mel 417 lubrication
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MEL 417 Lubrication. Reynold’s equation. Reynold’s equation for fluid flow. Assumptions: External forces are neglected (gravitational, magnetic etc.) Pressure is considered constant throughout the thickness of the film - PowerPoint PPT PresentationTRANSCRIPT
Reynold’s equation for fluid flow
Assumptions:• External forces are neglected (gravitational, magnetic etc.)• Pressure is considered constant throughout the thickness of
the film• Curvature of the bearing surfaces are large compared to the oil
film thickness• No slip at boundaries• Lubricant is Newtonian• Flow is laminar• Fluid inertia can be neglected• Viscosity is constant through the thickness of the film
Newtonian fluid: shear stress-shear strain relationship
dy
dudy
du
•Linear dependence
•Slope is 1/
Shear stress
Shea
r rat
e
Reynold’s eqn: Equilibrium of a fluid elementforces in one dimension
dydz)dx.x
pp(
pdydz
dxdy)dz.z
( xz
Fluid element
dxdyxz
yz
x
dx
dz
dy
x, y, z: Mutually perpendicular axes
p: pressure on left face, : shear stress on bottom face in x direction
dx, dy, dz: elemental distances
Shear force on top face
Shear force on bottom face
Pressure force on left face
Pressure force on right face
Fluid element- equilibrium equations
Forces on left should match forces on right.Therefore
Simplifying we get:
OR similarly
dxdy)dz.z
(pdydz dydz)dx.x
pp(dxdy
dxdydzx
pdxdydz
z
)1.....(x
p
zxz
)2.....(y
p
zyz
Substituting using Newton’s law of viscosity
In the z direction the pressure gradient is 0, therefore
According to Newton’s law for viscous flow and
Where u and v are the particle velocities in the x and y directions respectively is the coefficient of dynamic viscosity
0z
p
z
uxz
z
vyz
Pressure gradients
Therefore the pressure gradients in terms of only the viscosity and velocity gradients is
and
Assuming that the viscosity is constant
and
z
u
zx
p
z
v
zy
p
2
2
z
u
x
p
2
2
z
v
y
p
Conditions
• p and are independent of z (assumptions)• Therefore integrating
• we get
• Applying boundary conditions, u = U1 at z = h and u = U2 at z = 0 we get C2 = U2 and
so
x
p.
1
z
u2
2
21
2
CzC2
z.
x
p.
1u
21
2
1 UhC2
h
x
p.
1U
1
21 C2
h.
x
p.
1
h
UU
h
z=0
z=h
Substituting we get
And
Let the rate of flow (per unit width) in the x and y directions be qx and qy respectively
Therefore
and
Volume flow rate
h
UU
2
hz
x
p.
1
z
u 21
2212 U
h
z)UU()zhz(
x2
pu
)3.....(2
h)UU(
x
p.
12
hudzq 21
3h
0
x
)4.....(2
h)VV(
y
p.
12
hvdzq 21
3h
0
y
Reynold’s equation for fluid flow between inclined surfaces
0dx
dp
0dx
dp
Oil wedge
Bottom surface
pmaxTop surface
ho
Pressure profile
Bottom surface moves with velocity U
h
p = Pressure
Film thickness = h
When h = ho
p = pmax
therefore
Upper surface is stationary
2
hU
x
p.
12
hq
3
x
and
Reynold’s equation in one dimension
When p = pmax, dp/dx=0, and h = ho
Therefore
Substituting we get
If is the density of fluid, the mass flow rate in the x direction is
2
hUq o
x
)5.....(h
hhU6
dx
dp3
o
x
p.
12
h
2
hU.q.m
3
x
.
Flow rate after substitution
• Equation of continuity for 2 dimensions
• In most bearing systems there is no flow in the y direction, therefore V1=V2=0. If surface1 is stationary then U1 is also 0. Then equations 3 and 4 reduce to
0y
q
x
q yx
)4...(2
hU
x
p.
12
hq
3
x
)5...(
y
p.
12
hq
3
y
and
Reynold’s equation in 2 dimensions
Substituting into the continuity equation we get
Which gives
0y
p.
12
h
yx
p.
12
h
2
Uh
x
33
)6...(dx
dhU6
y
p.h
yx
p.h
x33
Velocity of flow at a fluid element
)dx.x
uu(
u
)dz.z
ww(
Fluid element
w
yz
x
dx
dz
dy
Velocity at top face
Velocity at bottom face
Velocity at left face
Velocity at right face
)dy.y
vv(
vVelocity at front face
Velocity at back face
Refer to book Principles of Lubrication by Cameron A
Balancing in and out flow rates
• The velocities entering the element are u, v, and w along x, y, and z directions respectively
• The velocities leaving are correspondingly , , and
Therefore the flow rates are:
In- udydz, vdxdz, and wdxdy
Out- , and
)dx.x
uu(
)dz.z
ww(
)dy.y
vv(
dydz)dx.x
uu(
dxdz)dy.y
vv(
dxdy)dz.z
ww(
Continuity equation in 3 dimensions• As there are no source or sinks for fluid flow within the element and the volume
remains constant, the total volume flowing in = total volume flowing out, per unit time
Therefore:
On simplifying we get:
Which is the continuity equation in three dimensionsIf we retain the volume terms, we get:
Where qx, qy, and qz are the flow rates per unit width in the x, y, and z directions respectively
dxdy)dzz
ww(dxdz)dy
y
vv(dydx)dx
x
uu(wdxdyvdxdzudydz
0z
w
y
v
x
u
0z
q
y
q
x
q zyx
17
Reynold’s equation- Infinitely long bearing (L>>D)
In this assumption, the pressure does not vary in the y direction
Therefore = 0 and the flow rate qy = 0
Assuming that only one surface moves, with a velocity U, we get (derived earlier)
and
where ho is the film thickness at max/min pressure
y
p
3o
h
hhU6
x
p
2
Uhq o
x LDiameter D
L>>D
18
Infinitely long bearing (L >> D)
3o
h
hhU6
x
p
Pressure p can be obtained from the equation
Provided h can be expressed in terms of x
Therefore
Where C is a constant of integration. Two boundary conditions are required to obtain the values for ho and C. This can be obtained from knowledge of the start and end points of the pressure curve where p = 0
The pressure curve in the figure below ranges from x = 0 to x = B
)10...()Ch
dxh
h
dx(U6p
x
0
x
03o2
x = 0 x = B
Pressure curve
LDiameter D
19
Reynold’s equation- Infinitely short bearing (D>>L)
• In this case the length of the bearing is considered much shorter than the diameter
• Therefore the pressure differential in the x – direction is considered 0 as it is much lower compared to the pressure differential in the y direction
• We therefore get
• The film thickness is assumed not to vary with x, therefore
• Reynold’s equation in two dimensions then becomes
2
Uhqy
2
233
dy
pdh
y
ph
y
32
2
h
dx/dhU6
dy
pd
L = length of bearing
Diameter D
20
Infinitely short bearing
On integration we get
Further integration gives
Where K1 and K2 are constants of integration These can be obtained by putting pressure = 0 at the edges of
the bearing and pressure gradient = 0 at the middle of the bearing (assuming symmetry)
13Ky
h
dx/dhU6
dy
dp
21
2
3KyK
2
y.
h
dx/dhU6p
pmax
-L/2 +L/2y
21
Infinitely short bearing
We therefore get and
The equation therefore becomes
If p = 0 other than when y = -L/2 or +L/2, either dh/dx=0 or h3 is infinite This fact is applied to journal bearings and dh/dx=0 at points of maximum
and minimum film thickness It is also applicable to narrow rotating discs It is not applicable to thrust bearings This theory is applicable when L/D<1/4 and infinitely long theory is
applicable when L/D>=4
0K1 4
L.
h
dx/dhU3K
2
32
)11...(4
Ly
h
dx/dhU3p
22
3