Question

Determine the cross section of the greatest hydraulic efficiency for a trapezoidal channel with side slope of 1 vertical to 2 horizontal if the design discharge is 250 m3/s. The channel slope is 0.001 and Manning’s roughness factor is 0.020.

Answer 1

Consider the rectangular channel having width b and depth y.

Calculate width and depth of the trapezoidal channel:

From the Manning’s equation for flow

…… (1)

Where, Flow rate

Manning’s roughness coefficients

Flow area

Hydraulic radius

Side slope

From the diagram,Angle made with the horizontal surface

Calculate angle:

From the table, geometric properties of common open-channel shapes

Area …… (2)

Wetted Perimeter

To obtain optimum condition, eliminate b

From equation (2)

Substitute the value of in wetted perimeter, we get

Differentiate the above equation with respect to y

…… (3)

Let, apply the optimum conditions, for equation (3)

…… (4)

Equate equations (2) and (4)

…… (5)

Substitute equation (5) in (1)

Now substitute the value of A in wetted perimeter, then

Calculate hydraulic radius,

Calculate depth of the trapezoidal section:

Substitute: for, for, for in (1)

Simplify the above equation

Calculate width of the channel:

Therefore, the depth of channel isand Width of the channel is.