Thermodynamic Characterization of Undefined Petroleum Fractions

Article pubs.acs.org/IECR

Thermodynamic Characterization of Undefined Petroleum Fractions using Group Contribution Methods Bernardo Carreón-Calderón,* Verónica Uribe-Vargas, Edgar Ramírez-Jaramillo, and Mario Ramírez-de-Santiago Instituto Mexicano del Petróleo, Dirección de Investigación y Posgrado; Programa de Aseguramiento de la Producción de Hidrocarburos, Eje Central Lázaro Cárdenas Norte 152, Col. San Bartolo Atepehuacan, Deleg. Gustavo A. Madero, C.P. 07730 México, D.F. ABSTRACT: Properties of petroleum fractions of unknown composition, the so-called undefined petroleum fractions, are estimated using an approach based on group contribution methods. Using liquid density and molecular weight as experimental data and classical thermodynamics as framework, functional groups were assigned to each undefined fraction by minimizing its free energy. Thus, methods requiring molecular structures were used directly in phase equilibrium simulations of petroleum fluids. The proposed procedure also allows the critical properties of such undefined fractions to be calculated employing no specific correlations. The obtained results show reasonable accuracy concerning phase equilibrium experiments.

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INTRODUCTION Petroleum is a complex mixture of hydrocarbons where most of components are unidentified. Although experimental analyses can be performed,1,2 a complete molecular description is not practical to be obtained because of the huge number of components in the mixture. For this reason, petroleum composition and molecular structures remain unknown for calculation purposes, giving rise to peculiar challenges for its numerical modeling. Petroleum composition is usually expressed in terms of two groups: defined components and undefined fractions. The former includes hydrocarbon components and nonhydrocarbon components such as nitrogen, carbon dioxide, and hydrogen sulfide. The second is composed of all unidentified components with more than six carbons: the heptane plus fractions (C7+). These undefined mixtures are responsible for the modeling problems in petroleum thermodynamic characterization, especially when molecular structures are required.3 Hence, a convenient set of components and undefined fractions are needed for petroleum thermodynamic characterization.4,5 An important aspect is that components forming undefined fractions do not have to be neither real nor present in the mixture. They can be either welldefined individual molecules or constructed ones.6−8 To obey with increasingly stringent environmental regulations, molecular-level characterizations have been suggested for undefined petroleum fractions. Apparently, this sort of characterization has become fundamental for refinery process modeling.9,10 In principle, a further detailed molecular description allows chemical and physical properties to be calculated with higher accuracy. However, more computing time and experimental data are required. On the other hand, when the undefined petroleum fractions are mathematically treated as individual components, some bulk properties are used for estimating the missing ones. One of the disadvantages of this approach is the use of specific correlations, which are limited to certain conditions and types of petroleum. This has given rise to different sets of correlations.11,12 Therefore, © 2012 American Chemical Society

thermodynamic characterization also involves selecting models and procedures for estimating bulk properties of undefined petroleum fractions.13,14 Attempts have been made to combine advantages from molecular-level characterization with the use of undefined fractions as individual components by the group contribution concept.15,16 There, chemical and physical properties are written in terms of functional groups (CH3−, −CH2−, etc.) either by correlations17,18 or by a trial and error procedure.19 In the latter case, a suggested chemical structure is adjusted so that the available experimental information coincides with calculations. These procedures require much less experimental information than molecular-level characterization. For example, molecular weight, density, true boiling point, and the amount of hydrocarbon families are commonly needed.20 The aim of this paper is to provide an approach to the thermodynamic characterization and generation of PVT data of petroleum by using group contribution methods, particularly the Joback−Reid group contribution method.21 The number of functional groups and the critical properties for undefined petroleum fractions are automatically determined by minimizing their Gibbs free energy.

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DETERMINATION OF MOLECULAR PSEUDOSTRUCTURE By definition, having undefined fluid mixtures implies mixtures of unidentified components with unknown chemical and physical properties. Thus, it appears that it does not make sense to seek molecular structures of components belonging to such mixtures. In this work, however, one molecular structure is determined for a given undefined petroleum fraction, which implies that we regarded such fraction as a hypothetical Received: Revised: Accepted: Published: 14188

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individual component. In contrast to actual components, undefined petroleum fractions are described by noninteger values of the number of functional groups.19 These noninteger numbers give rise to what we call molecular pseudostructures, which may not correspond to those coming from experimental analysis. However, it is expected that they allow bulk properties to be reproduced by using approaches where molecular structures are required.3,21 Critical Properties and Liquid Density from Group Contribution Methods. Group contribution methods for determining bulk properties take into account molecular interactions by the so-called functional groups, which are structural units forming molecules. In these methods, the molecule is considered as a mixture of these structural units. For example, the Joback−Reid method is recommended for predicting properties not only of pure components but also of undefined petroleum fractions.22 The Joback-Reid method (JR)21 includes 41 functional groups belonging to many types of organic molecules. Interactions between functional groups are not taken into account; hence, this method is unable to make a distinction between isomeric components. As will be shown in what follows, this characteristic is irrelevant in this work because pseudostructures will be constructed from bulk properties and not from properties at molecular level. In JR method, the critical properties are calculated by the following expressions: Tc = Pc =

0.584 +

Tb FG 0.965 ∑i = 1 υitci

−

FG (∑i = 1 υitci)2

f1 (Tbr) = −5.03365τ + 1.11505τ1.5 − 5.41217τ 2.5 − 7.46628τ 5 Tbr (6)

Tbr = Tb/Tc is the reduced normal boiling point and τ = 1 − Tbr. Liquid density is another important property in the characterization of undefined petroleum fractions; hence, its value is usually available experimentally. The liquid density of pure components can be predicted by group contribution method as well. For example, the group contribution method GCVOL24,25 and its extensions are applicable for a wide range of molecular weights and conditions. Liquid densities can be calculated for temperatures from the melting to normal boiling point for nonpolymeric components, and for temperatures from the glass transition to degradation point for polymeric components. In these methods the liquid densityρ is calculated using ρ=

FG

Δvi = αi + βi T + χi T 2 (1)

(2)

17.5 + ∑i = 1 υivci 106

(3)

Tc, Pc, and Vc are the critical temperature, pressure, and volume, respectively. Similarly, tci, pci, and νci represent contributions to Tc, Pc, and Vc because of the presence of the ith functional group where the number of functional groups of type i is given by υi. In the above equations, nA is the number of atoms in the molecule, Tb is the normal boiling point, and FG is the total number of types of functional groups. The units for temperature, pressure and volume are K, MPa, and m3/mol, respectively. As observed, calculation of the critical temperature requires the normal boiling point either measured or estimated; the former provides better results. In addition to critical properties, the acentric factor ω is another important property for phase equilibrium problems; however, in JR method, no expression is suggested. In this work, the following expressions are used together with eqs 1, 2, and 3:23 ω=−

gR g − gI ≡ = ln(φ) RT RT

(9)

where g is the molar Gibbs energy, the superscript I indicates ideal gas behavior, R denotes the gas constant, T is the system temperature, and φ is the fugacity coefficient. Equation 9 can be rewritten as ⎛P⎞ g gI μ 0 (T ) = ln(φ) + = ln(φ) + + ln⎜ 0 ⎟ ⎝P ⎠ RT RT RT

ln(Pc/1.01315) + f0 (Tbr) f1 (Tbr)

(8)

where αi, βi, and χi as parameters of GCVOL approach. Calculation of the Number of Functional Groups. In what follows, we intend to characterize thermodynamically an undefined petroleum fraction not only by estimating its critical properties but also by establishing one molecular pseudostructure. To achieve this goal, an undefined petroleum fraction is visualized as a solitary system formed by one component at equilibrium conditions for a fixed temperature and pressure. In addition, this system is considered as independent of the other undefined fractions. From the equilibrium general conditions, it is well-known that the Gibbs energy must have a minimum value for systems at constant temperature and pressure. Hence, to develop a methodology for determining one molecular pseudostructure, we start with the residual Gibbs energy gR

FG

Vc =

(7)

where MW is the molecular weight and Δνi is the group volume increment, which is expressed as a function of temperature using

0.1 (0.113 + 0.0032nA − ∑i = 1 υipci )2

MW MW = FG V ∑i = 1 υiΔvi

(10)

I

g is determined by the second and third terms on the righthand side, where μ is the chemical potential and P is the system pressure. The superscript 0 stands for properties at a reference state. The minimum value of any function corresponds to that point where its first partial derivatives with respect to all independent variables are equal to zero. As temperature and pressure remain fixed, these properties are not independent variables for minimization purposes. Hence, the second and third terms on the right-hand side can be discarded from eq 10, which reduces to

(4)

where f0 (Tbr) = −5.97616τ + 1.29874τ1.5 − 0.60394τ 2.5 − 1.06841τ 5 Tbr (5)

and 14189

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⎡ ∂Φj(υ , λ1 , λ 2) ⎤ ⎢ ⎥ ⎢⎣ ⎥⎦ ∂υk λ ,λ

(11)

Considering that an undefined petroleum fraction is treated as an individual component and temperature and pressure are kept constant, then the molecular structure of this hypothetical component is the only responsible for the system energy. At this point, we assume that such energy can be evaluated using functional groups. Thus, eq 11 may take the form e g / RT = φ(υ1 , υ2 , ..., υFG)

1

e

⎡ ∂Φj(υ , λ1 , λ 2) ⎤ ⎢ ⎥ ⎢⎣ ⎥⎦ ∂λ1 λ

(12)

FG

d(ng ) =

=

⎡ ∂(ng ) ⎤ ⎥ ⎣ ∂υi ⎦

∑ μi* dυi + μ dn

⎡ ∂(ng ) ⎤ μi* ≡ ⎢ ⎥ ⎣ ∂υi ⎦n , P , T , υ ; k ≠ i

(15)

k

(16)

where superscript l denotes the liquid phase. The former constraint establishes that the sum of all υi multiplied by its molecular weight must be equal to the molecular weight of the undefined petroleum fraction j. Similarly, the latter constraint establishes that the sum of all υ i multiplied by the corresponding group volume increment must be equal to the ratio of the molecular weight to the liquid density of the undefined petroleum fraction j. This latter constraint comes from rearranging eq 7. The corresponding Lagrangian Φ function for the minimization problem is

∑ μi* dυi i=1

+ λ1(∑ υi MWi − MW) j

(20)

⎡ ∂(ng ) ⎤ dυi + ⎢ dn ⎥ ⎣ ∂n ⎦ P , T , υ

(21)

(22)

(23)

This is one of the thermodynamic fundamental relationships, but in terms of the number of functional groups. Finally, to resolve the system of equations given by expressions 18−20, it is necessary to suggest a link between the fugacity coefficient and the critical properties. In this work, cubic equations of state are suggested because they are commonly used for phase equilibrium problems applied to multicomponent hydrocarbon mixtures.

i=1

⎛ FG MWj ⎞ ⎟ + λ 2⎜⎜∑ υiΔvi − l ⎟ ρ ⎝ i=1 j ⎠

=0

FG

d(ng ) =

FG

Φj(υ , λ1 , λ 2) =

ρjl

Although at first sight this is an unconventional way to define the chemical potential, eq 22 still represents energy changes due to mass changes. The only difference here is that mass changes are given by functional groups instead of components. In other words, instead of determining the amount of pure components, we seek the amount of functional groups that minimize the Gibbs energy of a given undefined petroleum fraction. This assumption is supported by the fact that the thermodynamic fundamental relationships remain valid irrespective of the system internal nature. Thus, if the total number of moles is constant, which is one assumption in the minimization process, eq 21 reduces to

MWj

φjl(υ)

MWj

where n is the total number of moles of the undefined petroleum fraction whose molecular pseudostructure is integrated by the noninteger numbers υi. Accordingly, the product ng represents the total Gibbs energy of a given undefined petroleum fraction. Thus, we may define the chemical potential μ*i of the functional group i in a mixture of functional groups as follows:

(14)

ρjl

n , P , T , υk ; k ≠ i

i=1

and

i=1

i=1

∑⎢ i=1

FG

∑ υiΔvi =

FG

∑ υiΔvi −

FG

subject to the following linear constraints

FG

(19)

Note that there are FG + 2 equations in the FG + 2 unknowns υ = [υ1, υ2,...,υFG], λ1 and λ2. Before going forward, it is worth highlighting from eq 12 that the total differential of the system energy is

φjl(Tcj(υ1 , υ2 , ..., υFG), Pcj(υ1 , υ2 , ..., υFG), ωj

i=1

∑ υi MWi − MWj = 0 i=1

2,υ

1

where critical properties link the number of functional groups to fugacity coefficients. It is important to take into account that undefined petroleum fractions are at liquid state and that experimental molecular weight and density of such fractions are usually known. Therefore, in the minimization process of eq 13, the fugacity coefficient has to correspond to a liquid phase and satisfy the experimental values of molecular weight and liquid density. In other words, to calculate the number of functional groups υi forming the pseudostructure of an undefined petroleum fraction j, the objective function to be minimized is

∑ υi MWi = MWj

(18)

FG

=

⎡ ∂Φj(υ , λ1 , λ 2) ⎤ ⎢ ⎥ = ⎢⎣ ⎥⎦ ∂λ 2 λ ,υ

(13)

(υ1 , υ2 , ..., υFG))

λ1, λ 2 , υl ; l ≠ k

and

= φ(Tc(υ1 , υ2 , ..., υFG), Pc(υ1 , υ2 , ..., υFG),

ω(υ1 , υ2 , ..., υFG))

+ λ1MWk

+ λ 2Δvk = 0; k = 1, ..., FG

This equation is not useful in practice because functional groups are employed to predict critical properties and not fugacity coefficients. Thus, it is convenient to rewrite eq 12 in terms of critical properties: g / RT

2 , υl ; l ≠ k

⎡ ∂φl(υ) ⎤ j ⎥ =⎢ ⎢ ∂υk ⎥ ⎣ ⎦

(17)

where λ 1 and λ 2 are the Lagrange multipliers. The corresponding first-order necessary conditions are 14190



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PVT PROPERTIES OF UNDEFINED PETROLEUM FRACTIONS USING MOLECULAR PSEUDOSTRUCTURES Cubic Equations of State. In general, equations of state are mathematical functions relating bulk properties to each other. Among these functions, cubic two-constant equations of state are widely used in the petroleum industry because of their good enough results and their relative mathematical simplicity. RT a P= − 2 (24) V−b V + ubV + wb2 The parameters u and w are specific for each equation of state. The parameters a and b are specific for a given component by its critical properties, in accordance with the following expressions: a=

have been developed for handling defined components should be capable of handling these undefined fractions without additional assumptions or considerations. To prove this assertion, phase equilibrium problems involving petroleum will be predicted by using mixing rules containing no specific regression parameters. The first mixing rule used in this work is the classical or van der Waals mixing rule (VDW), where, for an N-component mixture, the parameters a and b entering into eq 24 are found from26,27 N

A=

N

∑ ∑ xixj

Ai Aj (1 − kij) (29)

j=1 i=1

and N

Ωa(RTc)2 f Pc

B= (25)

∑ xjBj (30)

j=1

where

2

⎡ ⎛ f = ⎢1 + (r1 + r2ω + r3ω 2)⎜⎜1 − ⎢⎣ ⎝

T TC

⎞⎤ ⎟⎟⎥ ⎠⎥⎦

where, Aj = Paj/(RT) and Bj = Pbj/RT. xi is the mole fraction of the component i, and kij is the binary interaction parameter for the components i and j. In general, kij parameters are obtained by regression of phase equilibrium data; consequently, their values depend on the available data bank and the used equation of state. The computational efforts are reduced by assuming kij values equal to or close to zero for two different components of approximately the same polarity. For this reason, in the case of petroleum, kij is usually estimated only for interactions between hydrocarbons and components, such as nitrogen and carbon dioxide. In the current treatment, however, binary interaction parameters are predicted from expression29

2

(26)

and b=

Ω bRTc Pc

(27)

where Ωa, Ωb, and ri are specific parameters for each equation of state. In this work, the Peng−Robinson (PR)26 and Soave− Readlich−Kwong (SRK)27 equations of state are employed; their parameters are given in Table 1. Using eq 24 and

⎡ C(V V )1/6 ⎤ D ci cj ⎥ kij = 1 − ⎢ 1/3 ⎢⎣ Vci + Vcj1/3 ⎥⎦

Table 1. Equation-of-State Parameters parameter

PR

SRK

u w Ωa Ωb r1 r2 r3 η c1 c2

2 −1 0.45724 0.07780 0.37464 1.54226 −0.26992 −0.593 0.50033 0.25969

1 0 0.42747 0.08664 0.480 1.574 −0.176 −0.53 0.40768 0.29441

where i and j represent any component or undefined petroleum fraction. Although C and D are commonly used as adjustable parameters to minimize the error between predicted and experimental values, in this work they are set equal to 2 and 3 as suggested initially, because it has been confirmed that these values are suitable for nonpolar mixtures, which is the case of petroleum. In the case of undefined petroleum fractions, critical volumes are those obtained from the minimization process by eq 14 together with eq 3. In other words, binary interaction parameters depend on functional groups υ, as it has already been proposed using specific correlations17,18 or for defined mixtures.30 Note that none of the binary interaction parameters used in eq 29 is set equal to zero and that the same binary interaction parameters are used irrespective of the cubic equation of state. The second mixing rule used in this work is the modified Huron−Vidal mixing rule of one parameter (MHV1).31 This mixing rule belongs to the mixing rule family where excess Gibbs energy models are used to determine mixing parameters a and b entering into eq 28. The MHV1 mixing rule can be expressed as

associated expressions, the fugacity coefficient can be determined from the thermodynamic fundamental expression:28 V

ln(φ) =

∫∞

⎡1 ⎛ PV ⎞ 1 ⎛⎜ ∂P ⎞⎟ ⎤ ⎟ ⎢ − ⎥dV − ln⎜ ⎝ RT ⎠ ⎝ ⎠ ⎢⎣ V RT ∂n T , V ⎥⎦

(31)

(28)

This expression is the relationship that connects the fugacity coefficient to the critical properties, as required by expression 14. Thus, the minimization problem described previously can be resolved using eq 28 together with eq 24 and group contribution methods for critical properties. Mixing Rules. Once the number of functional groups υ is known through the previous minimization procedure, undefined petroleum fractions become defined components from the calculations point of view. Accordingly, all approaches that

A = B

N

⎧ ⎪A

∑ xj⎨⎪ j=1

j

⎩ Bj

+

⎫ ⎡ ⎛ B ⎞⎤⎪ 1⎢ ln γj + ln⎜⎜ ⎟⎟⎥⎬ η ⎢⎣ ⎝ Bj ⎠⎥⎦⎪ ⎭

(32)

where eq 30 is commonly proposed to calculate the B parameter. η is an parameter relying on equation of state as 14191



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⎛ α + β T + χ T2 ⎞ nm nm nm ⎟⎟ Ψnm = exp⎜⎜ − T ⎝ ⎠

shown in Table 1, and γj is the activity coefficient, which can be calculated from any excess Gibbs energy model. Considering that, we are interested in models using functional groups υ as data, UNIFAC method is proposed for calculating activity coefficients; its basic equation is32

ln γi = ln γiC + ln γi R

where αnm, βnm, and χnm are specific parameters for a couple of functional groups n and m. Unlike VDW mixing rule, MHV1 mixing rule combined with UNIFAC method is not routinely used in phase equilibrium problems involving petroleum because of the unknown numbers υ. However, it will be shown that this combination can be also used routinely for solving this sort of phase equilibrium problems, using the number of functional groups obtained from the minimization process proposed in this work. Liquid Molar Volumes. Cubic equations of state are widely recognized for their capability of predicting fluid phase equilibria but their ability to predict volumetric data is limited, especially for liquid phases. To overcome this deficiency and to keep their strengths, the volume translation cj has been introduced as follows:33

(33)

The former term on the right-hand side corresponds to the combinatorial contribution and the latter corresponds to the residual contribution. The combinatorial part is calculated by ⎛ ⎛ Φi ⎞ ⎛ θi ⎞ Φ ln γi = ln⎜ ⎟ + 5qi ln⎜ ⎟ + li − ⎜⎜ i x Φ ⎝ i⎠ ⎝ i⎠ ⎝ xi C

⎞ ∑ xjlj⎟⎟ ⎠ j=1 N

(34)

where li = 5(rj − qj) − (rj − 1)

(35)

N

xiqi

θi =

V = V EOS −

N

∑ j = 1 xjqj xiri N ∑ j = 1 xjrj

(37)

cj = c1j

(38)

i=1

(39)

Here again, υij represents the number of functional groups of the type i in a defined component j. In the case of undefined petroleum fractions, υ ij are those obtained from the minimization process by eq 14. On the other hand, the residual part is obtained using

(47)

if V jexp =

∑ υijΔvi i=1

where FG

∑ m=1

θm Ψim ⎤⎥ FG ∑n = 1 θn Ψnm ⎥⎦

(48)

which is the constraint given by eq 20. According to eq 47, the volume translation of undefined petroleum fractions is a function relying on temperature. This has already been suggested for phase equilibrium problems involving petroleum.14 Finally, it is important to underline that the volume found from eq 44 corresponds to the petroleum as a whole, whereas eq 20 represents the liquid volume of just one undefined fraction at fixed temperature and pressure. For this reason, the regression to an experimental value in the minimization process does not mean a regression to experimental values of petroleum liquid densities at different conditions. Therefore, eqs 44, 45, and 47 provide a predictive approach once the number of

(41)

and Q mX m FG ∑n = 1 Q nX n

∑ υijΔvi i=1

(40)

i=1

θm =

(46)

FG

∑ υij(ln Γi − ln Γij)

FG ⎡ ln Γi = Q i⎢1 − ln( ∑ θm Ψmi) − ⎢⎣ m=1

(45)

FG

cj = V jEOS −

FG

ln γjR =

(c 2j + 0.29056 − 0.08775ωj)

where superscript EXP indicates an experimental value. This last expression is useful when undefined petroleum fractions are involved because their experimental liquid densities are commonly known. It is interesting to note from eq 16 that eq 46 can be rewritten in the following form

FG i=1

Pcj

cj = V jEOS − V jexp

and

∑ υijQ i

RTcj

The constant parameters c1j and c2j are given in Table 1. Equation 45 is employed for calculating the volume translation of pure components such as nitrogen, methane, ethane, and so on. On the other hand, if liquid volume data is available, then the volume translation can be estimated from

FG

∑ υijR i

qj =

(44)

The superscript EOS indicates the molar volume calculated from eq 24, while the volume translation is obtained by the expression

θi is the area fraction and Φi is the segment fraction of the component i. Parameters qi and ri are, in turn, determined as the summation of the group volume and area parameters Rk and Qk: rj =

∑ xjcj j=1

(36)

and Φi =

(43)

(42)

Xm represents the mole fraction of the functional group m in the mixture. Finally, the group-interaction parameter Ψnm is given by 14192



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functional groups is calculated for each undefined fraction belonging to the petroleum under study.

Table 3. Properties Calculated in This Work for Petroleum Hexane Plus Groups37

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APPLICATIONS To illustrate the minimization process suggested previously, it is necessary first to decide what functional groups are suitable. In Table 2. Functional Groups Joback−Reid

GCVOL

−CH3 −CH2− (chain) >CH− (chain) >C< (chain) −CH2− (cyclic) >CH− (cyclic) >C< (cyclic) CH (cyclic) C< (cyclic)

−CH3 −CH2− (chain) >CH− (chain) C (chain) CH2 (cyclic) CH (cyclic) C (cyclic) ACH AC−C

the case of undefined petroleum fractions, Table 2 shows the nine functional groups suggested for each of the group contribution methods. As can be observed, they all consist of hydrocarbon segments from paraffinic, naphthenic and aromatic components, which are the most common components found in petroleum fluids. Once the set of functional groups has been defined, it is also necessary to decide the suitable numerical method for solving the minimization problem given by eq 14. In this work, a method developed by Spellucci is employed,34 which is a local solver. As any other optimization problem, the global minimum is required but this is a very time-consuming task generally. Hence, a less ambitious approach is to search for the greatest number of local minima using multiple initial estimates in the hope of finding the global minimum.35,36 As will be shown in the following applications, this approach is enough for the minimization problem proposed in this work requiring only few initial estimates. As first illustration, pseudostructures of the so-called petroleum hexane plus groups are calculated using average boiling point, liquid density, and molecular weight data.37 The calculated critical properties are shown in Table 3. Figure 1 shows normal boiling points calculated by solving the equality of chemical potentials with PR equation of state and using the critical properties calculated from the minimization process. The simulated results agree well with the experimental data;37 the average error is 0.2% even though noninteger values are obtained for υ from the minimization process. According to these results, the chemical nature of an undefined fraction is not required for the corresponding bulk properties to be estimated properly. As another example, pseudostructures and the associated critical properties are calculated for undefined petroleum fractions belonging to petroleum fluids.38 The original names have been retained in this work for simplicity: fluid 3, fluid 5, fluid 9, fluid 11, and fluid 12. With the purpose of illustrating the results, fluid 5 was chosen arbitrarily. The calculated functional groups are showed in Figure 2, where the x-axis represents molecular weights of each undefined petroleum fraction. Contrary to what one might expect, the results are not distributed randomly; some tendencies can be observed. For example, although included all together during calculations, some functional groups turn out to be equal to zero for some

carbon number

Tc (K)

Pc (MPa)

ω

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

501.20 532.51 559.65 585.27 614.23 637.33 674.83 693.34 710.47 728.57 751.50 766.50 776.77 786.88 798.53 808.60 818.48 828.96 838.35 847.62 857.16 866.69 875.55 883.46 890.46 899.03 906.69 914.69 921.80 930.51 938.36 946.58 956.42 965.43 974.85 983.46 992.51 1002.00 1013.18 1023.60

3.196847 2.946114 2.733030 2.486514 2.310609 2.153904 2.123419 1.974642 1.810129 1.673866 1.708543 1.600375 1.505825 1.432704 1.362081 1.274524 1.205742 1.147480 1.092680 1.037945 0.987467 0.937373 0.893587 0.852943 0.815151 0.780689 0.747809 0.717076 0.687656 0.660703 0.634787 0.611012 0.588004 0.566281 0.545785 0.526425 0.507661 0.490344 0.473938 0.458371

0.314548 0.364456 0.407252 0.458384 0.461061 0.485483 0.412565 0.437746 0.467294 0.495321 0.506800 0.534913 0.559086 0.579088 0.595549 0.615007 0.630304 0.642846 0.652985 0.661374 0.667090 0.669890 0.668935 0.664768 0.657285 0.647108 0.632827 0.615104 0.593201 0.568702 0.540147 0.509321 0.474647 0.437215 0.397344 0.355317 0.310558 0.265124 0.218483 0.170943

Vc (m3/mol) 3.6141 4.0819 4.5146 5.0692 5.4998 5.9646 6.2176 6.7188 7.1447 7.7197 8.2172 8.7448 9.2416 9.6648 1.0092 1.0665 1.1163 1.1624 1.2086 1.2584 1.3082 1.3615 1.4113 1.4612 1.5109 1.5603 1.6099 1.6595 1.7094 1.7588 1.8086 1.8580 1.9077 1.9575 2.0072 2.0569 2.1069 2.1565 2.2061 2.2556

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3

Figure 1. Normal boiling points of petroleum hexane plus groups34.

undefined petroleum fractions. As seen, the lightest fractions are represented with −CH3 and −CH2− groups only, that is, 14193



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example, the UNIFAC approach. The set of UNIFAC functional groups used in the MHV1 mixing rule is shown in Table 4.39 At this point, it is important to recall that the normal boiling point is required by JR method for the critical temperature to be calculated. Because no normal boiling points are provided for the undefined petroleum fractions,38 several correlations were employed, which are shown in Table 5.40−44 In this study were selected these five empirical correlations arbitrarily to establish the difference in the results as a function of estimated normal boiling point. For instance, the estimated critical properties using the correlation five are shown in Table 6 for each undefined fraction from Fluid 5. Average absolute deviation (AAD) between experimental and calculated data for flash separations is shown in Table 7, where experimental data come from the same reference.38 The results show AAD between 4 × 10−4 and 2 × 10−2 to 6 × 10−4 and 9 × 10−3 for vapor phase and for liquid phase, respectively. It is interesting to note that there is no significant difference between the simulated and experimental results irrespective of the used combination. Finally, as illustration, Figures 3 and 4 show molar composition of liquid and vapor coming from a separation flash at 288.15 K and 0.101325 MPa for fluid 5. To assess the prediction of liquid-phase densities, isothermal differential vaporizations were simulated for the five petroleum fluids using all available combinations of equation of state, mixing rules, and normal boiling point correlations. The average absolute percent error of results is given in Table 8, and some illustrations for fluid 5 are shown in Figure 5, where a good agreement between the experimental and calculated results is observed. As can be observed in Table 8, the maximum error was 7% using SRK-VDW-TB4 combination; even so, this value is lower than 10% value recommended for a petroleum characterization to be regarded as suitable.5 Finally, even though errors are similar to each other using all possible combinations, PR-MHV1-TB3 combination seems to be the best. Phase Envelope Calculations. In this section, it is demonstrated the capability of the proposed characterization method for being used in phase envelope calculations. Figure 6 shows the phase envelope of Fluid 5 using SRK equation of state with the two mixing rules used in this work. The used critical properties are those shown in Table 6. Although both

Figure 2. Number of functional groups for undefined petroleum fractions from fluid 5.

Table 4. UNIFAC Functional Groups chemical formula

main group

subgroup

CH3 CH2 CH C ACH AC CO2 CH4 N2 H2S

1 1 1 1 3 3 56 57 60 61

1 2 3 4 9 10 117 118 115 114

these fractions have a paraffinic nature essentially. The heavier petroleum fractions become, cyclic and aromatic groups are incorporated into the molecular pseudostructure, prevailing paraffinic and cyclic groups over aromatic ones. The previous qualitative results are in agreement with the general behavior expected in petroleum fluids. 4.1. Vapor−Liquid Equilibria. So far, we have calculated properties of individual undefined petroleum fractions, which are not useful by themselves. The significance of the proposed method comes from its capability of being employed together with approaches where chemical structures are required, for

Table 5. Semiempirical Correlation for Estimating Normal Boiling Point no.

normal boiling point

1

1/2.1962 ⎤ ⎡ 5 ⎛ MW × SG1.0164 ⎞ ⎥ Tb = ⎢⎜ ⎟ − 5 ⎥ 9 ⎢⎣⎝ 4.5673 × 10 ⎠ ⎦

2

Tb =

1/2.3776 ⎤ ⎡ 5 ⎢⎛ MW × SG0.9371 ⎞ ⎥ ⎟ ⎜ ⎥⎦ 9 ⎢⎣⎝ 1.4350476 × 10−5 ⎠

3

Tb =

5 0.3323 0.04609 ) ρ (97.58PM 9

Tb =

5 [6.77857 × MW 0.401673SG−1.58262 exp(3.7741 × 10−3MW 9

4

comments and limits

ref

311 K ≤ NBP ≤ 728 K

40

311 K ≤ NBP ≤ 728 K

41

42

200 ≤ MW ≤ 800

43

+ 2.984036SG − 4.25288 × 10−3SG)] 5

Tb =

5 (1928.3 − 1.695 9 −3

× 105PM−0.03522SG3.266e−4.922 × 10

PM − 4.7685SG + 3.462 × 10−3PM × SG

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44

) dx.doi.org/10.1021/ie3016076 | Ind. Eng. Chem. Res. 2012, 51, 14188−14198


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Table 6. Molar Compositions and Bulk Properties of Undefined Fractions Related to Fluid 538 critical properties calculated in this work data component component component component component component component component component component undefined fraction undefined fraction undefined fraction undefined fraction undefined fraction undefined fraction undefined fraction undefined fraction undefined fraction undefined fraction undefined fraction undefined fraction undefined fraction undefined fraction undefined fraction

hydrogen sulfide nitrogen carbon dioxide methane ethane propane i-butane n-butane i-pentane n-pentane C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+

0.383 0.450 2.070 26.576 7.894 6.730 1.485 3.899 1.937 2.505 3.351 4.311 4.133 3.051 2.033 2.635 2.285 2.364 2.038 1.752 1.589 1.492 1.263 0.812 12.962

MW

ρ (kg/m3) at 288.15 K

Tc (K)

Pc (MPa)

Vc (m3/mol)

86.2 92.6 109 120 138 149 163 177 191 205 219 234 248 263 450

679.4 726.2 750.9 771.6 787.8 803.7 815.4 827.0 841.2 858.8 862.7 858.6 864.8 877.1 956.0

528.3 531.8 563.5 583.8 612.7 636.4 683.6 703.5 722.4 741.8 756.1 769.7 782.1 795.6 911.1

3.137854 3.042714 2.708938 2.518928 2.242871 2.130054 2.287035 2.136269 2.005751 1.925963 1.804308 1.681590 1.580605 1.494292 0.808082

3.70 3.94 4.58 5.02 5.73 6.06 6.06 6.54 7.02 7.46 7.95 8.48 8.97 9.48 1.59

× × × × × × × × × × × × × × ×

ω

10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−4 10−3

0.3200 0.3606 0.4192 0.4639 0.5269 0.5136 0.4024 0.4292 0.4565 0.4830 0.5075 0.5318 0.5541 0.5799 0.6885

Table 7. Average Absolute Difference between Experimental and Calculated Data for Flash Separations fluid 3

fluid 5

fluid 9

fluid 11

fluid 12

EOS

MR

Tb−C

vapor

liquid

vapor

liquid

vapor

liquid

vapor

liquid

vapor

liquid

SRK

VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5

0.0011 0.0016 0.0013 0.0017 0.0013 0.0016 0.0015 0.0017 0.0011 0.0016 0.0009 0.0017 0.0014 0.0015 0.0016 0.0017 0.0015 0.0013 0.0017 0.0015

0.0039 0.0023 0.0042 0.0022 0.0039 0.0022 0.0042 0.0021 0.0037 0.0037 0.0041 0.0034 0.0040 0.0024 0.0043 0.0022 0.0042 0.0031 0.0047 0.0028

0.0030 0.0005 0.0033 0.0005 0.0028 0.0006 0.0031 0.0006 0.0019 0.0015 0.0020 0.0015 0.0008 0.0008 0.0029 0.0008 0.0021 0.0010 0.0023 0.0010

0.0029 0.0009 0.0032 0.0009 0.0026 0.0011 0.0030 0.0011 0.0019 0.0018 0.0021 0.0018 0.0024 0.0012 0.0027 0.0013 0.0022 0.0015 0.0024 0.0015

0.0015 0.0004 0.0017 0.0004 0.0014 0.0005 0.0016 0.0006 0.0011 0.0004 0.0013 0.0004 0.0013 0.0005 0.0016 0.0005 0.0012 0.0005 0.0014 0.0005

0.0018 0.0007 0.0022 0.0007 0.0015 0.0008 0.0019 0.0011 0.0018 0.0012 0.0022 0.0011 0.0015 0.0006 0.0019 0.0009 0.0025 0.0022 0.0027 0.0022

0.0010 0.0014 0.0016 0.0014 0.0009 0.0017 0.0012 0.0018 0.0008 0.0017 0.0010 0.0018 0.0006 0.0013 0.0009 0.0014 0.0008 0.0018 0.0011 0.0019

0.0014 0.0011 0.0016 0.0018 0.0010 0.0018 0.0014 0.0022 0.0009 0.0018 0.0013 0.0022 0.0012 0.0009 0.0016 0.0009 0.0009 0.0020 0.0012 0.0024

0.0175 0.0179 0.0176 0.0179 0.0181 0.0180 0.0175 0.0180 0.0181 0.0186 0.0180 0.0186 0.0175 0.0180 0.0175 0.0180 0.0174 0.0181 0.0174 0.0181

0.0084 0.0051 0.0088 0.0051 0.0056 0.0046 0.0084 0.0046 0.0056 0.0041 0.0057 0.0040 0.0078 0.0044 0.0082 0.0044 0.0068 0.0038 0.0072 0.0038

PR SRK PR SRK PR SRK PR SRK PR

phase envelopes rely on the correlation used for estimating the normal boiling point. This result reflects the dependency of the critical temperature on the normal boiling point in JR method according to eq 1. However, it was found that all correlations provides acceptable estimations taking into consideration the use of mixing rules containing no specific regression parameters. On the other hand, to establish which phase envelope corresponds to the real one in Figure 6, more data are

phase envelopes are comparable to each other and both provides a good estimation for the only experimental data available,38 they are not the same; the differences between them are smaller at low pressures than at high pressures. These results are in agreement with those in Table 7, where results are similar to each other irrespective of the used combination for separations flash at low pressures. Similar to the results in Tables 7 and 8, it is important to point out that the calculated 14195



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Figure 5. Isothermal differential vaporization at 394.25 K for fluid 5.

Figure 3. Molar compositions of liquid coming from a flash separation at 288.15 K and 0.101325 MPa for fluid 5.

Figure 6. Phase envelope for fluid 5 using different mixing rules. Figure 4. Molar compositions of vapor coming from a flash separation at 288.15 K and 0.101325 MPa for fluid 5.

developments into a single predictive characterization approach. Therefore, any combination of cubic equation of state and mixing rule may be used to solve phase equilibrium problems of petroleum fluids, as long as properties arises from the minimization problem given by expressions 14 to 16. However, additional applications will be necessary in order to generalize this result; especially, if more difficult problems are taken into account, for example, heavy oil characterization45 and minimum miscibility pressure.46

Table 8. Average Absolute Percent Error for Isothermal Differential Vaporization EOS

MR

Tb−C

fluid 3

fluid 5

fluid 9

fluid 11

fluid 12

SRK

VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1 VDW MHV1

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5

3.98 2.11 3.63 1.85 4.60 2.73 4.26 2.53 2.65 1.43 1.94 1.52 4.74 2.84 4.42 2.65 3.84 2.60 4.24 2.41

3.31 3.57 3.35 2.97 5.09 4.50 4.18 3.84 4.64 3.28 3.79 2.69 5.67 4.80 4.85 4.20 5.33 4.66 4.73 4.00

5.16 4.53 4.97 4.44 6.03 5.33 5.84 5.12 6.03 3.10 3.06 3.06 5.70 4.97 5.51 4.76 4.75 3.99 4.56 3.77

3.26 2.56 2.81 1.47 3.95 2.29 3.43 1.99 2.16 1.35 1.60 1.41 4.17 2.50 3.61 2.18 4.06 2.33 3.45 1.95

6.04 5.40 5.75 5.18 6.71 6.00 6.37 5.72 5.91 3.42 5.42 3.09 7.02 6.16 6.62 5.85 6.85 5.99 6.41 5.63

PR SRK PR SRK PR SRK PR SRK PR

■

CONCLUSIONS Functional groups and associated critical properties were assigned to undefined petroleum fractions by a minimization process of the Gibbs energy using group contributions methods. It was demonstrated that such properties can be directly used into models for predicting phase equilibria and liquid-phase densities, without additional assumptions or considerations. This generalized approach not only provides results in quantitative agreement with experiments, but also in qualitative agreement with the chemical nature expected for petroleum fluids. Finally, this work illustrates that microscopic properties may not be required for the thermodynamic characterization of an undefined petroleum fraction. The only needed properties are macroscopic.

■

necessary. However, this argument is about which mixing rule is appropriate for the fluid under study and not about the characterization method proposed in this work. The previous results stand out not only because of the agreement between calculated and experimental results, but also because of the combination of group contribution

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 14196



■

Article

ACKNOWLEDGMENTS The authors thank Mexican Institute of Petroleum for permission to publish this work. Founding for this research from grant D.00432 is gratefully acknowledged. We also express a special thanks to Dr. F. Garcia-Sanchez for his very fruitful advices.

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i = functional group i or pure component i j = undefined petroleum fractions j or pure component j

REFERENCES

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NOTATION AAD = absolute average deviation c = volume translation, m3/mol FG = number of functional groups types G = Gibbs free energy, J g = molar Gibbs free energy, J/mol JR = the Joback−Reid method k = binary interaction parameter MW = molecular weight MHV1 = Modified Huron-Vidal mixing rule of one parameter, eq 32 N = total number of components in the system n = total number of moles nA = number of atoms in eq 2 P = pressure, MPa Pc = critical pressure, MPa pc = contribution to Pc by functional groups PR = the Peng−Robinson equation of state, eq 24 R = the universal gas constant, 8.31441 J/mol K SG = specific gravity at 60/60 °F SRK = the Soave−Readlich−Kwong equation of state, eq 24 T = temperature, K Tb = boiling temperature at 0.101325 MPa, K Tb−C = boiling temperature correlation Tc = critical temperature, K tc = contribution to Tc by functional groups V = volume, m3/mol Vc = critical volume, m3/mol VDW = van der Waals mixing rule, eq 29 νc = contributions to Vc by functional groups wc = contributions to ω by functional groups x = mole fraction Zc = critical compressibility factor

Greek Letters

α = coefficient in eqs 8 and 43 β = coefficient in eqs 8 and 43 χ = coefficient in eqs 8 and 43 Δν = group volume increment, eq 8 Φ = Lagrangian function, eq 17 φ = fugacity coefficient γ = activity coefficient λ = Lagrange multiplier μ = chemical potential, J/mol ρ = mass density, kg/m3 υ = number of functional groups ω = acentric factor

Superscripts

EOS = equation of state EXP = experimental value GI = ideal gas R = residual property l = liquid phase 0 = reference state Subscripts

c = critical property 14197



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Thermodynamic Characterization of Undefined Petroleum Fractions

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