Reference Manual

3−2
Figure 3-1. Liquid Critical Pressure Ratio Factor for Water
equation. This is the mathematical statement of
conservation of the fluid mass. For steady flow
conditions (one-dimensional) this equation is
written as follows:
òVA + constant (3)
Using these fundamental equations, we can
examine the flow through a simple, fixed
restriction such as that shown in figure 3-1. We
will assume the following for the present:
1. The fluid is incompressible (a liquid)
2. The flow is steady
3. The flow is one-dimensional
4. The flow can be treated as inviscid (having no
viscosity)
5. No change of fluid phase occurs
As seen in figure 3-1, the flow stream must
contract to pass through the reduced flow area.
The point along the flow stream of minimum cross
sectional flow area is the vena contracta. The flow
processes upstream of this point and downstream
of this point differ substantially, thus it is
convenient to consider them separately.
The process from a point several pipe diameters
upstream of the restriction to the vena contracta is
very nearly ideal for practical intents and purposes
(thermodynamically isentropic, thus having
constant entropy). Under this constraint,
Bernoulli’s equation applies and we see that no
mechanical energy is lost — it merely changes
from one form to the other. Furthermore, changes
in elevation are negligible since the flow stream
centerline changes very little, if at all. Thus,
energy contained in the fluid simply changes from
pressure to kinetic. This is quantified when
considering the continuity equation. As the
flowstream passes through the restriction, the
velocity must increase inversely proportional to the
change in area. For example, from equation 4
below:
V
VC
+
(constant)
A
VC
(4)
Using upstream conditions as a reference, this
becomes:
V
VC
+ V
1
ǒ
A
1
A
VC
Ǔ
(5)
Thus, as the fluid passes through the restriction,
the velocity increases. Below, equation 2 has been