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# Venturi effect

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The Venturi effect is the reduction in that results when a moving fluid speeds up as it flows through a constricted section (or choke) of a pipe. The Venturi effect is named after its discoverer, the 18th-century Italian Giovanni Battista Venturi.

The effect has various engineering applications, as the reduction in pressure inside the constriction can be used both for measuring the fluid flow and for moving other fluids (e.g. in a ).

Background
In , an incompressible fluid's must increase as it passes through a constriction in accord with the principle of mass continuity, while its must decrease in accord with the principle of conservation of mechanical energy (Bernoulli's principle). Thus, any gain in a fluid may attain by its increased velocity through a constriction is balanced by a drop in pressure.

By measuring pressure, the flow rate can be determined, as in various devices such as Venturi meters, Venturi nozzles and .

Referring to the adjacent diagram, using Bernoulli's equation in the special case of steady, incompressible, inviscid flows (such as the flow of water or other liquid, or low-speed flow of gas) along a streamline, the theoretical pressure drop at the constriction is given by

$p_1 - p_2 = \frac{\rho}{2} (v_2^2 - v_1^2),$

where $\rho$ is the of the fluid, $v_1$ is the (slower) fluid velocity where the pipe is wider, and $v_2$ is the (faster) fluid velocity where the pipe is narrower (as seen in the figure).

Choked flow
The limiting case of the Venturi effect is when a fluid reaches the state of , where the approaches the local speed of sound. When a fluid system is in a state of choked flow, a further decrease in the downstream pressure environment will not lead to an increase in velocity, unless the fluid is compressed.

The mass flow rate for a compressible fluid will increase with increased upstream pressure, which will increase the density of the fluid through the constriction (though the velocity will remain constant). This is the principle of operation of a de Laval nozzle. Increasing source temperature will also increase the local sonic velocity, thus allowing increased mass flow rate, but only if the nozzle area is also increased to compensate for the resulting decrease in density.

Expansion of the section
The Bernoulli equation is invertible, and pressure should rise when a fluid slows down. Nevertheless, if there is an expansion of the tube section, turbulence will appear, and the theorem will not hold. In all experimental Venturi tubes, the pressure in the entrance is compared to the pressure in the middle section; the output section is never compared with them.

Experimental apparatus

Venturi tubes
The simplest apparatus is a tubular setup known as a Venturi tube or simply a Venturi (plural: "Venturis" or occasionally "Venturies"). Fluid flows through a length of pipe of varying diameter. To avoid undue , a Venturi tube typically has an entry cone of 30 degrees and an exit cone of 5 degrees.
(2024). 9783319089485, Springer.

Venturi tubes are often used in processes where permanent pressure loss is not tolerable and where maximum accuracy is needed in case of highly viscous liquids.

Orifice plate
Venturi tubes are more expensive to construct than simple , and both function on the same basic principle. However, for any given differential pressure, orifice plates cause significantly more permanent energy loss.

Instrumentation and measurement
Both Venturi tubes and orifice plates are used in industrial applications and in scientific laboratories for measuring the flow rate of liquids.

Flow rate
A Venturi can be used to measure the volumetric flow rate, $\scriptstyle Q$, using Bernoulli's principle.

Since \begin{align}

         Q &= v_1 A_1 = v_2 A_2 \\[3pt]
p_1 - p_2 &= \frac{\rho}{2}\left(v_2^2 - v_1^2\right)

\end{align}

then 

 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}}


A Venturi can also be used to mix a liquid with a gas. If a pump forces the liquid through a tube connected to a system consisting of a Venturi to increase the liquid speed (the diameter decreases), a short piece of tube with a small hole in it, and last a Venturi that decreases speed (so the pipe gets wider again), the gas will be sucked in through the small hole because of changes in pressure. At the end of the system, a mixture of liquid and gas will appear. See aspirator and for discussion of this type of .

Differential pressure
As fluid flows through a Venturi, the expansion and compression of the fluids cause the pressure inside the Venturi to change. This principle can be used in for gauges calibrated for differential pressures. This type of pressure measurement may be more convenient, for example, to measure fuel or combustion pressures in jet or rocket engines.

The first large-scale Venturi meters to measure liquid flows were developed by who used them to measure small and large flows of water and wastewater beginning at the end of the 19th century.Herschel, Clemens. (1898). Measuring Water. Providence, RI:Builders Iron Foundry. While working for the Holyoke Water Power Company, Herschel would develop the means for measuring these flows to determine the water power consumption of different mills on the Holyoke Canal System, first beginning development of the device in 1886, two years later he would describe his invention of the Venturi meter to in a letter dated June 5, 1888.

Compensation for temperature, pressure, and mass
Fundamentally, pressure-based meters measure density. Bernoulli's equation (used above) relates this to and volumetric flow:

$\Delta P = \frac\left\{1\right\}\left\{2\right\} \rho \left(v_2^2 - v_1^2\right) = \frac\left\{1\right\}\left\{2\right\} \rho \left\left(\left\left(\frac\left\{A_1\right\}\left\{A_2\right\}\right\right)^2-1\right\right) v_1^2 = \frac\left\{1\right\}\left\{2\right\} \rho \left\left(\frac\left\{1\right\}\left\{A_2^2\right\}-\frac\left\{1\right\}\left\{A_1^2\right\}\right\right) Q^2 = k\, \rho\, Q^2$

where constant terms are absorbed into k. Using the definitions of density ($m=\rho V$), molar concentration ($n=C V$), and ($m=M n$), one can also derive mass flow or molar flow (i.e. standard volume flow):

\begin\left\{align\right\}\Delta P &= k\, \rho\, Q^2 \\

&= k \frac{1}{\rho}\, \dot{m}^2 \\
&= k \frac{\rho}{C^2}\, \dot{n}^2 = k \frac{M}{C}\, \dot{n}^2.

\end{align}

However, measurements outside the design point must compensate for the effects of temperature, pressure, and molar mass on density and concentration. The ideal gas law is used to relate actual values to :

$C = \frac\left\{P\right\}\left\{RT\right\} = \frac\left\{\left\left(\frac\left\{P\right\}\left\{P^\ominus\right\}\right\right)\right\}\left\{\left\left(\frac\left\{T\right\}\left\{T^\ominus\right\}\right\right)\right\} C^\ominus$ $\rho = \frac\left\{MP\right\}\left\{RT\right\} = \frac\left\{\left\left(\frac\left\{M\right\}\left\{M^\ominus\right\} \frac\left\{P\right\}\left\{P^\ominus\right\}\right\right)\right\}\left\{\left\left(\frac\left\{T\right\}\left\{T^\ominus\right\}\right\right)\right\} \rho^\ominus.$

Substituting these two relations into the pressure-flow equations above yields the fully compensated flows:

\begin\left\{align\right\}\Delta P &= k \frac\left\{\left\left(\frac\left\{M\right\}\left\{M^\ominus\right\} \frac\left\{P\right\}\left\{P^\ominus\right\}\right\right)\right\}\left\{\left\left(\frac\left\{T\right\}\left\{T^\ominus\right\}\right\right)\right\} \rho^\ominus\, Q^2

&= \Delta P_{\max} \frac{\left(\frac{M}{M^\ominus} \frac{P}{P^\ominus}\right)}{\left(\frac{T}{T^\ominus}\right)} \left(\frac Q{Q_{\max}}\right)^2\\
&= k \frac{\left(\frac{T}{T^\ominus}\right)}{\left(\frac{M}{M^\ominus} \frac{P}{P^\ominus}\right) \rho^\ominus} \dot{m}^2
&= \Delta P_{\max} \frac{\left(\frac{T}{T^\ominus}\right)}{\left(\frac{M}{M^\ominus} \frac{P}{P^\ominus}\right)} \left(\frac{\dot{m}}{\dot{m}_{\max}}\right)^2\\
&= k \frac{M \left(\frac{T}{T^\ominus}\right)}{\left(\frac{P}{P^\ominus}\right) C^\ominus} \dot{n}^2
&= \Delta P_{\max} \frac{\left(\frac{M}{M^\ominus}\frac{T}{T^\ominus}\right)}{\left(\frac{P}{P^\ominus}\right)} \left(\frac{\dot{n}}{\dot{n}_{\max}}\right)^2.

\end{align}

Q, m, or n are easily isolated by dividing and taking the . Note that pressure-, temperature-, and mass-compensation is required for every flow, regardless of the end units or dimensions. Also we see the relations:

\begin\left\{align\right\}\frac\left\{k\right\}\left\{\Delta P_\left\{\max\right\}\right\} &= \frac\left\{1\right\}\left\{\rho^\ominus Q_\left\{\max\right\}^2\right\}\\

&= \frac{\rho^\ominus}{\dot{m}_{\max}^2}\\
&= \frac


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