Viscosity measurements of poly(ethyleneglycol) 400 [PEG 400] at temperatures from 293 K to 348 K and at pressures up to 50 MPa using the vibrating wire technique

Fluid Phase Equilibria 496 (2019) 7e16

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Viscosity measurements of poly(ethyleneglycol) 400 [PEG 400] at temperatures from 293 K to 348 K and at pressures up to 50 MPa using the vibrating wire technique Maria C.M. Sequeira a, Marta F.V. Pereira a, Helena M.N.T. Avelino a, b, *, ~o M.N.A. Fareleira a, ** Fernando J.P. Caetano a, c, Joa Centro de Química Estrutural, Instituto Superior T ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001, Lisboa, Portugal  Area Departamental de Engenharia Química, Instituto Superior de Engenharia de Lisboa, Instituto Polit ecnico Lisboa, R. Conselheiro Emídio Navarro, 1, 1959-007, Lisboa, Portugal c Departamento de Ci^ encias e Tecnologia, Universidade Aberta, Rua da Escola Polit ecnica, 147, 1269-001, Lisboa, Portugal a

b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 22 March 2019 Received in revised form 16 May 2019 Accepted 19 May 2019 Available online 25 May 2019

The article reports new measurements of the viscosity of Poly(ethyleneglycol) 400 [PEG 400] in the range (293e348) K and pressures up to 50 MPa. Complementary measurements of the density of the same sample of PEG 400 have been made covering the ranges of temperature and pressure, (293e353) K and (0.1e50) MPa, respectively. The viscosity measurements were performed using the vibrating wire technique in the forced mode of oscillation and the density measurements were carried out with an Anton Paar vibrating U-tube densimeter. The density raw data were corrected for viscosity effects. The overall uncertainty of the viscosity measurements is estimated to be less than ±2% for viscosities up to 68 mPa s and less than ±2.6% for higher viscosities. The densities have an estimated overall uncertainty of ±0.2%. The rheological behaviour of Poly(ethylene Glycol) 400 has also been studied, using a cone-plate Brookfield viscometer, in a temperature range between (293 and 333) K. The measurements were carried out at shear rates up to 20 s
1 and shear stresses up to 2.20 Pa and have evidenced Newtonian behaviour. The viscosity data obtained were correlated by means of a modified hard-sphere based correlation technique. The relative root mean square, rms, deviation of the experimental results from the correlation equations is 0.54%, and their bias is practically zero. The density data obtained were correlated using a Tait-type equation. As a complement of the present study, the surface tension of PEG 400 was measured by the pendant drop method. This study aims to be useful for viscosity measurements using capillary viscometers. As far as the authors are aware, the present viscosity measurements are the first results to be published for PEG 400 at pressures higher than atmospheric pressure. © 2019 Elsevier B.V. All rights reserved.

Keywords: Polyethylene glycol PEG Viscosity Vibrating wire High pressure

1. Introduction Liquid Poly(ethyleneglycols) [PEGs] are in general considered as green solvents. They are non-volatile; their toxicity is very low,

cnico, * Corresponding author. Centro de Química Estrutural, Instituto Superior Te Universidade de Lisboa. Av. Rovisco Pais, 1, 1049-001, Lisboa, Portugal. ** Corresponding author. E-mail addresses: [email protected] (H.M.N.T. Avelino), j.fareleira@ tecnico.ulisboa.pt (J.M.N.A. Fareleira). https://doi.org/10.1016/j.fluid.2019.05.012 0378-3812/© 2019 Elsevier B.V. All rights reserved.

such that they are being used as food additives [1]. PEGs have been found to be biodegradable by bacteria in soil or sewage, but the ability of bacteria to biodegrade PEG decreases with increasing molecular weight. Below 600 Da, PEGs can be used as solvents in their pure form. PEGs are highly soluble in water due to the hydration of the ether groups and the terminal hydroxyl group. The latter are comparatively more important in low molecular mass PEGs, which favours their water solubility [1]. Poly(ethyleneglycol) 400 [PEG 400] is a low-molecular-weight

8

M.C.M. Sequeira et al. / Fluid Phase Equilibria 496 (2019) 7e16

Table 1 Characterization of the liquids used in this work. PEG 400

CAS number

Supplier

Water content (mg,kg
1)

Lot number

Purity (mass fraction)

Polyethylene glycol 400

25322-68-3

Merck KGaA

S7378385

e

Toluene

108-88-3

Sigma-Aldrich

251.9 (*) 153.5 (**) 14

STBD1659V

99.8%

(*) Water content of the samples before viscosity and density measurements. (**) Water content of the samples before the surface tension measurements.

grade of poly(ethylene glycol). It is a clear, colourless, viscous liquid. Due in part to its low toxicity, PEG 400 is widely used in a variety of pharmaceutical formulations [1]. As a consequence of the increasing importance of PEGs, which mainly is itself a result of their characteristics as sustainable solvents a number of studies have been carried out by researchers from Centro de Química Estrutural [2e5]. Furthermore, their importance is also due to their use as model liquids, e.g., for situations involving the oil industry, like the study of CO2 saturated PEG 200 and PEG 400 mixtures. In the present article, the viscosity of PEG 400 is measured with high accuracy using the vibrating wire technique at elevated pressures. The same technique has recently been used to perform the viscosity measurement of three ethylene glycols, namely diethylene, triethylene and tetraethylene glycols, at moderately high pressures [6]. Also this experimental setup has been extensively used in our group to measure the viscosity of other fluids such as di-alkyl adipates [7,8] and n-tetradecane [9,10]. Furthermore, measurements have been performed to support the initial efforts to propose two high viscosity standard fluids, namely, diisodecylphthalate (DIDP) [11,12] and tris(2-ethylhexyl) trimellitate (TOTM) [13e16]. Those fluids have subsequently been the subject of international joint initiatives to formalize their proposition as high viscosity standard fluids at high pressures [14], under projects developed by the International Association for Transport Properties (http://transp.cheng.auth.gr/index.php/iatp/2010). The present work is part of a programme to study the thermophysical properties of ethylene glycols and PEGs. One of the aims of the work is to analyse the relation of the present results with those obtained before for CO2 saturated PEG400 mixtures [17]. 2. Materials and methods 2.1. Materials The PEG 400 was acquired from MercK KGaA. Samples of the fluid were dried with molecular sieves from SigmaeAldrich with a porosity of 0.4 nm and no further purification was performed. The water content was checked using a KarleFischer 831 KF Coulometer from Metrohm. Table 1 shows the characterization of the PEG 400 samples used in this work. The calibration of the vibrating wire sensor has been verified with toluene, supplied by Sigma Aldrich, having a nominal purity of 99.8%, without further treatment, except for drying to a water content of ca. 14 mg kg
1, measured with the above mentioned equipment, and degassing by helium spraying. The sample of PEG 400 used in this work has a mass-average molar mass of 392 g mol
1 according to the certificate of analysis indicated by the supplier [18]. Polyethylene glycol 400 (PEG) oligomers, HO(C2H4O)nH, were analysed by gas chromatography/mass spectrometry (GC/MS) in the Laboratory of Organic Chemistry of Instituto Superior de Engenharia de Lisboa, Portugal. The results do not show any significant disagreement with the certificate of analysis: mass-average molar mass, Mm ¼ 388 g mol
1, the

number-average molar mass, Mn ¼ 360 g mol
1 and the dispersity, Mm/Mn ¼ 1.08. 2.2. Methods 2.2.1. Vibrating U-tube density measurements The densities of PEG 400 were measured in the range (293e353) K and at pressures up to 50 MPa, along seven isotherms, using a DMA HP vibrating U-tube densimeter from Anton Paar, with model DMA 5000 as a reading unit. The experimental technique is the same used by the authors in previous works, as described by Brito e Abreu et al. [19]. The raw data obtained require a correction for the effect of the viscosity of the samples. The corrections were performed according to the procedure described by Diogo et al. [16]. The present density measurements have an estimated uncertainty of ±0.2% at a 95% confidence level. 2.2.2. Vibrating wire method The vibrating wire sensor, and the whole setup, including the ancillary equipment used in the present work for measuring the viscosity of PEG 400 has been fully described in a previous article by Diogo et al. [20]. The experimental technique and the particular electronic instrumentation used for the present measurements, have been described by Diogo et al. [21]. The technique is essentially based on the analysis of the effect of the viscosity of the fluid sample on the steady-state transverse oscillation of a metallic wire immersed in a permanent magnetic field. For this purpose, the frequency response of the oscillating wire is analysed in a range of frequencies containing the velocity resonance of the transverse oscillation. The wire radius has been determined by calibration with toluene, at 303.15 K and 0.1 MPa. The parameters of the sensor are shown in Table 2. The expanded uncertainty of the present vibrating-wire measurements, at a 95% confidence level, is estimated to be less than ±2%, for viscosities up to 68 mPa s and less than ±2.6% for higher viscosities. 2.2.3. Surface tension measurements The surface tension of the PEG 400 was determined using the pendant-drop method. Main details of the selected technique and Table 2 Parameters of the vibrating wire sensor at the reference temperatures, Tref. Tref/K Tungsten wire data Radiusa Densityb

303.15 293.15

Rs/m rs/(kg,m
3)

151.218 ✕10
6 19230

Resonance under vacuum Frequencyc Internal damping coefficientc

298.15 298.15

u/(rad,s
1) D0

6561 1.39 ✕10
4

a b c

By calibration. From Ref. [22]. Measured.

M.C.M. Sequeira et al. / Fluid Phase Equilibria 496 (2019) 7e16

2.2.4. Rheological studies A Brookfield DV-II þ PRO viscometer was used to characterize the rheological properties of PEG 400. All rheological tests were performed with a rotational viscometer equipped with a cone-plate spindle CPA-40Z and 0.5 ml of the sample. The rheometer's software (Rheocalc V. 3.0) was used to define the shear rate and to record the rheogram data. The equipment used has a nominal uncertainty of ±3%. Tests were performed in a temperature range of (293e333) K. The cup temperature was controlled using a Julabo FP40 circulating thermostatic bath. 3. Results and discussion 3.1. Density 3.1.1. Experimental results As indicated previously, density measurements were performed at pressures up to 50 MPa and at seven temperatures from (293e353) K and the results obtained are shown in Table 3. The measurements have an estimated uncertainty of ±0.2% at a 95% confidence level, this estimate is based on previous sensitivity studies [19], taking into account the particular characteristics of the present experiments. It should be noted that the density obtained by the oscillating U-tube technique must be corrected for viscosity effects [16]. For this purpose, a procedure has been used, which is described in the literature [6,16]. The corrections of the present results have a maximum of the order of 0.07% for the highest viscosities. This value, although lower than the overall uncertainty of the measurements is higher than their repeatability. In practice, the recommended procedure implies that the correction must be considered whenever it is higher than the repeatability of the density results. 3.1.2. Correlation of the density with temperature and pressure The corrected density data presented in Table 3 were correlated as a function of temperature and pressure through the modified Tait equation [25].






r ¼ r0 1
C ln

Dþp D þ p0


1 (1)

where r0 represents the density at a pressure p0 equal to 0.1 MPa, and is described by the polynomial

r0 ¼

2 X

bi T i

(2)

i¼0

where C and D are fitting parameters. In Eq. (1), it is assumed that C is independent of the temperature. The behaviour of each isotherm along the pressure is modelled by the parameter D, which exhibits a smooth dependency with temperature, described by Eq. (3).

D¼

3 X i¼0

di T i

(3)

The parameters of the fitting Eqs. (1)e(3) are listed in Table 4. The root mean square, rms deviation, s, of all the data is smaller than 0.005% and the bias is essentially zero. The statistical parameters, s and bias, are defined by Eqs. (4) and (5).

2

!2 31 2 N X Xexp;i 1 s¼4
1 5 N Xcalc;i =

the setup used have been described by Morgado [23]. The analysis of the drop shapes was made with the axisymmetric drop shape analysis (ADSA) method [24]. The repeatability of the present surface tension measurements is better than ±0.3%. Based on previous studies by Morgado [23], the uncertainty of the results is estimated to be ±0.5%.

9

(4)

i

N Xexp;i 1 X bias ¼
1 N Xcalc;i

! (5)

i

where N is the total number of experimental data points, the subscripts (exp,i and calc,i) stand for the ith experimental and calculated data points, respectively, and X stands either for the density or the viscosity. Fig. 1 shows the deviations of the density data obtained in the present work from the fitting Eqs. (1)e(3), with the parameters given in Table 4. The maximum relative deviation of the density data does not exceed ±0.006%. 3.1.3. Comparison with literature data Only one set of high pressure density data was found in the literature. Crespo et al. [26] measured the density of a sample of PEG 400 in a wide range of temperatures (283e363 K) and pressures (0.1e95) MPa, using an Anton Paar U-tube DMA-HPM instrument. Those results have not been corrected for viscosity effects. The data set reported by Crespo et al. [26] was compared with the present density results, as described by Eqs (1) e (3), along the isotherms (293 and 353) K and for pressures up to 50 MPa. The relative deviations are plotted in Fig. 2. The deviations of the data published by Crespo et al. [26] from the present results are all positive, the approximate minimum and maximum being 3.0% and 3.8%, respectively. The deviations are all slightly higher than the nominal uncertainty of their measurements. It is worth of note that the correction of the U-tube results for viscosity effects would, in principle, lower values for the density. However, it is clear that the corrections depend on the specific U-tube used, and no quantitative estimates can be made, at the present moment, about the data published by Crespo et al. [26]. 3.2. Viscosity 3.2.1. Experimental results The viscosity of compressed PEG 400 was measured with a vibrating-wire viscometer operated in the forced mode of oscillation, at pressures from (1e50) MPa and temperatures between (293 and 348) K. The expanded uncertainty of the present viscosity results at a 95% confidence level is estimated to be less than ±2% for viscosities up to 68 mPa s and less than ±2.6% for higher viscosities. These estimates are based on previous sensitivity studies [7,20,21]. The viscosity results are shown in Table 5. In Table 5, the density data used to compute the vibrating wire viscosity results were obtained using the corresponding Tait-type Eqs. (1)e(3) with parameters listed in Table 3 and incorporate the correction for the viscosity effect on the oscillating U-tube density measurements as described in section 2.2.1. 3.2.2. Correlation of the viscosity with the molar volume The viscosity data were correlated by the scheme proposed by Li et al. [27], which is based on the hard-spheres model developed by Dymond [28]. This powerful scheme allowed the correlation and prediction of the thermal conductivity and viscosity of a wide range

10

M.C.M. Sequeira et al. / Fluid Phase Equilibria 496 (2019) 7e16

Table 3 Density data of PEG 400 obtained with an Anton Paar DMA HP densimeter, either corrected, r, and uncorrected, rHP, and viscosity, h, calculated for T and p using Eqs (6)e(8). T/K

p/MPa

rHP/(kg,m
3)

h/(mPa,s)

r/(kg,m
3)

293.15

0.10 0.10 0.62 1.05 1.58 2.10 3.06 5.06 7.03 9.95 14.85 19.77 24.65 29.60 34.47 39.40 44.29 49.18 0.10 0.10 0.59 1.10 1.58 2.15 3.02 5.04 7.02 9.96 14.84 19.75 24.61 29.58 34.47 39.40 44.29 49.20 0.10 0.10 0.61 1.07 1.58 2.10 3.06 5.11 7.00 9.96 14.87 19.76 24.65 29.54 34.49 39.36 44.31 49.18 68.75 0.10 0.10 0.63 1.10 1.59 2.09 3.08 5.08

1126.3 1126.3 1126.6 1126.8 1127.0 1127.2 1127.7 1128.6 1129.5 1130.8 1133.0 1135.1 1137.2 1139.2 1141.3 1143.2 1145.2 1147.1 1117.9 1117.9 1118.2 1118.4 1118.7 1118.9 1119.3 1120.3 1121.2 1122.6 1124.8 1127.0 1129.2 1131.3 1133.5 1135.5 1137.5 1139.4 1109.6 1109.6 1109.9 1110.1 1110.4 1110.6 1111.1 1112.1 1113.0 1114.5 1116.8 1119.1 1121.4 1123.5 1125.8 1127.8 1129.9 1131.9 1139.6 1101.3 1101.3 1101.6 1101.8 1102.1 1102.3 1102.9 1103.9

114.5 114.6 115.3 115.9 116.6 117.3 118.7 121.6 124.4 128.8 136.6 144.8 153.4 162.6 172.3 182.5 193.5 204.8 67.5 67.5 67.9 68.2 68.6 69.0 69.7 71.2 72.8 75.0 79.1 83.3 87.7 92.3 97.2 102.1 107.5 112.8 43.0 43.0 43.2 43.4 43.7 43.8 44.3 45.2 46.1 47.4 49.8 52.2 54.8 57.4 60.3 63.0 66.0 69.0 82.2 29.0 29.0 29.2 29.3 29.4 29.6 29.9 30.4

1125.6 1125.6 1125.8 1126.0 1126.2 1126.5 1126.9 1127.8 1128.7 1130.0 1132.2 1134.3 1136.4 1138.5 1140.5 1142.4 1144.4 1146.2 1117.3 1117.2 1117.5 1117.7 1118.0 1118.2 1118.7 1119.6 1120.5 1121.9 1124.2 1126.3 1128.5 1130.6 1132.7 1134.7 1136.8 1138.7 1109.0 1109.0 1109.3 1109.5 1109.8 1110.0 1110.5 1111.5 1112.4 1113.8 1116.2 1118.4 1120.7 1122.9 1125.1 1127.2 1129.3 1131.2 1138.9 1100.8 1100.8 1101.0 1101.3 1101.5 1101.8 1102.3 1103.3

303.15

313.15

323.15

Expanded uncertainties: U(T) ¼ 0.05 K; U(p) ¼ 0.08 MPa; U(r) ¼ 0.2%.

T/K

333.15

343.15

353.15

p/MPa

rHP/(kg,m
3)

h/(mPa,s)

r/(kg,m
3)

6.99 9.97 14.84 24.64 29.61 34.45 39.40 44.28 49.20 0.10 0.10 0.62 1.10 1.58 2.12 3.05 5.04 6.99 9.96 14.83 19.71 24.65 29.55 34.47 39.41 44.27 49.16 0.10 0.10 0.62 1.11 1.55 2.12 3.04 5.06 7.01 9.96 14.86 19.78 24.63 29.58 34.46 39.40 44.29 49.21 0.10 0.10 0.65 1.11 1.58 2.14 3.06 5.08 7.02 9.93 14.90 19.79 24.64 29.56 34.43 39.36 44.26 49.21

1104.9 1106.4 1108.8 1113.6 1115.8 1118.1 1120.2 1122.4 1124.5 1093.1 1093.1 1093.4 1093.7 1093.9 1094.2 1094.7 1095.8 1096.9 1098.4 1101.0 1103.4 1105.9 1108.2 1110.6 1112.8 1115.1 1117.2 1085.1 1085.0 1085.4 1085.6 1085.9 1086.2 1086.7 1087.8 1088.9 1090.5 1093.2 1095.7 1098.3 1100.7 1103.2 1105.5 1107.8 1110.0 1077.0 1076.9 1077.3 1077.6 1077.8 1078.2 1078.7 1079.9 1081.0 1082.7 1085.5 1088.1 1090.7 1093.3 1095.8 1098.2 1100.6 1102.9

31.0 31.9 33.4 36.5 38.1 39.9 41.6 43.4 45.2 20.6 20.6 20.7 20.8 20.9 21.0 21.2 21.6 22.0 22.5 23.6 24.6 25.7 26.7 27.9 29.0 30.2 31.4 15.3 15.3 15.4 15.4 15.5 15.6 15.7 16.0 16.2 16.6 17.4 18.1 18.8 19.6 20.4 21.2 22.0 22.8 11.7 11.7 11.8 11.8 11.9 11.9 12.0 12.2 12.4 12.7 13.3 13.8 14.3 14.9 15.5 16.0 16.6 17.2

1104.3 1105.8 1108.2 1113.0 1115.2 1117.5 1119.6 1121.8 1123.8 1092.6 1092.6 1092.9 1093.2 1093.4 1093.7 1094.2 1095.3 1096.4 1097.9 1100.5 1102.9 1105.3 1107.7 1110.0 1112.3 1114.5 1116.6 1084.6 1084.6 1084.9 1085.2 1085.4 1085.7 1086.3 1087.3 1088.5 1090.0 1092.7 1095.2 1097.8 1100.2 1102.7 1105.0 1107.3 1109.5 1076.5 1076.5 1076.9 1077.1 1077.4 1077.7 1078.3 1079.4 1080.6 1082.2 1085.0 1087.7 1090.3 1092.8 1095.4 1097.7 1100.2 1102.5

M.C.M. Sequeira et al. / Fluid Phase Equilibria 496 (2019) 7e16

1395.216
1.004967 2.90458✕10
4

d0/MPa d1/(MPa,K
1) d2/(MPa,K
2) d3/(MPa,K
3)

663.67612
2.397169 3.32508✕10
3
1.934094✕10
6

C

0.081520

s/%

0.005
0.00003

bias/%

h * ¼ 6:035  108



1 MRT

1 2 =

b0/(kg,m
3) b1/(kg,m
3,K
1) b2/(kg,m
3,K
2)

of simple molecules, alkanes, aromatic hydrocarbons, alcohols [29] and has been widely applied by the authors with good results [6,8,9,14,20,30,31]. In this method, a dimensionless viscosity h* is defined such that, using SI units can be described by Eq (6)

2

h ðVm Þ

=

Table 4 Fitting parameters of Eqs. (1) e (3) for the density data, r, after correction for viscosity effects.

11

3

(6)

where Vm is the molar volume, R is the gas constant, and M is the molar mass. The reduced viscosity is then correlated as a function of (Vm/V0) as follows

Fig. 1. Deviations of the density, r, of PEG 400 obtained with the Anton Paar DMA HP densimeter and corrected for viscosity effects, from the correlation Eqs. (1)e(3): 303 K; , 313 K; , 323 K; , 333 K; , 343 K; , 353 K.

Fig. 2. Deviations of the density, r, of PEG 400 obtained by Crespo et al. [26], from correlation Eqs. (1)e(3):

, 293. K;

, 303 K;

, 313 K;

, 323 K; , 333 K;

, 293 K;

, 343 K;

,

, 353 K.

12

M.C.M. Sequeira et al. / Fluid Phase Equilibria 496 (2019) 7e16

Table 5 Viscosity data for PEG 400 measured with the vibrating wire technique at temperatures from (293e348) K and pressures up to 50 MPa. Density values were obtained by Eqs. (1)e(3). T/K

r/MPa

p/(kg,m
3)

h/(mPa,s)

T/K

p/MPa

r/(kg,m
3)

h (mPa,s)

T/K

p/MPa

r/(kg,m
3)

h/(mPa,s)

293.25

0.24 0.24 0.24 0.24 0.24 0.24 0.26 1.18 1.19 2.34 1.91 5.03 4.99 5.00 5.00 5.02 5.01 10.18 10.18 10.18 10.19 10.22 10.21 20.26 20.25 20.26 20.26 20.27 29.81 29.81 29.80 29.80 29.80 29.81 40.15 40.20 40.21 40.22 40.37 0.25 0.25 0.25 0.25 0.25 0.25 0.34 0.35 0.35 1.29 1.29 1.29 1.29 1.29 2.33 2.33 2.33 2.33 2.32 2.33 5.13 5.14 5.14 5.13 10.11 10.11 10.11 10.10 10.10 20.36 20.37 20.38 20.39 30.27

1125.6 1125.6 1125.6 1125.6 1125.6 1125.6 1125.6 1126.0 1126.0 1126.5 1126.3 1127.7 1127.7 1127.7 1127.7 1127.7 1127.7 1130.1 1130.1 1130.1 1130.1 1130.1 1130.1 1134.4 1134.4 1134.4 1134.4 1134.5 1138.5 1138.5 1138.5 1138.5 1138.5 1138.5 1142.6 1142.7 1142.7 1142.7 1142.7 1117.2 1117.2 1117.2 1117.2 1117.2 1117.2 1117.3 1117.3 1117.3 1117.7 1117.7 1117.7 1117.7 1117.7 1118.2 1118.2 1118.2 1118.2 1118.2 1118.2 1119.6 1119.6 1119.6 1119.6 1121.9 1121.9 1121.9 1121.9 1121.9 1126.5 1126.5 1126.5 1126.5 1130.8

113.6 113.3 114.7 113.7 113.3 114.7 114.7 115.7 116.6 117.2 116.5 121.0 121.0 121.3 120.1 120.3 120.7 128.6 128.7 128.1 127.9 128.1 128.5 145.0 145.0 145.3 145.2 144.9 161.8 162.3 161.5 162.0 161.4 162.7 182.7 182.6 183.3 182.8 183.4 67.57 67.26 67.54 67.41 67.44 67.62 66.90 66.74 67.49 68.08 67.64 68.40 68.23 68.33 68.50 68.37 69.21 68.62 68.47 68.93 71.44 70.80 71.34 70.21 74.44 75.06 75.41 74.99 74.90 83.24 82.72 83.69 83.67 91.90

313.23

5.14 5.15 5.15 5.15 10.17 10.17 10.17 10.17 10.17 10.17 20.17 20.17 20.17 20.17 20.17 20.17 30.07 30.07 30.07 30.07 30.07 30.07 40.17 40.17 40.17 40.16 40.17 40.18 50.21 50.21 50.20 50.20 0.25 0.26 0.26 0.26 0.27 0.27 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.98 0.98 0.98 0.97 0.98 0.98 2.11 2.10 2.10 2.11 2.11 2.12 5.07 5.07 5.06 5.06 5.06 5.06 5.06 5.06 5.50 5.49 5.49 5.49 5.49 5.49 5.49

1111.4 1111.4 1111.4 1111.4 1113.9 1113.9 1113.9 1113.9 1113.9 1113.9 1118.6 1118.6 1118.6 1118.6 1118.6 1118.6 1123.1 1123.1 1123.1 1123.1 1123.1 1123.1 1127.4 1127.4 1127.4 1127.4 1127.4 1127.4 1131.6 1131.6 1131.6 1131.6 1101.4 1101.4 1101.4 1101.4 1101.4 1101.4 1101.4 1101.4 1101.4 1101.4 1101.4 1101.4 1101.4 1101.4 1101.8 1101.8 1101.8 1101.8 1101.8 1101.8 1101.8 1101.8 1101.8 1101.8 1101.8 1101.8 1103.3 1103.3 1103.3 1103.3 1103.3 1103.3 1103.3 1103.3 1103.6 1103.6 1103.6 1103.6 1103.6 1103.6 1103.6

45.29 45.00 45.19 45.05 47.51 47.10 47.25 47.19 47.40 47.27 52.24 52.43 52.12 52.03 52.20 52.51 57.71 57.41 57.57 57.82 57.69 57.50 63.11 62.76 63.00 63.49 62.72 63.44 69.12 68.75 68.86 68.94 29.44 29.38 29.43 29.45 29.48 29.45 29.62 29.54 29.46 29.60 29.48 29.43 29.50 29.61 29.53 29.63 29.73 29.80 29.52 29.56 29.94 30.01 30.01 29.95 29.97 30.06 30.72 30.84 30.68 30.86 30.47 30.87 30.80 30.66 30.90 31.03 30.85 30.89 30.93 30.95 30.89

323.19

39.95 39.95 50.00 50.00 49.99 49.99 49.99 49.99 50.00 50.00 50.00 0.25 0.22 0.22 1.05 1.05 1.05 1.05 1.05 1.05 1.06 1.97 1.97 1.98 1.98 1.98 1.98 5.03 5.03 5.03 5.03 5.03 10.12 10.12 10.12 10.12 10.12 10.12 20.01 20.00 20.00 20.00 30.05 30.06 30.06 40.14 40.14 40.14 40.14 50.09 0.28 0.28 0.28 0.28 0.28 1.06 1.06 1.06 1.06 1.06 1.06 2.05 2.05 2.05 2.04 5.08 5.08 5.08 5.09 5.10 5.13 5.14 5.14

1119.9 1119.9 1124.2 1124.2 1124.2 1124.2 1124.2 1124.2 1124.2 1124.2 1124.2 1092.7 1092.7 1092.7 1093.2 1093.2 1093.2 1093.2 1093.2 1093.2 1093.2 1093.7 1093.7 1093.7 1093.7 1093.7 1093.7 1095.3 1095.3 1095.3 1095.3 1095.3 1098.0 1098.0 1098.0 1098.0 1098.0 1098.0 1103.0 1103.0 1103.0 1103.0 1107.9 1107.9 1107.9 1112.6 1112.6 1112.6 1112.6 1117.1 1081.0 1081.0 1081.0 1081.0 1081.0 1081.4 1081.4 1081.4 1081.4 1081.4 1081.4 1082.0 1082.0 1082.0 1082.0 1083.7 1083.7 1083.7 1083.7 1083.8 1083.8 1083.8 1083.8

41.79 41.99 45.83 46.21 45.96 46.06 45.80 46.13 45.89 46.00 46.10 20.74 20.87 20.79 20.80 20.79 20.77 20.75 20.74 20.76 20.75 20.94 21.02 20.87 20.90 20.89 20.91 21.50 21.58 21.49 21.56 21.56 22.52 22.53 22.50 22.52 22.52 22.53 24.58 24.61 24.56 24.52 26.75 27.09 27.03 29.21 29.50 29.40 29.47 31.89 13.44 13.56 13.53 13.59 13.48 13.61 13.51 13.65 13.50 13.55 13.59 13.68 13.65 13.62 13.73 13.97 14.09 14.02 14.06 14.06 14.15 14.07 14.08

303.26

323.19

333.15

347.74

M.C.M. Sequeira et al. / Fluid Phase Equilibria 496 (2019) 7e16

13

Table 5 (continued ) T/K

313.23

r/MPa

p/(kg,m
3)

h/(mPa,s)

30.27 30.27 39.90 39.88 39.88 39.87 50.01 50.01 0.25 0.25 0.25 0.25 0.25 0.25 0.29 0.30 0.31 0.32 1.06 1.06 1.06 1.07 1.06 1.05 1.99 1.98 1.98 1.97 1.97

1130.8 1130.8 1134.9 1134.9 1134.9 1134.9 1139.0 1139.0 1109.0 1109.0 1109.0 1109.0 1109.0 1109.0 1109.0 1109.0 1109.0 1109.0 1109.4 1109.4 1109.4 1109.4 1109.4 1109.4 1109.9 1109.9 1109.9 1109.9 1109.9

91.97 91.92 101.8 102.3 101.5 101.6 112.9 114.1 43.17 43.13 42.84 43.19 42.84 42.81 42.83 42.91 43.05 43.01 43.17 43.13 43.23 43.22 43.33 43.16 43.78 43.56 43.80 43.83 43.89

T/K

p/MPa

r/(kg,m
3)

h (mPa,s)

9.95 9.96 9.96 9.97 9.96 9.97 9.97 9.97 9.97 20.10 20.10 20.10 20.10 20.09 20.09 29.92 29.93 29.92 29.92 29.93 29.94 29.94 39.94 39.95 39.95 39.95 39.95 39.95 39.95

1105.8 1105.8 1105.8 1105.8 1105.8 1105.8 1105.8 1105.8 1105.8 1110.8 1110.8 1110.8 1110.8 1110.8 1110.8 1115.4 1115.4 1115.4 1115.4 1115.4 1115.4 1115.4 1119.9 1119.9 1119.9 1119.9 1119.9 1119.9 1119.9

32.24 32.25 32.22 32.09 32.28 32.25 32.17 32.25 32.31 35.51 35.48 35.58 35.37 35.62 35.39 38.90 38.67 38.63 38.71 38.62 38.72 38.51 41.62 41.97 41.98 42.26 42.01 42.05 41.95

T/K

p/MPa

r/(kg,m
3)

h/(mPa,s)

5.14 5.14 10.37 10.38 10.38 10.40 10.41 10.42 20.09 20.09 20.10 20.10 20.10 20.10 30.12 30.12 30.11 30.11 30.11 30.11 30.11 40.08 40.09 40.09 40.08 50.12 50.12 50.12 50.12

1083.8 1083.8 1086.7 1086.7 1086.7 1086.7 1086.7 1086.7 1091.9 1091.9 1092.0 1092.0 1092.0 1092.0 1097.1 1097.1 1097.1 1097.1 1097.1 1097.1 1097.1 1102.0 1102.0 1102.0 1102.0 1106.7 1106.7 1106.7 1106.7

14.16 14.12 14.66 14.67 14.67 14.76 14.69 14.70 15.97 15.84 15.87 15.91 16.04 15.95 17.34 17.41 17.36 17.25 17.31 17.25 17.39 18.73 18.67 18.68 18.79 20.29 20.00 20.25 20.30

Expanded uncertainties: U(T) ¼ ±0.05 K; U(p) ¼ ±0.08 MPa; U(r) ¼ 0.2%; U(h) ¼ ±2%, for viscosities up to 68 mPa s; U(h) ¼ ±2.6% for higher viscosities.

4 X 1 ¼ ai h* i¼0



Vm V0

i (7)

where V0 is a characteristic molar volume whose temperature dependence can be described by Eq (8).

V0 ðTÞ ¼ V0;ref þ l



T
Tref



þ m



T
Tref

2

(8)

A reference value of V0(Tref) for PEG 400 has been calculated at 313.15 K, assuming it would be equivalent to the volume of closepacking of hard-spheres, calculated from the hard-sphere diameter, according to the procedure described in Ref. [32]. The critical parameters of PEG 400, were estimated by the method by Marrero n and Pardillo-Fontdvila [33] as described in Ref. [34] and Marejo those were then used to estimate the Lennard-Jones potential parameters [35] and the close-packing volume of hard-spheres [36,37]. The value of Vo (313.15 K) is 299.4039  10
6 (m3 mol
1). The fitting coefficients l and m were obtained by simultaneous fitting of Eqs. (7) and (8) and are listed in Table 6, together with the rms deviation, s, and bias of the experimental data from the correlation.

The deviations of the vibrating wire viscosity data presented in Table 5 from correlation Eqs. (7) and (8) are shown in Fig. 3, with the parameters listed in Table 6, as a function of pressure. The maximum deviation of the viscosity data of PEG 400 from the correlation, does not exceed ±1.13%. No viscosity data for PEG 400 at pressures higher than atmospheric pressure could be found in the literature. 3.3. Surface tension measurements The results of the surface tension measurements of PEG 400 performed at atmospheric pressures and at four temperatures, from (304e331) K, are shown in Table 7 and Fig. 4. The repeatability of the surface tension measurements is better than ±0.15% and their estimated expanded uncertainty, at a 95% confidence level, is ±0.5%. The experimental data of the surface tension were correlated with temperature using Eq. (9) having a rms deviation, s, of 0.2% and a practically zero bias. The deviations of the results from the correlation are within ±0.3%, which is within the uncertainty of the measurements.

sð10
3 N mÞ ¼ 72:172
0:0993 TðKÞ

(9)

Table 6 Fitting parameters of Eqs. (7) and (8) for the viscosity data, h, obtained with the vibrating wire viscometer. l/(m3 mol
1 K
1) m/(m3 mol
1 K
2)


3.69035  10
7 1.07343  10
9

3.4. Rheological studies

a0/ a1/ a2/ a3/ a4/


0.276033 0.891552
1.058228 0.542095
0.099526

s/% bias/%

0.526
0.0002

Fig. 5 shows a plot of the shear-rate versus shear stress for PEG 400 at 293 and 303 K, for shear rates up to 20 s
1. For both temperatures the linearity between the shear stress and the shear rate evidences its Newtonian behaviour. These results corroborate our observations using different driving current intensities in the vibrating wire technique viscosity measurements. The slope of each isotherm indicated in the graph corresponds to the viscosity of PEG

14

M.C.M. Sequeira et al. / Fluid Phase Equilibria 496 (2019) 7e16

Fig. 3. Deviations of the viscosity, h, of PEG 400 obtained with a vibrating wire viscometer, from correlation Eqs. (6)e(8):

Table 7 Experimental (10
3 N m).

values

for

surface

tension,

T/K

10
3 s/(N m)

304.24 309.34 320.70 331.29

42.06 41.33 40.34 39.29

Expanded U(s) ¼ ±0.5%.

uncertainties:

s

U(T) ¼ ±0.05 K;

400 at the corresponding temperature, 113.3 mPa s and 64.4 mPa s at 293 and 303, respectively. These results compare with the viscosity obtained by the correlation equations Eqs. (7) and (8) of the

, 293 K;

, 303 K;

, 313 K; , 323 K;

, 333 K;

, 348 K.

present work within 0.8% and 4.3%. respectively, which are commensurate with the mutual uncertainty of the measurements. 4. Conclusions To the knowledge of the authors, the present viscosity results for PEG 400 are the first to be published at pressures above atmospheric pressure. Those measurements were carried out in the range of temperature (293e348) K and pressures up to 50 MPa. The method used was the vibrating wire technique in the forced mode of oscillation. On the same sample of PEG 400, complementary measurements of the density, using an Anton Paar U-tube densimeter, and of the surface tension, using the pendant drop method have been performed. The raw data of the density measurements were corrected for viscosity effects. The present results

Fig. 4. Surface tension data of PEG 400 obtained with the pendant drop technique, ; and the correlation of data,

; as a function of the temperature.

M.C.M. Sequeira et al. / Fluid Phase Equilibria 496 (2019) 7e16

Fig. 5. Shear stress of PEG 400 at 293 K,

and 303 K,

are deemed to be useful for the interpretation of the viscosity of mixtures of PEGs with CO2. [6]

Acknowledgements and Funding This work was supported by the Project UID/QUI/00100/2013 ~o para a and Project UID/QUI/00100/2019 funded by Fundaça ^ncia e a Tecnologia, Portugal. The authors are grateful to FCT, Cie Portugal, for its support.  nio Velez Marques, InstiThe authors are grateful to Prof. Anto tuto Superior de Engenharia de Lisboa, Portugal, for having performed the GC/MS tests. Appendix A. Supplementary data In the Supplementary Material, a tool e an Excel™ file e is available to easily interpolate the viscosity and the density in the ranges (0.1e50) MPa and (298e348) K. Supplementary data to this article can be found online at https://doi.org/10.1016/j.fluid.2019.05.012.

[7]

[8]

[9]

[10]

[11]

[12]

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