Volumetric flow rate

The volumetric flow rate in fluid dynamics and hydrometry, (also known as volume flow rate or rate of fluid flow) is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second [m3 s-1] in SI units, or cubic feet per second [cu ft/s]). It is usually represented by the symbol Q.

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.

Given an area A, and a fluid flowing through it with uniform velocity C with an angle θ away from the perpendicular direction to A, the flow rate is:

Q = A \cdot C \cdot \cos \theta .

In the special case where the flow is perpendicular to the area A, that is, θ = 0, the volumetric flow rate is:

Q = A \cdot C .

The equation above is commonly referred to as the continuity equation (for one-dimensional incompressible flows). If the velocity of the fluid through the area is non-uniform (or if the area is non-planar) then the rate of fluid flow can be calculated by means of a surface integral:

Q = \iint_{S} \mathbf{C} \cdot d \mathbf{S}

where dS is a differential surface described by:

d\mathbf{S} = \mathbf{n} \, dA

with n the unit surface normal and dA the differential magnitude of the area.

If a surface S encloses a volume V, the divergence theorem states that the rate of fluid flow through the surface is the integral of the divergence of the velocity vector field C on that volume:

\iint_S\mathbf{C}\cdot d\mathbf{S}=\iiint_C\left(\nabla\cdot\mathbf{C}\right)dV .


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