Page 55 - Fluid Power Engineering

32 Cha pte r T w o

The bulk modulus of pure oil is nearly constant when operating
at a certain temperature and pressure. However, when the oil in-
cludes bubbles of gases, air, or vapors, the bulk modulus of this mix-
ture decreases due to the high compressibility of gases. If the total
volume of mixture is V , the gas volume is αV , and the oil volume is
T T
(1 −α)V , an equivalent bulk modulus B of the mixture is deduced as
T e
follows:
The compression of gases is governed by
n
PV = const (2.49)
+
n
Then, VdP nV n−1 PdV = 0 (2.50)
dP
or B =− = nP (2.51)
g
/
dV V
Assuming that the oil-gas mixture is subjected to a pressure vari-
ation ΔP, then the variations of volumes are

α V
ΔV =− T ΔP (2.52)
g nP

α
(1 − ) V
ΔV =− T ΔP (2.53)
o B
Δ P
B =− (2.54)
e V + Δ
Δ ( V )/ V
o g T
nBP
Thus, B = nP −(1 α ) + Bα (2.55)
e
B nP
and B = e = (2.56)
B nP −(1 α ) + Bα

or 1 = α + 1 − α (2.57)
B nP B
e
where α= Ratio of gases volume to the total volume, at atmospheric
pressure
n = Polytropic exponent = 1 to 1.4
P = System absolute pressure, Pa
ΔV = Change in gas volume due to compressibility, m 3
g
ΔV = Change in oil volume due to compressibility, m 3
o
B = Equivalent bulk modulus of mixture, Pa
e