Extending the Gibbs tangent plane semicontinuous mixtures

Fluid Phase Equilibria 190 (2001) 1–13

Extending the Gibbs tangent plane semicontinuous mixtures Jorge E.P. Monteagudo a , Geraldo L. Rochocz b , Paulo L.C. Lage a , Krishnaswamy Rajagopal c,∗ a

c

Programa de Engenharia Qu´ımica-COPPE, Universidade Federal do Rio de Janeiro, C.P. 68502, Rio de Janeiro, RJ 21945-970, Brazil b Chemtec Serviços de Engenharia, Rua Nilo Peçanha 51, Sala 2106, Rio de Janeiro, RJ, Brazil Escola de Qu´ımica, Universidade Federal do Rio de Janeiro, C.P. 68542, Rio de Janeiro, RJ 21949-900, Brazil

Abstract The computation of phase equilibrium for semicontinuous mixtures results in a set of algebraic-functional equations which are commonly solved using the pseudocomponent approach or reducing the order of the functional equations to a set of algebraic equations by means of a Petrov–Galerkin approach. Both approaches can be taken as particular cases of the method of weighed residuals. In order to perform stability analysis of phase equilibrium, the classical tangent plane criterion algorithms can be used as the first approach. There is a lack of functional formulation for the stability criterion in the literature. In some special cases, such as mixtures containing continuous families of heavy-components, a new formulation of the tangent plane criterion is required. In the present work, functional extensions of the Gibbs tangent plane criterion are described. The use of the proposed criteria was exemplified by calculating solid–liquid phase equilibrium of a crude oil containing asphaltenes on titration with a precipitating agent, n-pentane. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Gibbs energy; Continuous thermodynamics; Solid–liquid equilibria; Liquid–liquid equilibria; Vapor–liquid equilibria; Equation of state

1. Introduction The computation of multiphase equilibrium can be done either by a direct minimization of Gibbs free energy [1] or by means of a phase stability analysis followed by a phase-split calculation [2,3]. The phase stability analysis is usually made with the Gibbs tangent plane criterion. The first application of this criterion, made by Baker et al. [2] and Michelsen [3] were related to mixtures of discrete components in which all of them were present in all the phases in the stability analysis. An extension of the tangent plane criterion is needed for semicontinuous mixtures and for mixtures in which some of the components or continuous families appear in negligible quantities in one of the phases. By semicontinuous mixtures we refer to mixtures formed by some discrete components and one or several continuous families. ∗

Corresponding author. E-mail address: [email protected] (K. Rajagopal). 0378-3812/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 1 ) 0 0 5 3 0 - 1

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The phase equilibrium of semicontinuous mixtures was studied many years ago by Briano and Glandt [4], Kehlent et al. [5] and Cotterman [6]. Normally, a continuous family is represented by a probability density function with one characterization variable such as the boiling point, molecular weight or refractive index. The classical thermodynamics was rigorously extend for continuous families by Kehlent et al. [5]. In that work, by applying notions of variational calculus, the state functions were generalized by functionals of state. Then, phase equilibrium calculation of a semicontinuous mixture results in a set of algebraic-functional equations. Restricting to single continuous family and simple thermodynamics models, such as ideal solutions and perfect gas mixtures: analytical solutions of these set of equations were obtained, i.e. a continuous version of Raoult’s law. For more rigorous calculations, numerical solutions are necessary. Cotterman [6] applied the generalized Laguerre–Gauss quadrature method to calculate the integrals appearing in the equations resulting from the phase equilibrium of a continuous mixture. Perhaps Shibata et al. [7] were the first to clearly state that the characterization of a semicontinuous mixture could be regarded as the correct selection of pseudocomponents. The roots of the orthogonal polynomials used in the quadrature method employed to numerically evaluate the involved integrals proved to be a good choice for the pseudocomponents. Thus, the computation of phase equilibrium for these mixtures could be performed with the same well-known algorithms of discrete thermodynamics. Several works have followed this approach (Wilman and Teja [8,9], Chegulet and Vera [10] Rochocz [11] Chachamowitz [12], Rochocz et al. [13]). Later, Luks et al. [14] made several observations concerning the use of quadrature for selecting pseudocomponents. He noted that some care should be taken when dealing with multistage operations of semicontinuous mixtures, such as in absorption or distillation columns, due to the change of the domain and the functional form of the continuous families along the stages. For this kind of problem it is better to use an order reduction method, such as the recent one developed by Marquardt and Watzdorf [15], and Watzdorf and Marquardt [16]. They compute liquid–vapor equilibria of continuous mixtures by reducing the system of integral equations to a set of algebraic equations by means of the Wavelet–Galerkin method. This approach finds the coefficients of the approximation function of the continuous families, which do not necessarily have to be the same in all the phases. In fact, both the pseudocomponent and the Wavelet–Galerkin approach can be seen as particular cases of the method of weighted residuals (MWR) [15]. In the first case, the weight function is a displaced Dirac’s delta function and, in the second one, the weight function is a member of a Wavelet basis which is also used for the trial function. The set of equations solved by Marquardt and Watzdorf [15] and Watzdorf and Marquardt [16] applying the Wavelet–Galerkin method, can be regarded as a phase-split calculation, but nothing is said regarding the stability analysis formulation for this kind of mixtures. Some recent works dealing with the stability of continuous or semicontinuous mixtures can be found on Browarzik and Kehlent [17]. Hu and Prausnitz [18] and Hu and Ying [19]. However all these works were more concerned with calculation of spinodal points using a functional approach. In the present work, we extend the tangent plane criterion to semicontinuous mixtures. It is true that the pseudocomponent approach allows us to use all the well-known algorithms developed for discrete molecular thermodynamics including stability analysis and phase-split calculation. Nevertheless heavy-component continuous families as polymer macromolecules and heavy fractions of oil mixtures give rise to some special cases in phase equilibrium computation. Usually, the vapor phase will contain very little of the heavy-components, while a liquid or solid phase formed in equilibrium will be almost free of the lighter components of the initial mixture [20,21]. The presence of solvent in the polymer-rich liquid or solid phase, or of heavy-components in the vapor phase even in negligible

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quantities can invalidate the thermodynamic models employed to compute their fugacities in the mixture [22], due to differences in size and/or chemical nature between the components. Therefore, in these cases it is useful for the sake of numerical computation to completely neglect them from one of the phases. An extension of the Gibbs tangent plane criterion is necessary when dealing with multicomponent phases where some of the components are not present in one of the phases, and we also present such an extension in this work. 2. Functional formulation of the tangent plane criterion for semicontinuous mixtures Consider a semicontinuous system, containing N moles of mixture, which is described by the moles of nd discrete components (n1 , n2 , . . . , nnd ), and the moles and normalized distribution functions of m continuous families (w1 , w2 , . . . , wm and F1 (I ), F2 (I ), . . . , Fm (I )). For a given temperature T and pressure P the Gibbs free energy of the system is given by the following expression: nd m 

0 ni µi + ws Fs (I )µ0s (I ) dI (1) G ≡ G(n) ≡ s

i

I

or in molar fraction composition   nd m 

0 0 zi µi + ηs Fs (I )µs (I ) dI G=N s

i

(2)

I

where zi and ηs are the discrete component and continuous family molar fractions, respectively. The variation of the Gibbs free energy, G, due to the appearance of a new incipient phase II can be expressed as (Appendix A)  nd  m 

II 0 II 0 G = ε yi (µi − µi ) + νs Ws (I )[µs (I ) − µs (I )] dI (3) i

s

I

where yi and ν s are the discrete component and continuous family in the incipient phase, respectively. Let us define the functional G (4) Ψ ( y) ≡ ε where ε is the number of moles of the incipient phase, and y = [yi |νs Ws (I )]T Thus, the sufficient and necessary criteria for the initial system to be stable is Ψ ( y) ≥ 0

(5)

for any trial composition of the incipient phase II. Following the ideas of Michelsen [3], it is not necessary to test all possible compositions of the incipient phase in order to determine the stability of the system. It is enough to calculate the stationary points of Eq. (4) and to test if this functional is positive in all the stationary points. This is equivalent to state that

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the distance of the hyperplane tangent to the Gibbs energy hypersurface at the initial composition will be always positive for all compositions. It is possible to demonstrate (Appendix B) that the condition for a given composition y be a stationary point of Ψ is µIIi − µ0i = κ,

i = 1, . . . , nd

µIIs (I ) − µ0s (I ) = κ,

s = 1, . . . , m

(6) (7)

where κ does not depend on component i or family s or the distribution variable I. In this way the value of Ψ in a stationary point is Ψsp =

nd


yi κ +

m 


i

s

I

νs Ws (I )κ dI = κ

(8)

Thus, the system will be stable if κ ≥ 0 at all stationary points except for the trivial solution where phase II is identical to the initial mixture (κ = 0). In order to use an equation of state it is convenient to express chemical potential differences as µIIi − µ0i = RT(ln yi + ln ϕiII − ln zi − ln ϕi0 ) µIIs (I ) − µ0s (I ) = RT(ln νs Ws (I ) + ln ϕsII (I ) − ln ηs Fs (I ) − ln ϕs0 (I ))

(9) (10)

where R is the gas constant. Following Michelsen [3], we define hi = ln zi + ln ϕi0

(11)

hs (I ) = ln ηi Fs (I ) + ln ϕs0 (I )

(12)

Thus, Eqs. (6) and (7) can be rewritten as ln yi + ln ϕiII − hi = κˆ

(13)

ln νs Ws (I ) + ln ϕiII (I ) − hs (I ) = κˆ

(14)

where κˆ =

κ RT

(15)

If we also define Yi = yi e−κˆ

(16)

Ys (I ) = νs Ws (I ) e−κˆ

(17)

then, stationary points of Ψ will be obtained from the solution of ln Yi + ln ϕiII − hi = 0

(18)

ln Ys (I ) + ln ϕsII (I ) − hs (I ) = 0

(19)

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Compositions of semicontinuous mixtures are related to the new variables used in Eqs. (18) and (19) by yi = nd i

Yi +

Yi m 

νs Ws (I ) = nd i

s

I Ys (I ) dI

Yi +

Ys (I ) m  s

I Ys (I ) dI

(20) (21)

The system formed by Eqs. (18) and (19) is algebraic-functional. If we calculate the integrals included in Eq. (19) using a quadrature method we are falling back to the pseudocomponent approach [6]. In that case it would not be necessary to develop this approach. However, if we pretend to solve functional equations by order reduction, using the Wavelet–Galerkin method [15,16], for example, then we have an useful set of equations for the stability analysis of semicontinuous mixtures. Rochocz [11] solved numerically this set of equations using the method of Hendriks [23] to reduce the order of the functional equations. The method was originally limited to two-parameter EOS and extended by Rochocz [11] to EOS of any number of parameters. The main assumptions are that the interaction parameters between a family and a component and between two distinct families are constant, i.e. they do not depend on the distribution variable, and that they are zero for two components of the same continuous family. The MWR approach [15,16] is not restricted to these considerations and its main goal is to determine the coefficients of the approximations for the distribution functions of the continuous families in each phase. The set of algebraic-functional equations presented in this work would allow to test the stability of a semicontinuous mixture and, thus, calculate the trial composition for a phase-split calculation. 3. Tangent plane criterion for special cases of semicontinuous mixtures Let us consider a solution consisting of a heavy-component continuous family dissolved in a mixture of nd discrete components. It is useful for the sake of numerical computation, to consider that the composition of the heavy-components will be negligible in the vapor phase and that they will be free of the other light component in the solid phase. A special extension of the tangent plane criterion has to be formulated for these cases as shown below. The variation of the Gibbs free energy can be obtained from Eq. (3) for both cases. Thus, when a heavy-component continuous family forms a solid or a new pseudoliquid phase free of light components the variation of the Gibbs free energy is  G = ε νW (I )[µII (I ) − µ0 (I )] dI (22) I

and when the vapor phase is considered free of heavy-components G = ε

nd
yi (µIIi − µ0i )

(23)

i

It should be noted that, in both cases, µ0 depends on the composition of the whole mixture, while µII depends on the composition of the incipient phase, which can be a solid free o solvent (light components) or a vapor free of heavy-components.

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Table 1 Titration of a Brazilian light oil — experimental data Dilution (cm3 of n-pentane/g oil)

Precipitated asphaltenes (wt.%)

1.0 2.5 5.0 10.0 20.0 30.0 40.0 50.0

0.125 0.255 0.325 0.425 0.430 0.420 0.420 0.430

The corresponding stability criteria and stationary point equations are  G νW (I )[µII (I ) − µ0 (I )] dI ≥ 0 Ψ ≡ ε I

(24)

Ψsp ≡ µII (I ) − µ0 (I ) = κ

(25)

for the first case, and G
= yi (µIIi − µ0i ) ≥ 0 ε i nd

Ψ ≡

Ψsp ≡ µIIi − µ0i = κ,

(26)

i = 1, . . . , nd

(27)

for the second one. If the continuous family is discretized in nq pseudocomponents using the method of Gaussian quadrature [6], the functional Eq. (24) is reduced to the equation G
Ψ [≡] yj [µIIj − µ0j ] ≥ 0 ε j =1 nq

(28)

Table 2 PARA characterization of a Brazilian light oil Component

Molecular weight

Tc (K)

Pc (MPa)

Accentric factor

Molar fraction

C3 C4–C6 C7–C8 C9–C1O C11–C14 OC15–C29 NC15-C29 Aromatics OC3O+ NC3O+ Resin + asphaltene

44.097 70.909 117.879 131.980 169.033 241.352 286.557 367.000 390.359 461.822 514.244

369.850 442.328 521.708 553.552 620.829 703.546 767.247 800.000 854.945 890.208 935.737

4.247 3.579 2.820 2.538 2.009 1.852 1.630 1.550 1.323 1.128 0.895

0.152 0.203 0.307 0.354 0.475 0.680 0.740 0.900 0.999 1.100 1.205

0.2979 14.8540 26.1338 17.9167 12.0130 12.1266 6.1589 5.7921 2.0452 0.7589 1.9029

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Fig. 1. Simulation of a Brazilian light oil titration.

and Eq. (25) reduces to Ψsp ≡ µIIj − µ0j = κ,

j = 1, . . . , nq

(29)

It must be pointed out that subscript i in Eqs. (26) and (27) corresponds just to the discrete light components and j in Eqs. (28) and (29) refers to the discretized heavy-components. However, in both cases µ0 depends on the composition of the whole mixture. The stability criterion stated in Eq. (28) was used by Monteagudo [20] to compute the precipitation of asphaltene by means of a polydisperse thermodynamic molecular model. It is common, in molecular thermodynamics models for asphaltene deposition, to admit that the asphaltenic phase will be free of solvent [24–27]. Gibbs tangent plane criterion is straightforward if we consider that the asphaltenic phase is formed by only one type of asphaltene. However, if asphaltenes are represented by pseudocomponents chosen from a continuous family, we should use the formulation presented here. A polydisperse thermodynamic model that uses this stability criterion with a solid–liquid split calculation, was formulated by Monteagudo [20] elsewhere. As an example we show the titration of a Brazilian unstable light oil with n-pentane as a precipitating agent. We show in Table 1 the experimental titration data and in Table 2 the PARA characterization [28] for this oil. Fig. 1 shows the results of the fitting of the model against experimental data showing a plausible matching. 4. Conclusions The Gibbs tangent plane criterion is extended for the two important cases of phase equilibrium calculations involving semicontinuous mixtures. For the order reduction methods of calculating phase equilibrium, the algebraic-functional formulation of equations developed in Section 2 should be used. For the pseudocomponent approach where some of the components of the discretized mixture are not considered to be present in one of the incipient phases, the criteria showed in Section 3 should be used. The application of these extensions is straight forward and proved to be effective in the example tested.

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List of symbols F(I) distribution function of the original mixture G Gibbs free energy I characterization variable m number of continuous families n composition of the mixture nd number of discrete components N total number of moles in the mixture P pressure T temperature W(I) distribution function of the trial phase y composition of trial phase z composition of initial phase Greek symbols ε composition of trial phase ϕ fugacity coefficient µ chemical potential Ψ functional related to Gibbs free energy of a semicontinuous mixture Subscript c critical state i discrete component j pseudocomponent s continuous family Superscript 0 initial mixture composition I phase I — original phase II phase II — incipient phase

Acknowledgements K. Rajagopal gratefully acknowledges the scholarships given by CNPq (Bolsa de Produtividade) and FAPERJ (Bolsa do Cientista do Estado). The financial support of PETROBRAS, Project PRONFX 124/96 (contract 4196087800 FINEP/MCT) and PADCT III/CNT for carrying out this work is also acknowledged. J.E.P. Monteagudo and P.L.C. Lage acknowledge the financial support given by the CNPq Grant nos. 190044/96-9 and 520660/98-6.

Appendix A. Variation of Gibbs free energy of an unstable semicontinuous mixture In this section it will be demonstrated the passage from Eqs. (2) to (3).

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The Gibbs free energy of a semicontinuous mixture of nd discrete components and m continuous families can be written nd m 

0 G ≡ G(n) = ni µi + ws Fs (I )µ0s (I ) dI (A.1) I

s

i

where n = [ni |ws ]T = N[zi |ηs Fs (I )]T

(A.2)

If a new incipient phase is forming, the variation in the Gibbs free energy can be stated as G = GI (n0 − ε) + GII (ε) − GI (n0 )

(A.3)

where n0 and ε represent the composition of the initial mixture and of the incipient phase II, in number of moles, therefore, n0 = [n0i |ws0 ]T = N[zi |ηs0 Fs (I )]T

(A.4)

ε = [εi |νs ]T = ε[yi |νs Ws (I )]T

(A.5)

Expanding GI by using series of Taylor and neglecting second-order terms GI (n0 − ε) = GI (n0 ) − ε · ∇GI |n0 where



∇GI |n0 = since µi =



∂G ∂ni

∂GI ∂ni



 T

δGI

= [µ0i |µ0s (I )]T



δn + sI T ,P ,I =I T ,P ,j =i

(A.6)

(A.7)

(A.8) T ,P ,j =i

and by using the concept of partial derivative of a functional [5]  δG µs (I ) = δnsI T ,P ,I + =I

(A.9)

The last term of (A.6) represents an extension of the scalar product concept. For a function of discrete variables the scalar product resulting from a Taylor series expansion corresponds to a summation of several parcels. In the case of a simple functional with no scalar variables and depending on one single function this summation will be transformed into an integral. In a more general case such as the semicontinuous mixtures in which the functional related to the Gibbs free energy depends on several discrete components and continuous families the scalar product represents a summation of the discrete parcels plus the summation of the integral parcels resulting from each continuous family. Therefore,  nd  m 

ε · ∇G|n0 = ε yi µ0i + νs Ws (I )µ0s (I ) dI (A.10) i

s

I

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and by substituting Eq. (A.8) into Eq. (A.4)  nd  m 

GI (n − ε) = G0 − ε yi µ0i + νs Ws (I )µ0s (I ) dI s

i

 nd  m 

II II yi µi + νs Ws (I )µs (I ) dI G (ε) = ε II

i

s

(A.11)

I

(A.12)

I

Substitution of (A.11) and (A.12) into (A.3) gives the Eq. (3)   nd m 

II 0 II 0 yi (µi − µi ) + νs Ws (I )[µs (I ) − µs (I )] dI G = ε s

i

(A.13)

I

Appendix B. Stationary points of Ψ In this section it will be demonstrated that the stationary point of functional Ψ (see Eqs. (3) and (4)) can be found by solving Eqs. (6) and (7). First we define the total differential of functional Ψ following [11] nd  m  

∂Ψ δΨ dΨ = dyi + δ(νs Ws (I )) dI (B.1) ∂y δν i sI I s i The variables (yi , ν s Ws (I)) are related; so, perturbations dyi and δ(ν s Ws (I)) are not independent, and must satisfy the following restriction nd m 

dyi + δ(νs Ws (I )) dI = 0 (B.2) i

s

I

At a stationary point dΨ = 0 any perturbation in the composition. Two different type of perturbation are applied in order to get Eqs. (6) and (7). B.1. 1st Perturbation Variation of the molar fraction of component yk (k = n) with compensation on molar fraction of component nd, ynd . Then dyk = −dynd = 0

(B.3)

and dΨ = 0 =

∂Ψ ∂Ψ dyk + dynd ∂yk ∂ynd

(B.4)

from Eqs. (B.3) and (B.4) it follows that ∂Ψ ∂Ψ = ∂yk ∂ynd

(B.5)

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deriving Eq. (A.13) with respect to yk  II
 II nd m 
∂µi ∂Ψ ∂µs (I ) II 0 = µk − µk + yi νs Ws (I ) + dI ∂yk ∂yk ∂yk s i

11

(B.6)

The extension of the Gibbs–Duhem equation at constant temperature and pressure for semicontinuous mixtures [6,11] is nd m 

yi dµi + νs Ws (I )δµs (I ) dI = 0 (B.7) i

I

s

Applying Eq. (B.7) into (B.6) ∂Ψ = µIIk − µ0k , ∂yk

k = 1, . . . , nd − 1

(B.8)

Analogously ∂Ψ = µIInd − µ0nd ∂ynd

(B.9)

Since, both derivatives in (B.8) and (B.9) are equal as shown in (B.5), then µIIk − µ0k = κ,

k = 1, . . . , nd

(B.10)

where κ does not depend on the component. In this way, the first condition has been demonstrated. B.2. 2nd Perturbation Local variation in one of the distribution functions with compensation on molar fraction of component nd, ynd , then δ(νs Ws (I )) = −dynd being this local variation defined as νs Ws I ∈ [I + , I + + I ] δ(νs Ws (I )) = 0 I∈ / [I + , I + + I ]

(B.11)

(B.12)

where I → 0 and νs Ws → 0

(B.13)

Eq. (B.11) can be rewritten as νs Ws = −dynd Then, the expression for this perturbation remains   ∂Ψ ∂Ψ dΨ = 0 = dynd + δ(νs Ws (I )) dI ∂ynd ∂νsI I

(B.14)

(B.15)

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as δ(ν s Ws (I)) is null in all the domain of I but [I+ , I + + I ] and I → 0, the integral of the above expression can be substituted by the value of the integrand at the point I+  ∂Ψ ∂Ψ dΨ = 0 = dynd + (B.16) νs Ws ∂ynd ∂νsI I + Substituting (B.14) into (B.16) we get  ∂Ψ ∂Ψ = ∂ynd ∂νsI I + Deriving (A.13) with respect to ν sI , at point I+   II  II nd m 

∂µi ∂µi (I ) ∂Ψ II + 0 + = yi + µs (I ) − µs (I ) + νI WI (I ) dI ∂νsI I + ∂νsI ∂νsI I i I

(B.17)

(B.18)

where the term µIIs (I + ) − µ0s (I + ) appears in process similar to the one used to derive Eq. (B.16). Again by applying Gibbs–Duhem relation (B.7) into (B.18) we get  ∂Ψ = µIIs (I + ) − µ0s (I + ) (B.19) ∂νsI I + Using Eqs. (B.17) and (B.9) into (B.19) we get µIIs (I + ) − µ0s (I + ) = µIInd − µ0nd

(B.20)

This expression is valid for all the families and at all the points I+ , and taking into account Eq. (B.10),we derive µIIs (I + ) − µ0s (I + ) = κ,

s = 1, . . . , m ∀I

(B.21)

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

R. Gautam, W.R. Seider, AIChE J. 25 (6) (1979) 991–999. L.F. Baker, A.C. Pierce, K.D. Luks, SPE J. 22 (1982) 731–742. M.L. Michelsen, Fluid Phase Equilib. 9 (1982) 1–9. J.G. Briano, E.D. Glandt, Fluid Phase Equilib. 14 (1983) 91–102. H. Kehlent, M.T. Ratzsch, J. Bergmann, AIChE J. 31 (1985) 1136–1147. R.L. Cotterman, Phase equilibria for complex fluid mixtures at high pressures. Development and application of continuous thermodynamics, Ph.D. Dissertation, University of Florida, FL, USA, 1981. S.K. Shibata, S.I. Sandler, R.A. Behrens, Chem. Eng. Sci. 42 (1987) 1977–1987. B. Wilman, A.S. Teja, Ind. Eng. Chem. Res. 26 (1987) 948–952. B. Wilman, A.S. Teja, Ind. Eng. Chem. Res. 26 (1987) 953–957. F.L. Chegulet, J.H. Vera, Can. J. Chem. Eng. 69 (1991) 1374–1381. C.L. Rochocz, Fquilibrio liquido–vapor em misturas semicontinuas, M.Sc. Thesis, COPPE/IUFRJ, Rio de Janeiro, Brazil, 1990. J.C. Chachamowitz, Simulaco de processos de separacao de misturas semicontinuas, M.Sc. Thesis, PEQ/COPPE/UFRJ, Rio de Janeiro, Brazil, 1993. C.L. Rochocz, M. Castier, S.I. Sandler, Fluid Phase Equilib. 139 (1997) 137–153. K. Luks, E.A. Turek, T.K. Kragas, Ind. Eng. Chem. Res. 32 (1993) 1767–1771.

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[15] W. Marquardt, R. Watzdorf, Reduced Modelling of Complex Fluid Mixtures by the Wavelet–Galerkin Method, RWTH Aachen University of Technology, Report No. 1995-06, Germany, 1995. [16] R. Watzdorf, W. Marquardt, Comput. Chem. Eng. 21 (1997) S811–S816. [17] D. Browarzik, H. Kehlent, Fluid Phase Equilib. 123 (1996) 17–28. [18] Y. Hu, J.M. Prausnitz, Fluid Phase Equilib. 130 (1997) 1–18. [19] Y. Hu, X. Ying, Fluid Phase Equilib. 127 (1997) 21–27. [20] J.E.P. Monteagudo, Modelo termodinâmico molecular polidisperso para a precipitação de asfaltenos em óleo vivo, M.Sc. Thesis, PEQ/COPPE/UFRJ, Brazil, 1998. [21] B. Bungert, C. Sadowski, W. Arlt, Fluid Phase Equilib. 139 (1997) 349–359. [22] H. Orbey, C.P. Bokis, C. Chen, Ind. Eng. Chem. Res. 37 (1998) 1567–1573. [23] E.M. Hendriks, Fluid Phase Equilib. 33 (1987) 207–221. [24] A. Hirschberg, L.N.J. DeJong, B.A. Schipper, J.G. Meijer, SPE J. 24 (1984) 283–293. [25] S.L. Kokal, J. Najman, S.G. Sayegh, A.E. George, J. Can. Petroleum Technol. 31 (4) (1992) 24–30. [26] L.X. Nghiem, M.S. Hassam, R. Nutakki, Efficient Modelling of Asphaltene Precipitation, in: Proceedings of the 68th Annual Technical Conference and Exhibition of the SPE, Houston, TX, 3–6 October 1993, pp. 375–384. [27] L.X Nghiem, D.A. Coombe, SPE J. 2 (1997) 170–176. [28] K.J. Leontaritis, PARA-Based (Paraffin–Aromatic–Resin–Asphaltene) Reservoir Oil Characterization, SPE 37252, in: Proceedings of the International Symposium on Oilfield Chemistry, Houston, TX, 18–21 February, 1997.