Page 44 - Fluid Power Engineering

Hydraulic Oils and Theor etical Backgr ound 21

Considering the shown fluid element in the radial clearance, and neglect-
ing the minor losses at the inlet and outlet, and assuming a concentric
stationary spool, an expression for the leakage flow rate can be deduced
as follows. In the steady state, the fluid element speed is constant and the
forces acting on it are in equilibrium. These forces are the pressure forces
and the friction forces acting on the internal and external surfaces of the
fluid element.
r DdP
The pressure force is F = 2 π (2.10)
P
The friction force is F = 2 π Ddxτ (2.11)
τ
r = 05 c − y then du =− du (2.12)
.
dy dr
du du
For Newtonian fluid, the shear stress is τ = μ = − μ (2.13)

dy dr

Since F = F then du =− rdP or du =− rdP dr (2.14)
P τ μ
dr μ dx dx
The pressure gradient dP/dx is constant.
dP = Δ P where ΔP = P − P (2.15)
dx L 1 2
The velocity distribution in the radial clearance is found by inte-
grating Eq. (2.14).

rdP
∫
u = − μ dx dr a+=− 2 r 2 μ dP (2.16)
a
dx
If the fluid velocity at the boundaries is zero, then
u = 0 for r =± c/2 (2.17)

By substitution from Eqs. (2.15) and (2.17) into Eq. (2.16), the
following expression for the velocity distribution is obtained:

1 Δ P c ⎛ 2 ⎞
u = − r 2 (2.18)
⎜
2μ L ⎝ 4 ⎟ ⎠
The leakage flow rate, Q , is then found as follows:
L
L ∫
Q = − c/2 u Ddr = π Dc L 3 Δ P (2.19)
π
μ
12
c/2