ABSTRACT
The Bernoulli equation in ßuid mechanics states that when a ßow is steady and incompressible, and when frictional loss is negligible, the total energy of a ßuid particle of unit mass is constant along a streamline, namely,
constant (along a streamline) (10.1)
where
V
,
p
,
r
, and
z
are the velocity, pressure, density, and elevation, respectively, of the ßuid particle at any given location along a streamline, and
g
is the gravitational acceleration. From this equation, it can be proved that the discharge of incompressible ßow through a venturi is
(10.2)
where subscripts 1 and 2 represent the upstream tap and the downstream tap (the throat), respectively. Note that a factor
C
is included in Equation 10.2 to take into account the energy loss not included in the Bernoulli equation. For turbulent ßow through a venturi with smooth interior,
C
is approximately 0.98. When the venturi is horizontal, the quantity (
z
–
z
) is zero, and Equation 10.2 reduces to
(10.3)
Equation10.3 is the counterpart of Equation 3.53 for compressible ßow treated in Chapter 3. For gas ßow through a venturi at relatively high speed, the density of the gas changes signiÞcantly from point 1 to point 2 of the venturi, and the compressible ßow solution given by Equations 3.53 and 3.54 should be used to determine the mass ßow rate, which is
r
A
V
. For a given venturi in a given pipe, the values of
C
,
A
A
,
z
,
z
,
g,
and
r
are all known and constant, and so Equation 10.2 (or Equation 10.3) is reduced to
where (10.4)
where
C
is a constant. The above equation shows that the discharge
Q
through a venturi is proportional to the square root of the piezometric head difference,
D
h
. This relationship is true for most ßowmeters operating in the turbulent-ßow range.
