fluid_mechanics

3.6 Examples of Use of the Bernoulli Equation 119
(1) (2)
Orifice
Nozzle
Venturi
(1) (2)
F I G U R E 3.18 Typical devices
for measuring flowrate in pipes.
conduits and devices to measure flowrates in open channels. In this chapter we will consider
“ideal” flow meters—those devoid of viscous, compressibility, and other “real-world” effects.
Corrections for these effects are discussed in Chapters 8 and 10. Our goal here is to understand
the basic operating principles of these simple flow meters.
An effective way to measure the flowrate through a pipe is to place some type of restriction
within the pipe as shown in Fig. 3.18 and to measure the pressure difference between the
low-velocity, high-pressure upstream section 112, and the high-velocity, low-pressure downstream
section 122. Three commonly used types of flow meters are illustrated: the orifice meter, the nozzle
meter, and the Venturi meter. The operation of each is based on the same physical principles—
an increase in velocity causes a decrease in pressure. The difference between them is a matter of
cost, accuracy, and how closely their actual operation obeys the idealized flow assumptions.
We assume the flow is horizontal 1z 1 z 2 2, steady, inviscid, and incompressible between
points 112 and 122. The Bernoulli equation becomes
p 1 1 2rV 2 1 p 2 1 2rV 2 2
1The effect of nonhorizontal flow can be incorporated easily by including the change in elevation,
z 1 z 2 , in the Bernoulli equation.2
If we assume the velocity profiles are uniform at sections 112 and 122, the continuity equation
1Eq. 3.192 can be written as
Q
The flowrate varies
as the square root
of the pressure difference
across the
flow meter.
Q ~ Δp
Δp = p 1 – p 2
Q A 1 V 1 A 2 V 2
where A 2 is the small 1A 2 6 A 1 2 flow area at section 122. Combination of these two equations results
in the following theoretical flowrate
Q A 2 B
21p 1 p 2 2
r31 1A 2A 1 2 2 4
(3.20)
Thus, as shown by the figure in the margin, for a given flow geometry 1A 1 and A 2 2 the flowrate
can be determined if the pressure difference, p 1 p 2 , is measured. The actual measured flowrate,
Q actual , will be smaller than this theoretical result because of various differences between the “real
world” and the assumptions used in the derivation of Eq. 3.20. These differences 1which are quite
consistent and may be as small as 1 to 2% or as large as 40%, depending on the geometry used2 can
be accounted for by using an empirically obtained discharge coefficient as discussed in Section 8.6.1.