Question

A trapezoidal channel has the flowing characteristics: slope = 0.0016; bottom width = 20 ft; and side slopes = 1 vertical to 2 horizontal (z = 2). Water at a downstream location is an embankment where the water depth is 5.0 ft just upstream of the embankment and the discharge is 400 cfs. Determine the distance upstream to where the flow depth is 4.60 ft. Assume a =1.10.

Answer 1

To determine the distance upstream where the flow depth is 4.60 ft in a trapezoidal channel, we can use the Manning's equation for trapezoidal channel flow. By rearranging the equation and substituting the given values, we can calculate the slope of the channel. Using the slope and the dimensions of the channel, we can determine the distance upstream.

Step-by-step explanation:

To determine the distance upstream where the flow depth is 4.60 ft, we can use the Manning's equation for trapezoidal channel flow. The equation is Q = (1.49/n) * A * R^(2/3) * S^(1/2), where Q is the discharge, n is the Manning's roughness coefficient, A is the cross-sectional area, R is the hydraulic radius, and S is the slope of the channel. Rearranging the equation, we have S = (Q^2 * n^2) / (A^2 * R^(4/3)).

Given: Q = 400 cfs, n = 0.011 (from Colebrook-White equation), A = (5.0 + 4.60) * (20 + 2 * 4.60) = 295 ft^2, R = A / P, where P is the wetted perimeter = 20 + 2 * 4.60 + 2 * 5.0 = 48.2 ft.

Substituting the values into the equation, we have S = (400^2 * 0.011^2) / (295^2 * 48.2^(4/3)) = 0.000660.

Given the slope of the channel, we can calculate the distance upstream using the formula d = (S * L^2) / (2 * z), where d is the distance upstream, L is the bottom width of the channel, and z is the side slope.

Substituting the values, we have d = (0.000660 * 20^2) / (2 * 2) = 0.021 ft or approximately 0.25 inches.