Fluid Phase Equilibria 408 (2016) 58–71

An evolved cubic equation of state with a new attractive term

Pouya Hosseinifara, Saeid Jamshidib,*aMAPSA Technology Center, Tehran, IranbDepartment of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran

A R T I C L E I N F O

Article history:Received 20 May 2015Received in revised form 12 August 2015Accepted 13 August 2015Available online 17 August 2015

Keywords:Vapor pressureEquation of statea-FunctionPeng–Robinson EOSReservoir fluidsPetroleum fractions

A B S T R A C T

A large variety of modifications have been presented for the temperature dependent function (a) existing inthe attractive term of cubic equations of state (CEOS). Most of a-functions attempted to modify the vaporpressure prediction of polar components while other modifications have focused on both polar and non-polar compounds and other relations have considered an expansion of polynomials in the acentric factor (v)and reduced temperature (Tr) to predict vapor pressure more accurately. In most cases such as Soave andPeng–Robinson equations of state, the suggested a-functions do not show a limiting behavior whentemperature increases infinitely. In addition, the incompetency of many a-functions in describing thesupercritical behavior of fluids have been modified by defining specific parameters for some differentcomponents or switching a-functions. Such approaches can create difficulties for the calculation ofproperties in which, there are equations of state derivatives. In this work, a new a-function has been putforward to improve the attractive term of Peng–Robinson EOS. The proposed model is capable of predictingthe saturation vapor pressure of light gases, polar, nonpolar and heavy hydrocarbon compounds. Thenormal boiling point (Tb) is used as an additional parameter along with acentric factor (v) and reducedtemperature (Tr) to enhance the ability of the proposeda-function in the prediction of fluid phase behaviorespecially for petroleum fractions and heavy cuts existing in reservoir fluids samples. Moreover, twodiscriminating indexes (l1, l2) are introduced, to handle all pure substances and classify them underdistinct groups for the allocation of special coefficients to each group. Experimental vapor pressure of31 different pure components, 11 binary mixtures and 12 reservoir oil samples were collected, in order toevaluate the performance of the proposed model compared to the original Peng–Robinson. A comparisonbetween the proposed model and other powerful equations of state such as modified PR, PSRK and PC-SAFTmodel was also performed, and the results of which indicate that the new proposed model has betteroutcomes over the mentioned methods, especially for complex mixtures and reservoir fluids. Furthermore,the proposed a-function shows a reasonable limiting behavior as the temperature approaches infinity andis capable of describing the fluid phase behavior at supercritical region.

ã 2015 Elsevier B.V. All rights reserved.

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1. Introduction

Equations of state are of specific importance in chemicalengineering design, especially in the study of the phase equilibriaand describing thermodynamic properties of pure substances andfluid mixtures. Initially, equations of state were applied for puresubstances but they have improved rapidly in the prediction of phasebehavior in non-polar and polar mixtures. The merits of usingequations of state is their capability to calculate phase equilibria overwide ranges of temperature and pressure in mixtures of differentcomponents, from light gases to heavy liquids. They can also be used

* Corresponding author. Fax: +98 21 66022853.E-mail addresses: hosseinifar.pouya@yahoo.com, jamshidi@sharif.edu,

jamshid.s@gmail.com (S. Jamshidi).

http://dx.doi.org/10.1016/j.fluid.2015.08.0130378-3812/ã 2015 Elsevier B.V. All rights reserved.

for the prediction of vapor–liquid, liquid–liquid and supercriticalfluid phase equilibria. A large number of equations of state have beensuggested in the literature with either an empirical, semi-empiricalor theoretical basis. In spite of the recent development of molecular-based equations of state such as statistical association fluid theory(SAFT) [1–2], the simplicityand popularity of cubic equations of state(CEOS) lead to the continuation of their improvements. Van derWaals equation of state [3] was the first equation to predict vapor–liquid coexistence. Although Van der Waals equation was notaccurate for most applications, it attracted great attentions to thisfield because it was the first effort to describe the real PVT behavioroffluids. Later on, Redlich–Kwong equation of state [4] improved theprecision of van der Waals equation by modifying the attractive termthrough defining a temperature dependent function. Soave [5] andPeng and Robinson [6] proposed considerable modifications for

Nomenclature

a1, a2, a3, a4, b1, b2,b3, c1, c2 and d

adjustable variables used in Eqs. (9)–(11)

a attraction parameterac constant parameter defined by Eq. (6)A dimensionless parameter defined by

Eq. (26)AAD% average absolute percent deviationb van der Waals co volumeB dimensionless parameter defined by

Eq. (27)Bo oil formation volume factorCP specific heat capacityGOR gas oil ratioMw molecular weightOF objective functionP pressureR gas constantSG specific gravityT temperaturen specific molar volumex mole fractionZ compressibility factor

Greek lettersa temperature dependent function in the attractive termb temperature dependent function defined by Eq. (10)g temperature dependent function defined by Eq. (11)l1 index of discriminationl2 index of discriminationj a variable which can be substituted by Mw or Tb in

Eq. (16)r density’ fugacity coefficientF a variable which can be substituted by any of the critical

properties in Eq. (15)v acentric factor

SubscriptsA attractionb normal boiling point (for temperature) and bubble

point (for pressure)br reduced normal boiling pointc critical propertyi, j property of component i, jL liquidmax maximumr reduced propertyR repulsionsat saturation

Superscriptscalc calculatedexp experimental0 first derivative00 second derivative* ideal gas state^ partial property in the mixture

P. Hosseinifar, S. Jamshidi / Fluid Phase Equilibria 408 (2016) 58–71 59

Redlich–Kwong equation to improve the predictions of voluetric,thermodynamic, and phase equilibrium properties such as vaporpressure, liquid density, and equilibria ratios. Other modifications

have also been presented for the repulsive term by Carnahan andStarling [7], Guggenheim [8] and Boublik [9]. Christoforakos andFranck [10] improved both the attractive and repulsive terms of vander Waals equation of state. Capability of a cubic equation of state tocalculate phase equilibria of mixtures depends upon the accuracy ofpredicting pure component vapor pressures. A significant part ofattempts for improving the precision of cubic equations of state hasbeen devoted to the enhancement of the temperature dependentattractive term, through the amending a-function. A variety ofmodifications have been proposed for the temperature-dependentfunction a (Tr, v) existing in the attractive term of the SRK and PRequations to improve correlations and predictions of vapor pressurefor diverse fluids. Table 1 presents a summary of expressionspreviously reported in the literature for temperature dependentfunction. Most of the proposed a-function models are related tomodifying the vapor pressure estimation of polar components.Some modifications have focused on both polar and non-polarcompounds while other developments have applied an expansionof polynomials in the acentric factor (v) or reduced temperature topredict vapor pressure more accurately. In most cases, the proposeda-functions do not abide by a limiting behavior when thetemperature increases infinitely (such as Soave and Peng–Robinsona-functions). Furthermore many a-functions suffer from incom-petency to describe supercritical behavior precisely. This problemhas been remedied by presenting specific parameters for differentcomponents or changing a-functions. Such methods can lead todifficulties in derivative properties. Hence, a generalized function ismore desirable than defining component-dependent parameters ina-function. On the other hand the bubble pressure of complexmixtures containing ambiguous fractions with non-distinguishablenature (such as petroleum fractions in reservoir fluids) is deeplyaffected by the accurate estimation of a-function for puresubstances and petroleum fractions. A petroleum fraction includesa large number of various unspecified hydrocarbon componentswith unknown composition. Several correlations have beenproposed by Twu [43], Kesler and Lee [44], Riazi and Daubert[45], and Hosseinifar and Jamshidi [46], to characterize petroleumfractions for the estimation of their physical and critical properties(critical temperature, critical pressure and acentric factor). Apply-ing an accurate correlation to predict the acentric factor forpetroleum fractions can have a major influence on the prediction ofbubble pressure in the mixture in those methods using the acentricfactor to estimate a-function. In this study, we concentrate ourattention on the attractive term of Peng–Robinson equation of stateto overcome all the aforementioned defects. The main purpose forthe development of the new suggested model is obtaining ageneralized and enhanced expression for the temperature depen-dent function existing in PR EOS, which is capable of covering alltypes of pure substances. One of the major problems with theformer expressions for the prediction of a-function is their inabilityto cover wide ranges of miscellaneous pure substances. Due to thisrestriction, several functions have been proposed in which themodel adjustable parameters are tuned to the experimental data ofa particular group. In the proposed model, two useful discriminat-ing indexes (l1, l2) are introduced to handle all pure substances andclassify them under distinct groups. In the next step, the adjustableparameters of the proposed model are determined for eachparticular group using an appropriate optimization method. Basedon this approach, the proposed model will be applicable to differentsubstances in a wider range of temperatures and pressures.Moreover, the normal boiling point temperature (Tb) is used asan additional parameter along with the acentric factor and reducedtemperature to improve the prediction of a-function. Supercriticaldata of pure substances are also employed to enhance theperformance of equation of state at supercritical region.

Table 1Various modifications of the temperature dependent function (a) in the attractiveterm for cubic EOS.

Functions References

1ffiffiffiffiffiTr

p [11]

Tr þ 1:57 þ 1:62vð Þ 1 � Trð Þ [12]

1 þ c1 þ c2v þ c3v2� �1 � Tr

0:5� �h i2 [13]

1 þ 1 � Trð Þ m þ nTr

� �[14]

1 þ c1 1 �ffiffiffiffiffiTr

p� �þ c2

1Tr

� 1� � 2 [15]

1 þ c1lnTr þ c2 1 � lnTrð Þ2h i2

; Tr > 1ð Þ [15]

exp c 1 � Trn� �� �

[16]

1 þ c1 1 �ffiffiffiffiffiTr

p� �þ c2 1 � Trð Þ 0:7 � Trð Þ

h i2 [17]

1 þ mð1 � ffiffiffiffiffiTr

p Þ� �2; Tr < 1

exp cð1 � Tdr Þ

h i; Tr > 1

m ¼ ðc1 þ c2v þ c3v2Þ; d ¼ 1 þm2; c ¼ 1 � 1

d

8>>><>>>:

[18]

1 þ k 1 � ffiffiffiffiffiTr

p� �� �2k ¼ c1 þ c2v þ c3v2 þ c4v3 þ k1 1 � ffiffiffiffiffi

Trp� �

0:7 � Trð Þ

( [19–20]

1 þ c1 1 � ffiffiffiffiffiTr

p� �þ c2 1 � ffiffiffiffiffiTr

p� �2 þ c3 1 � ffiffiffiffiffiTr

p� �3h i2 [21]

1 þ c1 1 � Tr23

� �þ c2 1 � Tr

23

� �2þ c3 1 � Tr

23

� �3 [22]

1 þ c1 1 �ffiffiffiffiffiTr

p� �þ c2 1 � Trð Þ [23]

exp c1 1 � Trð Þ þ c2 1 �ffiffiffiffiffiTr

p� �2 [24]

10 c1 a0þa1Trþa2Tr2ð Þ 1�Trð Þ½ � [25]

Trc3 c2�1ð Þexp c1 1 � Tr

c2c3� �� �

[26]

1 þ c1 1 � Trð Þ þ c2 1 � Trð Þ2 [27]

K þ v TrA2 exp B2 1 � Tr

C2

� �h i� K

� �; K ¼ Tr

A1 exp B1 1 � TrC1

� �h i[28]

Tr 1 þ mTr

� �[29]

mTr

þ n

Tr2

[30]

1 þ m 1 � Trð Þ2 1 �ffiffiffiffiffiTr

p� �2 [31]

1 þ n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � Tr

0:7

rþ m 1 �

ffiffiffiffiffiTr

p� �" #2 [32]

m1 þm2

Trþ m3

Tr2

[33]

m1 þm2

Trþ m3

Tr2 þ

m4

Tr3

[34]

1 þ m 1 � ffiffiffiffiffiTr

p� �� �2Tr

[35]

1 þ m 1 þ Trn� �� �2 [36]

1 þ m 1 �ffiffiffiffiffiTr

p� �h i2þ n Tr � 0:6ð Þ2 [37]

1 þ m 1 �ffiffiffiffiffiTr

p� �� p 1 � Trð Þ q � Trð Þ

h i2 [38]

1 þ n � exp �kð Þ1 þ n � exp �kTr

2m� �

0@

1ATr

2m�1exp 1 � Tr2m

� �k [39]

exp p 1 � Trð Þj1 � Tr jG�1 þ q Tr�1 � 1

� �h i[40]

1 þ m 1 �ffiffiffiffiffiur

p� �h i2; ur ¼ T � Tpt

Tc � Tpt

[41]

1 þ A þ Bv þ Cv2� �1 � TD

r

� �h i1E [42]

exp A þ BTrð Þ 1 � TCþDvþEv2

r

� �h i[42]

60 P. Hosseinifar, S. Jamshidi / Fluid Phase Equilibria 408 (2016) 58–71

2. Method description

In most cases, the original Peng–Robinson equation of stateexpresses more accurate results in prediction of liquid density thanthe original Soave–Redlich–Kwong equation of state [5]. One of theproblems with the PR EOS is its large error in prediction of vaporpressure of heavy hydrocarbons [47]. Peng–Robinson EOS was thenselected to modify its attractive term in estimation of sufficientlyaccurate a-function, which is capable of predicting the vaporpressure for polar, non-polar, heavy hydrocarbons and the bubblepressure of complex mixtures such as reservoir fluids. The originalPeng–Robinson EOS [6] is expressed as:

P ¼ PR þ PA ð1Þ

PR ¼ RTn � b

ð2Þ

PA ¼ �a Tð Þg nð Þ ð3Þ

g nð Þ ¼ n n þ bð Þ þ b n � bð Þ ð4Þ

a Tð Þ ¼ aca Tr; vð Þ ð5Þ

ac ¼ 0:45724R2T2

c

Pcð6Þ

b ¼ 0:0778RTc

Pcð7Þ

a Tr; vð Þ ¼ 1 þ �0:176v2 þ 1:574v þ 0:48� �

1 �ffiffiffiffiffiTr

p� �h i2ð8Þ

where P is pressure, PR is the repulsion pressure, PA is the attractionpressure and g nð Þ is a function of molar volume and constant brepresenting the size of the hard spheres. The parameter a(T) candescribe the intermolecular attraction forces. This parameter isconstant solely in van der Waals equation of state. For SRK and PRequations a dimensionless factor a Tr; vð Þ is used demonstratingthe temperature dependence of the parameter a(T). The parameterac is a constant relating to the critical properties same as theparameter b, which can be obtained by setting equal the first andsecond derivatives of pressure with respect to volume, to zero atthe critical point. Also T is temperature, n the molar volume, R thegas constant, Tr the reduced temperature, Tc the critical tempera-ture, Pc the critical pressure and v is the acentric factor.

2.1. Data collection

All pure substances used in this study can be categorized asnon-polar [48–52], heavy normal paraffins [53–56], light gases[57–63] and polar components [64–65]. Collected data include thesaturation vapor pressures and supercritical data for puresubstances in wide ranges of temperature and pressure. Singlephase (pressure–temperature–density) data at supercritical regionwere also selected ranging from low temperatures and pressuresup to 1000 K and 300 MPa for pure components, respectively.

P. Hosseinifar, S. Jamshidi / Fluid Phase Equilibria 408 (2016) 58–71 61

2.2. Development of the new (a) function

Many efforts have been carried out to modify a-function for theoriginal Peng–Robinson equationof state by relating it tothe reducedtemperature (Tr) and acentric factor (v). Most relations haveparticular adjustable parameters that should be fitted to experi-mental vapor pressure data for the component of interest andtherefore cannot present acceptable results for other substances. Onthe other hand, there exist pseudo components in the analysis ofreservoir fluids called petroleum fractions that are hardly describ-able. Since, the exact details of such unknown mixtures cannot bespecified, they should be characterized based on their measurablephysical properties such as molecular weight and normal boilingpoint. Those a-functions that have been developed based onreduced temperature (Tr) and acentric factor (v), cannot exactlyidentify petroleum fractions. As normal boiling point (Tb) is aneffective property in the characterization of pseudo components, Tbwas considered as an additional parameter besides the reducedtemperature and acentric factor in the proposed model, enablingthe model to gather more information about the substances,especially for pseudo components and petroleum fractions. Normalboiling point is also a commonly used property which can easily bemeasured for both pure substances and unknown mixtures withoutprior knowledge about mixture composition. Furthermore, it canpartly encompass structural information about a molecule or apseudo component, and can also be regarded as a measure of thestrength of intermolecular forces. Hence, the normal boiling pointwill be capable of enhancing the performance of a-function in theprediction of attraction forces between molecules in diversesubstances. The first step in developing the suggested model, wasobtaining experimental a values. The equality of fugacities criterionat equilibrium state was employed to obtain experimental values ofa at desired temperatures and their corresponding saturation vaporpressures. The values of a were once plotted versus the reducedtemperature for all different components existing in the databaseand a large scattering appeared as shown in Fig. 1. Once again theacentric factor (v) and normal boiling point (Tb) were separatelyapplied apiece to associate with the reduced temperature for theobviation of the dispersing behavior of a values, when they aresolely plotted versus the reduced temperature. According to theaccomplished investigations, it was found that the reducedtemperature to the power of a polynomial expansion in acentricfactor can have a significant influence on concentrating the data

Fig. 1. The experimental a values obtained from applying the equality of fugacities criexisting in the data base.

toward a linear behavior as illustrated in Fig. 2. On the other hand,Plots ofa values against the productof reduced temperature and thereciprocal of normal boiling point, also indicated a fairly good linearbehavior as shown in Fig. 3. These trends can be enhanced throughan appropriate optimization method in which a suitable fitnessfunction has been employed. Several examinations were carried outto derive the most acceptable fitness function, comprising twoterms containing the acentric factor and normal boiling point,previously pointed in Figs. 2 and 3. Finally, among the variousfunctions created by combination of the two achieved trends, it waspointed out that the simplest a-function which exhibits moreflexibility for the error reduction in the tuning process can beexpressed as:

a Tr; Tbr; vð Þ ¼ c1b þ c2gð Þd ð9Þ

b Tr; Tbrð Þ ¼ j1 � Ta1r ja2

j1 � Ta3br ja4

ð10Þ

g Tr; vð Þ ¼ Tb1v3þb2v2þb3vþb4ð Þ

r ð11Þwhere Tr is the reduced temperature, Tbr is the reduced normal

boiling point, v is the acentric factor, b is a function which dependson the reduced temperature and reduced normal boiling point andg is another function related to the reduced temperature andacentric factor. a1, a2, a3, a4, b1, b2, b3, c1, c2, and d are the adjustablevariables which can be tuned to the experimental vapor pressuredata.

2.3. Discriminating indexes

As real gas behavior approaches ideal gas at high temperatures,the attractive forces must be decreased. So, the temperaturedependent function (a) in the attractive term descends byincreasing the temperature. The uppermost a value appears inthe vicinity of the triple point and its value reduces to unity at thecritical point for each substance. All substances will then havesimilar a values at high temperatures which cannot easily bedistinguished, while the most sensible difference between a valuesis shown at the triple point for various components, so they canclearly discern in comparison with one another. Therefore, the

terion to the VLE data versus the reduced temperature for all different substances

Fig. 2. The influence of applying the acentric factor along with the reduced temperature in obviating the dispersing behavior of a values compared to the situation solelyplotted versus reduced temperature.

Fig. 3. The influence of applying the normal boiling point along with the reduced temperature in obviating the dispersing behavior of a values compared to the situationsolely plotted versus reduced temperature.

62 P. Hosseinifar, S. Jamshidi / Fluid Phase Equilibria 408 (2016) 58–71

maximum a value at the triple point called amax, was chosen as ayardstick to measure the strength of attractive intermolecularforces in diverse compounds. Evaluating amax values of all puresubstances existing in the database, indicated a recognizabledistribution around 1.15 up to 2.22, for different components. Inaccordance with the molecular structure of pure substances, it wasfigured out that those components which have amax nearly greaterthan 1.35, contain specific features in their intermolecular forcesthat lead to an increase in amax value. Such an increase could becaused either by specific forces between molecules (such ashydrogen bonds) or strong dispersion forces formed between largemolecules with high molecular weights. These two factors are thereasons for the enlargement of amax value up to 1.35 for acomponent. In spite of having small molecular weights, somecomponents produce high amounts of amax (more than 1.35), dueto the existence of specific forces such as hydrogen bonds. The

proposed index using for the diagnosis of such compounds is calledl1, and indicates the induction forces between a permanent dipole(sometimes quadruple) or association and solvation resulting fromthe specific and chemical forces. It can be defined as follows:

l1 ¼ Tb

Mw

� �v ð12Þ

where Tb (K), Mw (g/mol) and v are normal boiling point, molecularweight and acentric factor, respectively. This index expresses therelevance between the strength of specific attractive forces, shapeand size of a molecule. For instance, H2O and CH4 as two differentmolecules with approximately the same molecular weights, have asignificant difference in their normal boiling points, because of theas mentioned specific forces. Hydrogen bonds between watermolecules lead for the normal boiling point to raise considerablycompared to methane. Hence, the ratio of the normal boiling point

Fig. 4. The attractive forces detected by discriminating indexes for water andeicosane molecules. High amount of l1 for water molecules indicates the existenceof hydrogen bonds and the high value of l2 illustrates the strong dispersion forcesbetween eicosane molecules.

P. Hosseinifar, S. Jamshidi / Fluid Phase Equilibria 408 (2016) 58–71 63

to the molecular weight given in Eq. (12) for water is greater thanthat of methane, which indicates the stronger attractive forces andhigher amax values of water molecules. The influence of non-sphericity of molecules on amax values was also verified using theacentric factor, and it was found that molecules with greateracentric factors can lead to larger amounts of amax. As expressed inEq. (12), l1 index is defined by the multiplication of two factors: theratio of normal boiling point to molecular weight and acentric factor,which both cause to increase amax value. In fact l1 is a quantitativeindex to measure the strength of specific forces in the role of thenormal boiling point, respect to the size and shape of a component,represented by its molecular weight and acentric factor, respec-tively. Since, high values of l1 are equivalent to great amax values(more than 1.35), an optimum boundary value should bedetermined for l1 index, which corresponds to (amax� 1.35).Based on the comparison carried out between the calculatedvalues of l1 index for different substances and their correspondingamax values, the optimum boundary value of l1 was considered as1. Therefore, those substances which satisfy the criterion of l1 > 1,are able to create values more than 1.35 for amax and the specificattractive forces such as hydrogen bonds would exist betweentheir molecules. There are another group of substances that,despite the lack of specific forces and hydrogen bonds, can have alarge value of amax (higher than 1.35) due to the existence of strongdispersion forces. Such compounds may also have l1 greater than1, but the increase in their amax is not caused by the specific forces.The second proposed index used to detect these compounds iscalled l2 and represents dispersion forces between non-polarmolecules, as expressed below:

l2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiMwTb

pð13Þ

Similarly, l2 is another quantitative index to evaluate, what themolecules can create a high value of amax due to largeintermolecular forces resulting from the dispersion forces notthe specific forces. According to Eq. (13), non-polar componentswith high molecular weight and high normal boiling point canproduce a large l2 value, which leads to great attractive forces andamax > 1.35. Through the same procedure as the method used todetermine the optimum boundary value of l1, boundary value forl2 was considered to be 150. It was previously pointed that, allsubstances which satisfy l1 > 1, contain specific forces andhydrogen bonds, but provided that they abide by l2 � 150. Indeed,substances satisfying both criteria of l1 > 1 and l2 > 150, canproduce amax values greater than 1.35 because of the strongdispersion forces usually formed in non-polar compounds such aseicosane. Fig. 4 shows a schematic of attractive forces identifiedthrough the discriminating indexes (l1, l2) for water and eicosanemolecules. The index of l1 can recognize hydrogen bonds betweenwater molecules and l2 is able to detect strong dispersion forcesbetween eicosane molecules. Generally, these two indexes indicatehow much the a-function can be affected by the strength ofintermolecular force.

2.4. Optimization of coefficients

As already mentioned, adjustable parameters of the formera-functions fail to fit with the experimental data of miscellaneoussubstance groups. In the proposed model, two discriminatingindexes have been introduced, used to rectify this difficulty andclassify all various types of substances under specific groups.Applying the discriminating indexes for available experimentaldata, led to divide them into four distinct categories:

1) Tr> 1: all substances above the critical temperature.

2) Tr� 1, l1 > 1 and l2 > 150: substances with large attractiveforces only due to the existence of strong dispersion forcesamong their molecules, below the critical temperature (mainlyheavy non-polar compounds).

3) Tr� 1, l1 > 1 and l2� 150: substances with large attractiveforces only due to the existence of specific forces such ashydrogen bonds in their molecules, below the critical tempera-ture (mainly polar compounds).

4) Tr� 1 and l1�1: all other substances with no considerablylarge attractive forces below the critical temperature.

Accordingly, the experimental data of each specific category canbe employed in the tuning process, in order to assess the modeladjustable parameters for that category, exclusively. The proposeda-function given in Eqs. (9)–(11), was implemented in anoptimization algorithm in order to determine the adjustableparameters. The Nelder and Mead [67] optimization method wasapplied to minimize the objective function (OF) defined as:

OF ¼ 1Nexp

XNexp

i¼1

aexpi � acalc

i

aexpi

!2

ð14Þ

where a is the temperature dependent function and Nexp is thetotal number of experimental points. The coefficients a1, a2, a3, a4,b1, b2, b3, c1, c2 and d are all adjustable parameters given in Eqs. (9)–(11), which were separately regressed using experimental data foreach category, the results of which are shown in Table 2. Besidesthe role of suggested indexes for data partitioning in the parametertuning process to determine model constants, they are also appliedto recognize what group of the tuned parameters in Table 2 can beused for a particular substance to estimate its a-function. In thisregard, a comparison was made on the values of two indexes (l1,l2), for three diverse components, the results of which arepresented in Table 3. For instance, the value of amax for methane isequal to 1.15 and according to the aforementioned explanations, itshould be placed in the fourth category, because the criterion ofl1 ¼ 0:08 � 1 is satisfied and there exists no significant attractiveforce between its molecules. Water molecules with ðamax ¼1:64Þ > 1:35 can be placed in the third category because ofsatisfying the criteria of ðl1 ¼ 6:82Þ > 1 and ðl2 ¼ 82Þ � 150, andforming hydrogen bonds between its molecules. Eicosane mole-cules have amax ¼ 1:94, which is higher than 1.35 and satisfy boththe criteria of ðl1 ¼ 1:91Þ > 1 and ðl2 ¼ 417:54Þ > 150 and belong

Table 2The optimized values of the proposed a-function coefficients for pure substances.

a-Function a1 a2 a3 a1 a1 a4 b1 b2 b3 b4 c1 c2 d

Tr> 1 – – �0.69560 0.786752 0.710102 0.140024 �1.07370 1.604866 �0.93029 -0.44313 �0.56996 1 0.594998Tr� 1 l1 > 1 l2 > 150 0.432948 1.146441 �1.01574 1.185891 �0.38446 1.240158 �2.49074 -0.16104 1.042102 1 0.526882Tr� 1 l1 > 1 l2 > 150 0.229145 0.878492 3.279808 -0.6464 4.956359 1.730576 �3.10611 -0.62314 2.048654 1 0.414679Tr� 1 l1�1 – 1.646198 1.138750 �1.98296 0.940935 �0.26017 2.624602 �3.36394 -0.44360 0.661815 1 0.428991

Table 3Comparison of discriminating indexes for three different substances.

Components Tb Mw v amax l1 l2

Water 373.15 18.020 0.3290 1.64 6.82 82.000Eicosane 617.00 282.56 0.8751 1.94 1.91 417.54Methane 111.63 16.040 0.0113 1.15 0.08 42.310

64 P. Hosseinifar, S. Jamshidi / Fluid Phase Equilibria 408 (2016) 58–71

to the second category, due to the existence of large intermoleculardispersion forces. Moreover, Figs. 5 and 6 illustrate a distribution ofl1 andl2 indexes values among various compounds. Thecontribution of each component depends upon the type andstrength of its intermolecular forces. Fig. 5 allocates the maximumcontribution to water molecules for l1 index and Fig. 6 shows themaximum contribution of l2 index for Octacosane molecules.According to the comparison results, the high amount of l1 orl2

indexes is an indication of large amax value, because of eitherspecific forces between mainly polar compounds or strongdispersion forces between non-polar components, respectively.

3. Results and discussion

Performance of the new proposed a-function was evaluatedusing experimental a values obtained from the equality of thefugacities criterion at given temperatures and their correspondingpressures. Fig. 7 shows the percent of average absolute errors in theestimation of a-function through Peng–Robinson and our pro-posed model for pure substances used in this study from the triple

Fig. 5. The contribution percent of each subs

point up to the critical point. As shown in Fig. 7, the predicteda-function values of the proposed model are more accurate thanthe results predicted by Peng–Robinson EOS. Three fundamentalnecessities must be considered for the temperature dependenta-function: first, values of a-function should be positive and finitefor all temperatures. Second, the value of a-function should beequal to 1 at critical point. Third, a-function should approach afinite value when temperature increases infinitely. The proposeda-function satisfies all the mentioned provisos and exhibits areasonable limiting behavior as the temperature approachesinfinity. Fig. 8 illustrates the variations of propane a-functionversus the reduced temperature calculated both by Peng–RobinsonEOS and the proposed model. It is evident from the figure thatunlike the proposed model, Peng–Robinson a-function valuebegins to rise up at a reduced temperature of approximately eightðTr � 8Þ and does not approach a finite value by increasing thetemperature.

3.1. Reliability of model for the pure substances

To evaluate the performance of the proposed model in theprediction of pure component saturation vapor pressures, experi-mental VLE data of 31 pure substances from different types werecollected in various temperatures and corresponding saturationpressures [48–66]. The original Peng–Robinson EOS, a modifiedversion of Peng–Robinson EOS (MPR) [68], predictive Soave–Redlich–Kwong EOS (PSRK) [69] and perturbed hard chain statisticalassociation fluid theory (PC-SAFT) [2], were selected to comparetheirresults against theproposedmodel.Calculatedsaturationvapor

tance in creating the value of l1 index.

Fig. 6. The contribution percent of each substance in creating the value of l2 index.

Fig. 7. The values of a-function predicted by the proposed model in comparison to the original Peng–Robison EOS for various compounds from the triple point up to thecritical point.

P. Hosseinifar, S. Jamshidi / Fluid Phase Equilibria 408 (2016) 58–71 65

pressures through the proposed model and all other mentionedmethods were compared against the experimental vapor pressuredata at specified reduced temperature ranges and the percent ofaverage absolute deviations of which, is presented in Table 4 foreach component. As can be seen in this table, the original PR exhibitslarge deviations in predicting the saturation pressure of heavyhydrocarbons and polar compounds, and also does not give goodresults for some light gases. The predictive SRK (PSRK) is a groupcontribution equation of state developed based on the original SRKEOS and uses the UNIFAC method to calculate the parameter of EOSforamixture.Thisversion ofSRK EOSyields sufficiently precisevaporpressurefornonpolarcompoundsandslightly polarcompounds, dueto applying the Mathias and Copeman [21] expression as tempera-ture dependent function. It is shown in Table 4 that PSRK EOS has asensible deviation for light hydrocarbon compounds, presentsgood results for heavy hydrocarbons up to eicosane (for higherhydrocarbons has large deviations) and predicts saturation pressure

of polar compounds more accurate than the original PR EOS. Themodified Peng–Robinson EOS (MPR) gives acceptable results inpredictingthesaturationpressureofnonpolarcompoundsespeciallyfor heavy hydrocarbons, but a considerable deviation occurs inprediction of polar compound saturation pressures. As indicated inTable 4, PC-SAFT model yields fairly accurate results for light gases,nonpolar and non-associative fluids. PC-SAFT model requires threeparameters for non-associative fluids and two additional parametersshould be considered to improve the saturation pressure predictionof polar compounds and associative fluids, which was ignored in thisstudy. It is clear that, the proposed model could noticeably enhancethe estimation power of PR EOS for light gases, heavy substances,polar and nonpolar compounds. The percent of average absoluteerrors for all substances are 13.54%, 7.17%, 10.36% and 3.16% fororiginal Peng–Robinson, modified PR, PSRK and PC-SAFT model,respectively which has been reduced down to 1.76% by the proposemodel.

Fig. 8. Comparison of the proposed model limiting behavior to the original Peng–Robinson EOS in the prediction of propane a-function as the temperature approachesinfinity.

Table 4The percent of average absolute deviation (AAD%) in the prediction of the saturation vapor pressure for all pure substances.

Components Saturation pressure (AAD%) Tr range No. of data points References

Original PR Modified PR PSRK PC-SAFT Proposed model

Methane 5.77 2.12 6.84 0.86 0.95 0.42–1 45 [48]Ethane 15.5 3.42 18.4 0.94 0.78 0.30–1 95 [49]Propane 13.6 2.25 20.5 1.29 1.41 0.35–1 25 [50]n-Butane 8.46 3.99 11.9 0.75 1.69 0.35–1 56 [50]i-Butane 5.78 6.36 4.29 1.35 2.14 0.34–1 53 [50]i-Pentane 5.33 4.29 3.87 0.71 1.34 0.55–1 45 [51]n-Hexane 5.49 3.54 4.21 0.65 1.57 0.35–1 45 [52]Decane 6.04 2.48 1.43 0.24 1.21 0.52–1 16 [53–56]Dodecane 6.82 2.21 1.12 2.11 0.93 0.53–1 13 [53–56]Tetradecane 5.49 2.37 0.96 4.83 0.71 0.53–1 16 [53–56]Hexadecane 7.37 1.89 0.55 4.87 0.27 0.54–1 20 [53–56]Octadecane 9.52 1.46 0.71 4.99 0.34 0.55–1 17 [53–56]Nonadecane 11.2 1.61 0.69 5.08 0.98 0.55–1 16 [53–56]Eicosane 12.6 1.39 1.21 2.26 0.43 0.56–1 16 [53–56]Docosane 18.3 1.98 23.2 2.61 0.26 0.57–1 12 [53–56]Tetracosane 20.4 1.74 28.6 1.49 0.36 0.56–1 13 [53–56]Octacosane 48.3 1.86 53.2 1.93 1.42 0.57–1 14 [53–56]Argon 3.87 3.32 5.01 0.84 1.21 0.55–1 20 [57]Nitrogen 6.11 4.85 5.47 0.33 2.37 0.50–1 16 [58–61]Oxygen 12.6 11.8 15.1 3.71 3.24 0.35–1 16 [57]Hydrogen 5.84 3.73 4.43 2.25 0.22 0.42–1 16 [57]Carbon dioxide 3.98 3.36 5.56 2.78 1.88 0.71–1 16 [62]Hydrogen sulfide 3.64 2.88 1.98 1.82 1.34 0.71–1 25 [63]Water 14.8 9.48 10.5 7.47 4.39 0.42–1 20 [64]Fluorine 11.9 10.5 12.7 2.94 2.31 0.37–1 16 [65]Ethylene 9.83 9.12 12.6 2.89 4.41 0.36–1 16 [57]Benzene 9.55 8.12 8.97 3.47 5.34 0.49–1 16 [66]Ammonia 6.89 5.29 5.71 0.92 1.18 0.48–1 16 [65]Acetone 7.76 6.47 5.58 1.73 1.24 0.51–1 16 [65]Methanol 56.9 46.9 17.3 10.7 4.49 0.34–1 16 [65]Ethanol 60.1 51.4 28.6 18.3 4.28 0.29–1 16 [65]

66 P. Hosseinifar, S. Jamshidi / Fluid Phase Equilibria 408 (2016) 58–71

3.2. Reliability of model for binary mixtures comprising of well-definedcomponents

There exist several elements that can influence mixtureproperties. For example the bubble pressure estimation for amixture regardless of the capability of EOS, depends upon the

ability of mixing rules and combining rules used in the structure ofan equation of state, or the estimate precision of the temperaturedependent function (a) for the pure substances existing in themixture. According to this fact, a portion of errors produced in thebubble pressure estimation is related to the inabilities of mixingrule and combining rules, and the rest of error contributions can be

P. Hosseinifar, S. Jamshidi / Fluid Phase Equilibria 408 (2016) 58–71 67

attributed to the weakness of the a function. Hence, in addition tothe role of a proper mixing rule for prediction of mixtureproperties, the ability and precision of a-function in describingpure component behavior, can have a sensible effect on theprediction of mixture behavior. Experimental bubble pressures of11 binary mixtures in various temperature were selected [70–74]to be compared against results calculated by the proposed modeland other equations of state as shown in Fig. 9. The chosen binarysamples include the bubble pressure of light gases, polar-nonpolarand nonpolar–nonpolar binary mixtures at the various composi-tions from 0 up to 1, and the temperature range for each binarymixture is also given in Fig. 9. The percent of average absoluteerrors for all binary mixtures are 21.51%, 13.43%, 10.51%, 9.51% and6.71% for original Peng–Robinson, modified PR, PSRK, PC-SAFT andproposed model, respectively. It can be seen that the proposedmodel could improve bubble pressure estimations of binarymixtures and yields better prediction results for the consideredmixtures.

3.3. Reliability of model for reservoir fluids containing well-definedcomponents and petroleum fractions

Reservoir fluids include some well-defined components such as(N2, H2S, CO2, C1, C2, C3, i-C4, n-C4, i-C5, and n-C5) and some pseudocomponents known as petroleum fractions (C6, C7, C8 . . . C12+). Thereliability of proposed model was previously checked for well-defined and pure components using their experimental vaporpressures, but this is not applicable for petroleum fractions,because experimental data regarding bubble pressures of petro-leum fractions are not available, exclusively. Accordingly, experi-mental bubble pressures of reservoir oil samples were used, inorder to evaluate the model reliability for petroleum fractions. Forthis purpose, 12 oil samples were chosen at different temperaturesand compositions. Conditions and specifications of oil samples aregiven in Table 5. Experimental data of reservoir fluids containbubble pressure, liquid phase density (rL), oil formation volumefactor (Bo) and gas oil ratio (GOR), measured in differentialliberation test (DL test). It has to be noted that a wide variety of

Fig. 9. A comparison between the proposed model and other equations in predicting thepressure of all binary mixtures at the various compositions from 0 up to 1. The temperatu[298–348] K, (6, 7, 8, 9) [313] K, (10) [298–318] K, (11) [313–353] K, respectively.

compounds can be found in reservoir fluids analysis from lightgases such as N2, CO2 and H2S to heavy components such aspetroleum fractions. Reservoir fluid temperature may be lowerthan the critical temperature of some components and exceeds therest. So, the a-function must have enough precision for allcomponents of the mixture either upper or lower than their criticaltemperature, in order to estimate the more accurate bubblepressure of mixture. Peng–Robison EOS fails to predict an accuratea-function in supercritical temperatures, especially for nitrogen,carbon dioxide, hydrogen sulfide and methane at the reservoirfluid temperature. On the other hand, the lack of detailedinformation about the ingredients of petroleum fractions led theirnature to be unknown. Characterization of such compounds areusually performed by some physical properties such as molecularweight and normal boiling point. Employing the normal boilingpoint as an additional input parameter in the proposed a-functioncould enhance its ability to get more structural information aboutpetroleum fractions. The critical temperature and acentric factorusing for a calculation must also be estimated for petroleumfractions. An appropriate correlation was selected to predict thecritical properties and acentric factor of petroleum fractions asfollows:

F f SGð Þ; jð Þ ¼ a � f SGð Þbjc þ d � f SGð Þejf� �g

ð15Þ

f SGð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 þ 2 � SG3 � SG

rð16Þ

where F indicates any of critical properties or acentric factor, SG isthe specific gravity and j is substituted by either molecular weightor normal boiling point. Coefficients a, b, c, d, e, f and g can be foundin the work of Hosseinifar and Jamshidi [46]. Table 6 presents theaverage absolute errors in the prediction of bubble pressure (Pb),liquid phase density (rL), oil formation volume factor (Bo) and gasoil ratio (GOR) obtained from the proposed model in comparison tooriginal Peng–Robinson EOS for all chosen oil samples.

bubble pressure of different binary mixtures. Experimental data include the bubblere ranges of the mixtures are (1) [230–270] K, (2) [210–270] K, (3) [278–293] K, (4, 5)

Table 5The specifications of reservoir fluids used in this study.

Samples Reservoir temperature (K) Bubble point pressure (bar) Reservoir pressure (bar) Sampling depth (m)

Oil 01 365.5 95.900 300.94 2730Oil 02 368.0 192.78 272.27 2286Oil 03 377.2 141.53 304.11 2842Oil 04 373.2 212.40 297.22 2717Oil 05 350.3 15.390 207.27 1765Oil 06 385.2 257.70 326.38 3212Oil 07 389.8 234.74 328.10 3266Oil 08 341.0 132.73 168.94 1457Oil 09 372.7 122.19 272.68 2331Oil 10 375.8 81.810 275.71 2371Oil 11 362.0 188.57 249.04 2286Oil 12 340.8 34.910 187.21 1742

Table 6The percent of average absolute deviations (AAD%) in the prediction of bubblepressure (Pb), oil formation volume factor (Bo), liquid phase density (rL) and gas oilratio (GOR), for selected oil samples using the proposed model and original Peng–Robinson EOS.

Samples AAD% (original PR) AAD% (proposed model)

Pb Bo rL GOR Pb Bo rL GOR

Oil 01 22.51 12.66 3.540 22.38 9.55 8.570 1.09 11.71Oil 02 4.790 2.960 8.890 7.460 1.11 2.240 8.33 5.320Oil 03 29.71 8.430 7.010 77.08 9.31 6.130 5.27 44.19Oil 04 23.03 2.240 2.870 11.49 7.29 1.120 2.14 3.980Oil 05 21.94 1.440 6.390 15.68 6.66 0.550 7.05 14.06Oil 06 20.31 11.16 9.460 28.91 3.48 10.77 9.32 17.37Oil 07 17.22 4.670 2.550 18.52 3.16 4.180 2.29 15.94Oil 08 22.16 2.790 1.940 11.55 6.36 1.990 2.06 7.920Oil 09 29.99 5.680 4.540 19.48 2.36 2.770 2.22 8.560Oil 10 26.16 22.19 16.22 44.18 8.47 19.73 15.4 17.95Oil 11 24.53 7.390 6.840 14.82 7.56 8.040 4.29 13.62Oil 12 33.14 21.45 3.080 32.11 9.01 19.85 2.29 19.87

Table 7A comparison of all methods in prediction of bubble pressure (Pb), oil formationvolume factor (Bo), liquid phase density (rL) and gas oil ratio (GOR) for five chosenoil samples.

EOS Property Calculated properties (AAD%)

Oil 02 Oil 04 Oil 06 Oil 09 Oil 12

Original PR Pb 4.79 23.0 20.3 29.9 33.1Bo 2.96 2.24 11.1 5.68 21.4rL 8.89 2.87 9.46 4.54 3.08GOR 7.46 11.4 28.9 19.4 32.1

Modified PR Pb 3.94 12.5 14.4 13.1 16.6Bo 2.51 2.11 10.5 4.43 20.8rL 8.75 2.54 9.73 3.81 2.97GOR 7.09 9.41 22.6 17.2 25.1

PSRK Pb 3.61 16.4 13.5 19.9 21.7Bo 2.34 2.01 11.2 3.56 18.9rL 9.03 2.17 8.98 3.22 2.51GOR 6.11 7.55 19.7 18.3 21.1

PC-SAFT Pb 2.06 7.81 4.26 3.39 8.81Bo 1.98 2.27 9.84 4.06 20.2rL 6.44 1.28 7.37 2.62 1.33GOR 4.95 4.37 15.8 9.53 19.2

Proposed model Pb 1.11 7.29 3.48 2.36 9.01Bo 2.24 1.12 10.7 2.77 19.8rL 8.33 2.14 9.32 2.22 2.29GOR 5.32 3.98 17.3 8.56 19.8

68 P. Hosseinifar, S. Jamshidi / Fluid Phase Equilibria 408 (2016) 58–71

It is observed from Table 6 that major differences betweenresults predicted by the proposed model and Peng–Robinson EOScan be seen in the cases of bubble pressure and gas oil ratio values.Average absolute errors calculated by the original Peng–RobinsonEOS are 22.95% for bubble pressure, 8.58% for oil formation volumefactor (Bo), 6.11% for liquid phase density (rL) and 25.31% for gas oilratio (GOR). The corresponding results obtained from the proposedmodel are 6.19% for bubble pressure, 7.16% for oil formation volumefactor (Bo), 5.15% for liquid phase density (rL) and 15.04% for gas oilratio (GOR), respectively. Furthermore, another comparison wasalso performed between results predicted by the proposed modeland other methods for five chosen oil samples and the results ofwhich are presented in Table 7. It is observable that, the proposedmodel could significantly improve the bubble pressure predictionsfor reservoir fluids and gives better results in the prediction of gasoil ratio (GOR) compared to the original PR, modified PR and PSRKequations of state, while the prediction accuracy of PC-SAFT modelis almost similar to the proposed model. Although, in thesecomparisons it cannot explicitly be distinguished how much of theuncertainties are caused by the petroleum fractions, it isobservable from Table 4 that the average absolute errors inpredicting the saturation pressure of all well-defined componentsexisting in reservoir fluids analysis (N2, H2S, CO2, C1, C2, C3, i-C4, n-C4, i-C5, and n-C5) are 7.54%, 3.72%, 8.75%, 1.2% and 1.5% for originalPR, modified PR, PSRK, PC-SAFT and proposed model, respectively.On the other hand, Table 7 indicates that the average absolute

errors in predicting the bubble pressure of all selected oil samples(including well-defined components and petroleum fractions) are22.25%, 12%, 15.02%, 5.26% and 4.65% for original PR, modified PR,PSRK, PC-SAFT and proposed model, respectively. Therefore, thefirst group of AAD% obtained from Table 4 is an indicative of thewell-defined component deviations existing in reservoir fluids andthe second group of AAD% shows the total component deviations(well-defined and petroleum fractions). The difference betweenthese two groups are 14.67%, 8.38%, 6.26%, 4.06% and 3.15% fororiginal PR, modified PR, PSRK, PC-SAFT and proposed model,respectively which can be attributed to the petroleum fractions.Hence, it can then be concluded that the proposed model is capableof providing more accurate results for petroleum fractions existingin reservoir fluids, rather than the other methods. It should benoted that, oil formation volume factor (Bo) and liquid phasedensity (rL) were less affected by the temperature dependentfunction (a) modifications. The improvement of a-functioninfluences chiefly the vapor pressure estimation, and the liquidphase density can be corrected using the volume translationmethods [75–77].

P. Hosseinifar, S. Jamshidi / Fluid Phase Equilibria 408 (2016) 58–71 69

3.4. Heat capacity calculations using the first and second derivatives ofa-function

The first and second derivatives of the new proposeda-function were determined as below:

a0 ¼ dad�1dð Þ c1b

0 þ c2g 0

Tc

� �ð17Þ

b0 ¼ �a1a2bTa1�1ð Þr

1 � Ta1r

� � ð18Þ

g 0 ¼ gTr

lnglnTr

� �ð19Þ

a00 ¼ d � 1d

� �a02

aþ da

d�1dð Þ c1b

00 þ c2g 00

T2c

!ð20Þ

b00 ¼ b0 a1 � 1Tr

þ b0 a2 � 1ð Þba2

ð21Þ

g 00 ¼ g 0 g 0

g� 1Tr

� �ð22Þ

Experimental data of isobaric heat capacity at 50 bar andvarious temperatures were collected [49] to evaluate the perfor-mance of proposed a-function in the estimation of specific heatcapacity at constant pressure (Cp). The residual molar heat capacityis defined as:

Cp � C�p ¼ �R þ T

Z n

1

@2P

@T2

!n

dn � T @P=@Tð Þ2n@P=@nð ÞT

ð23Þ

where R is the gas constant, n is the specific molar volume, T is thetemperature, P is the pressure, Cp is the specific molar heatcapacity and C�

p is the specific molar heat capacity for ideal gas

Fig. 10. Experimental data, proposed model and Peng–Robinson EOS pr

which can be calculated from the relation suggested by Aly and Lee[78]. The residual molar heat capacity given in Eq. (23) wasdetermined for the proposed a-function as expressed below:

Cp � Cp� ¼ �R þ ART2a00

2ffiffiffi2

pBln

Z þ B 1 þffiffiffi2

p� �Z þ B 1 �

ffiffiffi2

p� �24

35

�RT2 1

Z�Bð ÞT � Aa0

Z2þ2BZ�B2

� �22Aa ZþBð Þ

Z2þ2BZ�B2ð Þ2 �1

Z�Bð Þ2

ð24Þ

Z ¼ PnRT

ð25Þ

A ¼ acP

RTð Þ2ð26Þ

B ¼ bPRT

ð27Þ

Molar heat capacity (Cp) can be calculated from Eq. (24) bysubstituting a0 and a00 from the Eqs. (17)–(22). Fig. 10 illustratesspecific heat capacity predictions through the proposed model andPeng–Robinson EOS for ethane. The results are also compared toexperimental data of isobaric heat capacity at 50 bar. Calculatedaverage absolute errors is 13.35% for Peng–Robinson EOS and is4.39% for the proposed model which indicate the proposed modelpresents more accurate results in the prediction of specific heatcapacity at constant pressure. Finally, a number of applicablerelations required to calculate desirable thermodynamic proper-ties are derived and presented in Table 8. The scrutiny of the aboveresults obtained from a-functions, reveal the proposed model canimprove the attractive term of Peng–Robinson EOS so that it willyield more acceptable results in the prediction of saturation vaporpressure of pure substances, bubble pressure of mixtures (such asreservoir fluids), isobaric specific heat capacity and all VLEcalculations relying on the temperature dependence of a-function.

edictions for the isobaric specific heat capacity of ethane at 50 bar.

Table 8A list of required relations for the calculation of different thermodynamic properties.

The EOS in terms ofA,B and Z

Z3 � 1 � Bð ÞZ2 þ A � 3B2 � 2B� �

Z � AB � B2 � B3� �

¼ 0

Z ¼ PnRT

; A ¼ acP

RTð Þ2; B ¼ bP

RT

Molar enthalpy departureh � h� ¼ RT Z � 1ð Þ þ ART a0T � að Þ

2ffiffiffi2

pB

lnZ þ B 1 þ

ffiffiffi2

p� �Z þ B 1 �

ffiffiffi2

p� �24

35

Molar entropy departures � s� ¼ Rln Z � Bð Þ þ ARTa0

2ffiffiffi2

pBln

Z þ B 1 þffiffiffi2

p� �Z þ B 1 �

ffiffiffi2

p� �24

35

Molar internal energy departureu � u� ¼ ART a0T � að Þ

2ffiffiffi2

pB

lnZ þ B 1 þ

ffiffiffi2

p� �Z þ B 1 �

ffiffiffi2

p� �24

35

Fugacity coefficientln’ ¼ Z � 1 � ln Z � Bð Þ � A

2ffiffiffi2

pBln

Z þ B 1 þffiffiffi2

p� �Z þ B 1 �

ffiffiffi2

p� �24

35

Fugacity coefficient in mixtureln’̂i ¼

bib

Z � 1ð Þ � ln Z � Bð Þ � A

2ffiffiffi2

pB

2P

ixiaija

� bib

� �ln

Z þ B 1 þffiffiffi2

p� �Z þ B 1 �

ffiffiffi2

p� �24

35

Molar heat capacity departure at constant pressureCp � Cp

� ¼ �R þ ART2a00

2ffiffiffi2

pBln

Z þ B 1 þffiffiffi2

p� �Z þ B 1 �

ffiffiffi2

p� �24

35�

RT2 1Z�Bð ÞT � Aa0

Z2þ2BZ�B2

� �22Aa ZþBð Þ

Z2þ2BZ�B2ð Þ2 �1

Z�Bð Þ2

Molar heat capacity departure at constant volumeCn � Cn

� ¼ �R þ ART2a00

2ffiffiffi2

pBln

Z þ B 1 þffiffiffi2

p� �Z þ B 1 �

ffiffiffi2

p� �24

35

First derivative of a-functiona0 ¼ da

d�1dð Þ c1b

0 þ c2g 0

Tc

� �

b0 ¼ �a1a2bTa1�1ð Þr

1 � Ta1r

� � ; g 0 ¼ gTr

lnglnTr

� �

Second derivative of a-functiona00 ¼ d � 1

d

� �a02

aþ da

d�1dð Þ c1b

00 þ c2g 00

T2c

!

b00 ¼ b0 a1 � 1Tr

þ b0 a2 � 1ð Þba2

; g 00 ¼ g 0 g 0

g� 1Tr

� �

70 P. Hosseinifar, S. Jamshidi / Fluid Phase Equilibria 408 (2016) 58–71

4. Conclusion

The temperature dependent function (a) existing in theattractive term of cubic equations of state can have an importantrole in the detection of attractive intermolecular forces. In thepresent paper, a new a-function was proposed to enhance thecapability of Peng–Robinson EOS in the prediction of saturationvapor pressure for polar, nonpolar, light gases and heavyhydrocarbon compounds. The proposed model can also improvethe bubble pressure and gas oil ratio (GOR) of complex mixturessuch as reservoir fluids including petroleum fractions and heavycuts. The normal boiling point (Tb) was employed as an additionalparameter, in order to modify the performance of a-function. Thismodel has an acceptable precision in representing fluid phasebehavior at supercritical region and also shows a reasonablelimiting behavior when the temperature increases infinitely. Acomparison was performed between the proposed model andother equations of state, the results of which indicate that theproposed model gives better outcomes compared to the originalPeng–Robinson and other methods for pure components and fluidmixtures.

Acknowledgment

We would like to express our deepest gratitude to MAPSATechnology Center for their technical and financial support.

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