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Predicting Binary-Interaction Parameters of Cubic Equations of State for Petroleum Fluids Containing Pseudo-components Xiaochun Xu,† Jean-Noel̈ Jaubert,*,† Romain Privat,† Pierre Duchet-Suchaux,‡ and Francisco Braña-Mulero§ †

Université de Lorraine, Ecole Nationale Supérieure des Industries Chimiques, Laboratoire Réactions et Génie des Procédés (UMR CNRS 7274), 1 rue Grandville, 54000 Nancy, France ‡ Total S.A., 2 place Jean Millier, La Défense 6, 92078 Cedex Paris La Défense, France § Schneider Electric, 10900 Equity Drive, Houston, Texas 77041, United States S Supporting Information *

ABSTRACT: Cubic equations of state (EoS) are widely used for the prediction of thermodynamic properties of petroleum fluids containing both well-defined and pseudo-components. Such EoS require as input parameters the critical temperature (Tc), the critical pressure (Pc), and the acentric factor (ω) of each compound. For well-defined components, such properties are known from experiments and easily obtained. For pseudo-components they are routinely estimated using one of the numerous characterization methods (CM) available in the open literature. A CM is nothing more than a set of correlations which makes it possible to estimate Tc, Pc, and ω of a pseudo-component (PC) from the knowledge of its normal boiling point (NBP), molecular weight (MW), or specific gravity (SG). Regarding the binary-interaction parameters (BIP) kij (where i and/or j are/is a pseudo-component(s)) which appear in classical mixing rules, they are either set to zero or estimated by a specific correlation. Most of the proposed correlations are however purely empirical and usually only make possible the estimation of the kij between light components (H2S, CO2, N2, C1, C2, and C3) and a pseudo-component. The full kij matrix is thus beyond reach and the BIP are usually temperature-independent. In this work, the PPR78 model is used to predict BIP suitable for the Peng−Robinson EoS whereas the PR2SRK model is used to predict BIP suitable for any other cubic EoS. Since these models can be seen as groupcontribution methods (GCM) to estimate the kij, one needs to access the chemical structure of each PC. The chemical structure of PC is however too complex to be precisely determined. For this reason, it was assumed that each PC was made of only three groups: CPAR, CNAP, and CARO in order to take into account their paraffinic, naphthenic, and aromatic characters, respectively. The occurrences (N) of the three aforementioned groups are determined from the knowledge of Tc,CM, Pc,CM, and ωCM (issuing from a CM). To reach this goal, GC methods aimed at estimating Tc, Pc, and ω of hydrocarbons were developed. Such methods have the ability to consider only three elementary groups: CPAR, CNAP, and CARO. In the end, the three known properties (Tc,CM, Pc,CM, and ωCM) can be expressed as functions of NPAR, NNAP, and NARO (the occurrences of the groups) and we thus only need to solve a system of three equations with three unknowns. To check its validity, the present approach is applied to the prediction of the phase behavior of real petroleum fluids containing pseudo-components. The test results show the pertinence of the proposed method to predict the kij when i and/or j is a pseudo-component.

1. INTRODUCTION

employed distribution functions of molar composition against molecular weight for the characterization of pseudo-components in petroleum fluids before using correlations to estimate Tc, Pc, and ω. Avaullée et al.12−14 also developed an original characterization method to estimate cubic EoS parameters for pseudo-components by defining an equivalent carbon number, which can be determined from a group-contribution method. However, although many characterization methods have been developed to predict {Tc, Pc, and ω} for pseudo-components, the binary-interaction parameters (BIP) kij of a pair involving a pseudo-component are either set to zero or estimated by a specific correlation. Most of the proposed correlations are however purely empirical and usually only make possible the

Cubic equations of state (EoS) are widely used for modeling thermodynamic properties of petroleum fluids involving both well-defined and pseudo-components. The latter are usually illdefined C7 plus fractions. Such EoS require as input parameters the critical temperature (Tc), the critical pressure (Pc), and the acentric factor (ω) of each component. For well-defined components, such properties are known from experiments and easily obtained. For pseudo-components they are routinely estimated using one of the numerous characterization methods (CM) available in the open literature. A CM is nothing more than a set of correlations which makes it possible to estimate Tc, Pc, and ω of a pseudo-component (PC) from the knowledge of two properties among the normal boiling point (NBP), the molecular weight (MW), and the specific gravity (SG). The most well-established are certainly those published by Cavett,1 Lee and Kesler,2,3 Riazi and Daubert,4,5 and Twu.6 We also can cite the works by Whitson et al.7,8 and Pedersen et al.9−11 who © 2015 American Chemical Society

Received: Revised: Accepted: Published: 2816

December 18, 2014 February 24, 2015 February 24, 2015 February 24, 2015 DOI: 10.1021/ie504920g Ind. Eng. Chem. Res. 2015, 54, 2816−2824

Article

Industrial & Engineering Chemistry Research

2. A FEW WORDS ON THE PPR78 AND PR2SRK MODELS

estimation of the kij between light components (H2S, CO2, N2, C1, C2, C3) and a pseudo-component. The full kij matrix is thus beyond reach and the BIP are usually temperature-independent. The PPR78 and PR2SRK methods,15,16 which have been shown to be accurate and reliable for the phase-equilibrium prediction of many systems15−33 make possible the estimation of kij parameters for any cubic EoS from the mere knowledge of {Tc, Pc, ω} and chemical structures of components. Therefore, to calculate k ij (where i and/or j are/is a pseudocomponent(s)) from the PPR78 or PR2SRK methods, the chemical structure of the pseudo-component is required. The objective of this work is to propose an approach to predict the BIP between a pseudo-component and a real component or another pseudo-component with the PPR78 (or PR2SRK) methods. The approach is divided into three steps. • First, a so-called characterization method (CM) is employed to estimate the critical parameters and the acentric factor of the considered pseudo-component from available experimental data like its MW, NBP, or SG. The calculated parameters are denoted Tc,CM, Pc,CM, and ωCM where subscript CM indicates that such properties are estimated from a characterization method. • Second, we need to access the chemical structure of the PC in order to be able to use the PPR78 model (for the Peng−Robinson EoS) or the PR2SRK model (for any other cubic EoS) to predict the BIP. Since the chemical structure of PC is too complex to be precisely determined, it was assumed that each PC was made up of only three groups: CPAR, CNAP, and CARO in order to take into account its paraffinic, naphthenic, and aromatic characters, respectively. The possible sulfur, nitrogen, and oxygen contents of a PC are thus neglected. From our experience such a hypothesis is certainly acceptable. The occurrences N = (NPAR, NNAP, NARO) of the three aforementioned groups are determined from the knowledge of Tc,CM, Pc,CM, and ωCM by using GC methods aimed at estimating Tc, Pc, and ω of hydrocarbons that were specifically developed for this study. Such GC methods have the ability to consider only three elementary groups: CPAR, CNAP, and CARO. As a consequence, the three known properties (Tc,CM, Pc,CM, and ωCM) can be expressed as functions of NPAR, NNAP, and NARO, and we thus only need to solve the following system of three equations to determine N.

In 1978, Peng and Robinson published an improved version of their well−known equation of state, referred to as PR78 in this paper. For a pure component, the PR78 EoS is P=

ai(T ) RT − v − bi v(v + bi) + bi(v − bi)

(1)

and ⎧ R = 8.314472 J· mol−1· K−1 ⎪ ⎪ 3 3 ⎪ X = −1 + 6 2 + 8 − 6 2 − 8 ⎪ 3 ⎪ ≈ 0.253076587 ⎪ RTc, i X ⎪ ⎪bi = Ωb P with: Ωb = X + 3 ≈ 0.0777960739 c,i ⎪ ⎪ 2 2 ⎪ a = Ω R Tc, i α(T ) with: Ω = 8(5X + 1) ⎪ i a a 49 − 37X Pc, i ⎨ ⎪ ≈ 0.457235529 ⎪ 2 ⎡ ⎛ ⎞⎤ ⎪ T ⎟⎥ ⎢ ⎜ ⎪ and α(T ) = ⎢1 + mi⎜1 − Tc, i ⎟⎠⎥⎦ ⎝ ⎣ ⎪ ⎪ ⎪ if ωi ≤ 0.491 mi = 0.37464 + 1.54226ωi ⎪ 2 ⎪ − 0.26992ωi ⎪ ⎪ if ωi > 0.491 mi = 0.379642 + 1.48503ωi ⎪ − 0.164423ω 2 + 0.016666ω 3 ⎩ i i

(2)

where P is the pressure, R is the gas constant, T is the temperature, ai and bi are the EoS parameters of pure component i, v is the molar volume, Tc,i is the critical temperature, Pc,i is the critical pressure, and ωi is the acentric factor of pure i. To apply this EoS to a mixture, mixing rules are necessary to calculate the values of a and b of the mixture. Classical Van der Waals one-fluid mixing rules are used in the PPR78 model: N N ⎧ ⎪ a(T , z) = ∑ ∑ zizj ai(T ) · aj(T ) [1 − kij(T )] ⎪ ⎪ i=1 j=1 ⎨ N ⎪ ⎪b(z) = ∑ zibi ⎪ ⎩ i=1

⎧Tc,CM = Tc,GC(N) ⎪ ⎪ ⎨ Pc,CM = Pc,GC(N) ⎪ ⎪ ω = ω (N ) ⎩ CM GC

(3)

where zi represents the mole fraction of component i and N is the number of components in the mixture. The kij(T) parameter, whose estimation is difficult even for the simplest systems, is the so-called binary interaction parameter (BIP) characterizing the molecular interactions between molecules i and j. Although the common practice is to fit kij to reproduce the vapor-liquid equilibrium data of the mixture under consideration, the predictive PPR78 model calculates the kij value, which is temperature-dependent, with a group-contribution method using the following expression:

The subscript “GC” indicates that the properties are estimated from a group-contribution method. • Third, for each pseudo-component i, BIP kij (where j is either another pseudo-component or a real component) are calculated from the knowledge of {Tc,i,CM, Pc,i,CM, ωi,CM, Ni} using the PPR78 or PR2SRK approaches.15,16 In this last step, it is assumed that groups CPAR, CARO, and CNAP (assigned to the pseudo-components) are equivalent to groups CH2, CHaro, and CH2,cyclic defined in the PPR78/PR2SRK models. 2817

DOI: 10.1021/ie504920g Ind. Eng. Chem. Res. 2015, 54, 2816−2824

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Industrial & Engineering Chemistry Research ⎧ N N ⎪ 1⎡ g g kij(T ) = ⎨ − ⎢∑ ∑ (αik − αjk)(αil − αjl) ⎪ 2 ⎢⎣ k = 1 l = 1 ⎩ ⎛ a (T ) ⎥−⎜ i − ⎜ ⎥⎦ ⎝ bi

(Bkl / Akl − 1)⎤

⎛ 298.15 ⎞ ⎟ Akl ⎜ ⎝ T /K ⎠

⎧ a (T )·a (T ) ⎪ i j /⎨2 ⎪ bi ·bj ⎩

3

[(Pc − Pca)/MPa]−0.5 − (Pcb/MPa) =

i=1

(7)

2⎫ aj(T ) ⎞ ⎪ ⎟⎬ bj ⎟⎠ ⎪ ⎭

⎫ ⎪ ⎬ ⎪ ⎭

⎛ω⎞ exp⎜ ⎟ ⎝ ωa ⎠

⎧ E SRK (T ) − (δiSRK − δjSRK )2 ⎪ k SRK(T ) = ij ij ⎪ 2δiSRKδjSRK ⎪ ⎨ ⎪ EijSRK (T ) = ξEijPPR78(T ) ⎪ ⎪ ξ ≈ 0.807341 ⎩

(4)

∑ Niωi

(8)

i=1

⎛ f EXP − f GC ⎞2 ∑ ⎜⎜ i EXP i ⎟⎟ fi ⎠ i=1 ⎝ N

(9)

in which f i is either Tc, Pc, or ω of compound i. Superscripts EXP and GC stand for experimental value and value calculated from the group-contribution method, respectively. In this study a databank containing 174 hydrocarbons (paraffinic, naphthenic, and aromatic compounds) for which the experimental values of Tc, Pc, and ω are known was built. The experimental values originate either from the DIPPR databank or from the handbook by Yaws.38 The carbon atom number of the selected hydrocarbons varies from 6 to 24 in accordance with carbon atom numbers of most pseudo-components encountered in petroleum fluids. The fitted universal constants and group contributions are listed in Tables 1 and 2, respectively. The Table 1. Universal Constants To Be Used in the Proposed Group-Contribution Methods

(5)

3. DEVELOPMENT OF GC METHODS (GCM) TO ESTIMATE {TC, PC, ω} OF HYDROCARBONS BY CONSIDERING ONLY THREE ELEMENTARY GROUPS: CPAR, CNAP, AND CARO The group-contribution concept has been widely applied for estimating the properties of pure compounds. In such methods, a physical property of a pure molecule is estimated from the knowledge of its chemical structure. The molecule is split in atomic groups (groups of atoms) and its property is calculated knowing the occurrence and the contribution of each group to the property. Gani’s research group34−37 reported groupcontribution methods for accurately estimating the critical properties and the acentric factor of compounds. It was thus decided to use similar mathematical expressions to relate Tc, Pc, and ω to the occurrences of the three groups considered in this study. The following equations were used:

constants

values

Tca/K Pca/MPa Pcb/MPa−1/2 ωa ωb ωc

238.2735 0.335196 0.370214 0.597969 0.451450 1.393320

Table 2. Group Contributions to Estimate Tc, Pc, and ωa

a

group

Pc,i/MPa

Tc,i/K

ωi

CPAR CNAP CARO

0.036416 0.027210 0.018064

1.304497 1.601452 1.828457

0.093981 0.066826 0.083756

See eqs 6−8.

comparisons between the experimental and estimated Pc, Tc, and ω are shown in Figure 1 and make it possible to conclude that although rather accurate, the proposed GCM do not allow distinguishing isomers since only three elementary groups are considered. As a consequence, the largest deviations between experimental and predicted results are obtained for a molecule in a series of isomers. For each of the 174 compounds, the

3 i=1

3

− ωc =

1 OF = N

Note that the value of the ξ parameter is theoretically justified in the original paper presenting the PR2SRK model.16

∑ Ni × (Tc,i/K)

ωb

In these equations, Tca, Pca, Pcb, ωa, ωb, and ωc are universal constants. Tc,i, Pc,i, and ωi are the contributions of group i to Tc, Pc, and ω, respectively. Ni is the occurrence of group i in the considered molecule. In this work, only three elementary groups (CPAR, CARO, and CNAP) were considered. Here, we mean that group CPAR stands for all the groups that can be found in paraffinic molecules (CH3, CH2, CH, or C). In the same way, group CARO stands for all the groups which can be found in aromatic compounds (CHaro, Caro, or Cfused aromatic rings) and CNAP stands indifferently for CH2,cyclic, CHcyclic, or Ccyclic. The universal constants and the group contributions were determined by minimizing the following objective function:

In eq 4, T is the temperature. The ai and bi values are given in eq 2. The Ng variable is the number of different groups defined by the group-contribution method (for the time being, 21 groups are defined, and Ng = 21). The αik variable is the fraction of molecule i occupied by group k (occurrence of group k in molecule i divided by the total number of groups present in molecule i). The group-interaction parameters, Akl = Alk and Bkl = Blk (where k and l are two different groups), were determined in our previous papers32 (Akk = Bkk = 0). To be really exhaustive, let us recall that the PPR78 model can also be seen as the combination of the PR EoS with a Van Laar type activity coefficient (gE) model under infinite pressure. The PR2SRK model16 makes it possible to deduce the kij of any cubic EoS (like the SRK EoS) knowing those of the PR EoS. In other words, the group-interaction parameters (Akl and Bkl) initially developed for the PR EoS can be used to predict the kij of any other cubic EoS combined with any alpha function. As an example, BIP for the SRK EoS are given by

⎛ T /K ⎞ exp⎜ c ⎟= ⎝ Tca/K ⎠

∑ Ni × (Pc,i/MPa)

(6) 2818


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4. INVERSION OF THE DEVELOPED GCM TO DETERMINE GROUP OCCURRENCES FROM THE KNOWLEDGE OF TC, PC, AND ω Our idea is to use, in a reverse way, the group-contribution methods developed in the previous section to determine the simplified structure (NPAR, NNAP, NARO) of a pseudocomponent. Indeed, if the critical properties and the acentric factor of a pseudo-component are provided by a characterization method, eqs 6−8 could, in principle, be used to estimate the group occurrences N = (NPAR, NNAP, NARO). Before applying this methodology to a pseudo-component, it was decided to check its consistency on the 174 pure components used to develop the GCM. For each component, knowing the experimental values of Tc, Pc, and ω, the group occurrences (NPAR, NARO, NNAP) were determined from eqs 6−8 and compared to the expected values. In some cases, the solved NPAR, NNAP, or NARO values are likely to be negative. However, as shown in Figure 2, the

Figure 2. Parity plots showing the predicted carbon atom numbers (nc,pred) of 174 hydrocarbons by inverse GCM against the real values (nc,real).

predicted carbon atom number of the molecule (nc,pred = NPAR + NARO + NNAP) is consistent with the real carbon atom number (nc,real). So NPAR , N NAP, and NARO could be renormalized with the following equations which keep nc,pred unchanged. 3 ⎧ ∑i = 1 Niold ⎪ N new = N old (if Niold > 0) ⎪ i i ∑ N old > 0 Niold ⎨ i ⎪ new ⎪N = 0 (if Niold < 0) ⎩ i

Figure 1. Parity plots showing the calculated properties (Tc, Pc, and ω) of 174 hydrocarbons by GC against the experimental values.

(10)

The mean errors on the prediction of NPAR, NNAP, and NARO are 1.37, 1.54, and 0.93, respectively, and the max errors on the prediction of NPAR, NNAP, and NARO are 6.60, 6.85, and 4.39, respectively. In order to quantify the impact of such errors on the kij values it was decided to consider a ternary system made up of n-hexane + n-decane + n-eicosane. The kij values were thus calculated through the PPR78 model assuming they were either well-defined components or pseudo-components. The obtained results are summarized in Table 3 where three temperatures (0, 20, and 50 °C) were considered. This table highlights that, regardless of the temperature, the kij value between two n-alkanes depends very little on the fact that

deviations between experimental and predicted values of Tc, Pc, and ω by the developed GCM as well as the group occurrences are listed in Table S1 of the Supporting Information. We can thus conclude that it will be possible to use such GCM in a reverse way in order to determine the numbers (NPAR, NNAP, and NARO) of CPAR, CNAP, and CARO groups in a pseudo-component from the knowledge of its critical temperature, critical pressure, and acentric factor. This point is discussed in the next section. 2819


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Industrial & Engineering Chemistry Research Table 3. Predicted Binary-Interaction Parameters between nHexane, n-Decane, and n-Eicosane at (a) 273.15, (b) 293.15, and (c) 323.15 Ka

compositions of the well-defined light components (H2S, N2, CO2, methane, ethane, propane, i-butane, n-butane, i-pentane, n-pentane, and n-hexane) were determined by chromatography. For each blend, a TBP distillation was done so that the composition, the molecular weight, and the density of each distillation cut, from C7 to C20+, could be measured. The results of the compositional analyses can be found in Table 4.

a compounds n-hexane n-decane n-eicosane

(1) (2) (1) (2)

n-hexane

n-decane

−0.00044 −0.00239 −0.00894 −0.01497

−0.00443 −0.00497

Table 4. Compositions of the Three Petroleum Fluids Analyzed by TOTAL mole fraction (%)

b n-hexane

compounds n-hexane n-decane n-eicosane

(1) (2) (1) (2)

n-decane

−0.00055 −0.00248 −0.00956 −0.01560

−0.00466 −0.00524

n-hexane

n-decane

−0.00069 −0.00261 −0.01049 −0.01657

−0.00500 −0.00565

c compounds n-hexane n-decane n-eicosane

(1) (2) (1) (2)

a

(1) BIP predicted from actual occurrences of groups in compounds (the n-alkanes are considered as well-defined components). (2) BIP predicted from the estimated occurrences of groups in compounds (the n-alkanes are considered as pseudo-components).

alkanes are considered as pseudo-components or well-defined components which validates our approach for this simple system. The detailed results for each of the 174 compounds used in this study can be found in Table S2 of the Supporting Information.

5. CHECK OF THE VALIDITY OF THE PROPOSED APPROACH As discussed in former sections, it is assumed that the critical properties (Tc,i,CM, Pc,i,CM) and the acentric factor (ωi,CM) of a pseudo-component are obtainable from a given characterization method. Moreover, eqs 6, 7, 8, and 10 make it possible to determine a simplified structure (NPAR, NNAP, and NARO) of the considered pseudo-component. Consequently, by making the hypothesis that groups CPAR, CNAP, and CARO (assigned to the pseudo-component) are respectively equivalent to groups CH2, CH2,cyclic, and CHaro defined in the PPR78/PR2SRK models, it becomes possible for a pseudo-component i to calculate kij (where j is either another pseudo-component or a real component) with eq 4. In this section, the relevance of the proposed approach to determine kij (where i is a pseudo-component) from the knowledge of Tc,CM, Pc,CM, and ωCM is investigated. On the other hand, the efficiency of the characterization method used to generate Tc,CM, Pc,CM, and ωCM is beyond the scope of this paper and will not be discussed. The verification process involves four steps. Step 1. Generation of Detailed Compositions for Three Different Petroleum Fluids. Three PVT reports established by the French company TOTAL on three petroleum fluids (a crude oil, a heavy oil, and a gas condensate) were randomly selected from their huge databank. The

component

crude oil

heavy oil

gas condensate

H2S N2 CO2 methane ethane propane i-butane n-butane i-pentane n-pentane n-hexane C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+

1.49 0.32 2.80 45.30 9.11 5.50 1.06 3.07 1.24 1.82 2.38 2.84 2.83 2.49 2.15 1.74 1.42 1.31 1.15 1.00 0.88 0.76 0.69 0.62 6.03

0.18 0.21 1.88 32.36 7.32 7.35 1.80 4.73 2.19 3.11 2.11 2.35 4.61 4.17 3.72 3.11 2.52 2.17 1.84 1.62 1.32 1.12 0.94 0.82 5.71

0.00 0.50 1.55 80.62 8.57 4.24 0.70 1.40 0.44 0.49 0.22 0.24 0.32 0.27 0.16 0.08 0.07 0.04 0.04 0.02 0.02 0.01 0.01 0.01 0.01

Starting from such analytical data, a very detailed composition of the three blends was generated. The aim of this step is to characterize in detail the chemical structure (i.e., the group occurrences) of each cut from C7 to C20+ since they will be considered each as a pseudo-component. To generate such a detailed composition, it was assumed that each distillation cut (from C7 to C19) was a mixture of a paraffinic (P), a naphthenic (N), and an aromatic (A) compound. For a cut Cn (7 ≤ n ≤ 19), the P, N, and A components were selected so that their NBP’s stand between the NBP of the n-alkane having (n − 1) carbon atoms and the n-alkane having n carbon atoms. The selected components along with their NBP, MW, and SG are listed in Table S3 of the Supporting Information. The internal compositions of the P, N, and A compounds in each cut were determined from the equations: MWexp = x P·MWP + x N·MWN + xA ·MWA (11) SGexp =

1 wP SGP

+

wN SG N

1 = x P + x N + xA

+

wA SGA

(12) (13)

where xP, xN, and xA denote the mole fractions of the paraffinic, naphthenic, and aromatic components in each cut, and wP, wN, 2820


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Industrial & Engineering Chemistry Research and wA are the corresponding weight fractions. MWexp and SGexp are the experimental molecular weight and specific gravity of cut Cn determined during the distillation process. In the case where the resulting xP, xN, or xA were out of the range 0−1, they were readjusted using the following equations: ⎧ xiold new ⎪ (if xiold ≥ 0) ⎪ xi = old ∑x old > 0 xk ⎨ k ⎪ ⎪ x new = 0 (if xiold < 0) ⎩ i

Table 5. Detailed Molar Composition of Each Cut (from C7 to C20+) for the Three Studied Petroleum Fluids cut C7

C8

(14)

For the residue of distillation (cut C20+), the process was slightly different. The C20+ cut was assumed to be a mixture of naphthalene and a heavy n-alkane ranging from n-C35 to n-C45. For each n-alkane (35 ≤ n ≤ 45), the mole fractions of naphthalene (xnap) and heavy n-alkane (xp) were calculated by exp ⎧ ⎪ MW C20+ = xnap· MWnap + x p· MWp ⎨ ⎪ ⎩1 = xnap + xp

C9

C10

C11

(15)

where MWnap and MWp are the molecular weights of naphthalene and testing n-alkane. The selected n-alkane is the one which minimizes the difference between the experimental and calculated SG of the C20+ cut. That is, it minimizes the difference:

C12

C13

1

SGCexp20+ −

wnap SGnap

+

wp SGp

(16)

C14

where wnap and wp are weight fractions of naphthalene and nalkane, respectively. In our case, n-C35H72 is selected to model the residue of the gas condensate and n-C45H92 is selected for the residue of both the crude oil and the heavy oil. For the three blends, the detailed composition, in terms of mole fractions, of each cut can be found in Table 5. From this table, it is thus possible to define for each cut (from C7 to C20+), a detailed decomposition in elementary groups. Step 2. Estimation of {Tc, Pc, and ω} of Each Cut for Each Petroleum Fluid and Definition of a Reference Case (Case 1: Calculation of kij from Detailed Group Decomposition for Each Cut and Values of {Tc, Pc, ω}cut Stemming from Twu’s CM). It was decided to consider each cut (from C7 to C20+) as a pseudo-component and to use the characterization method developed by Twu to provide {Tc,CM, Pc,CM, and ωCM}. We thus started by calculating the NBP and the SG of each cut (considered as a pseudo-component) from the experimental values compiled in Table S3 of the Supporting Information and from the compositions given in Table 5. The following equations were used:

C15

C16

C17

C18

C19

C20+

compound 2,4-dimethylpentane cyclohexane benzene 2,2,3-trimethylpentane 1-trans-2-cis-3trimethylcyclopentane toluene 2,2,3,3-tetramethylpentane 1-trans-3,5trimethylcyclohexane m-xylene n-decane isobutylcyclohexane isobutylbenzene n-undecane cis-decahydronaphthalene 3-ethyl-o-xylene n-dodecane 1-methyl-2pentylcyclohexane p-tert-butylethylbenzene n-tridecane 2-butyl-1,1,3trimethylcyclohexane 1,2,3-trimethylindene n-tetradecane dicyclohexylmethane 1,2,3,4-tetraethylbenzene n-pentadecane 1,1-dicyclohexylethane 1,1-diphenylethane n-hexadecane 2,2-dicyclohexylpropane 2-butylnaphthalene n-heptadecane decylcyclohexane n-decylbenzene n-octadecane dodecylcyclopentane n-undecylbenzene n-nonadecane tetradecylcyclopentane n-dodecylbenzene n-pentatetracontane n-pentatriacontane naphthalene

crude oil

heavy oil natural gas

gas condensate

0.5858 0.1562 0.2579 0.7589 0.0000

0.5957 0.2856 0.1187 0.7418 0.0000

0.4675 0.0000 0.5325 0.5379 0.4621

0.2411 0.8456 0.0000

0.2582 0.8497 0.0000

0.0000 0.8667 0.0000

0.1544 0.5195 0.0000 0.4805 0.6104 0.0734 0.3163 0.4691 0.0000

0.1503 0.5069 0.0000 0.4931 0.3669 0.0000 0.6331 0.4261 0.0000

0.1333 0.5629 0.0000 0.4371 0.6790 0.1935 0.1275 0.4661 0.0000

0.5309 0.0000 1.0000

0.5739 0.0000 1.0000

0.5339 0.0000 1.0000

0.0000 0.4400 0.5600 0.0000 0.0828 0.9172 0.0000 0.3072 0.6928 0.0000 0.3449 0.0000 0.6551 0.0000 1.0000 0.0000 0.0000 0.7547 0.2453 0.6570 0.0000 0.3430

0.0000 0.4310 0.5690 0.0000 0.0000 1.0000 0.0000 0.3501 0.6499 0.0000 0.3669 0.0000 0.6331 0.0000 1.0000 0.0000 0.1142 0.6994 0.1864 0.6471 0.0000 0.3529

0.0000 0.5893 0.4107 0.0000 0.0000 1.0000 0.0000 0.3522 0.6478 0.0000 0.4230 0.0000 0.5770 0.0930 0.9070 0.0000 0.4851 0.5149 0.0000 0.0000 0.5588 0.4412

3

NBPcutCn =

∑ xi NBPi i=1

SGcutCn =

component its detailed chemical structure (see Table 5) and Tc, Pc, and ω. Knowledge of such a detailed chemical structure, i.e., knowledge of actual occurrences of the PPR78 groups (CH3, CH2, CH, C, CHARO, CARO, CH2cyclic, CHcyclic, etc.) and the estimated values of Tc, Pc, and ω (stemming from the CM) of each pseudo-component in each petroleum fluid makes it possible to calculate BIP kij of any pair (i)/(j) (where i and/or j are/is a pseudo-component(s)) using the PPR78 model. The (P, T) phase envelope of each blend (natural gas, heavy, and crude oils) can then be plotted. Calculations performed in such a way constitute the reference case (see case 1 in Figures 3−5).

(17)

1 3

w

∑i = 1 SGi

i

(18)

where xi and wi are the mole and weight fractions of component i (see Table 5) in the cut, respectively. The Twu characterization method,6 which makes it possible to provide Tc, Pc, and ω from the mere knowledge of NBP and SG, has been used in this study but other methods could have been selected. From now on, we know for each pseudo 2821


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Figure 3. (P, T) phase envelopes of the crude oil. Case 1 is the reference case in which all the kij were estimated knowing the detailed structure of each pseudo-component. Case 2 (to be compared with case 1) is the present approach: the kij (when i is a pseudocomponent) are determined from the knowledge of Tc, Pc, and ω and from a simplified structure. In case 3 all the kij were set to zero.

Figure 5. (P, T) phase envelopes of the gas condensate. Case 1 is the reference case in which all the kij were estimated knowing the detailed structure of each pseudo-component. Case 2 (to be compared with case 1) is the present approach: the kij (when i is a pseudocomponent) are determined from the knowledge of Tc, Pc, and ω and from a simplified structure. In case 3 all the kij were set to zero.

Step 3. Implementation of the Proposed Approach (Case 2: for Each Cut, Calculation of kij from Guesstimated Values of {NPAR, NARO, NNAP} and Values of {Tc, Pc, ω}cut Stemming from Twu’s CM). Knowing Tc,CM, Pc,CM, and ωCM (see step 2) of each cut considered as a pseudocomponent, it is possible to apply the methodology developed in this paper to determine N (NPAR, NARO, NNAP) for each pseudo-component, and then to predict all kij for the three petroleum fluids using this simplified structure and the PPR78 model. The corresponding (P, T) phase envelopes are shown in Figures 3−5 (see the curve labeled case 2: present approach). Note that contrary to case 1 in which BIP were estimated by splitting each pseudo-component into elementary detailed groups (CH3, CH2, CH, ...), case 2 involves BIP calculated by considering simplified-group occurrences (NPAR, NARO, NNAP) estimated from the developed GC methods. Step 4. Traditional Approach (Case 3: kij = 0 and Values of {Tc, Pc, ω}cut Stemming from Twu’s CM). The (P, T) phase envelopes of the three petroleum fluids were also simulated by setting all kij to zero. This extreme case could be encountered in some process simulation software. The results can be seen in Figures 3−5 (see case 3). As shown in Figures 3−5 the predicted (P, T) phase envelopes of the three petroleum fluids with the present approach are very close to those obtained with the reference case. It indicates that the feasibility of the present method to predict kij has been proven. Moreover, the obvious difference between cases 1 and 3 (with kij = 0) highlights the importance of estimating BIP when one wants to predict the phase behavior of petroleum fluids. This is particularly true during the design of an enhanced oil recovery project39−41 where the critical points need to be accurately predicted.

Figure 4. (P, T) phase envelopes of the heavy oil. Case 1 is the reference case in which all the kij were estimated knowing the detailed structure of each pseudo-component. Case 2 (to be compared with case 1) is the present approach: the kij (when i is a pseudocomponent) are determined from the knowledge of Tc, Pc, and ω and from a simplified structure. In case 3 all the kij were set to zero.

Note that in this section, we attempted to check the capability of the proposed approach to estimate BIP in mixtures involving pseudo-components, regardless of the considered CM (as previously mentioned, the efficiency of the CM is not the point of this article). To do so, it was decided to build several cases only differing by the way used to estimate BIP. Therefore, the same CM was incorporated to the reference case as well as all the subsequent cases. For all the cases, the same numerical values of critical properties and acentric factors of pseudo-component (stemming from Twu’s CM) were thus considered. This is also the reason for which the reference case is neither a calculated phase envelope of a mixture containing only welldefined compounds, nor an experimental phase envelope.

6. CONCLUSION Commercial process simulators like PRO/II contain many correlations to estimate Tc, Pc, and ω of a pseudo-component from the mere knowledge of its molecular weight, normal boiling point, or specific gravity. Estimation of kij (where i is a 2822


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(3) Lee, B. I.; Kesler, M. G. A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States. AIChE J. 1975, 21, 510−527. (4) Riazi, M. R.; Daubert, T. E. Simplify Property Predictions. Hydrocarbon Processing 1980, 59, 115−116. (5) Riazi, M. R. Characterization and Properties of Petroleum Fractions; ASTM: Philadelphia, PA, 2005. (6) Twu, C. H. An internally consistent correlation for predicting the critical properties and molecular weights of petroleum and coal-tar liquids. Fluid Phase Equilib. 1984, 16, 137−50. (7) Whitson, C. H. Characterizing Hydrocarbon Plus Fractions. SPE J. 1983, 23, 683−694. (8) Whitson, C. H.; Andersen, T. F.; Søreide, I., C7+ characterization of related equilibrium fluids using the gamma distribution. In C7+ fraction characterization characterization; Chorn, L. G., Mansoori, G. A., Eds.; Taylor & Francis New York Inc.: New York, 1989; pp 35−56. (9) Pedersen, K. S.; Thomassen, P.; Fredenslund, A. Characterization of gas condensate mixtures. In C7+ Fraction Characterization; Chorn, L. G.; Mansoori, G. A., Eds.; Taylor & Francis New York Inc.: New York, 1989; Vol. 1, pp 137−152. (10) Pedersen, K. S.; Fredenslund, A.; Thomassen, P. Properties of Oils and Natural Gases. Contributions in Petroleum Geology and Engineering; Gulf Publishing: Houston, 1989; Vol. 5. (11) Pedersen, K. S.; Blilie, A. L.; Meisingset, K. K. PVT calculations on petroleum reservoir fluids using measured and estimated compositional data for the plus fraction. Ind. Eng. Chem. Res. 1992, 31, 1378−1384. (12) Avaullee, L.; Trassy, L.; Neau, E.; Jaubert, J. N. Thermodynamic modeling for petroleum fluids I. Equation of state and group contribution for the estimation of thermodynamic parameters of heavy hydrocarbons. Fluid Phase Equilib. 1997, 139, 155−170. (13) Avaullee, L.; Neau, E.; Jaubert, J.-N. Thermodynamic modeling for petroleum fluids II. Prediction of PVT properties of oils and gases by fitting one or two parameters to the saturation pressures of reservoir fluids. Fluid Phase Equilib. 1997, 139, 171−203. (14) Avaullee, L.; Duchet-Suchaux, P.; Durandeau, M.; Jaubert, J. N. A new approach in correlating the oil thermodynamic properties. J. Pet. Sci. Eng. 2001, 30, 43−65. (15) Jaubert, J.-N.; Mutelet, F. VLE predictions with the PengRobinson equation of state and temperature dependent kij calculated through a group contribution method. Fluid Phase Equilib. 2004, 224, 285−304. (16) Jaubert, J.-N.; Privat, R. Relationship between the binary interaction parameters (kij) of the Peng-Robinson and those of the Soave-Redlich-Kwong equations of state: Application to the definition of the PR2SRK model. Fluid Phase Equilib. 2010, 295, 26−37. (17) Jaubert, J.-N.; Vitu, S.; Mutelet, F.; Corriou, J.-P. Extension of the PPR78 model (predictive 1978, Peng-Robinson EOS with temperature dependent kij calculated through a group contribution method) to systems containing aromatic compounds. Fluid Phase Equilib. 2005, 237, 193−211. (18) Mutelet, F.; Vitu, S.; Privat, R.; Jaubert, J.-N. Solubility of CO2 in branched alkanes in order to extend the PPR78 model (predictive 1978, Peng-Robinson EOS with temperature-dependent kij calculated through a group contribution method) to such systems. Fluid Phase Equilib. 2005, 238, 157−168. (19) Vitu, S.; Jaubert, J.-N.; Mutelet, F. Extension of the PPR78 model (Predictive 1978, Peng-Robinson EOS with temperature dependent kij calculated through a group contribution method) to systems containing naphthenic compounds. Fluid Phase Equilib. 2006, 243, 9−28. (20) Plée, V.; Jaubert, J. N.; Privat, R.; Arpentinier, P. Extension of the E-PPR78 equation of state to predict fluid phase equilibria of natural gases containing carbon monoxide, helium-4 and argon. J. Pet. Sci. Eng. 2015, (accepted for publication). (21) Privat, R.; Jaubert, J.-N.; Mutelet, F. Use of the PPR78 model to predict new equilibrium data of binary systems involving hydrocarbons and nitrogen. Comparison with other GCEOS. Ind. Eng. Chem. Res. 2008, 47, 7483−7489.

pseudo-component and j is a well-defined component or another pseudo-component) is currently a tricky issue, and in this paper, it was decided to propose a solution to this specific problem. In particular, we have shown that the knowledge of Tc, Pc, and ω of a pseudo-component is enough to determine its simplified structure. It thus becomes possible to use the PPR78 (or the PR2SRK) model to predict all the kij involving a pseudo-component. To verify the feasibility of the present approach, the (P, T) phase envelopes of three petroleum fluids were predicted following the proposed approach and compared to those obtained by considering a very detailed structure of each pseudo component. The results were extremely close which validates the present approach.

■

ASSOCIATED CONTENT

S Supporting Information *

Detailed results for the 174 pure compounds used in this study (accuracy of the developed GCM to estimate Tc, Pc, and ω and accuracy of the reverse GCM to estimate the structure); components used to model the distillation cuts. This material is available free of charge via the Internet at http://pubs.acs.org/.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +33 3 83 17 50 81. Fax: +33 3 83 17 51 52. Notes

The authors declare no competing financial interest.

■

LIST OF SYMBOLS a = attractive parameter of the PR78 EoS Akl = group-contribution parameters in the PPR78 model b = molar covolume of the PR78 EoS Bkl = group-contribution parameters in the PPR78 model kij = binary-interaction parameter MW = molecular weight NBP = normal boiling point NARO = number of aromatic groups NNAP = number of naphthenic groups NPAR = number of paraffinic groups Ng = number of groups defined in the PPR78 model Ni = occurrence of group i P = Pressure Pc = critical pressure Pca, Pcb = universal constants to estimate Pc by group contribution R = gas constant, 8.314472 J·mol−1·K−1 SG = specific gravity T = temperature Tc = critical temperature Tca = universal constant to estimate Tc by group contribution v = molar volume ω = acentric factor ωa, ωb, ωc = universal constants to estimate ω by group contribution

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REFERENCES

(1) Cavett, R. H. In Physical Data for Distillation Calculations, VaporLiquid Equilibria; Proceeding of 27th API meeting, API Division of Refining, 1962; pp 351−366. (2) Kesler, M. G.; Lee, B. I. Improve Prediction of Enthalpy of Fractions. Hydrocarbon Processing 1976, 55, 153−158. 2823


Article

Industrial & Engineering Chemistry Research (22) Privat, R.; Jaubert, J.-N.; Mutelet, F. Addition of the sulfhydryl group (-SH) to the PPR78 model (predictive 1978, Peng-Robinson EOS with temperature dependent kij calculated through a group contribution method). J. Chem. Thermodyn. 2008, 40, 1331−1341. (23) Privat, R.; Jaubert, J.-N.; Mutelet, F. Addition of the Nitrogen Group to the PPR78 Model (Predictive 1978, Peng Robinson EOS with Temperature-Dependent kij Calculated through a Group Contribution Method). Ind. Eng. Chem. Res. 2008, 47, 2033−2048. (24) Privat, R.; Mutelet, F.; Jaubert, J.-N. Addition of the Hydrogen Sulfide Group to the PPR78 Model (Predictive 1978, Peng-Robinson Equation of State with Temperature Dependent kij Calculated through a Group Contribution Method). Ind. Eng. Chem. Res. 2008, 47, 10041−10052. (25) Jaubert, J. N.; Coniglio, L.; Denet, F. From the correlation of binary systems involving supercritical CO2 and fatty acid esters to the prediction of (CO2−fish oils) phase behavior. Ind. Eng. Chem. Res. 1999, 38, 3162−3171. (26) Jaubert, J. N.; Coniglio, L. The group contribution concept: a useful tool to correlate binary systems and to predict the phase behavior of multicomponent systems involving supercritical CO2 and fatty acids. Ind. Eng. Chem. Res. 1999, 38, 5011−5018. (27) Vitu, S.; Privat, R.; Jaubert, J.-N.; Mutelet, F. Predicting the phase equilibria of CO2 + hydrocarbon systems with the PPR78 model (PR EOS and kij calculated through a group contribution method). J. Supercrit. Fluids 2008, 45, 1−26. (28) Jaubert, J.-N.; Privat, R.; Mutelet, F. Predicting the phase equilibria of synthetic petroleum fluids with the PPR78 approach. AIChE J. 2010, 56, 3225−3235. (29) Privat, R.; Jaubert, J.-N. Addition of the sulfhydryl group (SH) to the PPR78 model: Estimation of missing group-interaction parameters for systems containing mercaptans and carbon dioxide or nitrogen or methane, from newly published data. Fluid Phase Equilib. 2012, 334, 197−203. (30) Qian, J.-W.; Jaubert, J.-N.; Privat, R. Prediction of the phase behavior of alkene-containing binary systems with the PPR78 model. Fluid Phase Equilib. 2013, 354, 212−235. (31) Qian, J.-W.; Jaubert, J.-N.; Privat, R. Phase equilibria in hydrogen-containing binary systems modeled with the Peng-Robinson equation of state and temperature-dependent binary interaction parameters calculated through a group-contribution method. J. Supercrit. Fluids 2013, 75, 58−71. (32) Qian, J.-W.; Privat, R.; Jaubert, J.-N. Predicting the Phase Equilibria, Critical Phenomena, and Mixing Enthalpies of Binary Aqueous Systems Containing Alkanes, Cycloalkanes, Aromatics, Alkenes, and Gases (N2, CO2, H2S, H2) with the PPR78 Equation of State. Ind. Eng. Chem. Res. 2013, 52, 16457−16490. (33) Qian, J.-W.; Privat, R.; Jaubert, J.-N.; Duchet-Suchaux, P. Enthalpy and heat capacity changes on mixing: fundamental aspects and prediction by means of the PPR78 cubic equation of state. Energy Fuels 2013, 27, 7150−7178. (34) Constantinou, L.; Gani, R. New group contribution method for estimating properties of pure compounds. AIChE J. 1994, 40, 1697− 710. (35) Constantinou, L.; Gani, R.; O’Connell, J. P. Estimation of the acentric factor and the liquid molar volume at 298 K using a new group contribution method. Fluid Phase Equilib. 1995, 103, 11−22. (36) Marrero, J.; Gani, R. Group-contribution based estimation of pure component properties. Fluid Phase Equilib. 2001, 183−184, 183− 208. (37) Hukkerikar, A. S.; Sarup, B.; Ten Kate, A.; Abildskov, J.; Sin, G.; Gani, R. Group-contribution+ (GC+) based estimation of properties of pure components: Improved property estimation and uncertainty analysis. Fluid Phase Equilib. 2012, 321, 25−43. (38) Yaws, C. L. Thermophysical Properties of Chemicals and Hydrocarbons; William Andrew Publishing: Norwich, NY, 2008. (39) Jaubert, J. N.; Wolff, L.; Avaullée, L.; Neau, E. A very simple multiple mixing cell calculation to compute the minimum miscibility pressure whatever the displacement mechanism. Ind. Eng. Chem. Res. 1998, 37, 4854−4859.

(40) Jaubert, J. N.; Avaullée, L.; Pierre, C. Is it still necessary to measure the minimum miscibility pressure? Ind. Eng. Chem. Res. 2002, 41, 303−310. (41) Jaubert, J. N.; Avaullée, L.; Souvay, J. F. A crude oil data bank containing more than 5000 PVT and gas injection data. J. Pet. Sci. Eng. 2002, 34, 65−107.

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