Page 57 - Fluid Power Engineering

34 Cha pte r T w o

FIGURE 2.18 Single oil lump, subjected to oil compressibility.

Variation of volume of oil in line due to inlet and outlet flow
rates, ΔV , is
Q
ΔV = (∑ Q − ∑ Q ) dt (2.58)
Q ∫
in
out
Variation of volume of oil in the line due to the compressibility
effect, ΔV , is
c
V
ΔV =− Δp (2.59)
C
B
dV V dP
Then, referring to Eq. (2.48), C =− (2.60)
dt B dt

ΔV + ΔV = ΔV (2.61)
Q C L

The variation of volume of line ΔV depends on the line material,
L
wall thickness, diameter and system pressure: ΔV = (). Assuming
f P
L
rigid wall boundaries, then ΔV = 0.
L
dV dV
or Q + C = 0 (2.62)
dt dt
in ∑
∑ Q − Q out − B dt = 0
Then, V dP (2.63)

The application of Eq. (2.63) to the single lump flow in a pipe (see
Fig. 2.18) yields

V dP
Q − Q − = 0 (2.64)
1 2 B dt
B
B
i ∫
Q dt or P =
Q dt
or ΔP = ∫ V ( Q − ) P + V ( Q − ) (2.65)
2
1
2
1