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Volumetric flow rate

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Volume of fluid which passes per unit time

Not to be confused withVolumetric flux.

Volume flow rate
Common symbols	Q, ${\dot {V}}$
SI unit	m³/s
Dimension	${\mathsf {L}}^{3}{\mathsf {T}}^{-1}$

Thermodynamics

The classicalCarnot heat engine

Branches

Equilibrium / Non-equilibrium

Laws

Systems

State
Equation of state Ideal gas Real gas State of matter Phase (matter) Equilibrium Control volume Instruments
Processes
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Note:Conjugate variables initalics

Process functions
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Work Heat
Functions of state
Temperature / Entropy (introduction) Pressure / Volume Chemical potential / Particle number Vapor quality Reduced properties

Material properties

Property databases

Specific heat capacity

c=

$T {\displaystyle T}$	$\partial S$
$N {\displaystyle N}$	$\partial T$

Compressibility

\beta =-

$1 {\displaystyle 1}$	$\partial V$
$V {\displaystyle V}$	$\partial p$

Thermal expansion

\alpha =

$1 {\displaystyle 1}$	$\partial V$
$V {\displaystyle V}$	$\partial T$

Equations

Potentials

Internal energy
$U(S,V)$
Enthalpy
$H(S,p)=U+pV$
Helmholtz free energy
$A(T,V)=U-TS$
Gibbs free energy
$G(T,p)=H-TS$

History
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Category

Inphysics andengineering, in particularfluid dynamics, thevolumetric flow rate (also known asvolume flow rate, orvolume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbolQ (sometimes ${\dot {V}}$ ). ItsSI unit iscubic metres per second (m³/s).

It contrasts withmass 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. Inhydrometry, the volumetric flow rate is known asdischarge.

The volumetric flow rate across a unit area is calledvolumetric flux, as defined byDarcy's law and represented by the symbolq. Conversely, the integration of a volumetric flux over a given area gives the volumetric flow rate.

Units

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TheSI unit iscubic metres per second (m³/s). Another unit used isstandard cubic centimetres per minute (SCCM). InUS customary units andimperial units, volumetric flow rate is often expressed ascubic feet per second (ft³/s) orgallons per minute (either US or imperial definitions). Inoceanography, thesverdrup (symbol: Sv, not to be confused with thesievert) is a non-SI metric unit of flow, with1 Sv equal to 1 million cubic metres per second (35,000,000 cu ft/s);^[1]^[2] it is equivalent to the SI derived unit cubichectometer per second (symbol: hm³/s or hm³⋅s⁻¹). Named afterHarald Sverdrup, it is used almost exclusively inoceanography to measure the volumetric rate of transport ofocean currents.

Fundamental definition

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Volumetric flow rate is defined by thelimit^[3]

Q={\dot {V}}=\lim \limits _{\Delta t\to 0}{\frac {\Delta V}{\Delta t}}={\frac {\mathrm {d} V}{\mathrm {d} t}},

that is, the flow ofvolume of fluidV through a surface per unit timet.

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 flowsafter 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.

IUPAC^[4] prefers the notation $q_{v}$ ^[5] and $q_{m}$ ^[6] for volumetric flow and mass flow respectively, to distinguish from the notation $Q {\displaystyle Q}$ ^[7] for heat.

Alternative definition

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Volumetric flow rate can also be defined by

Q=\mathbf {v} \cdot \mathbf {A} ,

where

v =flow velocity,

A =cross-sectional vector area/surface.

The above equation is only true for uniform or homogeneous flow velocity and a flat or planar cross section. In general, including spatially variable or non-homogeneous flow velocity and curved surfaces, the equation becomes asurface integral:

Q=\iint _{A}\mathbf {v} \cdot \mathrm {d} \mathbf {A} .

This is the definition used in practice. Thearea required to calculate the volumetric flow rate is real or imaginary, flat or curved, either as a cross-sectional area or a surface. Thevector area is a combination of the magnitude of the area through which the volume passes through,A, and aunit vector normal to the area, ${\hat {\mathbf {n} }}$ . The relation is $\mathbf {A} =A{\hat {\mathbf {n} }}$ .

Derivation

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The reason for thedot product is as follows. The only volume flowingthrough the cross-section is the amount normal to the area, that is,parallel to the unit normal. This amount is

Q=vA\cos \theta ,

whereθ is the angle between the unit normal ${\hat {\mathbf {n} }}$ and the velocity vectorv of the substance elements. The amount passing through the cross-section is reduced by the factorcosθ. Asθ increases less volume passes through. Substance which passes tangential to the area, that isperpendicular to the unit normal, does not pass through the area. This occurs whenθ =⁠π/2⁠ and so this amount of the volumetric flow rate is zero:

Q=vA\cos \left({\frac {\pi }{2}}\right)=0.

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

Relationship with mass flow rate

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When themass flow rate is known, and the density can be assumed constant, this is an easy way to get $Q {\displaystyle Q}$ :

Q={\frac {\dot {m}}{\rho }},

where

ṁ =mass flow rate (in kg/s),

ρ =density (in kg/m³).

Related quantities

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

\int L\,\mathrm {d} \theta ={\frac {RT}{2\pi }}(\cos \theta _{2}-\cos \theta _{1})+{\frac {rT}{2\pi }}(\theta _{2}-\theta _{1}),

whereT is the time per revolution,R is the distance from the camshaft centreline to the cam tip,r is the radius of the camshaft (that is,R −r is the maximum lift),θ₁ is the angle where opening begins, andθ₂ 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

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References

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^"Glossary".Ocean Surface Currents.University of Miami Rosenstiel School of Marine, Atmospheric, and Earth Science. Archived fromthe original on 2019-07-24. Retrieved2019-04-15.
^"Sverdrups & Brine".Ecoworld. Archived fromthe original on 20 January 2011. Retrieved12 August 2017.
^Engineers Edge, LLC."Fluid Volumetric Flow Rate Equation". Engineers Edge. Retrieved2016-12-01.
^International Union of Pure and Applied Chemistry;https://iupac.org
^"Volume flow rate, qv".The IUPAC Compendium of Chemical Terminology. 2014.doi:10.1351/goldbook.V06642.
^"Mass flow rate, qm".The IUPAC Compendium of Chemical Terminology. 2014.doi:10.1351/goldbook.M03720.
^"Heat, q, Q".The IUPAC Compendium of Chemical Terminology. 2014.doi:10.1351/goldbook.H02752.

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