fluid_mechanics

3.6 Examples of Use of the Bernoulli Equation 111
h
(1)
d
h
(2)
(3)
d h
(2)
(1)
a
d j
d
(3)
a
(a)
(b)
F I G U R E 3.12 Horizontal flow from a tank. F I G U R E 3.13 Vena
contracta effect for a sharp-edged orifice.
h
V = 0
V = 2gh
(1)
V = √2gh
The diameter of a
fluid jet is often
smaller than that of
the hole from
which it flows.
h
(2)
As learned in physics or dynamics and illustrated in the figure in the margin, any object
dropped from rest that falls through a distance h in a vacuum will obtain the speed V 12gh,
the same as the water leaving the spout of the watering can shown in the figure in the margin. This
is consistent with the fact that all of the particle’s potential energy is converted to kinetic energy,
provided viscous 1friction2 effects are negligible. In terms of heads, the elevation head at point 112
is converted into the velocity head at point 122. Recall that for the case shown in Fig. 3.11 the pressure
is the same 1atmospheric2 at points 112 and 122.
For the horizontal nozzle of Fig. 3.12a, the velocity of the fluid at the centerline, V 2 , will be
slightly greater than that at the top, V 1 , and slightly less than that at the bottom, V 3 , due to the differences
in elevation. In general, d h as shown in Fig. 3.12b and we can safely use the centerline
velocity as a reasonable “average velocity.”
If the exit is not a smooth, well-contoured nozzle, but rather a flat plate as shown in Fig. 3.13,
the diameter of the jet, d j , will be less than the diameter of the hole, d h . This phenomenon, called
a vena contracta effect, is a result of the inability of the fluid to turn the sharp 90° corner indicated
by the dotted lines in the figure.
Since the streamlines in the exit plane are curved 1r 6 2, the pressure across them is
not constant. It would take an infinite pressure gradient across the streamlines to cause the
fluid to turn a “sharp” corner 1r 02. The highest pressure occurs along the centerline at 122
and the lowest pressure, p 1 p 3 0, is at the edge of the jet. Thus, the assumption of uniform
velocity with straight streamlines and constant pressure is not valid at the exit plane. It
is valid, however, in the plane of the vena contracta, section a–a. The uniform velocity assumption
is valid at this section provided d j h, as is discussed for the flow from the nozzle
shown in Fig. 3.12.
The vena contracta effect is a function of the geometry of the outlet. Some typical configurations
are shown in Fig. 3.14 along with typical values of the experimentally obtained contraction
coefficient, C c A jA h , where A j and A h are the areas of the jet at the vena contracta and the
area of the hole, respectively.
F l u i d s i n t h e N e w s
Cotton candy, glass wool, and steel wool Although cotton candy
and glass wool insulation are made of entirely different materials
and have entirely different uses, they are made by similar processes.
Cotton candy, invented in 1897, consists of sugar fibers. Glass wool,
invented in 1938, consists of glass fibers. In a cotton candy machine,
sugar is melted and then forced by centrifugal action to flow through
numerous tiny orifices in a spinning “bowl.” Upon emerging, the
thin streams of liquid sugar cool very quickly and become solid
threads that are collected on a stick or cone. Making glass wool insulation
is somewhat more complex, but the basic process is similar.
Liquid glass is forced through tiny orifices and emerges as very fine
glass streams that quickly solidify. The resulting intertwined flexible
fibers, glass wool, form an effective insulation material because the
tiny air “cavities” between the fibers inhibit air motion. Although
steel wool looks similar to cotton candy or glass wool, it is made by
an entirely different process. Solid steel wires are drawn over special
cutting blades which have grooves cut into them so that long, thin
threads of steel are peeled off to form the matted steel wool.