Volumetric Flow Rate - Useful Definition

Useful Definition

Volumetric flow rate can also be defined by:

where:

• v = velocity field of the substance elements flowing,
• A = cross-sectional vector area/surface,

The above equation is only true for flat, plane cross-sections. In general, including 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, A, and a unit vector normal to the area, . The relation is .

The reason for the dot product is as follows. The only volume flowing through the cross-section is the amount normal to the area; i.e., parallel to the unit normal. This amount is:

where θ is the angle between the unit normal and the velocity vector v of the substance elements. The amount passing through the cross-section is reduced by the factor . As θ increases less volume passes through. Substance which passes tangential to the area, that is perpendicular to the unit normal, doesn't pass through the area. This occurs when θ = π⁄₂ 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.