Calculation of Critical Points in Gasâ Condensate Mixtures under the

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Ind. Eng. Chem. Res. 2010, 49, 7610–7619

Calculation of Critical Points in Gas-Condensate Mixtures under the Influence of Magnetic Fields Wilson A. Canas-Marin Instituto Colombiano del Petro´leo, ECOPETROL S.A., Piedecuesta-Santander, Colombia

Carlos Lira-Galeana* Instituto Mexicano del Petro´leo, IMP, Ciudad de Me´xico, Me´xico

The mathematical conditions representing the gas-liquid critical points of multicomponent mixtures are used to compute the gas-liquid critical points of diamagnetic gas-condensate fluids of interest to the oil industry, where the effect of a static magnetic field on calculated critical points is studied for the first time. Magneto-chemical fugacity derivatives, required to fulfill the quadratic and cubic forms of the Helmholtz free energy at the critical points, have been obtained from a recently developed theory of phase equilibria in diamagnetic multicomponent fluids and numerical differentiation (Canas-Marin, W. A.; Ortiz-Arango, J. D.; Guerrero-Aconcha, U. E.; Lira-Galeana, C. AIChE J. 2006, 52 (8), 2887-2897). Results in diamagnetic synthetic gas-condensate mixtures show that a magnetic field can increase or decrease the critical points depending on mixture composition and the extent of the applied magnetic field. For mixtures rich in heavy diamagnetic substances (i.e., hydrocarbon oils), the critical points have an initial tendency to decrease up to a certain value of the field and then to increase with the applied magnetic field. If high amounts of light diamagnetic components are present (i.e., gases), the critical properties always increase with the applied field. The effect of the applied magnetic field on the stability limit of these mixtures is also studied in detail. Introduction The study of the effect of magnetic fields on the phase behavior of different types of systems has attracted the attention of the scientific community mainly from the theoretical point of view.1-6 Different opinions about the effect of a magnetic field on the phase transitions of physical systems have been obtained. For instance, phase diagrams were calculated for model spin systems within the mean field approximation.3,4 It was found that for fluids of hard spheres carrying Ising spins, an external magnetic field decreases the temperature of the gas-liquid critical point (Tc). However, the presence of isotropic van der Waals attractions between molecules can lead to the inverse effect; in other words, it could produce an increase in the gas-liquid critical temperature.3 Lomba et al.5 also studied the effect of strong external fields on the gas-liquid critical point of classical fluids using the Monte Carlo and integral equation methods. These authors concluded that an external magnetic field favors the phase separation, i.e., the application of the external field increases the gas-liquid critical temperature. Sokolovska and Sokolovskii6 pointed that the results of Lomba et al.5 could be related to the quite strong fields applied. They concluded that the effect of strong magnetic fields consists in a considerable increase of Tc. Strong external fields spread the gas-liquid coexistence region on the phase diagram, in other words, favors the gas-liquid phase separation. In this work, we apply a recently developed theory of phase equilibria in diamagnetic multicomponent fluids under the influence of an applied static magnetic field.7 Results showed that a magnetic field can increase or decrease the critical point depending on mixture composition and the magnitude of the * To whom correspondence should be addressed. E-mail: clira@ imp.mx.

applied magnetic field. For mixtures rich in heavy diamagnetic substances (i.e., hydrocarbon oils), the critical points have an initial tendency to decrease up to a certain value of the field and then to increase with the applied magnetic field. If high amounts of light diamagnetic components are present (i.e., gases), the critical properties always increase with the applied field. The effect of the applied magnetic field on the stability limit of these mixtures is also studied; in this last case, a magnetic field always tends to increase the size of the twophase region, favoring the vapor-liquid splitting. This last effect is higher for oils than for gas-condensates. Modeling Recently, Canas-Marin et al.7 developed a framework based on classical thermodynamics to model the effect of an applied magnetic field on the gas-liquid-solid phase behavior of (nonferromagnetic) hydrocarbon mixtures. In this work, the main goal is to apply this thermodynamic framework for predicting the effects of a magnetic field on the critical properties and the gas-liquid region size of diamagnetic mixtures. To the best of our knowledge, no previous thermodynamic framework for this purpose has been reported in open literature. The critical point of a mixture is a stable point on the stabilitylimit curve (or spinodal curve). Heidemann and Khalil9 formulated an alternative mathematical description for critical points based on the use of the Helmholtz free energy. The Helmholtz free energy in the presence of a magnetic field is given by7 NC

dA ) -S dT - P dV + H d(JV) +

∑ µ˜ dn i

i

(1)

i)1

where J is known as the intrinsic induction (also called magnetic polarization), H is the magnetic field strength, and µ˜ i is the magneto-chemical potential.

10.1021/ie900939y  2010 American Chemical Society Published on Web 07/19/2010

Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010

In the Heidemann-Khalil method the critical phase must fulfill the following two criteria: Q∆n ) 0

(2)

Table 1. Compositions of Three n-Paraffin Synthetic Mixtures feed composition (mol fraction) component

M1

M2

M3

nC10 nC18 nC19 nC20 nC21 nC22 nC23 nC24 nC25 nC26 nC27 nC28 nC29 nC30

0.8000

0.8000 0.0334 0.0283 0.0241 0.0206 0.0177 0.0152 0.0130 0.0112 0.0096 0.0082 0.0072 0.0061 0.0052

0.8000 0.0249 0.0236 0.0224 0.0213 0.0203 0.0193 0.0183 0.0174 0.0166 0.0159

and NC

C)

NC

(

NC

∑∑∑ i)1 j)1 k)1

NC

∂3A ∂ni∂nj∂nk NC

)

T,V,nl*i,j,k

( ) ∂2 ln fi

NC

∑ ∑ ∑ RT ∂n ∂n

∆ni∆nj∆nk )

j

i)1 j)1 k)1

∆ni∆nj∆nk ) 0 (3)

k T,V,nl*j,k

where ∆n ) [∆n1, ∆n2, ..., ∆nNC]T ∆nT∆n ) 1

0.0642 0.0439 0.0300 0.0205 0.0140 0.0096 0.0066 0.0045 0.0031 0.0021 0.0015

Table 2. Compositions of Two Synthetic Gas-Condensate Mixtures14

and

feed composition (mol fraction)

( ) ∂2A ∂ni∂nj

Q)

) RT

T,V,nl*i,j

( ) ∂ ln fi ∂nj

component

SHF1

SHF2

C1 C2 nC4 C7H8 nC8 nC16 C14H10

0.7581 0.1132 0.0488 0.0199 0.0268 0.0230 0.0102

0.7472 0.1097 0.0442 0.0272 0.0387 0.033

i, j ) 1 ... NC

;

T,V,nl*j

(3a)

Equation 2 represents the diffusional stability limit (spinodal curve). For a fluid in presence of a magnetic field, the total Helmholtz energy is here expressed as the sum of two contributions, A ) Anonmag + Amag

(4)

was derived by Canas-Marin et al.7 and its mathematical evaluation is widely discussed in that paper.

Equation 4 is derived for obtaining the fugacity, ln fi ) ln

finonmag

+ ln

fimag

ln fimag ) -

(5)

Here ln finonmag ) ln fi(EoS)

(6)

Then, the nonmagnetic contribution to fugacity can be calculated using a cubic equation-of-stare in the usual way. In this work, the cubic Peng-Robinson equation-of-state (PR-EoS) was always used.10 The criticality criteria in presence of a magnetic field are then evaluated as Q∆n ) 0

1 2RT

[ ( ) NC

∑ λ¯ x

j j

j)1

nT

∂Vjj ∂ni

T,P,H,ns*i

]

+ Vjiλ¯ i H2

(9)

where λji ≡ µ0κ(i) with κ(i) the volumetric susceptibility for the component i and µ0 the vacuum magnetic permeability. xi and Vji are the mole fraction and partial molar volume, respectively. Also the term (nT(∂Vjj/∂ni))T,P,Hb,ns*i is the partial volume derivative for the component j with respect to number of moles of the component i, and it can be evaluated using an equation of state.10 The magnetic field also contributes to the total pressure of the system, and then, from eq 4, the total pressure can also be expressed as a combination of a magnetic and a nonmagnetic part as

(7)

P ) Pnonmag + Pmag

(10)

Pnonmag ) P(EoS)

(11)

with

with Q)

( ) ∂2A ∂ni∂nj

) RT T,V,H,nl*i,j

( ) ∂ ln fi ∂nj

;

i, j ) 1 ... NC

T,V,H,nl*j

The magnetic part is given by (see Appendix),

and NC

C)

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NC

NC

∑∑∑ i)1 j)1 k)1

(

NC

∆ni∆nj∆nk )

3

∂A ∂ni∂nj∂nk NC

NC

)

Pmag ) -

T,V,H,nl*i,j,k

( ) 2

∂ ln fi

∑ ∑ ∑ RT ∂n ∂n i)1 j)1 k)1

j

∆ni∆nj∆nk ) 0

k T,V,H,nl*j,k

(8) In other words, the magnetic field is also kept constant during the derivation of fugacity with respect to mole number. A mathematical expression for the magnetic contribution to the fugacity of a component i in a nonferromagnetic mixture

κ(m) 105V

µ0H2

(12)

Here κ(m) is the volumetric susceptibility of the mixture, whose derivation and calculation procedures have also been widely discussed in the original paper.7 In eq 12, the 105 term is used for expressing the pressure in bar. The first and second-order derivatives of fugacity with respect to mole numbers can then be obtained by using eqs 5, 6, and 9. In this work, these derivatives were evaluated numerically by following the procedure suggested by Michelsen11 and also applied by Stockfleth and Dohrn.12 The procedure is described in the Appendix.

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Figure 1. n-Paraffin molar susceptibility as a function of molar mass at 298.15 K.15

Figure 2. M1 mixture: critical temperature versus magnetic field intensity.

Figure 3. M1 mixture: critical pressure versus magnetic field intensity.

Results Three synthetic oils (M1-M3) from Stamataki et al.13 and two synthetic gas condensates from Ungerer et al.14 (SHF1SHF2) were chosen for studying the computed trends of calculated critical properties and the extent of the two-phase region when a wide range of magnetic field strengths is applied.

M1-M3 fluids are synthetic mixtures formed mainly by n-paraffins whose compositions are shown in Table 1. The compositions for SHF1 and SHF2 are shown in Table 2. Figure 1 shows the variation of molar susceptibility (cgs units), δ, for the n-paraffins homologous series, as a function of molar mass. This variation fits almost perfectly to a straight line. Data


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were taken from Lide,15 where all the tabulated values correspond to room temperature, atmospheric pressure, and the most stable aggregation state. Ambient temperature was taken as 298.15 K.

At this temperature, most normal alkanes in the nC5-nC17 range are liquids but the heavier are solids.16 From Figure 1, the following correlation was obtained,

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Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010 4π × 10-6 (-0.8187MWj - 3.0404) S 1000Vj0

(13)

4π × 10-6 (-0.8187MWj - 3.0404) 1000VLj0

(14)

k(j) )

or k(j) )

where VSj0 and VLj0, for solid n-paraffins, are calculated from correlations given by Nichita17 VSj0 )

MWj 836.933 17 + 0.014 46MWj -

1.568 56 × 104 MWj

(15) VLj0 )

RTci [1+(1-Tri)2/7] (0.309 816 - 0.006 325 18V1/3 (16) ci ) Pci

The correlation obtained in Figure 1 indicates that each group -CH2- contributes approximately -11.4618 () -0.8187 × 14) to the averaged diamagnetic susceptibility, and the constant -3.0404 represents the contribution of the two hydrogen atoms at the extremes of the n-paraffin. In addition,


Figure 8. SHF1 mixture: critical temperature versus magnetic field intensity.

99.99% of the susceptibility variance may be explained from the carbon chain-length variation. For SHF1and SHF2, the magnetic susceptibilities for pure components were taken directly from Lide.15 Figures 2 and 3, Figures 4 and 5, and Figures 6 and 7 for M1-M2 and M3 respectively show the trend of the critical properties in a wide range of magnetic field intensity from 0 to 1000 T predicted by our model. This very wide range was used with the exclusive purpose of exploring the theoretical trend of the phase diagrams and critical properties with huge magnetic fields. The critical properties for M1-M3 (oils) generally increase with the applied field. For these mixtures, the critical properties at first exhibit a decreasing trend with the applied field for field values from 0 up to ∼2-3 T and then a clear increasing behavior is shown. Both the original Heidemann-Khalil algorithm but with multiple initial guesses and the recently modified version developed by Hoteit et al.18 were used for calculating critical points. Figures 8-11 show the critical properties behavior for the gas-condensates SHF1-SHF2. In contrast to mixtures M1-M3, these fluids always show an increasing trend of their critical properties in a complete range of applied fields. We found the magnetic contribution to the critical pressure


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Figure 9. SHF1 mixture: critical pressure versus magnetic field intensity.

Figure 10. SHF2 mixture: critical temperature versus magnetic field intensity.

Figure 11. SHF2 mixture: critical pressure versus magnetic field intensity.

completely negligible. This contribution increases only from 0 to approximately 1 × 10-2 bar when the field is varied from 0 up to 1000 T.

Finally, the effect of magnetic fields on the gas-liquid coexistence region, directly related with the variation of the respective spinodal curves, is presented. Figures 12-14 for mixtures M1-M3

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Figure 12. M1 mixture: spinodal curve variation (gas-liquid region) with magnetic field intensity.



and Figures 15 and 16 for SHF1-SHF2 clearly show that a strong magnetic field tends to increase the gas-liquid region favoring

the gas-liquid phase separation. However, this last effect is higher in oil than in gas- condensate mixtures.


Figure 15. SHF1 mixture: spinodal curve variation (gas-liquid region) with magnetic field intensity.

Figure 16. SHF2 mixture: spinodal curve variation (gas-liquid region) with magnetic field intensity.

Both the increase in critical temperature to strong magnetic fields and the fact that the model predicts an increasing gas-liquid coexistence region as the magnetic field magnitude also increases are in close agreement with the results obtained using different theoretical models of physics, particularly the mean field theory (MFT). Conclusions A thermodynamic framework for describing the phase equilibria for multicomponent (nonferromagnetic) hydrocarbon mixtures under the influence of a static magnetic field was used for predicting the tendency of the critical properties and the gas-liquid region behavior for three synthetic oil mixtures and two synthetic gas condensates. For the oil mixtures, the critical properties first exhibit a decreasing behavior for fields from 0 up to ∼2-3 T and then these properties increase monotonously. For the gas condensates mixtures, a monotonic increasing behavior was always found. Also, it was clearly predicted that a strong magnetic field tends to increase the gas-liquid region favoring the gas-liquid phase separation. These results are in close concordance with the results obtained by using different theoretical models of physics. Acknowledgment The authors are thankful to the authorities of both ECOPETROL S.A. and the IMP for permission to publish this work.

Nomenclature A ) Helmholtz energy C ) definition in eq 2 f ) fugacity H ) magnetic field strength, Amperes/meter κ ) volumetric susceptibility (dimensionless) n ) moles NC ) number of components m ) mixture property P ) pressure Q ) definition in eq 1 R ) universal gas constant T ) temperature, transpose V ) total volume V ) molar volume νj ) partial molar volume x ) mole fraction Greek Letters δ ) molar susceptibility λj ) definition in eq 8 of ref 7 µ ) permeability Subscripts i, j, k ) component indexes m ) mixture 0 ) vacuum Superscripts

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EoS ) equation-of-state mag ) magnetic contribution nonmag ) nonmagnetic contribution

V ) V*nt* ) Vnt S V

Appendix: Evaluation of Numerical Derivatives Michelsen2 suggested an alternative form to numerically calculate the triple sum, eq 3, using directional derivatives, NC

NC

( ) 2

NC

∂ ln fi RT ∂nj∂nk k)1

∑∑∑ i)1 j)1

T,V,H,nl*j,k NC

∆ni∆nj∆nk )

NC

∑∑ i)1 j)1

[

NC

∆ni∆nj

∑ k)1

( ) ∂qij ∂nk

]

∆nk

T,V,H,nl*k

( ) ∂ ln fi ∂nj

ln fi[-2h] - 8 ln fi[-1h] + 8 ln fi[-1h] - ln fi[+2h] (A8) 12h where the following abbreviation is introduced

(

∂ ln fi ∂nj

(A1) T,V,H,nl*j

The sum over k is numerically determined using the following abbreviate relation, here s is the directional increment or step. NC

∑ k)1

=

T,V,H,nl*j

ln fi[kh] ) ln fi T, V* )

( )

( ) ∂qij ∂nk

∆nk ) lim sf0

T,V,H,nl*k

(

)

∂[qij(n + s∆n, T, V)] ≡ qij* ∂s (A2)

then NC NC NC

∑∑∑ i)1 j)1 k)1

( )

RT

∂2 ln fi ∂nj∂nk

V 1 + kh′

(

ni*(k) ) ni + kε NC NC

∑ ∑ ∆n ∆n q * ) 0 (A3) i

j ij

i)1 j)1

Following to Stockfleth and Dohrn,12 it is clear that fugacity derivatives with respect to mole numbers are required, keeping temperature total volume and the mole number of other species constant. However, the fugacity is a function of intensive properties: temperature, molar volume, and molar fractions. Then for evaluating these derivatives, numerically, first the mole fractions, zi, must be converted to mole numbers ni by multiplying them by an arbitrary total mole number, nt (i.e., nt ) 1). ni ) ntzi

(A4)

(A5)

Producing modified molar compositions given by ni* zi* ) nt + h

zi*(k) )

)

(A9)

)

∆ni n nt t

(A10)

ni*(k) ni*(k)

(A11)

with NC

nt*(k) )

∑ n *(k)

(A12)

i

i)1

The constant total volume condition produces again a modified molar volume given by V*(k)nt*(k) ) Vnt S V*(k) ) V

nt nt*

(A13)

The directional derivative in eq A3 is evaluated using again a four-point scheme, qij* )

(A6)

Here h was chosen such that the analytical and numerical derivatives without magnetic field contribution given are practically the same numerical values, with h ) 0.000 01 was enough. As the total volume is taken as a constant in the derivation, then if the total moles are increased by h the respective molar volume will decrease as

n* 1 + kh

where k is a counter for numerical differentiation and takes on the values -2, -1, 1, and 2. The modified molar fractions are then given as

The mole number of component i is perturbed by a small finite increment in order to obtain a derivative with respect to ni. ni* ) ni + h

z* )

For evaluating the triple sum represented by the directional derivatives in eq A1, a similar procedure to that followed by Stockfleth and Dohrn12 was used. However, in this last case the mole numbers of all components are varied simultaneously. Again, the mole fractions, zi, are converted to mole numbers by multiplying them by an arbitrary total mole number as in eq A4. Here the mole numbers are varied by adding the respective change ∆ni and multiplying by a small number ε. Here ε was also chosen to reduce the difference between the analytical and numerical second derivatives with no magnetic field contribution. The ε ) 0.0001 value was enough. The modified mole numbers are calculated as

T,V,H,nl*j,k

∆ni∆nj∆nk )

(A7)

Finally, the four-point scheme suggested by Stockfleth and Dohrn12 to reduce the error in the numerical derivation was used here,

where qij ≡

nt nt )V nt* nt + h

)

(

∂[qij(n + s∆n, T, V)] ∂s

)

qij[-2] - 8qij[-1] + 8qij[+1] - qij[+2] (A14) 12ε

with qij(k) ) qij[T, V*(k), n*(k)] )

∂ln fi (k) ∂ni

(A15)


In the derivation of eq 12, from eq 1,

1 Pmag = - µ0k(m)H2 2

NC

∑ µ˜ dn

dA ) -S dT - P dV + H d(JV) +

i

(A16)

i

i)1

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(A27)

With the use of meter, kilogram, and second (MKS) units, eq A27 is in Pascal.

then Literature Cited

NC

dA ) -S dT - P dV + H(J dV + V dJ) +

∑

µ˜ i dni

i)1

(A17) or NC

dA ) -S dT - P dV + HJ dV + HV dJ +

∑ µ˜ dn i

i

i)1

(A18) With the use of dJ ) µ0k(m) dH in eq A18, NC

dA ) -S dT - (P - HJ) dV + HVµ0k(m) dH +

∑ µ˜ dn i

i

i)1

(A19) At constant T, V, n dA ) HVµ0k(m) dH

(A20)

with integration,

∫A dA ) ∫ A

H

H)0

0

(A21)

Vµ0k(m)H dH

then 1 A ) A0 + Vµ0k(m)H2 2

(A22)

A0 ≡ Helmholtz free energy without magnetic field. Derivation with respect to V, ∂A ( ∂V )

)

T,H,n

( ) ∂A0 ∂V

T,n

( )]

[

∂k(m) 1 + µ0 k(m) + V 2 ∂V

T,n

( )]

[

∂k(m) 1 -P ) -P0 + µ0 k(m) + V 2 ∂V

T,n

H2

H2

(A23) (A24)

P0 ≡ pressure without magnetic field.

[

( )]

∂k(m) 1 P ) P0 - µ0 k(m) + V 2 ∂V

H2

T,n

(A25)

ReceiVed for reView June 8, 2009 ReVised manuscript receiVed May 10, 2010 Accepted June 14, 2010

where

[

( )]

∂k(m) 1 Pmag ) - µ0 k(m) + V 2 ∂V

T,n

H2

(1) Omelyan, P.; Mryglod, I. M.; Folk, R.; Fenz, W. Ising fluids in an external magnetic field: An integral equation approach. Phys. ReV. E 2004, 69, 061506(1). (2) Lado, F.; Lomba, E. Heisenberg Spin Fluid in an External Magnetic Field. Phys. ReV. Lett. 1998, 80, 3535. (3) Kawasaki, T. Effects of Magnetic Interactions on Phase Separation in Binary Alloys. Prog. Theor. Phys. 1977, 58, 1357. (4) Shinagl, F.; Iro, H.; Folk, R. Magnetic fluids in an external field. Eur. Phys. J. B 1999, 8, 113. (5) Lomba, E.; Weiss, J. J.; Almarza, N. G.; Bresme, F.; Stell, G. Phase transitions in a continuum model of the classical Heisenberg magnet: The ferromagnetic systems. Phys. ReV. E 1994, 49, 5169. (6) Sokolovska, T. G.; Sokolovskii, R. O. Effect of an external magnetic field on the gas-liquid transition in the Heisenberg spin fluid. Phys. ReV. E 1999, 59, R3819. (7) Canas-Marin, W. A.; Ortiz-Arango, J. D.; Guerrero-Aconcha, U. E.; Lira-Galeana, C. Thermodynamics of Wax Precipitation under the Influence of Magnetic Fields. AIChE J. 2006, 52, 2887. (8) Canas-Marin, W. A.; Ortiz-Arango, J. D.; Guerrero-Aconcha, U. E. Prediction of Magnetic Susceptibilities in Diamagnetic Binary Fluid Mixtures. Int. J. Thermophys. 2008, 29, 634. (9) Heidemann, R. A.; Khalil, A. M. Calculation of Critical Points. AIChE J. 1980, 26, 769. (10) Peng, D. Y.; Robinson, D. B. A new two constant equation. Ind. Eng. Chem. Fundam. 1976, 15, 59. (11) Michelsen, M. L. Calculation of phase envelopes and critical points for multicomponent mixtures. Fluid Phase Equilib. 1980, 4, 1. (12) Stockfleth, R.; Dohrn, R. An algorithm for calculating critical points in multicomponent mixtures which can easily be implemented in existing programs to calculate phase equilibria. Fluid Phase Equilib. 1998, 145, 43. (13) Stamataki, S.; Magoulas, K.; Tassios, D. Prediction of Phase Behavior and Physico-Chemical Properties of HT-HP Reservoir Fluids. SPE International Symposium on Oilfield Chemistry, Houston, TX, February 18-21, 1997; SPE Paper 37294. (14) Ungerer, P.; Faissat, B.; Leibovici, C.; Zhou, H.; Behar, E. High Pressure-High Temperature Reservoir Fluids: Investigation of Synthetic Condensate Gases Containing a Solid Hydrocarbon. Fluid Phase Equilib. 1995, 111, 287. (15) Lide, D. R. Handbook of Chemistry and Physics; CRC Press: New York, 2003. (16) Firoozabadi, A. Thermodynamics of Hydrocarbon ReserVoirs; McGraw Hill: New York, 1999. (17) Nichita, D. V.; Goual, L.; Firoozabadi, A. Wax precipitation in Gas Condensate Mixtures. SPE Prod. Facil. 2001, 250. (18) Hoteit, H.; Santiso, E.; Firoozabadi, A. An Efficient and Robust Algorithm for the Calculation of Gas-Liquid Critical Point of Multicomponent Petroleum Fluids. Fluid Phase Equilib. 2006, 241, 186.

(A26)

IE900939Y

Calculation of Critical Points in Gasâ Condensate Mixtures under the

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