Question

A 4 m wide rectangular channel, having bed slope of 0.001 carries a discharge of 16 m³/s.

Considering Manning's roughness coefficient = 0.012 and g = 10 m/s², the category of the channel slope is:

1. Horizontal
2. Mild
3. Critical
4. Steep

Answer 1

Correct Answer - Option 2 : Mild

Concept:

For rectangular channel section

Side slope = 1 : m (m = horizontal and 1 = vertical)

Area of flow = A = BY

Hydraulic radius = R = Y/2

Wetted perimeter = P =  B + 2Y

Where,

B = width of the rectangular channel, Y = depth of the rectangular channel

Discharge through channel

\(Q = A \times V = A \times \frac{1}{n} \times {R^{\frac{2}{{3\;}}\;}} \times {S^{\frac{1}{2}}}\;\)

Where,

n = manning’s roughness coefficient, R = Hydraulic radius,S = bed slope

Calculation:

Given,

B = 4 m, S = 0.001, Q = 16 m³/s, n = 0.012

g = 10 m/s²

\(Q = A \times V = A \times \frac{1}{n} \times {R^{\frac{2}{{3\;}}\;}} \times {S^{\frac{1}{2}}}\;\)

\(16 = \left( {BY} \right) \times \frac{1}{{0.012}} \times {\left( {\frac{Y}{2}} \right)^{\frac{2}{{3\;}}\;}} \times {\left( {0.001} \right)^{\frac{1}{2}}}\)

\(16 = \left( {4Y} \right) \times \frac{1}{{0.012}} \times {\left( {\frac{Y}{2}} \right)^{\frac{2}{{3\;}}\;}} \times {\left( {0.001} \right)^{\frac{1}{2}}}\)

Y = depth of rectangular channel = 1.69 m

Critical depth of channel is given by,

\({y_c} = {\left( {\frac{{{q^2}}}{g}} \right)^{\frac{1}{3}}} = {\left( {\frac{{{4^2}}}{{9.81}}} \right)^{\frac{1}{3}}} = 1.169\)

y > y_c

Depth of channel is greater than critical depth of channel

∴ Flow is sub – critical and channel slope is mild.