Problem Statement
Find the rate of change of depth of water in a rectangular channel of 12 m wide and 2 m deep, when the water is flowing with a velocity of 1.5 m/s. The flow of water through the channel of bed slope 1 in 3000, is regulated in such a way that energy line is having a slope of 1 in 8000.
Given Data & Constants
- Channel width, \(B = 12 \, \text{m}\)
- Depth of flow, \(d = 2 \, \text{m}\)
- Velocity of flow, \(V = 1.5 \, \text{m/s}\)
- Bed slope, \(S_0 = 1 \text{ in } 3000 = \frac{1}{3000}\) (*Corrected from 1 in 300*)
- Energy line slope, \(S_f = 1 \text{ in } 8000 = \frac{1}{8000}\)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)
Solution
1. Calculate the Froude Number (\(Fr\))
The Froude number is needed to determine the flow regime (subcritical or supercritical).
$$ Fr = \frac{V}{\sqrt{g d}} = \frac{1.5}{\sqrt{9.81 \times 2}} \approx 0.3386 $$2. Apply the Gradually Varied Flow Equation
The rate of change of depth with respect to the distance along the channel (\(dy/dx\)) is given by the gradually varied flow equation.
$$ \frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2} $$ $$ \frac{dy}{dx} = \frac{\frac{1}{3000} - \frac{1}{8000}}{1 - (0.3386)^2} $$ $$ \frac{dy}{dx} = \frac{0.000333 - 0.000125}{1 - 0.1146} = \frac{0.000208}{0.8854} $$ $$ \frac{dy}{dx} \approx 0.000235 $$The rate of change of depth of water is approximately 0.000235.
(This means the water depth is increasing at a rate of about 0.235 mm for every 1 meter traveled downstream).
Explanation of Gradually Varied Flow
This problem describes a state of **Gradually Varied Flow (GVF)**. This occurs when the depth of flow changes gradually along the length of the channel. It is different from uniform flow, where the depth is constant.
- Slopes: In GVF, the slope of the channel bed (\(S_0\)) is not equal to the slope of the energy line (\(S_f\)). The energy line slope represents the rate at which energy is lost due to friction.
- Equation: The gradually varied flow equation, \( \frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2} \), describes how the depth (\(y\)) changes with distance (\(x\)).
- Interpretation: In this case, the bed slope (\(1/3000\)) is steeper than the energy loss slope (\(1/8000\)), and the flow is subcritical (\(Fr < 1\)). The positive result for \(dy/dx\) indicates that the water surface is rising, creating a "backwater curve." This typically happens when a downstream obstruction, like a dam or weir, forces the water level to increase.