fluid_mechanics

112 Chapter 3 ■ Elementary Fluid Dynamics—The Bernoulli Equation
d j
d h
C C = 0.61
C C = 1.0
(a) Knife edge
C C = A j /A h = (d j /d h ) 2
(b) Well rounded
C C = 0.61
C C = 0.50
(c) Sharp edge
(d) Re-entrant
F I G U R E 3.14 Typical flow patterns and contraction coefficients
for various round exit configurations. (a) Knife edge, (b) Well rounded, (c) Sharp
edge, (d) Re-entrant.
The continuity
equation states that
mass cannot be created
or destroyed.
V 2 = 2V 1
A 2
Q
A 1 = 2A 2
(1)
V 1
3.6.2 Confined Flows
In many cases the fluid is physically constrained within a device so that its pressure cannot be prescribed
a priori as was done for the free jet examples above. Such cases include nozzles and pipes
of variable diameter for which the fluid velocity changes because the flow area is different from
one section to another. For these situations it is necessary to use the concept of conservation of
mass 1the continuity equation2 along with the Bernoulli equation. The derivation and use of this
equation are discussed in detail in Chapters 4 and 5. For the needs of this chapter we can use a
simplified form of the continuity equation obtained from the following intuitive arguments. Consider
a fluid flowing through a fixed volume 1such as a syringe2 that has one inlet and one outlet
as shown in Fig. 3.15a. If the flow is steady so that there is no additional accumulation of fluid
within the volume, the rate at which the fluid flows into the volume must equal the rate at which
it flows out of the volume 1otherwise, mass would not be conserved2.
The mass flowrate from an outlet, m # 1slugss or kgs2, is given by m # rQ, where Q 1ft 3 s or m 3 s2
is the volume flowrate. If the outlet area is A and the fluid flows across this area 1normal to the area2
with an average velocity V, then the volume of the fluid crossing this area in a time interval dt is VA dt,
equal to that in a volume of length V dt and cross-sectional area A 1see Fig. 3.15b2. Hence, the volume
flowrate 1volume per unit time2 is Q VA. Thus, m # rVA. To conserve mass, the inflow rate
must equal the outflow rate. If the inlet is designated as 112 and the outlet as 122, it follows that m # 1 m # 2.
Thus, conservation of mass requires
r 1 A 1 V 1 r 2 A 2 V 2
If the density remains constant, then r 1 r 2 , and the above becomes the continuity equation for
incompressible flow
A 1 V 1 A 2 V 2 , or Q 1 Q 2
(3.19)
For example, if as shown by the figure in the margin the outlet flow area is one-half the size of the
inlet flow area, it follows that the outlet velocity is twice that of the inlet velocity, since