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Fluid Phase Equilibria 252 (2007) 66–73

Vapour–liquid interfacial tension of water and hydrocarbon mixtureat high pressure and high temperature conditions

A. Bahramian a,∗, A. Danesh b, F. Gozalpour b, B. Tohidi b, A.C. Todd b

a Institute of Petroleum Engineering, University of Tehran, Tehran, Iranb Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh EH14 4AS, UK

Received 3 May 2006; received in revised form 5 November 2006; accepted 20 December 2006Available online 5 January 2007

bstract

Interfacial tension values of two liquid–liquid–vapour systems, a ternary water–decane–methane and a quaternary water–cyclo-exane–decane–methane, were measured using a pendant drop technique at 150 ◦C and high pressures up to 28.1 MPa. The tested ternary mixture

howed a maximum liquid–liquid interfacial tension by increasing pressure whereas it only slightly increased liquid–liquid interfacial tension in theuaternary system. The predicted interfacial tension values by some models are compared and the application of liquid–liquid interfacial tensionodels for predicting the liquid–vapour interfacial tension at high pressure conditions is discussed.2007 Elsevier B.V. All rights reserved.

cane;

caa

2

aTcdsstnarp

eywords: Interfacial tension; Liquid–liquid equilibrium; Water; Methane; De

. Introduction

The interfacial tension (IFT) values of water–hydrocarbonystems affect important features of petroleum reservoir pro-esses, such as water–oil contact movement, water alternatingas drive (WAG) [1], trapping of hydrocarbons by water floodnd oil recovery by depressurisation [2]. Interfacial tension datare also required in calculations of transition zones in reservoirs.lthough data on water–hydrocarbon fluids at two-phase con-itions have been reported in the literature [3–23], such datat three-phase conditions, particularly at high pressure–highemperature (HPHT) conditions, are very scarce. Such data areecoming more valuable as deeper, hence higher pressure andemperature reservoirs are being exploited. At HPHT conditions,he mutual solubility of water and hydrocarbon phases is higher;herefore, the IFT of hydrocarbon–water systems at HPHT con-itions can be significantly different from those in conventionaleservoirs.

In this work, novel measured IFT data on two liquid–liquid–apour systems of equilibrated water–hydrocarbon mixtures,

ternary water–decane–methane and a quaternary water–

∗ Corresponding author. Tel.: +98 21 88632975; fax: +98 21 88632976.E-mail address: abahram@ut.ac.ir (A. Bahramian).

oudamid

378-3812/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2006.12.013

Cyclohexane; Experimental data

yclohexane–decane–methane are presented. Predictive modelsre employed and evaluated against the measured liquid–liquidnd liquid–vapour IFT data.

. Experiments

Fig. 1 shows the configuration of the HPHT rig incorporatingpendant drop facility for measurement of interfacial tension.he pendant drop method for measuring IFT at high pressureonditions is a well-established and documented procedure, withetails available in the literature [24]. The schematic diagramhows a droplet of liquid suspended from the tip of a stainlessteel dropper, which was made oil-wet prior to being installed inhe window cell. The drop is viewed remotely by a camera con-ected to a video monitor, where the drop is dimensioned usingvideo scale device. The equilibrium phase densities are also

equired in the calculation of IFT. These are measured by trap-ing each of the phases, in turn inside the removable pressure celln the left of the diagram and by measuring the weight and vol-me of the sample its density can readily be calculated. Densityata were measured for the three equilibrium phases at lowest

nd highest test pressures (3.4 and 28.1 MPa) and 150 ◦C. Theeasured density data were employed to adjust parameters of an

n-house phase behaviour model (Appendix A) for calculatingensity at other pressures.

A. Bahramian et al. / Fluid Phase E

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ig. 1. Schematic diagram of the HPHT equipment configured to measure inter-acial tension by the pendant drop technique.

Tests fluids were composed of decane (Sigma-Aldrich,99%) cyclohexane (Sigma-Aldrich, +99.9%, HPLC grade),ethane (Air Products, 99.995%) and triple distilled water.ecane and cyclohexane were purified separately by passing

ach six times through basic (as opposed to acidic) aluminiumxide to remove surface-active components. Initial tests onecane–water resulted in a data matching those in the literature22] within the error bands of the measurements.

All the components were individually added gravimetricallyo a previously cleaned and evacuated pressure vessel. Typicaloading of each compound was about 20–30 g with a resolutionf 0.001 g using a Mettler Toledo PR5003 mass comparator. Thisrovided compositional data on the molar fraction of compo-ents with a deviation of ±0.0001. The vessel was then mountedn the HPHT facility. The sample was pumped back and forthetween the high pressure cells to facilitate mixing and achiev-ng equilibrium at the test pressure before attempting to performny measurements.

The oil-wet pendant dropper was used to form droplets ofydrocarbon-rich liquid and water-rich liquid in the equilibriumapour phase for liquid–vapour IFT data measurements. TheFT of hydrocarbon-rich liquid/water-rich liquid was determinedy suspending a droplet of water in the liquid hydrocar-

on phase. It should be noted that all three phases (vapourhase, hydrocarbon-rich liquid phase and water-rich liquidhase) were present and they were at equilibrium during thexperiment.

Ttii

quilibria 252 (2007) 66–73 67

.1. Water–decane–methane three-phase system

The three-phase interfacial tension of water–decane–ethane system was measured at 150 ◦C and pressures of 3.4,

3.7, 24.0 and 28.1 MPa. Additional measurements were alsoade on hydrocarbon-rich liquid–vapour IFT, at three phase

onditions, at some pressure intervals, as shown in Table 1.The interfacial tension between vapour (V) and hydrocarbon-

ich liquid (O) increases sharply from 0.088 mN m−1 at8.1 MPa to 10.5 mN m−1 at 3.4 MPa. Similarly, the interfa-ial tension between vapour and water-rich liquid (W) increasesrom 30.1 to 41.7 mN m−1 over the above pressure range. Pres-ure reduction, however, initially increases the IFT between thewo liquid phases, reaching to a maximum value before reducingy further pressure reduction. However, changes in OW IFT areelatively small in comparison with the other pairs. This is inine with the effect of pressure on properties of liquids, whichs small, and the changes are considered to be mainly due toompositional variations of the hydrocarbon-rich liquid phase.

A main application of three-phase IFT data is in determina-ion of fluid distribution in rock pores of petroleum reservoirs,here the relative location of fluid phases is controlled by the

haracteristics of rock wetting and fluid spreading, described byhe spreading coefficient, S, defined as:

= σWV − (σVO + σOW) (1)

A negative spreading coefficient would allow a three-phaseuid contact, whereas full spreading where a liquid phase com-letely separates the second liquid from the vapour phase wouldccur at S = 0. The latter is equivalent to Antonow’s rule [25,26],hat is in a three-phase system at equilibrium, the sum of IFTalues of the two pairs more proximate phases is equal tohat of the largest one. The experimental data indicates thathe reduction of hydrocarbon-rich liquid/vapour IFT, σVO, byncreasing pressure is almost equal to the increase in water-richiquid/vapour IFT, σWV, with minimal changes in hydrocarbon-ich liquid/water IFT, σOW. The spreading coefficient remainslmost equal to zero, within the error band of IFT measurementsver the tested conditions.

.2. Water–cyclohexane–decane–methane three-phaseystem

The sample was gravimetrically prepared and the resul-ant mixture had the following molar composition: methane3.13 ± 0.01, decane 5.370 ± 0.004, cyclohexane 8.43 ± 0.01nd water 23.07 ± 0.01. The tests were performed at 150 ◦C andt four pressures, 13.7, 17.1, 20.5 and 24.0 MPa using the HPHTacility. The results are given in Table 2.

The hydrocarbon-rich liquid in equilibrium with the vapourhase shows a wide range of interfacial tension from.22 mN m−1 at 24.0 MPa rising to 3.7 mN m−1 at 13.7 MPa.

he water-rich liquid phase/vapour IFT increased only from 30.5

o 32.4 mN m−1 over the above pressure range. The trend in thenterfacial tension between the liquid phases was opposite wheret reduced with increasing pressure. No IFT value was recorded

68A

.Bahram

ianetal./F

luidP

haseE

quilibria252

(2007)66–73

Table 1Phase density and IFT of the water–decane–methane three-phase system at 150 ◦C

Pressure(MPa)

Vapour density(g cm−3, ±0.005)

Hydrocarbon-rich liquiddensity (g cm−3, ±0.005)

Water-rich liquid density(g cm−3, ±0.0001)

Vapour–hydrocarbon-richliquid IFT (mN m−1)

Vapour–water-richliquid IFT (mN m−1)

Liquid–liquid IFT(mN m−1)

S (mN m−1)

3.4 0.025 0.619 0.9179 10.5 ± 0.3 41.7 ± 0.7 28.5 ± 0.6 2.7 ± 1.66.8 0.047 0.595 8.0 ± 0.2

13.7 0.126 0.548 0.9214 4.1 ± 0.1 34.9 ± 0.6 31.1 ± 0.6 −0.3 ± 1.320.5 0.227 0.493 1.6 ± 0.124 0.256 0.462 0.9254 0.80 ± 0.04 32.8 ± 0.6 30.0 ± 0.5 2.0 ± 1.1

27.4 0.253 0.426 0.17 ± 0.0128.1 0.248 0.417 0.9270 0.088 ± 0.01a 30.1 ± 0.5 29.6 ± 0.5 0.412 ± 1.01

a The IFT value was measured by the meniscus height technique, described in ref. [51] as at this pressure the droplet was too small to be measured by the pendant drop method.

Table 2Phase density and IFT of the water–cyclohexane–decane–methane three-phase system at 150 ◦C

Pressure (MPa) Vapour density(g cm−3, ±0.005)

Hydrocarbon-rich liquiddensity (g cm−3, ±0.005)

Water-rich liquid density(g cm−3, ±0.0001)

Vapour–hydrocarbon-richliquid IFT (mN m−1)

Vapour–water-richliquid IFT (mN m−1)

Liquid–liquid IFT(mN m−1)

S (mN m−1)

13.7 0.105 0.584 0.9237 3.7 ± 0.1 32.4 ± 0.6 – –17.1 0.139 0.549 0.9255 2.4 ± 0.1 31.8 ± 0.6 27.6 ± 0.5 1.8 ± 1.220.5 0.178 0.503 0.9273 1.11 ± 0.05 30.5 ± 0.5 29.0 ± 0.5 0.39 ± 1.0524.0 0.225 0.423 0.9291 0.22 ± 0.01 30.5 ± 0.5 29.8 ± 0.5 0.48 ± 1.01

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t 13.7 MPa because the water droplet could only be partiallyeen as the refractive indices of the two phases became similar.he spreading coefficient over the tested range almost remainsqual to zero, within the accuracy of IFT measurements, similaro the previous case.

. Prediction

There are a number of theoretical based equations and empir-cal correlations to predict IFT of vapour–liquid or liquid–liquidystems [27–40]. The methods for vapour–liquid systems areenerally applicable at low-pressure conditions as the propertiesf the vapour phase are generally neglected. The methods usedor liquid–liquid systems are mostly reliable for simple mix-ures dominated by short range molecular forces. The reliabilityf some of the most widely used methods for hydrocarbon–waterpolar compound) mixtures at three-phase HPHT conditions isvaluated by comparing their predictions against the measuredata in this study.

.1. Liquid–vapour IFT

The most widely used correlation in the petroleum industry ishe parachor method [27–31], which is based on the observationsf Macleod [29] and the later work of Sugden [30]. The latteras extended to multi-component systems by Weinaug and Katz

31]:

σ��)1/4 =

c∑i=1

[Pix�i ρ� − Pix

�i ρ�] (2)

here σαβ is the interfacial tension between two phases � and, x the mole fraction, Pi the parachor value of the compound ind ρ is the molar density.

Firoozabadi and Ramey [32] demonstrated that IFTetween water and pure hydrocarbons, over a wide range ofemperature–pressure, can be related to the phase density differ-nce by a single curve, using the following IFT function:

FT function ≡ (σhw)0.25

(T/T hc )

0.3125

ρw − ρh (3)

here σhw is the hydrocarbon (vapour or liquid)–water-richiquid IFT and T h

c is the hydrocarbon phase critical temperature.The reliability of correlation was demonstrated for various

ompounds ranging from methane to dodecane. The curve cane represented almost by the following equation [41]:

hw = 111(ρw − ρh)1.024

(T

T hc

)−1.25

(4)

here IFT is in mN m−1, ρw and ρh are the water and hydrocar-on phase density in g cm−3, respectively.

.2. Liquid–liquid IFT

Amongst all the available methods to predict the liquid–liquidFT [34–40], only few of them relate IFT to mutual solubil-

oati

quilibria 252 (2007) 66–73 69

ty [34,36,39,40]. The Donahue–Bartel correlation [34] is theldest, which is based on their observation that plotting theiquid–liquid IFT of binary systems versus the logarithm of theum of mutual solubility results in a linear relationship:

OW = a − b log(xwo + xo

w) (5)

here σOW is the oil–water IFT, xwo and xo

w stand for moleraction of oil in the water-rich phase and water in the oil-richhase, respectively. Constants a and b, found by regression ofxperimental data [42], are −3.31 and 15.61, respectively. Ithould be noted that Donahue–Bartel, and the other models usedn this study, can also be used in non-aqueous systems.

Eq. (5) was modified later by Treybal [36] for ternary systems:

OW = a − b log

(xw

o + xow + xw

3 + xo3

2

)(6)

The compound 3 in the above equation is the hydrocarbonhat is more soluble in the water-rich phase. Eqs. (5) and (6)re the simplest equations of the kind but are limited to binarynd ternary systems. However, they have been found [42,43] toe the most accurate methods as well as the Fu et al. equation,hich is as follow [39]:

OW = KRTX

Aw0 exp(X)(xwo qO + xo

wqW + xr3q3)

(7)

here,

= ln(xwo + xo

w + xr3) (8)

Here the compound 3 is defined similar to that in Eq. (6), andhe letter r denotes the phase which is richer in compound 3.w0 is the van der Waals area of a standard segment, q the pureompound area parameter defined by the UNIQUAC model,temperature, R the universal gas constant and the empirical

arameter, K, found to be 0.9414 by regression using experimen-al IFT of binary systems [39]. Eq. (7) has the same limitationf Eq. (6), that is, it cannot be used for systems with more thanhree components.

Recently, employing the lattice theory and regular solutionssumptions, Bahramian and Danesh [40] proposed a simplequation for predicting liquid–liquid IFT of multi-componentystems. The equation does not include any empirical parametersnd has no limitation with the number of components:

c

i=1

[(xo

i xwi )0.5 exp

( ai

RTσOW

)]= 1 (9)

here c is the number of components and ai stands for the partialolar surface area of the ith compound at the interface.

.3. Results

The reviewed IFT models were evaluated against the mea-ured liquid–liquid–vapour IFT data. The required composition

f equilibrated phases for IFT predictions was calculated usingn in-house phase behaviour thermodynamic model based onhe Peng–Robinson equation of state with non-random mix-ng rules, to account for polarity of water (Appendix A). The

70 A. Bahramian et al. / Fluid Phase Equilibria 252 (2007) 66–73

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mwTtsdmovttLalpavi

T

wwh

u

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W

W

ig. 2. Measured and predicted liquid–liquid interfacial tension values of theater–decane–methane system at 150 ◦C.

odel and its reliability in predicting the mutual solubility ofater–hydrocarbons have been described elsewhere [44,45].he predicted compositions were employed then to calculate

he liquid–liquid and liquid–vapour IFT of the both investigatedystems using the described IFT models. The required phaseensities were determined using the in-house phase behaviourodel with its parameters adjusted to match a limited number

f measured density data of equilibrium phases. The parachoralues can be found in the literature [41] or calculated fromhe surface tension of the pure components. The area parame-er for methane in Eq. (9) can be calculated from the reportedennard–Jones size parameter [46] and those of the others,ccording to Bahramian and Danesh [40], from their molar pureiquid volumes, except for water, which is suggested to be 32◦ A2

er molecule [40,47,48]. The parameters in Eq. (7), that is Aw0nd q, are available in the literature [39,49]. The molar averagealue of the critical temperature of phase components was usedn Eq. (4).

hc =

∑i�=j

(xi∑i�=jxi

Tci

)(10)

here Tci is the critical temperature of component i, and j is

ater component. In other words, the critical temperature ofydrocarbon phase is calculated on a dry basis (water free).

Figs. 2 and 3 show the experimental liquid–liquid IFT val-es of the investigated systems as well as the predicted results.

potT

able 3verage absolute percentage deviation (AAD%) of the predicted interfacial tension v

Bahramian–Danesh [40] Firo

ater–decane–methaneVapour–hydrocarbon-rich liquid 673 –Vapour–water-rich liquid 21 5Water-rich liquid–hydrocarbon-rich liquid 3 43

ater–cyclohexane–decane–methaneVapour–hydrocarbon-rich liquid 68 –Vapour–water-rich liquid 8 29Water-rich liquid–hydrocarbon-rich liquid 6 24

ig. 3. Measured and predicted liquid–liquid interfacial tension values of theater–cyclohexane–decane–methane system at 150 ◦C.

hile the Bahramian–Danesh model can reliably predict the IFTalues of the ternary and quaternary systems, with an averagebsolute percentage deviation (AAD%) from the experimentalata of 3% and 6%, respectively, the deviations of the Trey-al (AAD = 77%) and Fu et al. (AAD = 57%) equations for theernary system indicate their lack of applicability at HPHT con-itions (Table 3). Their poor results, however, would be expecteds their empirical parameters have been calculated by matchingFT values at 25 ◦C and atmospheric pressure. The predictionsy the model of Fu et al., approach the measured value athe lowest tested pressure of 3.4 MPa. Fig. 3 does not includeesults of the Treybal and Fu et al. equations, owing to theirimitation for systems with more than three components. Theesults of the Firoozabadi and Ramey model, as indicated inigs. 2 and 3, show a monotonically decrease in liquid–liquidFT of the investigated ternary system (AAD = 43%), whereasor the quaternary system it shows a minimum with increasinghe pressure (AAD = 24%).

The unacceptable predicted trends by Fu et al. and Trey-al models (Fig. 2) can be attributed to the limitations in theorrelated parameters of their models where they used datat atmospheric pressure and 25 ◦C. However, the deficiencyf the Firoozabadi–Ramey model is mainly due to its depen-ency to the solubility of the heavy component in the water

hase. The solubility of the heavy hydrocarbons in the aque-us phase is increased by the pressure increment. This resultshe phase density difference to increase while reducing T h

c .herefore, according to Firoozabadi–Ramey model the IFT may

alues from the measured data

ozabadi–Ramey [32] Treybal [36] Fu et al. [39] Parachor [31]

903 103 5196 98 4877 57 –

– – 16– – 42– – –

A. Bahramian et al. / Fluid Phase Equilibria 252 (2007) 66–73 71

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Fw

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is

wW(rmpecitaa[iad

ig. 4. Measured and the predicted hydrocarbon-rich liquid–vapour interfacialension values of the water–cyclohexane–decane–methane system at 150 ◦C.

ncrease or decrease by increasing pressure (as it is evident fromigs. 2, 3, 5 and 6).

At high pressure conditions the characteristics of vapourpproach that of liquid. Indeed at the critical point the prop-rties of the vapour and the liquid become similar. Hence, itould be reasonable to expect that a reliable liquid–liquid IFTodel could produce reliable results for vapour–liquid systems

t high pressure conditions.Fig. 4 shows the measured and the predicted hydrocarbon-

ich liquid–vapour IFT values of the investigated quaternaryystem. The parachor method predicts the results reliably overhe full range with AAD = 16%. The Bahramian–Danesh modelredicts the trend reasonably well and it approaches the mea-ured value at the highest tested pressure. Such predictions ofapour–liquid IFT by a liquid–liquid model are very encourag-ng.

The predicted values for water-rich liquid–vapour IFT of theuaternary system are shown in Fig. 5. The predicted resultsy the Bahramian–Danesh (liquid–liquid) model are more reli-

ble (AAD = 8%) than those of the parachor (AAD = 42%) andiroozabadi–Ramey (AAD = 29%) correlations. However, theahramian–Danesh model predicts a small increase in IFT by

ig. 5. Measured and the predicted water-rich liquid–vapour IFT values of theater–cyclohexane–n-decane–methane system at 150 ◦C.

m

4

sTl

BFap

vs

sct

ig. 6. Measured and the predicted water-rich liquid–vapour interfacial tensionalues of the water–decane–methane system at 150 ◦C.

ncreasing the pressure contrary to the observed trend of mea-ured data.

Fig. 6 compares the measured and predicted values of theater-rich liquid–vapour IFT of the investigated ternary system.hile at low pressure the Bahramian–Danesh model prediction

AAD = 21%) is unreliable (as expected), it predicts the resultseliably at high pressure conditions. The other two liquid–liquidodels fail to give a reasonable prediction as their empirical

arameters have been determined at conditions vastly differ-nt from those of the test. The parachor method (AAD = 48%)an only predict the data at low-pressure conditions. It isnteresting that the predicted IFT results by the parachor andhe Firoozabadi–Ramey (AAD = 5%) models show a minimumround 24.7 MPa. Such a trend has been experimentally obtainednd reported for binaries of water–methane and water–propane50]. For some systems, e.g. water–helium, even a continuousncrease of liquid–vapour IFT with pressure at constant temper-ture has been reported [50]. The average absolute percentageeviations of the predicted interfacial tension values by all theentioned models are also presented in Table 3.

. Conclusions

Measured data on three-phase IFT of two water–hydrocarbonystems at high pressure–high temperature have been presented.he measured data were used to evaluate the capability of some

eading IFT models in the literature.Amongst the predictive liquid–liquid IFT models, the

ahramian–Danesh model produced the most reliable results.urthermore, the model could also predict vapour–liquid IFTt high pressure conditions, where the behaviour of the vapourhase approaches to the liquid phase.

The vapour–liquid parachor model reasonably predicted theapour–hydrocarbon-rich liquid IFT, but it failed to give a rea-onable prediction to the vapour–water-rich liquid IFT data.

Considering a zero spreading coefficient value at high pres-ure conditions for water–hydrocarbon three-phase systems, aombination of the Bahramian–Danesh liquid–liquid model,he parachor vapour–hydrocarbon-rich liquid model and the

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ntanow rule can provide reliable IFT data for all three pairs ofuids at HPHT conditions.

ist of symbolsconstant in Eq. (5)

i partial molar surface area of the ith compounda Eq. (A7)c Eq. (A2)w0 van der Waals area of a standard segment

constant in Eq. (5) and (A3)number of compounds

pi non-random binary interaction coefficient betweenpolar and hydrocarbon compound

ij random binary interaction coefficientconstant in Eq. (7)number of compounds in Eq. (A4)pressure

i parachor value of the compound iUNIQUAC pure compound area parameterideal gas constantspreading coefficienttemperature (K)

c critical temperature (K)molar volumemole fractionEq. (8)

reek lettersa function representing temperature dependency of theattractive term in Eq. (A2)molar density (mol cm−3) in Eq. (2) and mass density(g cm−3) in Eqs. (3) and (4)interfacial tension

ac constant in Eq. (A2)b constant in Eq. (A3)

ubscriptsith compoundhydrocarbonpolar compoundthe richer phase in compound 3 at ternary systemwaterthe more soluble hydrocarbon in the water-rich phaseat ternary system

uperscriptshydrocarbonhydrocarbon-rich phase (i.e. oil)water-rich phase

W interface of water-rich liquid and hydrocarbon-rich liq-uid

V interface of vapour and hydrocarbon-rich liquid

W interface of vapour and water-rich liquid

phase �phase �

� phase �� (interface of � and �)

a

n

quilibria 252 (2007) 66–73

cknowledgements

The research work on which this paper is based was equallyponsored by the UK Dept. of Trade and Industry, Total,

arathon International (GB) Ltd., Norsk Hydro, Gaz de France,chlumberger Oilfield UK Plc., and Shell US Exploration androduction Ltd., which is gratefully acknowledged. The authorsish to thank Mr. K. Bell for making the measurements, andr. J. Pantling and Mr. C. Flockhart for manufacturing andaintenance of the experimental facilities.

ppendix A

The van der Waals equation of state (EOS) has the followingeneric form:

= RT

ν − b− ac

ν2 + ubν + wb2 (Al)

here P and ν are the pressure and the molar volume, respec-ively. The parameter a (ac) is called the (conventional) attractiveerm and b is the repulsive term. The u and w symbols are con-tants in two-parameter forms of EOS. They are related to a thirdarameter or some other property in three-parameter form EOS.

c = ΩacαR2T 2

c

Pc(A2)

= ΩbRTc

Pc(A3)

here Ωac and Ωb are EOS constants. Tc and Pc are the criticalemperature and the critical pressure of compound, respectively.arameter α represents the temperature dependency of the attrac-

ive term. The van der Waals conventional random mixing ruleso determine ac and b parameters of EOS for multi-componentystems are:

c =N∑

i=1

N∑j=1

xixj(aci a

cj)0.5(1 − kij) (A4)

=N∑

i=1

xibi (A5)

here xi and N are the molar composition of component i andhe number of components in the mixture, respectively, kij is theonventional (random) binary interaction parameter (BIP). Inhis study to simulate the interaction between water (polar) andydrocarbons (non-polar), an asymmetric (non-random) termaa) was added to the conventional (random) attractive term (ac)f EOS:

= ac + aa (A6)

here:

a =∑

x2p

∑xiapilpi (A7)

The subscript “p” refers to polar components, and lpi is theon-random BIP between polar and hydrocarbon compounds.

hase E

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[[

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A. Bahramian et al. / Fluid P

he conventional (random) (kij) and the non-random (lpi) binarynteraction parameters (BIP’s) for water–hydrocarbon haveeen determined by matching the mutual solubility data ofydrocarbon–water systems.

eferences

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