Question

A rectangular brick-lined channel (n = 0.016) of 4.0-m width is laid on a bottom slope of 0.0009. It carries a discharge of 15 m3 /s and the fl ow is non-uniform. If the depth at a Section A is 2.6 m, calculate the depth at section B, 500 m downstream of A, by using (a) only one step, and (b) two steps.

Answer 1

To calculate the depth at section B, 500 m downstream of section A, we can use the Chezy formula:

V = C*R^(1/2)

Where V is the velocity, C is the Chezy coefficient, and R is the hydraulic radius.

a) Using only one step:

Since the flow is non-uniform, the velocity at section B can be assumed to be the same as at section A. Therefore, the depth at section B can be calculated using the same Chezy coefficient and hydraulic radius as at section A.

Hydraulic Radius (R) = A/P = (width * depth) / 2

R_A = (4 * 2.6) / 2 = 5.2 m

R_B = R_A = 5.2 m

Chezy coefficient (C) = (V^2 * n) / (2 * g * R^(1/2))

C = (15^2 * 0.016) / (2 * 9.81 * 5.2^(1/2)) = 1.94

Now we can use the Chezy formula to calculate the depth at section B

V = C*R^(1/2)

V = 1.94 * 5.2^(1/2) = 3.23 m/s

b) Using two steps:

First, we can calculate the velocity at section B using the continuity equation:

Q = A1 * V1 = A2 * V2

15 = (4 * 2.6 * 3.23) = (4 * y * V2)

V2 = (15 / 4) / y = 3.75/y m/s

Next, we can use the Chezy formula and the velocity at section B to calculate the depth at section B:

V = C*R^(1/2)

y = V^2 * n / (C^2 * g)

y = (3.75/y)^2 * 0.016 / (1.94^2 * 9.81)

y = 2.34 m

So, the depth at section B is 2.34 m by using two steps.

Note: The above calculations are based on the assumption that the slope is uniform along the channel and the flow is steady. In practice, other factors such as channel roughness and boundary conditions may also have an impact on the depth of flow.