Influence of the number of CH2CH2O groups on the viscosity of polyethylene glycol dimethyl ethers at high pressure

Fluid Phase Equilibria 222–223 (2004) 331–338

Influence of the number of CH2 –CH2 –O groups on the viscosity of polyethylene glycol dimethyl ethers at high pressure M.J.P. Comuñas a,b , A. Baylaucq b,∗ , F. Plantier b , C. Boned b , J. Fernández a a

b

Laboratorio de Propiedades Termof´ısicas, Departamento de F´ısica Aplicada, Facultad de F´ısica, Universidad de Santiago de Compostela, E-15706 Santiago de Compostela, Spain Laboratoire des Fluides Complexes, UMR CNRS 5150, Faculté des Sciences, BP 1155, F-64013 Pau, France Available online 21 July 2004

Abstract This work reports new measurements of the viscosity of liquid diethylene glycol dimethylether (DEGDME) upto 100 MPa at eight temperatures ranging from 293.15 to 353.15 K. The measurements at atmospheric pressure have been performed with an Ubbelohde-type glass capillary tube viscometer with an uncertainty of ±1%. At pressures upto 100 MPa the viscosity was determined with a falling body viscometer with an uncertainty of ±2%. Using previous polyalkylene glycol dimethylether viscosity and density data (Triethylene glycol dimethyl ether and tetraethylene glycol dimethyl ether) obtained at the laboratory, present measurements were used to study the dependence of the viscosity and of the molecular parameters of hard-sphere scheme and free-volume model on the number of CH2 –CH2 –O groups. © 2004 Elsevier B.V. All rights reserved. Keywords: Falling-body viscometer; High pressure; Polyethylene glycol dimethylethers; Viscosity

1. Introduction In the last years measurements of the viscosity of polyalkylene glycol dimethylethers, CH3 O–((CH2 )2 O)n – CH3 , under pressure have become important because of the use of these data in studying the reliability of these fluids as lubricants in refrigeration compressors and as absorbents for absorption systems. The use of these compounds has been also suggested for cleaning of exhaust air and gas streams from industrial production plants with the simultaneous recovery of valuable materials [1]. However, the database for transport properties of these compounds, at present, is scarce and at very limited temperature and pressure conditions. Most measurements have been made at atmospheric pressure and at temperatures in the range from 293.15 to 323.15 K. Thus, while there are abundant data, for these compounds, describing the temperature dependence of the viscosity at 0.1 MPa [2–5], studies versus pressure are less frequent. Only in our previous works [6,7] the dynamic viscosity at high pressure of triethylene gly-

∗

Corresponding author. E-mail address: [email protected] (A. Baylaucq).

0378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2004.06.034

col dimethylether (TriEGDME) and tetraethylene glycol dimethylether (TEGDME) have been measured. Respect to other thermophysical properties, density values of monoethylene glycol dimethyl ether (MEGDME) have been published by Senger [8] between 288.0 and 473.0 K and upto 37 MPa, and also by Sharipov and Bairamova [9] between 298.15 and 328.15 K and upto 202 MPa. These last authors have also reported densities of diethylene glycol dimethyl ether (DEGDME) in the same pressure and temperature ranges. Comuñas et al. [6,10] have measured the densities of MEGDME, DEGDME, TriEGDME and TEGDME from 293.15 to 353.15 K and at pressures upto 60 MPa, determining the isobaric thermal expansivity, αp , the isothermal compressibility, κT , and the internal pressure, π. Recently, Lopez et al. have measured the speed of sound of TriEGDME and TEGDME from 293.15 to 353.15 K and pressures from atmospheric upto 100 MPa, calculating the isentropic and isothermal compressibilities at the same conditions. To complete our experimental [6,10–13] and theoretical studies [7] of some physical properties of polyethylene glycol dimethyl ethers, in this paper we report new viscosity data of DEGDME at temperatures from 293.15 to 353.15 K and at pressures upto 100 MPa. These experimental values

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together with those obtained previously for TriEGDME and TEGDME are used to test the correlation ability of the hard-sphere scheme and free-volume model.

2. Experimental 2.1. Measurement techniques At atmospheric pressure the kinematic viscosities, ν, were measured with an Ubbelohde-type glass capillary tube viscometer with a Schott–Geräte automatic measuring unit (model AVS 350) in a thermostated bath, which regulates the temperature to within an accuracy of ±0.01 K. After multiplication by the density at atmospheric pressure, the dynamic viscosity, η, is obtained with an uncertainty lower than 1%. A falling body viscometer, of the type of the one designed by Ducoulombier et al. [14], was used to measure the viscosity of the compressed liquids. In this apparatus, a stainless steel cylinder falls through a fluid of unknown viscosity at given conditions of temperature and pressure. The viscosity is a function of the falling time, of the density difference between the cylinder and the fluid and of one constant obtained by calibrating the viscometer with a substance of known viscosity and density. We have used toluene [15] as calibrating fluid except for the 283.15 K isotherm for which we have used water [16]. Decane [17,18] has been used to verify the calibration. For the high pressure viscosity measurements temperature and pressure have accuracies of ±0.5 K and ±0.05 MPa, respectively. Each measurement of the falling time was repeated six times at thermal and mechanical equilibrium, and it is reproducible is better than 1%. The final value is an

average of these measurements. The total uncertainty of the viscosity values obtained was estimated to be within 2% which is comparable to the one estimated by other authors for similar devices as it has been discussed in previous papers [19,20]. In order to measure the viscosity upto 100 MPa, the density values are needed at all the pressure and temperature conditions, so, the experimental density values previously reported [10] from 293.15 to 353.15 K and at pressures upto 60 MPa have been used. At pressures higher than 60 MPa and upto 100 MPa, the Hogenboom modification of the Tait equation [21] was used, as in the previous papers [6], for density extrapolation. We have compared the extrapolated densities with the data reported by other authors in order to confirm that the error due to the extrapolation does not affect the viscosity measurements. In this sense we have estimated that an error in density of ±1 × 10−2 g cm−3 generates an error of only 1/800 in viscosity [19]. 2.2. Materials Diethylene glycol dimethyl ether, DEGDME (molar mass 134.18 g mol−1 ) was obtained from Aldrich with a purity of 99.5%. This product was subjected to no further purification.

3. Experimental results In order to verify that the error due to the extrapolation was lower than ±1 × 10−2 g cm−3 , the extrapolated density values for DEGDME upto 100 MPa obtained by using the Hogenboom modification [21] have been compared with the literature values reported by Sharipov and Bairamova [9]. In Fig. 1a, the deviations between both data sets can

Fig. 1. (a) Deviations between the extrapolated densities of DEGDME upto 100 MPa and the literature values reported by Sharipov and Bairamova [9]. (b) Deviations between the kinematic viscosities determined in this work and (䉫) those reported by Ku and Tu [2] and (䊊) by Conesa et al. [4].

M.J.P. Comuñas et al. / Fluid Phase Equilibria 222–223 (2004) 331–338 Table 1 Experimental dynamic viscosity of DEGDME vs. temperature T and pressure p p (MPa)

0.1 20 40 60 80 100

T (K) 293.15

303.15

313.15

323.15

333.15

343.15

353.15

1.1039 1.2669 1.4576 1.6671 1.8890 2.1195

0.9309 1.0725 1.2243 1.3899 1.5671 1.7539

0.8134 0.9290 1.0595 1.2020 1.3529 1.5100

0.6953 0.8118 0.9402 1.0720 1.2019 1.3265

0.6329 0.7439 0.8532 0.9654 1.0793 1.1935

0.5612 0.6500 0.7485 0.8531 0.9607 1.0692

0.4860 0.5769 0.6668 0.7566 0.8440 0.9276

be observed. For pressure between 60 and 100 MPa, the extrapolated values agree with the literature ones with an average absolute deviation of 0.05%, and with maximum absolute deviation of 0.15%. We have seen that the density extrapolation from 60 to 100 MPa yielded an error markedly less than ±1 × 10−2 g cm−3 . The experimental dynamic viscosity values for DEGDME at different pressures and temperatures are listed in Table 1. To our knowledge there are no literature viscosities for this fluid at pressures different from the atmospheric value. Ku and Tu [2] have published experimental measurements at 0.1 MPa and at different temperatures between 288.15 and 343.15 K. The average deviation of our measurements from their values is 1.7%. Our experimental values agree with those published by Conesa et al. [4] with an average deviation of 1.3%. These deviations are consistent with the uncertainty of the experimental apparatus at 0.1 MPa, and are plotted in Fig. 1b. Figs. 2–4 illustrate the variations of the dynamic viscosity versus temperature and pressure of DEGDME along with the values previously obtained [10] on triethylene glycol dimethylether and tetraethylene glycol dimethylether. It can be observed that the viscosity decreases with the temperature and increases with the pressure for each of the three compounds. In addition, the viscosity increases as the number n of CH2 –CH2 –O groups of the polyalkylene glycol dimethylether increases.

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Both previous equations can be combined in order to fit the viscosity as a function of temperature and pressure. We then propose the following equation [6]:
  E(T) + p (3) η(p, T) = η0 (T) exp D ln E(T) + 0.1 (MPa) where η0 (T) is the temperature dependence of the viscosity at the reference pressure (here 0.1 MPa) and is given by Eq. (2). Therefore, from Eqs. (2) and (3) one obtains the following equation:  
  B p + E(T) η(p, T) = A exp exp D ln (4) T −C 0.1 + E(T) where D was assumed to be temperature independent and E(T) a second-order polynomial. Using this equation (involving seven parameters) we have obtained an average absolute deviation of 1.16% for DEGDME. The maximum deviation is 4.62% at 353.15 K and 100 MPa. The previous results on TriEGDME and TEGDME provide absolute average deviations of 0.69 and 1.05%, and maximum deviations of 2.71 and 2.80%, respectively. 4.2. The hard-sphere scheme 4.2.1. Presentation of the model Recently [22,23] a scheme has been developed for the simultaneous correlation of self-diffusion, viscosity and thermal conductivity of dense fluids. The transport coefficients of real dense fluids expressed in terms of Vr = V/V0 , with V0 the close-packed volume and V the molar volume, are assumed to be directly proportional to the values given by the exact hard-sphere theory. The proportionality factor, described as a roughness factor Rη accounts for molecular roughness and departure from molecular sphericity. Universal curves for the viscosity were developed and are expressed as:  ∗    i 7 ηexp 1 log = (5) aηi Vr Rη i=0

4. Viscosity representations 4.1. Andrade representation The experimental data of viscosity on an isotherm, η(p) can be fitted for each isotherm to the following Tait-like equation:
  C+p η(p) = A exp B ln (1) C + 0.1 (MPa) and on an isobar η(T) can be fitted for isobar to the following Andrade’s equation [22]:   B η(T) = A exp (2) T −C

with η∗exp = 6.035 × 108 (1/MRT)1/2 η exp V 2/3 . It seems [24] that the coefficients aηi are universal and independent of the chemical nature of the compound. For alkanes [23] and aromatics [25] the authors give correlation formulas relative to V0 and Rη . For a given compound, parameters V0 are temperature-dependent, whereas Rη is a constant. 4.2.2. Results In our case, parameters V0 and Rη are not available in the literature and this is the reason why we have applied the hard-sphere model keeping constant the universal coefficients aηi of Assael et al. [23,25] and calculating the coefficients V0 (for each temperature) and Rη for DEGDME and also for TriEGDME and TEGDME. We have fitted the

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Fig. 2. Variation of dynamic viscosity of several polyethers, CH3 O–((CH2 )2 O)n –CH3 , vs. temperature for various pressures. This work: n = 2; [6]: n = 3,4; (—): Andrade’s representation.

parameters minimizing the absolute average deviation over a set of 42 experimental viscosity data for each compound. The obtained parameters values and the results for this model are presented in Table 2. Very good results have been obtained for each pure compound as it can be observed that

the absolute average deviations are below than the experimental uncertainty. In order to make an analysis of the influence of CH2 –CH2 –O groups, the next modeling step has then been to represent the variation of the dynamic viscosity as a

Fig. 3. Variation of dynamic viscosity of several polyethers, CH3 O–((CH2 )2 O)n –CH3 , vs. pressure for various temperatures. This work: n = 2; [6]: n = 3, 4; (—): Andrade’s representation.

M.J.P. Comuñas et al. / Fluid Phase Equilibria 222–223 (2004) 331–338

Fig. 4. Variation of dynamic viscosity of several polyethers, CH3 O–((CH2 )2 O)n –CH3 , vs. n at 0.1 and 100 MPa (䊉: 293.15, 䊏: 313.15, 䉱: 353.15 K) and at 293.15 and 353.15 K (䊊: 20, 䊐: 60 and 䉫: 100 MPa). (—): polynomial correlation.

function number of CH2 –CH2 –O groups, n which is equal to 2 for DEGDME, 3 for TriEGDME and 4 for TEGDME. The parameters V0 and Rη have then been re-estimated, expressing them with a simple linear function, on the overall data set (126 points). The correlations used are the following: V0 (T) = aV (T)n + bV (T) Rη = aR n + bR Table 2 Results obtained on the three compounds with the hard-sphere scheme

335

: 333.15 and

The results are presented in Table 3 along with the new parameters which have been reduced from 24 to 16 for the three polyethers. A good representation is obtained as the absolute average deviations are around 1% which is still below the experimental uncertainty and with maximum deviations lower than 6%. It can be observed that coefficients aV are all positive and are decreasing with the temperature increase while no particular trend can be observed for coefficients bV . 4.3. The free-volume viscosity model

Parameters aηk

T (liquid) K V0 (10−4 m3 mol−1 )

Universal parameters Liquid

DEGDME TriEGDME TEGDME

1.0945 −9.2632 71.0385 −301.9012 797.6900 −1221.9770 987.5574 −319.4636 Rη

0.9725 0.9631 0.9571 0.9512 0.9468 0.9396 0.9300

1.2981 1.2882 1.2794 1.2692 1.2609 1.2545 1.2453

1.6228 1.6118 1.5987 1.5869 1.5737 1.5660 1.5546

1.6006

1.9221

2.3039

Results Deviations (%) AAD (%) Bias (%) MD (%) N

293.15 303.15 313.15 323.15 333.15 343.15 353.15

Overall 0.93 −0.14 4.52 126

DEGDME TriEGDME TEGDME 0.88 0.74 1.17 −0.14 −0.17 −0.10 4.53 3.85 4.43 42 42 42

4.3.1. Presentation of the model An approach, based on the free-volume concept, has been very recently proposed [26,27] in order to model the viscosity of Newtonian fluids in both gaseous and dense states. The total viscosity η can be separated into a dilute gas term η0 and an additional η term, such as η = η0 + η. The η term characterizes the passage in a dense state. This approach connects the term η to molecular structure via a representation of the free volume fraction. The viscosity, in this theory, appears as being the product of the fluid modulus ρRT/M and the mean relaxation time of the molecule defined by Na L2 ζ/(RT) and the viscosity can be expressed as η = ρNa ζL2 /M where Na is the Avogadro number, ζ the friction coefficient of a molecule, and L an average characteristic molecular quadratic length. The friction coefficient ζ is related to the mobility of the molecule and to the diffusion

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Table 3 Results obtained on the three compounds with the hard-sphere scheme and the correlated parameters Rη

T(K)

aR

bR

0.488302764

0.498837627

Deviations (%)

Overall

AAD (%) Bias (%) MD (%) N

1.02 −0.14 5.64 126

V0 av (10−4 m3 mol−1 )

bv (10−4 m3 mol−1 )

293.15 303.15 313.15 323.15 333.15 343.15 353.15

3.1817 3.1632 3.1281 3.0927 3.0361 3.0315 3.0166

3.4223 3.3712 3.3853 3.3962 3.4822 3.4211 3.3675

DEGDME, n = 2

TriEGDME, n = 3

TEGDME, n = 4

0.97 0.12 3.50 42

0.81 −0.36 4.11 42

process (diffusion of the momentum for viscosity). Moreover the free volume fraction fv = vf /v (vf = v − v0 ), v is the specific molecular volume and v0 the molecular volume of reference or hard volume) is at temperature T defined by fv = (RT/E)3/2 assuming that the molecule is in a state such that the molecular potential energy of interaction with its neighbors is E/Na . It has been assumed [26] that E = αρ + PM/ρ where the term PM/ρ = PV is connected to the energy necessary to form the vacant vacuums available for the diffusion of the molecules and where E0 = αρ is connected to the barrier energy that the molecule must cross to diffuse. R is the gas constant. Using the empirical relation of Doolittle η = A exp (B/fv ) where B is the characteristic of the free volume overlap, the combination of both equations leads to write ζ in the form ζ = ζ0 exp (B/fv ) and thus: η =

ρNa L2 ζ0 exp (B/fv ) M

But it has been demonstrated that:  M E ζ0 = Na bf 3RT where bf is the dissipation length of the energy E. Finally     αρ + (PM/ρ) 3/2 ρ!(αρ + (PM/ρ)) η = η0 + exp B √ RT 3RTM

1.29 −0.18 5.64 42

The model valid for dense states has been tested [24] using a database of 41 compounds of very different chemical families: alkanes (linear and ramified, light and heavy), alkylbenzenes, cycloalkanes, alcohols, fluoroalkanes (refrigerant), carbon dioxide and water. For the pressure range P < 110 MPa and density range ρ > 200 kg m−3 (dense fluids) there are 3012 points in the database [26], giving an AAD of 2.8%. 4.3.2. Results The parameters values and the results are displayed in Table 4. It must be noticed that the absolute average deviations (AAD) for each of the compounds are lower than the experimental uncertainty with maximum absolute deviations (MD) lower or equal to 5.33% (obtained for the TEGDME). The overall results are then very good, with an absolute average deviation of 1.67%, if we consider that the calculations involve only three parameters for each one of the three polyether. The procedure presented for the hard-sphere scheme can also be used to reduce the number of the parameters. Each main parameter !, E0 and B have then been correlated against the number of CH2 –CH2 –O groups, n with linear equations of the type x = a · n + b (with x = !, E0 or B) so that only six parameters are used, for the three polyethers. The latter results and parameters are shown in Table 5. The results are

(6) where ! = L2 /bf is homogeneous with a length. This equation involves three physical parameters !, α and B which are characteristic of the molecule. It should be noticed that for dense fluids such as those considered here, η0 could be neglected and in a first approximation, E0 can be considered constant, obtaining       E0 + (PM/ρ) E0 + (PM/ρ) 3/2 η = ρ! exp B √ RT 3RTM (7)

Table 4 Results obtained on the three compounds with the free-volume viscosity model Overall ! (Å) E0 (kJ mol−1 ) B AAD (%) Bias (%) MD (%) N

1.67 −0.19 5.33 126

DEGDME

TsriEGDME

TEGDME

0.27475 105.5948 0.0088675

0.26092 141.011 0.006905

0.21154 182.735 0.0055883

1.39 −0.33 4.64 42

1.84 −0.55 5.33 42

1.78 0.30 3.91 42

M.J.P. Comuñas et al. / Fluid Phase Equilibria 222–223 (2004) 331–338 Table 5 Results obtained on the three compounds with the free-volume viscosity model and the correlated parameters

! (Å) E0 (kJ mol−1 ) B

a

b

−0.02652 28.7776 −0.0008095

0.33638 55.7014 0.0093358

Overall AAD (%) Bias (%) MD (%) N

1.93 −0.12 7.82 126

DEGDME 2.20 0.18 7.82 42

TriEGDME 1.41 −0.60 4.93 42

TEGDME 2.18 0.07 7.48 42

slightly of a lower quality but the absolute average is still of the order of the experimental uncertainty. The overall deviations obtained with the hard-sphere scheme representing the variation of the dynamic viscosity as a function number of CH2 –CH2 –O groups, n (AAD = 1.02% and MD = 5.64%) are lower than those obtained with the free volume model (AAD = 1.93% and MD = 7.82%), but the number of parameters used are 16 and 6, respectively. For the free volume model the reduction of the number of parameters from 9 to 6 increases considerably as the obtained MD for DEGDME and TEGDME rising from 3.90 to 7.82% and from 5.33 to 7.48%, respectively. In the case of the hard-sphere scheme the obtained deviations are very similar if 24 or 16 parameters are taken into account.

5. Conclusion

Greek letters η viscosity η0 dilute gas viscosity ρ density ζ free-volume friction coefficient Acknowledgements This work was carried out under the research project PPQ2001-3022 MCYT, Spain and the Spanish–French Joint Action HF 2001-0101 and 04238PG. References [1] [2] [3] [4] [5]

[7] [8] [9] [10] [11] [12] [13] [14] [15]

List of symbols B characteristic of the free-volume overlap bf dissipative length E energy E0 barrier energy fv free-volume fraction L2 average characteristic molecular quadratic length ! = L2 /bf characteristic molecular length M molecular weight

avogadro’s constant pressure gas constant roughness factor for viscosity temperature molar volume free volume reduced molar volume hard-core volume

Na P R Rη T v vf vr v0

[6]

The dynamic viscosity of diethylene glycol dimethylether has been measured upto 100 MPa in the temperature range 293.15 to 353.15 K. The experimental uncertainty for the viscosity measurements is less than 2%, except at 0.1 MPa where the uncertainty is 1%. It follows from the discussion that some simple viscosity approaches with a strong physical and theoretical background (the hard-sphere scheme, the free-volume model) are able to model the viscosity of this compound as well as triethylene glycol dimethylether and tetraethylene glycol dimethylether within the experimental uncertainty. The correlation of the parameters of these models against the number of CH2 –CH2 –O groups has been successfully considered.

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