Page 45 - Fluid Power Engineering

22 Cha pte r T w o

12μ L
or ΔP = Q = R Q (2.20)
π Dc 3 L L L
where a = Constant, m/s
c = Radial clearance, m
D = Spool diameter, m
F = Pressure force acting on the fluid element, N
P
F = Shear force acting on the fluid element, N
τ
L = Length of leakage path, m
3
Q = Leakage flow rate, m /s
L
r = Radial distance from the midpoint of clearance, m
R = Resistance to leakage, Ns/m 5
L
y = Distance between the element side surface and solid
boundary, m
u = Oil speed in the clearance, m/s
ΔP = Pressure difference across the radial clearance, Pa
It is important to note that the leakage is inversely proportional to
the viscosity, μ, and directly proportional to the cube of radial clearance.
If the radial clearance is doubled due to wear, the internal leakage
increases eight times. The power loss due to leakage is given by

π Dc 3 ΔP 2 ⎛12μ L⎞
ΔN = Q ΔP = ΔP = or ΔN = Q = R Q 2 (2.21)
2
2
L 12 μ L R ⎜ ⎝ π Dc ⎠ ⎟ L
3
L L L
The internal leakage reduces the effective flow rates and increases
the power losses. The dissipated power ΔN is converted to heat and
leads to serious oil overheating problems. Therefore, it is important to
keep the oil viscosity within the predetermined limits over the whole
operating temperature range. This is fulfilled by using hydraulic oils
of convenient viscosity index and implementation of oil coolers.

Fluid Flow in an Eccentric Mounting Radial Clearance In the case of
eccentric mounting, the radial clearance thickness is not constant (see
Fig. 2.6). The flow rate through a narrow radial clearance is given by

π Dc 3 ⎡ 3 ⎛ ⎞ ε 3 ⎤
Q = ⎢1 + ⎜ ⎟ ⎥Δ P (2.22)
12 μ L ⎢ ⎣ 2 c ⎝ ⎠ ⎥ ⎦

Fluid Flow in a Long-Thin-Slot Orifice In the case of a long-thin-slot
orifice, the fluid flow is laminar. The following expression gives the
fluid flow rate in this orifice (see Fig. 2.7).

bh 3
Q = Δ P (2.23)
12μ L