Editing Venturi effect (section)

==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]], <math>\scriptstyle Q</math>, using [[Bernoulli's principle]].

Since
<math display="block">\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}</math>

then
<math display="block">
  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>
<!-- The equation for flow needs a term for time which is not accounted for as it is written. -->
<!-- The explanation of how to mix fluids needs help; specifically the system used. -->
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 (pump)|aspirator]] and [[pressure head]] for discussion of this type of [[siphon]].

===Differential pressure===
{{main|Pressure head}}
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 [[metrology]] 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 [[Clemens Herschel]] who used them to measure small and large flows of water and wastewater beginning at the end of the 19th century.<ref>Herschel, Clemens. (1898). ''Measuring Water.'' Providence, RI:Builders Iron Foundry.</ref> 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.<ref>{{cite journal|page=254|volume=136|issue=3433|date=August 17, 1935|journal=Nature|doi=10.1038/136254a0|title=Invention of the Venturi Meter|bibcode=1935Natur.136Q.254.|doi-access=free}}</ref>

=== Compensation for temperature, pressure, and mass ===

Fundamentally, pressure-based meters measure [[kinetic energy]] density. [[Bernoulli's equation]] (used above) relates this to [[mass density]] and volumetric flow:

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

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

<math>\begin{align}\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}</math>

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|design values]]:

<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} \frac{P}{P^\ominus}\right)}{\left(\frac{T}{T^\ominus}\right)} \rho^\ominus.</math>

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

<math>\begin{align}\Delta P &= k \frac{\left(\frac{M}{M^\ominus} \frac{P}{P^\ominus}\right)}{\left(\frac{T}{T^\ominus}\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}</math>

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

<math>\begin{align}\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}.
\end{align}</math>