An extended scaled equation for the temperature

Fluid Phase Equilibria 172 (2000) 169–182

An extended scaled equation for the temperature dependence of the surface tension of pure compounds inferred from an analysis of experimental data C. Miqueu a,∗ , D. Broseta b , J. Satherley c , B. Mendiboure a , J. Lachaise a , A. Graciaa a a

b

Laboratoire des Fluides Complexes (Groupe des Systèmes Dispersés), Université de Pau et Des Pays de l’Adour, B.P. 1155, 64013 Pau Cedex, France Institut Français du Pétrole, 1&4 Avenue de Bois-Préau, 92582 Rueil-Malmaison Cedex, France c Department of Chemistry, University of Liverpool, Liverpool L69 7ZD, UK Received 7 April 2000; accepted 5 June 2000

Abstract We have made a literature survey and performed a critical analysis of the available experimental surface tension data for the most volatile compounds in petroleum fluids: nitrogen, methane, ethane, propane, i-butane, n-butane, n-pentane, n-hexane, n-heptane and n-octane. Including the selected data with those for oxygen, xenon, krypton and those obtained recently for 16 partially halogenated hydrocarbons (refrigerants), we propose the following extended scaled equation to represent the surface tension of these substances: σ = kTc

NA Vc

2/3 (4.35 + 4.14ω)t 1.26 (1 + 0.19t 0.5 − 0.25t)

where t≡1−T/Tc is reduced temperature, k, NA , Vc , and ω are the Boltzmann constant, Avogadro number, the critical volume and the acentric factor, respectively. This equation, which only differs slightly from that proposed by Schmidt et al. [J.W. Schmidt, E. Carrillo-Nava, M.R. Moldover, Fluid Phase Equilibria 122 (1996) 187–206] for refrigerants, yields values for σ within 3.5% of the experimental values for all these compounds. Available data for other compounds (refrigerants) are in agreement with this relation; in the light of that we also examine some compounds (carbon dioxide and argon) for which there exist conflicting datasets. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Surface tension; Alkanes; Refrigerants; Nitrogen; Data; Scaling law

∗ Corresponding author. Tel.: +33-5-59-92-30-41; fax: +33-5-59-92-30-93. E-mail address: [email protected] (C. Miqueu).

0378-3812/00/$20.00 © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 0 ) 0 0 3 8 4 - 8

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C. Miqueu et al. / Fluid Phase Equilibria 172 (2000) 169–182

Nomenclature AAD a2 g k NA Np ng ri T Tc t u(a2 ) u(σ ) Vc

absolute average deviation (%) capillary length (cm2 ) acceleration due to gravity (cm/s2 ) Boltzmann constant Avogadro number number of experimental points not given capillary-bore radii temperature (K) critical temperature (K) reduced temperature uncertainty of capillary rise measurement uncertainty in surface tension critical volume (cm3 /mol)

Greek letters δh uncertainty in height measurement δri uncertainty in bore radii 1ρ density difference between liquid and vapour phases (g/cm3 ) σ surface tension (mN/m) σi fitting parameters for surface tension ω acentric factor

1. Introduction Liquid/vapour surface tensions need to be accurately predicted because they control processes in which the liquid phase is finely dispersed. For instance, the parameters characterising the transport of vapour (gas) or liquid (oil) in a porous medium, such as capillary pressure, relative permeabilities and the residual liquid saturation, are strongly dependent on the surface tension. Surface tension values are, thus, necessary to simulate compositional and gas injection processes in petroleum recovery. To test the reliability of the methods used to calculate the surface tension of complex mixtures, one must first examine how well they perform with pure components. For many compounds, surface tension data can be found in reviews by Jasper [1] and by Vargaftik [2]. However, for substances of interest for petroleum, two reasons have prompted us to resume this work: firstly, these reviews date from the early 1970s and more recent work covering wider ranges of temperature has been published, and, secondly, in some datasets we found errors or misprints. We have first performed a literature survey and a critical analysis of published experimental data in order to provide a reliable database of the surface tensions for the most volatile components in petroleum fluids: nitrogen, methane, ethane, propane, i-butane, n-butane, n-pentane, n-hexane, n-heptane and n-octane.


171

Having selected a set of reliable data for each compound, we aim at proposing a correlation in the form of an extended scaled equation to represent the data. Then, inspired by the work from Schmidt et al. [3] on partially halogenated hydrocarbons (refrigerants), we present a simple general correlation for the temperature dependence of the surface tension of the lower alkanes, nitrogen, oxygen, xenon, krypton and those refrigerants from near the triple point to the critical temperature. Finally, other sets of data are examined in the light of this correlation.

2. Review and analysis of surface tension data of lower alkanes and nitrogen Our search has been limited to the most volatile components of petroleum mixtures for which experimental surface tension data are available in the literature: methane, ethane, propane, i-butane, n-butane, n-pentane, n-hexane, n-heptane, n-octane and nitrogen. For other substances, such as the normal paraffins with a carbon number greater than 9, naphthenic and aromatic compounds, experimental data are scarce but some values can be found in the reviews by Jasper [1], Vargaftik [2], and Korosi and Kovatz [4]. For each compound several sets of experimental values have been published. The selected data were obtained by using the following procedure. We first estimated the uncertainty in the published values by using the information available in the papers (how this estimation was obtained is detailed below, see Section 2.1). Whenever experimental values from two different sets agreed — within the estimated uncertainty — in some overlapping temperature interval, these two datasets were selected. It turned out that for each one of the compounds, the available datasets agreed with the exception of one or two sets: these discordant datasets were rejected. In general, we have not attempted to trace the origin of this discordance. In all the references, the values for surface tension, σ are deduced from measurements of the squared capillary length a2 and the density difference 1ρ between the liquid and vapour phases σ = 21 a 2 g 1ρ

(1)

where g is the acceleration due to gravity. The squared capillary length is derived from 1. differential capillary rise measurements (this is the most often used technique), and 2. contour analysis of pendant drops (e.g. [5]). The density difference and capillary length should ideally be measured simultaneously, as done for instance by Rathjen and Straub [6] for CO2 , SF6 , CF3 Cl and CBrF3 and Schmidt et al. [3], for a series of refrigerants, using refractometry. For the compounds considered in this article, we are not aware of such simultaneous measurements of a2 and 1ρ. To derive σ from their own capillary length measurements, the authors have used values of the phase densities determined separately and interpolated at the temperatures of the capillary length measurements. Whenever the authors gave their capillary length values, we used the most recent density measurements to recalculate σ , but found values for σ very close to the published values. An analysis has been carried out to obtain the uncertainty associated with each value of σ . 2.1. Uncertainty analysis To compare the data and to carry out a non-linear least-squares analysis we had first to estimate the uncertainty associated with each datum point from the information given by the authors. We have followed the procedure given by Schmidt et al. [3], which we briefly recall here.

172


When obtained from separate capillary rise and density measurements, the uncertainty in the surface tension, u(σ ), arises from the uncertainties in the determinations of the squared capillary length a2 and density difference 1ρ ( )1/2 2 u(1ρ) 2 u(a 2 ) + (2) u(σ ) = σ a2 1ρ The uncertainty in a2 , u(a2 ), arises from the uncertainties in 1. The cathetometer height measurements, δh (usually in the range of 5–20 ␮m) √ 2δh uh (a 2 ) ≈ (1/ri ) − (1/rj ) for a differential measurement with two capillaries i and j with internal radii ri and rj . 2. The measurement of the capillary internal radii, δri (usually in the range of 0.1–5 ␮m)  !2 1/2 2 2 a  δri + δrj  ur (a 2 ) ≈ |(1/ri ) − (1/rj )| ri2 rj2

(3)

(4)

The squared total uncertainty u2 (a2 ) is the sum of the above squared uncertainties. It can be shown that it has the following dependence on a2 : u2 (a 2 ) ≈ Aa4 + B

(5)

where A and B are simple functions of the uncertainties δri and δh. These uncertainties were not provided by all of the authors. For the cases where this information was not provided conservative estimates for δri and δh, and therefore for A and B, were adopted. For the uncertainties in 1ρ, we carried out a detailed analysis for only a few compounds (methane, ethane, propane, i-butane, n-butane, nitrogen) for which sufficient information was given. The outcome of this analysis was that this source of uncertainty was negligible compared to that originating from the determination of a2 . The uncertainty associated with each datum point is, therefore, given by u(σ ) = σ

u(a 2 ) a2

(6)

2.2. Sets of data The selected data for the lower alkanes and nitrogen are shown in Tables 1–10. These tables give the references to the original work, the number of experimental points per dataset, the temperature and surface tension ranges, the measured parameters and the stated purity of the compounds. For a number of the compounds the surface tension has been calculated from capillary length data from one author and density data from another one. Where this has been carried out is indicated in the tables. In the case of methane, propane and n-butane some additional explanation about the sets is required and is found below otherwise the tables are self-explanatory.


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Table 1 Surface tension references for methane Reference

Np

T range (K)

σ range (mN/m)

Measured parameter

Purity (%)

[16]a [18] [17]

15 11 26

159.83–190.27 96.29–109.86 90.95–114.65

3.81–0.009 13.15–6.28 16.98–12.43

a2 σ σ

99.995 99.95 99.99

a

Calculated with densities from [19].

Table 2 Surface tension references for ethane Reference

Np

T range (K)

σ range (mN/m)

Measured parameter

Purity (%)

[20]a [18] [8]a,b [9]a [21]

46 15 7 6 8

89.87–04.93 117.8–136.67 272.4–299.0 133.15–183.15 113.15–183.15

33.54–0.013 27.08–23.92 3.14–0.28 24.48–16.18 28.08–16.31

a2 σ a2 a2 σ

99.99 99.997 ng ng ng

a b

Calculated with densities from [22]. Temperatures translated so that the critical point and the correct one do correspond.

Table 3 Surface tension references for propane Reference a

[23] [5]a [24] a

Np 30 23 10

T range (K) 273.15–364.88 308.15–369.26 143.15–233.15

σ range (mN/m)

Measured parameter 2

Purity (%)

10.19–0.28 5.77–0.009 27.80–15.15

a a2 σ

ng 99.99 ng

σ range (mN/m)

Measured parameter

Purity (%)


Table 4 Surface tension references for n-butane Reference a

[5] [25]a [26] a

Np 41 13 11

T range (K) 310.93–422.04 237.05–302.85 173.15–273.15

2

10.35–0.11 23.21–12.46 27.20–14.84

a a2 σ

99.91 ng ng

σ range (mN/m)

Measured parameter

Purity (%)


Table 5 Surface tension references for i-butane Reference a

[23] [25]a [26] a

Np 30 13 8

T range (K) 273.15–403.96 236.85–296.45 173.15–243.15


13.05–0.15 21.48–10.3 25.2–16.48

2

a a2 σ

ng ng ng

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Table 6 Surface tension references for n-pentane Reference

Np

T range (K)

σ range (mN/m)

Measured parameter

Purity (%)

[27]a [28] [29] [26]

25 11 3 8

144.18–469.67 302.72–467.26 283.15–303.15 253.15–313.15

33.76–0.002 14.76–0.072 17.15–14.94 20.5–13.8

a2 σ σ σ

99.90 98.66 ng ng

a

Calculated with liquid densities from [30] and vapour ones from [2].

Table 7 Surface tension references for n-hexane Reference

Np

T range (K)

σ range (mN/m)

Measured parameter

Purity (%)

[31]a [27]a [26] [29] [32]

42 24 13 6 24

273.91–463.28 175.12–507.39 223.15–333.15 283.15–333.15 273.15–493.15

20.59–2.68 31.74–0.002 25.8–14.33 19.42–14.32 20.56–0.58

a2 a2 σ σ σ

>97.0 99.9 ng ng ng

Measured parameter

Purity (%)

a


Table 8 Surface tension references for n-heptane Reference a

[27] [32] [28] [26] [29] a

Np 23 23 13 14 9

T range (K) 183.21–540.08 293.15–513.15 303.15–537.59 243.15–363.15 283.15–363.15

σ range (mN/m) 32.28–0.001 20.86–1.29 19.02–0.061 25.30–13.60 21.12–13.28

2

a σ σ σ σ

99.89 ng 98.413 ng ng


For methane, the data listed in Table 1 differ from those compiled from Vargaftik [2] and Jasper [1]. Jasper [1] claims that he used measurements from Fuks and Bellemans [7] but the values he quotes are very different from the ones that appear in the original article. Consequently, their data have not been included in the table. In the case of propane as well, the data listed in Table 3 are not in agreement with Table 9 Surface tension references for n-octane Reference

Np

T range (K)

σ range (mN/m)

Measured parameter

Purity (%)

[27]a [32] [31]a [26] [29]

22 23 43 11 12

218.51–568.79 293.15–513.15 274.53–463.15 298.15–393.15 283.15–393.15

9.26–0.0005 21.82–3.10 23.52–6.70 21.26–12.60 22.57–12.11

a2 σ a2 σ σ

99.89 ng >99.0 ng ng

a



175

Table 10 Surface tension references for nitrogen Reference

Np

T range (K)

σ range (mN/m)

Measured parameter

Purity (%)

[16]a [1] [24]

14 7 7

92.74–125.87 78–90 68–90

5.32–0.014 8.75–6.03 11.0–6.16

a2 σ σ

99.9992 ng ng

a


those from Katz and Saltman [8] and Jasper [1]. Jasper claims that he uses the data from Leadbetter et al. [9] but in this article only data for ethane and nitrous oxide are presented. In their article, Katz and Saltman [8] did observe discrepancies with previous data from Maass and Wright [10] but explain that ‘no error or reason could be found’. Consequently, we have not included these data in Table 3. For the same reason we have rejected the data for n-butane from Katz and Saltman [8]. These tables provide a consistent and reliable database of surface tension data of light components of petroleum fluids and hence, are a useful tool to test different correlations for calculating surface tension or to obtain parameters appearing in such correlations. The sets of surface tensions and estimated uncertainties are available on request. 3. Surface tension correlation 3.1. Theoretical background The expression used for representing the surface tension data is derived from our current understanding of the critical behaviour of fluids. The asymptotic behaviour (divergence or vanishing) of physical properties when approaching the critical point can be represented by scaling laws with universal exponents and substance-dependent parameters. Thus, the vanishing of surface tension on approaching the critical point (t=1−T/Tc → 0) is described by σ = σ0 t µ

(7)

where µ is an universal exponent, equal to 1.26. Actually, from Widom’s scaling law for the interfacial tension [11]: µ=2ν, ν=0.63 being the exponent for the divergence of the correlation length, and σ 0 is a substance-dependent parameter. The above equation holds very near the critical point (t≈0) and is well verified experimentally for near-critical liquid–vapour systems. Corrections for this asymptotic behaviour remote from the critical point are accounted for by the so-called Wegner’s [12] expansion in powers of t∆ [13–15], where ∆≈1/2 is the correction-to-scaling exponent σ = σ0 t 1.26 (1 + σ1 t 0.5 + σ2 t + · · · )

(8)

3.2. Application to the surface tension data of alkanes and nitrogen The surface tension versus temperature data collected in Section 2 have been fitted to the previous expression (8) truncated after the first order in t 0.5 σ = σ0 t 1.26 (1 + σ1 t 0.5 )

(9)

176


Table 11 Surface tension fitting parameters to Eq. (9) and deviation of calculated σ from experimental data Substance

σ0

σ1

AAD (%)

N2 C1 C2 C3 iC4 nC4 nC5 nC6 nC7 nC8

25.9 35.3 56.3 60.4 55.6 58.2 55.2 52.4 56.3 57.9

0.210 0.146 −0.102 −0.191 −0.112 −0.123 −0.011 0.056 −0.041 −0.091

1.4 1.5 2.2 1.1 1.3 0.6 3.2 4.8 2.2 4.0

and after the second order in t σ = σ0 t 1.26 (1 + σ1 t 0.5 + σ2 t)

(10)

The fitting parameters σ i , obtained by non-linear least-squares minimisation, are listed in Table 11 (Eq. (9)) and Table 12 (Eq. (10)) for the lower alkanes and nitrogen. According to Eq. (9) the reduced surface tension σ /t1.26 is a monotonic function either decreasing (if σ 1 is negative) or increasing (if σ 1 is positive) with t. In reality, the experimental reduced surface tension typically increases at low t, goes through a slight maximum for t in the range 0.05–0.2 mN/m and then decreases at large t. This behaviour can only be described by an equation of the form (10) in which σ 1 is positive and σ 2 is negative, as it is the case for most of the compounds (see Table 12). Eq. (9) will, however, prove to be sufficient for representing surface tension data in a range of temperature not too close to the critical temperature. For each compound, Eqs. (9) and (10) with the coefficients listed in Tables 11 and 12 represent the experimental data very precisely (in fact within experimental uncertainty) over the whole range of the temperature corresponding to two phase coexistence. As a rule, deviations between experimental surface tensions and values calculated with Eqs. (9) and (10) are very small far from the critical point and increase, due to the experimental uncertainties, when approaching the critical point. Table 12 Surface tension fitting parameters to Eq. (10) and deviation of calculated σ from experimental data Substance

σ0

σ1

σ2

AAD (%)


25.4 32.3 49.7 59.8 50.9 59.2 53.2 48.9 55.0 52.1

0.344 0.828 0.324 −0.156 0.238 −0.194 0.157 0.399 0.072 0.366

−0.171 −0.779 −0.343 −0.033 −0.332 0.068 −0.169 −0.342 −0.116 −0.447

1.3 1.1 2.6 1.2 0.3 0.6 2.6 3.8 2.0 2.9


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Table 13 Critical parameters and acentric factor for lower alkanes and nitrogen Substance

ω

Tc (K)

Vc (cm3 /mol)


0.039 0.0115 0.099 0.153 0.183 0.199 0.251 0.299 0.349 0.398

126.19 190.56 305.34 369.85 407.85 425.16 469.7 507.6 540.2 568.8

89.41 98.63 145.45 200.00 259.07 255.10 304.00 370.00 432.00 492.00

Among the three constants σ i , σ 0 has the dimension of a surface tension while σ 1 and σ 2 are pure numbers. From a simple corresponding-states argument, σ 0 should, thus, be proportional to kTc (NA /Vc )2/3 , where Tc and Vc are the critical temperature and volume, k the Boltzmann constant and NA is the Avogadro number. Anticipating some deviation from the corresponding states principle, σ0 /kTc (NA /Vc )2/3 is not constant but taken to vary (in a first approximation) linearly with the acentric factor ω=−1−log10 (Ps /Pc ), where Ps is the vapour pressure for a reduced temperature T/Tc =0.7. From Tables 11 and 12, it appears that there is no obvious trend for σ 1 and σ 2 . Our purpose is actually to represent the data with an expression similar to the one proposed by Schmidt et al. [3] NA 2/3 1.26 t (1 + σ1 t 0.5 + σ2 t) (11) σ = (a + bω)kTc Vc where a, b, σ 1 and σ 2 form a unique set of constants. Using the data and uncertainties determined in Section 2, we obtain from non-linear least squares fitting the following values: NA 2/3 σ = kTc (4.49 + 4.12ω)t 1.26 (1 + 0.083t 0.5 − 0.163t) (12) Vc The values of acentric factor, critical temperature and volume are listed in Table 13 for each substance. Eq. (12) yields values of σ within an average of 3.7% of the measured values for all components. The deviations between the measured and calculated values (using Eq. (12)) are plotted as a function of t in Fig. 1, where it is clearly apparent that these deviations grow near the critical point. 3.3. Generalisation to other substances In a second stage of this study, we have included additional compounds for which surface tension data over a wide temperature range are available. These data cover some gases: O2 , Xe, Kr and the refrigerants considered by Schmidt et al. [3]. These compounds are listed in Table 14, together with the references to the capillary rise and densities measurements.

178


Fig. 1. Deviations ((σexp − σ(12) )/σexp ) of experimental surface tensions from Eq. (12) vs. reduced temperature t=1−TR for the lower alkanes+nitrogen.

A non-linear least squares fitting to Eq. (11) of the whole sets of data, including the latter compounds and those considered in Section 2, yields the following equation: NA 2/3 (4.35 + 4.14ω)t 1.26 (1 + 0.19t 0.5 − 0.25t) (13) σ = kTc Vc This equation yields values for σ within an average absolute deviation (AAD) of 3.5% of the measured values for all 29 compounds. Parameters only differ slightly from the ones in Eq. (12) and the estimation Table 14 Densities and surface tension references

O2 Xe Kr R32 R123 R123a R124 R125 R134 R134a

ρ ref

σ ref

[34] [36] [17] [38] [39] [35] [3] [38] [35] [39]

[16] [37] [17] [38] [39] [35] [3] [38] [35] [39]

R141b R142b R143a R152a R236ea R236fa R245ca R245fa E125

ρ ref

σ ref

[35] [35] [3] [35] [3] [3] [3] [3] [3]

[35] [35] [3] [35] [3] [3] [3] [3] [3]


179

Fig. 2. Experimental reduced surface tensions (circles) for the 29 components as a function of acentric factor, ω, and reduced temperature t=1−TR . The surface is Eq. (13).

of the surface tension for alkanes and nitrogen is almost as good as with Eq. (12). The results of Eq. (13) are plotted in Fig. 2 and Fig. 3 displays deviations of the data to the calculated values. Schmidt et al. [3] obtained for their refrigerants the following scaled equation to represent the experimental surface tensions: NA 2/3 (4.498 + 3.513ω)t 1.26 (1 + 0.348t 0.5 − 0.487t) (14) σ = kTc Vc

Fig. 3. Deviations ((σexp − σ(13) )/σexp ) of experimental surface tensions from Eq. (13) vs. reduced temperature t=1−TR for the 29 components.

180


Fig. 4. Experimental and calculated Eq. (13) (- - - -) surface tensions for CF3 Cl (䉬), CBrF3 (䊉) and SF6 (䉱) [6].

which described the data of the refrigerants with an absolute average deviation of nearly 5.5%. Eq. (14) has been modified, compared to the one in [3], to account for the conventional definition of the acentric factor ω = −1 − log10 (Ps /Pc )Tr =0.7 (while in [3] the natural logarithm was used to define ω). Eq. (13) provides an absolute average deviation of 5.9% when applied solely to the refrigerants. The relationship of Schmidt et al. [3], Eq. (14), is indeed better when applied only to the refrigerants but is less suitable than Eq. (13) when considering a more general list of components as it provides an AAD of 4.5% when it is applied to the 29 components we have used. Indeed, these compounds span a wider range of acentric factors from 0.005 (Kr) to 0.398 (nC8 ) than the refrigerants considered in [3]: 0.213 (r141b)–0.387 (r245fa). We have applied Eq. (13) to other substances that have not served to establish it to test if it could be used in a predictive way. We only found three compounds for which surface tension data were available in a wide range of temperature: CClF3 , CF3 Br, SF6 [6]. As shown in Fig. 4, the surface tensions calculated with Eq. (13) are in very good agreement with the experimental data of these three substances, which demonstrates again the generality of Eq. (13). Finally, we have applied Eq. (13) to sets of conflicting published data concerning carbon dioxide and argon, for which we have no means to discriminate between these data. The data for carbon dioxide are from Gielen et al. [16] and Rathjen and Straub [6] and the ones for argon are from Gielen et al. [16] and Nadler et al. [17]. We observed for each substance that one of the datasets could be described by Eq. (13) but not the other (see Fig. 5 for carbon dioxide). For argon, the data from Nadler et al. can be correctly reproduced by Eq. (13) with an AAD of 1.9%, unlike the data from Gielen et al. that are systematically too low by around 4%. If we suppose again that Eq. (13) should be valid for every substance that obeys critical scaling laws and the extended corresponding states principle — which is the case for argon and carbon dioxide — then we have to select the data obtained for carbon dioxide by Rathjen and Straub [6] and for argon by Nadler et al. [17].


181

Fig. 5. Experimental and calculated Eq. (13) (- - - -) surface tensions for carbon dioxide: [16] (䉬), [6] (䉱).

4. Conclusion We have collected and selected sets of reliable experimental surface tensions for some of the most volatile components of petroleum fluids — nitrogen, methane, ethane, propane, i-butane, n-butane, n-pentane, n-hexane, n-heptane and n-octane — and provided for each an equation suitable to accurately represent these data. Gathering these data together with those available for other substances of industrial interest (gases and refrigerants), we propose a correlation suitable to describe surface tension as a function of temperature and the component’s critical temperature, volume and acentric factor. This correlation, Eq. (13), is able to represent the surface tension of 31 substances up to the critical point with an absolute average deviation of less than 3.5%. This equation should be valid for all normal (i.e. non-polar and non-hydrogen-bonded) fluids.

Acknowledgements We are very grateful to J. Schmidt from NIST for kindly providing us with the surface tension data and the associated uncertainties for the refrigerants. This work has been supported by Elf Exploration Production (EEP), TotalFina, Institut Français du Pétrole et Gaz de, France, as well as by a joint EEP-University of Liverpool Marie Curie fellowship (Contract no. JOF3-CT98-5009). We thank S. Bickerton for assistance with preparing the manuscript. C. Miqueu is grateful to the University of Liverpool and the High Pressure Laboratory of the Chemistry Department for hosting her visit. She thanks EEP for financing her PhD. The authors are very grateful to F. Montel (EEP) for suggesting this joint project, for his helpful comments and for the interest he has taken in this study.

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