The specific energy for a 6 m wide rectangular channel is to be 5 kg-m/kg. If the rate of flow of water through the channel is 24 m³/s, determine the alternate depths of flow.

Alternate Depths Calculation

Problem Statement

Given Data & Constants

Width of channel, $B = 6 \, \text{m}$
Specific Energy, $E = 5 \, \text{kg-m/kg} = 5 \, \text{m}$
Discharge, $Q = 24 \, \text{m}^3/\text{s}$
Acceleration due to gravity, $g = 9.81 \, \text{m/s}^2$

Solution

1. Calculate Discharge per Unit Width (q)

$$ q = \frac{Q}{B} = \frac{24 \, \text{m}^3/\text{s}}{6 \, \text{m}} = 4 \, \text{m}^2/\text{s} $$

2. Set up the Specific Energy Equation

The specific energy equation relates the depth, velocity, and specific energy. We can express velocity in terms of the discharge per unit width ($V = q/d$).

$$ E = d + \frac{V^2}{2g} = d + \frac{q^2}{2gd^2} $$ $$ 5 = d + \frac{4^2}{2 \times 9.81 \times d^2} = d + \frac{16}{19.62 d^2} $$ $$ 5 = d + \frac{0.8155}{d^2} $$

Rearranging this gives a cubic equation in terms of depth (d).

$$ 5d^2 = d^3 + 0.8155 $$ $$ d^3 - 5d^2 + 0.8155 = 0 $$

3. Solve for the Alternate Depths (d)

This cubic equation must be solved by trial and error or using a numerical solver. The equation will have two positive roots, which are the alternate depths.

By solving the equation, we find the two depths:

$$ d_1 \approx 4.96 \, \text{m} \quad (\text{Subcritical Flow Depth}) $$ $$ d_2 \approx 0.41 \, \text{m} \quad (\text{Supercritical Flow Depth}) $$

Final Result:

The alternate depths of flow are approximately 4.96 m and 0.41 m.

Explanation of Alternate Depths

For any given discharge in an open channel, there is a certain amount of specific energy ($E$). Unless the flow is at the critical depth (where specific energy is at a minimum), there are always two possible depths at which the flow can have that same amount of specific energy. These are known as **alternate depths**.

Subcritical Flow ($d_1$): This is a deep, slow-moving flow. The potential energy (depth) is high, and the kinetic energy (velocity) is low.
Supercritical Flow ($d_2$): This is a shallow, fast-moving flow. The potential energy (depth) is low, and the kinetic energy (velocity) is high.

A flow can transition between these two states through a hydraulic jump (from supercritical to subcritical) or over a smooth weir (from subcritical to supercritical).