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Ind. Eng. Chem. Res. 1999, 38, 316-323
Assessment of Activity Coefficient Models for Predicting Solid-Liquid Equilibria of Asymmetric Binary Alkane Systems Ekaterini N. Polyzou,† Panagiotis M. Vlamos,‡ Georgios M. Dimakos,§ Iakovos V. Yakoumis,† and Georgios M. Kontogeorgis*,† IGVP & Associates Engineering Consultants Ltd., 35 Kifissias Avenue, 115 23 Ampelokipi, Athens, Greece, Department of Mathematics, National Technical University of Athens, Zographos Campus, 15780 Athens, Greece, and Department of Primary Education, University of Athens, 13A Navarinou Street, 10680 Athens, Greece
Hydrocarbon fluids play an important role in today’s society, being essential components of oil and fuels. Phase equilibria knowledge of such mixtures is important in designing equipment and processes. Among the least studied equilibriums is that between solid and liquid phases (SLE) which is required in, e.g., wax formation studies. Still, a number of models have been proposed for estimating the activity coefficient but are rarely investigated against SLE data. In this study, a number of “well-established” and “up-to-date sophisticated” activity coefficient models are tested for their capability in predicting solid-liquid equilibria of alkane systems. A large database covering 63 alkane/alkane systems for which SLE data are available is considered. The models include several of the popular approaches proposed during the past 30 years covering the gap of a comprehensive literature review. The capabilities and limitations of the models are discussed and those that provide successful predictions are recommended at various levels of accuracy and complexity. Introduction Numerous activity coefficient models have been proposed over the past 30 years. The activity coefficient approach has been employed as an alternative to the more rigorous equation of state approach in several difficult cases whenever polar components or condensed (solid, liquid) phases or equilibria such as solid-liquid and liquid-liquid are involved. Most attention is paid to vapor-liquid (VLE) and to a certain degree to liquidliquid equilibria (LLE), while solid-liquid (SLE) is often neglected. Still, SLE is important in the paraffin deposition commonly known as wax formation and in crystallization studies as well as in the design of such processes and operations which highly depend on the accuracy of the data employed1 and in applications related to food product and process design.2 Many efforts concentrated in suggesting and developing the various models and less in testing them over extended database or different cases. Often the models are tested over a limited range of conditions while “extreme” cases such as polymers and infinite dilution conditions at both ends (solute and solvent) are rarely covered. Moreover, limited comparisons between the various literature models make it difficult to assess the actual improvement of each suggested procedure over previous ones. This is sometimes understood in the sense of the time devoted in the development of each new model but can mask the actual range of capability and application of every suggested approach. In particular, activity coefficient models are usually employed for VLE calculations together with an equa* Corresponding author. Tel.: 30 1 6982455. Fax: 30 1 6982314. E-mail:
[email protected]. † IGVP & Associates Engineering Consultants Ltd. ‡ National Technical University of Athens. § University of Athens.
tion of state for the vapor phase and for LLE calculations. These models are usually tested against vapor phase compositions and pressure in VLE or binodal curves in LLE and sometimes directly against experimental activity coefficient data. However, the models are rarely tested against SLE compositions directly and when SLE examples are involved usually include activity coefficient data at solute infinite dilution obtained from extrapolation of the original SLE data. However, such activity coefficients may are in up to 30% error,3 and thus consideration of the actual SLE data for comparison should be preferred. In this work, a large number of activity coefficient models are considered for direct SLE calculations. The models and database are discussed in brief in the next section while the database is presented in some detail in the Appendix (Table 6). Due to the nature of the models and since the systems involve only alkanes, all the calculations are straight predictions (they do not involve adjustable parameters) thus permitting a fair and suitable comparison between the various models as well. Following the database section, a thorough discussion of the capabilities and limitations of the models is provided ending with the recommendations and the conclusions. Activity Coefficient Models and Database for SLE The principles of SLE calculations are discussed in numerous well-established thermodynamic monographs4 and will, thus, not be repeated here. To perform SLE the essential inputs involve a number of properties of the solute (solid) with the melting point (Tm) and the enthalpy of fusion (∆Hm) being the most important. Whenever single or multiple transition points exist in the solid phase, the temperatures and enthalpies of these phase transitions are required as well (Ttr, ∆Htr). These data are given in the Appendix (Table 7) for the
10.1021/ie9803826 CCC: $18.00 © 1999 American Chemical Society Published on Web 12/08/1998
Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 317
solutes involved in this study. Only n-hexatriacontane includes two transition points prior to melting temperature. The general equation after some simplifying assumptions is
ln x2 ) -
(
∆Hm2 Tm2 RTm2 T
)
-1 +
)
1 -
∑ T tr
(
∆Cpm
T
+
Tm2
R Tm2 T ∆Htr2 Ttr2 - 1 - ln γ2 (1) RTtr2 T ln
( )
where x2 is the solubility and γ2 the solute activity coefficient which needs to be determined from a model. ∆Cpm is the heat capacity of melting at the melting point. The rest of the properties of eq 1 also correspond to the solute and were mentioned previously. All the terms in eq 1 are not of the same importance but all have some effect in the calculated solubilities or, given the solubility values, in the calculated experimental activity coefficients. For example, if the solid transition term is not considered, then for some systems, e.g. heptane/nonadecane, the calculated “experimental” activity coefficients from eq 1 are erroneously obtained to be higher than unity,5 while it is well-known that in these systems negative deviations from Raoult’s law are typically observed. If the transition term is included, this inconsistency is eliminated. In some cases, it is possible to perform SLE calculations using the above information and the assumption of ideal solubility in the liquid phase (activity coefficient equal to unity). Indeed this might be correct for alkane solutions of similar size. In many cases, however, the two alkanes differ substantially in size/shape, and/or very small solute concentrations are involved and therefore large negative deviations from Raoult’s law are observed (activity coefficients much below unity), while other effects such as order/disorder phenomena further complicate the solution’s behavior.6 In these cases, a thermodynamic model, e.g. equation of state or activity coefficient, is required for modeling the nonideality and for performing reliable SLE calculations. Traditionally, the Flory-Huggins approach and the solubility parameter theory have been employed in the past in wax formation studies but with limited success and often at the cost of adjustable parameters tailored to each specific system.1 In this work we focus on activity coefficient models of the general type
+ ln γres ln γi ) ln γcomb-fv i
(2)
The two terms comb-fv (combinatorial-free volume) and res (residual) represent the major contributions expected to arise from solution nonidealities, those from differences in size/shape and energetic forces, respectively. In activity coefficient terminology, the “energetic” effects are often called “residual”, though dominating in several cases (e.g. in the presence of polar compounds) in accordance to the established importance of combinatorial (size/shape) effects in the liquid state. The comb-fv and res terms may be considered equivalent to the repulsive and attractive terms of equation of state.7 In alkane solutions the energetic effects can be approximately neglected and thus the total activity coefficient in eq 2 equals only the comb-fv contribution.
Table 1. Description of the Models Considered in This Study model
Si
ref
Flory-Huggins UNIFAC Flory-Huggins/vol modified UNIFAC SUPERFAC ASOG Dortmund UNIFAC Dortmund UNIFAC-SG Masuoka Masuoka without SG R-UNIFAC R-UNIFAC without SG Sheng
ri ri + SG Vi ri2/3 ri2/3 + SG Nci (number of carbon atoms) ri3/4 + SG ri3/4 ri, 0.6583r2 + SG ri, 0.6583r2 riR, R ) 0.9 (1 - (r1/r2)) riR, R ) 0.9 (1 - (r1/r2)) riPi, Pi ) 1 - [1/(0.8(i - 1)0.5 + 1)] i ) ri1/3(qi/ri) V2/3 Vfi, Vfi ) Vi - Vwi VfiPi, Pi ) 1 - (r1/r2) Vfi, Vf ) (V1/3 - Vw1/3)3C, c ) 1.1 Vfi, Vi, Pi ) 1 - (2/(r2 - r1)) ci ) ri/qi
20 9 20 21 22 23 24
V2/3 entropic-FV p-FV Flory-FV chain-FVa
a
(
ln γchain-FV ) ln i
) ( ) (
φifv φivol φivol +1+ ci ln vol + xi xi φi cmix
25 14 19 18 12 12 13
)
φivol φifv . xi xi
Since this study involves alkane systems, we focus here on the evaluation of various functions for the combinatorial/free volume term covering “old” and “new” concepts and theories (models). Most models have the general form
φi φi ln γcomb-fv ) ln + 1 - ( SG i xi xi
(3)
where the concentration function is given by the equation
∑j xjSj
φi ) xiSi/
(4)
An exception to the general form (eq 3) is the chainFV model described in detail previously,13 where both volume and free-volume fractions are employed. A system-dependent exponent is used in the free-volume fractions. The SG (Staverman-Guggenheim) correction term, added in many models, is believed to take into account the effect of differences in shape of the molecules:
(
)
φseg φseg i i z +1SG ) - qi ln 2 ϑi ϑi
(5)
where
) xiri/ φseg i
∑j xjrj;
∑j xjqj
ϑseg ) xiqi/ i
(6)
z, the coordination number, is typically taken to be equal to 10; ri and qi are the van der Waals volume and surface area, respectively, estimated from the group increments given originally by Bondi8 and available in all UNIFAC tables for practically all groups.9 Table 1 gives a summary of the various models belonging to the type of eq 3 for various values of the
318 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 Table 2. Special Features of the Models Considered in This Studya model Flory-Huggins UNIFAC Flory-Huggins/vol modified UNIFAC SUPERFAC ASOG Dortmund UNIFAC Dortmund UNIFAC-SG Masuoka Masuoka - SG R-UNIFAC R-UNIFAC-SG Sheng V2/3 Entropic-FV p-FV Flory-FV chain-FV a
includes includes system includes applied to applied to agreed V/FV exponent dependent exponent SG polymers (dev < 50%) multicomp systems with simuln N N Y N N N N N N N N N N Y Y Y Y Y
N N N Y Y N Y Y N N Y Y Y Y N Y Y Y
N N N N Y Y Y N Y N Y
N Y N N Y N Y N Y N Y N N N N N N N
N N Y N N N N N Y Y Y N Y Y Y Y
Y Y Y Y Y Y Y Y Y Y N N Y Y Y N Y N
N N N N N N N N Y Y N Y Y Y Y
N ) no, Y ) yes.
parameter Si. Table 2 gives an account of the models presented in Table 1 in a different manner, i.e., considering a number of key elements which differentiate the various types of models and theories: (a) whether the model includes volume and/or freevolume terms; if volume terms are included, the volumetric properties for the low molecular weight alkane solvents are calculated from the DIPPR compilation10 while for the heavy alkane solutes from the GCVOL group contribution method;11 (b) whether the model includes exponent in the parameter Si and if this exponent is constant or system dependent; (c) whether the model includes the StavermanGuggenheim (SG) correction term (eq 5); (d) whether the model can be successfully applied in the limiting case of polymers as established in previous related research;12-14 (e) whether the model can be rigorously extended to multicomponent systems as established in previous related research;15 (f) whether the model is in relative agreement with recent molecular simulation studies for model alkane systems16 and their interpretation by Kontogeorgis et al.17 The above key elements, discussed extensively in previous studies, are important since they differentiate the various models and theories, despite their superficial similarity, and provide a number of criteria for choosing the best model as well as for assessing the capabilities and limitations of each approach over a range of conditions and applications. SLE Database The SLE database employed in this study is given in tabular form in the Appendix (Table 6). For each system are shown the temperature and composition range with respect to the heavy alkane (solute) as well as the source of the experimental data. The references of the data are provided at the end of Table 6. The database involves solutes from n-hexadecane to n-hexatriacontane and small alkane solvents including n-alkanes, and cyclo and branched alkanes. Results and Discussion To better evaluate and compare the performance of the models both against the experimental data, each
Figure 1. Experimental and calculated solid-liquid equilibrium curves for n-C32/n-pentane.
other and against the ideal solution assumption, Table 3 gives a list of the systems examined where besides the conditions are also shown the values of experimental activity coefficients and the ratio of the van der Waals volumes of the components (R2/R1). The experimental activity coefficients obtained from SLE are calculated from the experimental solubilities using eq 1. In addition, in Table 3 each system is assigned a characterization identification depending on its asymmetry (with respect to size, concentration range, and activity coefficient values). Basically, in most cases the following distinction is followed: S (symmetric): activity coefficient values >0.8; MA (medium asymmetric): activity coefficient values >0.7; A (asymmetric): activity coefficient values
