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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, in particular
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, 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 (sometimes ). 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 Discharge may refer to Expel or let go * Discharge, the act of firing a gun * Discharge, or termination of employment, the end of an employee's duration with an employer * Military discharge, the release of a member of the armed forces from ser ...
''. Volumetric flow rate should not be confused with volumetric flux, as defined by Darcy's law and represented by the symbol , with units of m3/(m2·s), that is, m·s−1. The integration of a
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
over an area gives the volumetric flow rate. The SI unit is
cubic metres per second A cubic metre per second (m3s−1, m3/s, cumecs or cubic meter per second in American English) is the unit of volumetric flow rate in the International System of Units (SI) equal to that of a stere or cube with sides of in length exchanged or ...
(m3/s). Another unit used is
standard cubic centimetres per minute Standard cubic centimeters per minute (SCCM) is a unit used to quantify the flow rate of a fluid. 1 SCCM is identical to 1  cm³STP/min. Another expression of it would be Nml/min. These standard conditions vary according to different reg ...
(SCCM). In US customary units and
imperial units The imperial system of units, imperial system or imperial units (also known as British Imperial or Exchequer Standards of 1826) is the system of units first defined in the British Weights and Measures Act 1824 and continued to be developed th ...
, volumetric flow rate is often expressed as cubic feet per second (ft3/s) or gallons per minute (either US or imperial definitions). In
oceanography Oceanography (), also known as oceanology and ocean science, is the scientific study of the oceans. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynami ...
, the
sverdrup In oceanography, the sverdrup (symbol: Sv) is a non- SI metric unit of volumetric flow rate, with equal to . It is equivalent to the SI derived unit cubic hectometer per second (symbol: hm3/s or hm3⋅s−1): 1 Sv is equal to 1 hm3/s. It is u ...
(symbol: Sv, not to be confused with the sievert) is a non- SI metric unit of flow, with equal to ; it is equivalent to the SI derived unit cubic hectometer per second (symbol: hm3/s or hm3⋅s−1). Named after Harald Sverdrup, it is used almost exclusively in
oceanography Oceanography (), also known as oceanology and ocean science, is the scientific study of the oceans. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynami ...
to measure the volumetric rate of transport of
ocean current An ocean current is a continuous, directed movement of sea water generated by a number of forces acting upon the water, including wind, the Coriolis effect, breaking waves, cabbeling, and temperature and salinity differences. Depth conto ...
s.


Fundamental definition

Volumetric flow rate is defined by the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
: : Q = \dot V = \lim\limits_\frac= \frac That is, the flow of
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of fluid through a surface per unit time . 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. IUPAC prefers the notation q_v and q_m for resp. volumetric flow and mass flow, to distinguish from the notation Q for heat.


Useful definition

Volumetric flow rate can also be defined by: :Q = \mathbf v \cdot \mathbf A where: * = flow velocity * =
cross-sectional Cross-sectional data, or a cross section of a study population, in statistics and econometrics, is a type of data collected by observing many subjects (such as individuals, firms, countries, or regions) at the one point or period of time. The analy ...
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 In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one ...
: :Q = \iint_A \mathbf v \cdot \mathrm d \mathbf A This is the definition used in practice. The
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
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, , and a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
normal to the area, \hat. The relation is \mathbf A = A\hat . The reason for the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
is as follows. The only volume flowing ''through'' the cross-section is the amount normal to the area, that is,
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
to the unit normal. This amount is: :Q = v A \cos\theta where is the angle between the unit normal \hat and the velocity vector of the substance elements. The amount passing through the cross-section is reduced by the factor . As increases less volume passes through. Substance which passes tangential to the area, that is
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
to the unit normal, does not pass through the area. This occurs when and so this amount of the volumetric flow rate is zero: :Q = v A \cos\left(\frac\right) = 0 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 Q. :Q = \frac Where: * = mass flow rate (in kg/s). * =
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
(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: :\int L \, \mathrm d \theta = \frac \left(\cos\theta_2 -\cos\theta_1\right) + \frac\left(\theta_2-\theta_1\right) where is the time per revolution, is the distance from the camshaft centreline to the cam tip, is the radius of the camshaft (that is, is the maximum lift), is the angle where opening begins, and 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 Cardiac physiology or heart function is the study of healthy, unimpaired function of the heart: involving blood flow; myocardium structure; the electrical conduction system of the heart; the cardiac cycle and cardiac output and how these interact a ...
: the
cardiac output In cardiac physiology, cardiac output (CO), also known as heart output and often denoted by the symbols Q, \dot Q, or \dot Q_ , edited by Catherine E. Williamson, Phillip Bennett is the volumetric flow rate of the heart's pumping output: th ...
* In
hydrology Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is call ...
:
discharge Discharge may refer to Expel or let go * Discharge, the act of firing a gun * Discharge, or termination of employment, the end of an employee's duration with an employer * Military discharge, the release of a member of the armed forces from ser ...
** List of rivers by discharge ** List of waterfalls by flow rate ** Weir § Flow measurement * In dust collection systems: the air-to-cloth ratio


See also

*
Flow measurement Flow measurement is the quantification of bulk fluid movement. Flow can be measured in a variety of ways. The common types of flowmeters with industrial applications are listed below: * a) Obstruction type (differential pressure or variable area ...
* Flowmeter * Mass flow rate * Orifice plate * Poiseuille's law * Stokes flow


References

{{Authority control Fluid dynamics Temporal rates