volumetric flow rate

TheInfoList

OR:

In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
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 specializ ...
, 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) and ...
, 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 of measurement, unit is kilogram per second in SI units, and Slug (unit), slug per second or pound (mass), pound per second in US cust ...
, 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''. Volumetric flow rate should not be confused with volumetric flux, 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 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 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 Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
is cubic metres per second (m3/s). Another unit used is standard cubic centimetres per minute (SCCM). In
US customary units United States customary units form a system of Units of measurement, measurement units commonly used in the United States and Territories of the United States, U.S. territories since being standardized and adopted in 1832. The United States cust ...
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 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, Wind wave, waves, and geophysical flu ...
, the
sverdrup In oceanography, the sverdrup (symbol: Sv) is a non-International System of Units, SI Metric_units#Volume_flow_rate, metric unit of volumetric flow rate, with equal to . It is equivalent to the SI derived unit cubic hectometer per second (symbol ...
(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 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, Wind wave, waves, and geophysical flu ...
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 Wind is the natural movement of atmosphere of Earth, air or other gases relative to a planetary surfac ...
s.

# Fundamental definition

Volumetric flow rate is defined by the limit: :$Q = \dot V = \lim\limits_\frac= \frac$ 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 In continuum mechanics the flow velocity in 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 ''aer ...
* = cross-sectional
vector area In 3-dimensional geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in t ...
/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 Integral, integration over surface (differential geometry), surfaces. It can be thought of as the double integral analogue of th ...
: :$Q = \iint_A \mathbf v \cdot \mathrm d \mathbf A$ 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 o ...
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 Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in t ...
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_(mathematics_and_physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
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 (mathematics), scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidea ...
is as follows. The only volume flowing ''through'' the cross-section is the amount normal to the area, that is, parallel 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 Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works i ...
to the unit normal, does not pass through the area. This occurs when and so this amount of the volumetric flow rate is zero: :$Q = v A \cos\left\left(\frac\right\right) = 0$ 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 of measurement, unit is kilogram per second in SI units, and Slug (unit), slug per second or pound (mass), pound per second in US cust ...
is known, and the density can be assumed constant, this is an easy way to get $Q$. :$Q = \frac$ 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 of measurement, unit is kilogram per second in SI units, and Slug (unit), slug per second or pound (mass), pound per second in US cust ...
(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 language, Greek letter Rho (letter), rho), although the Latin letter ''D'' ca ...
(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: :$\int L \, \mathrm d \theta = \frac \left\left(\cos\theta_2 -\cos\theta_1\right\right) + \frac\left\left(\theta_2-\theta_1\right\right)$ 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; Cardiac muscle, myocardium structure; the electrical conduction system of the heart; the cardiac cycle and cardiac output and how ...
: 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: tha ...
* 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 ** List of waterfalls by flow rate ** Weir § Flow measurement * In dust collection systems: the air-to-cloth ratio