Design and Modeling of Fluid Power Systems
ME 597/ABE 591 - Lecture 7
Dr. Monika Ivantysynova
MAHA Professor Fluid Power Systems
MAHA Fluid Power Research Center
Purdue University
Content of 6th lecture
The lubricating gap as a basic design element of displacement machines
Gap flow calculation. Gap with a constant gap height
Gap with variable gap height (slipper)
Gap between piston and cylinder, numerical solution of gap flow equations
Non-isothermal gap flow
Aim:
Derivation of basic equations for gap flow and calculation of
Gap flow parameters (pressure and velocity distribution, load ability,
gap flow and viscous friction)
Basic knowledge about the gap design and simulation models
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Lubricating Gap
Basic Design Element of Displacement Machines
Design & operating
parameter
Gap height
Gap flow
Load ability
Leakage
Performance & Losses
Viscous friction
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Lubricating Gaps -examples
Swash plate axial piston machine
Gaps seal the displacement chamber
QSK
Movable Piston
QSG
QSB
Rotating cylinder block
pi
Qr
Valve plate
(fixed)
Displacement chamber
Gaps between:
piston and cylinder
slipper and valve plate
cylinder block and valve plate
Swash plate (fixed or with adjustable angle)
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap Flow Calculation
Aim:
High load carrying ability
Low friction
Low leakage flow
Gap height lies in a range of some micrometers, whereas all other
dimensions are in a range of millimeters
Laminar flow of an incompressible fluid in the gap can be described using
Navier-Stokes-Equation:
y
z
h=f(x,y)
movable surface
Assumptions:
p2
p
1
- neglecting mass forces
h
- assuming steady state flow,
- change of fluid velocity only in direction of gap height
x
- pressure is not a function of gap height h
fixed surface
Pressure Force + Viscosity Force = 0
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap with constant gap height
Forces applied on a fluid element:
Pressure field
with
∂v
τ = µ⋅
∂z
after double integration velocity yields:
h
1 ∂p 2
v=
⋅ ⋅ z + c1 ⋅ z + c 2
2 ⋅ µ ∂x
boundary conditions:
z=0 … v=0
z=h … v=0
p
1 ∂p 2
v=
⋅ ⋅ (z − h ⋅ z)
2 ⋅ µ ∂x
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap with constant gap height
1 ∂p
Q = b ∫ v ⋅ dz = −
⋅ ⋅ b ⋅ h3
12µ ∂x
0
Gap flow:
p2 − p1
for pressure:
p=
⋅ x + p1
l
∂v z = 0h
Shear stress on surfaces: τ = µ ⋅
∂z z = 0h
Δp h
Δp h
τ z= h = − ⋅
τ z= 0 =
⋅
l 2
l 2
with:
Viscous friction:
Fv = τ ⋅ b ⋅ x
Power loss due to gap flow:
© Dr. Monika Ivantysynova
h
h
l
1 ∂p 2
v=
⋅ ⋅ (z − h ⋅ z)
2 ⋅ µ ∂x
2
1
Δp
PSQ = Q ⋅ Δp =
⋅ b ⋅ h3 ⋅
12 ⋅ µ
l
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap with constant gap height
z
p1
p1=p2
v0
h
boundary conditions:
z=0 … v=0
x=0 … p=p1
z=h … v=v0
x=l … p=p2
p2
x
l
b >> h
Flow velocity:
Shear stress on surfaces:
h
v0
Q
=
b
v
⋅
dz
=
b
⋅
Gap flow:
∫
h
0
Viscous friction:
τ = µ⋅
h
v0 h 2
v0
z
⋅
dz
=
b
⋅
⋅
=
b
⋅
⋅h
∫
h 2
2
0
∂v
∂z
v0
Fv = τ ⋅ l ⋅ b = µ ⋅ ⋅ l ⋅ b
h
l⋅b 2
PSv = Fv ⋅ v 0 = µ ⋅
⋅ v0
h
Power loss due to viscous friction:
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap with constant gap height
Boundary conditions:
z=0 … v=0
x=0 … p=p1
z=h … v=v0
x=l … p=p2
p1
Flow velocity:
1 ∂p 2
v0
v=
z − h ⋅ z) + ⋅ z
(
2 ⋅ µ ∂x
h
z
p1
v0
+
h
∂p
∂ 2v
= µ⋅ 2
∂x
∂z
p2
l
Gap flow:
p2
x
1 ∂p
h
b >> h
Q = b ∫ v ⋅ dz = −
⋅ ⋅ b ⋅ h 3 +v 0 ⋅ ⋅ b
12µ ∂x
2
0
∂v p2 − p1 1
v0
τ = µ⋅ =
⋅ (2 ⋅ z − h ) + ⋅ µ
Shear stress on surfaces:
∂z
l
2
h
h
Power losses:
Ps = PSQ + PSv = Q ⋅ Δp + Fv ⋅ v 0
2
1 Δp 2
v
Ps =
⋅ b ⋅ h3 + µ 0 ⋅ b ⋅ l
12 ⋅ µ l
h
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Radial Gap with constant gap
height
r
h
r
p2
p1
d
x
l
Gap width:
b=π⋅d
Equations from gap with constant gap height can be applied:
π ⋅ d ∂p 3
Q = b ∫ v ⋅ dz = −
⋅h
12µ ∂x
0
h
Gap flow:
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap with variable gap height
Hydrodynamic force
Boundary conditions:
z=0 … v=v0
z=h … v=0
with
z
p1=p2=p0= 0
Flow velocity:
x=0 … p=p1
x=l … p=p2
∂v
τ = µ⋅
∂z
∂p
∂ 2v
= µ⋅ 2
∂x
∂z
vv00
1 ∂p 2
− ⋅⋅zz+ v 0
v=
z − h ⋅ z) +
(
hh
2 ⋅ µ ∂x
Gap flow:
1 ∂p
h
3
Q = b ∫ v ⋅ dz = −
⋅ ⋅ b ⋅ h +v 0 ⋅ ⋅ b
12µ ∂x
2
0
h
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap with variable gap height
Pressure distribution in x-direction:
F
p
(1)
from
x
1 ∂p
h
= b ∫ v ⋅Qdz = −
⋅ ⋅ b ⋅ h 3 +v 0 ⋅ ⋅ b (2)
12µ ∂x
2
0
h
we obtain:
∂p
12µ ⋅ Q 6µ ⋅ v 0
=−
+
3
∂x
b⋅ h
h2
v0
(3)
⎛ 6µ ⋅ v 0 12µ ⋅ Q ⎞
p( x ) = ∫ ⎜ 2 −
dx
3 ⎟
⎝ h
b⋅ h ⎠
© Dr. Monika Ivantysynova
(4)
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with:
(5)
Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap with variable gap height
⎛ 6µ ⋅ v 0 12µ ⋅ Q ⎞
p( x ) = ∫ ⎜ 2 −
dx
3 ⎟
⎝ h
b⋅ h ⎠
(4)
using:
(5)
with:
(6)
(7)
in our case:
where n=-2 for the first term in Eq. (4) and n= -3 for the second term in Eq. (4)
and after integration:
6µ ⋅ v 0
1
12µ ⋅ Q
1
p( x ) =
⋅
−
⋅
2 + c (8)
h2 − h1
h2 − h1
h2 − h1
⋅ (−1)
⋅ x + h1 b ⋅
⋅ (−2) ⎛⎜ h2 − h1 ⋅ x + h ⎞⎟
1
l
l
l
⎝ l
⎠
h
Boundary conditions: x=0 p=p0
© Dr. Monika Ivantysynova
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h
Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap with variable gap height
6µ ⋅ v 0
1
12µ ⋅ Q
1
p( x ) =
⋅
−
⋅
2 +c
(8)
h2 − h1
h2 − h1
h2 − h1
⎛
⎞
h
−
h
⋅ (−1)
⋅ x + h1 b ⋅
⋅ (−2) ⎜ 2 1 ⋅ x + h ⎟
1
l
l
l
⎝ l
⎠
h
6µ ⋅ v 0 ⋅ l
6µ ⋅ Q ⋅ l
c = p0 +
−
2 (9)
(11)
h
−
h
⋅
h
b
⋅
h
−
h
⋅
h
( 2 1) 1
( 2 1) 1
6µ ⋅ l ⎛ v 0
Q
v0
Q ⎞
p( x ) =
⋅ ⎜− +
+ −
+ p0 (10)
2
2⎟
( h2 − h1 ) ⎝ h b ⋅ h h1 b ⋅ h1 ⎠
l
h2 − h1 = ( h − h1 ) ⋅ (12)
x
(13)
finally we get:
Boundary conditions: x=0 p=p0
6µ ⋅ x ⎛
h + h1 Q ⎞
p( x ) =
⋅ ⎜v 0 −
⋅ ⎟ + p0
h ⋅ h1 b ⎠
( h ⋅ h1 ) ⎝
© Dr. Monika Ivantysynova
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(14)
Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap with variable gap height
6µ ⋅ x ⎛
h + h1 Q ⎞
p( x ) =
⋅ ⎜v 0 −
⋅ ⎟ + p0
h ⋅ h1 b ⎠
( h ⋅ h1 ) ⎝
(14)
F
p
x
p1=p2=p0=0
v 0 ⋅ b ⋅ h2 ⋅ h1 (15)
Q=
h2 + h1
v0
h2
p1
h
h1
Using boundary conditions for x=l is
p=p0 and h=h2 from Eq. (14) follows :
p2
x
Substituting Eq. (15) into Eq. (14) the pressure p(x) yields:
6 ⋅ µ ⋅ x ⋅ v 0 h − h2
p( x ) =
⋅
+ p0
2
h
h2 + h1
when h=h1=h2
© Dr. Monika Ivantysynova
(16)
p(x)=p0
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No load ability!
Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap with variable gap height
Load capacity due to hydrodynamic
pressure field generated in the gap:
F=
l
(17)
b
⋅
p
−
p
⋅
dx
∫ ( 0)
6 ⋅ µ ⋅ x ⋅ v 0 h − h2
p( x ) =
⋅
+ p0 (16)
2
h
h2 + h1
0
h − h2
F = b ∫ 6 ⋅ µ ⋅ v0 ⋅ 2
x ⋅ dx
h ( h2 + h1 )
0
F
p
l
(18)
x
6 ⋅ µ ⋅ b ⋅ l 2 ⎡ h1
h1 − h2 ⎤
F=
−2⋅
⎥ ⋅ v 0 (19)
2 ⋅ ⎢ln
h1 + h2 ⎦
( h1 − h2 ) ⎣ h2
p1
v0
h2
h
h1
p1=p2=p0=0
x
Maximum pressure force for h1/h2=2.2 yields:
Fmax
6 ⋅ µ ⋅ l2 ⎡
2.4 ⎤
0.16 ⋅ µ ⋅ l 2
= 2 2 ⋅ ⎢ln2.2 −
⋅ v0 =
⋅ v0
⎥
2
1.2 ⋅ h2 ⎣
3.2 ⎦
h2
© Dr. Monika Ivantysynova
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p2
(20)
Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Example
A gap between two circular plain and parallel surfaces with a central inflow
as shown in Fig.1 is given. Both surfaces are fixed. The inner radius is R1
and the outer radius is R2.
Calculate the flow Q through the gap for a given gap height h, when the
pressure at the inner radius is p1 and the pressure at the outer radius is p2.
And determine the pressure on the radius r= 15 mm.
The following parameters are given:
p1=20 MPa
R1=12 mm
p2=0.5 MPa
R2=25 mm
h = 20 µm
p1
p2
Dynamic viscosity of the fluid:
µ =0.0261 Pa· s
Boundary conditions:
z=0 …
z=h …
v=0
v=0
r
r=R1 … p=p1
r=R2 … p=p2
© Dr. Monika Ivantysynova
p2
r
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Fig.1
Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Example
From force balance on a fluid element follows:
∂p ∂τ
=
∂r ∂z
∂v
with τ = µ ⋅
∂z
Boundary conditions:
After double integration the velocity yields:
1 ∂p 2
v=
z − h ⋅ z)
(
2 ⋅ µ ∂r
z=0 …
z=h …
v=0
v=0
r=R1 … p=p1
r=R2 … p=p2
(1)
3
π
⋅
r
⋅
h
∂p
Gap flow: Q = ∫ 2 ⋅ π ⋅ r ⋅ v ⋅ dz = −
⋅ (3)
6⋅ µ
∂r
0
h
Therefore:
and
∂p
6⋅ µ⋅ Q
=−
∂r
π ⋅ r ⋅ h3
(4)
6⋅ µ⋅ Q
dp
=
−
∫
∫ π ⋅ r ⋅ h 3 ⋅ dr
p1
R1
p2
p2
p1
p2
r
R2
(5)
(6)
© Dr. Monika Ivantysynova
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r
Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Example
(6)
(7)
From Eq. (6) follows for the gap flow:
For the pressure distribution in radial direction we obtain integrating Eq. (4):
(7)
(4)
And using boundary conditions for c follows:
(8)
© Dr. Monika Ivantysynova
(9)
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Example
The pressure distribution in radial direction yields:
(9)
p1
Substituting the volume flow equation Eq. (7)
into Eq. (9) we obtain:
p2
p2
r
(7)
r
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap between piston & cylinder
Axial and radial piston motion
p2
p1
cylinder
p1>p2
RZ
vp
RK
ω·Rz
Gap height
Unrolled gap in Cartesian co-ordinates system
p2
p1
Gap c
ircum
ferenc
Gap height due to inclined piston position
e
© Dr. Monika Ivantysynova
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th
g
en
l
p
Ga
Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap height function
cylinder
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap between piston & cylinder
After double integration flow velocity yields:
Boundary conditions:
z=0 … vx=0
z=h … vx=ω·Rz
vx =
vy=0
vy=vp
1 ∂p 2
z
⋅ ⋅ (z − h⋅ z) + ω ⋅ Rz ⋅
2 ⋅ µ ∂x
h
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap between piston & cylinder
Considering the continuity equation for incompressible fluid:
After substituting the mean velocities the Reynolds-equation can be derived:
and taking into account time dependent change of gap height
the Reynolds equation becomes:
This partial differential equation has to be solved numerically
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Numerical solution of gap flow
j
i
Applying the method of finite differences the
Reynolds equation can be written:
i, j
Δx
(1)
(2)
Δy
Different methods available:
-finite differences
-finite volumes
(4)
-finite elements
(3)
Substituting Eq. (1), (2), (3) and (4) into the Reynolds- equation we obtain:
(5)
Iterative solution of Eq. (5) to calculate the pressure pi,j
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Numerical solution of gap flow
The gap grid
Gap height
Gap height
Gap circumference
© Dr. Monika Ivantysynova
Gap
length
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Gap Parameter
Pressure field between piston and cylinder
p [MPa]
Load capacity of the gap
0
Ga
pl
en
gth
e
re n c
e
f
ircum
c
Gap
with:
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Contact between piston and swash
plate
Increase of volumetric losses
Support by slippers
Hydrodynamically balanced
Hydrostatically balanced
Additional hydrodynamic effect usable
Requires oil filled case
Special surface geometry
can support hydrodynamic
effect
Lower loading of piston-cylinder
Suitable for high pressure and high speed
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Slipper Design
BG… slipper balance (0.95 – 0.99)
Gap flow assuming constant
gap height
FGF
(1)
Flow from displacement chamber to slipper pocket
Through a small orifice
or laminar throttle
(3)
(2)
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Slipper Design
Load ability FGF due to pressure field under the slipper
DG/dG= 1.2 ÷ 1.3
(1)
FGF
(2)
DG/dK= 1.2 ÷ 1.3
Viscous friction force:
Losses due to gap flow:
© Dr. Monika Ivantysynova
Losses due to friction:
Power loss:
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Slipper Motion
Gap flow simulation requires the determination
of gap height
HP
HP
HP
Pressure field
LP
LP
LP
Operating parameter:
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Slipper
down
device
Slipperhold
hold down
using
springs
Outside the piston
Inside the
piston
Slipper hold down device
Slipper hold down by cylinder block spring
Cylinder block spring
Slipper hold down spring
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Slipper
hold down device
Slipper hold down using springs
Slipper hold down device
Slipper hold down spring
Slipper hold down spring
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591
Slipper hold down device
Fixed slipper hold down
Slipper hold down spring
Spacer
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power
Systems, ME 597/ABE 591