4. Flow Through Open Channels

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Subject: Hydraulics (CE 4001 Same as CC/CV 4001)

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Flow Through Open Channels: Unit 4 of CE 4001

In fluid mechanics, open channel flow refers to the movement of a fluid within a channel that is not pressurized, such as rivers, streams, and canals. Understanding the principles of flow through open channels is crucial for designing irrigation systems, flood control, and water transportation networks. In this blog, we will break down Unit 4 of the Hydraulics course (CE 4001) for 4th-semester mechanical engineering students at Rajasthan Polytechnic, focusing on key concepts such as flow geometry, discharge equations, velocity measurement, and important devices for measuring flow.

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4. Flow Through Open Channels

When fluids flow in an open channel, gravity is the primary force driving the flow. The analysis of such flows involves understanding various channel properties, calculating discharge, and measuring velocity.

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4.1 Geometrical Properties of Channel Section

To analyze open channel flow, understanding the geometrical properties of the channel is crucial. These properties help in determining flow efficiency and hydraulic performance.

4.1.1 Wetted Area

The wetted area is the portion of the channel cross-section that is in contact with the flowing fluid. It includes the area under the surface of the water, from the channel bed up to the water surface. In a rectangular channel, the wetted area is simply the width multiplied by the depth of flow.

Formula:

$$A = b \times d$$

Where:

• $A$ = Wetted area (m²)
• $b$ = Bottom width of the channel (m)
• $d$ = Flow depth (m)

4.1.2 Wetted Perimeter

The wetted perimeter is the length of the boundary between the flowing fluid and the channel surface. For a rectangular channel, the wetted perimeter is the sum of the channel bottom width and twice the depth of the flow. For trapezoidal channels, the wetted perimeter includes the sloped sides as well.

Formula:

• For a rectangular channel: $$P = b + 2d$$
  
• For a trapezoidal channel: $$P = b + 2d \left( \frac{1 + \tan \theta}{\cos \theta} \right)$$
  

Where:

• $P$ = Wetted perimeter (m)
• $\theta$ = Angle of side slope (degrees)

4.1.3 Hydraulic Radius for Rectangular and Trapezoidal Channel Section

The hydraulic radius (R) is a measure of how effectively a channel section allows water to flow. It is defined as the ratio of the wetted area to the wetted perimeter.

Formula:

$$R = \frac{A}{P}$$

Where:

• $A$ = Wetted area (m²)
• $P$ = Wetted perimeter (m)

For a rectangular channel, the hydraulic radius is:

$$R = \frac{b \times d}{b + 2d}$$

For a trapezoidal channel, the hydraulic radius is:

$$R = \frac{A}{P}$$

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4.2 Determination of Discharge by Chezy’s and Manning’s Equations

Chezy’s Equation

Chezy’s equation is used to calculate the velocity of flow in an open channel. The discharge is the product of the velocity and the cross-sectional area of the flow.

Formula:

$$V = C \sqrt{R \times S}$$

Where:

• $V$ = Flow velocity (m/s)
• $C$ = Chezy’s coefficient (depends on channel roughness)
• $R$ = Hydraulic radius (m)
• $S$ = Slope of the channel bed (dimensionless)

Manning’s Equation

Manning’s equation is another widely used formula for calculating the velocity of flow in an open channel, especially in natural channels like rivers and streams. It is preferred for practical applications due to its simplicity.

Formula:

$$V = \frac{1}{n} R^{2/3} S^{1/2}$$

Where:

• $n$ = Manning’s roughness coefficient (depends on the type of surface)
• $R$ = Hydraulic radius (m)
• $S$ = Slope of the channel bed (dimensionless)

The discharge (Q) is then calculated as:

$$Q = A \times V$$

Where:

• $A$ = Wetted area (m²)
• $V$ = Flow velocity (m/s)

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4.3 Conditions for Most Economical Rectangular and Trapezoidal Channel Section

To optimize flow and reduce construction costs, it's important to design channels with the most efficient cross-section. The economical cross-section minimizes the cost while maximizing flow.

Most Economical Rectangular Channel Section

For a rectangular channel, the most economical section is when the ratio of the width to the depth of flow is optimal. This is when the hydraulic radius is maximized. It can be derived that the most economical condition for a rectangular channel is when:

$$b = 2d$$

Where $b$ is the width and $d$ is the depth of the channel.

Most Economical Trapezoidal Channel Section

For a trapezoidal channel, the most economical section occurs when the side slope angle $\theta$ is such that the channel minimizes the perimeter and maximizes the hydraulic radius. The angle $\theta$ can be determined based on the desired flow and channel conditions.

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4.4 Discharge Measuring Devices

Discharge measurement is vital for managing water flow in open channels. Various devices are used to measure discharge, with notches being one of the simplest and most accurate methods.

4.4.1 Triangular Notch

A triangular notch is often used to measure discharge in small open channels. The flow over a triangular notch is related to the head of water above the notch.

Formula:

$$Q = C \times H^{3/2}$$

Where:

• $Q$ = Discharge (m³/s)
• $H$ = Head of water above the notch (m)
• $C$ = Calibration constant (depends on notch geometry)

4.4.2 Rectangular Notch

A rectangular notch is another type of weir used for measuring discharge. It has a simple linear relationship between the flow rate and the head of water.

Formula:

$$Q = C \times H^{3/2}$$

Where:

• $Q$ = Discharge (m³/s)
• $H$ = Head of water above the notch (m)
• $C$ = Calibration constant (depends on notch width)

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4.5 Velocity Measurement Devices

Velocity measurement is crucial in open channel flow analysis to calculate discharge and assess flow conditions. Several devices are used to measure the velocity of water.

4.5.1 Current Meter

A current meter is a device used to measure the velocity of the water at various depths and locations within a channel. The average velocity is then computed from the readings.

4.5.2 Floats

A float is a simple method where a floating object (such as a ball) is tracked to determine the surface velocity of the water. The velocity is calculated based on the time it takes the float to travel a certain distance.

4.5.3 Pitot’s Tube

A Pitot tube is used to measure the velocity of flow by determining the pressure difference between the static pressure and the dynamic pressure of the fluid. This method is particularly useful in high-velocity flow.

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4.6 Froude Number

The Froude number (Fr) is a dimensionless number that characterizes the type of flow in an open channel. It helps in determining whether the flow is subcritical, critical, or supercritical.

Formula:

$$Fr = \frac{V}{\sqrt{g \times d}}$$

Where:

• $V$ = Flow velocity (m/s)

• $g$ = Acceleration due to gravity (9.81 m/s²)

• $d$ = Depth of flow (m)

• Subcritical flow (Fr < 1): Flow is slow, with smooth flow patterns.

• Critical flow (Fr = 1): The flow is at the transition point between subcritical and supercritical.

• Supercritical flow (Fr > 1): Flow is fast and turbulent, typically in steep channels.

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Sample Questions for Practice

1. Calculate the discharge for a rectangular channel with a width of 3 meters, depth of 2 meters, and a slope of 0.001. Use Manning’s equation and assume $n = 0.015$.
2. What is the hydraulic radius for a trapezoidal channel with a bottom width of 5 meters, side slope angle of 30°, and a flow depth of 2 meters?
3. Determine the most economical section for a trapezoidal channel. Assume a channel with side slopes of 1:1.
4. Explain the working of a triangular notch and how it is used for discharge measurement.
5. Calculate the Froude number for a channel with a flow velocity of 4 m/s and a flow depth of 3 meters.

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Conclusion

This unit on Flow Through Open Channels provides essential knowledge for designing and analyzing open channel systems. From geometrical properties to measuring devices, it covers all the key aspects necessary for efficient water flow management in channels. Understanding these principles will help engineers optimize flow conditions and improve the design of water conveyance systems.

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