Energy & Fuels 2008, 22, 775–783
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Pseudocomponent Delumping for Multiphase Systems with Waxy Solid Phase Precipitation† Dan Vladimir Nichita,*,‡ Jerome Pauly,‡ Francois Montel,§ and Jean-Luc Daridon‡ Laboratoire des Fluides Complexes, UMR CNRS 5150, UniVersité de Pau, BP 1155, 64013 Pau Cedex, France, and TOTAL, CSTJF, AVenue Larribau, F-64018 Pau, France ReceiVed July 25, 2007. ReVised Manuscript ReceiVed September 21, 2007
Modeling of wax precipitation from hydrocarbon mixtures requires extended compositional data for the heavy fractions; as a result, the number of components in the mixture is usually large and phase equilibrium calculations are computationally expensive and may be prohibitive. A lumping procedure which reduces the dimensionality of phase equilibrium calculations without affecting the location of solid phase transitions is used. By lumping into pseudocomponents, some information on mixture behavior is lost; this information is recovered by a delumping (inverse lumping) procedure. Delumping acts as an interface between simulation tools and results involving different fluid representation levels. The key in the delumping procedure is to relate equilibrium constants of the detailed fluid to some quantities evaluated from lumped fluid flash results. A general form of a two-parameter cubic equations of state is used for vapor and liquid phases, and heavy components are assumed to precipitate in a single solid solution. The proposed lumping/delumping procedures are successfully tested for two synthetic mixtures and a typical reservoir fluid (gas-condensate). Very good agreement is obtained between phase distributions and component mole fractions of detailed and delumped systems. This is the first time that a delumping procedure is proposed for multiphase systems with solid phase precipitation.
Introduction Modeling of wax precipitation from hydrocarbon mixtures requires extended compositional data (including the paraffin–naphtene–aromatic distribution) for the heavy fractions; the full composition of the heavy normal-paraffins have to be accounted for to properly model the onset of wax precipitation.1 As a result, the number of components in the mixture is usually large and phase equilibrium calculations are computationally expensive (computer time increases at least with the square of component number). Modern characterization techniques can provide extended compositions up to high carbon numbers; using such a detailed mixture description leads, however, to prohibitive computer times and calculations may be affected by computational problems specific to large systems. It is a common practice to group together (lump) individual components (based on certain proximity criteria2,3) some (most, if not all) of the components into pseudocomponents; for the lumped mixture, the dimensionality of the phase equilibrium problem and thus the computational effort are reduced, but the thermodynamic behavior of the mixture is altered by lumping. Generally, a compromise between accuracy and computational speed needs to be accepted. The phase composition of the detailed mixture can be later inferred from flash calculations † Presented at the 8th International Conference on Petroleum Phase Behavior and Fouling. * Corresponding author. E-mail:
[email protected]. ‡ Université de Pau. § TOTAL, CSTJF. (1) Daridon, J. L.; Pauly, J.; Coutinho, J. A. P.; Montel, F. Energy Fuels 2001, 15, 730–735. (2) Montel, F.; Gouel, P. SPE 1984, 13119. (3) Lin, B.; Leibovici, C. F.; Jørgensen, S. B. Comput. Chem. Eng.,published online May 8, http://dx.doi.org/10.1016/j.compchemeng.2007.04.021.
performed on a lumped mixture via so-called delumping (or inverse lumping) procedures.4–6 The key in the delumping procedure is to relate equilibrium constants of the detailed fluid to some quantities evaluated from lumped fluid flash results. By lumping into pseudocomponents, some information is lost; this information is (at least partially) recovered by delumping. Delumping acts as an interface between simulation tools and results involving different fluid representation levels, such as reservoir compositional simulators (for which the acceptable number of components is limited, usually less than ten) and surface facilities simulators (for which extended descriptions of mixture composition are required). Recently, based on a reduction method,7 we have extended the analytical delumping procedure of Leibovici et al.4 to systems with nonzero binary interaction parameters (BIPs) in the equation of state (EoS)8 and then applied the new delumping approach to hydrocarbon reservoir fluids subject to various processes9 and finally to multiphase vapor–liquid–liquid systems.10 In all cases, very good agreement was obtained between equilibrium constants, mole fractions, and phase distributions of detailed and delumped fluids. (4) Leibovici, C. F.; Stenby, E. H.; Knudsen, K. Fluid Phase Equilib. 1996, 117, 225–232. (5) Rabeau, P.; Gani, R.; Leibovici, C.F. Ind. Eng. Chem. Res. 1997, 36, 4291–4298. (6) Leibovici, C. F.; Barker, J. W.; Waché, D. SPE J. 2000, 5, 227– 235. (7) Nichita, D. V.; Minescu, F. Can. J. Chem. Eng. 2004, 82, 1225– 1238. (8) Nichita, D. V.; Leibovici, C. F. Fluid Phase Equilib. 2006, 245, 71–82. (9) Nichita, D. V.; Broseta, D.; Leibovici, C. F. J. Pet. Sci. Eng. 2007, 59, 59–72. (10) Nichita, D. V.; Broseta, D.; Leibovici, C. F. Comput. Chem. Eng. 2006, 30, 1026–1037.
10.1021/ef700439q CCC: $40.75 2008 American Chemical Society Published on Web 12/04/2007
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To the best of our knowledge, a delumping procedure for phase equilibrium calculations involving a solid phase has not been reported yet in the literature; such an approach is more challenging than delumping vapor–liquid or vapor–liquid–liquid flashes because different thermodynamic models are used for the fluid and solid phases. Moreover, available lumping procedures may not be appropriate for grouping components which can precipitate. Improper lumping of heavy components can dramatically influence solid phase boundaries, i.e., wax appearance temperatures (WAT), or cloud points. In this work, we focus on delumping the results of threephase vapor–liquid–solid flash calculations performed on lumped mixtures in which precipitating components are grouped together. A general form of a two-parameter cubic EoS (containing the Peng–Robinson11 and Soave–Redlich–Kwong12 EoS) is used for vapor and liquid phases, and heavy components are assumed to precipitate in a single solid solution. The paper is structured as follows: we first present the thermodynamic model for wax precipitation, and then, the lumping and delumping procedures for equilibrium states with a waxy solid phase are described. The proposed lumping/ delumping procedures are successfully tested for two synthetic mixtures and a typical reservoir fluid (gas-condensate), and finally, conclusions of this work are drawn. Thermodynamic Model. A model based on the formation of a solid solution that uses cubic equations of state with the LCVM mixing rule for the description of the fluid phases and the Wilson equation or the UNIQUAC model13–15 for describing the solid solution nonideality has been developed and was shown to accurately predict the onset of crystallization as well as the amount of solid precipitate as a function of temperature and pressure in both synthetic systems16,17 and reservoir fluids.1 Unfortunately, this last model, which calculates with a good accuracy the fluid–solid phase equilibrium, is difficult to implement in multiflash algorithms used in the petroleum industry due to the use of GE mixing rule as well as group contribution. To overcome these shortcomings, a model with the same predictive capacity but easier to implement in conventional petroleum simulators has been developed.18 From this perspective, the GE mixing rule has been replaced by the quadratic mixing rule. This change of mixing rule modifies however the fluid-phase fugacity leading to a shift of the calculated wax appearance conditions. To handle this situation, the solid fugacity was also changed to keep the quality of the liquid–solid equilibrium description by introducing a new parameter, ξ, in the GE model used for the description of the nonideality of the solid phase. Model Development. Liquid–solid, liquid–vapor, and liquid–vapor–solid phase equilibria under pressure may be described by the equality of the fugacity of individual component in all the different phases (11) Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59–64. (12) Soave, G. Chem. Eng. Sci. 1972, 27, 1197–1203. (13) Coutinho, J. A. P.; Gonçalves, C.; Marrucho, I. M.; Pauly, J.; Daridon, J.-L. Fluid Phase Equilib. 2005, 233, 28–33. (14) Coutinho, J. A. P.; Pauly, J.; Daridon, J.-L. Braz. J. Chem. Eng. 2001, 18, 411–422. (15) Coutinho, J. A. P.; Mirante, F.; Pauly, J. Fluid Phase Equilib. 2006, 247, 8–17. (16) Pauly, J.; Daridon, J.-L.; Coutinho, J. A. P.; Lindeloff, N.; Andersen, S. I. Fluid Phase Equilib. 2000, 167, 145–159. (17) Pauly, J.; Daridon, J.-L; Coutinho, J. A. P. Fluid Phase Equilib. 2004, 224, 237–244. (18) Sansot, J.-M.; Pauly, J.; Daridon, J.-L.; Coutinho, J. A. P. AIChE J. 2005, 51, 2089–2097.
Nichita et al.
fVi (T, P, xVi ) ) fLi (T, P, xLi ) ) fSi (T, P, xSi );
i ) 1, nc
(1)
Using the liquid phase as the reference phase, the equilibrium ratios KiV and KiS can be defined as KVi )
KSi )
φLi (P, T, xL)
(2)
φVi (P, T, xV) φLi (P, T, xL)
(3)
φSi (P, T, xS)
where φiV, φiL, and φiS correspond respectively to the fugacity coefficients in the vapor, liquid, and solid phase. Among the available cubic equations of state, the Peng–Robinson EoS11 which provides a good description of high pressure vapor–liquid equilibria of asymmetric light/heavy hydrocarbons systems with only two parameters a and b is used for the estimation of liquid and vapor fugacity coefficients. The classical mixing rules (quadratic for a and linear for b) were used with binary interaction parameters kij evaluated by the correlations proposed by Pan et al.19 The critical properties required in the evaluation of the EoS parameters were obtained from the compilation of Reid et al.20 for light gases, whereas the Twu correlation21 is used to predict the critical properties of the heavy components. To correctly describe the volume of each phase, the volumetric properties calculated by the original Peng–Robinson (PR) EoS are corrected by using a volume translation introduced by Peneloux et al.22 V ) V′ + Ci
(4)
where V′ represents the volume calculated from the PR EoS and Ci is a translation parameter estimated at atmospheric pressure.23 Solid Phase Modeling. The variation with pressure of the fugacity fiwof a heavy paraffin in the waxy phase during an isothermal process can be evaluated by integration of the partial molar volume Vwi between the atmospheric pressure P0 and the pressure P
∫
1 P S V dP (5) RT P0 i The evaluation of this term requires the knowledge of molar ln fSi (P) ) ln fSi (P0) +
volume Vwi . As previously demonstrated, this quantity can be taken as proportional to the liquid molar volume for pure components VSi ) VSi ) βVLi 0
(6)
with a proportionality coefficient β assumed to be pressureindependent and evaluated to be equal to 0.90 for mixtures.16 The Poynting correction can thus be rewritten in terms of pure liquid fugacity at the pressure P as 1 RT
∫
P
P0
VSi dP )
β RT
∫
P
P0
VLi 0 ) β ln
fLi 0(P) fLi 0(P0)
(7)
(19) Pan, H.; Firoozabadi, A.; Fotland, P. SPE Prod. Facil. 1997, 12, 250–258. (20) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (21) Twu, C. H. Fluid Phase Equilib. 1984, 16, 137–150. (22) Peneloux, A.; Rauzy, E.; Frèze, R. Fluid Phase Equilib. 1982, 8, 7–23. (23) Elbro, H. S.; Fredenslund, Aa.; Rasmussen, P. Ind. Eng. Chem. Res. 1991, 30, 2576–2582.
Pseudocomponent Delumping for Multiphase Systems
Energy & Fuels, Vol. 22, No. 2, 2008 777
In eq 5, the fugacity of a paraffin i in this hypothetical waxy phase at the reference pressure P0 is therefore evaluated in terms of activity coefficients γSi fSi (P0) ) xSi γSi fSi 0 (8) S0 where fi , which represents the fugacity of the pure solid at the same temperature and pressure P0, is related to the pure subcooled liquid fugacity fL0 i from the change of free energy between the pure solid and the liquid at temperature T and reference pressure P0.24 ln
fSi 0 fLi 0
)-
(
)
(
∆HSL0 ∆GSL0 T ∆Htr0 T )1 - SL 1 - tr RT RT RT T 0 T 0
)
(9) where and represent respectively the fusion temperature and the enthalpy of fusion whereas Ttr0 is the solid–solid transition temperature and ∆Htr0 the enthalpy of solid–solid transition. Solid Activity Coefficient Model. The atmospheric activity coefficient γiw in eq 8, which takes into account the deviation from the ideal behavior, is calculated using the predictive Wilson equation25,26 which provides a good description of orthorhombic solid solutions phase behavior.27 xkΛki ln γSi ) 1 - ln xjΛij (10) j k xjΛkj TSL0
∆HSL0
∑
∑
∑ j
with
[
]
λij - λji (11) RT where the interaction parameter λii between two identical molecules is estimated from the enthalpy of sublimation, ∆HSV0. Since the interaction between two molecules arises only on the contact surface between them, it has been considered in a previous work25 that the interaction parameter λij is equal to the interaction parameter λii between two shorter molecules. However, this approach does not take into account the end effects, i.e., the interactions arising between chain ends. These can be accounted for by introducing a bending parameter (12) λij ) λii(1 - ξij) In this work, these bending parameters are replaced by a single average parameter ξ λij ) λii(1 - ξ) (13) considered as an empirical parameter used to fit the model to the wax appearance measurements performed at the lowest available pressure. Finally, by combining eqs 5, 7, 8, and 9, the expression of the solid fugacity in solid phase (of composition xS) at pressure P is given by Λij ) exp -
[
fSi (P, xS) ) xSi γSi (xS)[fLi 0(P0)]1-β[fLi 0(P)]βexp -
)
(
∆HSL0 1RT tr0
(
T ∆H T 1 - tr RT TSL0 T 0 Lumping Procedure
)]
equilibrium calculations. A lumping procedure consists in (i) choosing a clustering scheme and (ii) assigning properties to pseudocomponents. It is clear that lumping always affects mixtures thermodynamic behavior, and lumping procedures have to be designed in such a was that differences between flash results on detailed and lumped mixtures are minimized. However, classical lumping procedures based on proximity criteria2,3 may not be appropriate for grouping heavy components which can precipitate. Improper lumping of heavy components can dramatically influence phase boundaries, mainly WATs. Fusion temperature of n-paraffins is increasing exponentially with carbon number, thus the heaviest component controls the WAT. In this work, the heaviest n-paraffin is always kept as an individual component in the lumped mixture. For all other precipitating components, the lighter the components, the larger the number of components in the corresponding group (pseudocomponent). The nonparaffin heavy components (heavy naphtenic, iso-paraffinic, and aromatic components) can be grouped following a classical scheme, or in a single cut as described by Daridon et al.1 This lumping scheme is chosen to preserve both phase boundaries (WATs) and phase distributions. Pseudocomponent critical temperatures and pressures, acentric factors, and BIPs are assigned according to Leibovici’s method,28 which ensures consistency (for a two-parameter EoS, a(lumped) ) a(detailed), and b(lumped) ) b(detailed)); by further applying a delumping method, full consistency is kept, i.e. a(lumped) ) a(detailed) and b(lumped) ) b(detailed).4,8–10 The component properties involved in the solid phase model, that is fusion (Tm and ∆Hm) as well as solid transition (Ttr and ∆Htr) properties can be correlated with molecular mass or carbon number. Pseudocomponent properties are assigned using a polynomial interpolation on data from Pauly et al.16 Note that a high-quality lumping is a prerequisite for a good delumping. Delumping Procedure As a rule, the delumping equations are obtained by expressing the equilibrium constants (KVi ) xVi /xLi and KSi ) xSi /xLi ; i ) 1,nc) of components in the detailed mixture
(24) Prausnitz, J. M., Molecular thermodynamics of fluid phase equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1969. (25) Coutinho, J. A. P.; Knudsen, K; Andersen, S. I.; Stenby, E. H. Chem. Eng. Sci. 1996, 51, 3273–3282. (26) Wilson, G.M. J. Am. Chem. Soc. 1964, 86, 127–130. (27) Pauly, J.; Dauphin, C.; Daridon, J.L. Fluid Phase Equilib. 1998, 149, 191–207.
(15a)
ln KSi ) ln φLi - ln φSi
(15b)
in terms of quantities evaluated from lumped fluid flash results. Vapor–Liquid Equilibrium Constants. KVi are expressed as a function of the reduction parameters.29 A set of nr ) 2m + 2 reduction parameters is defined for each phase (m is the number of components having nonzero BIPs with the remaining ones). The vectors of reduction parameters are Qk ) (Qk,1, ..., Qk,M)T ;
(14)
Lumping individual components into pseudocomponents is a commonly used technique to reduce dimensionality of phase
ln KVi ) ln φLi - ln φVi
k ) L, V
(16)
with nc
Qkj )
∑q x ; k ij i
j ) 1, nr
(17)
i)1
where qij (i ) 1, nc; j ) 1, nr) are the elements of the reduction matrix29 (see the work of Nichita and Minescu7 for the reduction method used in this work); all necessary details on reduction (28) Leibovici, C. F. Fluid Phase Equilib. 1993, 87, 189–197. (29) Hendriks, E. M. Ind. Eng. Chem. Res. 1988, 27, 1728–1732.
778 Energy & Fuels, Vol. 22, No. 2, 2008
Nichita et al.
Table 1. Pseudocomponent Properties for Mixture B PC 1 2 3 4 5
zi
Tci (K)
pci (bar)
ωi
Mi (g/mol)
Tfi (K)
Ttri (K)
Hfi (J/mol)
Htri (J/mol)
k1j
0.054752 0.023600 0.015029 0.006809 0.002709
762.52 796.96 821.03 838.92 848.67
11.65 9.82 8.64 7.82 7.39
0.8678 0.9945 1.0956 1.1782 1.2263
273.33 323.23 365.31 401.24 422.82
306.64 320.13 329.27 335.45 338.70
296.63 312.00 322.70 330.20 334.21
44922.53 52815.42 59470.47 65151.66 69196.00
15518.82 21071.98 25754.23 29751.31 31523.00
0.0805 0.0902 0.0982 0.1044 0.1080
parameters and elements of the reduction matrix can be found in our previous papers on delumping.8–10 The fugacity coefficient of component i from a general form of two-parameter cubic EoS (including PR and SRK EoS) can be expressed as ln φki (Qk) ) h0k(Qk) + hRk(Qk)Ri + hBk(Qk)Bi + m
∑h
i ) 1, nc ;
γjk(Qk)γji ;
k ) L, V (18)
close, due to the regularity of all quantities involved in their calculation (molar distribution and component properties). This is the key of our approach for delumping flashes for multiphase systems with solid precipitation. Information from the less costly flash of the lumped system is used to infer the required quantities for the detailed mixture. Usually, a linear dependence is observed (if the composition of precipitating components obeys a certain distribution function, which is true in most cases), thus eq 20 reads
j)1
where Ri ) (Ai)1/2 and γji ) Ri(l – kji); j ) 1, m; i ) j + 1, nc, and Ai ) aiP/(RT)2, Bi ) biP/RT. Expressions for h functions can be found in refs 8–10. In eq 18, the last terms (under summation) contain the nonzero BIPs. If volume translation is used, the term CiP/RT should be added to eq 18. At given pressure and temperature conditions, component fugacity coefficients depend only on reduction parameters (directly and by means of the compressibility factor) and not on phase composition. This is the basis of reduction methods, with application to a variety of phase equilibrium problems.30–35 If np phases are present at equilibrium, reduction reformulates the problem, from an (np – 1)nc-dimensional compositional hyperspace to an (np – 1)(nr + 1)-dimensional hyperspace defined by the reduced variables and phase mole fractions, with nr
