Volumetric flow rate


In physics and engineering, in particular fluid dynamics, the volumetric flow rate is the volume of fluid which passes per unit time; usually it is represented by the symbol . Its SI unit is cubic metres per second.
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.
The volumetric flow rate across a unit area is called volumetric flux, as defined by Darcy's law and represented by the symbol. Conversely, the integration of a volumetric flux over a given area gives the volumetric flow rate.

Units

The SI unit is cubic metres per second. Another unit used is standard cubic centimetres per minute. In US customary units and imperial units, volumetric flow rate is often expressed as cubic feet per second or gallons per minute. In oceanography, the sverdrup is a non-SI metric unit of flow, with equal to ; it is equivalent to the SI derived unit cubic hectometer per second. Named after Harald Sverdrup, it is used almost exclusively in oceanography to measure the volumetric rate of transport of ocean currents.

Fundamental definition

Volumetric flow rate is defined by the limit
that is, the flow of volume 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 volumetric flow and mass flow respectively, to distinguish from the notation for heat.

Alternative definition

Volumetric flow rate can also be defined by
where
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:
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,, and a unit vector normal to the area,. The relation is.

Derivation

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
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 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.

Relationship with mass flow rate

When the mass flow rate is known, and the density can be assumed constant, this is an easy way to get :
where

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, is the angle where opening begins, and is where the valve closes. This has to be factored by the width of the valve throat. The answer is usually related to the cylinder's swept volume.

Some key examples