Find the discharge through a rectangular channel 3 m wide, having depth of water 2 m and bed slope as 1 in 1500. Take the value of K = 2.36 in Bazin’s formula.

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Rectangular Channel Flow (Bazin's Formula)

Problem Statement

Find the discharge through a rectangular channel 3 m wide, having depth of water 2 m and bed slope as 1 in 1500. Take the value of K = 2.36 in Bazin's formula.

Given Data & Constants

  • Width of channel, \(B = 3 \, \text{m}\)
  • Depth of water, \(d = 2 \, \text{m}\)
  • Bed slope, \(i = 1 \text{ in } 1500 = \frac{1}{1500}\)
  • Bazin's constant, \(K = 2.36\)

Solution

1. Calculate Geometric Properties

$$ \text{Area of flow, } A = B \times d = 3 \times 2 = 6 \, \text{m}^2 $$ $$ \text{Wetted Perimeter, } P = B + 2d = 3 + 2 \times 2 = 7 \, \text{m} $$ $$ \text{Hydraulic Mean Depth, } m = \frac{A}{P} = \frac{6}{7} \approx 0.857 \, \text{m} $$

2. Calculate Chezy's Constant (C) using Bazin's Formula

The metric version of Bazin's formula is used to find Chezy's constant from the roughness coefficient K.

$$ C = \frac{87}{1 + \frac{K}{\sqrt{m}}} $$ $$ C = \frac{87}{1 + \frac{2.36}{\sqrt{0.857}}} = \frac{87}{1 + \frac{2.36}{0.9258}} \approx \frac{87}{1 + 2.549} $$ $$ C = \frac{87}{3.549} \approx 24.51 $$

3. Calculate Velocity and Discharge

Now we use Chezy's formula with the calculated value of C to find the velocity and discharge.

$$ V = C \sqrt{m \cdot i} $$ $$ V = 24.51 \times \sqrt{0.857 \times \frac{1}{1500}} = 24.51 \times \sqrt{0.0005713} $$ $$ V = 24.51 \times 0.0239 \approx 0.586 \, \text{m/s} $$ $$ Q = A \times V = 6 \, \text{m}^2 \times 0.586 \, \text{m/s} \approx 3.516 \, \text{m}^3/\text{s} $$
Final Result:

The discharge through the rectangular channel is approximately \(3.52 \, \text{m}^3/\text{s}\).

Explanation of Bazin's Formula

While Chezy's formula (\(V = C \sqrt{m \cdot i}\)) provides a general relationship for open channel flow, the Chezy constant 'C' is not a true constant; it depends on the channel's roughness and its hydraulic mean depth. Several empirical formulas were developed to calculate 'C' more accurately.

Bazin's formula is one such method. It calculates 'C' based on the hydraulic mean depth (\(m\)) and a roughness coefficient (\(K\)) specific to the channel's surface material. This approach provides a more refined estimate of the flow velocity compared to using a single, fixed value for 'C' across all flow conditions.

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