Venturi Effect

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A pair of Venturi tubes on a light aircraft, used to provide airflow for air-driven gyroscopic instruments

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The upstream static pressure (1) is higher than in the constriction (2), and the fluid speed at "1" is lower than at "2", because the cross-sectional area at "1" is greater than at "2".

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A flow of air through a pitot tube Venturi meter, showing the columns connected in a manometer and partially filled with water. The meter is "read" as a differential pressure head in cm or inches of water.

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Video of a Venturi meter used in a lab experiment

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Idealized flow in a Venturi tube

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Venturi tube demonstration apparatus built out of PVC pipe and operated with a vacuum pump

Venturi Effect

The Venturi effect describes how fluid pressure decreases when a fluid flows through a constricted section of pipe. Named after Giovanni Battista Venturi, it has applications in measuring flow rates and mixing fluids.

Physical Principle

As an incompressible fluid passes through a constriction, it speeds up and loses static pressure in accordance with Bernoulli’s principle.

Flow Rate

A Venturi can be used to measure the volumetric flow rate, <math>Q</math>, using Bernoulli's principle and the continuity equation.

Since: <math> Q = v_1 A_1 = v_2 A_2 </math> and <math> p_1 - p_2 = \frac{\rho}{2} (v_2^2 - v_1^2) </math>

Then the flow rate can also be expressed as: <math> Q = A_1 \sqrt{ \frac{2}{\rho} \cdot \frac{p_1 - p_2}{\left(\frac{A_1}{A_2}\right)^2 - 1} } = A_2 \sqrt{ \frac{2}{\rho} \cdot \frac{p_1 - p_2}{1 - \left( \frac{A_2}{A_1} \right)^2} } </math>

This relationship allows precise determination of flow rate from pressure differentials across two sections of known cross-sectional area.

Differential Pressure

As fluid flows through a Venturi, the changes in velocity lead to corresponding changes in pressure. This differential pressure is central to flow measurement applications and can be expressed as:

<math> \Delta P = \frac{1}{2} \rho (v_2^2 - v_1^2) = \frac{1}{2} \left( \left( \frac{A_1}{A_2} \right)^2 - 1 \right) v_1^2 = \frac{1}{2} \left( \frac{1}{A_2^2} - \frac{1}{A_1^2} \right) Q^2 = k \rho Q^2 </math>

Where <math>k</math> is a constant incorporating geometric and flow characteristics.

Compensation for Temperature, Pressure, and Mass

Flow calculations must account for changes in temperature, pressure, and molar mass, especially when operating outside design conditions. The following relationships integrate ideal gas law corrections and offer various forms of compensated flow expression:

Density and concentration relationships: <math> C = \frac{P}{RT} = \frac{\left(\frac{P}{P^\ominus}\right)}{\left(\frac{T}{T^\ominus}\right)} C^\ominus </math> <math> \rho = \frac{MP}{RT} = \frac{\left(\frac{M}{M^\ominus} \cdot \frac{P}{P^\ominus}\right)}{\left(\frac{T}{T^\ominus}\right)} \rho^\ominus </math>

Substituting into flow-pressure equations yields the fully compensated pressure drop:

<math> \Delta P = k \cdot \frac{\left(\frac{M}{M^\ominus} \cdot \frac{P}{P^\ominus}\right)}{\left(\frac{T}{T^\ominus}\right)} \rho^\ominus Q^2 = \Delta P_{\max} \cdot \frac{\left(\frac{M}{M^\ominus} \cdot \frac{P}{P^\ominus}\right)}{\left(\frac{T}{T^\ominus}\right)} \left(\frac{Q}{Q_{\max}}\right)^2 </math>

<math> = k \cdot \frac{\left(\frac{T}{T^\ominus}\right)}{\left(\frac{M}{M^\ominus} \cdot \frac{P}{P^\ominus} \cdot \rho^\ominus\right)} \dot{m}^2 = \Delta P_{\max} \cdot \frac{\left(\frac{T}{T^\ominus}\right)}{\left(\frac{M}{M^\ominus} \cdot \frac{P}{P^\ominus}\right)} \left(\frac{\dot{m}}{\dot{m}_{\max}}\right)^2 </math>

<math> = k \cdot \frac{M \left(\frac{T}{T^\ominus}\right)}{\left(\frac{P}{P^\ominus}\right) C^\ominus} \dot{n}^2 = \Delta P_{\max} \cdot \frac{\left(\frac{M}{M^\ominus} \cdot \frac{T}{T^\ominus}\right)}{\left(\frac{P}{P^\ominus}\right)} \left(\frac{\dot{n}}{\dot{n}_{\max}}\right)^2 </math>

Design Point Normalization

The relationship between constants and maximum design parameters can be summarized as:

<math> \frac{k}{\Delta P_{\max}} = \frac{1}{\rho^\ominus Q_{\max}^2} = \frac{\rho^\ominus}{\dot{m}_{\max}^2} = \frac{{C^\ominus}^2}{\rho^\ominus \dot{n}_{\max}^2} = \frac{C^\ominus}{M^\ominus \dot{n}_{\max}^2} </math>

These expressions allow normalization across various system parameters to ensure precision and safety under changing environmental or operational conditions.