WO2009135639A1 - Determining hydrostatic leakage - Google Patents

Abstract

A method of predicting the hydrostatic leakage (Q) between a seal and a counter face, wherein the seal comprises a first surface portion and the counter face comprises a second surface portion and, in use, the first surface portion contacts the second surface portion and allows hydrostatic leakage (Q) of a fluid between a first region and a second region separated by the seal, the method comprising the steps of : determining a value of S_vm, wherein S_vm is the depth of the lowest valley in a measured roughness profile of the first surface portion relative to a mean line of the roughness profile, or wherein S_vm is the average depth of two or more of the lowest valleys in the roughness profile relative to the mean line; predicting the hydrostatic leakage (Q) based on the value of S_vm.

Description

Determining Hydrostatic Leakage

Technical field

The present invention relates to the field of seals. More specifically, the present invention relates to a method for predicting the hydrostatic leakage of a seal and a method for manufacturing a seal that permits a desired amount of hydrostatic leakage.

Background

Seals are used to prevent leakage between two environments. Seals can be used, for example, to retain a fluid, separate fluids or to prevent the transmission of particulate contaminants from one environment to another. A static seal would completely prevent leakage if the contacting surfaces were perfectly smooth or if the asperities in contact are heavily deformed and sufficiently flattened.

Seals can also be used in non-static devices such as rolling element bearings, to seal an annular gap between an inner and an outer ring of the bearing. The seal serves to retain lubricant, and to prevent the ingress of water and particulate contamination that would reduce the life of the bearing. Elastomeric radial lip seals are often applied to seal bearings and shafts, whereby the seal provides hydrostatic sealing when there is no relative motion between the seal lip and its counterface (e.g. a shaft surface) and provides hydrodynamic sealing when there is relative motion between the seal lip and its counterface (i.e. when the shaft or bearing is running) . To prevent excessive friction and wear of the lip during hydrodynamic operation, the seal relies on an extremely thin elasto-hydrodynamic lubrication film between the seal lip and the moving counterface. The presence of lubricant between the seal lip and the counterface is particularly important on start-up, when slow movement leads to large frictional forces and the seal is most prone to wear.

Thus, although one of the primary functions of a seal is to retain lubricant and reduce leakage, a small amount of hydrostatic leakage is desirable to ensure that lubricant is present between the contacting surfaces when the seal counterface starts to rotate. Furthermore, determination of hydrostatic leakage is an important factor that may be used to enhance the accuracy of predicting when a component such as a bearing should be relubricated. Various models for determining leakage have been proposed previously, but these often depend on particularly complex calculations and lack accuracy due to the many degrees of freedom encompassed in the models.

Accordingly, there is a need for an accurate model for predicting the hydrostatic leakage of a seal and a method to provide a seal that permits a desired amount of leakage.

Summary

The present invention seeks to address at least some of the problems associated with the prior art. In a first aspect, the present invention provides a method of predicting the hydrostatic leakage (Q) between a seal and a counterface. The seal comprises a first surface portion which, in use, contacts a second surface portion of the counterface. The seal allows some hydrostatic leakage between one side of the seal and the other and this leakage passes between the first and second surface portions where they are in contact . The first aspect predicts the hydrostatic leakage by determining a value of a valley parameter of the first surface portion Sv_n,, wherein S™ is the depth between the mean line of a measured roughness profile of the first surface and the lowest valley on the measured profile, or is an average depth between the mean line of the measured roughness profile and two or more of the lowest valleys on the measured profile. Hydrostatic leakage is then calculated on the basis of the determined value of Svm-

In a further development of the first aspect, hydrostatic leakage is determined on the basis of a composite valley parameter for the first surface portion and the second surface portion. Here, a first roughness profile is measured over a length L for the first surface portion of the seal and a second roughness profile is measured over the length L for the second surface portion of the counterface. The first and second roughness profiles are then added to obtain a composite roughness profile and the mean line of the composite profile is determined. The composite valley parameter S_vm' is defined as the depth between the mean line of the composite roughness profile and lowest valley on the composite roughness profile, or is the average depth between the mean line of the composite roughness profile and two or tnore of the lowest valleys on the composite roughness profile. The advantage of the further development is enhanced accuracy of the determination of hydrostatic leakage when the seal is contact with a rough surface.

According to a second aspect, the present invention provides a method of manufacturing a seal. The seal comprises a first surface portion which, in use, contacts a second surface portion of a counterface to thereby allow a desired hydrostatic leakage (Q) of a fluid between a first region and a second region separated by the seal . The method comprises modifying the surface of the first surface portion so that it has a value of S_vn,, defined above, that provides the desired hydrostatic leakage (Q) of the fluid.

According to a third aspect, the present invention provides a lip seal. The lip seal comprises a first surface portion which, in use, contacts a second surface portion of a counterface to thereby allow a desired hydrostatic leakage (Q) of a fluid between a first region and a second region separated by the lip seal. The first surface portion is textured such that it has a value of Svm/ defined above, that provides a desired hydrostatic leakage (Q) .

According to a fourth aspect, the present invention provides the use of the hydrostatic leakage (Q) predicted by performing the method of the first aspect, to determine a suitable relubrication interval for a component or a system of components that comprises one or more seals to retain lubricant within the component or system of components. An example of such a component is a rolling element bearing. An example of such a system of components is a machine or a battery of machines in which the rolling element bearings are lubricated by means of a central lubrication system such as an oil circulation system

According to a fifth aspect, the present invention provides a lubrication dispensing apparatus. The apparatus is configured to perform the calculation described in relation to the first aspect of the present invention, to determine the predicted hydrostatic leakage (Q) of one or more seals and thereby determine a suitable relubrication interval for a component or system of components that comprises the one or more seals.

Brief description of the drawings

The present invention will now be described further with reference to the accompanying drawings, provided by way of example, in which:

Figure 1 shows an example of a radial lip seal.

Figure 2 shows the relationship between contact pressure and the deformed aperture. The solid line shows the pressure at that point of the surface, the dashed line shows the actual surface topography and the shaded regions show the nominal contact force.

Figure 3 shows the real area of contact versus the nominal contact load for a number of surfaces under load. The percolation threshold in each direction is marked as black and white circles and the number corresponds to the respective surface. The dashed lines mark the minimum and maximum real area of contact at percolation threshold.

Figure 4 shows images of the contact spots (black regions) at the percolation threshold in the horizontal

(left) and vertical (right) direction for some surfaces. The corresponding nominal load in MPa is given in each subcaption.

Figure 5 shows nominal contact load versus roughness height parameters at the percolation thresholds in both directions. In the graph representing W versus S_pm, a linear expression for the load is shown.

Figure 6 shows leakage volume for three different rough surface test specimens, including a comparison between measured data, and data simulated with the homogenized and direct methods by using surface roughness measurement data from the same surface test specimen.

Figure 7 shows hydrostatic leakage as functions of nominal contact load for all surfaces in the xl -direction.

Figure 8 shows hydrostatic leakage (kg/h/m) in both directions for all surfaces as functions of nominal load

(Pa) with the surfaces scaled to the same roughness height parameter. The value of the common roughness parameter, which is the mean value of all surfaces, is written out in each figure. The axis limits and values are the same as in Figure 7. Figure 9 shows non-dimensional leakage as functions of non-dimensional contact load for all surfaces in both directions. The flow is scaled with Sv_m to the left, and with the same parameter but for the deformed surfaces, Svmd/ to the right .

Figure 10 shows leakage as functions of nominal contact load for all surfaces in both directions for direct numerical simulation solutions together with the analytical expression.

Figure 11 shows a schematic sketch of a test cell for measuring actual hydrostatic flow between parallel surfaces.

Figure 12 shows a schematic cross-section of the test cell of Figure 11, in an unloaded condition (Figure 12 a) and in a loaded condition (Figure 12b) .

Figure 13 shows a flow-chart representing an embodiment of the first aspect of the present invention.

Detailed Description

The present invention will now be described further. In the following passages different aspects/embodiments of the invention are defined in more detail . Each aspect/embodiment so defined may be combined with any other aspect/embodiment or aspects/embodiments unless clearly indicated to the contrary. In particular, any feature indicated as being preferred or advantageous may be combined with any other feature or features indicated as being preferred or advantageous . In the first aspect, the present invention provides a method of predicting the hydrostatic leakage (Q) between a seal and a counterface, wherein the seal comprises a first surface portion and the counterface comprises a second surface portion and, in use, the first surface portion contacts the second surface portion and allows hydrostatic leakage (Q) of a fluid between a first region and a second region separated by the seal, the method comprising the steps of: determining a value of S™, and predicting the hydrostatic leakage (Q) based on the value of Svm-

S_vm is a surface roughness parameter which characterises a surface in terms of the depth of one or more valleys on the surface relative to a mean line of surface. It is also referred to as a valley parameter. In one example, S_vm is the depth of the lowest valley in a measured roughness profile relative to the mean line of the roughness profile. In another example, S_vn, is the average depth of two or more of the lowest valleys in the measured roughness profile relative to the mean line of the roughness profile.

A valley is defined as a local minimum point that has neighbouring points whose external boundaries all have higher values than the local minimum. When a roughness profile is measured, the local minimums or the nadirs of the valleys can be identified as those points that have eight neighbouring points with a higher value. Identification on the basis of eight 'higher neighbours' has been found to provide good accuracy. Other numbers can also be used. When, relative to the mean line, the distance to the nadirs of more than one valley is measured in order to determine S_vm, the values are averaged. The term nadir is used to refer to the lowest point in any given valley. Accordingly, when two or more nadirs are measured they are necessarily taken from different valleys on the surface of the seal, rather than merely being directly adjacent lowest points. To avoid time-consuming analysis of the entire surface texture, it has been found that sampling and averaging the five lowest valleys provides an effective measure of S_vm which can be used to characterise the first surface portion as a whole .

S_vm is a measure of the surface roughness and can be measured using conventional techniques for investigating the topography of a surface. Such techniques include microscopy techniques including optical methods, interferometry, confocal microscopy and electrical capacitance and electron microscopy and physical methods such as atomic force microscopy or the use of a profilometer . The foregoing methods can be used to measure both the lowest valleys and to determine the mean line of the first surface portion.

According to the second aspect, the present invention provides a method of manufacturing a seal, wherein the seal comprises a first surface portion which, in use, contacts a second surface portion of a counterface to thereby allow a desired hydrostatic leakage (Q) of a fluid between a first region and a second region separated by the seal, the method comprising: modifying the surface of the first surface portion so that it has a value of S_vm (defined above) which, under the intended conditions of use, provides the desired hydrostatic leakage (Q) of the fluid.

The term "modifying the surface" as used herein, includes any known method for affecting the surface morphology of the surface of the seal . Such techniques include surface texturing using a laser or an abrasive and for resilient materials can include techniques such as shot-peening.

Preferred embodiments of each of the methods of the first and second aspects of the present invention are intended to be combined and used interchangeably. For example, the value of S_vm can be calculated for either method using the equation A set out below and any teaching regarding the parameters therein applies equally to either embodiment.

In one example, the seal referred to in the methods according to the invention is a shaft seal such as shown in

Figure 1. The seal 1 comprises a casing 2 to which an elastomeric lip 4 is bonded. The seal may be used, for example, to seal an annular gap between a shaft 5 and a bore of a housing (not shown) , whereby the housing contains one or more lubricated bearings that support the shaft. The seal casing 2 is mounted in the housing bore and the lip 4 has a surface 6 that bears against a counterface 7 on the shaft 5. In this example, the seal 1 further comprises a garter spring 3, which urges the lip surface 6 against the counterface 7. The seal is preferably formed of a deformable material and more preferably an elastic material. The seal may comprise an elastomer and may be reinforced by a spring or tensioned/resilient component. Preferred elastomers for seals include acrylate rubber, fluoro rubber, nitrile rubber, hydrogenated nitrile rubber, or mixtures of two or more thereof .

The counterface is the surface against which the seal operates and is not particularly limited. For example, it may be a surface of a shaft, as shown in Figure 1. The counterface can also be a surface of a bearing inner ring, whereby the seal casing is mounted to the bearing outer ring. The seal can also be a cartridge-type seal, which incorporates its own counterface in the form of e.g. a flinger, whereby the cartridge is mounted to seal the annular gap between a bearing outer ring and a bearing inner ring. Depending on the application and strength requirements, the counterface may comprise any suitable material. For example, a plastic, a synthetic or a metal counterface may be used.

The seal 1 provides hydrostatic sealing when the shaft 5 is stationary and provides dynamic sealing when the shaft rotates. Thus during rotation of the shaft 5, the lip surface 6 is in sliding contact with the counterface 7. The sliding contact produces friction, and to reduce the friction and associated wear of the lip surface 6, the sliding contact is lubricated such that a thin lubricant film forms between the lip surface 6 and the counterface 7. The lubricant can be a grease provided specifically to lubricate the sliding contact, or the shaft bearings may be lubricated via an oil bath in the housing, which oil also lubricates the sliding contact between the lip surface 6 and the counterface 7. When the shaft 5 is stationary it is important that a small amount of lubricant remains present between the lip surface 6 and the counterface 7, so that upon start-up, the seal lip is not subjected to excessive friction and wear.

In other words, it is desirable if the seal 1 permits an amount of hydrostatic leakage that sufficiently lubricates the interface between the lip surface 6 in contact with the counterface 7. Needless to say, the amount of hydrostatic leakage permitted must not be so great that the seal loses its function as a means to retain lubricant.

Thus, a means to predict the hydrostatic leakage permitted by a seal can be used to optimise the design of a seal in terms of its hydrostatic sealing performance. Furthermore, such a prediction would be useful in order to calculate how much lubricant is likely to be lost during hydrostatic conditions, to obtain a more accurate determination of the necessary lubrication interval for e.g. a bearing and shaft assembly comprising one or more seals.

The term "hydrostatic leakage" as used herein means leakage of a fluid past a seal when there is no relative motion between the seal and a surface against which the seal bears.

According to the first aspect of the invention, a method of predicting the hydrostatic leakage permitted by a seal is provided. The step of predicting the hydrostatic leakage (Q) is based on the hydrostatic conditions of the seal in use. The value of S_vm selected in the method of the second aspect, relating to the manufacture of a seal, is also based on the hydrostatic conditions that the seal will experience in use.

The hydrostatic conditions include the nominal 'contact load¹ between the first portion and the second surface, W. This is measured in Pascals and reflects the force that drives the surface of the seal towards the counterface . The greater the pressure the more any asperities on the seal are flattened and the lower the likely leakage. The nominal contact load can be determined using conventional methods, for example determining the force between the seal and the counterface and measuring the area of the seal in contact with the counterface.

The specific seal and counterface selected will each have a Young's modulus which can be used to calculate the hydrostatic leakage. Soft deformable materials, such as rubber, have a low Young's modulus in the region of 0.01 GPa. Hard materials, such as steel, have a high Young's modulus in the region of 200 GPa. The Young's modulus of particular materials can often be determined from literature, although the Young's modulus can also be determined by conventional methods of measuring the tensile stress/tensile strain of a material. The hydrostatic leakage is preferably calculated using a composite Young's modulus of the first surface portion of the seal and second surface portion of the counterface, E' . The composite Young's modulus is preferably calculated according to: E'

with E₁ and E₂ being Young's modulus of elasticity for the first and second surface portions respectively and V₁ and V₂ being the Poisson ratio for the first and second surface portions respectively. It can be seen that where the counterface has a very high Young's modulus in relation to the seal, the composite Young's modulus is primarily dependent on the Young's modulus of the softer seal. In some embodiments, when the Young's modulus of the counterface is greater than the Young's modulus of the first surface portion of the seal by a factor of at least ten, the calculation can be based solely on the Young's modulus of the seal; that is E' = E₁.

The pressure difference p_r across the seal is also important. That is, the difference in pressure that would drive a fluid from one side of the seal to another if the seal was removed. This pressure difference drives the hydrostatic leakage. The pressure is measured in Pascals and can be measured using a conventional pressure gauge.

The viscosity η of the fluid at the operational conditions is another factor that affects leakage. The more viscous the fluid, the lower the amount of hydrostatic leakage that would be expected. Viscosity is measured in Pas. The measurement of viscosity is well known in the art and various rheometers are known. Alternatively, viscosity values can be obtained from e.g. product data sheets supplied by lubricant manufacturers. According to a preferred embodiment of the present invention, the hydrostatic leakage permitted by a seal is calculated using the following equation (Equation A) :

Q is the hydrostatic leakage (kg/s/m) ; W is the nominal contact load (Pa); E' is the composite Young's modulus (Pa) ; P_r is the pressure difference across the seal (Pa) ; η is the viscosity of the fluid (Pa. s); a is a positive coefficient having a value of from 1 to 20; and Jb and c are coefficients.

Preferably a has a value of from 10 to 14, more preferably from 11 to 13, and most preferably 12. Preferably b has a value of from 0.02 to 0.03, more preferably from 0.02 to 0.025, and most preferably 0.023. Preferably c has a value of from 0.9 to 1.3, more preferably from 1.0 to 1.2 and most preferably 1.1. In a most preferred embodiment, a is 12, b is 0.023 and c is 1.1.

In one embodiment of the invention, the second surface portion of the counterface is deemed to be essentially smooth. The counterface may be a curved surface, such as a shaft surface (i.e. a rotating part), or in another embodiment, a flat surface for a sliding part. By smooth it is meant that the surface against which the seal bears has a low surface roughness, preferably a surface roughness at least an order of magnitude smaller than that of the seal . When the counterface has a smooth surface, the value of Svm can be considered dependent only on the surface roughness profile of the seal. This is a preferable approximation as it simplifies the calculation and is often an accurate assumption. Accordingly, S™ is the distance between the nadir of the lowest valley on the measured roughness profile of the seal and the mean line of the measured profile, or is the average distance between the mean line and the nadirs of two or more of the lowest valleys on the measured roughness profile. Hence, the hydrostatic leakage can be determined without consideration of the counterface surface.

When the seal is in contact with a rough counterface, an embodiment of the method of the invention for determining hydrostatic leakage does allow the surface roughness of the counterface to be taken into consideration. Specifically, the step of determining the valley parameter comprises determining a composite valley parameter Sv_n,¹.

In this embodiment, a first roughness profile is measured over a length L for the first surface portion of the seal and a second roughness profile is measured over the length L for the second surface portion of the counterface. The first and second roughness profiles are then added to obtain a composite roughness profile and the mean line of the composite profile is determined. The composite valley parameter S_vm' is defined as the depth of the lowest valley in the composite roughness profile relative to the mean line of the composite profile, or is the average depth of two or more of the lowest valleys in the composite profile relative o the mean line. To determine the hydrostatic leakage Q, the parameter Sv_m is replaced with Svm' in equation A.

Preferably, the depth of two or more of the lowest valleys is measured and an average depth is calculated in relation to the mean line of the roughness profile for the first surface portion or the mean line of a composite roughness profile. When the value of Sv_n, or Sv_n/ is based solely on the depth between the mean line and the nadir of the lowest valley on the (composite) measurement profile, the predicted hydrostatic leakage based on Equation A is accurate for the measurement domain. A mean value for the valley parameter S_vm or S_vm' provides a better characterisation of the (composite) surface as a whole, and is therefore preferred. It has been found that the average of the five lowest valleys relative to the mean line of the (composite) roughness profile results in a value for the valley parameter that adequately characterises the first surface portion as a whole or the composite surface as a whole. The greater the number of lowest valleys that are considered, the greater the accuracy of the mean value used.

According to the third aspect of the present invention there is provided a lip seal comprising a first surface portion, which, in use, contacts a second surface portion of a counterface to thereby allow a desired hydrostatic leakage (Q) of a fluid between a first region and a second region separated by the lip seal, wherein the first surface portion is surface modified such that it has a value of Sv_m (defined above) that provides a desired hydrostatic leakage (Q) .

According to the fourth aspect, the present invention provides for the use of the predicted hydrostatic leakage (Q) determined by the method as herein described, to determine a suitable relubrication interval for a bearing or bearing assembly. By determining the hydrostatic leakage of a seal it is possible to determine the rate of loss of lubricant retained by a seal, such as the lubricant retained within a bearing or a bearing housing. Accordingly, the hydrostatic leakage can be used to determine a suitable relubrication schedule to ensure that sufficient lubricant is available and thereby prevent bearing damage due to insufficient lubrication. The method can also be used to determine a suitable relubrication interval for other components, e.g. gears in a sealed gearbox.

According to a fifth embodiment, the present invention provides an automated lubrication dispensing machine. The machine or apparatus preferably comprises a computerised system configured to perform the method described herein, to determine the predicted hydrostatic leakage (Q) of one or more seals. Once the leakage has been determined a suitable relubrication interval for a component or system of components that comprises the one or more seals can be identified and the lubricant can be timely applied. Again, the components referred to can be bearings or gears.

The present invention also provides a method of designing a seal, wherein the seal comprises a first surface portion, which, in use, contacts a second surface portion of a counterface to thereby allow a desired hydrostatic leakage (Q) of a fluid between a first region and a second region separated by the seal, wherein the method comprises determining a value of Svm that provides the desired hydrostatic leakage (Q) , and designing a surface morphology for the first surface portion having the value of Svm- The method of designing a seal for e.g. a bearing provides a design on the basis of the hydrostatic conditions of the seal in use. These parameters are the same as those set out above in other aspects of the present invention.

Similarly, the present inventions provide a method of making a mould for producing a surface-modified seal, whereby the method comprises: determining an optimal value of S™ for a contact surface of a seal that provides a desired hydrostatic leakage (Q) of a fluid between a first region and second region separated by the seal, and modifying the surface of a mould so that it imparts a desired surface texture having the theoretical S_vn, value to a seal manufactured using the mould.

The mould produced according to this aspect of the present invention can be textured by laser texturing, shot peening or any other texturing method known in the art . The value of Svm desired can be determined as described above.

An example for measuring the actual leakage of a seal will now be described. The fluid flow at the interface of a seal and a counterface can be measured in a test rig. For example, measurements can be taken of the in-plane parallel flow to measure the fluid flow between a rough surface of a test specimen seal against a polished counter surface. Figure 11 shows a schematic sketch of a test cell for hydrostatic flow between parallel surfaces, while Figure 12 shows the test cell in cross-section in an unloaded condition (Figure 12a) and in a loaded condition (Figure 12 b) . The test cell shown in Figure 11 is designed as a holder 100 for a rectangular test specimen 110 with a thickness between 1 and 5 mm, whereby the edges perpendicular to the flow are sealed off to prevent side leakage. The location of the edge seal is indicated by the broken line 125' . The test specimen 110 used is rectangular in shape and has dimensions of 15 x 100 mm. During measuring the test specimen 110 is placed on top of a thin rubber seal 130 in the holder 100 with the rough surface facing upwards. The polished and flat counter surface 140 is placed on top of the test specimen 110. This is also sealed with a thin rubber seal 130 against a closing cap 150.

When the test cell is loaded, Figure 12 (b) , the closing cap 150 and seals 125, 130 prevent any fluid flow from passing out of the cell except from inside the interface between the test specimen 110 and the counterface 140. The cap 150 is loaded against the holder 100 with a hydraulic piston in order to ensure a high enough compressive force. The compressive force deforms the edge seal 125, whereby the amount of deformation is limited by two rigid spacers on either side of the edge seal 125.

One difficulty with this method is obtaining a precise fit between the test cell and the specimen 110. A specimen misfit can result in leakage or suppressed flow along the sides thereof, which will influence the measured fluid flows. Therefore, the centre bulk flow 180 (white arrows in Figure 11) is separated from the edge flow 170 (black arrows in Figure 11) so that any possible side leakage will not influence the bulk flow 180 measurement. The flow is obtained from measurements of the centre bulk flow 180 covering the middle 60% of the specimen width.

The inlet pressure is measured with a pressure transducer (WIKA type 891.13.500) placed next to an inlet hole 160 and the outlet pressure is ambient pressure, measured with a barometer. Temperature is measured at the inlet with a thermometer (GEFRAM PTlOO) to calculate the density and the viscosity of the fluid. In an example, the fluid used was ethanol and, hence, the viscosity and density at the observed temperature were well known. The mass of the fluid transported through the interface is measured with a balance (Precisa 3100D) that measures the collected fluid of the bulk flow 180 with a precision of +/-0.1 gram. The measurement time is 10 minutes and the last 5 minutes are used in comparison to the simulations to ensure a steady state flow through the interface when comparing. The measured mass flow reaches steady state when no air is present between the surfaces and all the pipes leading to the balances are filled with fluid.

The total load on the cap 150 is sampled during measurements. However, this force is a sum of the contact load applied to the test specimen 110 and the force required to deform the seals 125, 130 and also an extra force to ensure that the cap 150 is evenly supported by the spacers. The contact load on the test specimen 110, which is the load of interest, depends on the height of the spacers 120 and on the elasticity of the rubber seals 125, 130. One way to estimate the isolated contact load is to dismount the edge seal 125 around the test specimen 110. Thus the hydrostatic leakage of a test specimen under a specific contact load can be measured.

Examples

The effect of the present invention is demonstrated by the following non-limiting examples.

With reference to Figure 13, the method of predicting the hydrostatic leakage will now be described further. A new seal, intended in use to be mounted on a journal of a locomotive axle, is obtained. The first step 100 is to measure the roughness profile of the seal surface that will bear against a counterface. The surface roughness profile can be measured using a profilometer or other suitable device. The first step further comprises determining a mean line of the measured roughness profile and then determining a valley parameter S_vn, of the roughness profile, being the average distance between the mean line and the five deepest valleys on the roughness profile. Let us assume that a value for S_vm of 0.8 μm is measured

With reference to Figure 13, the second step 200 is to determine the hydrostatic conditions that the seal will experience in use. Let us assume that seal will bear against the locomotive journal with a contact load of 1000 Pa and be subject to a pressure difference across the seal of 200,000 Pa. The journal seal, in use, retains a lubricant within a journal bearing arrangement that supports the locomotive wheel. The viscosity of the lubricant fluid at an expected temperature of 323K, under hydrostatic conditions, is 0.01 Pa. s. The second step further comprises determining the relevant material parameters for the seal. The Young's modulus of the seal material is determined by conventional methods or can be obtained from a product data sheet supplied by the seal manufacturer. For example, a seal made of a fluoro rubber has a Young's modulus of approximately 2GPa. In use, the seal will contact the journal of the locomotive wheel. The journal is made of steel, which has a Young's modulus of 200 GPa and is assumed to have an essentially smooth surface (the surface roughness is at least 2 orders of magnitude finer than that of the rubber seal) .

The third step 300 is to determine the hydrostatic leakage of the seal. This is calculated on the basis of the above equation A, using the material parameters and expected condition parameters described under the previous steps. The value of E ' may be approximated to the Young ' s modulus of the seal (2GPa) . S™ is approximated to the valley surface roughness of the seal only, as the journal surface is substantially smooth. On the basis of the assumed values stated above, the journal seal will permit a hydrostatic leakage of approximately 0.0005 kg/h/m.

As a further fourth step 400 (optional) , a relubrication interval can be determined for the journal bearing assembly, which takes into account the length of time that the locomotive will be non-operational in a particular period and the predicted hydrostatic leakage during that period.

In another embodiment of the present invention, a seal for a locomotive journal bearing is to be produced. To ensure that a sufficient amount of lubricant is present between the seal and a counterface against which it bears, it is determined that for the seal dimensions in question, a hydrostatic leakage of 0.0003 kg/h/m is desirable

The corresponding value of S_v1n, to produce this rate of leakage is then calculated using equation A, taking into account the hydrostatic conditions that the seal will experience in use, as previously described. A pattern may then be provided on the surface of the seal that is in contact with the counterface, whereby the pattern is such that the distance between the mean line of a roughness profile of the seal surface and the averaged depth of the five lowest valleys on the roughness profile is equal to the desired value of S_vm- The pattern can be provided on the seal surface or, preferably, a corresponding 'reverse' pattern is provided on a seal mould by means of e.g. laser etching, meaning that seal is produced with the required pattern.

The present invention will now be described further, by way of example, in relation to the underlying theory behind the invention.

The following glossary is provided to help clarify the following theoretical discussion on the subject-matter of the present invention.

A_n Nominal contact area m²

Xi Global coordinate m η Fluid viscosity Pas p Fluid pressure Pa

Pd Surface contact pressure Pa W Nominal surface contact load Pa Q Fluid flow kg/h/tn

~Q Dimensionless flow E¹ Composite Young's modulus Pa of surface material u Surface displacement m h Roughness aperture m

Di Subscript indicating coordinate directions (1, 2) CIO Subscript indicating homogenized quantity

Roughness parameters

S_a Arithmetic mean of absolute deviation m from mean line

S_q Standard deviation of surface roughness m S_t Distance between maximum and mean line m

S_z Distance between average of five highest m peaks and five lowest valleys Sp Distance between the highest peak m and the mean line of surface S_pi_n Distance between average of the five highest m peaks and mean line

S_v Distance between minimum and mean line m

S_vπ, Distance between the average of the five lowest m valleys and the mean line of surface

Acronyms

DNS Direct numerical simulation HNS Homogenized numerical simulation ACM Acrylate rubber FKM Fluoro rubber NBR Nitrile rubber HNBR Hydrogenated nitrile rubber

This section presents the description of the rough aperture (relative gap between the surfaces) and its elastic displacement due to the contact load as well as the approach to calculate the leakage through the deformed aperture.

The aperture between two surfaces in a seal application is critical to the amount of fluid leakage or percolation through the interface. The aperture is described by the combined shape and roughness of the two interacting surfaces. To minimize the leakage in a seal, the surfaces are pressed together with a load leading to surface displacement and asperity flattening.

Since the periodic displacement will be calculated, it suffices to consider a periodic sub-domain Φ of the total leakage domain Ω. Let us therefore define a global coordinate as x e Φ c Q and the aperture between the sealing surfaces as:

h(x) = hi + hi + u - g₀₀, (F.I)

where hi is the combined roughness of the interacting surfaces, which is periodic on Φ and with an arithmetic mean of hi . The normal displacement of the surfaces due to contact load is described by u, and the rigid-body movement (interference) is described by g_Oo • Due to the subtraction of the rigid body movement, the definition above ensures that the aperture is always zero at contact spots. Here we consider nominally flat and parallel surfaces. For a more general description of the aperture, h could be replaced by a global geometry shape.

This theory section considers the deformation of elastomer surfaces. However, it would be understood that any purely elastic material with elastic displacement and no or minimal visco-elastic behaviour could also be modelled in this way. The elastic displacement may, according to the Boussinesq- Cerruti theory, be calculated from the linear convolution of a kernel and the contact pressure:

u = K(x) * pd(x) (F.2)

where ^λ*' is the convolution operator. The pressure at one point will affect the surface deflection at all other points. The convolution kernel, K, which describes the elastic behaviour of the material, is written as:

where E¹ is the composite elastic modulus expressed as:

with E being Young's modulus of elasticity and v being the Poisson ratio for each surface. A deformed rough aperture will have a contact area that is less than the total nominal area, A_n, i.e. unless completely deformed demonstrating 100% contact area, the aperture consists of patches with and without contact.

We may illustrate a loaded aperture as in Figure 2, which shows the relation between the pressure (contact pressure P_d) and the deformed aperture. The total load W carried by the deformed aperture is indicated by the shaded areas. The surfaces in the aperture are purely elastically deformed.

The corresponding contact mechanics system to be solved may be written as the following set of equations and inequalities, where the contact load, W, governs the pressure distribution and thus the amount of displacement: h(x) = hi + h_\{x) H- u(x) — goo, (F.5a) h(x) > 0, p_d{x) = 0. x £ x_c; (F.5b) h(x) = 0, p_d(x) > 0, x € x_c\ (F.5c) p_d(x) > 0, Vx € Φ, (F.5d)

W = - _Pd(x)dx, (F.5e)

Ω

where x_c corresponding to the local contact spots. The rigid-body movement g_Oo is a constant associated with the location of the contact plane. Because the above system is determined by W, g_Oo may be removed from the system in the solution process. The system, Eq. (F.5), can be solved. The discrete cyclic convolution, Eq. (F.2), evaluated by FFT, requiring periodic input in terms of K and P_d and produces periodic output u. The periodicity is a demand of the model adopted herein.

This section describes the method of calculating leakage through the aperture described by Eq. (F.5a) . Let us now introduce a parameter ε > 0 that describes the wavelength of the surface roughness. Now, let the aperture h parameterized in ε be described by hε . The Reynolds equation describing incompressible and iso-viscous flow applied on the parameterized aperture function may be written as

λ - - V ■ (JifVft) = Q, on Ω, (F.6)

where λ = 6ηU. The leakage through static apertures is considered, i.e. that the- velocity, U, is zero and thus reduces the problem to Poiseuille flow only.

Since the aperture does not include a global variable and therefore considers parallel surfaces, it will suffice to solve the hydrostatic Reynolds equation on a periodic sub domain Φ c^; Ω, with Dirichlet boundary conditions in the flow direction and periodic boundary conditions in the transverse direction, according to the following:

V • (hlV_Pε) = 0, Vi 6 Φ (F.7a)

p_ε periodic at X2 = 0, X₂ = £_2? (F.7c) where Φ = [0, εj_.]². To mimic the flow condition for the whole domain, Ω, the boundary pressure p_a is linearly- reduced by the number of periods, ω, contained in Ω. The pressure from Eq. (F.7a) only depends on the gradient of h³ and not on the absolute value of the roughness amplitude. Moreover, the solution for one set of Dirichlet boundary conditions contains all possible solutions through a linear scaling of the pressure.

Even though the above system may be solved for only a small part of the total component to be simulated, note that it only applies to the particular case of completely parallel surfaces with no influence from the global geometry shape. If considering the global geometry of the seal, the amount of degrees of freedom would increase and the direct numerical simulation approach becomes inefficient.

Therefore, we will also consider a flow factor method based on the homogenized Reynolds equation. The result of the homogenization process is that as ε → 0 , the Reynolds equation (F.6) becomes the following homogenized Reynolds equation:

V ^■ (A(i)Vpo) = 0, on Ω, (F.8a)

PG periodic at X₂ = 0, X2 = L_y. (F.8c)

with similar boundary conditions as in Eq. (F.7), but with the homogenized pressure p₀ as output. The homogenized coefficient A is written as:

where y e Y = [0, I]². The solutions to the cell problems, i.e. χl and χ2 , are given by

V_y • (h³ [a + VyXi)) = 0, on F, i = l,2, (F.10)

The homogenized equation describes the roughness influence on fluid flow in the limit of a vanishing wavelength ε → 0. Of course, this does never occur in reality, where the roughness wavelength always remains finite. However, it will be shown that the homogenized solution (HNS) mimics direct numerical solutions (DNS) of the same roughness with ε ≠ 0. The main benefit of the HNS approach is the possibility to calculate the effects of the surface roughness as flow factors before simulating the operating conditions in an application.

Leakage through the parameterized aperture may be calculated as :

where the subscript ε once again indicates the parameterisation, and i the flow direction. Analogously with Eq. (F.6), the integrated flow, Eq. (F.11), the corresponding homogenized flow becomes:

in the i direction. The flow is completely determined by coefficient A, which in turn only depends on the precise roughness of the aperture and the global shape of the application, e.g. a lip seal. Global displacement may be coupled to local roughness displacement through the homogenized approach.

The flow, Q, may be scaled into a dimensionless form, ~Q, through the following:

where h_r is a roughness height scale parameter, and the pressure scale factor is preferably set to the boundary pressure, i.e. p_r = p_a. Hence, ~Q is independent of viscosity, boundary pressure and the absolute roughness amplitude parameter.

With the current numerical scheme, the hydrostatic flow is solved for every grid node even at the contact spots. To avoid numerical problems with oscillating solutions near the borders of the contact spots, the aperture gap is increased with the small constant ε = 10^~8 m. Because of this, the flow will not exactly be zero when the percolation threshold is reached, possibly producing misleading results when the leakage in a small aperture is investigated. Thus, to improve the leakage results for extremely thin apertures, the flow calculated at the specific load corresponding to the percolation threshold will be subtracted from all leakage readings for that specific surface.

The surface roughness data in the leakage simulations comes from ten different surfaces. Table 1

# Type % S_p μτn Sy μτn S_a μm S_q μm S_t μm.

1 ACM 7050 B 100 0.588 0.376 0.089 0.111 0.964

Adapted 0.551 0.360 0.087 0.108 0.911

2 ACM 7050 A 98 2.581 2.648 0.483 0.641 5.229

Adapted 2.485 2.309 0.451 0.598 4.794

3 FKM 7323 B 100 0.554 0.474 0.085 0.111 1.027

Adapted 0.505 0.413 0.077 0.100 0.919

4 FKM 7323 A 100 0.998 1.156 0.190 0.247 2.154

Adapted 0.800 1.089 0.174 0.224 1.890

5 NBR 3147 B 100 0.889 0.540 0.068 0.100 1.429

Adapted 0.869 0.458 0.063 0.090 1.327

6 NBR 3147 A 100 1.712 1.302 0.311 0.399 3.014

Adapted 1.476 1.161 0.298 0.383 2.637

7 HNBR 7609 B 100 0.324 0.260 0.060 0.079 0.584

Adapted 0.322 0.216 0.058 0.077 0.538

8 HNBR 7609 A 99 1.374 2.364 0.336 0.423 3.738

Adapted 1.184 2.181 0.312 0.391 3.365

9 New lip seal 39 0.470 0.423 0.089 0.114 0.894

Adapted 0.337 0.360 0.078 0.099 0.697

10 Worn lip seal 72 0.278 0.703 0.078 0.100 0.980

Adapted 0.244 0.583 0.067 0.086 0.827

Table 1 shows data of the original unadapted surface measurements and the corresponding adapted surfaces. The surface number (#) is the unique name for the particular surface. The material for each surface is shown in the "Type" column. The ratio of valid measurement points is shown in the third column (%) . The parameters are calculated for a roughness measurement with spatial domain size of 42 x 42 μm, except surface #9 and #10 with domain size 25 x 53 μm.

Table 1 shows both the original data from the surface measurement and the corresponding data adapted to suit the computations from all surfaces. All surfaces are elastomers where surfaces #1-8 are used for tribological sliding tests. The material for these surfaces are displayed in Table 1, with the letter B indicating that the measurement is made before a test and A after. From the table, the roughness parameters are completely different before and after the test. Note that the measurements are not taken at the exact same location before and after the test.

Surface #9 (new roughness) and #10 (worn roughness) are the measured roughness from a lip seal (from SKF ERC) . Note the low ratio of valid points for these surfaces, which may lead to unreliable results. In the simulations, the elastomer materials are all considered completely linear elastic with a Young's modulus of E = 10 MPa and a Poisson ratio v = 0.5. A smooth steel counter surface is considered with E = 210 GPa and v = 0.3, which will have negligible effect on the total deformation.

The surfaces are measured by a non-contact optical surface profiler (Wyko NTIlOO) . The apparatus uses optical phase shifting and white light vertical scanning interferometry with sub nanometer vertical resolution. Measurements of the surface roughness are performed with a spatial resolution (measurement array) of 736 x 480 grid points.

The resulting roughness data will contain a certain amount of noise, much depending on the optical properties of the measured surface, such as colour and roughness slopes. Also, some points in the measurement array will not receive any values, i.e. invalid points. Table 1 shows the percentage of valid measurement points for each surface. To use the surface measurements in computations, the data needs to be adapted. The first significant modification to the measurement data is to remove points that can be assumed to be noise, unrepresentative peaks and valleys, or both in the extreme values of the surface. This is done by using the cumulative distribution of height values (Abbot curve) for the surface. The procedure is to split up the roughness measurement height values into 1000 bins. The roughness data are then cut at the height value of the first bin from the top and the bottom containing at least 10 measurement points.

A periodic roughness function is expected in the contact mechanics and the hydrodynamic two-scale approach. Therefore, the next step in the modification of the roughness signal is to render the edges of the data more periodically smooth. A detrimental effect of not having such a periodic match between the boundaries is that the discontinuities at the edges could form a wall, restricting all fluid from passing the edges. Another example on the effects of the sharp non-periodic edges is the ringing or rippling effect that may occur throughout the domain when performing Fourier filtering or convolution.

The edge modification is done using a technique of blurring the data edges. The procedure is to first apply a Gaussian low-pass filter to the data in the frequency domain. A Gaussian 2D transfer function is used where the appropriate size and standard deviation can be chosen for the particular type of roughness. The shape of the Gaussian distribution used in this paper was determined through experimentation. The edge modified data is calculated as the weighted sum of the original and the blurred data. The weighting function is based on the auto-correlation function of the transfer function. This procedure renders the modified data identical to the input data at the central part of the domain and equal to the blurred version at the edges. The modified data near the edges is arranged to get a smooth transition between the periodic pairs.

In this section, simulation results of the deformation and leakage in the rough apertures from the surface measurement samples will be shown. The percolation threshold will be discussed and the influence on some roughness height parameters on the leakage will be shown.

The percolation threshold is reached for a specific contact load when the surface is sufficiently deformed, according to Eq. (F.5) , so that no open path can be found for a fluid particle to travel from one edge to the other. This threshold may be found without any knowledge of the actual leakage from the aperture for lower loads and can give useful information about the behaviour of specific surface roughness in the aperture. The percolation threshold in each flow direction corresponds to a contact load and an area ratio of contact spots versus total nominal area, i.e. the real area of contact . In Figure 3 the real area of contact is plotted against contact load for all surfaces. Each number corresponds to the respective surface. As expected, the rougher surfaces must be loaded more before reaching the percolation threshold. Also, the real area of contact is nearly linear with respect to the load, as may be seen from the plots in Figure 3. The circles in Figure 3 represent the percolation threshold in the respective directions and the dashed lines mark the extreme values of A_r. The percolation threshold is contained within 33-55% of real contact area for all surfaces.

According to Eq. (F.5), the rough surfaces are elastically deformed through a series of contact loads. The percolation threshold is determined by processing images of the contact spots and finding closed contact regions. Since the surfaces are periodic in both directions, the fluid may travel across boundaries in the perpendicular flow direction. This means that connectedness of such regions across the boundaries in the perpendicular flow direction must be accounted for. Images of the contact spots for different rough apertures that are loaded to the percolation threshold in each direction are shown in Figure 4. The arrows indicate the flow direction in the sub captions together with the corresponding load. For most surfaces, the percolation threshold is reached for different loads in each direction. The more homogeneous the roughness structures, the more likely the percolation threshold will be reached for similar loading conditions. A significant difference between the directions can, e.g., be seen for surface #6. The surface has a longitudinal waviness in the vertical direction which means that the surface needs to be loaded heavier to reach the percolation threshold in that direction.

The percolation threshold obviously depends on the particular roughness. In Figure 5, eight different roughness height parameters from all surfaces are plotted versus the contact load at the percolation threshold in each direction. A clear trend is seen in all figures, i.e. the rougher the surface the more load is needed to reach the percolation threshold. The largest spread is achieved with the valley parameters, S_v (mean to largest valley) and S™ (mean to the average of five largest valleys) . From Figure 5, the information from the asperity peaks is clearly important with respect to the percolation threshold. The smallest spread is achieved with the peak parameters, i.e. S_p and S_pm. The simulated elastic displacement only depends on the specific surface data and the ratio between the contact load and the composite Young's modulus: W/E ' . The relatively small spread in the graph showing W versus S_pm makes it tempting to find a linear expression for the W = W(Sp_m/E') . A simple linear approximation of the contact load becomes:

W = 3.6 x 10⁴E¹Sp_1n, (F.14)

which is shown in the graph representing S_pm in Figure 5.

This section shows results of hydrostatic leakage simulations between parallel rough apertures from the surface roughness data. The specific roughness is loaded with a contact load, W, against a flat counter surface.

The results will be shown in both dimensional form for the sake of clarity and non-dimensional form for the sake of generality, since the leakage is proportional to the pressure drop and fluid viscosity. In the dimensional case, the simulated fluid has a constant density p = 10³ kg/m³ and a constant viscosity of η = 10^"3 Pas. The domain is considered to be completely parallel with no global geometry, making a DNS approach possible. This is because it is sufficient to consider only one cell (roughness data) with periodic boundary conditions perpendicular to the flow direction and Dirichlet conditions in the flow direction. The domain size is 1 x 1 mm and with a pressure drop of 1 MPa in the flow direction. Instead of elastomers milled and ground steel surfaces can be used and because of high local contact pressures, perfectly plastic displacement was considered in addition to the linear contact model. A comparison between measurements and simulations both DNS and HNS, is shown in Figure 6.

In Figure 7, the leakage as a function of contact load is plotted for all surfaces with surface number displayed for each curve. The DNS and HNS solutions are shown in the same plot for the xl-direction.

The results are similar even though a limited number of wavelengths may fit the domain. This shows that the assumption of a wavelength approaching zero, used in the HNS method, is valid for wavelengths far from zero for hydrostatic conditions, i.e. HNS is the approach to choose when considering a global geometry.

The roughness parameters from Table 1 are reflected in the flow in Figure 7, where the roughest surfaces are seen to admit more leakage. This is true at least for smaller contact loads. Leakage properties for higher contact loads may alter between the surface, e.g. surface #5. This behaviour may be explained by the set of extremely high peaks of surface #5. When the surface is sufficiently deformed, the fluid flow will experience a rapid drop at a level where the underlying roughness will start to affect the flow.

A large spread in flow between the surfaces is common to all curves in Figure 7. However, the small difference in leakage between the two directions indicates that the roughness height information could be useful in determining the flow. In an attempt to investigate the effects from the roughness parameters, the leakage may be scaled in such way that a specific roughness parameter is kept constant for all surfaces, see Eq. (F.13) . This is done in Figure 8 where the leakage is plotted as a function of contact load for all surfaces in both directions. The roughness height parameters from the undeformed surfaces are used and the surfaces are scaled to the average value of the specific parameter from all surfaces. The same scaling as in Figure 7 is used. There is less spread between the results for the surfaces compared to the curves in Figure 7, in particular when scaling with the valley parameters S_v and S_v1n. These parameters are the most pronounced to describe the flow, which is especially clear for a limited load.

Figure 9 shows the dimensionless flow versus the non- dimensional contact load, see Eq. (F.13) . This is the same graph as in lower right Figure 8, but with dimensionless values. The flow is scaled with the roughness height scale parameter, hr = S_vm/ to the left, and with the same parameter but for the deformed surfaces, hr = S_{vmd /} to the right. Results from all surfaces in both flow directions are shown. The left graph illustrates that the non-dimensional leakage for all surfaces converges towards 1 as the load decreases towards 0. This property is found only for the valley parameters among the roughness parameters studied in Figure 8 and suggests that a good approximation of the leakage for low loads would be :

Q « ² (F.is)

Low loads are considered to be those at which the hydrostatic leakage is substantially constant. These are loads less than 10,000 Pa, more preferably less than 5,000 Pa for the materials used in the simulation. The load range at which the hydrostatic leakage remains constant is, however, material dependent. A non-material dependent value can be derived from the dimensionless value of W/E¹ and is less than 10^"3, more preferably less than 10^"4. That is, a low load (at which the predictive accuracy of equation A is greatest) is related to the composite Young's modulus of the surfaces .

As seen to the right in Figure 9, the valley parameter for the deformed surfaces, S_vmd # strongly controls the flow as well as when the surfaces are deforming. Here, the non- dimensional leakage is close to constant and unity for a wide range of contact loads. We may write a simple approximate expression describing this relation as:

Q = (F.16)

The parameter Svm_d depends on the contact load and the undefortned roughness, i.e. S_v1n. An approximate expression for Svm_d is found to be:

With this information we may write a final approximate expression for the dimensional leakage through a rough aperture :

Ideally, the above expression would determine the leakage for all possible variations of boundary conditions, surface roughness configuration, elastic material properties through the composite Young's modulus, fluid properties through the viscosity, and all contact loads.

A comparison between the simulated flow in both directions and Eq. (F.18) for all surfaces is shown in Figure 10. Note that the leakage is independent on direction in Figure 10. The expression correlates well for some surfaces and less well for others. The approximation seems to give more reliable results for lower contact loads which is expected by inspecting the right-hand side of Figure 9. However, some surfaces show good agreement for the whole load range, from zero to the percolation threshold. The constants in Eq. (F.18) are compromises to fit all surfaces with the same expression. The agreement with numerical simulations could be made more accurate by tweaking the constants. The agreements for higher loads may be enhanced by using the information for the percolation threshold, Eq. (F.14) .

The hydrostatic leakage through a set of measured rough apertures was simulated with both direct numerical simulations (DNS) and with homogenized numerical simulations (HNS) . Real roughness measurements from elastomer surfaces were used as input to the simulations. The surfaces were elastically and periodically deformed for a broad range of loads and the leakage was simulated for the deformed rough apertures. Ten different surface measurements with significantly different characteristics were used, illustrating that the leakage properties are significantly different between the surfaces.

A specific surface roughness height parameter, i.e. Sv and Svm (representing the valley information) , is found to be of specific importance in characterizing the leakage. By scaling the surfaces to the same valley roughness parameter, the leakage becomes similar for all surfaces. A closed form expression is found that approximates the leakage as a function of contact load, Young's modulus, Svm, pressure drop and fluid viscosity.

It is shown that the roughness peak parameters, i.e. Sp and Spm, are significant to the percolation threshold. The percolation threshold appears for different contact loads in different directions and the threshold values for all surfaces are contained between 33 and 55% real area of contact. A closed form expression is found that approximates the contact load required to reach percolation threshold as function of Young's modulus and Spm.

The results from the HNS simulations agree well with DNS simulations. Thus, HNS may be used to reduce the required number of degrees of freedom when considering a global geometry.

The results show that the current simulation model merging contact mechanics and flow is useful as an engineering design and research tool for seals and other devices where leakage is of interest.

Furthermore, the present invention has been predominantly described with reference to the hydrostatic leakage of seals that are adapted for dynamic use. However, the method for determining hydrostatic leakage can equally be applied to static seals such as 0-rings and gaskets.

Also, the inventors believe that the Equation A used to calculate hydrostatic leakage in the methods of the present invention will also provide an adequate approximation of the leakage permitted by a seal at low rates of relative motion between the seal and the counterface against which it bears. Low rates of relative motion should be understood as rates of less than O.lm/s, preferably less than 0.01m/s. Thus the methods of the present invention may also be used to determine the leakage of and optimise the surface roughness profile of a seal designed for low speed applications.

The foregoing detailed description has been provided by way of explanation and illustration, and is not intended to limit the scope of the appended claims. Many variations in the presently preferred embodiments illustrated herein will be apparent to one of ordinary skill in the art, and remain within the scope of the appended claims and their equivalents .

Claims

Claims ;

1. A method of predicting the hydrostatic leakage (Q) between a seal and a counterface, wherein the seal comprises a first surface portion and the counterface comprises a second surface portion and, in use, the first surface portion contacts the second surface portion and allows hydrostatic leakage (Q) of a fluid between a first region and a second region separated by the seal, the method comprising the steps of:

^■ measuring a roughness profile of the first surface portion;

^■ determining a value of S™, wherein S-™ is the depth of the lowest valley in the measured profile relative to a mean line of the measured profile, or wherein Svm is the average depth of two or more of the lowest valleys in the measured profile relative to the mean line;

^■ predicting the hydrostatic leakage (Q) based on the value of Svm-

2. A method according to claim 1, wherein the step of predicting the hydrostatic leakage (Q) is carried out using the hydrostatic conditions of the seal when in use, wherein: a nominal contact load between the first portion and the second surface is W (Pascals) ; a composite Young's modulus of the first portion and second surface is E¹ (Pascals) ; a pressure difference between the first and second regions is p_r (Pascals) ; and a viscosity of the fluid at the operational conditions is η (Pascal seconds) , and whereby the hydrostatic leakage Q (kg/s/m) is calculated using the following equation:

wherein a is a positive coefficient having a value of from 1 to 20; and b and c are coefficients.

3. A method of predicting the hydrostatic leakage (Q) between a seal and a counterface, wherein the seal comprises a first surface portion and the counterface comprises a second surface portion and, in use, the first surface portion contacts the second surface portion and allows hydrostatic leakage (Q) of a fluid between a first region and a second region separated by the seal, the method comprising the steps of: ■ measuring a roughness profile of the first surface portion;

^■ measuring a roughness profile of the second surface portion;

^■ adding the roughness profiles of the first and second surface portions to obtain a composite roughness profile;

^■ determining a composite value of Svm' , wherein Sv_n,' is the depth of the lowest valley in the composite profile relative to a mean line of the composite profile, or wherein S_vm' is the average depth of two or more of the lowest valleys in the composite profile relative to the mean line,- ^■ predicting the hydrostatic leakage (Q) based on the value of Sv_n/ ■

4. A method according to claim 3, wherein the step of predicting the hydrostatic leakage (Q) is carried out using the hydrostatic conditions of the seal when in use, wherein: a nominal contact load between the first portion and the second surface is W (Pascals) ; a composite Young's modulus of the first portion and second surface is E¹ (Pascals) ; a pressure difference between the first and second regions is p_r (Pascals) ; and a viscosity of the fluid at the operational conditions is η (Pascal seconds) , and whereby the hydrostatic leakage Q (kg/s/m) is calculated using the following equation:

wherein a is a positive coefficient having a value of from 1 to 20; and b and c are coefficients.

5. A method of manufacturing a seal, wherein the seal comprises a first surface portion which, in use, contacts a second surface portion of a counterface to thereby allow a desired hydrostatic leakage (Q) of a fluid between a first region and a second region separated by the seal, the method comprising; ^■ modifying the surface of the first surface portion so that it has a value of S_vn, that provides the desired hydrostatic leakage (Q) of the fluid, wherein S_vn, is the depth of the lowest valley in a measured roughness profile of the first surface portion relative to a mean line of the measured profile, or wherein S_vm is the average depth of two or more of the lowest valleys in the measured roughness profile relative to the mean line of the measured profile.

6. A method according to claim 3, wherein, the value of S_vπ, is calculated using the following equation:

wherein, W is a nominal contact load between the first surface portion and the second surface portion;

E¹ is a composite Young's modulus of the first surface portion and the second surface portion; p_r is a pressure difference between the first and second regions; η is a viscosity of the fluid; a is a positive coefficient having a value of from 1 to

20 ; and b and c are coefficients.

7. A method according to claim 2, claim 4 or claim 6, wherein at least one of the following is true: (i) a has a value of from 10 to 14 (ii) b has a value of from 0.02 to 0.03; (iii)c has a value of from 0.9 to 1.3.

8. A method according to claim 2, claim 4 or claim 6, wherein b has a value of 0.023, c has a value of 1.1, and a has a value of 12.

9. A method according to claim 1, claim 2, or any of claims 5 to 8 , wherein S_vm is the average depth of the five lowest valleys in the measured roughness profile of the first surface portion relative to the mean line of the measured profile.

10. A method according to claim 3 or claim 4, wherein Sv_1n' is the average depth of the five lowest valleys in the composite roughness profile relative to the mean line of the composite roughness profile.

11. A method according to claim 2, claim 4 or claim 6, wherein the Young's modulus of the second surface portion is greater than the Young's modulus of the first surface portion by a factor of at least ten, and the composite Young's modulus E' can be approximated as the Young's modulus of the first surface portion.

12. A method according to any one of the preceding claims, wherein the seal is a radial lip seal.

13. A method according to any one of the preceding claims, wherein the seal comprises an elastomer that is preferably selected from acrylate rubber, fluoro rubber, nitrile rubber, hydrogenated nitrile rubber, or mixtures of two or more thereof .

14. A lip seal comprising a first surface portion which, in use, contacts a second surface portion of a counterface to thereby allow a desired hydrostatic leakage (Q) of a fluid between a first region and a second region separated by the lip seal, wherein the first surface portion is textured such that it has a value of S_vm that provides a desired hydrostatic leakage (Q) , wherein S_vm is the depth of the lowest valley in a measured roughness profile of the first surface portion relative to a mean line of the measured profile, or wherein S_vm is the average depth of two or more of the lowest valleys in the measured roughness profile relative to the mean line of the measured profile.

15. The use of the hydrostatic leakage (Q) predicted by the method of any of claims 1 to 4 for one or more seals to determine a suitable relubrication interval for a component or system of components comprising the one or more seals.

16. A lubrication dispensing apparatus configured to perform the step of calculating the hydrostatic leakage (Q) of one or more seals according to the formula given in claim 2 or claim 4 and to thereby determine a suitable relubrication interval for a component or system of components that comprises the one or more seals.