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Static and Dynamic Compressibility of Lubricants Under High Pressures
Lubricants and Lubrication / D. Dowson et al. (Editors) 0 1995 Elsevier Science B.V. All rights reserved.
189
STATIC AND DYNAMIC COMPRESSIBILITY OF LUBRICANTS UNDER HIGH PRESSURES P. Vergne Laboratoire de Mhnique des Contacts, URA CNRS 856 INSA, BAtiment 113,20 avenue Einstein, 69621 Villeurbanne cedex, France This paper presents an experimental investigation conducted on several synthetic lubricants of various chemical nature: silicone fluids, polyalphaolefins and a diester. Volume losses are first reported at different temperatures. Then, isothermal secant and tangent compressibility modulus values are deduced. They are compared to isentropic bulk moduli, calculated from ultrasonic measurements. Finally, a discussion on the correlation between the relative value of these moduli and the lubricants physical state completes the paper. 1. INTRODUCTION
During the last decades, the lubricants compressibility under high pressure has been the rheological parameter which has certainly been less studied. Nevertheless, the compressibility role appears to be significant in many circumstances, as for instance: - in highly loaded lubricated contacts as those occurring between gear teeth, - and also in most of lubricated metal forming processes where small amounts of lubricant are compressed between the tool and the metal sheet, trapped by surface asperities. The compressibility contribution in the running of E.H.D.contacts has been first described by Dowson and Higginson [l] and recently developed by Jacobson et al. [2,3] for instance. The reference work in the fluid compressibility domain has been brought by Hayward who proposed at first [4] an adequate terminology and methods of expressing results. His contribution has concerned also compressibility equations [4,5], testing methods [4,6,7], experimental investigations in various liquids [4,8]. To our knowledge, most of compressibility equations are empirical relationships which have frequently been experimentally verified in narrow pressure ranges. Hayward [5] and later on Yusa, Mathur and Stager [9] have reviewed and tested
some of them, but due to the experimental limits already mentioned, any clear tendency emerges. In lubrication, the more frequently used densitypressure equation found in literature is the Dowson and Higginson [l] relationship which is written as follows: p/po = 1 + 0.6P/( 1 + 1.7P )
(1)
where p is the density under pressure P, and p,, is the density at ambient pressure. Concerning experimental results on lubricants, two different approaches could be mentioned. Very accurate experiments have been run in a restricted domain (typically 700 bars) and therefore values for higher pressures are extrapolated. This method has been applied by Klaus and OBrien [lo], Wright [ 111 and Tichy and Winer [ 121 on various kind of fluids: mineral base oils, polyphenyl ether, silicone fluids and diesters. The second approach consists to conduct experiments in a pressure domain as large as possible. The limit generally corresponds to the maximum admissible stress of the high pressure cell material or to the appearance of leakage problems. Since Galvin et a1.[13] who reach 350 MPa, special devices have been developed to work under 1 GPa or more[2,3,14,15].
190
DOS
Di 2 ethyl hexyl
Chemical
Index Density at 20°C Kin. Viscosity at 40°C Kin. Viscosity at 100°C VIE a in l/GPa a in l/GPa
S1L2
dimethyl poly siloxane -5OOC
POL 1
methylchlorophe polyalphaolefin . . nyl hlysiloxane -75°C -40°C
POL2 polyalphaolefin _ _ _ -57°C
1.449
1.497
1.416
1.470
0.913
1.066
1.016
0.848
0.848
11.7 cst
77 cst
52 cst
408 cst
33 cst
3.5 cst 199 12.5 at 20°C 12.1 at 25°C
22.2 cst 314 18.2 at 60°C 10.4 at 150°C
20.9 cst 414 12.8 at 60°C 10. at 150°C
41.6 cst 154 12.6 at 60°C 9.7 at 150°C
7 cst 181 12.6 at 60°C 8.7 at 15OOC
I I
SIL 1
I
Table 1 :Lubricants properties (ais the secant pressure-viscositycoefficient, calculated at 400 MPa). 2. TESTED MATERIALS AND DEVICE 2.1. Tested materials
Five products have been studied: one diester, two silicone fluids and two polyalphaolefins respectively named DOS, SIL1, SIL2, POL1 and POL2. Most of these lubricants are pure synthetic fluids: the exception is POL2 which contains an additive package. Their properties are summarised in Table 1: apart from pour point temperatures, the results reported in Table 1 have been collected on our own devices. -Refractive index has been measured according to the A.S.T.M. D1218 standard. -Density variations versus temperature have been determined with a Mettler density kit set up on an electronic balance. -Kinematic viscosity changes versus temperature have been deduced from dynamic viscosity results obtained with a Couette viscometer and from density data. -VIEhas been determined according to the A.S.T.M. D2270 standard. -Pressure-viscosity coefficients have been calculated from experiments run on our high pressure falling body viscometer. We have deliberately chosen to give the secant coefficients computed for a 400 MPa pressure increase instead of usual coefficients which
serve for EHD film thickness calculations. Here, the aim is to report a parameter which represents the viscosity increase in a wide pressure domain. The fluids are all characterised by low pour points, low viscosity at ambient pressure and temperature (except for POL1) and high viscosity index. SIL2 exhibits the greater stability: pressure and temperature influence is very weak compared to the other lubricants. This is the consequence of its chemical structure which contains phenyl groups and chlorine atoms. 2.2. Experimental device Reliable bulk moduli data are difficult to obtain on a fluid [7,11]: following the advice of previous authors, a maximum of caution has been taken during the experimental phase. The measurements have been performed in the high pressure (up to 700 MPa), wide temperature range (from -25OC to 160OC) vessel of our falling body viscometer. To avoid dispersion due to unhomogeneous strains, the high pressure densimeter works under perfect hydrostatic pressure and temperature. It is made of a cylindrical stainless steel cell filled by the sample: its top end is closed by the ceramic disk of an ultrasonic transducer and its bottom end is sealed by a metallic piston
191 supported by O-rings. This feature allows to accommodate volume changes in the fluid column by a translating motion of the piston. Consequently the distance between the ceramic disk and the piston changes as a function of pressure and temperature. These distance variations are recorded by means of an ultrasonic technique which requires the preliminary measurement of the sound velocity in the sample. This previous work is run at similar ceramic-reflector distance to avoid possible linearity fluctuations of the electronic unit. We use an impulsion method (3 periods maximum by pulse) with longitudinal waves: the transducer works as an emitter/receiver. To limit energy dissipation in the fluid under pressure, we have chosen a high ultrasonic frequency (1 MHz) and a low emission one (125 Hz):this means that the ratio of the time during fluid molecules undergo ultrasonic waves to the real time is negligible. The experiments are computer controlled: recorded data are directly converted in length and thus in volume. Elastic deformations caused by hydrostatic pressure and temperature fields are directly taken into account: their influence is very
experience in the field, we consider that our volume losses are obtained with an accuracy of +/- 3 %. The best means to validate this p i n t is to compare our results to previously published data. Here we have chosen di 2 ethyl hexyl sebacate @OS in Table 1) results from the A.S.M.E. Pressure Report [16]. The comparison at 25OC is presented Figure 1, where the relative volume reduction (-AVN,) is plotted versus pressure. We observe a good agreement in the low and in the high pressure domains: nevertheless a sensible discrepancy (close to 7%) is noted between 2 and 3 kbar. Even if this kind of deviation could appear acceptable for volume losses, it should be still considered with care for bulk moduli determination,especially for tangent bulk moduli. The comparison with Dowson and Higginson relationship [ 11 shows a good agreement at low and at high pressure. However the analytical curve is flater than experimental ones which implies significant differences in the moduli. The discussion on the validity of this relationship is presented in the next section.
Weak.
To avoid uncertainties in the low pressure domain, the first pressure step is 25 MPa for DOS and 50 MPa for the other fluids. Hayward [7] has cleverly suggested that during compressibility experiments, the first pressure step should be at least one tenth of the maximum pressure to minimize the errors, especially on the moduli. To purge the air entrapped in the fluid, the densimeter is placed in a tank filled with an excess sample volume, to allow the handling. Then all the parts (metallic surfaces + sample volume) are degassed under vacuum (pressure < 10' mbar ). Then the densimeter is closed, always in the tank, filled this time by air-free lubricant. As it has been reported above, compressibility measurements are very difficult to conduct because of the low volume changes with pressure: a high accuracy is required. In our case pressure measurements are run with a precision of +/- 0.25% and the temperature is known to within +/- 0.5 %. These values suggest that the main cause of potential deviation in our results comes mostly from volume measurements. Taking into account our
I
kii
d > 10
5
0
0
1000
u)(30
3000
4000
5000
Figure 1 :Volume losses at 25OC in DOS.
3. VOLUME LOSSES RESULTS
All the results obtained at 60, 100 and l5OoC are reported in Figures 2 to 5, respectively for SIL1, SL2, POL1 and POL2.
192
-
IOOOC
P
0.10-
0,05*
IF
u,w-
0
Figure 2 :Volume loss versus pressure in SILL
PRESSURE (MPa) 100
200
300
400
500
Figure 4 :Volume loss versus pressure in POL1.
z 025
Eg
2
w
3
0,20 0,15
P 0,IO 0.05 0,oo
Figure 3 :Volume loss versus pressure in SIL2.
Most of the results concerns a maximum pressure increase of 500 MPa, except for SIL1: its viscosity shows a strong pressure dependence at 6 O O C . Consequently compressibility tests are impossible to run under this condition. Normally the pressure steps are equal to 50 MPa but due to ultrasonic problems (signal fluctuations), some points have been voluntary omitted.
0
100
200
300
400
500
Figure 5 : Volume loss versus pressure in POL2.
After the initial pressure steps, SILl curves suggest a linear increase of volume losses with pressure as usually volume reduction found at the highest pressure levels tends to decrease. Although data at 6OoC are close to those given by equation (l), the gap between experimental and calculated values increases continuously. At 15OoC, the
193 experimental curve is very different from the analytical one. SIL2 behavior is very specific: it shows a large compressibility as the volume losses reach or exceed 20% at 500 MPa at all tested temperatures. Furthermore, we observe a very large volume reduction since the first pressure steps: at 150 MPa and 6OoC, 10% is reached as in SIL1, this reduction is found at 250 MPa. In the highest pressures domain (typically when P > 300 MPa), we discern the same tendency that for S L l . The total volume loss increases linearly with pressure (up to 500 MPa): the slopes found with SIL2 are lower than those observed on SIL 1. Polyalphaolefins show results in better agreement with usual tendencies. POLl data at 60°C and equation (1) are very close. The maximum volume reduction attains 16% at 150°C as it was greater than 22% for SIL2. At 15OoC,we have obtained very similar results for POLl and POL2 as at lower temperatures, the curves differ.
Figure 6 :Volume losses comparison at 100°C.
To illustrate the above mentioned tendencies, we have reported all the volume losses measured at 100°C in Figure 6. The curves confirm that SIL2 is more compressible than the other fluids. It is also shown that the behavior of each chemical structure is typical: polyalphaolefin curves tend to an horizontal asymptote as silicone curves suggest the continuation of a linear increase.
Concerning the validity of equation (l), our results show that the fluid chemical nature, pressure and temperature govern the lubricants compression behavior. Correlation with diester results near ambient temperature is acceptable and comparison with polyalphaolefins data has shown that a shift occurs during the first pressure steps and after the deviation remains constant. Even for SIL2, we can consider that the data could be fitted by a similar equation as the curvatures are very close. All these observations suggest that equation (1) could be used on a wide experimental domain if the two constants (0.6 and 1.7) are changed by more appropriate ones. Temperature influence is more important with SIL2 and the less significant with pOL2. Volume reductions found on POL2 are more temperature dependent than those obtained with SIL1. A detailed analysis of the curves shows that temperature influence takes place essentially during the two first pressure steps. Beyond 100 MPa, we could consider for each lubricant that the curves obtained at 60, 100 and 150°C are roughly parallel. These observations on the temperature influence are consistent with literature [ 10-131. Comparisons with previously published result is not obvious. From our bibliographical analysis, it appears that silicone fluids are typically very compressible [4,10,12,13], compared to mineral oils for instance. Klaus and OBrien [ 101 have compared predicted and measured (from A.S.M.E. [ 161) isothermal secant bulk moduli for a similar fluid that SILl. At 100°C, they found 8.2% and 11.1% at 152 and 242 MPa and our more approaching data are respectively equal to 7.7% and 10.9% at 150 and 250 MPa. Galvin et al. [ 131 have reported data at 100°C on a silicone fluid which fits very well our results on SIL2. Concerning polyalphaolefins, Jacobson and Vinet [2,3] have reported some data but only for pressures greater than 425 MPa, making the comparison impossible. 4. BULK MODULI RESULTS 4.1. Isothermal secant bulk moduli
Isothermal secant bulk moduli 6) are directly derived from volume losses by the following equation:
194
K, = AP / (-AV/V,)
T = constant
Fluid T"C
(2)
SILl 100 150 60 60 1774 1612 1349 953 2.62 2.57 2.73 3.25
K,, A,
Equation (2) means that K, refers to the pressure references. Measurements are run under steady state pressure and temperature conditions. The results are reported in Figure 7 for silicone fluids and in Figure 8 for polyalphaolefins. The initial value at P = 0.1 MPa has been determined following an analytical method presented in the next section.
100
200
300
Figure 7 :Isothermal secant bulk moduli in silicone fluids SILl and SIL2.
In our pressure domain, we note that the isothermal secant bulk moduli increase roughly linearly with pressure. In agreement with the remarks reported on silicone fluids volume losses, the correlations between secant bulk moduli variations and linear fittings is very good. For these fluids, the results differ essentially in their initial value, i. e. their slopes are almost equal and not temperature dependent. Table 2 presents the linear fitting results of the isothermal secant bulk moduli variations versus pressure for SILl and SIL2. K, =K,,+A,.P
(3)
where A, is the slope, K,, is the initial isothermal bulk modulus (in MPa), and P the pressure (in MPa).
3.36
300
200
400
500
Figure 8 :Isothermal secant bulk moduli in polyalphaolefins POLl and POL2.
500
400
150 585
Table 2 : Results of linear regressions on K, (in Mpa) versus pressure for SILl and SIL2.
100
0
I
sIL2 100 776 3.33
Concerning POLl and POL2 isothermal secant bulk moduli, the tendencies are more temperature dependent. At 60, 100 and 150°C for POLl and at 60°C for POL2 a second degree polynomial gives a better fitting than a simple linear regression. Table 3 presents the values of this correlation: K, = K,, + B, . P + 10-3.c, . ~2
(4)
where B, and C, are the polynomial coefficients, K,, is the initial isothermal bulk modulus (in MPa), and P the pressure (in MPa).
L
Fluid T"C
K ,
B,
C,
I
I
60 I 1663 2.16 4.70
POL1 100 I 1285 2.99 2.13
1 1
I
POL2
100 I 150 1103 1580 1256 1023 3.12 1.98 3.15 3.62 1.18 2.52 0.630 0.147 150
60
Table 3 : Results of degree 2 curve fitting on the isothermal secant bulk moduli for POLl and pOL2.
195 4.2. Isothermal tangent bulk moduli
This parameter is the thermodynamically correct isothermal bulk modulus. It represents the true rate of volume change at the pressure of interest. The main difference with Ks is that the tangent modulus does not refer to the initial volume V,,. When P tends to the atmospheric pressure, Ks and Ic, become equal. K, is given by:
K,
=
- V . ( 6 P/ 6V )
at T = constant (4)
As V(P) is known point by point, the K, determination is not direct. At first, the volume losses have been fitted by a polynomial function. Then these functions have been derived and K, values have been deduced.
Figures 9 and 10 present the isothermal tangent bulk moduli found respectively in silicone fluids and in polyalphaolefins. These moduli are always greater than secant ones. Concerning SILl and SIL2 (Figure 9), the effects previously mentioned conduct to weakly curved slopes. The variations are more significant with POLl and POL2 (Figure 10). The curves shows that high values (> 10 GPa) have been reached at 400 or 500 MPa. It is also expected that under higher pressures, the moduli can be multiplied by a factor of 10 at least. This means that volume losses should be very limited, actually close to zero.
*
-0Y
--f
sIL11wc SILllSoOC sIL2wc SlLZlWC
M
U
0
100
200
300
400
500
Figure 10 :Isothermal tangent bulk moduli in polyalphaolefins POLl and POL2.
4.3. Isentropic tangent bulk moduli It is the volumetric tangent modulus of elasticity under conditions of constant entropy. Under normal conditions it is greater than the isothermal tangent bulk modulus by the ratio of the fluid specific heats. When data are available, this parameter is used under conditions where pressure changes are rapid, with little opportunity for the temperature to come in equilibrium. This definition underlines the interest to use this parameter in lubrication problems where dynamic high pressures occur. One of the methods developed to run isentropic compressibility experiments is based on ultrasonic velocity measurements. It is supported by the following equation:
-
53 5000
k, = V. ( 6 P/6V ) = p . c2 at S = constant ( 5 )
E
where kT is the isentropic tangent bulk modulus, p is the density, c is velocity of sound and S the entropy.
ca
9
b
$0 Y
0
o
100
200
300
400
500
Figure 9 :Isothermal tangent bulk moduli in silicone fluids SILl and SIL2.
The validity of equation ( 5 ) has been verified by Hayward [6] and Noonan [17]. Due to our peculiar device which requires the ultrasonic velocity measurements before running compressibility tests, we have been able to determined k, under the same conditions as the other moduli.
196
where DT is the slope, kTois the initial isentropic bulk modulus (in m a ) , and P the pressure (in MPa).
100
0
200
300
400
500
Figure 11 : Isentropic tangent bulk moduli in silicone tluids SILl and SIL2. Table 4 :Results of linear fitting on k,
,
At the opposite of isothermal tangent bulk moduli, all the fluids give similar curves: they differ by the initial value at P = 0.1 MPa as their slopes are very close.
v)
o . . - . 0
100
PRESSURE (MPa) 200
300
400
500
Figure 12 :Isentropic tangent bulk moduli in polyalphaolefins POL1 and POL2.
The results are plotted on Figure 11 for silicone fluids and on Figure 12 for polyalphaolefins. The curves are almost strait lines: a weak curvature towards the pressure axis is observed. This implies that isentropic tangent bulk moduli increase almost linearly with pressure. Apart for initial values, a linear regression performed on these results gives an acceptable agreement (see Table 4).
The explanation could come from the fluids viscoelastic behavior in compression. First of all, it is important to remind that longitudinal waves could be splitted in pure compression waves plus pure shear waves and that longitudinal velocities are also frequency dependent [18]. The limit is the infinite frequency velocity which is 20 to 40% greater than the ultrasonic one, in the liquid domain. If we apply time (frequency) pressure equivalence principle, there any reason for the ultrasonic velocity to increase suddenly in our experimental conditions. As in parallel the density increase is continuous, this could not lead to discontinuous results or to very large variations. From the lubricant physical state point of view, we could consider that the material is not in thermodynamically equilibrium when submitted to ultrasonic waves. As the output analysis is also made during a very short time, only a part of the fluid response could appear during this kind of experiment.
197 Concerning the ratio of the isentropic tangent modulus to the isothermal one, our results confrm the usual rule i. e. isentropic values are greater, but only in the first half of our pressure domain. For higher pressures, isothermal tangent moduli becomes greater. Consequently this is true also for heat capacities. 5. CONCLUSIONS
Compressibility experiments have been conducted under high pressure on several lubricants of various chemical composition. Volumes losses have been found to vary as a function of pressure, temperature and of the fluids chemical nature. Among the tested lubricants, silicone fluids have been found more compressible than polyalphaolefins. Dowson and Higginson [l] relationship should be used to fit compressibility results if its parameters are determined before. Isothermal secant bulk moduli increase almost linearly as a function of pressure, with slopes weakly temperature dependent. Isothermal tangent bulk moduli are greater than secant ones. Their increase with pressure is more pronounced, especially under the highest pressures where they can reach or exceed 10 GPa. Isentropic tangent bulk moduli have been deduced from both compressibility and ultrasonic measurements. We have observed a nearly linear increase with pressure. This means that under high pressures, isentropic tangent bulk moduli becomes lower than isothermal ones, in opposition to the comments found in literature. This discrepancy has been explained by the fluids viscoelastic behavior in compression.
ACKNOWLEDGMENTS
This research was partly supported by the C.E.A. (Centre du Ripault). G.Roche from L.M.C. is thanks for his contribution to the experiments.
REFERENCES
1. Dowson D. and Higginson G.R., "Elastohydrodynamic lubrication", Pergamon Press Ltd., SI Edition 1977. 2. Jacobson Bo 0. and Vinet P., "A model for the influence of pressure on the bulk modulus and the influence of temperature on the solidification pressure for liquids lubricants", presented at the A. S.M.E./A.S.L.E. Joint Tribology Conference, Pittsburgh, Pa., October 20-22 1986, Paper no 86Trib-63. 3. Jacobson Bo O., "Shear strength and compressibility of lubricants at high pressure and temperature", presented at the 6th International Colloquium "Industrial Lubricants: Properties, Application, Disposal" held at the Technixhe Akademie Esslingen, Germany, January 12-14 1988, paper 12.2. 4. Hayward A.T.J., "The compressibility of hydraulic fluids", Journal of the Institute of Petroleum, Vol. 51, N0494, p 35-52, February 1965. 5. Hayward A.T.J., "Compressibility equations for liquids: a comparative study", Brit. J. Appl. Phys., 1967, Vol. 18, p 965-977
6. Hayward A.T.J., "Experimental verification at high pressure of the relationship between compression, density and sonic velocity", Nature, Vol. 221, March 15th 1969, p 1047.
7. Hayward A.T.J., "How to measure the isothermal compressibility of liquids accurately", J. Phys. D: Appl. Phys., 1971, Vol. 4, p 938-950. 8. Hayward A.T.J., "Precise determination of the isothermal compressibility of mercury at 2OoC and 192 bar", J. Phys. D: Appl. Phys., 1971, Vol. 4, p 951-955. 9. Yusa M., Mathur G.P. and Stager R.A., "Viscosity and compression of ethanol-water mixtures for pressures up to 40000 psig", J. of Chem. and Eng. Data, Vol. 22, NO1,1977, p 32-35.
198 10. Klaus E.E. and O'Brien J.A., "Precise measurement and prediction of bulk modulus values for fluids and lubricants", Transactions of the A.S.M.E., J. of Basic Engineering, September 1964, p 469-474. 11. Wright W.A., "Prediction of bulk moduli and pressure volume temperature data for petroleum oils", A.S.L.E. Transactions, Vol. 10, 1967, p 349356.
15. Ohno N., Kuwano N. and Hirano F., "Diagrams for estimation of the solidified film thickness at high pressure in EHD contacts", Proceedings of the 20th Leeds-Lyon Symposium on Tribology "Dissipative Processes in Tribology", Lyon, France, September 7-10 1993. 16. Pressure-Viscosity Report, Research Committee on Lubrication, The American Society of Mechanical Engineers, New York, 1953.
12. Tichy J.A. and Winer W.O., "A correlation of bulk moduli and P-V-T data for silicone fluids at pressures up to 500,000 psig", A.S.L.E. Transactions, Vol. 11, 1968, p 338-344.
17. Noonan J.W., "Ultrasonic determination of the bulk modulus of hydraulic fluids", Mat. Research & Standards, December 1965, p 615-621.
13. Galvin G.D., Naylor H. and Wilson A.R., "The effect of pressure and temperature on some properties of fluids of importance in elastohydrodynamiclubrication", Proc. Instn. Mech. Engrs. 1963-64, Vol. 178 Pt 3N, p 283-290.
G.,Vergne P., "Viscmlastic parameters of 5P4E as
14. Goldman I.B., Ahmed N., Venkatesan P.S. and Cartwright J.S., "The compressibility of selected fluids at pressures up to 230,000 psi", Lubrication Engineering, October 1971, p 334-34 1.
18. Bezot P., Hesse-Bezot C., Berthe D., Dalmaz
a function of pressure and temperature by light scattering technique", J. of Tribology, 1986, Vol. 108, p 579-583.
189
STATIC AND DYNAMIC COMPRESSIBILITY OF LUBRICANTS UNDER HIGH PRESSURES P. Vergne Laboratoire de Mhnique des Contacts, URA CNRS 856 INSA, BAtiment 113,20 avenue Einstein, 69621 Villeurbanne cedex, France This paper presents an experimental investigation conducted on several synthetic lubricants of various chemical nature: silicone fluids, polyalphaolefins and a diester. Volume losses are first reported at different temperatures. Then, isothermal secant and tangent compressibility modulus values are deduced. They are compared to isentropic bulk moduli, calculated from ultrasonic measurements. Finally, a discussion on the correlation between the relative value of these moduli and the lubricants physical state completes the paper. 1. INTRODUCTION
During the last decades, the lubricants compressibility under high pressure has been the rheological parameter which has certainly been less studied. Nevertheless, the compressibility role appears to be significant in many circumstances, as for instance: - in highly loaded lubricated contacts as those occurring between gear teeth, - and also in most of lubricated metal forming processes where small amounts of lubricant are compressed between the tool and the metal sheet, trapped by surface asperities. The compressibility contribution in the running of E.H.D.contacts has been first described by Dowson and Higginson [l] and recently developed by Jacobson et al. [2,3] for instance. The reference work in the fluid compressibility domain has been brought by Hayward who proposed at first [4] an adequate terminology and methods of expressing results. His contribution has concerned also compressibility equations [4,5], testing methods [4,6,7], experimental investigations in various liquids [4,8]. To our knowledge, most of compressibility equations are empirical relationships which have frequently been experimentally verified in narrow pressure ranges. Hayward [5] and later on Yusa, Mathur and Stager [9] have reviewed and tested
some of them, but due to the experimental limits already mentioned, any clear tendency emerges. In lubrication, the more frequently used densitypressure equation found in literature is the Dowson and Higginson [l] relationship which is written as follows: p/po = 1 + 0.6P/( 1 + 1.7P )
(1)
where p is the density under pressure P, and p,, is the density at ambient pressure. Concerning experimental results on lubricants, two different approaches could be mentioned. Very accurate experiments have been run in a restricted domain (typically 700 bars) and therefore values for higher pressures are extrapolated. This method has been applied by Klaus and OBrien [lo], Wright [ 111 and Tichy and Winer [ 121 on various kind of fluids: mineral base oils, polyphenyl ether, silicone fluids and diesters. The second approach consists to conduct experiments in a pressure domain as large as possible. The limit generally corresponds to the maximum admissible stress of the high pressure cell material or to the appearance of leakage problems. Since Galvin et a1.[13] who reach 350 MPa, special devices have been developed to work under 1 GPa or more[2,3,14,15].
190
DOS
Di 2 ethyl hexyl
Chemical
Index Density at 20°C Kin. Viscosity at 40°C Kin. Viscosity at 100°C VIE a in l/GPa a in l/GPa
S1L2
dimethyl poly siloxane -5OOC
POL 1
methylchlorophe polyalphaolefin . . nyl hlysiloxane -75°C -40°C
POL2 polyalphaolefin _ _ _ -57°C
1.449
1.497
1.416
1.470
0.913
1.066
1.016
0.848
0.848
11.7 cst
77 cst
52 cst
408 cst
33 cst
3.5 cst 199 12.5 at 20°C 12.1 at 25°C
22.2 cst 314 18.2 at 60°C 10.4 at 150°C
20.9 cst 414 12.8 at 60°C 10. at 150°C
41.6 cst 154 12.6 at 60°C 9.7 at 150°C
7 cst 181 12.6 at 60°C 8.7 at 15OOC
I I
SIL 1
I
Table 1 :Lubricants properties (ais the secant pressure-viscositycoefficient, calculated at 400 MPa). 2. TESTED MATERIALS AND DEVICE 2.1. Tested materials
Five products have been studied: one diester, two silicone fluids and two polyalphaolefins respectively named DOS, SIL1, SIL2, POL1 and POL2. Most of these lubricants are pure synthetic fluids: the exception is POL2 which contains an additive package. Their properties are summarised in Table 1: apart from pour point temperatures, the results reported in Table 1 have been collected on our own devices. -Refractive index has been measured according to the A.S.T.M. D1218 standard. -Density variations versus temperature have been determined with a Mettler density kit set up on an electronic balance. -Kinematic viscosity changes versus temperature have been deduced from dynamic viscosity results obtained with a Couette viscometer and from density data. -VIEhas been determined according to the A.S.T.M. D2270 standard. -Pressure-viscosity coefficients have been calculated from experiments run on our high pressure falling body viscometer. We have deliberately chosen to give the secant coefficients computed for a 400 MPa pressure increase instead of usual coefficients which
serve for EHD film thickness calculations. Here, the aim is to report a parameter which represents the viscosity increase in a wide pressure domain. The fluids are all characterised by low pour points, low viscosity at ambient pressure and temperature (except for POL1) and high viscosity index. SIL2 exhibits the greater stability: pressure and temperature influence is very weak compared to the other lubricants. This is the consequence of its chemical structure which contains phenyl groups and chlorine atoms. 2.2. Experimental device Reliable bulk moduli data are difficult to obtain on a fluid [7,11]: following the advice of previous authors, a maximum of caution has been taken during the experimental phase. The measurements have been performed in the high pressure (up to 700 MPa), wide temperature range (from -25OC to 160OC) vessel of our falling body viscometer. To avoid dispersion due to unhomogeneous strains, the high pressure densimeter works under perfect hydrostatic pressure and temperature. It is made of a cylindrical stainless steel cell filled by the sample: its top end is closed by the ceramic disk of an ultrasonic transducer and its bottom end is sealed by a metallic piston
191 supported by O-rings. This feature allows to accommodate volume changes in the fluid column by a translating motion of the piston. Consequently the distance between the ceramic disk and the piston changes as a function of pressure and temperature. These distance variations are recorded by means of an ultrasonic technique which requires the preliminary measurement of the sound velocity in the sample. This previous work is run at similar ceramic-reflector distance to avoid possible linearity fluctuations of the electronic unit. We use an impulsion method (3 periods maximum by pulse) with longitudinal waves: the transducer works as an emitter/receiver. To limit energy dissipation in the fluid under pressure, we have chosen a high ultrasonic frequency (1 MHz) and a low emission one (125 Hz):this means that the ratio of the time during fluid molecules undergo ultrasonic waves to the real time is negligible. The experiments are computer controlled: recorded data are directly converted in length and thus in volume. Elastic deformations caused by hydrostatic pressure and temperature fields are directly taken into account: their influence is very
experience in the field, we consider that our volume losses are obtained with an accuracy of +/- 3 %. The best means to validate this p i n t is to compare our results to previously published data. Here we have chosen di 2 ethyl hexyl sebacate @OS in Table 1) results from the A.S.M.E. Pressure Report [16]. The comparison at 25OC is presented Figure 1, where the relative volume reduction (-AVN,) is plotted versus pressure. We observe a good agreement in the low and in the high pressure domains: nevertheless a sensible discrepancy (close to 7%) is noted between 2 and 3 kbar. Even if this kind of deviation could appear acceptable for volume losses, it should be still considered with care for bulk moduli determination,especially for tangent bulk moduli. The comparison with Dowson and Higginson relationship [ 11 shows a good agreement at low and at high pressure. However the analytical curve is flater than experimental ones which implies significant differences in the moduli. The discussion on the validity of this relationship is presented in the next section.
Weak.
To avoid uncertainties in the low pressure domain, the first pressure step is 25 MPa for DOS and 50 MPa for the other fluids. Hayward [7] has cleverly suggested that during compressibility experiments, the first pressure step should be at least one tenth of the maximum pressure to minimize the errors, especially on the moduli. To purge the air entrapped in the fluid, the densimeter is placed in a tank filled with an excess sample volume, to allow the handling. Then all the parts (metallic surfaces + sample volume) are degassed under vacuum (pressure < 10' mbar ). Then the densimeter is closed, always in the tank, filled this time by air-free lubricant. As it has been reported above, compressibility measurements are very difficult to conduct because of the low volume changes with pressure: a high accuracy is required. In our case pressure measurements are run with a precision of +/- 0.25% and the temperature is known to within +/- 0.5 %. These values suggest that the main cause of potential deviation in our results comes mostly from volume measurements. Taking into account our
I
kii
d > 10
5
0
0
1000
u)(30
3000
4000
5000
Figure 1 :Volume losses at 25OC in DOS.
3. VOLUME LOSSES RESULTS
All the results obtained at 60, 100 and l5OoC are reported in Figures 2 to 5, respectively for SIL1, SL2, POL1 and POL2.
192
-
IOOOC
P
0.10-
0,05*
IF
u,w-
0
Figure 2 :Volume loss versus pressure in SILL
PRESSURE (MPa) 100
200
300
400
500
Figure 4 :Volume loss versus pressure in POL1.
z 025
Eg
2
w
3
0,20 0,15
P 0,IO 0.05 0,oo
Figure 3 :Volume loss versus pressure in SIL2.
Most of the results concerns a maximum pressure increase of 500 MPa, except for SIL1: its viscosity shows a strong pressure dependence at 6 O O C . Consequently compressibility tests are impossible to run under this condition. Normally the pressure steps are equal to 50 MPa but due to ultrasonic problems (signal fluctuations), some points have been voluntary omitted.
0
100
200
300
400
500
Figure 5 : Volume loss versus pressure in POL2.
After the initial pressure steps, SILl curves suggest a linear increase of volume losses with pressure as usually volume reduction found at the highest pressure levels tends to decrease. Although data at 6OoC are close to those given by equation (l), the gap between experimental and calculated values increases continuously. At 15OoC, the
193 experimental curve is very different from the analytical one. SIL2 behavior is very specific: it shows a large compressibility as the volume losses reach or exceed 20% at 500 MPa at all tested temperatures. Furthermore, we observe a very large volume reduction since the first pressure steps: at 150 MPa and 6OoC, 10% is reached as in SIL1, this reduction is found at 250 MPa. In the highest pressures domain (typically when P > 300 MPa), we discern the same tendency that for S L l . The total volume loss increases linearly with pressure (up to 500 MPa): the slopes found with SIL2 are lower than those observed on SIL 1. Polyalphaolefins show results in better agreement with usual tendencies. POLl data at 60°C and equation (1) are very close. The maximum volume reduction attains 16% at 150°C as it was greater than 22% for SIL2. At 15OoC,we have obtained very similar results for POLl and POL2 as at lower temperatures, the curves differ.
Figure 6 :Volume losses comparison at 100°C.
To illustrate the above mentioned tendencies, we have reported all the volume losses measured at 100°C in Figure 6. The curves confirm that SIL2 is more compressible than the other fluids. It is also shown that the behavior of each chemical structure is typical: polyalphaolefin curves tend to an horizontal asymptote as silicone curves suggest the continuation of a linear increase.
Concerning the validity of equation (l), our results show that the fluid chemical nature, pressure and temperature govern the lubricants compression behavior. Correlation with diester results near ambient temperature is acceptable and comparison with polyalphaolefins data has shown that a shift occurs during the first pressure steps and after the deviation remains constant. Even for SIL2, we can consider that the data could be fitted by a similar equation as the curvatures are very close. All these observations suggest that equation (1) could be used on a wide experimental domain if the two constants (0.6 and 1.7) are changed by more appropriate ones. Temperature influence is more important with SIL2 and the less significant with pOL2. Volume reductions found on POL2 are more temperature dependent than those obtained with SIL1. A detailed analysis of the curves shows that temperature influence takes place essentially during the two first pressure steps. Beyond 100 MPa, we could consider for each lubricant that the curves obtained at 60, 100 and 150°C are roughly parallel. These observations on the temperature influence are consistent with literature [ 10-131. Comparisons with previously published result is not obvious. From our bibliographical analysis, it appears that silicone fluids are typically very compressible [4,10,12,13], compared to mineral oils for instance. Klaus and OBrien [ 101 have compared predicted and measured (from A.S.M.E. [ 161) isothermal secant bulk moduli for a similar fluid that SILl. At 100°C, they found 8.2% and 11.1% at 152 and 242 MPa and our more approaching data are respectively equal to 7.7% and 10.9% at 150 and 250 MPa. Galvin et al. [ 131 have reported data at 100°C on a silicone fluid which fits very well our results on SIL2. Concerning polyalphaolefins, Jacobson and Vinet [2,3] have reported some data but only for pressures greater than 425 MPa, making the comparison impossible. 4. BULK MODULI RESULTS 4.1. Isothermal secant bulk moduli
Isothermal secant bulk moduli 6) are directly derived from volume losses by the following equation:
194
K, = AP / (-AV/V,)
T = constant
Fluid T"C
(2)
SILl 100 150 60 60 1774 1612 1349 953 2.62 2.57 2.73 3.25
K,, A,
Equation (2) means that K, refers to the pressure references. Measurements are run under steady state pressure and temperature conditions. The results are reported in Figure 7 for silicone fluids and in Figure 8 for polyalphaolefins. The initial value at P = 0.1 MPa has been determined following an analytical method presented in the next section.
100
200
300
Figure 7 :Isothermal secant bulk moduli in silicone fluids SILl and SIL2.
In our pressure domain, we note that the isothermal secant bulk moduli increase roughly linearly with pressure. In agreement with the remarks reported on silicone fluids volume losses, the correlations between secant bulk moduli variations and linear fittings is very good. For these fluids, the results differ essentially in their initial value, i. e. their slopes are almost equal and not temperature dependent. Table 2 presents the linear fitting results of the isothermal secant bulk moduli variations versus pressure for SILl and SIL2. K, =K,,+A,.P
(3)
where A, is the slope, K,, is the initial isothermal bulk modulus (in MPa), and P the pressure (in MPa).
3.36
300
200
400
500
Figure 8 :Isothermal secant bulk moduli in polyalphaolefins POLl and POL2.
500
400
150 585
Table 2 : Results of linear regressions on K, (in Mpa) versus pressure for SILl and SIL2.
100
0
I
sIL2 100 776 3.33
Concerning POLl and POL2 isothermal secant bulk moduli, the tendencies are more temperature dependent. At 60, 100 and 150°C for POLl and at 60°C for POL2 a second degree polynomial gives a better fitting than a simple linear regression. Table 3 presents the values of this correlation: K, = K,, + B, . P + 10-3.c, . ~2
(4)
where B, and C, are the polynomial coefficients, K,, is the initial isothermal bulk modulus (in MPa), and P the pressure (in MPa).
L
Fluid T"C
K ,
B,
C,
I
I
60 I 1663 2.16 4.70
POL1 100 I 1285 2.99 2.13
1 1
I
POL2
100 I 150 1103 1580 1256 1023 3.12 1.98 3.15 3.62 1.18 2.52 0.630 0.147 150
60
Table 3 : Results of degree 2 curve fitting on the isothermal secant bulk moduli for POLl and pOL2.
195 4.2. Isothermal tangent bulk moduli
This parameter is the thermodynamically correct isothermal bulk modulus. It represents the true rate of volume change at the pressure of interest. The main difference with Ks is that the tangent modulus does not refer to the initial volume V,,. When P tends to the atmospheric pressure, Ks and Ic, become equal. K, is given by:
K,
=
- V . ( 6 P/ 6V )
at T = constant (4)
As V(P) is known point by point, the K, determination is not direct. At first, the volume losses have been fitted by a polynomial function. Then these functions have been derived and K, values have been deduced.
Figures 9 and 10 present the isothermal tangent bulk moduli found respectively in silicone fluids and in polyalphaolefins. These moduli are always greater than secant ones. Concerning SILl and SIL2 (Figure 9), the effects previously mentioned conduct to weakly curved slopes. The variations are more significant with POLl and POL2 (Figure 10). The curves shows that high values (> 10 GPa) have been reached at 400 or 500 MPa. It is also expected that under higher pressures, the moduli can be multiplied by a factor of 10 at least. This means that volume losses should be very limited, actually close to zero.
*
-0Y
--f
sIL11wc SILllSoOC sIL2wc SlLZlWC
M
U
0
100
200
300
400
500
Figure 10 :Isothermal tangent bulk moduli in polyalphaolefins POLl and POL2.
4.3. Isentropic tangent bulk moduli It is the volumetric tangent modulus of elasticity under conditions of constant entropy. Under normal conditions it is greater than the isothermal tangent bulk modulus by the ratio of the fluid specific heats. When data are available, this parameter is used under conditions where pressure changes are rapid, with little opportunity for the temperature to come in equilibrium. This definition underlines the interest to use this parameter in lubrication problems where dynamic high pressures occur. One of the methods developed to run isentropic compressibility experiments is based on ultrasonic velocity measurements. It is supported by the following equation:
-
53 5000
k, = V. ( 6 P/6V ) = p . c2 at S = constant ( 5 )
E
where kT is the isentropic tangent bulk modulus, p is the density, c is velocity of sound and S the entropy.
ca
9
b
$0 Y
0
o
100
200
300
400
500
Figure 9 :Isothermal tangent bulk moduli in silicone fluids SILl and SIL2.
The validity of equation ( 5 ) has been verified by Hayward [6] and Noonan [17]. Due to our peculiar device which requires the ultrasonic velocity measurements before running compressibility tests, we have been able to determined k, under the same conditions as the other moduli.
196
where DT is the slope, kTois the initial isentropic bulk modulus (in m a ) , and P the pressure (in MPa).
100
0
200
300
400
500
Figure 11 : Isentropic tangent bulk moduli in silicone tluids SILl and SIL2. Table 4 :Results of linear fitting on k,
,
At the opposite of isothermal tangent bulk moduli, all the fluids give similar curves: they differ by the initial value at P = 0.1 MPa as their slopes are very close.
v)
o . . - . 0
100
PRESSURE (MPa) 200
300
400
500
Figure 12 :Isentropic tangent bulk moduli in polyalphaolefins POL1 and POL2.
The results are plotted on Figure 11 for silicone fluids and on Figure 12 for polyalphaolefins. The curves are almost strait lines: a weak curvature towards the pressure axis is observed. This implies that isentropic tangent bulk moduli increase almost linearly with pressure. Apart for initial values, a linear regression performed on these results gives an acceptable agreement (see Table 4).
The explanation could come from the fluids viscoelastic behavior in compression. First of all, it is important to remind that longitudinal waves could be splitted in pure compression waves plus pure shear waves and that longitudinal velocities are also frequency dependent [18]. The limit is the infinite frequency velocity which is 20 to 40% greater than the ultrasonic one, in the liquid domain. If we apply time (frequency) pressure equivalence principle, there any reason for the ultrasonic velocity to increase suddenly in our experimental conditions. As in parallel the density increase is continuous, this could not lead to discontinuous results or to very large variations. From the lubricant physical state point of view, we could consider that the material is not in thermodynamically equilibrium when submitted to ultrasonic waves. As the output analysis is also made during a very short time, only a part of the fluid response could appear during this kind of experiment.
197 Concerning the ratio of the isentropic tangent modulus to the isothermal one, our results confrm the usual rule i. e. isentropic values are greater, but only in the first half of our pressure domain. For higher pressures, isothermal tangent moduli becomes greater. Consequently this is true also for heat capacities. 5. CONCLUSIONS
Compressibility experiments have been conducted under high pressure on several lubricants of various chemical composition. Volumes losses have been found to vary as a function of pressure, temperature and of the fluids chemical nature. Among the tested lubricants, silicone fluids have been found more compressible than polyalphaolefins. Dowson and Higginson [l] relationship should be used to fit compressibility results if its parameters are determined before. Isothermal secant bulk moduli increase almost linearly as a function of pressure, with slopes weakly temperature dependent. Isothermal tangent bulk moduli are greater than secant ones. Their increase with pressure is more pronounced, especially under the highest pressures where they can reach or exceed 10 GPa. Isentropic tangent bulk moduli have been deduced from both compressibility and ultrasonic measurements. We have observed a nearly linear increase with pressure. This means that under high pressures, isentropic tangent bulk moduli becomes lower than isothermal ones, in opposition to the comments found in literature. This discrepancy has been explained by the fluids viscoelastic behavior in compression.
ACKNOWLEDGMENTS
This research was partly supported by the C.E.A. (Centre du Ripault). G.Roche from L.M.C. is thanks for his contribution to the experiments.
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