1. Red Sea University Faculty of Engineering Department of Mechanical Engineering
HYDROSTATIC
LUBRICATION
Applied Fluid Flow
Moataz Abdelgadir Ali Abdelgadir

2. General considerations
A hydrostatic system is composed by two surfaces:
one smooth,
the other having one or more pockets (or recesses),
One can characterize two regions:
a first one where the film thickness h is thin (AB and CD)
and a second one, composed by the pockets where the film thickness is larger (BC).

3. Surfaces
a) constant flow
b) constant pressure supply

4. The inlet of the external fluid is located in the recesses.
Fluid supply
The inlet of the external fluid is located in the recesses.
To feed the fluid inside the bearing, two methods can be used :
a constant flow ( valid for liquids only) (a). a pump with a constant output is used). When there are many pockets, every bearing recess can be supplied with a single pump or using constant output adjustment.
a constant pressure (b). In this config., a hydraulic resistance is located upstream of the pockets. The restrictors commonly used are capillary tubes and diaphragms.

5. The fundamental questions
Hydrostatic bearings have a wide range of characteristics and need to be carefully controlled for optimum effect.
The following questions summarize the potential problems that an engineer or tribologist might confront.
how can these films be controlled and produced when needed?
What are the practical applications of this type of lubrication?
What are the critical design parameters of hydrostatic bearings?
What is the bearing stiffness and how can it be controlled?

6. Advantages and limits
The main disadvantages are their price and their complexity. Nevertheless, very often, an existing pressure source can be used to supply the bearing.
Hydrostatic bearings, particularly those operating with liquids, have many advantages:
both surfaces are always separated by a fluid film, even when they are standing still, which means, theoretically, zero wear, and guarantees a long life. The stick-slip occurrence is avoided.

7. Advantages and limits …cont…
the pressure is distributed over a large surface, so there is no pressure peak.
since the load carrying capacity is not related to surface motion, the consequences of machinery errors are less important for liquids.

8. HYDROSTATIC BEARING ANALYSIS

9. Circular hydrostatic pad bearing
The analysis of hydrostatic bearings is much simpler than the analysis of hydrodynamic bearings. It is greatly simplified by the condition that the surfaces of these bearings are parallel.
Consider, a flat circular hydrostatic pad bearing with a central recess as shown

10. Circular h.p. bearing … cont…
Where its parameters are:
pr is the recess pressure [Pa];
h is the lubricant film thickness [m];
η is the lubricant dynamic viscosity [Pa s];
R is the outer radius of the bearing [m];
R0 is the radius of the recess [m];
Q is the lubricant flow [m3/s].
n is the speed of the bearing [rev/s]

11. Pressure distribution
The pressure distribution can be calculated by considering the lubricant flow in a bearing. For a bearing supplied with lubricant under pressure, the flow rate given by:
the flow through the elemental ring at radius ‘r’ is (circular bearing):
integration yields (h ≠ f(r)):

12. Pressure distribution …cont.
Boundary conditions are:
p = 0 at r = R
Substituting into equation (6.2), yields the constant ‘C’:
Hence the pressure distribution for this type of bearing in terms of lubricant flow, bearing geometry and lubricant viscosity is given by:
(1)

13. Lubricant Flow
By rearranging equation (1), the lubricant flow, i.e. the minimum amount of lubricant required from the pump to maintain film thickness ‘h’ in a bearing, is obtained:
Since at r = R0 , p = pr then:
By substituting Q (eq. 2), the pres-sure distribution (eq. 1) is expressed only in terms of the recess pressure and bearing geometry, i.e.:
(2)
(3)

14. Load Capacity
The total load supported by the bearing is obtained by integ-rating the pressure distribution over the specific bearing area:
From pressure distribution shown, the expression for total load is composed of two terms; one related to the recess area and the other to the bearing load area.
(4)
(5)

15. Load Capacity … cont.
Since the recess pressure is constant (5) is reduced to:
(6)

16. Load Capacity … cont. Substituting for pressure (eq. 3) into eq. (6)
Integrating by parts and sub-stituting gives
The expression for the total load that the bearing can support is:
where: W is the bearing load capacity [N].
(7)

17. Friction Torque
The frictional resistance of a rotating hydrostatic circular pad bearing consists only of friction torque (usually very small) can be cal-culated from:(6)
In a similar for load, the expression for total torque has two components; one related to the recess area and the other to the bearing load area:

18. Friction Torque …. Cont.
The frictional resistance of a rotating hydrostatic circular pad bearing consists only of friction torque (usually very small) can be cal-culated from:(6)
the shear stress is:
The friction force in its differential form ‘dF’, also has two components; one for the recess and the other to the bearing load (land) area,
(8)
(9)

19. Friction Torque… cont.
Substituting for U = 2πrn and Assuming constant viscosity and velocity and integrating yields: integration eq. (8) the Friction torque is:
The friction power loss which is transmitted through the operating surfaces is calculated from:
Hf = Tω = 2Tπn
where:
ω is the bearing angular velocity, ω = 2πn, [rad/s];
Hf is the friction power loss in the bearing [W].
(10)

20. OPTIMIZATION OF H. BEARING DESIGN
The parameters of a hydrostatic bearing, such as bearing area, recess area, lubricant flow rate, etc., can be varied to achieve:
either maximum stiffness, maximum load capacity for a given oil flow
or minimum pumping power.
The H. bearing is almost entirely under external control
It is possible to regulate the characteristics of such bearings to a far greater extent than for those of hydrodynamic bearings.

21. Non-dimensional parameters.
Equations (2) & (10) of flow & Load Capacity can be re-written in the following forms for flat circular pads (and flat square pads) in terms of non-dimensional load and flow times a non-dimensional scale factor
where:
A is the total pad area [m2];
Ā and B are non-dimensional load and flow coefficients defined as:
(11) _
(12)

22. Design coefficients for flat circular pad bearings

23. Minimization of Power
From H.P. bearing Figure, if the recess is made almost as large as the bearing diameter, then supply pressure is maintained over virtually the entire area of the bearing. This would ensure a higher load capacity than with a smaller recess but with the disadvantage of requiring a very high rate of lubricant supply pumping power.
The total power required is the sum of friction power (10) and the pumping power
(13)

24. Minimization of Power … cont.
Pumping power ‘Hp’ is defined as the product of the lubricant flow ‘Q’ and the recess pressure ‘pr’,
The total power describes the rate at which the friction and pumping energies are converted into heat in the bearing.
The heat dissipation rate is the product of mass flow rate, specific heat and temperature, thus:
Q : the lubricant flow [m3/s];
ρ : the density of the lubricant [kg/m3];
σ : the specific heat of the lubricant [J/kgK];
ΔT : the temperature rise [°C].
(14)
(15)

25. Ratio of friction to pumping power ζ
This ratio of friction power to pumping power ‘ζ’ (i.e. ζ = Hf/Hp) is used as a measure of the proportion of the hydrodynamic effects to the hydrostatic effects.
If the bearing is not rotating then ζ = 0 (because Hf = 0)
Bearings operating with ζ ≥ 1 are considered as ‘high speed bearings’ and with ζ « 1 as ‘low speed bearings’
When ζ ≥ 3 then the hydrostatic and hydrodynamic effects on load are of the same order

26. Low Speed Recessed Bearings
In low speed bearings Hf ≈ 0, thus Ht ≈ Hp and ζ = 0. The bearing geometry can be optimized so that at a given total bearing area, film thickness, viscosity and applied load the pumping power is a minimum.
This is achieved by calculating pumping power to load ratio, i.e.:

27. High Speed Recessed Bearings
If the bearing is forced to operate at high speeds, the effects of viscous shear due to the relative motion between the surfaces, may become significant.
The ratio of friction power to pumping power & the total power are expressed as:
Substituting and rearranging yields

28. High Speed Recessed …. cont.
For any value of ‘ζ’ it is still necessary to find a minimum value of the ‘H’ parameter to optimize the bearing geometry for maximum load and minimum power.
Other parameters such as  and h can also be optimized.
For example, the lubricant viscosity can be optimized by calculating power losses and load capacity for a range of viscosities, while maintaining all the other parameters at required design levels.

29. High Speed Recessed …. cont.
The optimum clearance is obtained when the power ratio ζ = 3.
The bearing gives the optimum performance when the power ratio ‘ζ’ is between 1 ≤ ζ ≤ 3
The effect of optimization is relatively small since the difference between the most and least effective procedures is only about 15% of total bearing power consumption

30. CFD Analysis

31. H. W. (1)
A circular hydrostatic pad is supporting a load of W = 1000 N, and the upper disk has rotational speed of 5000 RPM. The disk diameter is 200 mm, and the diameter of the circular recess is 100 mm. The oil is SAE 10 at an operating temp. of 70ºC, having a viscosity of  = 0.01 N-s/m2. The efficiency of the hydraulic pump system is 0.6 and that of the motor and drive system is 0.9. Optimize the clearance, h0, for minimum total power consumption.
Hint
Ignore friction loss at recess
Use excel to plot total power