2012-02-28_REVISION - M1980244 ‎(70)‎

• Permit No: M1980244
• IBM Index Class Name: REVISION
• Doc Date: 2/28/2012
• Doc Name: VOL 5, Appendix C: Design Calculations
• From: CC&V
• To: DRMS
• Type & Sequence: AM10
• Email Name: TC1
• Media Type: D
• Archive: No

4-2 HANDBOOK OF HYDRAULICS
approximately diameter downstream from the inner face of
the orifice plate. The area at the vena contracts, as in Fig.
4 -1, is related to the area of the orifice a as follows:
as '" C.a (4-1)
where C. is called the coefficient of contraction.
The contraction is caused by the fact that those elements of
the fluid with approach from the aide of the orifice have trans-
verse velocity components directed
toward the center of the orifice.
Therefore the nearness of bounda-
ries, as, for example, for orifices in
pipes (p. 4 -14), would tend to reduce
the amount of contraction and
increase C.. If one edge of an
orifice is flush with a wall, the con-
traction on that side will be entirely
eliminated. However, the con-
traction from the other side will
be increased by approximately
the same amount. Increasing the
roughness of the inner face of the
orifice plate would also reduce the
transverse velocity components
slightly. Rounding the inner edge
of the orifice reduces contraction,
and the contraction can be eliminated completely by shaping the
orifice to conform with the form of the contracting jet as shown
in Fig. 4-2.
Fundamental Equations. The Bernoulli equation (Sec. 3),
written from any point in the liquid, such as 1 in Fig. 4-1, to the
vena contracta (point 2), taking the datum plane through the
center of the orifice, is
Fro. 4-2. Bell- mouthed
orifice with rounded up-
stream edge.
and
vs
VII Or V: VI
2g w 2g w
(4-2)
`w w 2g
Point 2 being located where the jet has ceased to contract, its
pressure is that of the surrounding fluid. For discharge into
(4 -3)
ORIFICES, GATES, AND TUBES 4-3
the atmosphere, ps is therefore zero on the gage scale. For large
tanks, v, is so small that it may be neglected. Replacing pl/w
with h and dropping the subscript of vs, Eq. (4-3) may now be
written
v m V2g0 — h,) (4-4)
Neglecting energy losses, the equation for the theoretical
velocity becomes
(4-8)
(4-6)
or
Then
h
C l , ffi g
The expression v,'/2g is termed the velocity head. Table
4-1 gives values of v, for heads from 0 to 50 ft, and Table 4-2
gives h. for velocities from 0 to 50 ft per see.
It is convenient to take care of the energy loss by introducing
a coefficient of velocity C.. Thus, instead of subtracting the
energy loss as in Eq. (4-4), the expression for the velocity at the
yens contracts may be written
v m C. (4 -7)
The discharge is the product of the velocity and area at the
vena contracta as follows:
Q m alas (
Inserting the value of a from Eq. (4-1) and the value of v
from Eq. (4-7),
Q - CAC. ViTh (4-9)
The product of the coefficient of velocity C, and the coefficient
of contraction C. is called the coefficient of discharge C. Equa-
tion (4 -9) may then be written
Q m Ca 1,r2g7i (410)
The energy lose may be expressed in terms of C. by equating
the expressions for v given in Eqs. (4-4) and (4-7) as follows:
V2g(h hr) °° C, 1/
hr a (1 -- C.')h (4 -11)