In
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 r ...
and
engineering
Engineering is the use of scientific method, 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 rang ...
, 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) an ...
, 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
In physics and engineering, mass flow rate is the mass of a substance which passes per unit of time. Its unit is kilogram per second in SI units, and slug per second or pound per second in US customary units. The common symbol is \dot (''ṁ ...
, 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
Hydrometry is the monitoring of the components of the hydrological cycle including rainfall, groundwater characteristics, as well as water quality and flow characteristics of surface waters. The etymology of the term ''hydrometry'' is from el, ...
, the volumetric flow rate is known as ''
discharge''.
Volumetric flow rate should not be confused with
volumetric flux
In fluid dynamics, the volumetric flux is the rate of volume flow across a unit area (m3·s−1·m−2).
Volumetric flux has dimensions of volume/(time*area). The density of a particular property in a fluid's volume, multiplied with the volumetri ...
, as defined by
Darcy's law
Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of e ...
and represented by the symbol , with units of m
3/(m
2·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 ph ...
over an area gives the volumetric flow rate.
The
SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
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 ...
(m
3/s). Another unit used is
standard cubic centimetres per minute (SCCM). In
US customary units
United States customary units form a system of measurement units commonly used in the United States and U.S. territories since being standardized and adopted in 1832. The United States customary system (USCS or USC) developed from English 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 thro ...
, volumetric flow rate is often expressed as
cubic feet
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system, a crystal system ...
per second (ft
3/s) or
gallons per minute
The gallon is a unit of volume in imperial units and United States customary units. Three different versions are in current use:
*the imperial gallon (imp gal), defined as , which is or was used in the United Kingdom, Ireland, Canada, Austral ...
(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 dynamic ...
, 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 used ...
(symbol: Sv, not to be confused with the
sievert
The sievert (symbol: SvNot be confused with the sverdrup or the svedberg, two non-SI units that sometimes use the same symbol.) is a unit in the International System of Units (SI) intended to represent the stochastic health risk of ionizing radi ...
) is a non-
SI metric unit
The metric system is a system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the Internati ...
of flow, with equal to ; it is equivalent to the SI derived unit cubic
hectometer
The hectometre ( International spelling as used by the International Bureau of Weights and Measures; SI symbol: hm) or hectometer (American spelling) is a unit of length in the International System of Units (SI), equal to one hundred metres and ...
per second (symbol: hm
3/s or hm
3⋅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 dynamic ...
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 contours, s ...
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 ...
:
:
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). The de ...
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
and
for resp. volumetric flow and mass flow, to distinguish from the notation
for heat.
Useful definition
Volumetric flow rate can also be defined by:
:
where:
* =
flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
* =
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 In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an ''oriented area'' in three dimensions.
Every bounded surface in three dimensions can be associated with a ...
/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 may ...
:
:
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
A shape or figure is a graphics, graphical representation of an obje ...
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 In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an ''oriented area'' in three dimensions.
Every bounded surface in three dimensions can be associated with a ...
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 vecto ...
normal to the area,
. The relation is
.
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 algebra ...
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 of IBM ...
to the unit normal. This amount is:
:
where is the angle between the unit normal
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 can ...
to the unit normal, does not pass through the area. This occurs when and so this amount of the volumetric flow rate is zero:
:
These results are equivalent to the dot product between velocity and the normal direction to the area.
When the
mass flow rate
In physics and engineering, mass flow rate is the mass of a substance which passes per unit of time. Its unit is kilogram per second in SI units, and slug per second or pound per second in US customary units. The common symbol is \dot (''ṁ ...
is known, and the density can be assumed constant, this is an easy way to get
.
:
Where:
* =
mass flow rate
In physics and engineering, mass flow rate is the mass of a substance which passes per unit of time. Its unit is kilogram per second in SI units, and slug per second or pound per second in US customary units. The common symbol is \dot (''ṁ ...
(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. Mathematical ...
(in kg/m
3).
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:
:
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: t ...
* 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 calle ...
:
discharge
**
List of rivers by discharge
This is a list of rivers by their average discharge (that is, their water flow rate). Here, only those rivers whose discharge is more than are shown, as this list does not concern itself with rivers having a water flow rate of less than .
For c ...
**
List of waterfalls by flow rate
This list of waterfalls by flow rate includes all waterfalls which are known to have an Volumetric flow rate, average flow rate or Discharge (hydrology), discharge of at least . The waterfalls in this list are those for which there is verifiable i ...
**
Weir § Flow measurement
* In
dust collection system
A dust collection system is an air quality improvement system used in industrial, commercial, and home production shops to improve breathable air quality and safety by removing particulate matter from the air and environment. Dust collection syste ...
s: the
air-to-cloth ratio
The air-to-cloth ratio is the volumetric flow rate of air (m3/minute; SI m3/second) flowing through a dust collector's inlet duct divided by the total cloth area (m2) in the filters. The result is expressed in units of velocity.
: v = {V / \Delta ...
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
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) ...
*
Mass flow rate
In physics and engineering, mass flow rate is the mass of a substance which passes per unit of time. Its unit is kilogram per second in SI units, and slug per second or pound per second in US customary units. The common symbol is \dot (''ṁ ...
*
Orifice plate An orifice plate is a device used for measuring flow rate, for reducing pressure or for restricting flow (in the latter two cases it is often called a ').
Description
An orifice plate is a thin plate with a hole in it, which is usually placed in ...
*
Poiseuille's law
*
Stokes flow
Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion,Kim, S. & Karrila, S. J. (2005) ''Microhydrodynamics: Principles and Selected Applications'', Dover. . is a type of fluid flow where advective iner ...
References
{{Authority control
Fluid dynamics
Temporal rates