handbook of modern sensors

360 11 Flow Sensors
(A)
(B)
Fig. 11.1. Tube of flow (A) and flow of a medium through a plane (B).
Fig. 11.2. Profile of the velocity of flow in a pipe.
where v is the velocity of moving medium, which must be integrated over area A,
and x is the displacement of volume V . Figure 11.2 shows that the velocity of a
liquid or gas in a pipe may vary over the cross section. It is often convenient to define
an average velocity
∫
vdA
v a =
A . (11.3)
When measuring the velocity by a sensor whose dimensions are substantially
smaller than the pipe size, one should be aware of the possibility of erroneous detection
of either too low or too high velocity, whereas the average velocity, v a , is somewhere
in between. A product of the average velocity and a cross-sectional area is called the
flux or flow rate. Its SI unit is cubic meters per second. The U.S. Customary System
unit is cubic feet per second. The flux can be found by rearranging Eq. (11.3):
∫
Av a = vdA. (11.4)
What a flow sensor usually measures is v a . Thus, to determine the flow rate, the
cross-section area of tube of flow A must be known, otherwise the measurement is
meaningless.