Volumetric flow rate

The most fundamental quantity describing the rate at which fluid flows is the volumetric flow rate. It is defined as the volume of the fluid flowing through a given cross section in unit time:

\[I := \lim\limits_{\Delta t \rightarrow 0}\frac{\Delta V}{\Delta t} = \frac{\mathrm{d}V}{\mathrm{d}t}.\]

The SI unit of volumetric flow rate is the cubic metre per second:

\[\left[I\right] = 1\,\frac{\mathrm{m}^3}{\mathrm{s}} = 1000\,\frac{\mathrm{\ell}}{\mathrm{s}}.\]

Volumetric flow rate and flow speed

Let us assume that the fluid particles move parallel to each other with the same flow velocity \(\mathbf{v}.\) This way, in time \(\Delta t\) they travel a distance \(s = v \Delta t.\) From the perspective of fluid flow, this means that volume \(\Delta V\) has been carried through a given cross-section area \(A\) of the tube, where
\[\Delta V = A s = A v \Delta t.\]
Thus the volumetric flow rate of the flow is
\[I = \frac{\Delta V}{\Delta t} = \frac{A v \Delta t}{\Delta t} = A v.\]