Find the slope of the free water surface of a rectangular stream 15 m wide and 3 m deep. The slope of the stream bed is 1 in 5000. Total discharge is 29 m³/s. Assume the slope of the energy line = 1 in 21,700 and that the depth is increasing in the direction of the flow.

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BHEL Engineer Trainee Civil 24 Aug 2023 Official Paper

Answer 1

Concept:

The gradually varied flow (GVF) equation derived from energy slope and channel bed slope is:

$\frac{dy}{dx} = \frac{S_0 - S_f}{1 - F_r^2} $

Given:

Width of stream (b) = 15 m

Depth of stream (y) = 3 m

Discharge (Q) = 29 m³/s

Bed slope $S_0 = \frac{1}{5000} = 0.0002$

Energy slope $S_f = \frac{1}{21700} = 0.000046$

Step 1: Area and velocity

$ A = b \cdot y = 15 \cdot 3 = 45 \, m^2 $

$ V = \frac{Q}{A} = \frac{29}{45} = 0.644 \, m/s $

Step 2: Froude number

$ F_r = \frac{V}{\sqrt{g y}} = \frac{0.644}{\sqrt{9.81 \cdot 3}} = \frac{0.644}{5.427} \approx 0.1186 $

Step 3: Apply GVF equation

$\frac{dy}{dx} = \frac{0.0002 - 0.000046}{1 - (0.1186)^2} = \frac{0.000154}{1 - 0.0141} = \frac{0.000154}{0.9859} \approx 0.0001562$

Step 4: Find slope ratio

$ \frac{1}{0.0001562} \approx 6400 $