A Venturi meter is a differential pressure flow-measuring device used to determine the discharge of a fluid flowing through a closed conduit. It operates on the combined principles of the continuity equation and Bernoulli's theorem.

When a fluid flows through a pipe of varying cross-sectional area, its velocity and pressure change continuously. As the flow area decreases, the velocity increases and the pressure decreases. By measuring the pressure difference between two sections of the pipe, the rate of flow through the pipeline can be determined.

The Venturi meter is widely used because it provides accurate flow measurements with relatively small energy losses.

Everyday Intuition

The working principle of a Venturi meter can be observed in everyday situations.

• Water flows faster when the end of a garden hose is partially covered.
• A river flows more rapidly through a narrow channel than through a wide section.
• Air speeds up when passing through a narrow passage, causing a reduction in pressure.

In each case, the reduction in flow area increases the fluid velocity while decreasing the pressure. The Venturi meter uses this relationship to measure fluid discharge.

Experimental Relevance

The objective of the Venturi meter experiment is to determine the discharge of water flowing through a pipeline and to evaluate the coefficient of discharge of the Venturi meter.

As water enters the converging section,

• Flow area decreases,
• Velocity increases,
• Pressure decreases.

The pressure becomes minimum at the throat, where the area is smallest. The pressure difference between the inlet and the throat is measured using a differential manometer.

The theoretical discharge is calculated using Bernoulli's theorem and the continuity equation, while the actual discharge is determined by collecting water in a measuring tank.

The ratio of the actual discharge to the theoretical discharge gives the coefficient of discharge.

Mathematical Formulation

For steady, incompressible, and frictionless flow between two sections,

$$\frac{P_1}{\rho g} + \frac{V_1^2}{2g} + Z_1 === \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + Z_2.$$

For a horizontal Venturi meter,

$$Z_1=Z_2,$$

therefore,

$$\frac{P_1}{\rho g} + \frac{V_1^2}{2g} ====== \frac{P_2}{\rho g} + \frac{V_2^2}{2g}.$$

The continuity equation states

$$Q=A_1V_1=A_2V_2,$$

where

• $Q$ = Discharge,
• $A_1$ = Inlet area,
• $A_2$ = Throat area,
• $V_1$ = Inlet velocity,
• $V_2$ = Throat velocity.

Combining Bernoulli's equation with the continuity equation gives the theoretical discharge,

$$Q_t= \frac{A_1A_2} {\sqrt{A_1^2-A_2^2}} \sqrt{2gh},$$

where

$$h$$

is the differential head between the inlet and the throat.

Due to friction and other losses,

$$Q_a<CQ_t,$$

and the coefficient of discharge is defined as

$$C_d=\frac{Q_a}{Q_t},$$

where

• $Q_a$ = Actual discharge,
• $Q_t$ = Theoretical discharge.

A well-designed Venturi meter generally has a coefficient of discharge close to unity.

Application to the Venturi Meter Apparatus

The Venturi meter consists of three principal sections:

Converging Section

The flow area gradually decreases, increasing the velocity and reducing the pressure.

Throat

The throat is the section of minimum area where the velocity is maximum and the pressure is minimum.

Diverging Section

The area gradually increases, reducing the velocity and recovering part of the pressure energy.

Pressure taps are provided at the inlet and throat sections and connected to a differential manometer. The pressure difference obtained from the manometer is used to calculate the theoretical discharge.

Water passing through the Venturi meter is collected in a measuring tank to determine the actual discharge experimentally. Comparing the actual and theoretical values allows the coefficient of discharge to be evaluated.

Engineering Significance

Venturi meters are extensively used for flow measurement because they combine high accuracy with low head loss.

Important applications include:

• Water supply systems,
• Irrigation networks,
• Industrial pipelines,
• Hydraulic laboratories,
• Pumping stations,
• Chemical process industries,
• Hydroelectric installations.

The Venturi meter demonstrates the practical application of Bernoulli's theorem and the continuity equation for measuring fluid discharge and is one of the most widely used flow-measuring devices in hydraulic engineering.