Page 450 - Chemical process engineering design and economics

Design of Flow Systems 431

velocity rapidly increases. Simultaneously, the pressure drops rapidly, as illus-
trated in Figure 8.12. A further increase in velocity and decrease in pressure may
occur because of the formation of a vena contracta which is the contraction of the
jet flowing from the orifice, as illustrated in Figure 8.11. For liquids, if the pres-
sure reaches the vapor pressure of the liquid, vaporization will occur. After the
vena contracta, the pressure increases and the fluid velocity decreases, because of
an increase in the cross-sectional area of the valve. The pressure at the outlet of
the valve will not reach its value at the inlet because there is a pressure loss caused
by friction. If the pressure rise is rapid, any vapor bubbles formed in the valve will
collapse instantaneously releasing large amounts of energy in a small area, which
may be sufficient to dent the metal. This phenomena is called cavitation, i.e., cavi-
tation is the formation of vapor bubbles followed by their sudden collapse. Dis-
solved gases will also cause cavitation, such as air dissolved in water.
The problem that we must consider next is to relate the pressure drop across
the valve to flow rate and valve size. After applying Bernoulli's equation, Equa-
tion 8.2, across the valve we obtain, for an incompressible fluid,

(Pl~P2)
E = ———— (8.3)
P

because the change in kinetic energy and potential energy is small, and the work
done is zero.
The friction loss term, E, is given by the empirical expression
vo 2
E = K—— (8.4)
2g c
where, K, an experimentally determined factor, is the friction-loss factor for a
valve.
Combining Equations 8.3 and 8.4 to eliminate E and solving for, the fluid ve-
locity in the valve orifice, v 0, we find that
r2 (p - ) v /2
gc
1 P2
v 0 = I ——————— I (8.5)
I Kp )
Multiply Equation 8.5 by A to obtain the volumetric flow rate through the
o
valve, and let p = pil, where r| is the specific gravity of the fluid. If a valve
w
coefficient, Cy, is defined by 2
(2 gc r
C v = 7.48 (12) (60) AO | —— | (8.6)

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