Multicomponent phase equilibrium calculations for associating mixtures Georgios M. Kontogeorgis1,*, Iakovos V. Yakoumis1, Henk Meijer2, Eric Hendriks2 and Tony Moorwood3 1
IGVP & Associates Engineering Consultants Ltd.
35 Kifissias Ave., 115 23 Ampelokipi, Athens, Greece E-mail:
[email protected] 2
Shell Research and Technology Centre, Amsterdam
P.O. Box 38000, 1030 BN Amsterdam, The Netherlands 3
Infochem Computer Services Ltd.
South Bank Technopark, 90 London Road, London SE1 6LN, United Kingdom Keywords: equation of state, liquid-liquid equilibria, association, multicomponent, alcohol, water Abstract Prediction of phase equilibrium for multicomponent systems containing associating compounds (e.g. water and alcohols) is essential in a number of engineering applications (e.g. environmental technology, gas hydrate inhibition) and at the same time represents one of the most stringent tests for a thermodynamic model. Conventional models (e.g. cubic equations of state and excess Gibbs free energy models) provide rapid and often reliable estimates of phase equilibrium in many cases but extension to multicomponent systems, especially those containing water is often troublesome. On the other hand, novel association equations of state perform considerably better but are slower compared to conventional models. Furthermore, the extension of several of them to cross-associating systems (e.g. water-alcohols) exhibits problems. In this work, the performance of two well-known conventional models (SRK and NRTL) is compared for multicomponent systems to a recently proposed association
equation of state both in terms of accuracy of predictions and timing. The proposed model incorporates the Wertheim chemical association theory (employed previously in models such as SAFT) and the SRK equation. The model is applied in this work to multicomponent systems in such a way that the inclusion of the Wertheim theory does not give execution times much higher than conventional models. The model yields very satisfactory predictions of multicomponent equilibria for aqueous (both vapor-liquid and liquid-liquid equilibria) systems containing methanol, gases and hydrocarbons; the results are considerably better than those from SRK and NRTL equations.
Introduction The association model which is employed in this work is based on earlier work on pure associating compounds and binary VLE and LLE for alcohols and hydrocarbons ([1], [2], [3]). The model is acronymed by the authors who originally developed it as CPA (Cubic Plus Association). Recently the model has been extended to aqueous hydrocarbon systems ([4]). The CPA equation of state is described in detail in these publications and is not repeated here. There is a difference between the model as described previously and as applied in this work in the equation used for the radial distribution function (rdf) which is employed in the term containing the association strength. SAFT and CPA (in the previous publications) employ the hard-sphere radial distribution function: g (d ) =
2− y 2(1 − y )
3
,
y=
b 4V
(1)
which, in the case of CPA, is only approximate [1] since the latter employs the vdW and not the hard-sphere repulsive term (the latter is used in SAFT). For simplification and timing reasons, the CPA as employed in this work make use of a simplified-hard sphere (hs) rdf form first proposed by Elliott et al. ([5]): g( d ) =
b 1 , y= 4V 1 − 19 . y
(2)
The two rdfs are compared graphically in figure 1. It is observed that they yield very similar values, yet the simplified-hs rdf is much simpler (also in terms of computational time) and is adopted in this work. CPA with equation (2) will be hereafter denoted as sCPA (simplified CPA).
Finally, in accordance to the previous CPA studies, the two-site association scheme is employed for methanol, the four-site association scheme for water while hydrocarbons are considered to be inert (no association sites). The sCPA pure component parameters are listed in Table 1. The sCPA model is, extended in this work to cross-associating systems (e.g. water/alcohols) and to multicomponent equilibria for water/alcohol/hydrocarbons. The extension is accomplished in such a way that the inclusion of the Wertheim theory does not give execution times much higher than conventional models. This is accomplished by adopting a suitable previously proposed combining rule for the association strength of mixtures ([6]). The following combining rule is adopted for the association strength (hereafter referred for simplicity as the Elliott Rule): ∆ ij = ∆ i ∆ j
(3)
Results and Discussion Figures 2-6 present graphically results for characteristic binary systems with the sCPA equation of state. These results include binary vapor-liquid, liquid-liquid and gasliquid equilibria (in case of supercritical components). In all cases only a single interaction parameter per binary is used which is estimated from the specific binary data. The kij-parameter is the correction to the geometric mean rule for the cross-energy parameter :
(
a ij = a i a j 1 − k ij
)
(4)
Thus, the only binary parameter of sCPA is included in the physical term, while the extension of the Wertheim association term to mixtures does not require any specific mixing rules. The only assumption of this term is the combining rule for the association strength (equation 3) which substitutes the need for specific combining rules for the two association parameters (energy and volume) separately. Figures 7-9 show multicomponent results with sCPA and two conventional models (SRK and NRTL). The default versions of NRTL and SRK (a modified version of SRK) as present in the PRO/II program (version 4.12) of SimSci have been used in this study. In all cases the results can be considered as straight predictions since only binary parameters are employed and no fine tuning based on the ternary data has been
attempted. Finally, table 2 present timings for flash calculations with these models in a number of tests (water/methanol, methanol+29 hydrocarbons and water/ methanol/29 hydrocarbons). These timings show that while sCPA is slower than the SRK and NRTL models, the computing times are still reasonable and it would be feasible to use the sCPA method for engineering simulations. The following points summarize the most important conclusions of our study: 1. Excellent correlation is achieved with sCPA for all binary systems considered including methanol/alkanes, water/alkanes, water/methanol and water (or methanol) / gases. This implies that the extension of sCPA to cross-associating systems (e.g. water/methanol) and associating compound/gas systems is very satisfactory. Furthermore, this is accomplished with a single interaction parameter employed in the physical term of the Equation of State. 2. Excellent prediction of multicomponent equilibria is achieved with the sCPA equation of state. The three systems selected for testing are typical cases of interest to the petroleum, oil and gas industry. For example, the system water/methanol/ methane/n-heptane contains the main element compounds of a typical petroleum mixture: water, gas (simulated by the methane), oil (simulated by n-heptane) and methanol (injected for preventing hydrate formation). The sCPA results are considerably better compared to the conventional models, while further this is accomplished without significant cost with respect to timing provided that the Elliott rule is employed for the cross association strength.
Conclusions The sCPA equation of state is extended in this work to cross associating water/alcohol systems, methanol/supercritical fluids and multicomponent systems comprising of water, methanol and hydrocarbons. Excellent correlation of binary phase equilibria is obtained for all cases considered using a single interaction parameter in the physical term. Particularly impressive are the equilibrium calculations for the multicomponent systems which are considered as pure predictions since they are based exclusively on binary parameters. The predictions are considerably better than those of the conventional NRTL and SRK models.
The improved performance is attributed to the Wertheim association term as well as the successful extension of the equation of state to cross-associating systems employed here. Recent studies (e.g. [7]) indicate that the various traditional association theories (perturbation, chemical and lattice-fluid) are essentially equivalent. Thus, the good performance with the present model is, in principle, expected with other association models (e.g. SAFT, LFHB) as well, provided that a number of crucial ‘details’ are properly taken into account (pure-component parameter estimation and association scheme, cross-association effects, etc.).
List of symbols a: attractive parameter b: co-volume parameter β: association volume parameter c1: parameter of the attractive term Ä: association strength ε: association energy parameter g(d): radial distribution function kij: binary interaction parameter V: volume y: reduced density References [1] G. M. Kontogeorgis, E. Voutsas, I. Yakoumis, and D.P.Tassios, Ind. Eng. Chem. Res., 35 (1996) 4310-4318. [2] I. Yakoumis, G. M. Kontogeorgis, E. Voutsas, and D.P.Tassios, Fluid Phase Equilibria, 130 (1997) 31-47. [3] E. Voutsas, G. M. Kontogeorgis, I. Yakoumis, and D.P. Tassios, Fluid Phase Equilibria, 132 (1997) 61-75. [4] I. Yakoumis, G.M. Kontogeorgis, E. Voutsas, E. Hendriks, and D.P. Tassios, Ind. Eng. Chem. Res. (1998) submitted for publication. [5] J. R. Elliott, S. J. Suresh, and M. D. Donohue, Ind. Eng. Chem. Res., 29 (1990) 1476-1485. [6] S. J. Suresh, and J. R. Elliott, Ind. Eng. Chem. Res., 31 (1992) 2783-2794. [7] E. M. Hendriks, J. Walsh and A.R.D. van Bergen, 1997 J. Stat. Physics, 87: 12871306.
Table 1. sCPA pure component parameters (SI units). For inert compounds with critical temperature below 500K (i.e.: methane and propane) the critical properties and the acentric factor are employed in sCPA. a0 c1 ε β Compound b (*105) water 1.4515 0.12277 0.67359 16655 0.0692 methanol 3.0978 0.40531 0.43102 24591 0.0161 n-hexane 10.787 2.3678 0.83083 n-heptane 12.535 2.9178 0.9137 Table 2. Speed of various models (in hundredths of seconds on a 200 MHz Pentium PC for 100 identical flash calculations). In each case, the number of phases predicted to be present at equilibrium is shown in brackets after the time taken. Model Test 1 Test 2 Test 3 sCPA with Elliott’s rule
166 (2)
3361 (3)
3218 (3)
sCPA without Elliott’s rule
529 (2)
4938 (3)
5405 (3)
SRK (original)
28 (2)
830 (3)
900 (3)
NRTL (Ideal gas phase correction)
28 (2)
-
-
where: tests denote three different flash calculations as follows: Test 1: an equimolar mixture of water and methanol at 1bar and 350K. Test 2: a mixture of 29 hydrocarbons plus methanol at 10bar and 350K Test 3: a mixture of 29 hydrocarbons plus methanol and water at 10bar and 350K.
Figure 9. Prediction of the partitioning of methanol between the organic and the aqueous phase of the quaternary system: water/methanol/methane/n-heptane. Prediction with SRK is given for comparison purposes.
Figure 1. Comparison of the hard sphere radial distribution function (employed in SAFT and CPA) with the simplified radial distribution function (employed in sCPA).
Figure 3. Correlation of the water/methane vaporliquid equilibra at 75oC with the sCPA using a single binary parameter.
Figure 2. Correlation of the methanol/methane vaporliquid equilibria at 25oC with the sCPA using a single binary parameter.
Figure 4. Correlation of the methanol/n-hexane liquidliquid equilibria with the sCPA using a single binary parameter.
Figure 5. Correlation of the water/n-hexane liquidliquid equilibria with the sCPA using a single binary parameter
Figure 7. Prediction of the partitioning of methanol between the organic and the aqueous phase of the ternary system: water/methanol/propane
Figure 6. Correlation of the water/methanol vapor-liquid equilibria at 25oC with the sCPA using a single binary parameter
Figure 8. Prediction of the partitioning of methanol between the organic and the aqueous phase of the ternary system: water/methanol/n-hexane. Predictions with SRK and NRTL are given for comparison purposes
