Quality of Component- and Group-Interaction-Based Regression of

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Quality of Component- and Group-Interaction-Based Regression of Binary Vapor-Liquid Equilibrium Data Brian Satola, Juergen Rarey, and Deresh Ramjugernath Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b02118 • Publication Date (Web): 24 Aug 2017 Downloaded from http://pubs.acs.org on September 1, 2017

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Industrial & Engineering Chemistry Research

Quality of Component- and Group-Interaction-Based Regression of Binary Vapor-Liquid Equilibrium Data Brian J. Satola a, Jürgen Rarey a, b, c, *, Deresh Ramjugernath a a

Thermodynamics Research Unit, School of Chemical Engineering, University of Kwa-Zulu Natal, Durban 4041, RSA b

c

*

DDBST GmbH, Marie-Curie-Str. 10, 26129 Oldenburg, FRG

Industrial Chemistry, Carl von Ossietzky University Oldenburg, 26111 Oldenburg, FRG

Corresponding author: Email: [email protected]; Tel: +491795047519; Fax: +464417983330

Keywords: Activity coefficients, gE-models, correlation, prediction

Abstract To describe the behavior of real liquid mixtures, binary interaction parameters of component-based models like e.g. UNIQUAC are fitted to experimental data or estimated values from group contribution (GC) methods like UNIFAC and mod. UNIFAC. Since parameters of GC methods are based on a large number of datasets for many similar mixtures, they are typically less precise than models fitted directly to individual datasets for the binary mixture of interest. In some cases, however, it was reported that UNIFAC with group parameters independently regressed to the same datasets represents binary mixtures more accurately than UNIQUAC, which could be due to better localization of interactions. This possible advantage was analyzed using over 3700 datasets from the Dortmund Data Bank (DDB). Advantages and disadvantages of the group-based ACS Paragon Plus Environment

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approach are identified, that could be well explained by the change in concentration base—the results should also hold for segment-based models like NRTL-SAC and COSMO-RS.

Introduction Process simulators have become common place tools for the design, debottlenecking and troubleshooting of many chemical operations in industry, where their success and continued value relies on the sufficiently precise description of various thermophysical properties required for calculations. Liquid-phase activity coefficients, for instance, are particularly important for the successful description of many separation processes such as distillation or extraction where knowledge of component distribution between various phases is required. For the calculation of these activity coefficients, models like Wilson1, NRTL2 and UNIQUAC3 have long been used with great success, but they are hindered by the limited availability of suitable binary interaction parameters (BIP). In many cases, there is simply not enough experimental data (if any at all) of sufficient quality to fit the model binary interaction parameters. Out of this reason, group contribution (GC) methods like UNIFAC4 and mod. UNIFAC5 and, more recently, segment contribution models like NRTL-SAC6 and COSMO-RS7 have been developed. If VLE data is not available, BIPs can be estimated based on the results predicted by these methods. Group interaction parameters are regressed, however, to reproduce the behavior of a large number of different mixtures simultaneously—often more than 3000 experimental data points are used to obtain the parameters for a single group pair8—and so they are rarely capable of representing specific experimental information more accurately than component-based models fitted directly to individual datasets. Predictive methods, therefore, are typically only used as secondchoice data generators when no experimental data of sufficient quality is available. By regressing available experimental data or GC-model predictions using equations like Wilson, NRTL and UNIQUAC, practitioners are able to separately model the calculated behavior of specific componentbinaries in their mixture. Alternatively, the same thing cannot be easily accomplished using methods like UNIFAC, since the changing of any group interaction parameter will most often also affect the calculated ACS Paragon Plus Environment

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behavior of other component pairs in the mixture. Although most process simulation software offers the regression of e.g. UNIFAC parameters (where significantly fewer model parameters are mostly required for multicomponent mixtures), practitioners are encouraged to stay within the component-based approach in order to separately influence the behavior of component binaries. This is illustrated in Figure 1, where UNIQUAC parameters were independently regressed to individual datasets and compared against group contribution predictions based on the regression results for more than 2,200 consistent datasets. To simplify discussion only absolute deviations in vapor phase composition will be analyzed. Similar results were obtained for deviations in temperature and pressure.

Figure 1. Mean absolute deviation in vapor phase composition, temperature, and pressure for 2200 consistent binary VLE datasets using UNIQUAC, UNIFAC, and mod. UNIFAC (Do.).9 The separate regression of binary interaction parameters for each dataset using UNIQUAC yields the lowest deviation; the resulting mean deviation of 0.58% for all datasets can be attributed to the scatter of experimental data and the inability of the two-parameter model to fit the datasets more precisely. The slight increase to 0.88% obtained for modified UNIFAC includes the short-comings of the group contribution method i.e. group parameters that can be used to represent many different mixtures. Furthermore, it should be noted that data from different authors for the same mixture often do not match. The UNIQUAC regressions, for instance, were performed for each dataset individually, while mod. ACS Paragon Plus Environment

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UNIFAC (Do.), as well as all other group contribution methods, must be regressed to all datasets simultaneously. Discrepancies between the results of different authors, therefore, enter into the difference in deviation between UNIQUAC and the UNIFAC methods. The decrease in deviation from 1.41% obtained for UNIFAC to the 0.88% of mod. UNIFAC is a result of various improvements, e.g. the modification of the combinatorial contribution, modified group fragmentations, wider range and types of data used in regression, as well as the addition of temperature dependent parameters. While the deviations in the case of UNIFAC or mod. UNIFAC are only slightly higher than in the case of directly fitting the individual datasets, it is nevertheless the current practice to stay within the component-based approach in order to be able to independently influence component-binary mixtures. This could explain why the group interaction concept has not been widely investigated outside of this role, although group contribution models may give more realistic results as the different interactions are more localized and not “smeared” over the whole molecule. In limited cases this has been observed10, and it is desirable to test to what extent this advantage holds for specific binary systems. The effects of both solution environments, component-based and group-based, are analyzed here in detail.

Accounting for Molecular Interactions in Activity Coefficient Models Component based models determine the total excess Gibbs energy of a mixture based on the interactions between individual molecules, whereas group contribution methods treat interactions between molecular groups that make up the molecules. In case of binary mixtures, the number of parameters in both situations is often the same, but the group approach may have a more physically sound basis. Both approaches are depicted in Figure 2 below, using a mixture of hexane and ethanol for illustration purposes.

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Figure 2. The different treatment of solution environments is depicted for hexane and ethanol, with the aid of Discovery Studio11. By treating interactions between whole molecules as shown in Figure 2a, the component-based approach makes no physical distinction of which molecular sites are involved in which types of intermolecular forces. The strong hydrogen bonds formed between the hydroxyl groups (—OH) of the ethanol molecules are therefore lumped together with the weak-dispersive forces occurring between the alkyl groups (—CH2, —CH3) of the hexane and ethanol molecules. Specific properties are therefore averaged over the entire molecular surface for each component-component interaction. On the other hand, as shown in Figure 2b, the group contribution approach represents these same interactions as occurring between the molecular groups used to define these same molecules. Although the treatment of interactions between whole molecules is basically correct, as in the component-based approach, the strong H-bonding actually only exists between the hydroxyl groups of the ethanol molecules. By fragmenting molecules into a subset of molecular groups a more physically realistic and intuitive representation is obtained, where binary interactions between molecular groups (and not whole molecules) are localized to the sites involved in specific interactions. The group contribution approach, therefore, more closely resembles the detailed molecular models employed in modern molecular dynamics (MD) and Monte Carlo (MC) simulations, where the electrostatic potential of molecules is typically ACS Paragon Plus Environment

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determined as shown in Figure 2 (minus sign indicating a partial negative charged region and a positive sign for a partial positive charge). As an initial test of this assumption, experimental activity coefficients at infinite dilution for the mixture hexane (1) and 1-butanol (2) are plotted in Figure 3 together with the values calculated from UNIQUAC component interaction and UNIFAC CH2 – OH group interaction parameters regressed to several VLE datasets at different temperatures individually. The extrapolation of the group contribution based UNIFAC model yields much higher and more realistic values for the limiting activity coefficient of 1butanol in hexane. This can be attributed to the much lower concentration of OH-groups in the group mixture than 1-butanol in the component mixture. In case the components are used as groups, the UNIFAC model becomes identical to UNIQUAC, so that the localization of the interactions is the only good explanation for the much better results. In case of hexane in 1-butanol, where the CH2 group concentration approaches a fixed value as the hexane concentration becomes zero, the results are nearly identical. If the above argumentation is correct, similar behavior should be visible also in further mixtures of components where one contains only unpolar groups and the other both unpolar and a strongly polar group (e.g. OH).


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Figure 3. Experimental activity coefficient at infinite dilution data12 (,) for the mixture hexane – 1-butanol together with the values calculated from UNIQUAC component interaction (,) and UNIFAC CH2 – OH group interaction parameters (,) regressed to VLE data13,14—solid markers used for hexane.

Materials and Methods The direct regression of interaction parameters for both approaches was performed to see, on a large scale, which of the two methodologies represents the real behavior of mixtures more precisely. The binary interaction parameters between main groups were fitted using the original UNIFAC method for each individual dataset separately using the 2008 UNIFAC Consortium group parameter definitions15. This increased the number of systems which could be regressed using the UNIFAC method, since the ACS Paragon Plus Environment

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Consortium matrix includes 35 new subgroups and 15 additional main groups not found in the latest public version of the UNIFAC method16. The individual fitting results obtained using UNIFAC were then compared to the regression results made in turn using the Wilson, NRTL and UNIQUAC model for the same datasets for all cases where no immiscible region was predicted by either of the two models being compared. For the case of UNIFAC, the number of main groups in each mixture was initially restricted to two, resulting in the regression of 2 BIPs for the Wilson and UNIQUAC models. In the case of NRTL, however, three adjustable parameters were used for data regression, but since the third adjustable parameter often contributes negligibly to the overall fit of the model (except in cases of limited miscibility which have been ignored here) these regression results are being considered here comparable to those obtained using the Wilson, UNIQUAC and UNIFAC models. This comparison was later extended to binary mixtures which could be fragmented into three and four different main groups, resulting in six and twelve group interaction parameters respectively. Binary isothermal data—Px(T)—were used for the evaluation, where the entire composition range for each binary dataset (0 ≤ 𝑥1 ≤ 1) was divided into bins of 0.1 mole fractions each, and only those datasets having a minimum of 10 data points filling at least 5 of these bins were considered. The reason for choosing, incomplete, Px(T) data lies in the fact that these data are usually obtained from static measurements with high precision synthetic liquid composition and very precise total pressure measurement as recommended by Gibbs and Van Ness17. The number of datasets was further restricted to those for which the pure component vapor pressures of each mixture component was reported by the authors, so that the calculated pure component vapor pressures could be adjusted to the experimental values to avoid any inconsistencies between the published values and those calculated using databank parameters found in the Dortmund Data Bank12 (DDB). These restrictions removed many datasets of insufficient quality from the evaluation. The number of Px(T) datasets from the DDB meeting all of these requirements is given in Figure 4, where they have been arranged according to the number of unique main groups defining the mixture. ACS Paragon Plus Environment

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Figure 4. Number of Px(T) datasets for mixtures with different number of different main groups used for regression (data taken from the DDB12). Using the afore mentioned criteria, the parameters of the Wilson, NRTL, UNIQUAC and UNIFAC models were then fitted to each dataset individually, using the Simplex Nelder-Mead method18 by minimizing the relative mean-squared-deviation (MSD) in pressure for each dataset: 𝑛

𝑃𝑀𝑆𝐷

𝑃𝑒𝑥𝑝,𝑖 − 𝑃𝑐𝑎𝑙𝑐,𝑖 1 = min ∑ ( ) 𝑛 𝑃𝑒𝑥𝑝,𝑖

2

(1)

𝑖=1

where n is the number of data points for each dataset fitted. Since relative deviations of 0.001 in pressure are often acceptable for practical applications (i.e. the regression is sufficiently close to the experiment), a minimum relative squared deviation of 10-6 was imposed (√10−6 = 0.001). Any fitness values (MSD) less than 10-6 were set equal to the minimum value of 10-6. Although the Simplex Nelder-Mead method may not provide optimum results in case of higher numbers of parameters, it is nevertheless suitable for this work given that direct comparisons with UNIFAC are only relevant for regressions involving 2 parameters. For systems requiring 6 or 12 group contribution interaction parameters, the regression results were so far better than the results for the component based models, that the possible existence of a slightly


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better solution would not have influenced the results. All calculations were performed using the in-house program TRUx (Thermodynamic Research Utilities for Excel) and the Dortmund Databank (DDB).

Results The individual regression results obtained using the UNIFAC method are compared against the results using the Wilson equation in Figure 5, the NRTL equation in Figure 6 and the UNIQUAC equation in Figure 7. Each data point of each line in Figures 5-7 represents the ratio of the objective function (MSD) value achieved by UNIFAC relative to one of the component based models for an individual dataset. The log10 of these ratios is depicted on the ordinate of each graph, with values ordered from smallest to largest to obtain a continuous line for each comparison. The value on the abscissa then represents the percentage of datasets that have an MSD ratio equal to or less to the ordinate value. Separate curves represent the results for mixtures containing 2, 3 or 4 different UNIFAC main groups, as the number of main groups directly affects the number of UNIFAC group interaction parameters that can be regressed. The total number of datasets of each type (number of main groups) does not match that given in Figure 4 because pure component information (e.g. liquid molar volumes, pure component vapor pressure parameters, etc.) was missing for some components and because in several cases one of the models being compared predicted the formation of a miscibility gap and the dataset was excluded from the comparison. In addition, nearly ideal mixtures were excluded as the activity coefficients are in these cases only of minor importance for the real liquid mixture behavior. Mixtures are considered nearly ideal if the product of the two limiting activity coefficient values lay between 0.66 and 1.5. In very rare cases, where the two limiting activity coefficients are both above and below one, this could potentially lead to the unintended removal of the dataset from the test.


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Figure 5. The regression results obtained using the UNIFAC method compared against those obtained using the Wilson equation.


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Figure 6. The regression results obtained using the UNIFAC method compared against those obtained using the NRTL equation.


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Figure 7. The regression results obtained using the UNIFAC method compared against those obtained using the UNIQUAC equation. Since Figures 5-7 use a logarithmic ordinate, any value less than zero represents a case in which the group contribution approach using UNIFAC fitted the dataset better than the respective component-based approach. To account for the larger number of datasets where both models performed very similarly, the MSD of the model must differ by at least 1% in order to be considered different. In Figures 5-7, the numbers in the lower part represent the percentage of data where UNIFAC (left value) or the component based model (right value) performs better. The middle value shows that percentage of the datasets, where the performance of both models is very similar. When comparing UNIFAC against the component based models for mixtures composed of 2 main groups, Wilson performs better in 60% and NRTL in 54% of the datasets. In case of UNIQUAC, 57 % are identical while similar fractions (25% and 18%) favor one of the approaches. For mixtures with 3 or 4 main groups, UNIFAC becomes the best model in most cases, but it should be noted that this is attributed


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to the drastic increase in the number of parameters. The regressed group interaction parameters are quite definitely not physically realistic and should be considered mere fitting parameters for mixtures containing more than 2 main groups. As the Wilson model is not able to predict or reproduce limited liquid miscibility and performs only slightly better, NRTL is generally viewed as the best general gE-model for process simulation. This may be due to the additional (third) parameter or the functional form of the model. In order to evaluate the effect of the different concentration scales (group-based vs. component-based) the further discussion will be restricted to the comparison between UNIFAC and UNIQUAC which both share the same underlying equations for the calculation of group or component activity coefficients19. In case of mixtures comprised out of only two main groups, the number of parameters of both models is identical. Yet in 25% of the cases shown in Figure 7, UNIFAC outperforms UNIQUAC. In this subset, predominantly alcohol – hydrocarbon mixtures are found. UNIFAC seems to benefit from the fact that the alcohol group concentration approaches zero more rapidly than the component concentration of the alcohol. On the other hand, the group contribution concept also seems to carry certain disadvantages as in 25% of the cases it is outperformed by the component-based UNIQUAC. A typical example for this is shown in Figure 8. In the mixture methyl-tert-butyl-ether (MTBE) (1) and 4,4-dimethyl-1,3-dioxane (2) both components consist of several alkane- plus one or two ether-groups. Figure 8a shows, that although the data can be perfectly described by UNIQUAC, both prediction and regression using UNIFAC fail significantly. Figure 8b shows that the group concentrations of the alkaneand the ether-group cover only a small range as function of component concentration. In case of two components that contain two groups in same surface area proportion, UNIFAC results will always be very close to or identical with ideal behavior.


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Figure 8. Experimental Px(T) data20 () for the mixture methyl-tert-butyl-ether (MTBE) (1) – 4,4dimethyl-1,3-dioxane (2) at 323.15 Kelvin, together with the values calculated from UNIQUAC component interaction (---) and UNIFAC (—) CH2 – OH group interaction parameters (,) regressed to VLE data. To further illustrate the findings, typical mixtures for which either UNIFAC outperforms UNIQUAC or vice versa are listed in Table 1. Mixtures for which UNIFAC leads to better representation are composed of a hydrocarbon and a molecule with one polar group plus hydrocarbon groups. As shown in Figure 3 for a similar example, UNIFAC is in all cases better able to describe the behavior of the polar component at strong dilution, which can be attributed to the more realistic localization of the polar interaction. In all cases, the polar component has a sufficiently high vapor pressure, so that its activity is visible in the total pressure reported by the author. Among the cases where UNIQUAC outperforms UNIFAC, several alkane-alcohol systems can be found. In all of these, the difference between the models is rather small and the alcohol has a much lower vapor pressure than the hydrocarbon, which makes its contribution to the total pressure neglectable. All other mixtures consist of components with very similar group composition. While component mole fractions cover the full range from zero to one, group fractions only span a rather small range, which reduces the ability of UNIFAC to represent the mixture behavior. In the last case, the group composition of the xylenes is identical so that UNIFAC loses all ability to ACS Paragon Plus Environment

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correlate the data and will always predict ideal mixture behavior. This would also explain the shortcomings of UNIFAC and mod. UNIFAC that where observed in cases of multifunctional components (e.g. pharmaceuticals21), as with increasing number of different groups in the mixture, the possible concentration range for each group shrinks drastically. Table 1. Example mixtures. UNIFAC Outperforms UNIQUAC butane ethanol pentane ethanol hexane butanol hexane 2-methyl-1-propanol heptan pentanol decane N-methylacetamide isopropylamine hexane propionitrile heptane

UNIQUAC Outperforms UNIFAC hexane decanol tert-Butanol 2-methyl-1-propanol methyl acetate ethyl acetate methyl tert-butyl ether 4,4-dimethyl-1,3-dioxane methyl acetate propyl acetate isoprene trans-1,3-pentadiene cyclohexanone 2-methylcyclohexanone m-xylene o-xylene

Conclusions An examination of two popular approaches (group- and component-based) employed by models for the calculation of activity coefficients using a large number of VLE datasets for miscible liquid mixtures showed advantages and disadvantages of the group-based approach, that could be well explained by the change in concentration base. The argumentation and results should also be applicable to segment based models like NRTL-SAC or COSMO-RS. In general, although potentially more realistic by localizing interactions on specific parts of a molecule, the sometimes much smaller range of group concentrations is likely to affect the flexibility of a model in a negative way. Compared to the UNIQUAC model, both Wilson and NRTL resulted in overall better regression results. Given the limitation of the Wilson model to miscible liquid mixtures, this explains the popularity of NRTL. Based on these findings, further regression of published UNIFAC parameters to the specific mixtures of interest would show limited success. As a next step in this work, the authors evaluate the possibility to combine both group-and component based approaches into a hybrid approach to resolve the various limitations discussed here.


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Acknowledgements This work is based upon research supported by the National Research Foundation of South Africa under the South African Research Chair initiative of the Department of Science and Technology and data provided by DDBST GmbH.


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log10 (UNIFAC / Wilson)

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3.0 2 Main Groups (970) 2.0

3 Main Groups (1488)

1.0

4 Main Groups (262)

0.0 -1.0 -2.0 -3.0

20%

-4.0

57%

-5.0

2MG

60%

20%

18% 75%

25% 17%

3MG 8% 4MG

-6.0 0%

20%

40%

60%

Percentage of Data Sets


80%

100%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

log10 (UNIFAC / NRTL)


Page 22 of 23

3.0 2 Main Groups (924) 2.0


1.0

4 Main Groups (264)

0.0 -1.0 -2.0 -3.0

27%

-4.0

19% 61%

-5.0

2MG

54% 16% 16%

70%

3MG

23% 14%

4MG

-6.0 0%

20%

40%

60%



80%

100%

Page 23 of 23

3.0 2 Main Groups (864) log10 (UNIFAC / UNIQUAC)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.0


1.0

4 Main Groups (259)

0.0 -1.0 -2.0 -3.0

25%

-4.0

61%

-5.0

29%

66%

2MG

18%

57%

25%

10% 3MG 9%

4MG

-6.0 0%

20%

40%

60%



80%

100%

Quality of Component- and Group-Interaction-Based Regression of

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