Content uploaded by Philippe Vergne

Author content

All content in this area was uploaded by Philippe Vergne on Jun 06, 2018

Content may be subject to copyright.

STATIC AND DYNAMIC COMPRESSIBILITY OF LUBRICANTS UNDER HIGH

PRESSURES

P. Vergne

Laboratoire de Mécanique des Contacts, URA CNRS 856

INSA, Bâtiment 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 [1] 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 density-

pressure equation found in literature is the Dowson

and Higginson [1] relationship which is written as

follows:

ρ/ρ

0

= 1 + 0.6P/( 1 + 1.7P ) (1)

where ρ is the density under pressure P,

and ρ

0

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 O'Brien [10], Wright [11]

and Tichy and Winer [12] 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 al.[13] who reach 350 MPa, special

devices have been developed to work under 1 GPa

or more[2,3,14,15].

DOS SIL1 SIL2 POL1 POL2

Chemical

Structure

Di 2 ethyl hexyl

sebacate dimethyl

polysiloxane methylchlorophe

nyl polysiloxane

polyalphaolefin polyalphaolefin

Pour Point °C

-60°C -50°C -75°C -40°C -57°C

Refractive

Index

1.449

1.497

1.416

1.470

Density at

20°C

0.913

1.066

1.016

0.848

0.848

Kin. Viscosity

at 40°C

11.7 cst

77 cst

52 cst

408 cst

33 cst

Kin. Viscosity

at 100°C

3.5 cst

22.2 cst

20.9 cst

41.6 cst

7 cst

VI

E

199 314 414 154 181

α in 1/GPa

12.5 at 20°C 18.2 at 60°C 12.8 at 60°C 12.6 at 60°C 12.6 at 60°C

α in 1/GPa

12.1 at 25°C 10.4 at 150°C 10. at 150°C 9.7 at 150°C 8.7 at 150°C

Table 1 : Lubricants properties (α is the secant pressure-viscosity coefficient, 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.

-VI

E

has 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 -25°C to 160°C) vessel of our falling

body viscometer. To avoid dispersion due to un-

homogeneous 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 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

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

-1

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 experience

in the field, we consider that our volume losses are

obtained with an accuracy of +/- 3 %.

The best means to validate this point is to

compare our results to previously published data.

Here we have chosen di 2 ethyl hexyl sebacate (DOS

in Table 1) results from the A.S.M.E. Pressure

Report [16]. The comparison at 25°C is presented

Figure 1, where the relative volume reduction (-∆

V/V

0

) 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 [1] shows a good agreement at low and

at high pressure. However the analytical curve is

flatter than experimental ones which implies

significant differences in the moduli. The discussion

on the validity of this relationship is presented in the

next section.

3. VOLUME LOSSES RESULTS

All the results obtained at 60, 100 and 150°C are

reported in Figures 2 to 5, respectively for SIL1,

SIL2, POL1 and POL2.

Most of the results concerns a maximum pressure

increase of 500 MPa, except for SIL1: its viscosity

shows a strong pressure dependence at 60°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.

After the initial pressure steps, SIL1 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 60°C are close to those given by equation (1),

the gap between experimental and calculated values

increases continuously. At 150°C, the 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 60°C, 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 SIL1. The total volume loss

increases linearly with pressure (up to 500 MPa): the

slopes found with SIL2 are lower than those

observed on SIL1.

Polyalphaolefins show results in better agreement

with usual tendencies. POL1 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 150°C, we have obtained very

similar results for POL1 and POL2 as at lower

temperatures, the curves differ.

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 (1), 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 [10-13].

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 O'Brien [10] have compared

predicted and measured (from A.S.M.E. [16])

isothermal secant bulk moduli for a similar fluid that

SIL1. 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. [13] 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 (K

S

) are directly

derived from volume losses by the following

equation:

K

S

= ∆P / (-∆V/V

0

) T = constant (2)

Equation (2) means that K

S

refers to the

volumetric change with pressure from atmospheric

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.

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 SIL1 and SIL2.

K

S

= K

S0

+ A

S

. P (3)

where A

S

is the slope,

K

S0

is the initial isothermal bulk modulus (in MPa),

and P the pressure (in MPa).

F

luid

SIL1 SIL2

T

°C

60 100 150 60 100 150

K

S0

1774

1612

1349

953 776 585

A

S

2.62 2.57 2.73 3.25 3.33 3.36

Table 2 : Results of linear regressions on K

S

(in

Mpa) versus pressure for SIL1 and SIL2.

Concerning POL1 and POL2 isothermal secant

bulk moduli, the tendencies are more temperature

dependent. At 60, 100 and 150°C for POL1 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

S

= K

S0

+ B

S

. P + 10

-3

.C

S

. P

2

(4)

where

B

S

and

C

S

are the polynomial coefficients,

K

S0

is the initial isothermal bulk modulus (in MPa),

and P the pressure (in MPa).

F

luid

POL1 POL2

T

°C

60 100 150 60 100 150

K

S0

1663

1285

1103

1580

1256

1023

B

S

2.16 2.99 3.12 1.98 3.15 3.62

C

S

4.70 2.13 1.18 2.52 0.630

0.147

Table 3 : Results of degree 2 curve fitting on the

isothermal secant bulk moduli for POL1 and POL2.

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 K

S

is that the tangent modulus

does not refer to the initial volume V

0

. When P tends

to the atmospheric pressure, K

S

and K

T

become

equal. K

T

is given by:

K

T

= - V . ( δ P/ δV ) at T = constant (4)

As V(P) is known point by point, the K

T

determination is not direct. At first, the volume

losses have been fitted by a polynomial function.

Then these functions have been derived and K

T

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 SIL1 and SIL2 (Figure 9), the effects

previously mentioned conduct to weakly curved

slopes.

The variations are more significant with POL1

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.

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:

k

T

= - V. ( δ P/δV ) = ρ . c

2

at S = constant (5)

where k

T

is the isentropic tangent bulk modulus,

ρ is the density,

c is velocity of sound and S the entropy.

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

T

under the same

conditions as the other moduli.

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).

k

T

= k

T0

+ D

T

. P (3)

where D

T

is the slope,

k

T0

is the initial isentropic bulk modulus (in MPa),

and P the pressure (in MPa).

Fluid T °C k

T0

D

T

SIL1 60 1479 9.95

SIL1 100 1314 8.80

SIL1 150 1038 8.76

SIL2 60 1098 9.32

SIL2 100 921 9.08

SIL2 150 734 8.73

POL1 60 1591 9.56

POL1 100 1377 9.00

POL1 150 1112 8.78

POL2 60 1999 11.15

POL2 100 1652 11.15

POL2 150 1244 11.05

Table 4 : Results of linear fitting on k

T

.

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.

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.

Concerning the ratio of the isentropic tangent

modulus to the isothermal one, our results confirm

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 [1]

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 O. 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 n° 86-

Trib-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 Technische

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, N°494, 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 20°C 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, N°1, 1977, p 32-35.

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 349-

356.

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.

13. Galvin G.D., Naylor H. and Wilson A.R., "The

effect of pressure and temperature on some

properties of fluids of importance in

elastohydrodynamic lubrication", Proc. Instn. Mech.

Engrs. 1963-64, Vol. 178 Pt 3N, p 283-290.

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 334-341.

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.

17. Noonan J.W., "Ultrasonic determination of the

bulk modulus of hydraulic fluids", Mat. Research &

Standards, December 1965, p 615-621.

18. Bezot P., Hesse-Bezot C., Berthe D., Dalmaz

G., Vergne P., "Viscoelastic parameters of 5P4E as a

function of pressure and temperature by light

scattering technique", J. of Tribology, 1986, Vol.

108, p 579-583.