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| {{Infobox physical quantity
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| |bgcolour={default}
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| |name = Volume flow rate
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| |image =
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| |caption =
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| |unit = m<sup>3</sup>/s
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| |symbols = ''<math>\dot{V}</math>, <math>Q</math>''
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| |derivations =
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| }}
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| {{Thermodynamics}}
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| In [[physics]] and [[engineering]], in particular [[fluid dynamics]] and [[hydrometry]], the '''volumetric flow rate''', (also known as '''volume flow rate''', '''rate of fluid flow''' or '''volume velocity''') is the volume of fluid which passes through a given surface per unit time. The [[SI unit]] is m<sup>3</sup>/s ([[cubic meters per second]]). In [[US Customary Units]] and [[British Imperial Units]], volumetric flow rate is often expressed as ft<sup>3</sup>/s ([[cubic foot|cubic feet]] per second). It is usually represented by the symbol ''Q''.
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| Volumetric flow rate should not be confused with [[volumetric flux]], as defined by [[Darcy's law]] and represented by the symbol ''q'', with units of m<sup>3</sup>/(m<sup>2</sup>·s), that is, m·s<sup>−1</sup>. The integration of a [[flux]] over an area gives the volumetric flow rate.
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| ==Fundamental definition==
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| Volume flow rate is defined by the [[limit of a function|limit]]:<ref>http://www.engineersedge.com/fluid_flow/volumeetric_flow_rate.htm</ref>
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| :<math> Q = \lim\limits_{\Delta t \rightarrow 0}\frac{\Delta V}{ \Delta t}= \frac{{\rm d}V}{{\rm d}t}</math>
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| I.e., the flow of [[volume]] of fluid ''V'' through a surface per unit time ''t''.
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| 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.
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| ==Useful definition==
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| Volumetric flow rate can also be defined by:
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| :<math>Q = \bold{v} \cdot \bold{A} </math>
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| where:
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| *<math>\bold{v}</math> = [[flow velocity]] of the substance elements
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| *<math>\bold{A}</math> = [[Cross section (geometry)|cross-sectional]] [[vector area]]/surface
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| The above equation is only true for flat, plane cross-sections. In general, including curved surfaces, the equation becomes a [[surface integral]]:
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| :<math>Q = \iint_A \bold{v} \cdot {\rm d}\bold{A} </math>
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| 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, ''A'', and a [[unit vector]] normal to the area, <math>\bold{\hat{n}}</math>. The relation is <math>\bold{A} = A \bold{\hat{n}}</math>.
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| The reason for the dot product is as follows. The only volume flowing ''through'' the cross-section is the amount normal to the area; i.e., [[parallel (geometry)|parallel]] to the unit normal. This amount is:
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| :<math>Q = v A \cos\theta </math> | |
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| where ''θ'' is the angle between the unit normal <math>\bold{\hat{n}}</math> and the velocity vector '''v''' of the substance elements. The amount passing through the cross-section is reduced by the factor <math>\cos\theta </math>. As ''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 ''θ'' = {{frac|''π''|2}} and so this amount of the volumetric flow rate is zero:
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| :<math>Q = v A \cos\left(\frac{\pi}{2}\right) = 0</math>
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| These results are equivalent to the dot product between velocity and the normal direction to the area.
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| When the [[mass flow rate]] is known, this is an easy way to get <math>\dot{V}</math>.
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| <math>\dot{V} =
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| \frac{\dot{m}}{\rho}
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| </math>
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| Where:
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| *'''<math>\dot{m}</math>''' = [[mass flow rate]] ''(kg/s)''.
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| *<math>\rho</math> = [[density]] ''(kg/m<sup>3</sup>)''.
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| ==Related quantities==
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| Volumetric flow rate is really just part of mass flow rate, since mass relates to volume via density.
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| In internal combustion engines, the time.area integral is considered over the range of valve opening.
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| The time.lift integral is given by:
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| <math>\int \! L \, \mathrm{d} \theta = \frac{T}{2 \pi} ( - ( \cos{\theta_1}) \cdot R - r \cdot \theta_1) - \frac{T}{2 \pi} ( - ( \cos { \theta_2}) \cdot R - r \cdot \theta_2 ))</math>
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| where <math>T</math> is time per revolution, <math>R</math> is distance from camshaft centreline to cam tip, <math>r</math> is radius of camshaft (that is, <math>R - r</math> is the maximum lift), <math>\theta _{1}</math> is the angle where opening begins, and <math>\theta _{2}</math> is where valve closes (secs, mm, radians).
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| This has to be factored by the width (circumference) of the valve throat.
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| The answer is usually related to the cylinder swept volume.
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| ==See also==
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| *[[Air to cloth ratio]]
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| *[[Discharge (hydrology)]]
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| *[[Flow measurement]]
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| *[[Flowmeter]]
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| *[[Orifice plate]]
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| *[[Poiseuille's law]]
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| *[[Stokes flow]]
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| *[[:Category:Units of flow|Units of flow]]
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| ==References==
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| {{Reflist}}
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| [[Category:Fluid dynamics]]
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