Page 139 - Fluid Power Engineering

Hydraulic Pumps 113

FIGURE 4.24 Centrifugal force acting on a fl uid element.

Considering the pressure and centrifugal forces acting on an ele-
ment of fluid (see Fig. 4.24), the following relations are deduced neglect-
ing the term (drdξ) compared with (rdξ):

ξ
+
(P dP )brd =ξ Pbrd +ξ F or F = brd dP (4.33)
r
r
The centrifugal force F is given by
r
ξ
2
F = mrω 2 = ρ rd bdrrω (4.34)
r
Then, dP =ρ rω 2 dr (4.35)
∫ 0 Pc dP = ∫ 0 r ρω 2 rdr (4.36)

ω 2
or P =ρ r 2 (4.37)
C
2
The pump input pressure P should be greater than the centrifu-
i
gal forces pressure P . Therefore, the maximum pump speed should
C
be limited as follows:
ω = 2 πn (4.38)
P > P (4.39)
i C
1
or n < P /2ρ (4.40)
max π r i
where P = Pump inlet pressure, Pa
i
r = Gear addendum radius, m
ρ = Oil density, kg/m 3