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.
Read more about this topic: Volumetric Flow Rate
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