The Venturi effect is the reduction in fluid pressure 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 18thcentury Italian physicist 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 vacuum ejector).
By measuring pressure, the flow rate can be determined, as in various flow measurement 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 lowspeed flow of gas) along a streamline, the theoretical pressure drop at the constriction is given by
$$p\_1\; \; p\_2\; =\; \backslash frac\{\backslash rho\}\{2\}\; (v\_2^2\; \; v\_1^2),$$
where $\backslash rho$ is the density 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).
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.
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.
Since $$\backslash begin\{align\}$$
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 pressure head for discussion of this type of siphon.
The first largescale Venturi meters to measure liquid flows were developed by Clemens Herschel 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 William Unwin in a letter dated June 5, 1888.
$\backslash Delta\; P\; =\; \backslash frac\{1\}\{2\}\; \backslash rho\; (v\_2^2\; \; v\_1^2)\; =\; \backslash frac\{1\}\{2\}\; \backslash rho\; \backslash left(\backslash left(\backslash frac\{A\_1\}\{A\_2\}\backslash right)^21\backslash right)\; v\_1^2\; =\; \backslash frac\{1\}\{2\}\; \backslash rho\; \backslash left(\backslash frac\{1\}\{A\_2^2\}\backslash frac\{1\}\{A\_1^2\}\backslash right)\; Q^2\; =\; k\backslash ,\; \backslash rho\backslash ,\; Q^2$
where constant terms are absorbed into k. Using the definitions of density ($m=\backslash rho\; V$), molar concentration ($n=C\; V$), and molar mass ($m=M\; n$), one can also derive mass flow or molar flow (i.e. standard volume flow):
$\backslash begin\{align\}\backslash Delta\; P\; \&=\; k\backslash ,\; \backslash rho\backslash ,\; Q^2\; \backslash \backslash $
&= 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 Standard state:
$C\; =\; \backslash frac\{P\}\{RT\}\; =\; \backslash frac\{\backslash left(\backslash frac\{P\}\{P^\backslash ominus\}\backslash right)\}\{\backslash left(\backslash frac\{T\}\{T^\backslash ominus\}\backslash right)\}\; C^\backslash ominus$ $\backslash rho\; =\; \backslash frac\{MP\}\{RT\}\; =\; \backslash frac\{\backslash left(\backslash frac\{M\}\{M^\backslash ominus\}\; \backslash frac\{P\}\{P^\backslash ominus\}\backslash right)\}\{\backslash left(\backslash frac\{T\}\{T^\backslash ominus\}\backslash right)\}\; \backslash rho^\backslash ominus.$
Substituting these two relations into the pressureflow equations above yields the fully compensated flows:
$\backslash begin\{align\}\backslash Delta\; P\; \&=\; k\; \backslash frac\{\backslash left(\backslash frac\{M\}\{M^\backslash ominus\}\; \backslash frac\{P\}\{P^\backslash ominus\}\backslash right)\}\{\backslash left(\backslash frac\{T\}\{T^\backslash ominus\}\backslash right)\}\; \backslash rho^\backslash ominus\backslash ,\; Q^2$
&= \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 square root. Note that pressure, temperature, and masscompensation is required for every flow, regardless of the end units or dimensions. Also we see the relations:
$\backslash begin\{align\}\backslash frac\{k\}\{\backslash Delta\; P\_\{\backslash max\}\}\; \&=\; \backslash frac\{1\}\{\backslash rho^\backslash ominus\; Q\_\{\backslash max\}^2\}\backslash \backslash $
&= \frac{\rho^\ominus}{\dot{m}_{\max}^2}\\ &= \frac

