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Examining the effect of binary interaction parameters on VLE modelling using cubic equations of state M.K. Ikeda ∗ , L.A. Schaefer 323 Benedum Hall, 3700 O’Hara Street, University of Pittsburgh, Pittsburgh, PA 15261, USA

a r t i c l e

i n f o

Article history: Received 14 January 2011 Received in revised form 30 March 2011 Accepted 31 March 2011 Available online 8 April 2011 Keywords: VLE data prediction Binary interaction parameters Cubic equations of state

a b s t r a c t Vapor–liquid equilibrium (VLE) data are important in the optimization of thermodynamic cycles. As energy concerns continue to grow, improving the efﬁciencies of power and refrigeration cycles is increasingly important. Numerical simulations using empirical equations of state provide an excellent alternative to time consuming experimental measurement of VLE data. However, it is important to understand the limitations of using correlative equations for data prediction. In this study, a water–ethanol mixture is simulated with various VLE models. Non-optimal binary interaction parameters are considered and model accuracy is evaluated in terms of average absolute percent deviation (%AAD) between simulated and experimental bubble and dew point pressures. For this system, it is found that as the correlative accuracy of a model increases, the predictive ability decreases. Speciﬁcally, the temperature dependence of the binary interaction parameters is shown to be an important consideration for the water–ethanol system when more complex combining rules are implemented. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The need for extensive vapor–liquid equilibrium (VLE) data for the optimization of thermodynamic cycles and separation processes has been well-established [1–5]. For example, an understanding of the temperatures, pressures, and compositions at which mixtures are pure liquids or vapors is necessary to properly match operating conditions to working ﬂuids, or alternatively, to correctly select a cycle’s working ﬂuid based on operating parameters. Without this knowledge, it is likely that a cycle’s efﬁciency will be greatly diminished, or the cycle may even cease to function altogether. Furthermore, as energy efﬁciency, and consequently cycle efﬁciency, becomes increasingly important, the demand for VLE data will only increase [6]. Due to the time-consuming nature of experimental measurement, equations of state (EOSs) have become a signiﬁcant source of VLE data through various modelling approaches. The most commonly used methods rely on empirical cubic equations of state, implemented in conjunction with mixing and combining rules, to determine mixture properties [7,8]. The combining rules include experimental ﬁtting values, referred to as binary interaction parameters, which lead to largely correlative sets of equations. Consequently, it is important to consider the limits on the predictive calculation of data using these equations.

∗ Corresponding author. Tel.: +1 805 766 6895. E-mail addresses: [email protected] (M.K. Ikeda), [email protected] (L.A. Schaefer). 0378-3812/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ﬂuid.2011.03.029

Recent work has focused on the use of predictive combining rules that account for the temperature within the expression of the binary interaction parameters [9]. Of speciﬁc note is the success by Jaubert et al. and Soave et al. in predicting VLE data for hydrocarbons and related mixtures [10–20]. They have used group contribution methods to determine the temperature dependence of the binary interaction parameters implemented in the Peng–Robinson and Soave–Redlich–Kwong equations of state. In this study, VLE data simulation is based on more traditional VLE models in order to determine the extent of their predictive ability for the highly nonideal water–ethanol system. This allows a better evaluation of the strengths and shortcomings of these simple models for this system. To investigate this concept, an analysis of the dependence of various equation combinations on the accuracy of binary interaction parameters over a range of temperatures is considered. 2. Numerical method While innumerable equations of state, mixing rules, and combining rules exist, an assortment of some of the most common equations was chosen for this study, with the purpose of including very basic as well as somewhat complex forms. The simplest equation of state capable of simulating both liquid and vapor phases is the van der Waals EOS [21]: Z=

Vm a + − RTVm Vm − b

,

(1)

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Table 1 Modiﬁcations to the van der Waals attractive term for various EOSs [8,22–26]. Equation of state

Attractive term (−Zatt )

Soave–Redlich–Kwong (SRK)

a(T ) RT (Vm +b)

Peng–Robinson (PR)

a(T )Vm RT [Vm (Vm +b)+b(Vm −b)] a(T )Vm

Peng–RobinsonStryjek–Vera (PRSV)

2 +2bV −b2 ) RT (Vm m

where the ﬁrst and second terms represent the repulsive and attractive contributions to the compressibility, respectively. The equations of state considered in this study provide improvements to the van der Waals attractive term, various forms of which are listed in Table 1 [8,22–26]. It has been shown that the Peng–Robinson–Stryjek–Vera equation of state is speciﬁcally amenable to nonideal systems, such as the water–ethanol mixture studied here [26]. The a and b parameters shown in the EOSs are calculated for a mixture using mixing rules. Standard linear and quadratic mixing rules are used in this study, and their forms for an arbitrary parameter are shown in Eqs. (2) and (3), respectively [8,22,23]: =

N

zi ii ,

=

i

zi zj ij .

Combining rule

k12,optimal

k21,optimal

Arithmetic Conventional Margules form van Laar form

−0.015 −0.105 −0.08 −0.12

N/A N/A −0.12 −0.09

and 343.15 K. Optimal binary interaction parameters for each temperature and each equation set were determined by varying the parameter values until the average absolute percent deviation between experimental and simulated bubble and dew point pressures was minimized. This process was carried out using experimental data at ﬁve different temperatures, as tabulated in the Vapor Liquid Equilibrium Data Collection [30,31]. One set of optimized binary interaction parameter values is given in Table 3, for the Peng–Robinson–Stryjek–Vera equation of state at a temperature of 323.15 K.

(2)

3.1. Perturbation of optimal binary interaction parameters

(3)

The optimal binary interaction parameters were subsequently perturbed by ±15% to determine the effect of erroneous parameters on the simulated results. The perturbed values were calculated from the optimal values, such as those shown in Table 3, using Eq. (4):

i N N

Table 3 Optimal binary interaction parameters determined for a water–ethanol mixture at 323.15 K, modelled with the PRSV equation of state, the linear mixing rule for the b parameter, the quadratic mixing rule for the a parameter, and the speciﬁed combining rule for cross-interaction terms.

j

In these expressions zi represents the the mole fraction of the ith component, ii is the EOS parameter for a single, pure component and ij depends on mixture behavior and is determined using a combining rule. The combining rules implemented here are shown in Table 2, where kij and kji refer to the binary interaction parameters which are of particular interest in this study [8,22–26]. The arithmetic and conventional combining rules are one parameter rules, where ij = ji . The Margules and van Laar forms are two parameter models with kij = / kji . As is common, only the linear mixing rule was considered for the co-volume parameter in the equations of state [27]. This simpliﬁcation removes the dependence of the repulsive term on cross-interaction effects and therefore removes the need for a combining rule for its calculation. Conversely, the use of both of the mixing rules and all of the combining rules listed was permitted for the calculation of the attractive parameter. Fugacity coefﬁcients were calculated for the various combinations of the equations, and equilibrium was determined via the iterative − approach [22,28]. Laguerre’s method was implemented for the numerical solution of the compressibility equation in order to guarantee the convergence of the calculation [29]. 3. Results and discussion

kij,perturbed = (1 ± 0.15) × kij,optimal .

(4)

Fig. 1a illustrates the bubble and dew point curves for a mixture at 323.15 K. These particular results were modelled using the Peng–Robinson–Stryjek–Vera (PRSV) equation of state, the quadratic mixing rule for a, and the arithmetic combining rule. Data calculated using both the optimal (solid lines) and perturbed (dotted lines) binary interaction parameter are shown. Fig. 1 displays the same curves modelled with the same equation of state and mixing rule, but using the van Laar combining rule with both binary interaction parameters, k12 and k21 , perturbed equally. A comparison of these two ﬁgures shows qualitatively that the van Laar combining rule is more accurate than the arithmetic combining rule when provided optimal binary interaction parameters. However, the difference between the results using optimal parameters and those using perturbed parameters is also larger for the van Laar combining rule. This illustrates that the van Laar form is also more strongly dependent on the accuracy of the parameters. The average absolute percent deviations between experimental and simulated data (%AAD) for these results are shown in Table 4. The %AAD is determined by taking an average of the %AADs from the dew and bubble point data: 1 (%AADbubble + %AADdew ) , 2

In this study, a mixture of water and ethanol was considered with the temperature ﬁxed at different values between 298.15 K

%AAD =

Table 2 Combining rules considered in this study [8,22–26].

Table 4 Average absolute percent deviations (%AAD) using optimal and perturbed binary interaction parameters modelled with the PRSV equation of state, the linear mixing rule for the b parameter, the quadratic mixing rule for the a parameter, and the combining rule speciﬁed below, at 223.15 K.

Combining rule

ij

Arithmetic (A)

1 (1 2

Conventional (C)

(1 − k)

Margules form (M) van Laar form (vL)

− kij )(ii + jj )

ii jj

(1 − zi kij − zj kji )

1−

kij kji

zi kij +zj kji

ii jj

ii jj

(5)

Combining rule

Optimal

Perturbed

Arithmetic Conventional Margules van Laar

3.265 3.265 1.721 1.718

3.336 6.549 6.440 6.474

M.K. Ikeda, L.A. Schaefer / Fluid Phase Equilibria 305 (2011) 233–237

Table 5 Average absolute percent deviations (%AAD) between experimental and simulated data at 343.15 K using binary interaction parameters (BIPs) optimized at different temperatures.

35

P [kPa]

30

Equation of state

Combining rule

343.15

333.75

323.65

313.15

298.15

SRK

A C M vL

6.174 6.168 3.6 3.987

6.174 6.168 4.525 3.987

6.819 7.249 5.625 5.217

6.388 7.249 4.525 4.525

6.252 6.285 5.699 4.525

PR

A C M vL

3.696 3.723 3.085 2.986

3.696 3.723 4.329 3.484

4.419 5.557 5.202 4.701

3.901 4.586 4.329 4.701

3.696 4.586 5.202 6.073

PRSV

A C M vL

2.722 2.693 0.678 0.803

2.844 2.693 1.946 1.648

3.135 3.791 3.098 3.804

3.135 3.791 3.098 3.378

2.722 3.791 6.015 5.158

25

20

OP-BP PP-BP OP-DP PP-DP ED

15

10 0

0.2

0.6

0.4

0.8

1

Mole Fraction Ethanol: z 1 (a) Arithmetic Combining Rule

35

%AADs calculated using BIPs optimized using experimental data at a temperature of:

measured by its accuracy in simulating data at conditions that are different from those used to optimize the binary interaction parameters, as would be common in the practical implementation of VLE data simulation. In the following section, this effect is investigated.

30

P [kPa]

235

25

3.2. Variation of the experimental dataset used to determine binary interaction parameters 20

OP-BP PP-BP OP-DP PP-DP ED

15

10 0

0.2

0.6

0.4

0.8

1

Mole Fraction Ethanol: z1 (b) van Laar Combining Rule Fig. 1. Perturbation of binary interaction parameters from optimal using the PRSV equation of state, the quadratic mixing rule for a, and the combining rule speciﬁed, at 323.15 K (PP: perturbed parameter, OP: optimal parameter, BP: bubble point, DP: dew point, and ED: experimental data).

where the individual %AAD values are deﬁned by:

%AADbubble

i=1

and, %AADdew

N bubble bubble 1 Pi,calc − Pi,exp = × 100, N P bubble

N dew dew 1 Pi,calc − Pi,exp = × 100. N P dew i=1

(6)

i,exp

(7)

i,exp

bubble and P dew are the experimental bubble and dew point presPi,exp i,exp sures, respectively, corresponding to the ith experimental data bubble and P dew are the pressures calculated at the point, while Pi,calc i,calc same conditions as that data point, using a VLE model. It can be concluded that when the parameters are perturbed by ±15%, the more complex van Laar combining rule becomes less accurate than the simple arithmetic rule, for the PRSV equation of state. However, it has yet to be determined how this perturbation translates to a limit on the predictive ability of a VLE model. As stated above, the binary interaction parameters are optimized by simulating data at a set of conditions and minimizing the error between an experimental dataset at the same conditions. Therefore, the ability of a model to predict data can be

VLE data was simulated at a temperature of 343.15 K using the binary interaction parameters determined from experimental data not at 343.15 K, but at different temperatures. The deviations, averaged over the entire datasets, were then determined for each equation set. The results from this study are presented in Table 5. The column labelled 343.15 K contains the results calculated with optimal binary interaction parameters. It can be seen that these values yielded improved accuracy as equation complexity increased. When the VLE model implemented the Soave–Redlich–Kwong equation of state, the most complex combining rule considered was always the most accurate. However, as the distance between the simulated data and the experimental dataset used to determine the parameters increased, that trend disappeared for other equations of state. For example, when an experimental dataset at 298.15 K was used to calculate binary interaction parameters, the simplest combining rule, the arithmetic rule, provided the best results for the PRSV equation of state. Therefore, it can be concluded that the common practice of simply choosing a complex equation of state and a complex combining rule may not always yield the most accurate results. Instead, it is important to take into account the combination of equations that make up the VLE model, along with a knowledge of the available experimental data by which the binary interaction parameters will be determined. These results show that the binary interaction parameters carry with them an inherant temperature dependence. This premise, which was described by Coutinho et al. and has been the basis of much work by Jaubert et al. and Soave et al. is shown here to stand for the highly nonideal, polar water–ethanol system as well [9,10,20]. However, the degree of temperature dependence seems closely linked to the complexity and form of both the combining rule and the equation of state. For example, while the PRSV equation of state was developed by Stryjek and Vera to correlate nonideal mixture data extremely well using the Margules and van Laar form combining rules, this accuracy is shown to quickly break down when data prediction is desired [26]. For the system studied here, increases in VLE model complexity lead to

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an increase in the temperature dependence of the binary interaction parameters, and consequently, a decrease in predictive ability. It is hypothesized that this difference in dependence on equation complexity is directly related to the effect of binary interaction parameters on volume calculation. The quadratic mixing rule considered in these simulations is sensitive to disparities between the volumes of the molecules in a system. As the size differences between molecules increase, the rule becomes less accurate. In fact, a system with molecules of very different sizes, such as a water–decanol mixture, cannot be modelled with this mixing rule [32]. The water–ethanol system under consideration has a volume ratio within the usable limitation of this mixing rule. However, this assumes that the volumes calculated by the model maintain the same ratio as the actual volumes. If the volumes are calculated correctly, as is the case with complex combining rules using optimal binary interaction parameters, this ratio will clearly be satisﬁed. Conversely, when non-optimal binary interaction parameters are used, volume calculations may no longer be accurate. It is therefore surmised that when a simple arithmetic combining rule is used with the quadratic mixing rule and non-optimal binary interaction parameters, although the calculated volumes differ from the actual volumes, the ratio of the calculated volumes for the water and ethanol molecules still falls under the ratio limit of the mixing rule. When a complex combining rule is implemented, however, with non-optimal binary interaction parameters, it is possible that the ratio of the calculated volumes exceeds the limit and, as a result, the validity of the quadratic mixing rule breaks down. While further work is needed to conﬁrm this intuition, it is possible that this explains the maintenance of reasonable accuracy with simple combining rules and the abrupt failure of complex combining rules that occurs with parameter perturbation.

4. Conclusions and future work It can be concluded that while an increase in the complexity of an equation of state or combining rule can often lead to better correlative accuracy, during the prediction of data using empirical equations, it is important to consider the speciﬁc combination of equations being implemented, as well as the distance between the conditions of the simulated and experimental datasets. Future work would greatly beneﬁt from a more thorough analysis of these binary interaction parameters to determine how they affect the optimal choice of equations in the VLE model. It would also be of particular interest to optimize the binary interaction parameters over a large range of temperatures instead of a single temperature dataset. This could allow further insight into the importance of the data used for parameter determination and will be pursued in subsequent studies. Moreover, this study only considered a small number of equations under a restricted set of conditions for a water–ethanol mixture. The incorporation of additional equations and substances could lead to additional conclusions about the manner in which binary interaction parameters affect the potential for VLE data prediction.

Acknowledgements This work was supported by the NSF grant DGE-0504335 and the Mascaro Center for Sustainable Innovation at the University of Pittsburgh. Computational resources were provided by the Pittsburgh Supercomputing Center.

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