Volumetric flow rate

Fundamental definition

Volumetric flow rate is defined by the limit

${\displaystyle Q=\dot{V}=\underset{\mathrm{\Delta }t\to 0}{lim}\frac{\mathrm{\Delta }V}{\mathrm{\Delta }t}=\frac{\mathrm{d}V}{\mathrm{d}t},}$

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 prefers the notation ${\displaystyle q_v}$ and ${\displaystyle q_m}$ for volumetric flow and mass flow respectively, to distinguish from the notation ${\displaystyle Q}$ for heat.

Alternative definition

Volumetric flow rate can also be defined by

${\displaystyle Q=\mathbf{v}\cdot \mathbf{A},}$

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:

${\displaystyle Q={\iint }_A\mathbf{v}\cdot \mathrm{d}\mathbf{A}.}$

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

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

${\displaystyle Q=vA\cos \theta ,}$

where θ is the angle between the unit normal ${\displaystyle \hat{\mathbf{n}}}$ 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:

${\displaystyle Q=vA\cos \left(\frac{\pi }{2}\right)=0.}$

These results are equivalent to the dot product between velocity and the normal direction to the area.

Relationship with mass flow rate

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

${\displaystyle Q=\frac{\dot{m}}{\rho },}$

where

ṁ = mass flow rate (in kg/s),

ρ = density (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

${\displaystyle \int L\mathrm{d}\theta =\frac{RT}{2\pi }(\cos {\theta }_2-\cos {\theta }_1)+\frac{rT}{2\pi }({\theta }_2-{\theta }_1),}$

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

• 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