Viscosity measurements for squalane at high pressures to 350 MPa from T = (293.15 to 363.15) K

Viscosity Measurements for Squalane at High Pressures, to 350 MPa, between 293.15 K and 363.15 K M. J. P. Comuñas,1 X. Paredes,1 F. M. Gaciño,1 J. Fernández,1,* J. P. Bazile,2 C. Boned,2,* J. L. Daridon2, G. Galliero,2 J. Pauly2 and K. R. Harris3 1Laboratorio de Propiedades Termofísicas, Departamento de Física Aplicada, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain 2Laboratoire des Fluides Complexes et leurs Reservoirs (UMR-5150 with CNRS and TOTAL), Université de Pau et des Pays de l'Adour, BP 1155, F-64013 Pau Cedex, France 3School of Physical, Environmental and Mathematical Sciences, University College, University of New South Wales, P.O. Box 7916, Canberra BC, ACT 2610, Australia ____________________________________________ * Authors to whom correspondence should be addressed. Electronic mail: [email protected] (J.F.) [email protected] (C.B.) ABSTRACT Squalane is being recommended as a secondary reference material for viscometry at moderate to high pressure and at moderate viscosity. As part of this work, a correlation has been developed for atmospheric pressure (Comuñas et al, J. Phys. Chem. Ref. Data, 42, 033101, 2013). Here we report new experimental high pressure viscosities for squalane (176 data points obtained between 293.15 and 363.15K, at pressures up to 350 MPa with a maximum viscosity of 745 mPa·s. These have been determined with four different falling-body viscometers as well as a quartz crystal resonator viscometer. A preliminary high pressure viscosity correlation for squalane is proposed, based on our new data. At pressures up to 350 MPa, this correlation provides an absolute average deviation of 1.5 % with a maximum absolute deviation of 8.9 %. Comparison is made between the different instruments. In addition, we have also considered the validity of a thermodynamic scaling model. Keywords: Squalane; Viscosity; High Pressures; Reference material; Thermodynamic scaling 1. Introduction There is a requirement for a reference material for viscosity measurements at moderate to high pressures that can be used for the calibration of viscometers designed for such pressures, both for laboratory and industrial work. To this end, there have been several previous studies that have led to useful, though limited, correlations. The first is based on one of the commercially available certified viscometer calibration fluids, Cannon S20, which is a mixture of hydrocarbons. Kandil et al. [1] employed both vibrating wire (VW) and falling body (FB) techniques over a temperature range of 273 to 473 K at pressures to 275 MPa with an expanded uncertainty (k = 2) of 2.3 % . Both types of viscometer have well-defined and well-verified working equations. This work extended previous measurements by Lundstrom et al. [2] and Sopkov et al. [3] that each had an upper limit of 55 MPa. The atmospheric pressure values used to calibrate the high pressure viscometers derive from the standard viscosity [4] of water at 293.15 K and 0.1 MPa, either directly or indirectly, through the step-up procedure employing a chain of intermediate reference liquids and a series of master capillary viscometers. However, materials such as S20 slowly degrade, due to composition changes caused by evaporation, and so must be used within a specified time, and the absolute atmospheric pressure viscosities of individual lots may differ. It is necessary therefore to assume that the pressure dependence of different lots is the same for the high pressure measurements to be employed as a reference. The second study proposed diisodecyl phthalate (DIDP) as an interim reference material, and was put forward by the International Association of Transport Properties (http://transp.eng.auth.gr) [5]. This is also a careful and critical compilation based on measurements from a range of viscometers of different types (VW, FB, capillary (C) and surface light scattering (SLS)), but applies to only atmospheric pressure between 288 and 308 K for viscosities between 49 and 171 mPa·s. It has an expanded uncertainty of 2 % (k = 2). There are high pressure data [6; 7] available, but these have not been used for a reference correlation as DIDP is not available as pure substance, but only as a complex mixture of isomers (the effect of this is considered to be small, but cannot be determined exactly), and, more seriously, the main manufacturer has ceased production of high purity material. Other moderately viscous fluids that are available as single isomers, such as di(2- ethylhexyl) sebacate (DEHS), 2-ethylhexyl benzoate (EHB), and bis(2-ethylhexyl) phthalate (DEHP), have also been suggested for use as moderate viscosity reference fluids [8]. Harris [8], and Paredes et al. [9] have reported viscosities at high pressures for these compounds, including a correlation of the literature high pressure data for DEHS in the second study. However, the European Chemical Agency has listed DEHP as a “Substance of Very High Concern (SVHC)” (20th October 2008), classifying it as toxic to (human) reproduction [8; 10]. Consequently, this substance may not meet occupational safety and health (OSH) criteria in certain countries or organizations for use as a reference viscosity standard [8; 11]. Of the other compounds, EHB is quite an expensive material and its viscosities are the lowest among the possible candidates. DEHS has an important advantage in that it is a common material, being used as a power transmitting fluid, but its main drawback is that it is only available with 97 % purity at present. Given the constraints on the use of S20 and DIDP as moderate viscosity reference materials, and the drawbacks of DEHP, EHB and DEHS, attention has moved to squalane (C30H62, tetracosane, 2,6,10,15,19,23-hexamethyltetracosane or, for brevity, SQN). Its advantages are that it is moderately viscous, has a wide liquid range and a low vapor pressure, is relatively inexpensive and is available with mole purities above 99 % in a single isomeric form. As a hydrocarbon, it should be of particular usefulness as a reference material for work on hydrocarbon-based oils under the conditions met in oilfield reserves and on liquids that wet the solid surfaces of a viscometer in a similar way (unlike water). The available experimental data in the literature, together with new measurements from a variety of viscometer types, all at atmospheric pressure, have been critically examined recently and a reference correlation put forward, again under the auspices of IATP [12]. The next stage is to enlarge the available high pressure data. In this work, we have made additional experimental measurements of the viscosity of squalane as a function of temperature and pressure to extend and round out the range of state points to be covered. This has been done using falling body and quartz crystal resonator viscometers in three different laboratories at the Universities of New South Wales (UNSW), Pau (UPPA) and Santiago de Compostela (USC), thus providing a consistency check for the various instruments and techniques, as in the DIDP study [5]. These are all reported here as a single source of data. The data cover the temperature range 293.15 K and 363.15 K, and extend to 350 MPa pressure. A high pressure correlation for squalane, considering these new measurements and all the high pressure data from the literature [8; 13-22] as well as other two sets recently measured in other laboratories, is to be published in a separate work [23]. We will only give a preliminary correlation in this work, based only on our own data, just in order to check the consistency of our new measurements. 2. Experimental Section The provenance and purity of the three squalane samples (CAS 111-01-3) and some reference fluids, decane (CAS 124-18-5), diisodecyl phthalate sample (CAS 26761- 40-0) and Bis(2-ethylhexyl) phthalate (CAS 117-81-7), are provided in Table 1. The water content of squalane was measured at USC by Karl Fischer analysis, yielding a weight fraction of 12·10-6. A falling body viscometer was used at the University of New South Wales (FBUNSW). This viscometer has been described previously [24; 25]. In this viscometer, for laminar flow, the viscosity 𝜂 is related to the sinker’s fall time  t, by the following equation: ( ) ( ) ( ) 1 1 2 298.15 K 1 2 0.1MPa 3 LS t A T p   − =+ − − −         (1) where  L and  S are the liquid and sinker densities, respectively, A is the viscometer constant,  is the linear expansion coefficient and  is the linear compressibility of the sinker and the tube. The density values  L needed in equation (1) to determine the viscosities of squalane were calculated as described previously [8]; the density of the sinker at each temperature and pressure was determined using the equation given by Dymond et al. [26]. The viscometer constant A was determined by calibration with Cannon Instrument Company oils (N100, S200 and N4000, 6.0 mm sinker; N100, plus toluene and octane, 6.3 mm sinker [8]) at atmospheric pressure between 298.15 and 353.15 K. It is found to be independent of temperature and viscosity, and assumed independent of pressure, any pressure effect on the sinker clearance being taken into account by the compressibility term in the denominator of equation (1). The measurements have been performed over the temperature interval 313.15 to 363.15 K and up to 349.8 MPa. The expanded uncertainties (coverage factor k = 2) claimed [8] are 0.2 MPa for the pressure, 0.02 K for the temperature and the expanded uncertainty of the viscosity measurements performed with this viscometer and this procedure is 2.3 % [1]. The falling body viscometer of the University of Pau (FB-UPPA) was described previously [27]. The uncertainty in pressure, measured with a HBM P3M transducer, is  0.1 % of the full scale (200 MPa) and that of the temperature, measured with an AOIP PHP602 thermometer, is  0.05 K. The fall time is measured with a stop watch and is reproducible to better than 1 %. The viscosity was determined from the following relation: ( ) ( )   tTpKTp LS −=  ,, (2) where  S is the density of the sinker,  L the density of the liquid at pressure p and temperature T,  t the fall time and K(p,T) is a calibration parameter that takes into account the geometry of the apparatus and which is taken to be function of p and T. Very often,  S is supposed independent of p and T for the intervals of pressure and temperature considered. This is the case for this study. A sinker with density 7.717 g·cm-3 was used. The density values  L needed in equation (2) for the viscosities of squalane have been taken from Ciotta et al. [15] The term K(p,T) is usually determined using a reference compound of known viscosity. It is best to use a reference compound where the viscosity is not too different from that of the fluid studied, but this is not always possible, as in the present case. An alternative is to use a ‘self-reference’ method. Measuring precisely the viscosity at atmospheric pressure with a capillary viscometer [12], it is easy to determine K(0.1 MPa, T): ( ) ( ) ( )   tTMPa TMPa TMPaK squalaneS squalane − =,1.0 ,1.0 ,1.0   (3) Then it is assumed that the pressure has a limited influence on K. Strictly speaking this is untrue but we consider that any deformations in the experimental cell remain very small as the fluid is located both inside and outside the inner tube of the measuring cell. We have constrained our study to p < 200 MPa. Under this condition, we used K(p,T)  K(0.1 MPa,T). The expanded uncertainty (k = 2) of the measured viscosity is estimated at  3.5 % with this falling-body viscometer and procedure for pressures below 200 MPa, between 303.15 and 348.15 K. The quartz crystal resonator viscometer of the University of Pau (QCR-UPPA) has recently been developed by Daridon et al. [28] The pressure is measured with a gauge (Kulite HEM-375 M, uncertainty 0.05 MPa) inserted into the cell. A heat-carrying liquid is circulated around the cell from a thermostat bath circulator (Haake F6) which has a stability of 0.01 K. A platinum resistance thermometer inserted into the vessel is used to measure the temperature of the cell with an uncertainty of 0.1 K. The viscosity is determined from the damping of the quartz vibrations for different overtones and using the ratio cexp between the half-band half-width and the overtone number through the relation: 2 2 01        + = erfaceint exp mL R c C f    (4) where f0 is the resonance frequency,  L is the density of the liquid, Cm is the Sauerbrey coefficient [29], and finally Rinterface is a coefficient related to the quartz surface morphology that appears to be independent of pressure [30]. Two methods have been used in this work to obtain Rinterface by calibration and thus to determine the viscosity of squalane under pressure from 293.15 to 333.15 K and up to 80 MPa. One is based on the estimation of the parameter Rinterface by using decane as reference liquid (QCR-UPPA 1) and the second one consists in using the viscosities of squalane at atmospheric pressure as a self-reference (QCR-UPPA 2). We have found only small differences between these methods, with an average absolute deviation of 0.5 % and a maximum deviation of 1.1 %, both below the expanded uncertainty estimated to be 4% (k = 2). At the University of Santiago de Compostela two falling body viscometers (FBUSC VisLPT1 and FB-USC VisLPT2) were employed [31; 32] with maximum working pressures of 150 and 250 MPa respectively. In both USC viscometers, the temperature is regulated using a circulating thermostat and is measured with thermocouples located inside the cells and in contact with the sample with an uncertainty of ± 0.5 K. In the VisLPT1 instrument, the sample is compressed by means of a hydraulic compressor and in the VisLPT2 instrument by a pneumatic pump. The pressure is measured by digital transducers with an accuracy of ± 0.2 MPa in both viscometers. The measuring principle is based on the relationship between viscosity and the time (  t) that a solid takes to fall through a viscous fluid once it has reached its terminal velocity. The viscosity was determined from the following expression: ( ) ( ) 2 tctba  ++= (5) where a, b and c are calibration parameters and  =  S -  L is the difference between the density of the sinker and that of the liquid under study. Both USC viscometers were calibrated using DIDP and DEHP as reference fluids and the measurements were performed from 313.15 to 343.15 K for VisLPT1, 313.15 to 363.15 K for VisLTP2, and at pressures reaching 150 and 250 MPa for the VisLPT1 and VisLPT2 instruments respectively. The viscosities of these fluids were taken from the correlations given by Harris [8] and Peleties and Trusler [33] for DIDP, and by Harris [8] for DEHP. The densities of DIDP and DEHP have been obtained from Peleties et al. [34] and from Harris [8], respectively. The density values  L needed in equation (6) to determine the viscosities of squalane have been taken from Ciotta et al. [15]. A sinker with a density of 7.695 g·cm- 3 was used in the VisLPT1 apparatus and one with a density of 7.738 g·cm-3 in VisLPT2. Note that for relations (1), (3) and (6) an error of 1 % in  L (i.e. a very bad estimation of the density of the fluid) implies a relative error less than 0.2 % in  (  S is around 7.5 Although several authors [50; 55; 56] have proposed expressions for the function f(TV  ) in equation (9) for non-reduced quantities, to our knowledge there are no studies on the form of f for reduced quantities. The use of reduced transport properties is an alternative procedure based on the behaviour of the liquids which obey the inverse potential law and on the new concept of “isomorphic” curves in the phase diagram [57; 58]. Again, the fitting should be done in such a way that the reduced data collapse onto a single master curve. Firstly, we define the following reduced quantities: * exp r    = , c exp rT T T= , c exp r    = with 6/13/2* M/RT  = (10) where R is the gas constant, M the molar mass, Tc the critical temperature and  c the critical density. We define the thermodynamic quantity xr as (Tr /   r ). We have fitted the reduced viscosities determined from our new data set for squalane using a completely arbitrary empirical function that appears to behave satisfactorily, namely: ( ) 3 r3 2 r2r10r x/ax/ax/aaexp +++=  (11) The experimental critical temperature [59] of squalane is Tc = 822 K, the critical density from Monte Carlo simulations [60] is estimated at  c = 228 kg·m-3, and the molar mass M = 0.42283 kg·mol-1. The density data, necessary for xr, were calculated from the correlation given by Ciotta et al. [15] but this correlation has been established only for the temperature range 303.15 < T/K < 448.25 and for pressures below 176 MPa. To avoid density extrapolation, the database has been reduced in consequence to these pressure and temperature ranges. The number of data points from Tables 2 and 3 remaining for the scaling fit is 134 (with viscosities lying between 3.1 and 422.5 mPa·s). Furthermore, we have also used viscosity data previously reported by our three laboratories [12] for 0.1 MPa within the Ciotta temperature interval (27 points). Therefore the reduced database contains a total of 161 values. Equation (11) was used to determine the coefficient  by minimizing the AAD(ln(  r) is linear function of the ai but is not linear function of  ). The parameters ai and the coefficient  are given in Table 6. The fit yielded AAD = 1.5 %, Bias= -0.11 % and a MD =9.3 %, found for the FB-USC VisLPT1 experimental value at 313.15 K and 10 MPa (  exp = 16.8 mPa·s) Figure 3 shows the variation of the deviations against pressure. From a total of 161 data points, there are only seven with an absolute relative deviation above 5 %. Again good agreement for the data of this work is consistently obtained as can be seen in Table 7. The value of  obtained from this procedure is practically 4.15, very close to that found by Roland et al. [49] and Harris [8] for non-reduced transport properties (  = 4.2 and 4.16, respectively) and also to that found by Bair and Laesecke [56] (  = 3.92) who reduced only the volume, not the viscosity. Figure 4 very clearly shows there is indeed a single master curve of  r(xr) and that the scaling relation of Roland et al. [47; 49] can be used to determine the volume and temperature dependence of the reduced viscosity. Note that the curve in Figure 4 is independent of the form of equation (11), used for the fitting. The curve depends only on the  value, and it shows that  ≈ 4.15 is really a good estimate in order to get a master curve. It is worth mentioning that in fact, it is also possible to determine  without any information on the function f. Considering equation 9, it follows that on an isoviscous line, the product T  -  has to be constant, which leads to ln(T) = A + γ ln(ρ) (likewise for the reduced quantities). Thus, drawing ln(T) versus ln(ρ) at isoviscous conditions allows the determination of  from the slope of the line. This looks like a rather simple method, but this approach presents its own difficulties because it is quite impossible to get such isoviscous data directly from experiments. In practice, it is necessary to draw various isobaric (or isothermal) viscosity lines, then choose a fixed viscosity value and keep track of the corresponding (p,T) values, so as to be able to calculate, the density of the liquid at those (p,T) states with a suitable density correlation equation. In addition, there creates an additional uncertainty at each step. Furthermore, this procedure should be carried out for several viscosity values and the number of points needs to be sufficient along each viscosity line for the calculation to be significant. Note that in using a closely related method (Figure 5 of Bohling et al. [61]), it seems that for some liquids there are deviations from single power-law scaling such as equation (9);  increases with density. However, these authors insist on the fact that the power law density scaling is a useful approximation to isomorph scaling when only moderate density changes are considered. This is the case for our scaling study where the density varies by only around 12 % (between 758 and 863 kg.m-3). Assuming equation 9, we get for the viscosity AAD = 1.5% (around the experimental uncertainty) and MD = 9.3%. These low values seem to indicate, that at least in our density interval, the power law density scaling is valid and  is constant. 4. Conclusions In this paper we report data and preliminary correlation for the viscosity of squalane over a range of temperatures, pressures and viscosities. We have made new measurements in our three laboratories using falling body and quartz resonator viscometers. In particular, that has allowed us to check the validity of the different calibration procedures for falling body viscometers. These values were satisfactorily correlated as function of temperature and pressure by using a modified VFT equation (5 adjustable parameters) with an average absolute relative deviation of 1.5 %. Finally we have used thermodynamic scaling to determine the scaling parameter,  , which is related to the steepness of the repulsive part of the intermolecular potential. This model was confirmed only for pressures below 176 MPa due to the constraints of the available equation of state. A value of  around 4.15 was obtained, consistent with earlier estimates in the literature. Acknowledgments We are grateful to our colleagues from the International Association for Transport Properties for helpful discussions. This work was supported by Spanish Ministry of Science and Innovation and the UE FEDER Program under projects CTQ2011-2395 and PSE-420000-2008-4. 5. References [1] M.E. Kandil, K.R. Harris, A.R.H. Goodwin, K. Hsu, K.N. Marsh, J. Chem. Eng. Data 51 (2006) 2185-2196. [2] R. Lundstrom, A.R.H. Goodwin, K. Hsu, M. Frels, D.R. Caudwell, J.P.M. Trusler, K.N. Marsh, J. Chem. Eng. Data 50 (2005) 1377-1388. [3] T. Sopkow, A.R.H. Goodwin, K. Hsu, J. Chem. Eng. Data 50 (2005) 1732-1735. 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Chem. Phys. 134 (2011) 044504. [58] N. Gnan, T. Schrøder, U.R. Pedersen, N.P. Bailey, J.C. Dyre, J. Chem. Phys. 131 (2009) 234504. [59] E.D. Nikitin, A.P. Popov, Fluid Phase Equilib. 237 (2005) 16-20. [60] B. Neubauer, J. Delhommelle, A. Boutin, B. Tavitian, A.H. Fuchs, Fluid Phase Equilib. 155 (1999) 167-176. [61] L. Bøhling, T.S. Ingebrigtsen, A. Grzybowski, M. Paluch, J.C. Dyre, T.B. Schrøder, New Journal of Physics 14 (2012) 113035. TABLE 1 Liquids used in this work. IUPAC Name Lot number Supplier Mole Fraction Purity 2,6,10,15,19,23-hexamethyltetracosane (Squalane) 1418796 Sigma Aldrich (Fluka) > 0.990a Decane 17230LC-195 Sigma Aldrich 0.994 Bis(8-methylnonyl) phthalate (Diisodecyl phthalate) K22132622 Merck 0.998a Bis(2-ethylhexyl) phthalate S42805-258 Sigma Aldrich 0.997a aGas chromatographic analysis 24 TABLE 2 Experimental viscosities,  /mPa·s, of squalane at high pressures, obtained with falling body viscometers. T/K p /MPa  /mPa·s T /K p /MPa  /mPa·s FB-UNSWa 338.15 50.6 15.56 363.15 21.0 5.15 338.15 102.8 32.94 363.15 30.9 5.97 338.15 150.2 60.90 363.15 40.7 6.84 338.15 198.4 108.9 363.15 50.6 7.82 338.15 249.7 194.0 363.15 60.6 8.94 338.15 272.7 243.8 363.15 70.7 10.18 338.15 299.8 327.6 363.15 80.6 11.49 338.15 349.2 538.5 363.15 91.0 13.06 338.15 349.2 531.0 363.15 100.0 14.59 348.15 198.3 70.83 363.15 120.4 18.36 348.15 200.4 72.63 363.15 150.3 25.56 348.15 300.6 202.9 363.15 200.2 42.82 348.15 349.8 327.6 363.15 225.6 55.03 363.15 10.2 4.36 363.15 246.9 66.23 FB-UPPAb 303.15 20 34.78 323.15 140 102.35 303.15 40 51.64 323.15 160 135.10 303.15 60 74.96 323.15 180 175.94 303.15 80 107.09 323.15 200 228.18 303.15 100 146.47 348.15 20 7.45 303.15 120 210.53 348.15 40 10.15 303.15 140 305.40 348.15 60 13.51 303.15 160 422.48 348.15 80 17.79 323.15 20 15.77 348.15 100 23.08 323.15 40 22.36 348.15 120 29.61 323.15 60 31.17 348.15 140 37.63 323.15 80 42.71 348.15 160 48.09 323.15 100 57.75 348.15 180 60.38 323.15 120 77.38 348.15 200 74.91 FB-USC VisLPT1c 313.15 10 16.8 343.15 10 7.3 313.15 15 18.9 343.15 15 8.0 313.15 25 23.8 343.15 25 9.4 313.15 50 39.2 343.15 50 13.6 313.15 75 61.2 343.15 75 19.2 313.15 100 92.3 343.15 100 26.5 313.15 125 136.4 343.15 125 36.2 313.15 150 199.0 343.15 150 49.1 FB-USC VisLPT2d 313.15 50 36.3 343.15 150 52.0 313.15 75 59.6 343.15 175 69.9 313.15 100 91.3 343.15 200 92.8 25 313.15 125 134.8 343.15 225 122.1 313.15 150 194.1 343.15 250 159.6 313.15 175 275.3 363.15 100 14.2 313.15 200 386.2 363.15 125 19.9 313.15 225 537.7 363.15 150 26.8 313.15 250 744.7 363.15 175 35.1 343.15 75 18.7 363.15 200 45.0 343.15 100 27.2 363.15 225 56.7 343.15 125 38.1 363.15 250 70.4 aUncertainties: u(T) = 0.02 K, u(p) = 0.2 MPa and U(  ) = 2.3% (k=2). bUncertainties: u(T) = 0.05 K, u(p) = 0.2 MPa and U(  ) = 3.5% (k=2). cUncertainties: u(T) = 0.5 K, u(p) = 0.2 MPa and U(  ) = 5% (k=2). dUncertainties: u(T) = 0.5 K, u(p) = 0.2 MPa and U(  ) = 5% (k=2). 32 Figure 2. Percentage deviations of the viscosity data, given as a function of the temperature, from the values calculated from equation (6) (VFT model). FB-UNSW (), FB-UPPA (◼), FB- USCVisLPT1 ( ), FB-USCVisLPT2 ( ), QCRUPPA 1 () and QCRUPPA 2 (+). -10 -5 0 5 10 283 293 303 313 323 333 343 353 363 373 100 ((  calc -  exp )/  calc )a T/K 33 Figure 3. Percentage deviations of the viscosity data, given as a function of the pressure, from the values calculated with equation (11) (Thermodynamic scaling model). FB-UNSW (), FBUPPA (◼), FB-USCVisLPT1 ( ), FB-USCVisLPT2 ( ), QCRUPPA 1 () , QCRUPPA 2 (+) and [12] (). -10 -5 0 5 10 025 50 75 100 125 150 175 100 ((  calc -  exp ) /  calc ) a p/MPa 34 Figure 4. Experimental reduced viscosity ηr versus xr = Tr /   r (303.15 < T/K < 363.15, p < 176 MPa and  = 4.15) 0 1 2 3 4 5 6 7 8 9 1.35 1.85 2.35 2.85 3.35 107  r 103xr