slope, use Manning's equation and solve for S after calculating the cross-sectional area and hydraulic radius. With some assumptions about Manning's roughness coefficient, the slope can be computed.
Step-by-step explanation:
Calculating Channel Slope for Conveying Water
To determine the required channel bottom slope for a rectangular channel that conveys water at a rate of 25 m³/s, where the width is limited to 4.0 m and the maximum flow depth is 1.2 m, we use the Manning's equation for open channel flow:
Manning's equation: Q = (1/n) * A * R^(2/3) * S^(1/2)
Where:
- Q is the flow rate (25 m³/s)
- n is the Manning's roughness coefficient
- A is the cross-sectional area of flow
- R is the hydraulic radius (A/P, with P being the wetted perimeter)
- S is the slope of the channel bottom
First, we will calculate the cross-sectional area (A) and the hydraulic radius (R). Given the constraints, the rectangular cross-section has a width (b) of 4.0 m and a depth (d) of 1.2 m.
A = b * d = 4.0 m * 1.2 m = 4.8 m²
The wetted perimeter (P) for a rectangular section is P = b + 2d = 4.0 m + 2(1.2 m) = 6.4 m
R = A / P = 4.8 m² / 6.4 m = 0.75 m
To find the slope (S), we rearrange Manning's equation and solve for S:
S = { [Q * n / (A * R^(2/3))] }^2
Without a specified Manning's roughness coefficient, we cannot determine the precise value of S. However, typical values of n for a concrete channel range between 0.012 to 0.017. Assuming n = 0.013:
S = { [25 m³/s * 0.013 / (4.8 m² * (0.75 m)^(2/3))] }^2
After calculating, we would get the required channel bottom slope.