Volumetric Flow Rate - Wikipedia

This article needs attention from an expert in Physics or Engineering. The specific problem is: Lacks treatment of compressible flow. See the talk page for details. WikiProject Physics or WikiProject Engineering may be able to help recruit an expert. (May 2023)

Volume of fluid which passes per unit time Not to be confused with Volumetric flux.

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

Thermodynamics
The classical Carnot heat engine
Branches Classical Statistical Chemical Quantum thermodynamics Equilibrium / Non-equilibrium
Laws Zeroth First Second Third
Systems Closed system Open system Isolated system State Equation of state Ideal gas Real gas State of matter Phase (matter) Equilibrium Control volume Instruments Processes Isobaric Isochoric Isothermal Adiabatic Isentropic Isenthalpic Quasistatic Polytropic Free expansion Reversibility Irreversibility Endoreversibility Cycles Heat engines Heat pumps Thermal efficiency
System propertiesNote: Conjugate variables in italics Property diagrams Intensive and extensive properties Process functions 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 = {\displaystyle c=} T {\displaystyle T} ∂ S {\displaystyle \partial S} N {\displaystyle N} ∂ T {\displaystyle \partial T} Compressibility  β = − {\displaystyle \beta =-} 1 {\displaystyle 1} ∂ V {\displaystyle \partial V} V {\displaystyle V} ∂ p {\displaystyle \partial p} Thermal expansion  α = {\displaystyle \alpha =} 1 {\displaystyle 1} ∂ V {\displaystyle \partial V} V {\displaystyle V} ∂ T {\displaystyle \partial T}
Equations Carnot's theorem Clausius theorem Fundamental relation Ideal gas law Maxwell relations Onsager reciprocal relations Bridgman's equations Table of thermodynamic equations
Potentials Free energy Free entropy Internal energy U ( S , V ) {\displaystyle U(S,V)} Enthalpy H ( S , p ) = U + p V {\displaystyle H(S,p)=U+pV} Helmholtz free energy A ( T , V ) = U − T S {\displaystyle A(T,V)=U-TS} Gibbs free energy G ( T , p ) = H − T S {\displaystyle G(T,p)=H-TS}
History Culture History General Entropy Gas laws "Perpetual motion" machines Philosophy Entropy and time Entropy and life Brownian ratchet Maxwell's demon Heat death paradox Loschmidt's paradox Synergetics Theories Caloric theory Vis viva ("living force") Mechanical equivalent of heat Motive power Key publications An Experimental EnquiryConcerning ... Heat On the Equilibrium ofHeterogeneous Substances Reflections on theMotive Power of Fire Timelines Thermodynamics Heat engines Art Education Maxwell's thermodynamic surface Entropy as energy dispersal
Scientists Bernoulli Boltzmann Bridgman Carathéodory Carnot Clapeyron Clausius de Donder Duhem Gibbs von Helmholtz Joule Kelvin Lewis Massieu Maxwell von Mayer Nernst Onsager Planck Rankine Smeaton Stahl Tait Thompson van der Waals Waterston
Other Nucleation Self-assembly Self-organization Order and disorder
Category
v t e

In physics and engineering, in particular fluid dynamics, 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 Q (sometimes V ˙ {\displaystyle {\dot {V}}} ). 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.

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 m3/(m2·s), that is, m·s−1. The integration of a flux over an area gives the volumetric flow rate.

The SI unit is cubic metres per second (m3/s). Another unit used is standard cubic centimetres per minute (SCCM). In US customary units and imperial units, volumetric flow rate is often expressed as cubic feet per second (ft3/s) or gallons per minute (either US or imperial definitions). In oceanography, the sverdrup (symbol: Sv, not to be confused with the sievert) is a non-SI metric unit of flow, with 1 Sv equal to 1 million cubic metres per second (260,000,000 US gal/s);[1][2] 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 to measure the volumetric rate of transport of ocean currents.

Fundamental definition[edit]

Volumetric flow rate is defined by the limit[3]

Q = V ˙ = lim Δ t → 0 Δ V Δ t = d V d t , {\displaystyle 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 of volume of fluid V through a surface per unit time t.

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

Alternative definition[edit]

Volumetric flow rate can also be defined by

Q = v ⋅ A , {\displaystyle 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 a surface integral:

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

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, n ^ {\displaystyle {\hat {\mathbf {n} }}} . The relation is A = A n ^ {\displaystyle \mathbf {A} =A{\hat {\mathbf {n} }}} .

Derivation[edit]

The reason for the dot product 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 ⁡ θ , {\displaystyle Q=vA\cos \theta ,}

where θ is the angle between the unit normal n ^ {\displaystyle {\hat {\mathbf {n} }}} and the velocity vector v of the substance elements. The amount passing through the cross-section is reduced by the factor cos θ. 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 = π/2 and so this amount of the volumetric flow rate is zero:

Q = v A cos ⁡ ( π 2 ) = 0. {\displaystyle 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[edit]

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

Q = m ˙ ρ , {\displaystyle Q={\frac {\dot {m}}{\rho }},}

where

ṁ = mass flow rate (in kg/s), ρ = density (in kg/m3).

Related quantities[edit]

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

∫ L d θ = R T 2 π ( cos ⁡ θ 2 − cos ⁡ θ 1 ) + r T 2 π ( θ 2 − θ 1 ) , {\displaystyle \int L\,\mathrm {d} \theta ={\frac {RT}{2\pi }}(\cos \theta _{2}-\cos \theta _{1})+{\frac {rT}{2\pi }}(\theta _{2}-\theta _{1}),}

where T 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, and θ2 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[edit]

• In cardiac physiology: the cardiac output
• In hydrology: discharge
  • 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[edit]

• Bulk velocity
• Flow measurement
• Flowmeter
• Mass flow rate
• Orifice plate
• Poiseuille's law
• Stokes flow

References[edit]

1. ^ "Glossary". Ocean Surface Currents. University of Miami Rosenstiel School of Marine, Atmospheric, and Earth Science. Retrieved 2019-04-15.
2. ^ "Sverdrups & Brine". Ecoworld. Archived from the original on 20 January 2011. Retrieved 12 August 2017.
3. ^ Engineers Edge, LLC. "Fluid Volumetric Flow Rate Equation". Engineers Edge. Retrieved 2016-12-01.
4. ^ International Union of Pure and Applied Chemistry; https://iupac.org
5. ^ "Volume flow rate, qv". The IUPAC Compendium of Chemical Terminology. 2014. doi:10.1351/goldbook.V06642.
6. ^ "Mass flow rate, qm". The IUPAC Compendium of Chemical Terminology. 2014. doi:10.1351/goldbook.M03720.
7. ^ "Heat, q, Q". The IUPAC Compendium of Chemical Terminology. 2014. doi:10.1351/goldbook.H02752.

v t e Rivers, streams and springs
Rivers(lists)	Alluvial river Braided river Blackwater river Channel Channel pattern Channel types Confluence Distributary Drainage basin Subterranean river River bifurcation River ecosystem River source Tributary
Streams	Arroyo Bourne Burn Chalk stream Coulee Current Stream bed Stream channel Streamflow Stream gradient Stream pool Perennial stream Winterbourne
Springs(list)	Estavelle/Inversac Geyser Holy well Hot spring list list in the US Karst spring list Mineral spring Ponor Rhythmic spring Spring horizon
Sedimentary processesand erosion	Abrasion Anabranch Aggradation Armor Bed load Bed material load Granular flow Debris flow Deposition Dissolved load Downcutting Erosion Headward erosion Knickpoint Palaeochannel Progradation Retrogradation Saltation Secondary flow Sediment transport Suspended load Wash load Water gap
Fluvial landforms	Ait Alluvial fan Antecedent drainage stream Avulsion Bank Bar Bayou Billabong Canyon Chine Cut bank Estuary Floating island Fluvial terrace Gill Gulch Gully Glen Meander scar Mouth bar Oxbow lake Riffle-pool sequence Point bar Ravine Rill River island Rock-cut basin Sedimentary basin Sedimentary structures Strath Thalweg River valley Wadi
Fluvial flow	Helicoidal flow International scale of river difficulty Log jam Meander Plunge pool Rapids Riffle Shoal Stream capture Waterfall Whitewater
Surface runoff	Agricultural wastewater First flush Urban runoff
Floods and stormwater	100-year flood Crevasse splay Flash flood Flood Urban flooding Non-water flood Flood barrier Flood control Flood forecasting Flood-meadow Floodplain Flood pulse concept Flooded grasslands and savannas Inundation Storm Water Management Model Return period
Point source pollution	Effluent Industrial wastewater Sewage
River measurementand modelling	Baer's law Baseflow Bradshaw model Discharge (hydrology) Drainage density Exner equation Groundwater model Hack's law Hjulström curve Hydrograph Hydrological model Hydrological transport model Infiltration (hydrology) Main stem Playfair's law Relief ratio River Continuum Concept Rouse number Runoff curve number Runoff model (reservoir) Stream gauge Universal Soil Loss Equation WAFLEX Wetted perimeter Volumetric flow rate
River engineering	Aqueduct Balancing lake Canal Check dam Dam Drop structure Daylighting Detention basin Erosion control Fish ladder Floodplain restoration Flume Infiltration basin Leat Levee River morphology Retention basin Revetment Riparian-zone restoration Stream restoration Weir
River sports	Canyoning Fly fishing Rafting River surfing Riverboarding Stone skipping Triathlon Whitewater canoeing Whitewater kayaking Whitewater slalom
Related	Aquifer Aquatic toxicology Body of water Hydraulic civilization Limnology Riparian zone River valley civilization River cruise Sacred waters Surface water Wild river
Rivers by length Rivers by discharge rate Drainage basins Whitewater rivers Flash floods River name etymologies Countries without rivers

Authority control databases: National	Germany Czech Republic