Page 50 - Fluid Power Engineering

Hydraulic Oils and Theor etical Backgr ound 27

is taken into consideration by introducing the velocity coefficient C ,
v
ranging from 0.97 to 0.99, and defined as

C = Actual velocity at vena contracta (2.34)
v v
2
The flow rate through the orifice is thus given by the following
expression:

CA 2
Q = A C v = v 2 ( P − ) (2.35)
P
2 v 2 2 ρ 1 2
1 −( AA )
/
2 1
or Q = C A 2 ( P − ) (2.36)
P
d 0 ρ 1 2
where the discharge coefficient, C , is given by
d
/
CA A CC
C = v 2 0 or C = v c (2.37)
d 2 d
1 − ( A / A ) 1 − CA A )/ 2
2
(
2 1 c 0 1
where C = Contraction coefficient depends on the geometry of the hole
c
C = Discharge coefficient, typically = 0.6 to 0.65
d
C = Velocity coefficient, typically = 0.97 to 0.99
v
v = Average fluid velocity, m/s
The discharge coefficient depends mainly on the contraction coef-
ficient and the orifice geometry. For a round orifice, the contraction
coefficient can be calculated using the following expression given by
Merritt (1967):
⎧ 2 ⎛ Cd⎞ ⎛ Cd⎞⎫
⎪ ⎪
⎪
C 1+ ⎜ D − c ⎟ tan − 1 ⎜ c ⎟ ⎬ = 1 (2.38)
c ⎨
⎩ ⎪ π ⎝ Cd D ⎠ ⎝ D ⎠ ⎪
⎭
c
where D = Pipe diameter, m
d = Orifice diameter, m
The variation of the contraction coefficient with the diameter
ratio (d/D) is shown in Fig. 2.12. For a sharp-edged orifice, the friction
losses are negligible: C = 1. Therefore, if the orifice diameter is much
v
−1
less than the pipe diameter (d<<D), then tan (C d/D) = C d/D and
c c
Eqs. (2.37) and (2.38) yield:
C = C = ππ/( + ) = .611 (2.39)
0
2
d c