Dynamic Viscosity under Pressure for Mixtures of Pentaerythritol Ester

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Ind. Eng. Chem. Res. 2007, 46, 1826-1835

Dynamic Viscosity under Pressure for Mixtures of Pentaerythritol Ester Lubricants with 32 Viscosity Grade: Measurements and Modeling L. Lugo,* X. Canet,† M. J. P. Comun˜ as, A. S. Pensado, and J. Ferna´ ndez Laboratorio de Propiedades Termofı´sicas, Dpto. de Fı´sica Aplicada, Facultade de Fı´sica, UniVersidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain, and Laboratoire de Thermodynamique, Faculte´ Polytechnique de Mons, Belgium

The dynamic viscosity under pressure of three mixtures of pentaerythritol ester lubricants (PEs) has been measured using a rolling-ball viscometer for several temperatures with an experimental uncertainty of 3%. The first one is a multicomponent mixture of several PEs named in the present work as PEC5-C9 lubricant; the second one is a binary mixture of pentaerythritol tetra(2-ethylhexanoate), PEB8, and pentaerythritol tetraheptanoate, PEC7, with a PEB8 mole fraction of 0.6670; and the third one is another binary mixture of PEB8 and pentaerythritol tetrapentanoate, PEC5, with a PEB8 mole fraction of 0.6911. The two binary mixtures, xPEB8 + (1 - x)PEC7 and xPEB8 + (1 - x)PEC5, have been prepared with the same viscosity grade as the PEC5-C9 lubricant (VG32). A total of 1176 experimental measurements of the rolling time have been performed at pressures up to 60 MPa for the determination of 196 dynamic viscosity data points. The viscosities of these binary mixtures have been compared with the predicted values obtained by using several viscosity models (Grunberg-Nissan and Katti-Chaudhri mixing laws, self-referencing model, hard-sphere theory, and free-volume model). All methods predict dynamic viscosity values for the two binary mixtures that agree with the experimental data within an average mean deviation of 10% over the entire temperature and pressure ranges. The best predictions were found with the free-volume model, for which the average mean deviation for both mixtures is lower than 4%. Parameter values for the self-referencing model were determined from experimental viscosity data of several pure PEs. These parameters permit the estimation of viscosity values of PE lubricant of unknown composition, when a viscosity value at any temperature and pressure is available. This model predicts the viscosities of PEC5-C9 lubricant with an average deviation of 4%. Introduction People are paying increasingly close attention to the environment with the development and progress of society. Following the Kyoto conference on climate change, energy efficiency is becoming an important performance characteristic for lubrication and all refrigeration and air-conditioning systems. The appropriate selection of a lubricant can have a significant impact on the overall efficiency of operation of domestic appliances and other refrigeration and air conditioning systems. In the lubrication field, including automotive and marine engine oils, compressor oils, hydraulic fluids, gear oils, and grease formulations, greater attention is being placed on the use of synthetic fluids. It is imperative in the present situation to study green lubricants. It is estimated that ∼10% of global lubricating oil production is fully synthetic products.1 Synthetic fluids differ from mineral oil in that they have generally better defined chemical structures, but also a wider range of chemical functionalities. Although, in recent years, remarkable progress in green chemical technology has been made, some problems remain to be solved, such as the compatibility among base oils, thickeners, and additives, as well as the development of novel additives and methods to assess the biodegradability of green lubricants.2 Synthetic lubricants3,4 are manufactured from a number of differing chemical bases. Several classes of compounds have been developed to provide base stocks for commercial synthetic fluids such as polyalphaolefins, PAOs, polyalkylene glycols, PAGs and esters, especially polyol esters, POEs. Different studies point out that these kinds of synthetic lubricants can be really called green lubricants.5-8 * To whom correspondence should be addressed. Tel.: 34981563100, ext. 14046. Fax: 34981520676. E-mail: [email protected]. † Faculte´ Polytechnique de Mons.

Among these green lubricants, POEs seem to be the lubricants of choice for use with the natural refrigerant CO2 or with nonchlorinated refrigerants, such as HFCs, for reasons of miscibility, low toxicity, and excellent biodegradability, and also because of their inherently good lubricity.9,10 Polyol esters are made by reacting a multifunctional alcohol with a monofunctional acid. Neopentyl glycol and pentaerythritol esters are sometimes known as neopentyl polyols because their structures are based on the hydrocarbon neopentane.11 Polyol esters are used in a wide variety of applications, namely, refrigeration compressors, aviation, greases, air compressors, metal working, fire resistant and biodegradable hydraulic fluids, and chain oils. Polyol ester base oils combine both excellent performance, including hightemperature applications, and biodegradability.5,12-14 Generally, linear polyol esters tend to be used when high degrees of biodegradability are required.11 POEs are less toxic and tend to be more effective lubricants than mineral oils, and they can be obtained using a significant proportion of raw materials derived, or potentially derivable, from renewable resources,5,15,16 e.g., through the hydrolysis of fats and oils to produce the constituent fatty acids as raw materials for chemical synthesis. A wide variety of natural sources, including solid fats and low-grade or waste materials such as tallow from rendering of animal carcasses or tall oil from wood pulp processing, can be converted through controlled chemical processing into pure fatty acids of consistent quality. Saturated short-chain fatty acids are used to make high-stability polyol esters that are used in highperformance synthetic car engine oils, jet engine lubricants, and compressor oils. Other benefits include extended life, reduced maintenance and downtime, lower energy consumption, and reduced smoke and disposal. Pentaerythritol esters (PEs) are a family of polyol ester synthetic lubricants manufactured by

10.1021/ie061187r CCC: $37.00 © 2007 American Chemical Society Published on Web 02/20/2007

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reacting pentaerythritol with a mixture of organic acids, which can have different molecule size and branching.12 Accurate viscosity data of liquids over wide ranges of temperature and pressure are needed for the design, operation, and optimization of equipment in the lubricant industry. The viscosity of a lubricant has a marked effect on wear because of its relation with film thickness, and it is a key requirement for systems operating under hydrodynamic or elastohydrodynamic lubrication (EHL). The effectiveness of oil in the rolling-element bearings and gears, the energy losses due to viscous dissipation, the improvement of fatigue life, and the reduction in friction and wear are entirely dependent on its viscosity.17 Thus, the energy requirement of an appliance compressor can be reduced by decreasing the lubricant viscosity. However, if the lubricant viscosity is reduced too strongly, it is no longer high enough to ensure that a full fluid film is entrained and metal-to-metal contact occurs, so that some wear is unavoidable.6 Although experimental viscosity data of pure compounds are commonly available at atmospheric pressure, the values as a function of pressure are less frequent. In fact, rheological properties of a lubricant that determine the film thickness in an EHL contact are normally the dynamic viscosity and the pressure-viscosity coefficient, which is often defined as R ) (1/η)(∂η/∂p)T. Pitting in gears and bearings is an example of surface-breaking rolling contact that can be avoided by using a lubricant with both an adequate viscosity and a suitable pressure-viscosity coefficient. High-viscosity pressure coefficients can be beneficial in Elastohydrodynamic (EHD) contacts, for example, in rotary-vane compressors for air-conditioning applications. However, in the formulation of many fuel-economy enhancing “energy-conserving” engine oils, gear lubricants, and transmission fluids, a lessviscous film is desired because mechanical energy can be wasted if the film is too robust. In other applications such as continuously variable transmissions, where the fluid must provide high levels of traction, a very high R value is needed.18,19 Therefore, it is very important to know how the pressure affects the viscosity of pure and mixed green lubricants, such as pentaerythritol esters, in order to get better formulations. This is the main objective of the present work, in which dynamic viscosities under pressure of a mixture of several pentaerythritol esters, named here as PEC5-C9, and a binary mixture composed of pentaerythritol tetra(2-ethylhexanoate), PEB8, and pentaerythritol tetraheptanoate, PEC7, have been measured. This study has been performed over the temperature and pressure ranges 303.15-353.15 K and 0.1-60 MPa, respectively. Moreover, the viscosities of a second binary mixture composed of PEB8 and pentaerythritol tetrapentanoate, PEC5, for two isotherms (313.15 and 333.15 K) have been determined over the same pressure range. All the mixtures have the same viscosity grade20 (VG32). The viscosity grades (VGs) are classified by the International Organization of Standardization (ISO 3448) for industrial engine oils. The grading system is based on the kinematic viscosity of the fluid, ν, in cSt (1 cSt ) 10-6 m2‚s-1) at 313.15 K and 0.1 MPa. To fall into a given viscosity grade, the oil must be within 10% of the midpoint kinematic viscosity. Thus, VG32 means that the kinematic viscosity is in the interval 28.8-35.2 cSt, at 313.15 K and 0.1 MPa. Among a wide variety of industrial applications, VG32 synthetic oils are adopted for both rolling-element and sleeve bearings, provide adequate lubrication, and minimize the frictional energy consumed with rolling-element bearings. Because it is impossible to perform viscosity measurements for all lubricants, it is necessary to develop predictive models and to verify the suitability of existing ones. Nevertheless, to

Figure 1. Molecular structures for pure pentaerythritol esters (PEs): pentaerythritol tetrapentanoate (PEC5), pentaerythritol tetraheptanoate (PEC7), pentaerythritol tetranonanoate (PEC9), and pentaerythritol tetra(2-ethylhexanoate) (PEB8).

carry out this last task, experimental viscosity data of pure or mixed lubricants with known molecular structures and compositions are needed. Thus, the experimental values of the binary mixtures measured in this article are used in the present work to evaluate the suitability of the Grunberg and Nissan21 and Katti and Chaudhri22 mixing laws, the self-referencing model,23 the hard-sphere theory,24,25 and the free-volume model26,27 to predict the viscosities of this kind of mixture. Experimental Technique The multicomponent mixture of several PEs, named here as PEC5-C9, with VG32 was provided by Uniqema (United Kingdom). The PEC5-C9 mixture is 99.9% pure (ester) and contains 95%.28-30 The molecular structures of pure pentaerythritol ester lubricants (PEs) are presented in Figure 1. Accordingly, the studied mixtures (PEC5-C9, PEB8 + PEC7, and PEB8 + PEC5) have similar kinematic viscosities, ∼33 cSt at 313.15 K and 0.1 MPa. The high-pressure viscosity apparatus consists of a commercial rolling-ball viscometer (Ruska 1602-830) and a pressure line that requires the construction and setup of several pieces of equipment and peripherals. The scheme of the apparatus has been published recently.31 The measurements can be conducted with the unit inclined at an angle θ, of 23°, 45°, or 70°, with the horizontal. The angle and the sphere diameter must be selected taking into account the viscosity range and the uncertainty in the rolling time. Thus, for higher viscosities, a higher angle and the sphere with the lower diameter should be utilized, in order to avoid lengthy rolling times. The temperature is measured using a PT 100 thermometer with a precision of (0.1 K, and the pressure is determined using a manometer HBM PE300 with an uncertainty of 0.07 MPa. The dynamic viscosity, η, is a function of the time, t(θ), taken for the ball to roll from one end of the fluid-filled tube to the other at a fixed angle θ; of the density difference between the ball and the fluid, ∆F ) Fball - Ffluid; and of the apparatus parameters. In previous work31 for the low-viscosity range (0.234-0.803 mPa‚s), the working equation of the viscometer was taken to be a linear function of t∆F with a nonzero intercept. The instrument performance for low viscosities was tested in that study31 by comparing the experimental dynamic viscosity

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measured for toluene and the values obtained with the reference correlation recommended by the IUPAC, obtaining an absolute average deviation of 0.9% and a maximum deviation of 2%. Recently,32,33 the experimental viscosities of four pure pentaerythritol esters (PEB8, PEC5, PEC7, and PEC9) have been measured with this equipment from 6 to 213 mPa‚s. For these last cases and also in the present work, for medium and high viscosity values, a linear dependence in t∆F with zero intercept has been used in accordance to the following working equation:

η(p,T) ) K(T,p,θ)t(θ)∆F

(1)

In order to determine the calibration function K(T,p,θ), 1-phenyldodecane was used as the reference substance because of its well-known density and viscosity. Experimental densities of this fluid (a total of 45 points) have been published by EtTahir et al.34 with an uncertainty of 2 × 10-4 g‚cm-3 and from 298.15 to 363.15 K up to 40 MPa. Experimental dynamic viscosities with an experimental uncertainty of (2% in the temperature interval 298.15-363.15 K and up to 100 MPa have been reported by the same authors.34 The calibration procedure has been thoroughly detailed in a recent work.32 Taking into account temperature, pressure, and rolling-time accuracies as well as the density and viscosity uncertainties of the reference fluid (1-phenyldodecane), the overall uncertainty of the present measurements is estimated to be 3%. For high viscosity values (up to 74.3 mPa‚s), the performance of the instrument was tested recently32 by comparing the experimental dynamic viscosity measured with this equipment for squalane and the literature values.35-37 The comparison has been performed32 over a viscosity interval ranging from 9 to 56 mPa‚s, obtaining an average deviation of 1.7% and a maximum deviation of 3.2%. In addition, our values agree with the new recent measurements of Kumagai et al.38 with an AAD of 1.5%, a bias of -1.2%, and a maximum deviation of 3.2%. We must point out that, in a very recent paper, Harris and Bair39 and Bair40 have indicated that squalane may be a suitable candidate material for use as a high-viscosity reference liquid. Liquid mixtures were prepared by weight using a Sartorius LC 1200 S balance with an uncertainty of 1 × 10-3 g. The estimated uncertainty in the mole fraction is η(PEC5-C9) ≈ η(PEB8 + PEC7) ≈ η(PEB8 + PEC5) > η(PEC9) > η(PEC7) > η(PEC5). Viscosity-Pressure Coefficient. The viscosity-pressure coefficient, R, is widely used in the lubrication area. Several definitions of the viscosity-pressure coefficient can be found in the literature.43,44 Besides its technological importance, there is a scientific interest in the structural information given by this coefficient. Lubrication engineers often define the viscositypressure coefficient as the slope of lines on graphs of the logarithm of viscosity versus pressure. The pressure-viscosity coefficient can also be estimated from oil film thickness and other properties by using a transparent disk-on-ball apparatus. In the present work, the definition of the pressure-viscosity coefficient previously used by Dowson and Higginson45 has been employed:

R ) (1/η)(∂η/∂p)T

(2)

From this equation, the relation η ) η0 exp(Rp), known as the exponential model or the Barus relation, can be deduced if R is taken to be pressure independent. We should point out that, although the exponential model is know as the Barus relation, Barus46 found that the pressure-viscosity coefficient decreases with the pressure. The experimental viscosity data, for each mixture, at isothermal conditions, η(p), were fitted with a polynomial equation, 3

η(p) )

Aipi ∑ i)0

(3)

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Figure 2. Experimental dynamic viscosity against temperature at different pressures: ([) 0.1 MPa; ()) 10 MPa; (2) 20 MPa; (4) 30 MPa; (b) 40 MPa; (O) 50 MPa; (9) 60 MPa.

Figure 3. Experimental dynamic viscosity against pressure: (b) PEC5;33 (9) PEC7;33 ([) PEB8;32 (O) PEC5-C9; (0) PEB8 + PEC7; (]) PEB8 + PEC5. Table 1. Experimental Dynamic Viscosities, η, versus Temperature, T, and Pressure, p, for the Three Tested Mixtures η (mPa·s) at p ) T (K)

0.1 MPa

1 MPa

5 MPa

10 MPa

15 MPa

20 MPa

25 MPa

30 MPa

35 MPa

40 MPa

45 MPa

50 MPa

55 MPa

60 MPa

303.15 313.15 323.15 333.15 343.15 353.15

52.7 32.1 21.3 15.1 11.2 8.4

53.4 32.7 21.7 15.3 11.4 8.6

56.6 35.6 23.3 16.6 12.0 9.3

61.2 39.4 25.5 18.2 12.9 10.2

66.3 43.4 27.9 19.9 13.9 11.1

PEC5-C9 multicomponent mixture 72.1 78.6 85.9 47.8 52.5 57.7 30.4 33.2 36.1 21.7 23.6 25.6 14.9 16.1 17.4 12.0 12.9 13.8

94.3 63.0 39.2 27.7 18.8 14.7

103.6 68.8 42.6 29.9 20.3 15.7

114.2 75.0 46.2 32.2 21.9 16.6

126.1 81.7 50.0 34.7 23.7 17.5

139.5 88.9 54.2 37.2 25.7 18.5

154.7 96.5 58.6 39.9 27.8 19.4

303.15 313.15 323.15 333.15 343.15 353.15

51.7 32.6 20.7 14.2 10.9 7.1

52.7 33.3 21.1 14.4 11.1 7.2

57.4 36.1 22.9 15.4 11.7 7.7

63.7 39.9 25.2 16.7 12.7 8.3

70.4 43.8 27.6 18.1 13.7 9.1

xPEB8 + (1 - x)PEC7, x ) 0.6670 77.7 85.5 94.0 103.1 47.8 52.1 56.6 61.2 30.1 32.7 35.3 38.1 19.7 21.4 23.3 25.3 14.8 16.0 17.2 18.6 9.8 10.7 11.6 12.5

112.9 66.1 41.1 27.6 20.2 13.6

123.5 71.2 44.1 30.0 21.8 14.7

135.0 76.5 47.2 32.6 23.6 15.9

147.3 82.0 50.4 35.5 25.5 17.3

160.6 87.7 53.8 38.7 27.6 18.7

313.15 333.15

32.3 13.7

32.7 13.9

34.6 14.9

37.3 16.2

40.2 17.6

xPEB8 + (1 - x)PEC5, x ) 0.6911 43.5 47.2 51.3 55.9 19.1 20.7 22.4 24.3

61.0 26.3

66.7 28.4

73.1 30.7

80.1 33.2

88.0 35.8

where Ai are adjustable parameters. Hence, the (∂η/∂p)T derivative is given by the following expression:

() ∂η ∂p

T

3

)

iAipi-1 ∑ i)0

(4)

In parts a and b of Figure 4, this derivative has been plotted against pressure at several temperatures for the PEB8 + PEC7 and PEC5-C9 mixtures. As can be seen, the higher values are

reached for the PEB8 + PEC7 mixture. Taking into account the results of this work and those of our previous studies,32,33 the following trend is found: (∂η/∂p)T(PEB8) > (∂η/∂p)T(PEB8 + PEC7) > (∂η/∂p)T(PEC5-9) > (∂η/∂p)T(PEC9) > (∂η/∂p)T(PEC7) > (∂η/∂p)T(PEC5). Therefore, for the mixtures containing linear and branched PEs, the values of this derivative are between those of the pure linear esters and the branched esters. The pressure-viscosity coefficient, R, can easily be obtained after multiplication of the (∂η/∂p)T derivative and the inverse

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Figure 4. (∂η/∂p)T versus pressure for (a) PEC5-C9 and (b) PEB8 + PEC7: (9) 303.15 K, (b) 313.15 K, (2) 323.15 K, (0) 333.15, (O) 343.15 K, and (4) 353.15 K. Table 2. Pressure-Viscosity, r, at Some Pressures and Temperatures

mixture

R (GPa-1) at 10 MPa & 313.15 K

R (GPa-1) at 50 MPa & 313.15 K

R (GPa-1) at 10 MPa & 353.15 K

R (GPa-1) at 50 MPa & 353.15 K

PEC5-C9 PEC7 + PEB8

20.0 19.1

16.9 14.1

17.7 16.6

10.7 15.9

of the dynamic viscosity (Table 2). The influence of pressure on the pressure-viscosity coefficient R is illustrated in Figure 5 for the PEB8 + PEC7 and PEC5-C9 mixtures, together with the values for PEB8 and PEC7, where it can be seen that this coefficient decreases when the pressure rises, in the range studied. This behavior corresponds to that of a typical glassforming liquid at higher temperatures and at our studied pressure range, according to Bair et al.43 We should stress that it is often assumed in EHL simulations that R is pressure independent. Most organic fluids, including lubricant base oils, exhibit a reversible viscosity increase with the application of high pressure due to molecular mobility restrictions imposed by the forces being exerted.18 Hence, the (∂η/∂p)T derivative increases when the pressure rises, as can be seen in parts a and b of Figure 4. In addition, as the pressure increases, the viscosity, η, rises and η-1 diminishes. Taking into account that R ) (1/η)(∂η/∂p)T, it can be seen that, when the pressure rises, the decrease in R implies that the diminution of η-1 is enough to counteract the increase in (∂η/∂p)T. For the mixtures analyzed in the present work, the R values are in the range of 10-21 GPa-1 for the PEB8 + PEC7 mixture and 13-22 GPa-1 for the PEC5-C9 mixture. These data are comparable with literature values for other lubricants,47 for instance, trimethyolpropane esters (8.4-9.8 GPa-1 at 313.15 K and 6.9-9.5 GPa-1 at 333.15 K), pentaerythritol esters (8.312.2 GPa-1 at 313.15 K and 6.7-11.6 GPa-1 at 333.15 K), and diesters (6.6-6.8 GPa-1 at 313.15 K and 5.3-5.8 GPa-1 at 333.15 K). These last values have been determined by Chang et al.47 from measurements of the oil-film thickness considering the method of Foord et al.,48 which compares the film thickness generated by a test oil with that from a reference oil of known pressure-viscosity coefficient. Discrepancies on R values for the same lubricant are due to the different definitions of this coefficient43 and to the pressure range. In order to show the pressure effect on R values, the pressure-viscosity coefficient of two VG22 lubricants, pentaerythritol tetrahexanoate (PEC6) and squalane (paraffinic hydrocarbon), are plotted in Figure 6. These values have been determined from the viscosities of refs 40 and 43 by using eq 2. In this figure, it can be seen that there is a strong pressure dependence of the pressure-viscosity coefficient and that the R values for squalane are higher than

Figure 5. Pressure-viscosity coefficient, R, against pressure at 313.15 K: (9) PEC5-C9; (b) PEB8 + PEC7; (0) PEB8;32 (4) PEC7.33

Figure 6. Pressure-viscosity coefficient against pressure for (9) PEC643 and (0) squalane40 at 313.15 K.

those of PEC6. Furthermore, Gold et al.49 found an R value for a mineral oil ∼7 GPa-1 bigger than those of a biodegradable ester oil with the same viscosity grade (VG32). These authors found for this ester an R value ∼11 GPa-1 at 2000 bar for temperatures between 296.15 and 353.15 K, which is a value close to that found for PEC6 (Figure 6). This fact agrees with the conclusions of Randles,11 who indicated that ester lubricants generally have lower viscositypressure coefficients than mineral oils. From the analyses of Randles,11 Sharma and Stipanovic,18 and Anderin et al.,50 it can

Ind. Eng. Chem. Res., Vol. 46, No. 6, 2007 1831

be concluded that compounds whose molecules are large, inflexible, unsymmetrical, and stereochemically very bulky, and with low internal pressure, tend to present high viscositypressure coefficients. Gold et al.49 studied the behavior of several base lubricant types: mineral oils, polyalphaolefins, ester, polyalkylene glycol, and silicones. These authors found that, for a given type of oil, the R values increase with kinematic viscosity at atmospheric pressure.49 This is in agreement with the previous conclusions based on refs 11, 18, and 50 because kinematic viscosity is bigger for compounds whose molecules are large, rigid, and unsymmetrical. The R values of ester lubricants are influenced by the length of the side chains of the molecules (the longer the better) and the degree of branching (the more the better), as has been previously remarked by Randles.12 These dependencies agree with those found in our previous works,32,33 where it has been observed that the R values of pure PEs increase when the branching degree of the pentaerythritol ester increases; thus, PEB8 has the highest values over all the studied range. These observations are in agreement with the results found in the present work; thus, the multicomponent mixture PEC5-9 presents R values higher than those corresponding to the PEs with short chains (PEC5, PEC7) and lower than those of the heavier PEs. For the binary mixture PEC7 + PEB8, the R values are lower than those corresponding to PEB8, as can be observed in Figure 5. High-Pressure Viscosity Modeling As previously stressed, well-defined mixtures appear to be good to testing systems for the evaluation of existing viscosity models. Thus, the viscosity values of PEB8 + PEC5 and PEB8 + PEC7 binary mixtures, together with those of the PEB8, PEC5, and PEC7, can be used to verify the predictive ability of different models. Most of the models used employ only viscosity data for the pure compounds contained in the mixtures. In order to establish comparisons between experimental and theoretical values, the absolute average percentage deviation, AAD, the maximum percentage deviation, MD, and the average percentage deviation, bias, have been used according to their definitions, given in a previous paper.32 Ideal Mixing Laws. First, the ideal mixing law of Katti and Chaudhri22 has been used. This mixing rule only involves the viscosity data of the pure compounds and their mole fractions,

ln νm ) x1 ln ν1 + x2 ln ν2

(5)

where ν is the kinematic viscosity and x is the mole fraction. Subscript m refers to the mixture, whereas subscripts 1 and 2 refer to the pure compounds. In order to apply eq 5, the kinematic viscosity values of pure PEC5, PEC7, and PEB8 were determined from the dynamic viscosities measured in previous works32,33 and the densities of Fandin˜o et al.28-30,41 The results obtained for PEB8 + PEC7 and PEB8 + PEC5 mixtures are plotted in Figure 7. This mixing rule predicts the kinematic viscosities with an absolute average deviation (AAD) of 5% and a maximum deviation (MD) of 15% (at 353.15 K and 0.1 MPa) for the PEB8 + PEC7 mixture, whereas an AAD of 4% and an MD ) 7% (at 313.15 K and 0.1 MPa) are found for the PEB8 + PEC5 mixture. The mixing rule predicts kinematic viscosities higher than the experimental values (87 points of a total of 112 experimental data points). As can be seen in Figure 7, the relative deviations are, in most of the cases, lower than zero, which indicates that the experimental viscosities of these mixtures are slightly lower than those predicted by the Katti

and Chaudhri law. This fact may be the result of volume expansion (the excess molar volumes of these binary mixtures are positive41), which consequently leads to lower dynamic viscosity values. Nevertheless, according to the Eyring theory, eq 5 corresponds to the case in which the excess Gibbs activation energy is zero, and it is considered, in a certain sense, to be representative of the viscosity of ‘‘ideal” mixtures. Thus, it could be said that the viscosities of these mixtures show a quasi-ideal behavior. This fact could be expected because the components of the mixtures are all of the same chemical family: pentaerythritol esters. The mixing law of Grunberg and Nissan21 is also often used to predict the viscosity of mixtures,

ln ηm ) x1 ln η1 + x2 ln η2

(6)

This equation predicts the viscosity with an AAD of 5% and an MD ) 16% (at 353.15 K and 0.1 MPa) for the PEB8 + PEC7 binary mixture, whereas an AAD of 5% and an MD ) 7% (at 313.15 K and 0.1 MPa) are found for the PEB8 + PEC5 mixture. So, the use of kinematic viscosities instead of dynamic viscosities improves the quality of the predictions, but both mixing rules21,22 give satisfactory results for these mixtures, taking into account their simplicity and predictive features. Self-Referencing Method. Kanti et al.23 have developed an approach for modeling the viscous behavior of petroleum cuts, whose complex compositions are difficult to characterize. For commercial lubricants also, the composition is often unknown. For these kinds of fluids, it is very difficult to use equations based on physical properties such as molecular weight, critical parameters, or acentric factors, since they have to be known for each of the components. Kanti’s method has the advantage that it only requires one experimental determination at atmospheric pressure and temperature T0. For this reason, this approach is the named self-referencing method. It can be applied without restriction equally well to pure substances, to synthetic mixtures, or to complex systems such as petroleum cuts or lubricants. The formulation of this method is

ln

(

)

(

)

(p - pref) η(P,T) ) (ay2 + by + c) ln 1 + 2 + η(pref,T0) dy + ey + f (gy02 + hy0 + i)

( )

1 1 (7) T T0

where y ) y0 + (gy02 + hy0 + i)(1/T - 1/T0) and y0 ) ln[η(pref,T0)]. This equation involving nine parameters is used with p and pref in MPa and T0 and T in K. The viscosity value at the reference pressure and temperature, η(pref,T0), is utilized in mPa‚ s. The number of experimental viscosity data points considered for each pentaerythritol ester (PEC5,33 PEC7,33 PEC9,32 and PEB832) is 84. We subtract one reference point for each liquid, the viscosity value at the reference pressure 0.1 MPa and the reference temperature T0, which has been taken as 303.15 K. It is not advisable to apply the model using the coefficients adjusted by Kanti et al.,23 because the deviations are too high because of these parameters having been obtained with a database of linear alkanes and alkylbenzenes with a carbon number g 7, which is very different from those of pentaerythritol esters. Thus, in the present work, first, the parameters have been fitted for each pure PE (PEC5, PEC7, PEC9, and PEB8). The values of the parameters are reported in Table 3. The obtained AADs for each pure PE are 200 kg‚m-3) was proposed by Allal et al.26 This approach connects viscosity, η, to molecular structure via a representation of the free-volume fraction. In the recent version of this model,27 the viscosity has the following expression,

η ) η0 +

(

F l RF +

pM F

x3RTM

[( ) ]

) exp B RF + pMF RT

3/2

(12)

where M is the molar mass, F is the density, η0 is the dilutedgas viscosity term, for which we have used the expression proposed by Chung et al.,52 and l , R, and B are adjustable parameters for each fluid. These three characteristic parameters were determined for PEC5, PEC7, PEB8, and PEC9 using the viscosity and density values reported previously by Pensado et al.32,33 and Fandin˜o et al.,28-30 respectively. Table 4 shows the values obtained for these parameters. The AADs for the four PEs are always lower than the experimental uncertainty (3%). Furthermore, the free-volume model has been applied to predict the viscosities of the binary mixtures studied in this work, using the obtained parameters for pure PE. Among the several mixing rules proposed53,54 for this model, we have found the best results with the following expressions,

ln η0m ) x1 ln η01 + x2 ln η02

(13)

ln(lm) ) x1 ln(l1) + x2 ln(l2)

(14)

2

Rm )

∑ xixjRij

i,j)1

with Rij ) xRiRj

1/Bm ) x1/B1 + x2/B2

Table 4. Parameter Values and Deviations Obtained with the Free-Volume Model for Pure Pentaerythritol Esters pure fluid

l /Å

R/m5‚mol-1‚s-2

B

PEC5 PEC7 PEB8

0.154 0.138 4 0.027 7

449.03 606.15 818.13

0.002 187 0.001 541 0.001 416

deviations

AAD %

MD %

Bias %

PEC5 PEC7 PEB8

1.3 1.9 2.2

3.8 5.6 5.6

0.05 0.4 0.4

The results obtained for the dynamic viscosity predictions for the analyzed binary mixtures with the five different methods tested, namely, Katti and Chaudhri, Grunberg and Nissan, selfreferencing, hard-sphere, and free-volume, are compared in Figure 8. It is interesting to note that all methods predict dynamic viscosity values with AADs e 10% for the two binary mixtures over the entire temperature and pressure ranges, which is perfectly acceptable for engineering purposes. The maximum deviations for the PEB8 + PEC7 mixture range from 15% to 25%, whereas for PEB8 + PEC5, they are e15% for all methods. The free-volume approach is the model that best predicts the viscosities for both binary mixtures of PEs, followed by the ideal mixing law of Katti and Chaudhri. However, it should be stressed that the Grunberg-Nissan and KattiChaudhri mixing laws are entirely predictive for dynamic viscosities and kinematic viscosities, respectively, whereas from a practical point of view, for the other models, the knowledge of the densities for each composition, pressure, and temperature is often required. As previously mentioned, the free-volume model is interesting from a fundamental point of view because it gives some insight into the microstructure of these complex fluids. In the case of the self-referencing model, both utilized procedures have a similar prediction quality for the mixtures of known composition. The use of the parameters determined from the database containing the four pure PEs allows the determination of the viscosity of mixtures of unknown composition with good certainty.

(15) (16)

where 1, 2, and m subscripts are used for the pure compounds and the mixture, respectively. The predicted values with the free-volume model agree with the experimental data of Table 1 for PEB8 + PEC7 and PEB8 + PEC5 mixtures within an AAD of 4%, with the maximum deviations for these mixtures being 15% and 10%, respectively.

Figure 8. Deviations between the experimental values and the predicted data for the models analyzed: (black) AAD, (white) MD, and (gray) bias; (a) PEB8 + PEC7 and (b) PEB8 + PEC5.

1834

Ind. Eng. Chem. Res., Vol. 46, No. 6, 2007

Conclusions A total of 196 experimental dynamic viscosity measurements are reported for a multicomponent mixture of several PEs, PEC5-C9, and for two binary mixtures of pentaerythritol esters (PEB8 + PEC7 and PEB8 + PEC5). The following sequences have been observed for viscosity: η(PEB8) > η(PEC5-C9) ≈ η(PEB8 + PEC7) ≈ η(PEB8 + PEC5) > η(PEC7) > η(PEC5). The predictive capacities of different viscosity models have been analyzed for the binary systems. The use of kinematic viscosities instead of dynamic viscosities in the ideal mixing laws improves the quality of the predictions. The free-volume is the model that best predicts the viscosities for both binary mixtures of PEs, if adequate mixing rules for l , R, and B parameters are used, followed by the ideal mixing law of Katti and Chaudhri. Because of the similarity of the viscosity values, it can also be concluded that a binary mixture composed of two PEs with the same viscosity grade can be used to model a real lubricant. Parameters values for the self-referencing model are proposed for the estimation of viscosity values of PE lubricants of unknown composition, when a viscosity value at some temperature and pressure is available. Acknowledgment This work was supported by Spanish Science and Technology Ministry, FEDER PPQ2002-03262 and CTQ2005-09176-C0201/PPQ, and Xunta de Galicia (PGIDIT03PXIC20608PN). Equipment funding is also acknowledged from SXID, Xunta de Galicia. The participation of A.S.P. was made possible by a grant from SXID. We are very grateful to Dr. S. J. Randles (Uniqema) for his excellent advice about ester lubricants and for providing lubricant samples. The authors wish to express their gratitude to Prof. S. Bair (Georgia Institute of Technology, Atlanta, GA) for providing us with unpublished viscosity values for PEC6. Literature Cited (1) Willing, A. What lies Ahead? Challenges and Opportunities for the Lubricants Industry in the Next Decade. In 14th International Colloquium Tribology; Technische Akademie Esslingen: Esslingen, Germany, 2004, Vol. 1, pp 23-28. (2) Wang, D.; Xu, L.; Chang, M. Past, the Present and the Future of Green Lubricants. Runhuayou 2005, 20, 6. (3) Marsh, K. N.; Kandil, M. E. Review of Thermodynamic Properties of Refrigerants + Lubricant Oils. Fluid Phase Equilib. 2002, 199, 319334. (4) Rudnick, L. R.; Shubkin R. L. Synthetic Lubricants and HighPerformance Functional Fluids, 2nd ed.; M. Dekker: New York, 1999; Vol. 77. (5) Willing, A. Lubricants Based on Renewable Resources an Environmentally Compatible Alternative to Mineral Oil Products. Chemosphere 2001, 43, 89. (6) Boyde, S. Green Lubricants. Environmental Benefits and Impacts of Lubrication. Green Chem. 2002, 4, 293. (7) Yadav, G. D.; Doshi, N. S. Development of a Green Process for Poly-a-Olefin Based Lubricants. Green Chem. 2002, 4, 528. (8) Randles, S. J.; McTavish, S. J.; Dekleva T. W. A Critical Assessment of Synthetic Lubricant Technologies for Alternative Refrigerants. Presented at ASHRAE 2003 Winter Meeting, Chicago, IL, 2003. (9) Randles, S. J. Refrigeration Lubes. In Synthetic Lubricants and HighPerformance Functional Fluids, 2nd ed.; Taylor and Francis: New York, 1999. (10) Hinrichs, J. Lubricant screening for CO2 automotive AC-systems, aspects from a compressor manufacturer point of view. Presented at VDA Alternating Refrigerant Winter Meeting, Saalfelden, Austria, Feb 23-24, 2004. (11) Randles, S. J. Esters in Synthetic Mineral Oils and Bio-Based Lubricants: Chemistry and Technology; Taylor and Francis Ed.: Boca Raton, FL, 2006.

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ReceiVed for reView September 8, 2006 ReVised manuscript receiVed December 21, 2006 Accepted January 15, 2007 IE061187R