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{{Short description|Volume of fluid which passes per unit time}}
{{Distinguish|Mass flow rate}}
{{Infobox physical quantity
| bgcolour = {default}
| name = Volume flow rate
| image =
| caption =
| unit = m3/s
| symbols = {{mvar|Q}}, {{mvar|V̇}}
| derivations =
| dimension = wikidata
}}
{{Thermodynamics}}


In [[physics]] and [[engineering]], in particular [[fluid dynamics]], the '''volumetric flow rate''' (also known as '''volume flow rate''', or '''volume velocity''') is the volume of fluid which passes per unit time; usually it is represented by the symbol {{mvar|Q}} (sometimes {{mvar|V̇}}). It contrasts with [[mass flow rate]], which is the other main type of fluid flow rate. In most contexts a mention of ''rate of fluid flow'' is likely to refer to the volumetric rate. In [[hydrometry]], the volumetric flow rate is known as ''[[discharge (hydrology)|discharge]]''.

Volumetric flow rate should not be confused with [[volumetric flux]], as defined by [[Darcy's law]] and represented by the symbol {{mvar|q}}, with units of m3/(m2·s), that is, m·s−1. The integration of a [[flux]] over an area gives the volumetric flow rate.


The [[SI unit]] is [[Cubic metre per second|cubic metres per second]] (m3/s). Another unit used is [[standard cubic centimetres per minute]] (SCCM). In [[US customary units]] and [[imperial units]], volumetric flow rate is often expressed as [[cubic foot|cubic feet]] per second (ft3/s) or [[gallons per minute]] (either US or imperial definitions). In [[oceanography]], the [[sverdrup]] (symbol: Sv, not to be confused with the [[sievert]]) is a non-[[International System of Units|SI]] [[Metric_units#Volume_flow_rate|metric unit]] of flow, with {{nowrap|1 Sv}} equal to {{convert|1|e6m3/s|gal/s}};<ref>{{Cite web |url=https://oceancurrents.rsmas.miami.edu/glossary.html#S |title=Glossary |website=Ocean Surface Currents |publisher=[[University of Miami]] [[Rosenstiel School of Marine and Atmospheric Science]] |access-date=2019-04-15}}</ref><ref>{{Cite web |url=http://www.ecoworld.com/technology/sverdrups-brine.html |title=Sverdrups & Brine |website=Ecoworld |archive-url=https://web.archive.org/web/20110120155822/http://www.ecoworld.com/technology/sverdrups-brine.html |archive-date=20 January 2011 |url-status=dead |access-date=12 August 2017}}</ref> it is equivalent to the SI derived unit cubic [[hectometer]] per second (symbol: hm3/s or hm3⋅s−1). Named after [[Harald Sverdrup (oceanographer)|Harald Sverdrup]], it is used almost exclusively in [[oceanography]] to measure the volumetric rate of transport of [[ocean current]]s.

==Fundamental definition==
Volumetric flow rate is defined by the [[limit of a function|limit]]:<ref>{{cite web|author=Engineers Edge, LLC. |url=http://www.engineersedge.com/fluid_flow/volumeetric_flow_rate.htm |title=Fluid Volumetric Flow Rate Equation |publisher=Engineers Edge |access-date=2016-12-01}}</ref>

:<math> Q = \dot V = \lim\limits_{\Delta t \rightarrow 0}\frac{\Delta V}{\Delta t}= \frac{\mathrm d V}{\mathrm d t}</math>

That is, the flow of [[volume]] of fluid {{mvar|V}} through a surface per unit time {{mvar|t}}.

Since this is only the time derivative of volume, a scalar quantity, the volumetric flow rate is also a scalar quantity. The change in volume is the amount that flows ''after'' crossing the boundary for some time duration, not simply the initial amount of volume at the boundary minus the final amount at the boundary, since the change in volume flowing through the area would be zero for steady flow.

==Useful definition==
Volumetric flow rate can also be defined by:
:<math>Q = \mathbf v \cdot \mathbf A</math>

where:
*{{math|'''v'''}} = [[flow velocity]]
*{{math|'''A'''}} = [[Cross section (geometry)|cross-sectional]] [[vector area]]/surface

The above equation is only true for flat, plane cross-sections. In general, including curved surfaces, the equation becomes a [[surface integral]]:
:<math>Q = \iint_A \mathbf v \cdot \mathrm d \mathbf A</math>

This is the definition used in practice. The [[area]] required to calculate the volumetric flow rate is real or imaginary, flat or curved, either as a cross-sectional area or a surface. The [[vector area]] is a combination of the magnitude of the area through which the volume passes through, {{mvar|A}}, and a [[unit vector]] normal to the area, {{math|'''n̂'''}}. The relation is {{math|'''A''' {{=}} ''A'''''n̂'''}}.

The reason for the [[dot product]] is as follows. The only volume flowing ''through'' the cross-section is the amount normal to the area, that is, [[parallel (geometry)|parallel]] to the unit normal. This amount is:
:<math>Q = v A \cos\theta </math>

where {{mvar|θ}} is the angle between the unit normal {{math|'''n̂'''}} and the velocity vector {{math|'''v'''}} of the substance elements. The amount passing through the cross-section is reduced by the factor {{math|cos ''θ''}}. As {{mvar|θ}} increases less volume passes through. Substance which passes tangential to the area, that is [[perpendicular]] to the unit normal, does not pass through the area. This occurs when {{math|''θ'' {{=}} {{sfrac|π|2}}}} and so this amount of the volumetric flow rate is zero:
:<math>Q = v A \cos\left(\frac{\pi}{2}\right) = 0</math>

These results are equivalent to the dot product between velocity and the normal direction to the area.

When the [[mass flow rate]] is known, and the density can be assumed constant, this is an easy way to get <math>Q</math>.

:<math>Q = \frac{\dot m}{\rho} </math>

Where:
*{{mvar|ṁ}} = [[mass flow rate]] (in kg/s).
*{{mvar|ρ}} = [[density]] (in kg/m3).

==Related quantities==
In internal combustion engines, the time area integral is considered over the range of valve opening. The time lift integral is given by:

:<math>\int L \, \mathrm d \theta = \frac{RT}{2 \pi} \left(\cos\theta_2 -\cos\theta_1\right) + \frac{rT}{2 \pi}\left(\theta_2-\theta_1\right) </math>

where {{mvar|T}} is the time per revolution, {{mvar|R}} is the distance from the camshaft centreline to the cam tip, {{mvar|r}} is the radius of the camshaft (that is, {{math|''R'' − ''r''}} is the maximum lift), {{math|''θ''1}} is the angle where opening begins, and {{math|''θ''2}} is where the valve closes (seconds, mm, radians). This has to be factored by the width (circumference) of the valve throat. The answer is usually related to the cylinder's swept volume.

==Some key examples==
* In [[cardiac physiology]]: the [[cardiac output]]
* In [[hydrology]]: [[discharge (hydrology)|discharge]]
** [[List of rivers by discharge]]
** [[List of waterfalls by flow rate]]
** [[Weir#Flow measurement|Weir § Flow measurement]]
* In [[dust collection system]]s: the [[air-to-cloth ratio]]

==See also==
*[[Flow measurement]]
*[[Flowmeter]]
*[[Mass flow rate]]
*[[Orifice plate]]
*[[Poiseuille's law]]
*[[Stokes flow]]

==References==
{{reflist}}

{{Rivers, streams and springs}}
{{Authority control}}

[[Category:Fluid dynamics]]
[[Category:Temporal rates]]

Volumetric flow rate - Revision history

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