6.9 Venturi Meter | Fluid Mechanics | Physics 11

A Venturi meter is a device used to measure the flow rate (or discharge) of a fluid moving through a pipe. It works by narrowing the path of the fluid, which causes its velocity to increase and its pressure to decrease. By measuring this pressure difference, the flow rate can be calculated. The device operates based on Bernoulli's Equation and the Equation of Continuity.

Construction

The Venturi meter consists of three main parts:

Converging Section: A short pipe that gradually decreases in diameter, causing the fluid to accelerate.
Throat: The narrowest part of the meter, where the fluid velocity is at its maximum and the pressure is at its minimum.
Diverging Section: A longer pipe that gradually increases in diameter, allowing the fluid to slow down and recover its pressure.

Working Principle

The operation of a Venturi meter is based on two fundamental principles of fluid dynamics:

Equation of Continuity ( $A_{1} V_{1} = A_{2} V_{2}$ ): As the fluid flows from the wider inlet to the narrow throat, its velocity must increase to maintain a constant mass flow rate.
Bernoulli's Principle ( $P + \frac{1}{2} ρ v^{2} = constant$ for a horizontal pipe): As the fluid's velocity increases at the throat, its pressure must decrease.

This pressure difference between the inlet and the throat is measured, typically with a U-tube manometer, and is directly related to the fluid's flow rate.

Section	Cross-Sectional Area	Fluid Velocity	Fluid Pressure
Inlet	Large ( $A_{1}$ )	Lower ( $V_{1}$ )	Higher ( $P_{1}$ )
Throat	Small ( $A_{2}$ )	Higher ( $V_{2}$ )	Lower ( $P_{2}$ )

Mathematical Derivation

The formula for the flow rate is derived by combining Bernoulli's equation and the equation of continuity.

Step 1: Apply Bernoulli's Equation

For an ideal, incompressible fluid flowing horizontally through the meter ( $h_{1} = h_{2}$ ), Bernoulli's equation between the inlet (point 1) and the throat (point 2) is:

$P_{1} + \frac{1}{2} ρ V_{1}^{2} = P_{2} + \frac{1}{2} ρ V_{2}^{2}$

Rearranging to find the pressure difference:

$P_{1} - P_{2} = \frac{1}{2} ρ (V_{2}^{2} - V_{1}^{2})$

Step 2: Apply the Equation of Continuity

The mass flow rate is constant:

$A_{1} V_{1} = A_{2} V_{2}$

We can express the velocity at the throat ( $V_{2}$ ) in terms of the inlet velocity ( $V_{1}$ ):

$V_{2} = \frac{A _{1}}{A _{2}} V_{1}$

Step 3: Combine the Equations

Substitute the expression for $V_{2}$ from Step 2 into the Bernoulli's equation from Step 1:

$P_{1} - P_{2} = \frac{1}{2} ρ ((\frac{A _{1}}{A _{2}} V_{1})^{2} - V_{1}^{2})$

Factor out $V_{1}^{2}$ :

$P_{1} - P_{2} = \frac{1}{2} ρ V_{1}^{2} (\frac{A _{1}^{2}}{A _{2}^{2}} - 1)$

Step 4: Solve for the Velocity ( $V_{1}$ )

Rearrange the equation to solve for the inlet velocity, $V_{1}$ :

$V_{1}^{2} = \frac{2 ( P _{1} - P _{2} )}{ρ ( \frac{A _{1}^{2}}{A _{2}^{2}} - 1 )} = \frac{2 ( P _{1} - P _{2} ) A _{2}^{2}}{ρ ( A _{1}^{2} - A _{2}^{2} )}$

$V_{1} = A_{2} \frac{2 ( P _{1} - P _{2} )}{ρ ( A _{1}^{2} - A _{2}^{2} )}$

Step 5: Express in Terms of Manometer Height ( $h$ )

The pressure difference is often measured with a manometer, where $P_{1} - P_{2} = ρ g h$ . Substituting this into the equation:

$V_{1} = A_{2} \frac{2 ρ g h}{ρ ( A _{1}^{2} - A _{2}^{2} )}$

The fluid density $ρ$ cancels out, giving the final expression for the velocity at the inlet:

$V_{1} = \frac{A _{2} 2 g h}{A _{1}^{2} - A _{2}^{2}}$

The volume flow rate ( $Q$ ) is then found by multiplying the inlet velocity by the inlet area: $Q = A_{1} V_{1}$ .

Possible Questions and Answers

Q: Why is the diverging section of a Venturi meter longer than the converging section?

A: The longer, gradual diverging section is designed to ensure a smooth and gradual deceleration of the fluid. This minimizes energy losses due to turbulence and allows for maximum pressure recovery, making the meter more efficient.

Q: What are some real-world applications of the Venturi meter?

A: Venturi meters are used to measure the flow of liquids and gases in industrial pipelines, water treatment plants, and laboratories. The Venturi effect is also the principle behind carburetors in older engines and some types of aerosol sprayers.

Pressure Units→

Key Formula	Description
$V_{1} = \frac{A _{2} 2 g h}{A _{1}^{2} - A _{2}^{2}}$	Calculates the fluid velocity at the inlet based on the height difference ( $h$ ) in a manometer.

Construction

The Venturi meter consists of three main parts:

Converging Section: A short pipe that gradually decreases in diameter, causing the fluid to accelerate.
Throat: The narrowest part of the meter, where the fluid velocity is at its maximum and the pressure is at its minimum.
Diverging Section: A longer pipe that gradually increases in diameter, allowing the fluid to slow down and recover its pressure.

Working Principle

The operation of a Venturi meter is based on two fundamental principles of fluid dynamics:

Equation of Continuity ( $A_{1} V_{1} = A_{2} V_{2}$ ): As the fluid flows from the wider inlet to the narrow throat, its velocity must increase to maintain a constant mass flow rate.
Bernoulli's Principle ( $P + \frac{1}{2} ρ v^{2} = constant$ for a horizontal pipe): As the fluid's velocity increases at the throat, its pressure must decrease.

This pressure difference between the inlet and the throat is measured, typically with a U-tube manometer, and is directly related to the fluid's flow rate.

Section	Cross-Sectional Area	Fluid Velocity	Fluid Pressure
Inlet	Large ( $A_{1}$ )	Lower ( $V_{1}$ )	Higher ( $P_{1}$ )
Throat	Small ( $A_{2}$ )	Higher ( $V_{2}$ )	Lower ( $P_{2}$ )

Mathematical Derivation

The formula for the flow rate is derived by combining Bernoulli's equation and the equation of continuity.

Step 1: Apply Bernoulli's Equation

For an ideal, incompressible fluid flowing horizontally through the meter ( $h_{1} = h_{2}$ ), Bernoulli's equation between the inlet (point 1) and the throat (point 2) is:

$P_{1} + \frac{1}{2} ρ V_{1}^{2} = P_{2} + \frac{1}{2} ρ V_{2}^{2}$

Rearranging to find the pressure difference:

$P_{1} - P_{2} = \frac{1}{2} ρ (V_{2}^{2} - V_{1}^{2})$

Step 2: Apply the Equation of Continuity

The mass flow rate is constant:

$A_{1} V_{1} = A_{2} V_{2}$

We can express the velocity at the throat ( $V_{2}$ ) in terms of the inlet velocity ( $V_{1}$ ):

$V_{2} = \frac{A _{1}}{A _{2}} V_{1}$

Step 3: Combine the Equations

Substitute the expression for $V_{2}$ from Step 2 into the Bernoulli's equation from Step 1:

$P_{1} - P_{2} = \frac{1}{2} ρ ((\frac{A _{1}}{A _{2}} V_{1})^{2} - V_{1}^{2})$

Factor out $V_{1}^{2}$ :

$P_{1} - P_{2} = \frac{1}{2} ρ V_{1}^{2} (\frac{A _{1}^{2}}{A _{2}^{2}} - 1)$

Step 4: Solve for the Velocity ( $V_{1}$ )

Rearrange the equation to solve for the inlet velocity, $V_{1}$ :

$V_{1}^{2} = \frac{2 ( P _{1} - P _{2} )}{ρ ( \frac{A _{1}^{2}}{A _{2}^{2}} - 1 )} = \frac{2 ( P _{1} - P _{2} ) A _{2}^{2}}{ρ ( A _{1}^{2} - A _{2}^{2} )}$

$V_{1} = A_{2} \frac{2 ( P _{1} - P _{2} )}{ρ ( A _{1}^{2} - A _{2}^{2} )}$

Step 5: Express in Terms of Manometer Height ( $h$ )

The pressure difference is often measured with a manometer, where $P_{1} - P_{2} = ρ g h$ . Substituting this into the equation:

$V_{1} = A_{2} \frac{2 ρ g h}{ρ ( A _{1}^{2} - A _{2}^{2} )}$

The fluid density $ρ$ cancels out, giving the final expression for the velocity at the inlet:

$V_{1} = \frac{A _{2} 2 g h}{A _{1}^{2} - A _{2}^{2}}$

The volume flow rate ( $Q$ ) is then found by multiplying the inlet velocity by the inlet area: $Q = A_{1} V_{1}$ .

Possible Questions and Answers

Q: Why is the diverging section of a Venturi meter longer than the converging section?

Q: What are some real-world applications of the Venturi meter?

Pressure Units→

Key Formula	Description
$V_{1} = \frac{A _{2} 2 g h}{A _{1}^{2} - A _{2}^{2}}$	Calculates the fluid velocity at the inlet based on the height difference ( $h$ ) in a manometer.