lubrication quality of compressor rolling bearings with

Tribology OnlineJapanese Society of Tribologists

http://www.tribology.jp/trol/

Vol. 15, No. 1 (2020) 45-52.ISSN 1881-2198

DOI 10.2474/trol.15.45

Article

Lubrication Quality of Compressor Rolling Bearings with Oil-Refrigerant Mixtures

G. E. Morales-Espejel1, 2)* and R. Hauleitner3)

1) SKF Research & Technology Development, Houten, The Netherlands2) Université de Lyon, INSA-Lyon, CNRS, LaMCoS UMR5259, F69621, France

3) SKF Product Development, Steyr, Austria

*Corresponding author: G. E. Morales-Espejel ([email protected])

Manuscript received 16 December 2019; accepted 19 February 2020; published 31 March 2020

Abstract

In refrigerant compressor, rolling bearings are lubricated with a mixture of oil and refrigerant. This has always represented a challenge for the bearing lubrication quality estimation (e.g. kappa or lambda parameters). Even if the dilution rate of the refrigerant is known, the exact effect on the lubricant film thickness remains doubtful due to the unknown piezo-viscosity and compressibility of the refrigerant and their variation with pressure and temperature. In the current paper, existing mixing laws for viscosity and piezo-viscosity are examined and adapted to better represent actual measurements. The results are compared with published Daniel plots showing reasonable agreement. Once this is done a modification to the bearing lubrication quality parameter kappa is proposed to better reflect the effect of the refrigerant on the lubrication quality of compressor rolling bearings. This is a first step in the direction of predicting the bearing life for this challenging application.

Keywords

rolling bearings, refrigerant compressors, lubrication quality, oil-refrigerant mixtures

Copyright © 2020 Japanese Society of TribologistsThis article is distributed under the terms of the latest version of CC BY-NC-ND defined by the Creative Commons Attribution License. 45

1 Introduction

The estimation of the lubrication quality in rolling bearings of refrigerant compressors has always been a challenge due to many unknowns, e.g. the refrigerant dilution, the bearing temperature, the refrigerant piezo-viscosity and viscosity variation with pressure and temperature, the chemical effects of the refrigerant. In rolling bearings the lubrication quality parameter κ (kappa) is used, while in general machine design the parameter Λ-ratio is employed. A discussion and comparison between the two are presented in [1]. However, even if the effective viscosity of the oil-refrigerant mixture is known in an application, the classical way to calculate the bearing lubrication quality parameter κ, as described in [2, 3] cannot be applied because this method assumes the lubricating film thickness in the bearing as calculated only for oil. It is to say, the reduction of the piezo-viscosity in the mixture and the increase of the compressibility with the presence of the refrigerant are not considered. Therefore, in the past several researchers proposed modifications to the original model.

Wardle et al. [4] measured viscosity of mixtures of oils and refrigerants HFC-134a and HCFC-22 and also film thickness in a ball-on-disc configuration to validate modifications to the classical film thickness calculation equations. Meyers [5] introduced these modifications into the calculation of the

lubrication quality parameter in rolling bearings (κ). Then, he applied this parameter in the L10 life estimation for rolling bearings. But no endurance tests were presented to validate the methodology. He also describes in an internal communication, that Jacobson found that the reduced chlorine of the HCFC-22 refrigerant and the absence of chlorine in the HFC-134a refrigerant significantly increases the viscosity requirements for rolling bearing lubrication. He estimated that two times greater operating viscosity is need for an HCFC-22 / mineral oil lubricated bearing and three times greater viscosity is needed for an HFC-134a / polyol ester (POE) oil lubricated bearing compared to an air / mineral oil lubricated bearing. For the first time an “engineering” safety factor due to refrigerant chemistry (beyond the simple calculation of film thickness) was introduced in the calculation of the bearing required viscosity. In [6] Meyers summarizes the new model.

Much later after these developments, Morales-Espejel et al. [7] reviewed the progress in refrigerant compressor technology, where they propose as a complement of the L10 life estimation a life calculation based only on surface distress (primarily failure mode in these applications), with modifications to the previously published surface distress model [8]. The L10 estimation, however, remains a challenge in this application, perhaps an ideal candidate for a bearing life model that explicitly separates the surface from the subsurface, as in [9].

Tribology Online, Vol. 15, No. 1 (2020) /46Japanese Society of Tribologists (http://www.tribology.jp/)

G. E. Morales-Espejel and R. Hauleitner

However, the methodology suggested in [7] requires a correct film thickness estimation in the contact, which is only possible if the viscosity, piezo-viscosity and compressibility of the mixture can be estimated. Some researchers have tried to develop methodologies for this. Notably the work of Gunsel et al. [10] with several mixture measurements reported. Also, Akei et al. [11, 12] with proposed mixture laws for viscosity and piezo-viscosity. And the recent experimental work of Bair [13] and the full elastohydrodynamic lubrication (EHL) numerical calculations of film thickness and pressures for mixtures of Bair et al. [14].

Despite of the described progress very few publications exist with an engineering methodology on the estimation of oil-refrigerant mixture viscosity, piezo-viscosity, film thickness and required viscosity. The current paper describes a methodology to achieve this by independently applying the viscosity variation with pressure and temperature of the oil and of the refrigerant separately. Then the lubricating properties of the mixture are calculated with the use of modified mixture law equations. From this, Daniel plots can be reproduced and the estimation of the bearing required viscosity and lubrication quality parameter calculations are possible.

2 Mixture laws

2.1 Literature modelsAs discussed before, studies have been carried out in the

past to measure and to estimate the film thickness in EHL contacts lubricated with oil-refrigerant mixtures [10-14]. In the calculation of film thickness in EHL contacts lubricated with oil-refrigerant mixtures the estimation of the mixture properties is a key intermediate step. Diverse equations have been suggested to estimate viscosity and piezo-viscosity of oil-refrigerant mixtures. For example [12] uses the Eyring theory to derive equations for the piezo-viscosity coefficient and viscosity:

(1)

(2)

Where , being M the molecular mass of the

component. In an example [13] with polyol ester (POE) oil and HFC-134a a value of m = 4.5 is reported obtained as a backwards calculation using Eq. (2) with a known viscosity mixture and known dilution rate. Equations (1) and (2) can be plotted for different refrigerant dilution rates and molecular mass ratios, as shown in Fig. 1 as illustrative example.

Bair [13] offers the Grunberg-Nissan [15] mixing law as an alternative to Eq. (2). In this work this law was programmed but no advantages or better agreement with measurements of [13] were found when the value of m is known. Therefore, in this work, initially Eqs. (1) and (2) are used.

Next, the predictions of Eq. (2) are confronted with the measurements of Bair. Considering the results of Fig. 8 of the reference [13], the original plots were digitized and compared with the predictions of Eq. (2). The comparisons are shown in Fig. 2. However, instead of back-calculating the value of m as done in [13], here the molecular mass of the oil (Mlub ≈ 850g/mol) is estimated from extrapolating the value of Emkarate RL 68H (CPI-Lubrizol) in [16] page 76 with a value of 700 g/mol. It is known that higher viscosity oils will have higher molecular

αmix = + αlubsref (m − 1) + 1msref (αref − αlub)

ln (ηmix) = (ln ηref − ln ηlub) + ln ηlub( )(m − 1) sref + 1msref

m = Mref

Mlub

Fig. 1 Illustrative example of the variation of viscosity and piezo-viscosity coefficient in a mixture of oil and refrigerant as a function of the refrigerant dilution and the molecular mass ratio (m) between the oil and the refrigerant, following Eqs. (1) and (2)

(a) Viscosity

(b) Piezo-viscosity coefficient

Fig. 2 Comparison of calculations (cal) using Eq. (2) and measurements from Bair [13] (Bair ) for different refrigerant dilutions of HFC-134a in POE 100 at two different temperatures and pressures



masses. Thus m ≈ 8.33 was considered for this case.Figure 2 shows poor agreement with respect to the

measurements of Bair when Eq. (2) is used. The equation was also tested with other mixtures where experimental data exist, having always an important disagreement with respect to measurements. Therefore, it was decided to modify these equations to better predict the measurements.

2.2 Adapted modelEquations (1) and (2) are modified with constants (kal, ket)

multiplying the molecular mass ratio, as follows:

(3)

(4)

These constants are calibrated from measurements of viscosity of different oil-refrigerant mixtures, so calibration functions that depend on the oil viscosity and working temperature can replace the originally constant parameters (kal, ket). Thus,

kal = f (η0.lub,T )

ket = f (η0.lub, T )

With the improved model given by Eq. (4) the comparison of Fig. 2 is repeated, the results are given in Fig. 3. It can be seen that much better agreement is found with the adapted model, the main disagreement with the measurements remains for this case in the area around 20% dilution. This model was tested with several other combinations of oils and refrigerants in different temperatures and dilution rates and the agreement with measurements was always much better than the original model of Eqs. (1) and (2).

2.3 Reproduction of Daniel plotHaving the model of Eqs. (3) and (4) well calibrated it

can be attempted to reproduce a Daniel plot [17] previously published for a different combination of oil and refrigerant

than the one shown in [13]. As explained in [18] originally these graphs are built by regression using a few measured points. In the following, the combination HFC-134a and POE oil RENISO TRITON SEZ 68 (Fuchs) is considered. For this combination the adapted Daniel plot from Fuchs is shown in Fig. 4. The Daniel plot provides the mixture viscosity at different temperatures and mixing ratios at saturation pressure, which by itself is depending on temperature and mixing ratio.

The viscosity data of the pure liquids is the starting point of the calculation. For the oil (RENSIO TRITON SEZ 68) the data published in [14], and for the refrigerant HFC-134a the measurements from [13] were used. The saturation pressure for the distinct mixing ratios of 70, 80 and 90% of oil, which are common in compressor applications and, therefore, shown in this type of graphs, are adapted from the leaflet provided by the oil manufacturer [19]. Where needed, the available data was interpolated to have the properties of oil and refrigerant available at the same temperature and pressure. The adapted mixing law with calibrated constants (4) is used to determine the viscosity for the different dilution rates of the mixture. The variation of viscosity within the range of pressures shown in the Daniel plot (up to 3 MPa in the shown example) is almost negligible. The viscosity data of the refrigerant had to be extrapolated to lower pressures, since the published data is in the range of several Megapascals and higher.

Figure 5 compares the calculated results of the mixture properties with the original Daniel plot adapted from [19]. There is very good agreement between the reproduction and the original data.

3 Lubrication quality in rolling bearings

The minimum film thickness formula for an EHL contact given by Hamrock and Dowson [20] is:

hm = 3.63 Rx W−0.073 U0.68 G0.49 (1−e−0.68k) (5a)

And the central film thickness is

hc = 2.69Rx W−0.067 U0.67 G0.53 (1−0.61e−0.73k) (5b)

(5c)

αmix = + αlubsref (kal m − 1) + 1kal msref (αref − αlub)

ln (ηmix) = (ln ηref − ln ηlub) + ln ηlub( )(ket m − 1) sref + 1ket msref

Fig. 3 Comparison of calculations (cal) using Eq. (4) and measurements from Bair [13] (Bair ) for different refrigerant dilutions of HFC-134a in POE 100 at two different temperatures and pressures

Fig. 4 Daniel plot for the case of POE oil RENISO TRITON SEZ 68 and HFC-134a as adapted from Fuchs in [19]

k = 1.030.64( )Rx

Ry



Where:

(6)

Thus, the ratio of film thickness for two lubricants with different piezo-viscosity parameter α, is:

(7)

also from the ISO [21] the relationship between kappa and lambda is given by:

κ ≈ Λ1.3 (8)

For two equal bearings using (7) with A for refrigerants and B for lubricant and with Eq. (8) one can derive,

(9)

Now, for bearings lubricated with oil-refrigerant mixtures the methodology to calculate kappa needs to consider the refrigerant mass dilution fraction, and is as follows:

By knowing the mass dilution fraction of refrigerant in the oil sref and the mass ratio m, calculate the viscosity and piezo-viscosity of the mixture by using Eqs. (3) and (4).

Since the required viscosity of the bearing ν1 is only valid for oil (or grease) lubrication and not for oil-refrigerant mixture, the actual mixture viscosity in the bearing (so far calculated with Eq. (4) or obtained from a Daniel plot if known) needs to be adjusted for the reduced piezo-viscosity introduced by the refrigerant. Therefore:

(10)

Even this adjusted viscosity needs to be corrected by the safety factor, similar to the one introduced by Meyers [5, 6] (so-called “Jacobson” safety factor) f at a dilution sref.c which will smoothly become 1 when the dilution of refrigerant becomes zero refrigerant in the oil, a convenient empirical function reflecting this is:

(11)

The final adjusted actual viscosity for kappa calculation

is:

(12)

Further to this correction, if the compressibility behaviour of the refrigerant is known (e.g. refrigerant density at saturation conditions and at contact load) then an extra compensation factor ( fh.c

1.3 ) can be multiplied to the right-hand side of Eq. (12). See Appendix A.

An overall adjustment factor for the actual viscosity in rolling bearings as calculated for oil lubrication is then defined from Eq. (12) as

νadj = ν fadj (13)

Thus,

κoil/ref = κoil (fadj) (14)

With, κoil being the lubrication quality factor calculated as oil but with the viscosity of the mixture.

And,

(15)

Because of the lower pressure-viscosity coefficient, differences in compressibility and the differences in lubricity and chemistry of dissolved liquid refrigerant, the viscosity of the oil-refrigerant mixture cannot be used for calculation of kappa without the adjustment factor (fadj) of Eq. (15). Mathematically this factor has no limit and can reach dilution rates of 100% refrigerant but of course, steel-steel bearings will not survive this, as discussed in [7]. So it is limited for practical reasons to sref ≤ 0.3.

4 Application examples

First, a validation for the film thickness (piezo-viscosity) correction factor is carried out using the experiments in [22] for POE Emkarate RL 68H and HFC-134a. Table 1 shows central film thickness values calculated (as oil, but with the mixture viscosity) with Eq. (5b). The piezo-viscosity correction factor (7), the adjusted film thickness and the calculated value given in [22]. The case corresponds to a steel ball (diameter 19.05 mm) on a glass disc (E' = 1.1382 × 1011 Pa) with a contact force of 20 N, a temperature of 45°C, dilution rate of 30% and rolling speeds from 0.1 to 3 m/s. Thus, W = 9.2482 × 10−9, η0.mix = 0.0037 Pas, αmix = 16.36 × 10−9 1/Pa.

The adjusted central film thickness calculations agree very well with the measured ones.

Consider now the case of two bearings, one cylindrical roller bearing and one angular contact ball bearing, with the operating conditions given in Table 2.

The oil and refrigerant properties are taken from [13, 14, 16] and are summarised in Table 3.

With the data of Tables 2 and 3 the overall factor to correct the required viscosity calculated in rolling bearings for oil (fadj) is plotted as calculated from Eq. (15) in Fig. 6.

The calculations of Fig. 6(a) assume a safety factor (Jacobson factor) of f = 1.5, and Fig. 6(b) a safety factor of f = 2. Figure 6 shows almost no difference in the adjustment factor for the two bearing applications described in Table 2. As it can be seen there is very little difference in Hertzian pressure between the

Fig. 5 Original data (Daniel plot) and reproduced Daniel plot (cal) for mixtures of different oil content in the case of POE oil RENISO TRITON SEZ 68 and HFC-134a

W = ,E’Rx

F U = , G = αE’E’Rx

η0u

≈0.53( )αB

αAhm.A

hm.B

≈ ≈1.30.53( )( )αlub

αref 0.70( )αlub

αrefνref

νlub

νpiezo = νmix αoil

αmix0.70( )

fs = 1 + (f − 1) tanh( )sref.c

4sref

νadj = ( )fs

νpiezo

κ =( )ν1

νadj

fadj = ( fh.c )1.30.70( )[ ]αoil

αmix

fs

1



two cases, this explains the similarity. Figure 6 also shows that for the case of high dilutions of refrigerant (30%) the effective viscosity of the mixture lubricating the bearing, could be reduced to a minimum value of around 35% of the same case calculated as for oil but with the viscosity of the mixture (case of f = 2).

5 Discussions and conclusions

A complete methodology for the estimation of the rolling bearing lubrication quality parameter κ has been described. First, accurate mixing law equations for viscosity

and piezo-viscosity are developed. With these equations the corresponding Daniel plots can be derived if not available. Then these equations are used to estimate the viscosity of the mixture. Further, the mixture viscosity is adjusted for piezo-viscosity, compressibility and safety. The methodology derives an adjustment factor for the actual viscosity in the bearing when a mixture of oil and refrigerant is present. This factor when multiplied with the oil-calculated-method mixture viscosity will reduce its value to a safe level.

As shown in this paper the Jacobson safety factor f leaves some freedom to the engineer to adapt the safety of the pursued design, since there are still several unknown factors. One of

Speed, [m/s]

U [‐] 𝒉𝒉𝒄𝒄, Eq. (5b) [nm]

Piezo‐viscous adjustment factor, Eq. (7)

Adjusted film thickness, [nm]

Measured film thickness, [22], [nm]

0.1 1.7021x10‐13 14.59 0.8991 13.11 12

0.5 8.5105x10‐13 42.89 0.8991 38.56 38

1.0 1.7021x10‐12 68.24 0.8991 61.35 62

3.0 5.1063x10‐12 142.47 0.8991 128.1 125

Table 1 Comparison of calculated and measured central film thickness values

Designation Radial load [N]

Axial load [N]

𝒑𝒑𝒉𝒉 [GPa]

Speed [rpm]

Temp [°C]

Oil Refrigerant

NU 1014 8700 0 1.12 1800 30 POE 68 HFC‐134a

7311 0 10500 1.5 1800 30 POE 68 HFC‐134a

Table 2 Example bearings and operating conditions

Substance 𝜼𝜼𝟎𝟎 [mPas] 𝜶𝜶∗ [1/Pa] 𝑴𝑴 [g/mol] 𝝆𝝆𝟎𝟎 [kg/m3] Temp [°C]

POE 68 113.8 20x10‐9 700 977 30

HFC‐134a 0.2 5x10‐9 102 1206 40

Table 3 Oil and refrigerant properties [13, 14, 16]

(a) with f = 1.5 (b) with f = 2

Fig. 6 Adjustment factor for kappa for the two bearings of Table 1, with two values of the safety factor (Jacobson factor)



them already discussed in [7] is the chemical aggressiveness of the refrigerant towards the bearing steel, reducing its fatigue strength due to corrosion or poor lubricity. For example, refrigerants with very low global warming potential (GWP) and zero ozone depletion potential (ODP) might be considered as highly reactive/corrosive fluids, especially with the presence of moisture. In those cases engineers are encouraged to use f ≥ 2, otherwise 1.5 ≤ f ≤ 2 would be sufficient.

From the analysis carried out here, the following conclusions can be drawn:(1) The oil-refrigerant mixture law equations found in literature

need some adaptation functions dependent on lubricant viscosity and temperature to be accurate in the prediction of mixture viscosity and piezo-viscosity. After this Daniel plots can be calculated.

(2) The oil-calculated-method mixture viscosity in a rolling bearing still needs to be adjusted for piezo-viscosity, compressibility and safety before it can be used to calculate the lubrication quality of a bearing working in oil-refrigerant conditions. The factor fadj is introduced for this purpose.

(3) The Jacobson safety factor is an adaptable variable for the design of an application, highly reactive refrigerants (especially in the presence of moisture) will require larger values of this safety factor.

Acknowledgements

The authors would like to thank Mr. B. van Leeuwen, Director SKF Research and Technology Development for his kind permission to publish this article.

References

[1] Morales-Espejel, G. E., Gabelli, A., Ioannides, E. and Hooke, C. J., “Micro-Geometry Lubrication and Life Ratings of Rolling Bearings,” Proc. IMechE Part C: J. Mechanical Engineering Science, 224, 12, 2010, 2610-2626.

[2] International Standard. Rolling Bearings – Dynamic Load Rating and Rating Life, ISO 281, 2007.

[3] SKF Rolling Bearing Catalogue, 17000.[4] Wardle, F. P., Jacobson, B., Dolfsma, H., Hoglund, E. and Jonsson,

U., “The Effect of Refrigerants on the Lubrication of Rolling Element Bearings Used in Screw Compressors,” International Compressor Engineering Conference, Paper 843, 1992.

http://docs.lib.purdue.edu/icec/843[5] Meyers, K. E., “New Life Method for Rolling Bearings in

Compressors,” International Compressor Engineering Conference, Paper 1209, 1998.

http://docs.lib.purdue.edu/icec/1209[6] Meyers, K. E., “Creating the Right Environment for Compressor

Bearings,” SKF Evolution, 1997. http://evolution.skf.com/creating-the-right-environment-for-

compressor-bearings/[7] Morales-Espejel, G. E., Wallin, H. H., Hauleitner, R. and Arvidsson,

M., “Progress in Rolling Bearing Technology for Refrigerant Compressors,” Proc. IMechE Part C: J. Mechanical Engineering Science, 232, 16, 2018, 2948-2961.

[8] Morales-Espejel, G. E. and Brizmer, V., “Micropitting Modelling in Rolling–Sliding Contacts: Application to Rolling Bearings,” Tribol. Trans., 54, 4, 2011, 625-643.

[9] Morales-Espejel, G. E., Gabelli, A. and de Vries, A. J. C., “A Model for Rolling Bearing Life with Surface and Subsurface Survival—

Tribological Effects,” Tribol. Trans., 58, 5, 2015, 894-906.[10] Gunsel, S. and Pozebanchuk, M., “Elastohydrodynamic Lubrication

with Polyolester Lubricants and HFC Refrigerants,” Final Report, vol 1, Prepared for The Air Conditioning and Refrigeration Technology Institute under ARTI MCLR Project Number 670-54400, 1999, DOE/CE/23810-102.

[11] Akei, M., Mizuhara, K., Taki, T. and Yamamoto, T., “Evaluation of Film-Forming Capability of Refrigeration Lubricants in Pressurized Refrigerant Atmosphere,” Wear, 196, 1-2, 1996, 180-187.

[12] Akei, M. and Mizuhara, K., “The Elastohydrodynamic Properties of Lubricants in Refrigerant Environments,” Tribol. Trans., 40, 1, 1997, 1-10.

[13] Bair, S., “A New High-Pressure Viscometer for Oil/Refrigerant Solutions and Preliminary Results,” Tribol. Trans., 60, 3, 2017, 392-398.

[14] Bair, S., Habchi, W., Baker, M. and Pallister, D. M., “Quantitative Elastohydrodynamic Film-Forming for an Oil/Refrigerant System,” Trans. ASME, J. of Tribol., 139, 6, 2017, 061501.

[15] Grunberg, L. and Nissan, A. H., “Mixture Law for Viscosity,” Nature, 164, 1949, 799-800.

[16] Martz, W. L. and Jacobi, A. M., “Refrigerant-Oil Mixtures and Local Composition Modeling,” University of Illinois Report, ACRC TR-68, 1994.

[17] Daniel, C., “Use of Half-Normal Plots in Interpreting Factorial Two-Level Experiments,” Technometrics, 1, 4, 1959, 311-341.

[18] Barthel, A. J. and Majurin, J., “Understanding and Improving the 9-Coefficient Pressure Viscosity Temperature (PVT) Model,” IOP Conf. Ser.: Mater. Sci. Eng., 604, 2019, 012033.

[19] Fuchs Schmierstoffe GmbH, “Produkt Information RENISO TRITON SEZ 68,” PI 4-1332.

[20] Hamrock, B. J. and Dowson, D., “Ball Bearing Lubrication – The Elastohydrodynamics of Elliptical Contacts,” Whiley-Interscience, New York, 1981.

[21] ISO – Rolling Bearings – Explanatory Notes on ISO 281 – Part 2, Technical Report ISO TR/ 1281-2, 2007.

[22] Gunsel, S. and Pozebanchuk, M., “Elastohydrodynamic Lubrication with Polyolester Lubricants and HFC Refrigerants,” Final Report, vol 2, Prepared for The Air Conditioning and Refrigeration Technology Institute under ARTI MCLR Project Number 670-54400, 1999, DOE/CE/23810-102.

[23] Dowson, D. and Higginson, G. R., “Elastohydrodynamic Lubrication, The Fundamentals of Roller and Gear Lubrication,” 1st ed., Pergamon Press, Oxford, UK, 1966, 89.

[24] Vergne, P., Fillot, N., Bouscharain, N., Devaux, N. and Morales-Espejel, G. E., “An Experimental and Modeling Assessment of the HCFC-R123 Refrigerant Capabilities for Lubricating Rolling EHD Circular Contacts,” Proc IMechE, Part J, J. of Eng. Tribol., 229, 8, 2015, 950-961.

[25] Jacobson, B. O. and Morales-Espejel, G. E., “High Pressure Investigation of Refrigerants HFC245fa, R134a and R123,” International Compressor Engineering Conference, Paper 1789, 2006, 1-8.

[26] Toumas, R. and Isaksson, O., “Compressibility of Oil/Refrigerant Lubricants in Elasto-Hydrodynamic Contacts,” Trans. ASME, J. of Tribol., 128, 1, 2006, 218-220.

[27] Bair , S . , Baker, M. and Pall ister , D. M., “Revisi t ing the Compressibility of Oil/Refrigerant Lubricants,” Trans. ASME J. of Tribol., 139, 2, 2017, 024501.

[28] Vergne, P., Fillot, N., Bouscharain, N., Devaux, N. and Morales-Espejel, G. E., “An Experimental and Modeling Assessment of the HCFC-R123 Refrigerant Capabilities for Lubricating Rolling EHD Circular Contacts,” Proc IMechE, Part J, J. of Eng. Tribol., 229, 8, 2015, 950-961.



6 Appendix A - An adjustment factor for the effect of the compressibility of the refrigerant

Most authors agree that refrigerants in liquid state are more compressible than oils [23-25]. This can bring an extra reduction of the film thickness in Eqs. (5a) and (5b). Other authors, however, claim refrigerants are less compressible than oils [26]. However the consensus and measurements [27] seem to show that indeed refrigerant are more compressible than oils. To develop a multiplication adjustment factor for this equation due to the effect of higher or lower compressibility of the refrigerant, one needs to start from the effect of compressibility in the central film thickness. At the centre of the contact, the density change can be calculated as:

(A1)

Where (h0) is the initial film thickness (incompressible), (ρ0) is the ambient density, ρ(p) is the density at a pressure p and h(p) is the film thickness at the same pressure p.

Now, the density of a mixture of refrigerant and oil can be estimated by:

(A2)

From Eq. (A1), the film thickness of the mixture at any pressure is,

(A3)

Thus, an adjustment factor for the effect of refrigerant compressibility in the film thickness Eqs. (5a) and (5b) respect to oil can be obtained from Eq. (A3), it is:

(A4)

Equation (A4) includes the ratio of incompressible film thickness mixture/oil. This term is already accounted for in Eq. (7) and should be equal to one here. Thus finally, considering only the effect of the compressibility of the refrigerant:

(A5)

When transferred this correction into viscosity for Eq. (12) it would be:

(A6)

The contribution of refrigerant compressibility in oil-refrigerant mixture film thickness is in general very modest [14], in most engineering cases it can be ignored. Except, perhaps, in very high refrigerant dilution cases > 50%.

6.1 Compressibility of oils and refrigerantsFor oils the well-known compressibility law of Dowson and

Higginson [23] can be used. It reads,

(A7)

For refrigerant HCFC-123, Vergne et al. [28] suggest to use the Tait temperature-pressure equation, thus:

(A8)

With p expressed in GPa, V0 is the volume at ambient

pressure, , ρR is the density at the reference

state (ambient pressure and reference temperature, TR), av is the thermal expansibility defined for a linear volume increase with temperature = 0.0008 K−1, K'0 = 11 (the Tait parameter), K0 the isothermal bulk modulus, K0 = K0Re[bK(T+273.15)], bk = −0.0065 K−1 and K0R = 9 GPa.

Equation (A8) gives the liquid density of the refrigerant as a function of pressure and temperature, when the liquid density ρR at ambient pressure and a reference temperature TR are known.

ρ0=

ρ(p)h(p)h0

=ρmix(p) 1

ρlub(p)slub

ρref (p)sref+

=hmix(p)h0.mixρ0.mix

ρmix(p)

= =fh.c

hmix(p)hlub(p)

ρ0.mix

ρmix(p)h0.mix

h0.lub( )( ) ρlub(p)

ρ0.lub( )

= =fh.c

hmix(p)hlub(p)

ρ0.mix

ρmix(p)( ) ρlub(p)ρ0.lub

( )

= ( fh.c)1.3νadj

νpiezo

fs

Notice that Eq. (A5) tends to 1 for slub and

.

→ 1 (i.e. sref → 0)

→ 0 (i.e. sref → 1), then fh.c →ρ0.ref

ρref (p)( ) ρlub(p)ρ0.lub

( )for slub

ρ0= 1 +

ρ(p)1 + 1.7 × 10−9 p

0.6 × 10−9 p

ρ0=

ρ(p)

{V0 [1 − (1/(1 + K’0))] ln [1 + (1 + K’0)p ]}

1

K0

1( )

=V0

1 + av(T−TR) ρR



NomenclatureE' Combined Elasticity modulus [Pa]

F Contact force [N]

f Jacobson safety factor at maximum dilution [-]

fs Safety multiplication factor for the required viscosity in oil-refrigerant mixture [-]

G Dimensionless Dowson-Higginson material parameter, Eq. (5) [-]

hm Minimum lubricant film thickness [m]

m Molecular mass ratio between the oil and the refrigerant [-]

M Molecular mass of a substance [g/mol]

p Pressure in the contact [Pa]

Rx Equivalent radius of curvature along rolling direction [m]

Ry Equivalent radius of curvature across rolling direction [m]

U Dimensionless Dowson-Higginson speed parameter, Eq. (5) [-]

u1 Contact surface 1 velocity [m/s]

u2 Contact surface 2 velocity [m/s]

u Entrainment velocity, u = (u2 + u1)/2 [m/s]

Rq r.m.s. of the roughness [m]

s Fraction content in the mixture [-]

W Dimensionless Dowson-Higginson load parameter, Eq. (5) [-]

Greek Symbols

α Piezo-viscosity coefficient of lubricant [GPa−1]

α* Integrated piezo-viscosity coefficient, [GPa−1]

Λ Lambda ratio, Λ = hm/Rq [-]

ρo Lubricant density at atmospheric conditions (oil) or at saturation conditions (refrigerant) [kg/m3]

ρ Lubricant density at contact pressure conditions [kg/m3]

ηo Dynamic viscosity at atmospheric conditions (oil) or at saturation conditions (refrigerant) [Pas]

η Dynamic viscosity [Pas]

ν Actual kinematic viscosity used in the bearing [cSt]

ν1 Required kinematic viscosity of the lubricant [cSt]

κ Lubrication quality parameter for rolling bearings, κ = ν/ν1 [-]

Subscripts

1 Contact surface 1

2 Contact surface 2

A Lubricant A

B Lubricant B

c critical

h Hertzian

ref Refrigerant

lub Lubricant (oil)

oil Referes to calculation using oil method but using the viscosity of a mixture

oil/ref Refers to a calculation of a mixture oil/refrigerant

mix Mixture oil-refrigerant

piezo Adjusted by piezo-viscosity

adj Adjusted for a mixture oil/refrigerant

η(p)0

∞α* = dp∫[ ]η(p = 0) −1

lubrication quality of compressor rolling bearings with

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