High Pressure Volumetric Properties and Viscosity of Base Oils Used

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Thermodynamics, Transport, and Fluid Mechanics

High Pressure Volumetric Properties and Viscosity of Base Oils Used in Automotive Lubricants and Their Modeling James S. Dickmann, Mark Devlin, John C. Hassler, and Erdogan Kiran Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b03484 • Publication Date (Web): 21 Nov 2018 Downloaded from http://pubs.acs.org on November 25, 2018

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Industrial & Engineering Chemistry Research

High Pressure Volumetric Properties and Viscosity of Base Oils Used in Automotive Lubricants and Their Modeling James S. Dickmann1, Mark T. Devlin2, John C. Hassler1, and Erdogan Kiran1* 1. Department of Chemical Engineering, Virginia Tech, Blacksburg, Virginia 24061 2. Afton Chemical Corporation, Richmond, Virginia 23219

Abstract

Temperature and pressure effects on density and viscosity are reported for six base oils consisting of four mineral oils and two synthetic oils composed of poly(alpha olefins). Using a variable-volume view-cell, density data were collected at 298, 323, 348, 373, and 398 K from 10-40 MPa. The data were then fit to the Sanchez-Lacombe equation of state and used to determine isothermal compressibility, isobaric thermal expansion coefficient, and internal pressure. Compressibility and internal pressure were found to vary based on composition, specifically with cycloalkane content. Viscosity data were collected as a function of temperature, pressure, and rotational speed using a custom-built high pressure rotational viscometer. Data were collected at 298, 323, 348, and 373 K from 10-40 MPa at rotational speeds of 300-800 rpm. A free volume model was used to model the viscosity and relate the viscous effects to density.

*Corresponding Author: Erdogan Kiran, [email protected]

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Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

I.

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Introduction The sections below provide the background regarding the (a) base oils and lubricants, (b)

determination of the volumetric properties and their modeling, and (c) the determination of viscosity and its modeling.

I.1 Base oils and lubricants in the automotive industry

There is great interest in the automotive industry to increase fuel efficiency. While there are numerous approaches to solve this problem, a more targeted one is to improve upon the effectiveness of the lubricants used.1 It has been reported in the literature that using lower viscosity lubricants can have an impact on fuel efficiency.2-4 These lubricants are exposed to high pressure conditions, often times in the range of GPas, and high shear rates up to 105 s-1.2,5,6 To fully understand the effectiveness of a fluid as a lubricant for automotive uses, high pressure viscosity and thermodynamic property data, particularly compressibility and internal pressure, are needed. Fully formulated engine oils and automatic transmission fluids (ATFs) are mixtures of a base oil with ten or more additives used to modify the physical properties of the fluid while also allowing the fluid to control deposit formation and reducing wear in mechanical systems. Examples of potential additives include viscosity index modifiers, detergents, dispersants, friction modifiers, and anti-wear modifiers.7-10 Despite the wide range of components used, the primary component of these lubricants are the base oils. Base oils are classified by the American Petroleum Institute. Table 1 summarizes this classification system. Group I-III oils are all mineral oils that are categorized based on composition and viscosity index. Viscosity index is determined by comparing kinematic viscosities at 40 oC and 100 oC to known standards. The higher the viscosity index, the lower the effect temperature has on the viscosity of the oil in question.11 Group I oils are generally created through simple solvent extraction of crude oils. Group II oils receive further hydrotreating to reduce sulfur content and increase the percentage of saturates contained in the base oil. Group III oils go through hydrocracking to open up ring structures and increase the viscosity index of the resulting base oil.12 Group IV oils are all synthetic, composed of poly(alpha olefins) (PAOs), while Group V oils are those that do not fit


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the other four categories. The base oils used in this study were oils belonging to groups II, III, and IV.

Table 1. Base oil categories as laid out by the American Petroleum Institute guidelines Group I Group II Group III Group IV Group V

Sulfur Content > 0.03% < 0.03% < 0.03% 0% NA

Saturates < 90% > 90% > 90% 100% NA

Viscosity Index 80-120 80-120 > 120 NA NA

Additional Information

Synthetic oils composed of PAOs All oils that don't fit into Groups I-IV

I.2 Sanchez-Lacombe equation of state and derived thermodynamic properties

The determination of the derived thermodynamic properties requires knowledge on the effect of both temperature and pressure on density. A variety of techniques have been employed in the literature to generate this required PVT data, such as vibrating densitometers or variablevolume bellows or cell system loaded with a known mass.2,13,14 The present work uses a variable-volume view-cell to determine the density of the oils being studied. Once PVT data have been generated, they need to be fit to an equation of state. The Tait equation is commonly employed but has its limitations.15 Due to being semi-empirical in its derivation, the meaning of many of the fitting parameters have limited physical significance. Another option for modeling high pressure PVT data is through the use of a lattice fluid model. By treating the fluid as fitting into a fixed lattice of both molecule chain segments and empty space, the Sanchez-Lacombe equation of state (S-L EOS) can be derived:16,17 1

𝜌𝜌�2 + 𝑃𝑃� + 𝑇𝑇� �ln(1 − 𝜌𝜌�) + �1 − 𝑟𝑟 � 𝜌𝜌�� = 0

Eq. 1

where 𝜌𝜌�, 𝑃𝑃�, and 𝑇𝑇� are reduced values of density, pressure, and temperature, respectively. The

parameter r represents the number of lattice sites filled: 𝜌𝜌

𝜌𝜌� = 𝜌𝜌∗ ,

𝑃𝑃 𝑃𝑃� = 𝑃𝑃∗ ,

𝑇𝑇 𝑇𝑇� = 𝑇𝑇 ∗ ,

𝑟𝑟 =

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𝑀𝑀𝑀𝑀𝑃𝑃∗ 𝑅𝑅𝑇𝑇∗ 𝜌𝜌∗

Eq. 2 3


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where ρ is density, P is pressure, T is temperature, MW is molecular weight, R is the gas constant, and ρ*, P*, and T* are the characteristic parameters. The S-L EOS can be put in terms of the interaction energy between molecular segments in each lattice (ε*), the close-packed volume of each segment (v*), and the number of segments in each molecule, previously defined as r. In conjunction with parameter r, the S-L EOS parameters can be related to these molecular parameters:16,17

𝑇𝑇 ∗ =

𝜀𝜀 ∗

𝜀𝜀 ∗

𝑃𝑃∗ = 𝑣𝑣∗

𝑘𝑘

Eq. 3

where k is Boltzmann’s constant. The way this model is derived is particularly relevant to modeling polymeric systems, with r standing in as similar in nature to chain length. Also relevant in choosing an equation of state is its effectiveness in modeling mixtures. Mixing rules can be employed to expand the use of the S-L EOS to multicomponent systems. This potential to model mixtures is needed as most engine oils and ATFs are not pure base oils, but mixtures with many different additives, including polymers. The application of mixing rules allows for the model to be expanded from just correlating density to temperature and pressure, but also composition.17,18 With the PVT data fit to an appropriate equation of state, derived thermodynamic properties can be calculated: isothermal compressibility, isobaric thermal expansion coefficient, and internal pressure. Isothermal compressibility and isobaric thermal expansion coefficient are related to the partial derivatives of volume with respect to either pressure or temperature:16,19 1 𝜕𝜕𝜕𝜕

1 𝜕𝜕𝜕𝜕

𝜅𝜅𝑇𝑇 = − 𝑉𝑉 �𝜕𝜕𝜕𝜕� = 𝜌𝜌 �𝜕𝜕𝜕𝜕� = 1 𝜕𝜕𝜕𝜕

𝑇𝑇

1 𝜕𝜕𝜕𝜕

𝑇𝑇

1 1 𝑃𝑃�𝑇𝑇� 𝑣𝑣��−1+ �−2�

𝑃𝑃

1 1 𝑇𝑇�𝑇𝑇� 𝑣𝑣��−1+ �−2�

𝛽𝛽𝑃𝑃 = 𝑉𝑉 �𝜕𝜕𝜕𝜕 � = − 𝜌𝜌 �𝜕𝜕𝜕𝜕� = 𝑃𝑃

𝑃𝑃� 𝑣𝑣� 2

𝑣𝑣

𝑟𝑟

1+𝑃𝑃� 𝑣𝑣� 2 𝑣𝑣

Eq. 4 Eq. 5

𝑟𝑟

Based on compressibility and the expansion coefficient, internal pressure can be calculated:2 𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝛽𝛽

𝜋𝜋 = �𝜕𝜕𝜕𝜕 � = 𝑇𝑇 �𝜕𝜕𝜕𝜕 � − 𝑃𝑃 = 𝑇𝑇 �𝜅𝜅𝑃𝑃 � − 𝑃𝑃 𝑇𝑇

𝑉𝑉


Eq. 6

𝑇𝑇

4

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Internal pressure is a measure of the overall attractive and repulsive interactions in the system.2,20,21

I.3 Viscosity measurements and free volume

An important parameter for the determination of lubricant effectiveness is viscosity. While ambient pressure viscosity values are commonly determined for base oils used in automotive lubricants, very little high pressure data has been collected.2 Common techniques for the acquisition of viscosity data under pressure include rolling ball/falling body viscometers, capillary tube viscometers, and vibrational/oscillating techniques.22-25 While all of these do have a calculable shear rate associated with the viscosity measurements, due to limitations of design, controlled variation of shear rate across a wide range is not a simple task. For these measurements, rotational viscometers have an advantage. Unfortunately, building a rotational viscometer that can operate under high pressure conditions is difficult. While high pressure rotational viscometers have been used before in the literature, they are not a common experimental setup.26-28 To carry out these experiments, we have designed our own unique high pressure rotational viscometer capable of measuring viscosity up to 160 mPa s, at rotational speeds up to 800 rpm (shear rates of up to 1800 s-1), at temperatures up to 373 K, and pressures up to 60 MPa. Viscosity can be related to density effects through the concept of free volume. Understanding how fluid density effects viscosity as both temperature and pressure are changed is necessary for the formation of a complete understanding of viscous effects under pressure. One early relationship between viscosity and density by means of free volume is the Doolittle equation:29,30 𝐵𝐵

𝜂𝜂 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 �1−𝑉𝑉 𝜌𝜌�

Eq. 7

0

where A and B are constants and V0 is the close-packed volume of the fluid. The Doolittle equation is empirical in nature. A later model by Allal et al.30,31 provides a more rigorous



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treatment of the effect of free volume on viscosity. The model equation associates a greater degree of physical significance to the constants that are involved in describing the viscosity:

𝜂𝜂 = 𝜂𝜂0 +

𝑃𝑃𝑃𝑃𝑃𝑃 𝜌𝜌𝜌𝜌�𝛼𝛼𝛼𝛼+ � 𝜌𝜌

√3𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅

𝑒𝑒

3⁄2 𝑃𝑃𝑃𝑃𝑃𝑃 𝛼𝛼𝛼𝛼+ 𝜌𝜌 � 𝐵𝐵� 𝑅𝑅𝑅𝑅

Eq. 8

Here η0 is the viscosity of the fluid at infinite dilution, ι is the characteristic molecular length, α is needed to calculate intermolecular energy, and B represents free volume overlap. Since the oils that are being studied are in the liquid phase with relatively high viscosities, it has been assumed that η>>η0, reducing the equation to:

𝜂𝜂 =

𝑃𝑃𝑃𝑃𝑃𝑃 𝜌𝜌𝜌𝜌�𝛼𝛼𝛼𝛼+ � 𝜌𝜌

√3𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅

𝑒𝑒

3⁄2 𝑃𝑃𝑃𝑃𝑃𝑃 𝜌𝜌 � 𝑅𝑅𝑅𝑅

𝛼𝛼𝛼𝛼+

𝐵𝐵�

Eq. 9

The number of parameters needed to model viscosity is thus reduced to three (ι, α, and B). All of these fits assume either zero shear viscosity is used, or that the fluid being examined is Newtonian.

I.4 Objectives

In the present study we have generated comprehensive data sets on the volumetric properties and viscosity of mineral and synthetic oils of significance that are used as lubricants in the automotive industry. The data covers a range of temperatures from 298 to 398 K and pressures from 10-40 MPa. The objective was to develop and test the possibility of a holistic model that can describe their thermophysical properties and link with viscosity. For this purpose, density data were generated using a variable volume-view high pressure view-cell and modeled using the Sanchez-Lacombe equation of state which was then used to determine the derived thermodynamic properties, namely isothermal compressibility, isobaric expansivity, and internal pressure which provides a measure also of the solubility parameters. The viscosity data were generated as a function of temperature, pressure and shear rate (rotational speed) using a 6 ACS Paragon Plus Environment

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specially developed high-pressure rotational viscometer. The data were then described using a free volume model in combination with the Sanchez-Lacombe equation of state description of the volumetric data in assessing the free-volume fractions. The experimental data and the results of the models are presented in the following sections.

II.

Materials and Methods

II.1 Materials

Six base oils were provided by Afton Chemical Corp. The oils studied can be classified by group type, as laid out in Table 1. Two Group II oils, two Group III oils, and two Group IV oils: IIA with a 373 K kinematic viscosity, viscosity index, and average molecular weight of 4.1 cSt, 103, and 354 g/mol, respectively; IIB (6.4 cSt, 103-109, and 445 g/mol); IIIA (3.1 cSt, 112, 333 g/mol); IIIB (6.5 cSt, 131, 474 g/mol); PAO 4 (4.1 cSt, 126, 489 g/mol); PAO 8 (7.9 cSt, 139, 526 g/mol) as laid out in Table 2. All four of the Group II and Group III oils are mineral oils. Group IV oils are all synthetic oils composed of poly(alpha olefins). Kinematic viscosities at 373 K were determined by standard ASTM D445, as reported by the manufacturers. Viscosity indexes are reported as specified by the manufacturers. All six oils were used as received. Compositions and average molecular weights of all six oils were determined by gas chromatography-mass spectroscopy done by Triton Analytics with the data provided by Afton Chemical. This GC-MS technique has been previously reported in the literature.32 Figures 1-3 show the composition of all six oils in terms of paraffins, cycloalkanes, and various aromatics. Both IIA and IIB are composed of approximately 30 wt% paraffins, nearly 60 wt% cycloalkanes, and the remainder mono- and diaromatics. IIIA and IIIB are composed of approximately 50 wt% paraffins, approximately 46 wt% cycloalkanes, and the remainder mono- and diaromatics. PAO 4 and PAO 8 are composed of 100 wt% paraffins as they are synthetic oils composed completely of poly(alpha olefins).



Table 2. Characteristics of the base oils studied. Kinematic Viscosity at 373 K

Viscosity

(cSt)

Index

IIA

4.1

103

354

IIB

6.4

103-109

445

IIIA

3.1

112

333

IIIB

6.5

131

474

PAO 4

4.1

126

489

PAO 8

7.9

139

526

70

50 40 30 20

Average Molecular Weight (g/mol)

70

IIA

60

Mass Percent

60

Mass Percent

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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IIB

50 40 30 20

10

10

0

0

Figure 1. Composition of base oils IIA (left) and IIB (right).


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60

60

IIIA

IIIB 50

Mass Percent

Mass Percent

50 40 30 20

40 30 20

10

10

0

0

Figure 2. Composition of base oils IIIA (left) and IIIB (right).

120

120

PAO 4

PAO 8 100

Mass Percent

100

Mass Percent

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


80 60 40

80 60 40

20

20

0

0

Figure 3. Composition of base oils PAO 4 (left) and PAO 8 (right).

II.2 Variable-Volume View-Cell

A variable-volume view-cell was used to determine the densities as a function of both temperature and pressure for all six base oils. Although the details of this instrument have been 9 ACS Paragon Plus Environment


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more fully described in earlier publications,33,34 a brief description is provided here. Figure 4 shows the basic layout of the experimental apparatus. A pair of sapphire windows allows for visual observation or the use of optical techniques in determining if phase separation occurs during an experiment. By passing light through the cell to a photosensor on the other side, phase separation can be detected by looking at the change in transmitted light intensity. The instrument also contains a variable-volume portion. Back pressure is applied to a movable piston to control both the volume and pressure inside the view-cell. A magnetic core is attached to the piston, allowing its position to be determined by a linear variable differential transformer. This allows volume in the view-cell to be determined to within ± 0.1 cm3 across a range of 11 to 23 cm3. Temperature and pressure of the fluid are measured in the instrument by a J-type thermocouple (± 1.1 K) and a Dynisco diaphragm pressure transducer (± 0.07 MPa). Back pressure applied to move the piston was applied by a motorized pressure generator allowing for the continuous collection of data during a pressure scan. A typical run consists of collecting density data as pressure increases at a set temperature. These isothermal pressure scans ensure that hundreds of data points are generated at each temperature. The instrument is loaded by pumping in a sample from a secondary container. The mass is measured by a Mettler PM6100 balance with a 0.01 g accuracy. Around 12-15 g of sample are loaded in any given experiment. Density is determined based off this initial mass loading and the measured volume to within 1% uncertainty. Density data for oils IIA, IIB, IIIA, and IIIB were generated from 10-40 MPa at five different isotherms: 298, 323, 348, 373, and 398 K. Pressure scans were performed at a rate of 0.4 MPa/s at each isotherm. The density data for PAO 4 and PAO 8 have been previously reported by Grandelli et al.2 All density values in this study are from a single run for each oil.


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Figure 4. Diagram of the variable-volume view-cell. MPG = motorized pressure generator; LVDT/PPS = linear variable differential transformer/piston position sensor; VVP = variable volume portion; MP = movable piston; PS = photosensor; LS = light source; VC = view-cell; TV = transfer vessel; Itr = transmitted light intensity2

II.3 High Pressure Rotational Viscometer

A high pressure rotational viscometer has been developed to examine the effect of temperature, pressure, and rotational speed (shear rate) on the viscosity of high viscosity fluids such as the base oils discussed in this paper. Figures 5-7 show the basic layout of both the inside and outside of the experimental apparatus, including the inner geometries of the instrument. Similar to the variable-volume view-cell, two pairs of sapphire windows are used to visually observe phase behavior in a system being studied. Additionally, two pistons with LVDTs (to monitor piston position) are used to apply pressure to the measured sample. An Omega PX91N0-60KSV pressure transducer, rated for pressures up to 400 MPa, (±0.4 MPa) and a Jtype thermocouple (± 1.1 K) are used to measure pressure and temperature of the sample, respectively. The thermocouple is installed in the instrument with direct contact with the measured fluid. A 60,000 psi pressure generator (PGN) purchased from the High Pressure Equipment Company was used to apply back pressure to the pistons. Viscosity was measured by recording the torque required to rotate the shaft at a certain rotational speed in the fluid being examined. The internal rotating shaft is magnetically coupled to the torque transducer/motor combination external to the high pressure cell. Due to this magnetic coupling system, the need to seal across moving parts at high pressure is eliminated. 11 ACS Paragon Plus Environment


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The upper component of a ThermoFisher Haake Viscotester 550 was used as the torque transducer/motor (± 0.00015 N m). A rotating portion with two neodymium magnets was attached directly to the motor, which in turn is coupled to a single neodymium magnet in the internal shaft. Calibration was carried out using Cannon Instrument Company, located in State College, PA, standards RT5, RT10, RT50, and RT100, which are all silicone oils. Calibration was done at ambient pressures. While the basic equation used to convert torque to viscosity for these silicone oils is based off of known equations for a Newtonian fluid in a rotational viscometer, a correction factor was needed due to complex interactions with the partially nonuniform geometry of the cavity (arising from the inclusion of sapphire windows and pistons) and frictional effects. The equation used to generate viscosity data is as follows: 𝜏𝜏−𝐵𝐵

𝜂𝜂 = 𝐴𝐴 �

Ω

�

Eq. 10

where η is viscosity, τ is torque, and Ω is rotational speed. In addition to compensating for frictional effects, there are magnetic effects that need to be accounted for. Magnetic eddy currents are generated by the rotating magnetic field, which in turn generate magnetic fields detected by the torque transducer. This reading is sensitive to temperature and needs to be taken into account with a calibration run at each combination of sample and temperature isotherm. A run is performed at each temperature at low pressure (below 3.5 MPa) from 300-800 rpm for a duration of one minute at constant temperature and pressure for each rotational speed. A linear fit between average torque at each rpm versus rpm is generated and the intercept is subtracted from all torque values before conversion to viscosity using Equation 10. Viscosity is calculated with an uncertainty of ± 1.5 mPa s. Figure 8 shows this process for PAO 4 at 298 K and 500 rpm. All six oils were evaluated at pressures from 10-40 MPa at 300-800 rpm, which correspond to shear rates of between 480 to 1270 s-1. Shear rates were calculated using the diameters of the body and shaft that are present for the majority of the length of the cylindrical portion of the instrument. While there is a portion of the shaft that has a wider diameter due to the incorporated magnet, this portion is a fraction of the size of the rest of the cylindrical portions of the instrument. While the length of the shaft is 136.9 mm, there are portions that overlap with 12 ACS Paragon Plus Environment

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the piston and window cavities. With the length of these cavities removed, the overall interactive length of the shaft is 95.9 mm. Figure 7 shows these measurements. These runs were performed at isotherms of 298, 323, 348, and 373 K. Pressure scans were performed at a rate of 0.27 MPa/s. The instrument was heated by four electric heating cores imbedded in the main body of the instrument. An Omega Engineering CN78000 temperature controller was used to regulate the temperature of the system. The instrument was not run at 398 K as with the density experiments due to the instrument being limited to a maximum operating temperature of 373 K. At lower RPMs, the noise passes an acceptable threshold, which is a scatter of greater than ± 3 mPa s. This was ignored in the case of select 298 K runs where the magnets decoupled due to high torques needed to rotate the inner shaft. All viscosity values used in this study are from a single run for each oil.

Torque Transducer/Motor

LVDT Thermocouple PGN

J Sapphire Windows

LVDT

PGN

Pressure Transducer

Figure 5. External diagram of the high pressure rotational viscometer.



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Coupling Magnets Hole LVDT Thermocouple Piston

LVDT

Magnetic Core

Figure 6. Internal diagram of the high pressure rotational viscometer.

23.8 mm 23.6 mm 16.7 mm

20.5 mm

14.1 mm

152.4 mm 136.9 mm

54.0 mm

18.9 mm

18.7 mm

Figure 7. Internal geometries of the high pressure rotational viscometer.


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0.018

0.018

PAO 4 298 K

0.016 0.014 0.012 0.01 0.008

y = 2E-05x + 0.0004 R² = 0.9998

0.006 0.004

Original

0.016 Corrected

0.015 0.014 0.013

0.002 0

PAO 4 298 K 500 rpm

0.017

Torque (N m)

Torque (N m)

0

200

400

600

0.012

800

Torque Correction

0

Rotational Speed (rpm) 70

10

20

30

40

50

Pressure (MPa)

PAO 4 298 K 500 rpm

65

Viscosity (mPa s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


60 55 50 45 40 35 30

0

10

20

30

40

50

Pressure (MPa) Figure 8. Evaluation of a sample run in the high pressure rotational viscometer. Torque values are compared to rotational speeds at low pressures to generate a calibration plot (top left). This baseline is then subtracted from the torque values from a run (top right). Corrected torque is converted to viscosity (bottom).



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II.4 Density and Viscosity Modeling

Density data for all six oils were modeled with the Sanchez-Lacombe equation of state (Equations 4 and 5). A Python® program was written to perform the fits and generate the S-L parameters P*, T*, and ρ*. The R2 values were calculated for each oil. Additionally, Equation 9 was used to model the relationship between viscosity and density. Since the available experimental density data was not collected at the exact temperature and pressure points as the viscosity data, the model equation was tied to the S-L EOS to generate the densities used in the model fit. A Python® program was set up to solve for the model parameters based on the experimental viscosity values and S-L generated densities. Root mean squared deviation (RMSE), percent absolute average deviation (%AAD), and the quantity β as an evaluation of the distribution of data points around the fit were all calculated for the viscosity fits:

∑(𝜂𝜂𝑐𝑐,𝑖𝑖 −𝜂𝜂𝑖𝑖 )2

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = �

𝑑𝑑𝑖𝑖 = �1 −

𝜂𝜂𝑐𝑐,𝑖𝑖 𝜂𝜂

𝑛𝑛

� ∗ 100% 1

% 𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑛𝑛 ∑|𝑑𝑑𝑖𝑖 | 1

ℬ = 𝑛𝑛 ∑ 𝑑𝑑𝑖𝑖

Eq. 11 Eq. 12 Eq. 13 Eq.14

where ηc is the calculated viscosity and n is the number of data points involved with the fits. All viscosity fits used data from the 500 rpm runs (with the exception of runs at 298 K where viscosity is high and decoupling occurred at these high rotational speeds). Additionally, values for RSME, %AAD, and β were calculated for the S-L EOS fits of the density data. III.

Results and Discussion

III.1 PVT data

Density values for base oils IIA, IIB, IIIA, and IIIB were generated from 10-40 MPa at isotherms 298, 323, 348, 373, and 398 K. In addition, density values for PAO 4 and PAO 8 16 ACS Paragon Plus Environment

Page 17 of 32

across the same range of temperatures and pressures from the literature were used.2 Figure 9 shows the generated PVT data for the base oil IIB across the full temperature and pressure range. Density data for the five remaining oils are included in the supporting information. Data for all six oils were used to generate fits using the Sanchez-Lacombe equation of state. Table 3 shows the fitted parameters. All fits had R2 values of 0.993 or greater. Figure 9 shows the comparison of the S-L fit for IIB to the original data. The remaining fits are included in the supporting information. Figure 10 shows a comparison of all six oils and their respective S-L fits at both 323 and 373 K. Oils IIA and IIB have measured densities higher than the other four oils, with IIB having the highest values. IIIB has slightly higher densities than IIIA, PAO 4, and PAO 8.

Table 3. Sanchez-Lacombe parameters of base oils P* (MPa) T* (K) rho* (g/cm3) MW (g/mol) RMSE (g/cm3) % AAD %ℬ

IIA 506.42 553.4 0.95617 354 0.00140 0.130 -0.00000860

IIB 501.44 556.03 0.96258 445 0.00176 0.171 -0.104 0.93

IIIA 488.06 538.91 0.91726 333 0.00157 0.160 -0.000579

IIIB 499.53 548.92 0.91979 474 0.00156 0.156 0.0000785

PAO 4 433.62 544.10 0.91160 489 0.00198 0.201 -0.000177

PAO 8 418.46 548.24 0.90929 526 0.00198 0.204 0.000285

IIB

0.91 298 K

Density (g/cm3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.89

323 K

0.87

348 K

0.85

373 K

0.83

398 K

0.81 0.79

0

10

20

30

40

50

Pressure (MPa) Figure 9. Density versus pressure for the base oil IIB at isotherms 298, 323, 348, 373, and 398 K. Sanchez-Lacombe EOS fits are represented by black dots. 17 ACS Paragon Plus Environment


0.86

0.91

323 K

373 K

0.89

0.84

0.87

IIB

IIA

0.85

IIIB PAO 8 IIIA

0.83 0.81

PAO 4

0.82

IIIB PAO 8 PAO 4

0.8 0.78

IIIA

0.76

0.79 0.77

IIA IIB

Density (g/cm3)

Density (g/cm3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 32

0

10

20

30

40

50

0.74

0

Pressure (MPa)

10

20

30

40

50

Pressure (MPa)

Figure 10. Density versus pressure for six base oils at 323 K (left) and 373 K (right). SanchezLacombe EOS fits are represented by black dots.

III.2 Derived thermodynamic Properties

Once an equation has been fitted to the experimental PVT data, derived thermodynamic properties can be calculated using Equations 4-6. Isothermal compressibility, isobaric thermal expansion coefficient, and internal pressure were calculated for all six base oils. Figures 11 and 12 show the calculated properties across all temperatures and pressures (298-398 K and 10-40 MPa) for IIB. Figures of all other calculated data are included in the supporting information. For all six oils, in the measured range of densities, isothermal compressibility and isobaric thermal expansion coefficient increase with temperature and decrease with pressure. Internal pressure decreases with temperature and increases with pressure.


Page 19 of 32

0.0011

0.0014

IIB

IIB

0.0012

20 MPa

βP (1/K)

0.0009

0.0008

398 K 373 K

0.0006

348 K 323 K

0.0004

30 MPa 40 MPa

0.0008 0.0007

298 K

0.0006

0.0002 0

10 MPa

0.001

0.001

κT (1/MPa)

0

10

20

30

40

0.0005

50

250

300

Pressure (MPa)

350

400

450

Temperature (K)

Figure 11. Isothermal compressibility versus pressure (left) and isobaric thermal expansion coefficient versus temperature (right) for IIB as calculated from the S-L EOS.

480

IIB

460

Internal Pressure (MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


440

298 K

420

323 K

400

348 K 373 K

380

398 K

360 340 320 300

0

10

20

30

40

50

Pressure (MPa) Figure 12. Internal pressure versus pressure for IIB calculated from the S-L EOS.

When comparing isothermal compressibility across all six base oils, some trends begin to appear. Across all temperatures and pressures, the compressibilities are in the order of PAO 8 ≈ 19 ACS Paragon Plus Environment


PAO 4 > IIIA > IIIB > IIA > IIB with the differences between IIIB, IIA, and IIB being low. While the calculated compressibilities are higher than those reported by Guimarey et al., 35 the trend of oil order appears to be in agreement, with a poly(alpha olefin), PAO 6, having higher values for compressibility that the reference Group III oil used in that work. The discrepancies in values for compressibility may be due to the method of calculation, this work calculated values from an equation of state, while Guimarey et al. calculated their values from density and speed of sound measurements.35 Figure 13 shows the comparison of isothermal compressibility for all six oils at both 323 and 373 K. When the comparison is made between oils and their composition, specifically cycloparaffin content, is taken into account, it can be seen that compressibility appears to drop as the concentration of cyclic compounds in the oil increases. Figure 14 shows the effect of cycloparaffin content on compressibility at 373 K and 10 MPa.

0.0014

0.0009

323 K 0.0008

373 K

0.0013 0.0012

PAO 8

0.0007

κT (1/MPa)

PAO 4

κT (1/MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 32

IIIA IIA

0.0006 0.0005 0.0004

0

10

20

30

0.001

0.0008

IIB

0.0007 50

IIA

0.0009

IIIB

40

PAO 8 PAO 4 IIIA

0.0011

0.0006

IIIB IIB

0

Pressure (MPa)

10

20

30

40

50

Pressure (MPa)

Figure 13. Isothermal compressibility versus pressure for six base oils at 323 K (left) and 373 K (right).


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.0014

373 K 10 MPa

0.0013

κT (1/MPa)

Page 21 of 32

0.0012 0.0011 0.001 0.0009 0.0008

0

20

40

60

80

Mass % Cycloparaffin Figure 14. Isothermal compressibility versus mass percent cycloparaffin content at 373 K and 10 MPa for all base oils.

The effect of base oil composition on isobaric thermal expansion coefficient is not as noticeable with the number of data points available. With the exception of oil IIIA, which has higher expansion coefficients than the rest, expansion coefficients for the base oils studied here do not appear to vary as much as isothermal compressibility. The effect of temperature on fluid expansion does not appear to vary with composition. Figure 15 shows the isobaric thermal expansion coefficient for all six oils at 10 and 40 MPa. Figure 16 shows the effect of composition on expansion coefficients at 373 K and 10 MPa.



0.001

0.0012

40 MPa

10 MPa 0.0011

IIIA IIIB IIA PAO 8

0.0009

βP (1/K)

0.001

βP (1/K)

0.0009 0.0008

0.0008 0.0007 IIIB PAO 4

0.0006

IIB

0.0007 0.0006

IIIA IIA IIB PAO 8

PAO 4

250

300

350

400

450

0.0005

250

Temperature (K)

300

350

400

450

Temperature (K)

Figure 15. Isobaric thermal expansion coefficient versus temperature for six base oils at 10 MPa (left) and 40 MPa (right).

0.0011

373 K 10 MPa

0.00108 0.00106 0.00104

βP (1/K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 32

0.00102 0.001 0.00098 0.00096 0.00094 0.00092 0.0009

0

20

40

60

80

Mass % Cycloparaffin Figure 16. Isobaric thermal expansion coefficient versus mass percent cycloparaffin content at 373 K and 10 MPa for all base oils.


Page 23 of 32

Once compressibility and expansion coefficients have been calculated, values for internal pressure can be generated. The internal pressure shows noticeable changes across the six oils studied: IIA > IIB > IIIB > IIIA > PAO 4 > PAO 8 with the greatest drop occurring between IIIA to PAO 4. Figure 17 shows internal pressure for all six oils at both 323 and 373 K. When these variations in internal pressure are put in terms of composition, a trend appears with internal pressure increasing with cycloparaffin concentration. This effect can be seen in Figure 18. The trends as seen using the S-L EOS for pressure effects on internal pressure for PAO 4 and PAO 8 appear to be inverse of those seen in a previous work.2 Grandelli et al. used individual fits for each isotherm and isobar to generate compressibility and thermal expansivity.2 This method of fitting the data does not fully take into account both pressure and temperature effects simultaneously as a more holistic fit would. A high degree of sensitivity leading to issues with the observed trends reveals the importance of picking a model that fits well to the experimental PVT data.

440

400

420

IIIB IIB

400

IIIA

380 PAO 4

360

PAO 8

340 320

0

10

20

30

40

373 K

390

IIA

50


323 K Internal Pressure (MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


IIA IIIB

380 370

IIIA

IIB

360 350 340

PAO 4

330

PAO 8

320 310 300

0

Pressure (MPa)

10

20

30

40

50

Pressure (MPa)

Figure 17. Internal pressure versus pressure for six base oils at 323 K (left) and 373 K (right).



410


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 32

373 K 10 MPa

390 370 350 330 310 290 270 250

0

20

40

60

80

Mass % Cycloparaffins Figure 18. Internal pressure versus mass percent cycloparaffin content at 373 K and 10 MPa for all base oils.

Both isothermal compressibility and internal pressure are affected by composition, specifically in terms of cycloparaffins. The effect of cycloparaffin content on the thermal expansion coefficient appears to be negligible. Based on Equation 4, isothermal compressibility is proportional to the partial derivative of density with respect to pressure. Having ring structures in the oil mixtures appears to allow for greater packing, decreasing the ability of the oil to be compressed, potentially affecting the formation of a film under pressure. Temperature effects appear to be relatively unaffected by the increased packing of these cyclic structures based on Figure 16 compared to Equation 5. As internal pressure is inversely related to isothermal compressibility (Equation 6), the effect of cycloparaffin concentration on this property is inversed, with internal pressure increasing with cyclic structure content. As the intermolecular packing becomes more optimal, internal pressure increases, signifying that the overall attractive forces are increasing.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


III.3 Viscosity and free volume effects

High pressure viscosity data were collected for all six base oils at isotherms 323, 348, and 373 K across a pressure range of 10-40 MPa at rotational speeds of 300-800 rpm (shear rates of 480 to 1270 s-1). For the 298 K isotherms, data was collected at rotational speeds from 300-800 rpm for the oil IIIA, 300-700 rpm for the IIA and PAO 4 oils, and 100-400 rpm for IIB, IIIB, and PAO 8 oils. At low temperatures, the viscosities were high enough in specific base oils that the torque values needed to rotate the inner shaft at higher rotational speeds could not be achieved without the decoupling of the torque transferring magnets. Figure 19 shows an example of the collected viscosity data for IIB at 500 rpm (300 rpm at 298 K), which corresponds to a shear rate of 1130 s-1. Figures for the viscosity of the other oils used in this study at 500 rpm are shown in supporting information. All six oils were found to be Newtonian in behavior at the shear rates tested. Figure 20 shows an example of the Newtonian behavior observed for all six oils in IIB at 323 K. The viscosities determined at each rotational speed overlap with each other, demonstrating that viscosity is independent of shear rate.

200

IIB 500 rpm

180 160

Viscosity (mPa s)

Page 25 of 32

140 120 100

298 K (300 rpm)

80 60 40

323 K 348 K 373 K

20 0

0

10

20

30

40

50

Pressure (MPa) Figure 19. Viscosity versus pressure for base oil IIB at 500 rpm (300 rpm for the 298 K run).



45

0.35

IIB 323 K 300-800 RPM

35 30 25 20

IIB 323 K 10 MPa

0.3

Shear Stress (N/m2)

40

Viscosity (mPa s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 32

0.25 0.2

R² = 0.9995

0.15 0.1 0.05

0

10

20

30

40

50

0

0

Pressure (MPa)

500

1000

1500

Shear Rate (1/s)

Figure 20. Viscosity versus pressure (left) and average shear stress at 10 MPa versus shear rate (right) for IIB at 323 K. All data points at rotational speeds from 300 to 800 rpm are represented in the figure on the left. The linearity of shear stress versus shear rate shows the oil is Newtonian in behavior.

Viscosity data collected at 500 rpm (except certain runs at 298 K) were fitted to a free volume correlation (Equation 9). This free volume correlation relates viscosity to density through an equation first reported in the literature by Allal et al.30 The fits were generated using density values generated using the Sanchez-Lacombe EOS (Equations 1 and 2) and the parameters in Table 3. By coupling this viscosity model with an appropriate equation of state, a more holistic model for relating viscosity to temperature and pressure effects is generated. Both PVT and viscosity data can be modeled at any temperature and pressure within the appropriate range the models cover, allowing direct comparisons between viscosity, density, and various thermodynamic properties. Table 4 shows the parameters for all six base oils for this free volume correlation to viscosity. Also reported are the root mean squared deviation, percent absolute average deviation, and the distribution of the data points around the model fit. While the majority of the fits had values for %AAD at 10% or lower, two of the lower viscosity oils had particularly high errors: IIA and IIB. This could be due to issues with the 373 K data. At 373 K, the viscosity values of these two oils drop below 3 mPa s, which is at the limits of the


Page 27 of 32

sensitivity of the high pressure rotational viscometer. Figures 21 and 22 compare the model fits of Equation 9 to actual viscosity data of the six base oils used in this study. While composition does affect viscosity, there was no clear trend with regards to cycloparaffin content as with the derived thermodynamic properties. One potential explanation is that viscosity is more dependent on molecular weight with the six base oils having molecular weights ranging from 333 to 526 g/mol. In any case, viscous effects are not simply based on the same mechanisms as compressibility.

Table 4. Parameters for the free volume correlation to viscosity (Equation 9). IIA

IIB

IIIA

IIIB

PAO 4

PAO 8

L (cm) X 105

8.09

2.49

18.7

12.2

31.8

10.3

α (MPa*cm6/g*mol)

413600

1089000

447200

638600

728400

936200

B X 103

4.44

1.17

3.480

2.46

1.57

1.44

RMSE (mPa s)

0.845

1.21

0.735

0.922

1.06

1.79

% AAD

28.8

10.0

25.3

5.07

8.98

5.53

%ℬ

-23.5

8.15

-20.5

-0.885

5.28

4.15

200

IIB 500 rpm

180 160

Viscosity (mPa s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


140 120 100

298 K (300 rpm)

80 60 40

323 K 348 K 373 K

20 0

0

10

20

30

40

50

Pressure (MPa)



Figure 21. Viscosity versus pressure for base oil IIB at 500 rpm (300 rpm for the 298 K run). Free volume correlation fit is represented by black dots.

70

323 K 500 rpm

60

Viscosity (mPa s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 32

PAO 8

50

IIIB

40

IIB

30

IIA PAO 4 IIIA

20 10 0

0

10

20

30

40

50

Pressure (MPa) Figure 22. Viscosity versus pressure for six base oils at 323 K and 500 rpm. Free volume correlation fit is represented by black dots.

IV.

Conclusions

The determination of thermodynamic properties from PVT data requires the use of a model that accurately represents the experimental data. The Sanchez-Lacombe equation of state, a lattice fluid model, was found to fit density data generated through the use of a variablevolume view-cell for six different base oils. The absolute average deviations for these S-L fits were found to be less than or equal to 0.2 %. With a well-fitting model, it was possible to examine the effect of composition on isothermal compressibility, isobaric thermal expansion coefficient, and internal pressure. Compressibility and internal pressure were both affected by the presence of cyclic molecules in the oils, as cycloparaffin content increases, compressibility decreases while internal pressure increases. These trends suggest that the effect of pressure on density is in part determined by how these cycloparaffins pack with each other, and other paraffinic molecules. Thermal expansion did not appear to be as affected by oil composition.


Page 29 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


A unique high pressure rotational viscometer was developed to determine the effect of temperature, pressure, and rotational speed on viscosity. All six oils were found to be Newtonian in nature at the shear rates tested. Experimental viscosity data was related to density by using a free volume correlation in conjunction with the Sanchez-Lacombe equation of state. For the higher viscosity oils, absolute average deviations were found to be mostly below 10%. For two of the oils studied, at higher temperatures (373 K), viscosities were below the minimum values the instrument is capable of accurately measuring, leading to absolute average deviations above 25%. By utilizing two models together, both viscosity and density can be simultaneously examined. The compositional effects seen in the calculated thermodynamic properties were not observed in the viscosity values or in the model fitting parameters, indicating that the molecular effects on viscosity are more complex than the packing interactions seen in PVT properties.

Acknowledgment

This research was supported by Afton Chemical Corporation.

Supporting Information

A. PVT data and thermodynamic properties for IIA, IIIA, IIIB, PAO 4, and PAO 8. B. Viscosity data for IIA, IIIA, IIIB, PAO 4, and PAO 8. C. Density, isothermal compressibility, isobaric thermal expansion coefficient, internal pressure, and dynamic viscosity at selected temperatures and pressures as calculated by the S-L EOS and free volume model for base oil IIA, IIIA, IIIB, PAO 4, PAO 8.



V. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16. 17. 18. 19. 20.

21. 22. 23.

Page 30 of 32

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High Pressure Volumetric Properties and Viscosity of Base Oils Used

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